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Identification of frictional effects and structural dynamics for improved control of hydraulic manipulators Bilandi, Shahram Tafazoli 1997

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Identification of Frictional Effects and Structural Dynamics for Improved Control of Hydraulic Manipulators Shahram Tafazoli Bilandi B.Sc, Sharif University of Technology, Tehran, Iran, 1989 M.Sc, Sharif University of Technology, Tehran, Iran, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming jj^the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1997 © Shahram Tafazoli Bilandi, 1997 In present ing this thesis in partial fulfillment of the requirements for an advanced degree at the Universi ty of Brit ish Co lumb ia , I agree that the Library shal l make it freely avai lable for reference and study. I further agree that permiss ion for extens ive copy ing of this thesis for scholar ly purposes may be granted by the head of my department or by his or her representat ives. It is understood that copy ing or publ icat ion of this thesis for f inancial gain shal l not be a l lowed without my written permiss ion. Department of E lec± rVcc4 E*flmecr.n^ T h e Universi ty of Brit ish C o l u m b i a Vancouver , C a n a d a Date D E - 6 (2/88) Abstract Frictional terms and structural dynamics play an important role in achieving high quality performance of hydraulic manipulators. This thesis is mainly concerned with the identification of these effects and compensation thereof through control, for two different industrial applications. The first application pertains to a Cartesian electrohydraulic manipulator of a prototype fish processing machine which has been developed in the Industrial Automation Laboratory. Both model-based and observer-based approaches to friction estimation are investigated for this basic system. In the first approach, a friction model is used that is linear in parameters, and the friction is assumed to be a fixed function of velocity. The model parameters and the object mass are determined by applying the least squares estimation procedure to experimental data. A combination of Coulomb, viscous, and Stribeck components are observed in the identified friction model. In the second approach, a modified version of an available nonlinear observer is used for real-time estimation of the friction parameter. Convergence rate of the observer is analyzed and an approximate algorithm is developed for choosing its gains. Next, a novel technique is developed for friction-compensated tracking control of the manipulator. The technique incorporates the estimated frictional force in an acceleration feedback control law. Specifically, the model-based approach to friction estimation results in a fixed friction compensation algorithm and the observer-based approach to friction estimation results in an adaptive friction compensation algorithm. Experimental investigations show that the technique that is presented in this thesis considerably improves tracking performance of the manipulator. The second application is a mini excavator system which is a typical example of the human-operated mobile hydraulic machines. A new technique is presented in this thesis for indirect measurement of the joint torques of the backhoe links using load pin force sensors. Also , to be able to control the link motions electronically, the original pilot stage is modified by using on/off solenoid valves that are operated with D P W M (Differential-Pulse-Width-Modulation) current signals. Model ing and identification of the modified pilot stage is studied in the thesis in detail. According to the experimental results, the designed switching pilot system has a reliable performance that is linear. After the instrumentation phase, experiments are carried out with the mini excavator to determine the mass and inertia-related parameters of the links. To estimate the six mass-related (gravitational) ii parameters, sensor outputs are recorded in various static poses of the manipulator. The least squares estimation procedure is then applied to the decoupled form of the torque equations. The validation tests which we have carried out verify that the identified parameters can be used to estimate the static joint torques, with a good accuracy. A n efficient algorithm is then developed for real-time estimation of the bucket load under static conditions. Experimental results show that the bucket load can be estimated at an accuracy level of 5%. To study the structural dynamics of the system, the Euler-Lagrange equations are derived for the manipulator. The structural (dynamic) parameters are defined using these equations. It is shown in this thesis that the six gravitational parameters are a subset of the nine dynamic parameters. Joint friction coefficients are then added to the parameter set. Finally, the combination of the dynamic parameters and friction coefficients are estimated from the identification data that is obtained by simultaneous movement of the links. The results are consistent with those of the static experiments. In this research, it is found that the friction inside the actuator seals is quite significant. It follows that the joint torques can be measured more accurately by using load pins instead of pressure sensors. i i i Table of Contents Abstract ii List of Tables viii List of Figures ix Graphic Symbols xiii Acknowledgments xv Dedication xvi Chapter 1 Introduction 1 1.1 Preliminary Remarks 1 1.2 Motivation and General Objective 3 1.3 Previous Work 4 1.4 Thesis Overview 6 Chapter 2 Friction in Machines 9 2.1 Importance of Friction 9 2.2 The Classical Mode l of Friction 9 2.3 Friction as a Function of Velocity: Recent Observations f f 2.4 Frictional Memory and Dynamic Modeling of Friction 14 Part I The Cartesian Electrohydraulic Manipulator 17 Chapter 3 Analysis and Modeling of Friction in the Cartesian Manipulator 18 3.1 The Experimental System 18 3.2 Model ing of the Electrohydraulic System 19 3.3 Friction-Velocity Curve at L o w Velocities 22 3.4 Estimation of the Mass and Friction Coefficients 25 Chapter 4 Observer-Based Estimation of Friction in the Cartesian Manipulator 31 4.1 A Nonlinear Observer for Friction Parameter 31 iv 4.2 Simultaneous Estimation of Friction, Velocity, and Acceleration 33 4.3 Experimental Results 36 Chapter 5 Friction-Compensated Position Tracking of the Cartesian Manipulator 39 5.1 Methodology 39 5.2 Design of the Friction-Compensating Controllers 40 5.3 Experimental Results 44 Part II The Mini Excavator 51 Chapter 6 Instrumentation of the Mini Excavator 52 6.1 Structure of the M i n i Excavator 52 6.2 Installed Sensors 55 6.3 Joint Torque Measurement 56 6.4 Modification of the Pilot Stage 60 Chapter 7 Modeling and Identification of the Actuator Dynamics 64 7.1 Nonlinear Dynamic Mode l of the Actuator 64 7.2 Open Loop Step Response of the Actuator 70 7.3 The System Identification Experiment 74 7.4 Linear Identification of the Pilot Stage 76 7.5 Linear Identification of the M a i n Stage 80 7.6 Analysis of the Actuator Friction 84 Chapter 8 Estimation of the Gravitational Parameters and Bucket Load 86 8.1 Related Work 86 8.2 Static Torque Equations 88 8.3 Least Squares Estimation of the Gravitational Parameters 90 8.4 Estimation of the Actuator Torques for the No-Load Condition 92 8.5 Bucket Load Estimation 94 v 8.6 Actuator Static Friction 96 Chapter 9 Estimation of the Full Dynamic Parameters and Friction Coefficients 98 9.1 Background 98 9.2 Related Work . 99 9.3 Dynamic Torque Equations 100 9.4 Model ing of the Joint Friction 104 9.5 Parameterization of the Dynamic Equations 104 9.6 Least Squares Estimation of the Parameters 108 Chapter 10 Conclusions 116 10.1 Contributions of This Research 116 10.2 Suggestions for Further Work 117 Bibliography 119 Appendix A Specifications of the Cartesian Manipulator 126 A.l The System Configuration 126 A.2 The Interface Components 127 A.3 Sensor Specifications 128 A. 4 The Data Acquisition Board (Advantech Co.) 130 Appendix B Specifications of the Mini Excavator 131 B. l The Prototype Machine 131 B.2 The M a i n Valve System 132 B.3 The Pilot System 136 B.4 The Interface Circuit for DPWM-Operated Solenoid Valves 136 B.5 Sensor Specifications 139 B.6 Manipulator Kinematics 140 Appendix C Differentiation and Low-Pass Filtering 145 vi C.l Numerical Differentiation 145 C.2 Numerical Low-Pass Filtering 146 Appendix D Linear Least Squares Estimation 148 vii List of Tables Table 3.1 Physical parameters of the prototype manipulator 19 Table 3.2 Values of the design parameters 28 Table 3.3 Off-line estimation of the mass and friction coefficients 28 Table 5.f Parameters of the controllers 45 Table 7.1 Estimated parameters and their standard deviation for Hi(z) 77 Table 7.2 Estimated parameters and their standard deviation for H2(z) 8f Table 8.1 Estimated gravitational parameters using the decoupled static torque equations 92 Table 8.2 Standard deviation of the no-load torque estimation errors 93 Table 8.3 Static friction of the actuator seals 97 Table 9.t Estimated structural parameters and viscous friction coefficients 113 Table 9.2 Standard deviation of the dynamic torque estimation errors 113 Table B . t L i n k parameters for the mini excavator 141 Table B .2 The approximating polynomials for the x(6) functions f42 Vlll List of Figures Figure 1.1 A picture of the fish processing work cell 3 Figure 1.2 A picture of the mini excavator 4 Figure 2.1 The classical friction model 10 Figure 2.2 Generalized curve of friction as a function of velocity 11 Figure 2.3 Typical hysteresis effect in friction-velocity curve 14 Figure 3.1 The electrohydraulic positioning system (for one axis) 18 Figure 3.2 A simplified block diagram of the system 22 Figure 3.3 The quasi-static sinusoidal response of the open loop system 23 Figure 3.4 Friction-velocity curve at very low velocities 24 Figure 3.5 The estimated inertial force in the quasi-static experiment 24 Figure 3.6 The identification experiment for determining mass and friction coefficients 27 Figure 3.7 The identified friction model. 29 Figure 3.8 Actuator force F (solid line) and the estimated total friction Fj (dashed line) 30 Figure 4.1 System model with Coulomb friction 31 Figure 4.2 The Friedland-Park Coulomb friction observer 32 Figure 4.3 The velocity filter 34 Figure 4.4 The combined observer 35 Figure 4.5 Off-line estimation of dynamic friction using the Friedland-Park observer (p = 1.5, ka = 156) 37 Figure 4.6 Friction as a function of velocity obtained using the Friedland-Park observer (solid line) compared to the identified model of Chapter 3 (dashed line). . . . 38 Figure 4.7 Investigation of the convergence rate for the Friedland-Park observer 38 Figure 5.1 Block diagram of the acceleration feedback control system 4f Figure 5.2 Ideal response to the desired position trajectory 44 Figure 5.3 Position tracking without friction compensation (PD control) 46 ix Figure 5.4 Position tracking with friction compensation: model-based approach 47 Figure 5.5 Position tracking with friction compensation: observer-based approach 48 Figure 5.6 Comparison of the position tracking error profiles 49 Figure 5.7 Comparison of the velocity tracking error profiles 50 Figure 6.f Schematic of the mini excavator 52 Figure 6.2 The four pilot-operated main valves in the mini excavator 54 Figure 6.3 Variable-displacement pumps in the mini excavator 55 Figure 6.4 The bucket degree of freedom. 56 Figure 6.5 The curves x(9) obtained by fully extending and retracting each actuator. . . . 59 Figure 6.6 Approximation errors using 5th order polynomials 59 Figure 6.7 The functions J(0) for full angular range of each degree of freedom 60 Figure 6.8 The stick actuation system with the modified pilot stage 6f Figure 6.9 D P W M current signals for u = 1.5V 63 Figure 7.1 The new pilot stage with on/off switching valves 65 Figure 7.2 Overlap model of the main valve 66 Figure 7.3 Nonlinear model of a single-rod hydraulic cylinder driven by DPWM-operated pilot valves 69 Figure 7.4 Open loop experiment: input voltage and pilot pressure signals 71 Figure 7.5 Open loop experiment: input voltage, pump pressures, load pressures, and piston displacement. 72 Figure 7.6 Open loop experiment: input voltage, pilot pressure difference, estimated piston velocity, and calculated force 73 Figure 7.7 Pilot pressure difference (zoomed) 74 Figure 7.8 System block diagram 74 Figure 7.9 Open loop system identification experiment using the stick actuator 75 Figure 7.10 Measured (solid line) and simulated (dashed line) output (pilot pressure difference) for two different input signals 78 Figure 7.11 Residual analysis of the estimated model for the pilot stage 79 Figure 7.12 Result of the correlation analysis for the pilot stage 79 x Figure 7.13 Comparison of the Bode plots for the pilot stage (solid line = from spectral analysis, dashed line = from the estimated model) 80 Figure 7.14 Measured (solid line) and simulated (dashed line) output (piston velocity) for two different input signals 82 Figure 7.15 Residual analysis of the estimated model for the main stage 82 Figure 7.16 Result of correlation analysis for the main stage 83 Figure 7.17 Comparison of the Bode plots for the main stage (solid line = from spectral analysis, dashed line = from the estimated model) 84 Figure 7.18 Closed loop position control experiment (to study the actuator friction). . . . 85 Figure 8.1 Gravitational forces on the backhoe links 88 Figure 8.2 Measured (solid line) and predicted (dashed line) actuator torques for the static condition when there is no load in the bucket 93 Figure 8.3 The measured joint torques with load (o) and the estimated no-load torques (*) 95 Figure 8.4 Bucket actuator: (a) forces measured by the load pin and pressure sensors, (b) calculated static Friction 96 Figure 9.1 First-order high-pass and low-pass filters 108 Figure 9.2 Open loop dynamic experiment: joint angles and velocities 112 Figure 9.3 Low-pass-filtered joint torques (solid lines) and their estimation (dashed lines) using the identified parameters 114 Figure 9.4 Low-pass-filtered joint torques (solid lines) and their estimation (dashed lines) using the identified parameters (cross validation with a different input) 115 Figure A . l Schematic diagram of the basic planar manipulator 126 Figure A . 2 The interface circuit 129 Figure B . l Schematic diagram of the main valve bank in the mini excavator 133 Figure B .2 Simplified diagram of the main valve system of the mini excavator 134 Figure B.3 Schematic diagram of the original pilot system of the mini excavator 136 Figure B.4 Typical signals of the analog to D P W M converter circuit 137 xi Figure B.5 Analog to D P W M converter circuit 138 Figure B .6 M i n i excavator along with the assigned D - H coordinate frames 141 Figure B.7 Projection of the links onto the vertical plane 144 Figure C . l Digital differentiator f45 Figure C.2 Amplitude response of a typical off-line low-pass filtered differentiator. . . . f 46 Figure C.3 Digital low-pass filter 147 xii Graphic Symbols hydraulic line pilot line lines crossing lines connection hydraulic reservoir (tank) hydraulic accumulator, gas charged filter or strainer electric motor linear actuator with single-rod piston hydraulic reversible motor unidirectional fixed displacement pump variable displacement pump pressure-compensated var. disp. pump cooler fixed orifice variable flow control orifice check valve pressure reliefe valve manually operated (valve) solenoid operated (valve) spring centered pilot controlled (valve) two-position, two-way flow control valve three-position, four-way flow control valve three-position, six-way flow control valve logic valve manual shut off valve flow control servovalve Acknowledgments I would like to express my first and foremost gratitude to my supervisors, Professor P.D. Lawrence and Professor C.W. de Silva, for their guidance and encouragement in the course of this research. Their patience and support are sincerely appreciated. I express my special gratitude to Dr. S.E. Salcudean whose useful guidelines were enlightening in many stages of my thesis work. I am very thankful to the external examiner, Professor Richard Burton, for his careful reading of my thesis. The time and advice provided by the members of my Ph.D. examination committee are also gratefully acknowledged. My gratitude extends to the technical staff of the robotics group, in particular, Dan Chan, Simon Bachmann, and Icarus Chau. I am thankful to Pertti Peussa, the visiting engineer from Finland, for his valuable experimental contributions to my work. I also appreciate the assistance of our kind and hard-working secretaries Leslie Nichols and Doris Metcalf. My deepest gratitude goes to the members of my family for their love, encouragement, and support. In particular, 1 am heavily indebted to my parents as I believe that my academic achievements are the result of the high value they have always placed on education. Finally, I wish to thank my friends, Mahan Movassaghi, Masoud Khoshzaban, John Hu, Ajay Agrawal, Keyvan Hashtrudi-Zaad, Wing Poon, Mohammad Sameti, and Ying Cui for their generous help of various kinds in the course of my education at UBC. XV Dedication To my dear Farnaz, whose unconditional love, support, and encouragement made it all possible. xvi Chapter 1 Introduction 1.1 Preliminary Remarks The knowledge of manipulator dynamics is very crucial in most robot control schemes that are reported in the literature. The computed-torque approach, which yields a family of position control schemes that work well in practice, is a well-known example of such model-based controllers [39, 64]. The structural dynamics of a manipulator with rigid links is, in general, expressed in the form of nonlinear and coupled, multivariable differential equations. Assuming that the manipulator kinematics is known, its dynamic equations can be expressed in terms of the dynamic parameters such as mass and moments of inertia of the links and associated geometric parameters [8]. These parameters are also required for implementation of real-time payload estimation, force control, and master-slave force-reflecting teleoperated control. The dynamic parameters of a robot can be determined by employing one of the following three methods [45]: 1 . Using Computer-Aided-Design ( C A D ) models of the links 2. Direct physical measurement of the link parameters 3. Indirectly, from experimental data The first method is prone to modeling errors attributed to complex structural components such as bearings, bolts and pins, and also requires a precise knowledge of the link geometry. The drawback of the second method is that cross-coupling inertia parameters cannot be determined. A l so , the approach is tedious and requires disassembling of the manipulator. The third method which involves the use of planned manipulator motions to generate the identification data is the most preferred of the three methods, particularly from control point of view. This method is less time-consuming than the other two and has been used extensively in the literature [8, 36, 45, 40, 41, 42, 28]. This approach, in a general sense, w i l l be used in this thesis in several sections, as the present research concerns identification of the unknown dynamic parameters in hydraulic manipulators. Friction is present in virtually all mechanical control systems including electric, pneumatic, and hydraulic manipulators. It can pose a serious challenge to achieving high quality performance of 1 Chapter 1. Introduction these machines. Friction can lead to tracking errors, limit cycle oscillations, and undesirable stick-slip motion. In some mechanisms, friction may even dominate the forces of motion [2]. Importance of friction is often ignored in the development of control systems because it is generally thought to be unpredictable, insignificant, or unmodelable. Recent investigations on various manipulators show, however, that friction is both repeatable and predictable in carefully controlled environments [2, 57]. A s a result, friction modeling, estimation, and compensation has received considerable attention from researchers. Numerous experimental investigations reported in the literature show that significant performance improvement can be achieved by proper compensation for friction [49, 11, 12, 57, 75, 82, 59]. Hydraulic actuation systems are the preferred choice in powerful robots that are needed for heavy-duty tasks. They can be found in many mobile, airborne, and stationary applications. Hydraulic systems have many unique features compared to other actuation systems [50, 16]. Major advantages of hydraulic systems are as follows: • They can apply very large forces (and torques). • They have a large torque to inertia ratio and acceleration capacity. High loop gains and bandwidths are possible with these actuators in a servo loop. • They have high stiffness, that is, their speed does not drop too much as loads are applied. • They are self-cooling and can apply large forces for an extended duration of time. The hydraulic fluid carries away the heat generated by internal losses and also acts as a lubricant. Some disadvantages that limit the use of hydraulic systems are as follows: • Hydraulic power is not as easily available as that of electric power. The latter is more attractive in stationary applications. • Power conversion efficiency is typically lower than that of electric actuators. • The components are usually costly due to small allowable tolerances. • F lu id dirt and contamination can degrade their performance. 2 Chapter 1. Introduction • Control design procedure can be difficult due to the complicated nonlinear differential equations that govern these systems. 1.2 Motivation and General Objective Knowledge of the important manipulator parameters can be used to improve control of the machine, identify potential safety problems, and predict machine wear or possible failure. This thesis addresses the estimation of these parameters in two industrial machines that require improved control. One machine has a performance that is limited by friction, while the performance of the second machine is dominated by structural dynamics. Friction is inherently present in various parts of hydraulic actuation systems and can deteriorate their performance. The reported work on friction estimation and compensation has primarily focused on electric motor-driven robots and machine tools. To our best knowledge, there is no published work on friction modeling, estimation, and compensation for hydraulically-actuated manipulators. A planar Cartesian electrohydraulic manipulator is used in this thesis for experimental investigation of the friction related issues. This manipulator is an integral part of an automated fish processing machine which has been developed in our laboratory [17]. Due to its simple kinematic structure, which corresponds to a Cartesian manipulator, only frictional effects are studied for this system. A picture of the fish processing machine is shown in Figure 1.1. Technical specifications of the prototype machine can be found in Appendix A . Figure 1.1 A picture of the fish processing work cell. 3 Chapter 1. Introduction Next, identification of the structural parameters is studied for another industrial manipulator (a mini excavator) which is available in our laboratory. Unlike the Cartesian fish processing manipulator, structural dynamics play a very important role in the control of this machine. Thus, the ground work for experimental identification of both static and dynamic parameters of this manipulator is carried out in this thesis. Friction related issues are also studied for this second manipulator, but our emphasis w i l l be on the identification of the link (structural) parameters. The mini excavator is a typical heavy-duty mobile hydraulic machine. A picture of the machine is shown in Figure 1.2. Technical specifications of the prototype mini excavator can be found in Appendix B . Figure 1.2 A picture of the mini excavator. instrumentation and retrofitting of the original prototype machines, and related issues w i l l be addressed as well , in this thesis. 1.3 Previous Work Many algorithms have been formulated by researchers for experimental determination of the dynamic parameters for serial-link rigid-body manipulators. Khosla and Kanade [36] performed one of the early work in this area. They outlined some fundamental properties of the Newton-Euler formulation of robot dynamics from the viewpoint of parameter identification and showed that the dynamic model can be transformed into a form which is linear in parameters. They developed both on-line and off-line parameter estimation procedures and presented simulation results for identifying the dynamic parameters of both a cylindrical robot and the C M U Direct Drive A r m II. In [8], 4 Chapter 1. Introduction Atkeson et al. studied estimation of the inertial parameters of manipulator loads and links. They experimentally verified their algorithms using a P U M A 600 industrial robot and the M I T serial link direct drive arm. Categorization of parameters in a dynamic model of robots has been considered by Khosla in [35]. According to his work, all the dynamic parameters of a manipulator can be classified into three categories: uniquely identifiable, identifiable in linear combinations, and unidentifiable. Yoshida et al. [83] proposed two identification methods and examined them experimentally to estimate the dynamic parameters of a P U M A 560 robot. In [45], L u et al. presented a practical approach for determining the friction parameters along with the parameters related to moments of inertia and mass in a robot manipulator. They demonstrated their method experimentally on a closed loop direct drive arm. Identification of the gravitational (mass-related) parameters of robot links using static experiments can be found in the work by M a and Hollerbach [47] and Yoshida et al. [83]. Estimation of a manipulator payload has been addressed by Paul [54] and Atkeson et al. [8]. Similarly, a large number of references can be found in the literature on friction modeling, estimation, and compensation. For example, in [2], Armstrong-Helouvry introduced various frictional effects and reported experimental investigations on a P U M A 560 arm. In their landmark paper [5], Armstrong-Helouvry et al. presented a comprehensive survey on the control of machines with friction. De Si lva and Shibly [18] employed an experimentally-based model to represent friction in robotic joints and transmissions, for accurate control of robot manipulators. Among other works one can refer to Friedland et al. [27, 26, 23, 49], Canudas de Wit et al. [13, 12, 11], Phillips and Bal lou [57], Pu et al. [59], Ehrich [22], Yang and Chu [82], Dupont and Dunlap [21], and C a i and Song [10]. Computer-assisted control of heavy-duty mobile hydraulic machines has received considerable attention by researchers at the University of British Columbia ( U B C ) . Sepehri studied dynamic simulation and control of a Caterpillar 215B excavator in his Ph.D. thesis [61]. He discussed actuator dynamics for simulation purposes and also proposed a technique for teleoperated control of heavy-duty hydraulic manipulators with* coupled actuation system. A summary of his work can be found in [62]. In [34], Khoshzaban et al. investigated dynamic calibration of the typical mobile hydraulic machines. They used the Newton-Euler formulation to derive a comprehensive dynamic model of the system (as a hybrid open-and-closed-chain structure) and attempted to identify al l the unknown parameters by using the least squares estimation technique. The experiments reported in [34] were 5 Chapter 1. Introduction carried out on a Caterpillar 215B excavator. Zhu investigated master-slave force-reflecting resolved motion control of mobile hydraulic machines in his Master's thesis [85]. Further work in this area was performed by Lawrence et al. [38, 37] and Parker et al. [53]. 1.4 Thesis Overview This thesis is concerned with modeling and experimental investigation of the frictional effects and structural dynamics in industrial hydraulic manipulators. Our goal in this regard is to improve the performance of these manipulators through the insight gained in this manner, and by incorporating the identified effects in the controller design. The thesis consists of ten chapters and four appendices. The experimental investigations are divided to two parts. Part f is dedicated to the Cartesian electrohydraulic manipulator and consists of Chapters 3, 4, and 5. Part ff is dedicated to the work on the mini excavator and consists of Chapters 6 through 9. The least squares estimation technique is used widely in this thesis for identification of the inertia and friction related parameters for both manipulators. This can be considered as the principal connection between the two parts of the thesis. A brief description of each chapter is given in what follows. Chapter 2 Friction in Machines: Various frictional effects are explored in this chapter. Both classical and modern approaches to modeling of friction are presented. Behavior of friction in grease-lubricated metal-metal contacts is discussed in detail. In particular, four regimes that are typically observed in the friction-velocity curve for this type of junctions, are presented. Chapter 3 Analysis and Modeling of Friction in the Cartesian Manipulator: In this chapter, first the structure of the Cartesian electrohydraulic manipulator that has been developed in our laboratory for fish processing, is described. The friction-velocity curve at very low velocities is obtained next, by carrying out a quasi-static experiment. A friction model is adopted from literature, which is linear in parameters and is able to capture Coulomb, viscous, and Stribeck components of friction. Parameters of the model and mass of the moving object are then determined by applying the least squares estimation technique to experimental data. The results are compared with the general properties of friction discussed in Chapter 2. Chapter 4 Observer-Based Estimation of Friction in the Cartesian Manipulator: A nonlinear observer for the Coulomb friction parameter that is adopted from literature, is presented in this chapter. 6 Chapter 1. Introduction By introducing a modification, the observer is used for real-time estimation of friction in the Cartesian electrohydraulic manipulator. Convergence speed of the observer is analyzed and a methodology is developed for choosing its gains. The experimental results are compared to those obtained using the model-based approach of Chapter 3. Chapter 5 Friction-Compensated Position Tracking of the Cartesian Manipulator: A novel approach for friction compensation is presented in this chapter. Our approach combines friction estimation with acceleration feedback control. Both model-based and observer-based friction compensation techniques are tested on the system and their performance is compared to that of Proportional-Plus-Derivative (PD) control. Chapter 6 Instrumentation of the Mini Excavator: The displacement and force sensors installed on the machine are described in this chapter. An innovative technique is then presented for joint torque measurement. The technique employs load pin readings to measure the reaction force of the cylinders at their hinges. Polynomial approximation is then used to efficiently calculate the joint torques. The concept of Differential-Pulse-Width-Modulation (DPWM) is explained and modification of the original pilot stage using DPWM-operated solenoid valves is discussed. Chapter 7 Modeling and Identification of the Actuator Dynamics: In this chapter, nonlinear differential equations of the actuator dynamics are derived symbolically and the open loop step response of a single actuator is analyzed. System identification is then employed to obtain linear models for the pilot and main stages of the actuation system. Various validation tests are performed on each identified model. Actuator friction is also briefly investigated in this chapter. Chapter 8 Estimation of the Gravitational Parameters and Bucket Load: The gravitational parameters of the backhoe links are statically estimated in this chapter. A decoupled form of the static torque equations is used for this purpose. A novel approach for real-time bucket load estimation in the static condition is also presented. The algorithm is experimentally tested on the real system. Chapter 9 Estimation of the Full Dynamic Parameters and Friction Coefficients: The Euler-Lagrange formulation is used in this chapter to derive the symbolic differential equations that govern the motion of the backhoe links. The full dynamic parameters are then defined and the torque equations are converted into a form that is linear in these parameters. Subsequently, joint friction 7 Chapter 1. Introduction coefficients are included in the parameter set. Least squares estimation of the full parameter vector from experimental data is presented and the estimated gravitational parameters are compared with the static result of Chapter 8. Chapter 10 Conclusions: Contributions of the research reported in the thesis and suggestions for further work are summarized in this chapter. Chapter 2 Friction in Machines 2.1 Importance of Friction There are several important nonlinearities such as friction, backlash, dead-zone, and actuator saturation that are inherent in mechanical control systems. Friction is a complex, natural phenomenon which is not yet completely understood, for the purpose of accurate modeling. Although friction may be a desirable effect, as it is for brakes, it is generally a detrimental effect for servo control. In applying feedback control strategies to moving objects, friction is inevitably among the forces of motion. Contributions from other sources of motion, such as non-dissipative structural dynamics, have long been analyzed and incorporated in control design. But, the effects of frictional forces are often studied using simplified models. It is known, however, that a good understanding of friction is necessary to accurately predict, simulate, and compensate for its undesirable effects. Researchers in a number of different fields have studied the problem of identifying and modeling the mechanism and dynamics of friction. Many of these investigations fall within the area of "tribology" (Greek for the study of rubbing) which emerged in England in the 1930's [5]. Tribology has made considerable progress in explaining the microscopic details of friction and developing predictive models. Tribologists are mostly interested in the consequences of steady-state rubbing in order to develop means to reduce wear and aging of mechanical components, while control engineers are typically interested in the effects of friction on dynamic behavior of the system under study. There has been a recent effort by control engineers to study and model friction dynamics [2, 4, 5, 13, 21, 22], which has proved to be quite enlightening. 2.2 The Classical Model of Friction According to Leonardo da Vinci (1519), friction is a force that opposes the motion, is proportional to load and is also independent of the contact area [2]. Da Vinci's simplified model was further developed by Charles Augustin Coulomb in 1785 and was expressed in the following form: Fc = fcsgn(v) (2.1) 9 Chapter 2. Friction in Machines In this text, capital letter F is used for force, while small letter / represents the corresponding parameter. Thus, Coulomb friction Fc acts as a constant retarding force fc for nonzero velocities v and is discontinuous at zero velocity. Note that fc is proportional to the normal load. The idea of static friction or stiction was then introduced in 1833 by Arthur Morin. Static friction (Fs) simply reflects the fact that some force is required to initiate the sliding (or rolling) motion between two surfaces with friction. In f866, Reynolds laid the ground work for the viscous friction component that is caused by energy dissipation in fluid flow, using the governing equations. Linear viscous friction is represented by Fv = Bv (2.2) where Fv is the viscous friction force and B is the.viscous friction coefficient (damping coefficient). The combination of Coulomb, static, and viscous components form the classical model of friction which may be shown as in Figure 2.1. Static Friction Breakaway Force Level of Coulomb Friction •fc Slope = B Corresponds to Viscous Friction Static Friction Regime Figure 2.1 The classical friction model. The classical model is commonly used in engineering applications and can be expressed as follows: ( fcsgn(v) + Bv, v ^ 0 (2.3) {FS(F), V = 0 where Fj is the net friction and Fs is the static friction component which is equal to the applied force F under zero velocity conditions. 10 Chapter 2. Friction in Machines 2.3 Friction as a Function of Velocity: Recent Observations Feedback control is often applied to mechanical systems with the objective of achieving improved performance. These systems typically involve metal-to-metal contact with grease or oil lubrication. From the viewpoint of control, friction as a function of the system state is of primary interest. Dynamic friction results from relative motion between the sliding components and hence is a function of velocity. This section summarizes some results from tribology on modeling of friction as a function of velocity in lubricated metal-to-metal junctions. A fluid lubricant is used to provide a fluid barrier between rubbing metal parts, thereby converting dry friction into viscous friction, and as a result significantly reducing wear. Usually the fluid is drawn into the sliding interface by the motion of the parts, in a process called "hydrodynamic lubrication". The fluid film is maintained only when the velocity is above some minimum value. Below this velocity, solid-to-solid contact occurs. As Armstrong-Helouvry has elaborated in [2, 5], there are four regimes in a typical, lubricated, metal-to-metal contact. These regimes contribute to the dynamics that a controller confronts as the moving parts accelerate away from the initial rest condition. The regimes correspond to a more realistic model than what is shown in Figure 2.1, and are demonstrated in the generalized characteristic curve of Figure 2.2. The first regime corresponds to static friction and the rest contribute to the dynamic friction. • '• ' • v Figure 2.2 Generalized curve of friction as a function of velocity. 2.3.1 Regime I: static friction It is well known in mechanics that contacts are compliant in both the normal and tangential direction. Thus, the common assumption that there is no motion while in the static friction regime 11 Regime I Chapter 2. Friction in Machines is not quite true. Experimental investigations have shown that for very small motions, a junction in static friction behaves similar to a spring. Up to a critical force at which breakaway occurs, there is "presliding displacement" which is approximately a linear function of the applied force. The breakaway displacement occurs with deflections on the order of 2 to 5 microns for steel junctions. The level of static friction breakaway force increases with the.time a junction spends in the "stuck" condition, i.e., dwell time. This phenomenon is referred to as "rising static friction". Presliding displacement is a result of elastic deformation of junctions, while rising static friction is caused by plastic deformation of the boundary film and asperities. When static friction in a mechanism is indeed greater than dynamic friction, intermittent motion known as "stick-slip" occurs. It refers to the repeated movement into and out of the static regime. The undesirable chattering of mechanisms is often caused by stick-slip friction. In [33], Karnopp presented a model for describing the effect of stick-slip friction in mechanical dynamic system. In this model, he assumed that static friction persists for a range of very low velocities and dynamic friction applies to velocities outside this range. Karnopp used this model for computer simulation of stick-slip to avoid numerical stiffness problems. Karnopp's model has been widely used by researchers. It has been pointed out in the literature that static friction is not really a force of friction, as it is neither dissipative nor a consequence of sliding; but it is in fact a force of constraint. Thus, a better name for static friction would be "tangential force". 2.3.2 Regime II: boundary lubrication At very low velocities, fluid lubrication is not effective as the velocity is not high enough to build a fluid film between the sliding surfaces. Then, additive to the boundary layer can provide the lubrication. Due to the solid-to-solid contact in this regime, shearing occurs in the boundary lubricant and it is often assumed that friction in boundary lubrication (regime II) is higher than it is for fluid lubrication (regimes III and IV). This, however, is not always the case and certain boundary lubricants do reduce static friction to a level below Coulomb friction. Boundary lubricants are standard additives in machine grease or oil and typically constitute less than 2% of the total fluid. Boundary lubrication is important to the control engineer because of the role it plays in stick-slip. If the level of static friction is reduced below the Coulomb friction, there will be no destabilizing 12 Chapter 2. Friction in Machines negative viscous friction, and stick-slip will be eliminated. 2.3.3 Regime III: partial fluid lubrication As the fluid lubricant is brought into the load bearing region by either sliding or rolling, some of it is displaced by the load, pressure. But, viscosity prevents all of the lubricant from escaping and thus a fluid film is formed; the greater the fluid viscosity or motion velocity, the thicker the fluid film. When the fluid film is not thicker than the height of asperities, some solid-to-solid contact will result, thereby giving rise to partial fluid lubrication. Solid-to-solid contact decreases with the lubrication, reducing friction and increasing the acceleration of the moving part. Thus, in most cases, by increasing velocity, friction will drop gradually from the static friction level to the dynamic friction level, and not abruptly, as predicted by the classical friction model of Figure 2.f. The phenomenon of negative viscous friction is often called "Stribeck effect". From the system control viewpoint, a negative-sloped friction curve is undesirable because it is destabilizing; a small increase in velocity causes a decrease in the friction force, thereby further increasing velocity. 2.3.4 Regime IV: full fluid lubrication As the velocity is increased, solid-to-solid contact is eliminated and full fluid lubrication is achieved. In this regime, wear is reduced by orders of magnitude and friction is considered well-behaved. The main objective of lubrication engineering is in fact to maintain full fluid lubrication effectively and at low cost. This regime can be adequately modeled using a combination of Coulomb and viscous friction. Several models have been proposed for friction as a function of velocity that incorporate the Stribeck effect. These models have been derived to approximate empirical data. They are generally a combination of Coulomb friction Fc, viscous friction Fv, and a Stribeck friction term Fstr that also includes static friction: A variety of functions have been proposed in the literature for the Stribeck friction term. In [30], Hess and Soom employed the following function: F/(v) = fcsgn(v) + Bv + Fstr(v)sgn(v) (2.4) Fstr(v) -fsc (2.5) 1 + (V/Vsf 13 Chapter 2. Friction in Machines where fsc = fs — fc = difference between breakaway force (/s) and Coulomb friction (/c) vs = critical velocity related to Stribeck phenomenon In [2], Armstrong-Helouvry used the following Gaussian function: F,tr(v) = f s c e - ( v / v - r (2.6) In [12], Canudas de Wit et al. proposed a simple friction model that captures the Stribeck effect at low velocities and is linear in parameters. Their model is particularly suitable for use in adaptive control, and can be expressed as follows: Ff(v) = a0sgn(v) + a^v^2 sgn(v) + a2v (2.7) where a0, a\, and a2 are the model parameters. This function will be employed in Chapter 3 for modeling of friction in the Cartesian electrohydraulic manipulator. 2.4 Frictional Memory and Dynamic Modeling of Friction Considerable empirical evidence indicates that friction does not immediately respond to a change in velocity. There is a time lag between a change in velocity and the resulting change in friction to its new steady-state value. This phenomenon is called "frictional memory". The classical friction models that use static maps between velocity and friction ignore this phenomenon [13]. The classical models cannot explain the hysteresis behavior that has been observed in the friction-velocity curve for unsteady velocities. As shown in Figure 2.3, the friction-velocity curve for accelerating conditions is different from that of decelerating conditions. L Tnrrpasi'ncr \. v. Velocity Decreasing ——«• \ Velocity — Figure 2.3 Typical hysteresis effect in friction-velocity curve. 14 Chapter 2. Friction in Machines A friction model involving dynamics is necessary to represent this phenomenon. In [30], Hess and Soom used a simple time lag (pure delay) to explain their observations in experiments with fluid-lubricated contacts under unsteady sliding velocity. Their model can be expressed in the following form [5]: Ff(t) = Fvel(v(t - At)) (2.8) where Fj(t) = instantaneous frictional force Fvei(t) = friction as a function of steady-state velocity At = characteristic time lag Their careful measurements showed that At is between 3 and 9 milliseconds depending on the supported load and the lubricant that is used. According to Rice and Ruina [60] and Dupont and Dunlap [20, 19], evidence supports a state-variable model, at extremely low velocities. The fact that friction is dependent not only on the current sliding conditions but also on the previous sliding history, can be expressed using the following state-variable model: Ff = Ff(v,9) (2.9) where 6 represents the state of the surface. This state does not change instantaneously with velocity and we have 6 = G(6,v) (2.10) The steady-state friction-veiocity characteristic can be obtained by setting G equal to zero. The state-variable model can be identified from the frictional response to an unsteady velocity at the surface. A seven-parameter friction model has been proposed by Armstrong-Helouvry et al. in [5]. This empirically-motivated integrated model is able to describe all the frictional effects that were explained in this chapter. The two time-dependent properties of friction, i.e., rising static friction and frictional 15 Chapter 2. Friction in Machines memory do not play an important role in the applications that are studied in this thesis. As a result, only static models of friction are employed throughout this research. The models introduced in this chapter can be used for friction modeling, simulation, and compensation. However, it is important to note that in reality friction is a function of: • type, location, and age of the contact • type, condition, and temperature of the lubricant For example, friction inside the actuator seals varies with the actuator age, profile of the actuator barrel, and condition of the hydraulic fluid. Thus, in general, friction is a time-varying phenomenon. This has to be taken into account in the design of the controllers that intend to compensate for friction. 16 Part I The Cartesian Electrohydraulic Manipulator An Industrial Automation Laboratory has been established in the Mechanical Engineering Department primarily for research and development of advanced technology in automation of the fish processing industry [17]. An automated machine for mechanical processing of salmon has been developed. In this industrial prototype,' the fish are transported using a conveyor system. When a fish reaches the cutter blade, its image is taken by a digital camera. A computer analyzes the image and generates the desired position coordinates (x,y) for removing the head, with minimal wastage of meat. The control system of the two independent (X and Y) electrohydraulic actuators then performs precise positioning of the cutter blade assembly. In order to meet the process requirements, this motion has to be fast, accurate, smooth, and non-oscillatory. The cutter positioning mechanism is in fact a Cartesian electrohydraulic manipulator. The sliding surfaces in the two axes generate frictional forces. Since friction can degrade the system performance, it is undertaken to identify it, with the objective of compensation for its effects in closed loop control. Part I of the thesis addresses these issues for the Cartesian manipulator. Specifications of the prototype manipulator are given in Appendix A. 17 Chapter 3 Analysis and Modeling of Friction in the Cartesian Manipulator 3.1 The Experimental System The prototype electrohydraulic manipulator is of Cartesian type with two orthogonal prismatic joints. The purpose of the manipulator is to position a pneumatically actuated cutter blade with respect to each fish that is moved by the system conveyor. The desired (x,y) positions are determined by an on-line computer vision system. The positioning manipulator has two control inputs which are the currents applied to the electrohydraulic proportional valves for the X and Y directions. Since the two degrees of freedom in the manipulator are quite similar and also independent with regard to operation, only the X direction will be studied. Figure 3.1 shows the simplified schematic diagram of the manipulator, for the X axis. 4> Lubricated Metal Guideways •o-; Actuation Pressure Transducers ~ ~ — — — _ T Q, Cutter Blade and Carriage M Current ~v Flow Control Servovalve Q i Position Transducer -Hydraulic Piston and Cylinder Figure 3.1 The electrohydraulic positioning system (for one axis). The servovalve consists of a pilot stage with torque-motor-actuated flapper and a spring-centered boost stage with double sliding spool arrangement (see Appendix A for details). The differential pressure from the pilot is applied across both spools-of the boost stage and is balanced by the centering 18 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator springs, producing a spool position that is proportional to the differential pressure and therefore to the input current. The nominal full flow current of this two-stage servovalve is / = 42 mA. A Temposonics™ linear displacement transducer is installed at the head of the hydraulic cylinder. This industrial magnetostrictive sensor precisely senses the position of an external magnet, which is connected to the piston, from the time interval between an interrogation pulse and a return pulse. The specific sensor that is used has a resolution of 0.025 mm and its measurement update time is less than 1 ms. Two gage-pressure transducers are also installed on the head and rod sides of the cylinder, in order to measure the fluid pressures Pi and P2, respectively. Note that there is no velocity or acceleration sensor in the system. All the sensor outputs (including pressure transducers) are low-pass filtered by first order anti-aliasing RC filters. A sampling frequency of fs = lk Hz is used to implement the digital controllers; accordingly, the cut off frequency of the filters is set at 200 Hz. A 12 bit PC-based data acquisition system is used for data collection and graphical display. The programs have been written in C++. The physical parameter values of the prototype manipulator are listed in Table 3.1. Note that all the experiments reported in this thesis were carried out at the lab temperature (around 22 °Q. Table 3.1. Physical parameters of the prototype manipulator. Parameter Value Measured moving mass (M) 32.7Kg Head-side piston area (A\) 1.14 x 1(T3 m 2 Rod-side piston area (A2) 0.633 X IO"3 m 2 Maximum piston displacement (X) 50 x 10 _ 3 m Volume of each hose between the j servovalve and the cylinder (V/J 8 . 9 x l 0 _ 5 m 3 1 3.2 Modeling of the Electrohydraulic System A dynamic model is derived in this section for the electrohydraulic manipulator. This analysis is 19 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator intended primarily to emphasize the nonlinear nature of the actuator dynamics. Similar approaches to modeling of asymmetric hydraulic actuators have been reported in [24, 78]. Primarily, there are three types of nonlinearities in the servovalve system, namely, the basic flow equation through an orifice, the hysteresis resulting from electromagnetic characteristics of the torque motor, and the flow forces on the valve spools [50, 78]. Displacement of the valve spools from their null position creates a pressure difference across the single-rod hydraulic actuator and the resulting fluid flow causes a change in the position of the piston. The force applied by the actuator can be calculated from cylinder pressures. Newton's second law applied to the actuator yields: F = PXAX - P2A2 = Mx + Fj (3.1) where F = actuator force, Fj = opposing frictional force in the X direction, x = piston displacement, M — equivalent moving mass in the X direction. The amount of fluid flow to the head-side (Qi) and from the rod-side (Q2) of the cylinder is a function of both the valve spool position and cylinder pressures. Assuming that the valve is of critical-center type with matched and symmetrical ports (see [50] for definitions), the relationship can be expressed in the following form [24]: ' Qi = Kxv [s{xvyPs - Pi + s(-xv)y/Fi] (3.2) , Q2 = Kxv [s(xv)y/P2~ + S(-Xv)T/Ps - P2\ valve spool displacement, a fixed gain, supply pressure, where xv — K = P, = 20 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator and s(xv) is a switching function defined as below: S(Xy) = { ( 1 Xv>0 0 xv < 0 (3.3) Furthermore, using the continuity principle, and taking fluid compressibility into account, one can write [50]: Q 1 = A a i + ^ ± ^ A Q2 = A 2 x - ^ f ± ^ P 2 (3.4) where, ft is the effective bulk modulus of the fluid. Next, define a state vector for the system as follows: x = [a>i x2 x3 x4] =[x x Pi P2 ]J (3.5) By combining equations (3.1), (3.2), and (3.4), one arrives at the following nonlinear state-space model of the system: ' X\ = x2 x2 = — [Aix3 - A2x4 - Ff] x4 Mxi + vh -0 A2(L - xi) + Vh [Kxv(s(xv)y/Ps - x3 + s(-xv)y/x^) - Axx2] [Kxv(s(xv)y/xlJr s(-xv)^Ps - x4) - A2x2] (3.6) Note that due to the nonlinear dynamics of the system, we will avoid using the frequency response experiments in this thesis. A simplified block diagram of the hydraulic manipulator is shown in Figure 3.2. In this diagram, the "Actuator dynamics" block represents the nonlinear hydraulic actuation system. Servovalve dynamics can also be included in this block. According to the state-space equations, the actuator dynamics depends on the position and velocity of the piston. 21 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator Actuation Current Applied Force i Actuator .< Dynamics M 1_ S Friction Dynamics V Figure 3.2 A simplified block diagram of the system. 1_ S Piston Position X Vossoughi and Donath [78] have discussed various nonlinearities that are present in electrohy-draulic systems. They used feedback linearization to compensate for these effects in the control design. Knowledge of the manipulator dynamics is not required in the control strategy that is proposed in this thesis. 3.3 Friction-Velocity Curve at Low Velocities [68] In this section, the characteristic of friction at very low velocities is investigated. In a quasi-static experiment, when the piston moves very slowly, the inertial force Mx in equation (3.1) can be neglected compared to Ff . Then, the applied force is a good representation of the frictional force. A sinusoidal current with an amplitude of 2.5mA ( 7% of the rated current) and a period of 100 seconds was applied to the servovalve. The pressures and the piston position were sampled every 2 seconds. Note that in this system, piston movement is quite limited. In view of this, the applied current to the servovalve, in open loop experiments, should be small. To estimate velocity from position signal (see Appendix C), a digital filter was employed off-line, which is in fact a high order, delay-free, low-pass-filtered differentiator. Figure 3.3 shows the result of this test. According to this figure, there is a significant amount of static friction that causes deadband nonlinearity in the open loop system [50]. The effect of this nonlinearity on the closed loop response will be explained later. The measured force and the estimated velocity were used to obtain the steady-state, friction-velocity characteristic, and is plotted in Figure 3.4. Note that only one response cycle is used to obtain this curve. The curve was found to be slightly different for different cycles, which is due to the random characteristics of friction. Obviously, the friction-velocity curve of Figure 3.4 corresponds 22 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator to the static friction and boundary lubrication regimes that were discussed in Chapter 2. Also note that the transition from static friction to dynamic friction regime is not abrupt as suggested by the classical friction model. Servovalve Current, i 4_ _| 1 . 1 1 1 1^ 1 1 ^ - - i — i i i —i - i- r 1 -I 1 1 i 1 0 20 40 60 80 100 120 140 160 180 s Piston Displacement, x _2000' 1 1 1 1 1 1 1 1 1 1 0 20 40 60 80 100 120 140 160 180 200 s Time Figure 3.3 The quasi-static sinusoidal response of the open loop system. 23 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator u s-< O P i TS c o •43 u 'S PH 2000 1000 -1000 -2000 Velocity (mm/s) Figure 3.4 Friction-velocity curve at very low velocities. Finally, acceleration was estimated by applying a similar digital filter to the velocity signal (to obtain x). As shown in Figure 3.5, the inertial force (Mx) is quite negligible compared to the applied force (F). x10"3 Mx 4 i 1 1 1 1 1 1 1 1 r _4 I L _ . i I I I I I i i I 0 20 40 60 80 100 120 140 160 180 200 Time (s) Figure 3.5 The estimated inertial force in the quasi-static experiment. 24 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator 3.4 Estimation of the Mass and Friction Coefficients [69] The least squares estimation technique (see Appendix D) is employed in this section for experimental determination of the mass and friction coefficients of the moving parts. For this purpose, a friction model is required which is linear in parameters. The model proposed by Canudas de Wit et al. [12], is adopted here: where ao, cx\, and a2 are the friction coefficients, v is the velocity and Fd is the dynamic friction. This symmetric model can capture Coulomb, viscous, and Stribeck components of friction. With this friction model, the equation of motion for dynamic conditions would be: The least squares estimation technique can be applied to equation (3.8) for experimental determination of the friction coefficients ao, «i, cx2, and the mass M. When the manipulator is at rest (v ~ 0), static friction exists which is equal to the applied force. The following interpolating function is used to incorporate static friction in the model: Here Ff is the net friction and Dv is the threshold velocity. This empirical function is adopted from [5, 13], where it has been used for a different purpose. A small neighborhood of zero velocity is defined by Dv, as in the model proposed by Karnopp [33]. According to equation (3.9), outside this neighborhood, friction is mainly dynamic in nature (a function of velocity). Inside the neighborhood, velocity is considered negligible, and static friction is dominant (a function of the applied force). This formulation is used to account for noise and numerical inaccuracy of the estimated velocity. Only off-line estimation of the parameters is considered here. Numerical differentiation of the position signal gives the velocity profile (see Appendix C). For satisfactory estimation, portions of the data that correspond to the static regime (\v\ < Dv) are removed. Acceleration measurement which is required in equation (3.8), is avoided by using the filtering technique employed by Hsu et Fd(v) = a0 + ai\v\' + a2\v\ sgn(v) (3.7) F = Mv + a0sgn(v) + ai\v\ ' sgn(v) + a2v (3.8) Ff = Fd + (F-Fd)e-MD-r (3.9) 25 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator al. [31] and Lu et al. [45]. The technique will be explained in detail in Chapter 9 for a more general case, ft requires applying the following first order low-pass filter to equation (3.8): i(s> = rrW (3-10) where TV > 0 is the time constant of the filter. 3.4.1 The Closed Loop Identification Experiment A proportional feedback controller (with a gain of K = 2000 ^ ) was used to close the feedback loop, at a sampling frequency of / s = 1 kHz. Figure 3.6 shows the experimental result obtained with the following sinusoidal reference trajectory: xd(t) = 0.020 cos (5irt) (3.11) Note that the nonlinear functions sgn(v) and \v^2sgn(v) in equation (3.8) justify the application of a single sinusoidal reference input for identification of the parameters, ft is observed from Figure 3.6 that: • Apparently, the response (x) lags the desired trajectory (xj). This is primarily due to the closed loop dynamics. The time lag may be reduced either by increasing the controller gain or by using a PD controller. Improved control of the system is addressed in Chapter 5. • The peaks of the actual position waveform (x) are somewhat flat. This is in fact a backlash behavior in the closed loop system which is caused by the deadband nonlinearity (mainly due to static friction) in the open loop system [50]. In other words, when the piston stops (as the direction of motion reverses), it arrives in the static friction regime. The servovalve current has to pass beyond some threshold value in order to overcome the static friction breakaway force. As a result, the piston will stay in the zero velocity condition for a finite duration of time. Note that in the experiment outlined above, we have intentionally used a low feedback gain in order to observe these phenomena more clearly. 26 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator Actual and Desired Positions, x & x. 0.02 k Velocity, v Actuator Force, F 1000 500 0 -500 1000 n 1 1 r I L_ n i— 1 r _l . I I l l_ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 s Time Figure 3.6 The identification experiment for determining mass and friction coefficients. 27 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator 3.4.2 Least Squares Estimation of the Parameters The chosen values for the design parameters are listed in Table 3.2. The assigned time constant of the low-pass filter defined in equation (3.10) corresponds to a cut-off frequency of l/(2irTv) = 113.7 Hz , which is almost half of the bandwidth chosen for the off-line velocity filter. The threshold velocity Dv is chosen experimentally, using a trial and error approach. Table 3.2. Values of the design parameters. Parameter Value Bandwidth of the off-line velocity filter (BW) 250 Hz Time constant of the low-pass filter (Tv) 1.4 ms Threshold velocity (Dv) 0.02 m/s The estimated parameters and their absolute and relative standard deviations are listed in Table 3.3. According to this table, the estimated mass is close to its measured value (32.7 Kg). Investigations showed that mass estimation is sensitive to the filter bandwidths. Thus, the measured value will be used to represent the moving mass in our future applications. Indeed, the estimation precision depends on the degree of excitation of the input signal. Regarding the coefficients a,-, as pointed out by Canudas de Wit et al. in [12], they are not unique and several sets of the parameter values might exist leading to an equivalent approximation. Table 3.3. Off-line estimation of the mass and friction coefficients. Parameter Estimated Value Estimated Standard Deviation Relative Standard Deviation M 33.8 Kg 0.7 Kg 2.1% a0 587.6 N 35.8 N 6.1% ai -1419.9 P— 201.0 2— V m / s 14.2% a2 2317.6 - 4 -m/s 259.8 -A-m/s 11.2% 28 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator The identified friction-velocity curve for the velocity range of the experiment is plotted in Figure 3.7. The Stribeck effect (negative viscous friction) is clearly observed in this figure. At high velocities, viscous friction is dominant. Thus, the experiment corresponds mainly to the partial fluid lubrication and full fluid lubrication regimes. Note that the level of friction in the dynamic condition is much lower than what was observed in the quasi-static condition. Similar behavior has been reported by other researchers such as Armstrong-Helouvry [2] and Phillips and Ballou [57]. c o to 6 cn W 1000 500 0 -500 -1000 t 1 1 1 a k ~r - — \ i i _| -0.5 -0.25 0 0.25 0.5 Velocity (m/s) Figure 3.7 The identified friction model. Experimental investigations show that the frictional behavior does not vary too much in our system. Thus, the identified fixed model can be used to estimate friction with a good accuracy. The force applied by the actuator and the estimated net friction calculated using equation (3.9) are plotted in Figure 3.8. This plot shows that friction consumes considerable amount of the hydraulic force. Thus, compensation for the frictional effects in the system is quite important. This issue will be addressed in Chapter 5. 29 Chapter 3. Analysis and Modeling of Friction in the Cartesian Manipulator 1000 -1000' ' 1 1 1 1 1 1 1 1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s Time Figure 3.8 Actuator force F (solid line) and the estimated total friction Fj (dashed line). An alternative approach for measuring the friction characteristics (see [2]) is to use a closed loop position control system and input a ramp signal as the desired position (a step signal as the desired velocity). The steady-state friction-velocity characteristics can be obtained by applying different desired trajectories to the system. This approach is not suitable for the Cartesian manipulator due to its small displacement range (5 cm). However, we have employed it in Chapter 7, in order to study the actuator friction characteristics of the mini excavator. 30 Chapter 4 Observer-Based Estimation of Friction in the Cartesian Manipulator 4.1 A Nonlinear Observer for Friction Parameter In this section, we introduce a nonlinear observer for the Coulomb-friction parameter, which has been proposed by Friedland and Park. The convergence rate of the observer will be analyzed as well. 4.1.1 The Friedland-Park friction observer Since the force applied by the actuator can be easily calculated from pressure readings (see equation (3.1)), the nonlinear observer proposed by Friedland and Park in [27] for Coulomb friction, appears to be suitable in the present application. The observer is briefly discussed in the following. Note that their approach is reformulated here in order to include the object mass M. System dynamics is assumed to be as shown by the block diagram in Figure 4.1. In this figure, the block denoted by "NL" represents the nonlinear friction element. Applied Force Coulomb Friction M a Mac sgn(v) NL V Piston Position X Figure 4.1 System model with Coulomb friction. The frictional force is assumed to be of classical Coulomb type, as given by: Fc = Macsgn(v) (4.1) To estimate the Coulomb-friction parameter ac, Friedland and Park proposed the following nonlinear observer: za = kafJ>\v\^ 1u.sgn(v) (4.2) 31 Chapter 4. Observer-Based Estimation of Friction in the Cartesian Manipulator where, ac is the estimated Coulomb-friction parameter, za denotes the state variable of the observer and the design parameters are the gain ka > 0 and the exponent p, > 0. The variable u represents the estimated acceleration and is given by: F - F , u = - - ^ (4.3) where, Fc is the estimated Coulomb friction; i.e., Fc = Macsgn(v) (4.4) Figure 4.2 shows a block diagram of the Friedland-Park nonlinear observer. V I NL2 NL1 u.-2 \L\V\ sgn(v) u 1 1^ sgn(v) F. 1_ M Figure 4.2 The Friedland-Park Coulomb friction observer. Defining estimation error as ea = ac — ac, where ac is the true friction parameter, Friedland and Park showed that the error dynamics is given by: ea = ac - kap,\v (4.5) Therefore, for a fixed true parameter (d c = 0), the estimation error converges asymptotically to zero if ka > 0 and v is bounded away from zero. By simulation, Friedland and Park also confirmed that the observer works well for more complicated friction models (such as the friction model in our system) where ac is a nonlinear function of velocity. 32 Chapter 4. Observer-Based Estimation of Friction in the Cartesian Manipulator 4.1.2 Convergence rate of the Friedland-Park observer Convergence rate of the Coulomb friction observer proposed by Friedland and Park, is analyzed now. This is an original contribution of the thesis. According to equation (4.5), the estimation error has linear, time-varying dynamics. Assuming a fixed true parameter (dc = 0), explicit solution of equation (4.5) for an initial estimation error ea(0), gives: T ea(T) = eo(0)e ° (4.6) Thus, nonzero velocity drives the estimation error to zero. Equation (4.6) can be written in the following form, to establish an analogy with first order linear time-invariant observers: ea(T) = ea(0)ea^T with &(T) = - (4.7) It follows that, the approximate pole of the observer is located at: p = -kaiilvf'1 (4.8) where, (.) denotes the mean value of the corresponding variable. Equation (4.8) can be used to choose the gain ka and the exponent /J, for a desired rate of convergence. 4.2 Simultaneous Estimation of Friction, Velocity, and Acceleration In [26], Friedland and Mentzelopoulou developed a combined observer for friction and velocity in the absence of direct velocity measurement in the system. The authors employed an observer for velocity estimation and then used this estimate (v) in the Friedland-Park algorithm. They showed local asymptotic stability for both velocity and friction estimations. As discussed by Tafazoli et al. in [68], the observer proposed by Friedland and Menzelopoulou must be modified to accommodate the static friction component in our system. In the modified observer, velocity estimation is performed independently of the friction estimation. The velocity observer is in fact the low-pass-filtered differentiator that is represented by the transfer function given in Figure 4.3 (see Appendix C). Note that kv > 0, is the observer gain (Tv = 1/KV, is the time constant of the filter). The estimated velocity is then used in the Friedland-Park algorithm. 33 Chapter 4. Observer-Based Estimation of Friction in the Cartesian Manipulator Position Estimated velocity X V s + kv • Figure 4.3 The velocity filter. Since the estimation update is poor at low velocities, the observer estimates dynamic friction only. As before, static friction has to be taken into account to obtain the net friction (see equation (3.9)): Ff = Fc + (F - Fc)e-^D^ Thus, the net frictional force can be computed using the following equation: pf = Fc+(F-Fcy-&D»y Likewise, the estimated acceleration would be ^_F-Fj _ {F - Fc) - [F - Fc)e-^)2 (4.9) (4.10) M M (4.11) The variable u in equation (4.3) represents the acceleration signal provided that the velocity is away from zero. By combining equations (4.3) and (4.11), we obtain: a = u\ (4.12) Thus, the estimated acceleration a is an enhanced version of u, as static friction is taken into account in its calculation. Figure 4.4 shows the final form of our new observer for velocity, friction, and acceleration. The design parameters for the observer are kv, ka, fi, and Dv. 34 Chapter 4. Observer-Based Estimation of Friction in the Cartesian Manipulator Measured Position Measured Chamber Pressures Velocity Filter s + K l -e -(V/DS NL2 NL1 1 :K 1 •* 1^1 fc C *( t - N Applied \ Force M M F=P 1A 1-P 2A 2 Estimates V —• c \ The Friedland-Park Coulomb Friction Observer / Figure 4.4 The combined observer. 35 Chapter 4. Observer-Based Estimation of Friction in the Cartesian Manipulator 4.3 Experimental Results The Friedland-Park nonlinear observer can be used off-line to estimate friction from the identification data. The estimated friction can then be compared to the identified model of Chapter 3. fn the dynamic experiment that was discussed in Section 3.4.1, the desired trajectory was chosen as xd(i) = 0.02cos(57rt) m (4.13) which gives the following desired velocity profile: vd(t) = -0.3142sm(57rr) m/s (4.14) To study the bandwidth of the observer, assuming that the tracking error is not very large, one can replace the actual velocity (v) in equation (4.8) with the desired velocity (vd): p~ -ka^v^-1 (4.15) Since \vd\ < 1, in order to obtain a high speed of convergence, it is desirable to choose the exponent in the range 1 < p < 2. With p = 1.5, the expression for the observer pole becomes: p~ -1.5fc„Mt)|°- s (4.16) For the velocity profile defined by equation (4.14), this yields p ~ -0.64ka. With ka = 156, the approximate pole location would be p = -100. Figure 4.5 shows the result of applying the Friedland-Park observer (with ka = 156) to the recorded velocity and force data. The extracted friction-velocity curve is plotted in Figure 4.6, along with the identified model of the previous chapter. Note that, the curve shown in Figure 4.6 is obtained from two cycles of the data. A hysteresis behavior is present in this curve, which shows that friction is in fact a dynamic function of velocity. Thus, the static (steady-state) model of friction that was identified in the previous chapter is the result of a simplifying assumption. 36 Chapter 4. Observer-Based Estimation of Friction in the Cartesian Manipulator Off-line Estimate of Velocity, v Actuator Force, F 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s 1000 500 0 -500 1000 Estimated Dynamic Friction, F 7l i I i I 1 i i i 1 J i i I 1 i 1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 S Time Figure 4.5 Off-line estimation of dynamic friction using the Friedland-Park observer Oi = 1.5, ka = 156). 37 Chapter 4. Observer-Based Estimation of Friction in the Cartesian Manipulator o o T3 CD » CO E 1000 500 •B -500 co LU -1000 -0.5 Velocity (m/s) Figure 4.6 Friction as a function of velocity obtained using the Friedland-Park observer (solid line) compared to the identified model of Chapter 3 (dashed line). To check equation (4.6) for the speed of convergence, a test was carried out on the recorded data. Suppose that at time t = 0.25s the observer is reset to zero. Numerical solution of equation (4.6) yields: 0.329 Kap j {v^dt = 5 (4.17) 0.25 Thus, at £ = 0.329s the estimation error should reach 0.67% of its initial value, which can be neglected. As shown in Figure 4.7, computer simulation with the recorded data verifies this prediction. 800 600 400 h 200 -200 h -400 0 Estimation started at f=0.25s. - Estimation started at 1=0. 0.22 0.24 0.26 0.28 0.3 0.32 Time (s) 0.34 0.36 0.38 0.4 Figure 4.7 Investigation of the convergence rate for the Friedland-Park observer. 38 Chapter 5 Friction-Compensated Position Tracking of the Cartesian Manipulator 5.1 Methodology Experimental investigations of Chapter 3 and Chapter 4 which were carried out using the Cartesian electrohydraulic manipulator, revealed, that frictional effects are quite significant in this system. Thus, it is expected that position tracking performance of the manipulator can be improved by proper friction compensation. The compensation technique proposed in this chapter is based on acceleration feedback control. Acceleration-assisted tracking control has been discussed by de Jager in [15] where a performance improvement by a factor of up to 1.5 is reported in the particular application. According to de Jager, acceleration measurement in mechanical systems can be used in control to decrease the tracking error, improve the robustness for modeling errors, and reduce the effective friction. Studenny et al. [65] have successfully applied acceleration feedback control to robotic manipulators. An alternative method for friction compensation is to measure the transmitted force (torque) and close a force (torque) feedback loop. The approach, termed joint torque sensory feedback, has been employed for force control of robotic manipulators by Vischer and Khatib [77], Pfeffer et al. [56], and Luh et al. [46]. In our system, there is no sensor to directly measure the acceleration or the transmitted force. Instead of direct measurement, acceleration will be computed from the estimated friction. Thus, by combining friction estimation and acceleration feedback control, a new friction compensation technique is established. Two approaches will be taken for friction estimation: • Model-based approach [69]: friction is assumed to be a time-invariant phenomenon and is estimated using the identified model of Chapter 3. • Observer-based approach [70, 67]: friction is estimated using the nonlinear observer that was discussed in Chapter 4. The second approach can accommodate the time-varying nature of friction and will result in an adaptive friction compensation strategy. 39 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator 5.2 Design of the Friction-Compensating Controllers In DC motors, the generated magnetic torque is directly proportional to the applied current [16]. Unlike DC motors, the force generated by a hydraulic actuator (F) is not proportional to the current applied to its servovalve (i), as significant actuator dynamics and nonlinearity would be present. Consequently, friction compensation in a hydraulic actuation system is not as straightforward as it is in a DC-motor-driven system. In the case of DC motors, one is able to conveniently add a correcting fraction of the estimated friction to the control effort so as to compensate for friction [11, 27, 26, 23]. The technique proposed herein employs the estimated frictional force to calculate acceleration [70, 69, 67]. The acceleration signal is then used to further improve the tracking performance. 5.2.1 Acceleration feedback control The estimated acceleration can be used in the feedback loop of the motion controller to decrease the tracking error. The proposed controller is obtained by adding an acceleration error term to a simple PD position controller. The control law can be expressed as: i =-K[(a - ad) + a(v - vd) + (3(x - xd)] (5.1) where, i is the current applied to the servovalve, (.)d is the desired value of the corresponding signal and {ii', a, [3} is a fixed set of the controller gains. Note that from velocity control viewpoint, the control law of equation (5.1) corresponds to a PID (Proportional + Integral + Derivative) controller. It is in fact a state feedback controller and the acceleration term represents the internal state of the actuation system. Should there be no actuator dynamics, a simple PD position controller would form the state feedback; however, the actuator dynamics requires acceleration feedback, for satisfactory control. A block diagram of the proposed control system is shown in Figure 5.1. 40 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator Desired Position Servovalve Current Electrohydraulic System Velocity Filter Actual Position X Figure 5.1 Block diagram of the acceleration feedback control system. Acceleration feedback control has also been used by other researchers to obtain high performance hydraulic servomechanisms [25, 81, 48]. fn [25], FitzSimons and Palazzolo used the root locus method to find the controller gains for a single-rod hydraulic actuator and presented simulation results, fn [81], Welch discussed the quadratic resonance phenomenon that is generally observed in the transfer function relating the load velocity to the servovalve current. He then explained that acceleration feedback can be used to damp the hydraulic system so as to achieve a higher bandwidth. Welch assumed that the load is predominantly of inertial type (with no dissipation) and used pressure readings to calculate acceleration. By using linear analysis of the system dynamics, he verified the effectiveness of adding a pressure feedback term (as a minor feedback loop) to a conventional PD controller. He also suggested high-pass filtering of the pressure feedback signal, to allow full pressure feedback in the frequency range of the load resonance. Welch named his approach "Derivative Pressure Feedback", and presented experimental results using a hydraulic motor. A similar technique was employed by Matsuura et al. [48], for damping out the resonance of a hydraulic cylinder, in an industrial application. The approach presented by us can also be considered a pressure feedback technique (see Figure 5.f), with the following unique features: 41 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator • Frictional effects are taken into account. • Controller gains are chosen using a well-established heuristic method. In [65], Studenny et al. proposed an approximate method for choosing the gains of an acceleration feedback controller. Accordingly, in a stable closed loop system, as the gain K is made large, the dynamics of the position error (Ax — x — xj) may be approximately represented by Ax + aAx + f3Ax = 0, (5.2) which corresponds to the following second order characteristic equation for the closed loop system: s2 + as + j3 = 0 with a = 2£un, (3 = u\ (5.3) 5.2.2 Selection of controller gains The heuristic method of Studenny et al. [65] is used here to assign the gains a and /3. Controller tuning might be required then, to achieve further improvement in the response. By choosing the desired natural frequency and damping factor as u>n = 100 rad/s and £ = 0.7, equation (5.3) gives: a = 140, (3 = 104 (5.4) The gain K can be decided using the nominal current of the servovalve. The choice of K(5 = 10 mA/mm = 104 mA/m, (5.5) yields: K = 1 ^ 4- (5.6) m/s 5.2.3 Selection of observer gains A method is proposed in this subsection for choosing the observer gains in the observer-based approach to friction compensation. This task is not required, however, in the model-based approach. As a rule of thumb, both the velocity and friction observers should be made at least four times faster than the closed loop system. For the feedback gains chosen as in equation (5.4), the approximate closed loop poles are: ai,2 = - 7 0 ± j 7 0 (5.7) 42 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator Then, |observer poles| > 280 . For the velocity observer, the chosen gain is kv = 500 and for the friction observer, using equation (4.16), we choose: 1.5ka\vd\0-5 = 300 (5.8) Given a typical desired velocity profile, ka can be computed from equation (5.8). 5.2.4 Quintic polynomial approach According to Andersson [1], a desired trajectory with smooth position, velocity, acceleration, and jerk components can be generated using a quintic (5th order) polynomial. With this choice, the motion noise of the actuator is also reduced. Details on choosing the coefficients of the polynomials to meet system specifications and limitations are given in [1]. In the present application, it is required to move the object (cutter blade assembly) from its initial stationary position to the desired position which is also stationary. The transient trajectory is chosen as a quintic polynomial. One can also use other types of trajectories such as ramp, second-order, or sinusoidal waveform, as the desired input. 5.2.5 Ideal response Ideally, the closed loop system should behave as a second order linear system with the following transfer function: /? _ 104 d [ S ) ~ S 2 + 2as + (3 ~ 5 2 + 140s + 104 ( 5 > 9 ) Figure 5.2 shows the simulated response of the transfer function of equation (5.9) to the desired quintic position trajectory that will be used in the experiments. The simulation result shows that the chosen values for u>n and £ are suitable. According to Figure 5.2, in the ideal case, the closed loop system exhibits a time lag of 14 ms. This value was found by calculating the time shift of the response which would result in the minimum Root-Mean-Square (RMS) value of the tracking error (0.047 mm). This RMS position tracking error is defined as follows: 1/2 ERMS = ((xd(t) - x(t + td)f ) ' (5.10) where, td is the response lag. Note that the response lag is not important in our application, provided that the final time is within specifications. The most significant issue is the closeness of the shape of the actual trajectory to the desired trajectory. 43 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator 0.02 H 0.01 -0.01 h -0.02 h i r T r x (Ideal Response) * d (Desired Trajectory) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 " Time (s) Figure 5.2 Ideal response to the desired position trajectory. 5.3 Experimental Results Tracking performance of the electrohydraulic actuator with and without friction compensation is investigated now, through experimentation. The digital controllers are implemented at a sampling frequency of fs = IkHz. To reduce the steady-state position tracking error, the following correction term is also added to the control effort of the controllers: Ai = -Idsgn(x - xd) (5.11) where, Id = 3mA is the threshold value of the input current. This term is included to compensate for the deadband nonlinearity of the open loop system (partly due to the static friction) that was discussed earlier in Sections 3.3 and 3.4.1. Figures 5.3, 5.4 and 5.5 show the experimental results. Figure 5.3 corresponds to the control law without friction compensation (a well-tuned PD controller); as given by i = -Ki(x - xd) - K2(v - vd) - Idsgn(x - xd), (5.12) Figure 5.4 corresponds to the model-based approach and Figure 5.5 corresponds to the observer-based approach to friction-compensated tracking control (acceleration feedback control); as given by i = -K[(a - ad) + a(v - vd) + (5(x - xd)} - Idsgn(x - xd). (5.13) For the desired range of motions, typically we have |^ | 0 ' 5 > 0.17. Consequently, using equation (5.8), a value of ka = 1200 is chosen for the friction observer gain. The threshold velocity is chosen 44 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator equal to Dv = 0.02 m/s, using a trial and error approach. The parameters of the three controllers are listed in Table 5.1. Note that the tuned gain a of the observer-based controller is different from its initial value given by equation (5.4). According to the results, there is a small lag (12 rns) in the responses, which is primarily due to the closed loop dynamics of the system., As discussed earlier, ideally, the system response should have a time lag of 14 ms. Therefore, the proposed control technique results in a tracking performance which is very close to the ideal case. Table 5.1 Parameters of the controllers. PD Control Proportional gain K\ = 104 mA/m Derivative gain K2 = 110 •2i4-=> i m/s Model-Based Approach to Friction-Compensated Control State feedback gains K = 1^_, a = 140, (5 = 104 Threshold velocity Dv = 0.02 m/s Observer-Based Approach to Friction-Compensated Control State feedback gains K = l ^ L , a = 80, (3 = 104 Observer gains kv — 500, ka — 1200 Threshold velocity Dv = 0.02 m/s 45 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator (a) Desired and Actual Displacement 0.02 i i i 1 1 1 —> x (Actual) i i 0.01 if \\ J 1 \ 1 \ \ Y\ xd (Desired) 'i \\ \\ \\ \\ \\ v. E 0 -0.01 \\ i A '/ v\ '/ \ J -0.02 1 1 1 1 1 1 t i i 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 t s (b) Desired and Actual Velocity 0.51 1 1 1 1 1 1 CO E _Q 51 _ — i i i i , i i i i i I ' 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s (c) Input Current, i 40 p 1 1 1 1 1 1 1 r -40 b 1 i i i i i i i i d 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 S Time Figure 5.3 Position tracking without friction compensation (PD control). 46 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator (a) Desired and Actual Displacement ~i r ~t r i r _l L J I I 1_ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s 0.5 T r (b) Desired and Actual Velocity i r -0.5 _i i i i_ 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 40 (c) Input Current, "1 1 ~i 1 1 1 1 r Time Figure 5.4 Position tracking with friction compensation: model-based approach. 47 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator (a) Desired and Actual Displacement ~i r ~i r (b) Desired and Actual Velocity -40 _l I I L_ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 s Time Figure 5.5 Position tracking with friction compensation: observer-based approach. 48 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator By shifting the position signals through 12 ms to compensate for the time lag, one can compare the tracking errors quantitatively. Figure 5.6 shows the position tracking error signals. According to the RMS values shown in this figure, a factor of t .9 reduction in position error is achieved using the model-based approach and a factor of f .65 reduction is achieved using the observer-based approach to friction compensation. PD Control - i r _2' 1 1 1 1 i i i i i I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 S Time Figure 5.6 Comparison of the position tracking error profiles. 49 Chapter 5. Friction-Compensated Position Tracking of the Cartesian Manipulator Likewise, Figure 5.7 shows the velocity tracking error signals. According to the RMS values shown in this figure, a factor of 2.07 reduction in velocity error is achieved using the model-based approach and a factor of 2.3 reduction is achieved using the observer-based approach to friction compensation. Model-Based Approach 100 50 1 0 6 -50 r -100 Observer-Based Approach -i 1 r~ RMS = 11.56 mm/s i i_ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s Time Figure 5.7 Comparison of the velocity tracking error profiles. Thus, both model-based and observer-based approaches to friction compensation considerably improve the tracking performance of the manipulator. The observer-based approach results in an adaptive friction compensation strategy and is more desirable. 50 Part II The Mini Excavator The mini excavator is a heavy-duty human-operated mobile hydraulic machine [73]. This machine has a manipulator-like structure and is used for versatile construction operations such as digging, carrying loads, dumping loads, straight traction, and ground leveling. Typically, a human operator controls the main four links of the manipulator in joint-space coordinates through movements of two 2-DOF mechanical hand levers. Considerable improvements in the performance of these types of machines can be achieved by computer-assisted control [62]. Indeed, human factor experiments on similar machines have shown that resolved-mode endpoint velocity control leads to faster task completion time [38, 37]. In such an approach, the two hand levers are replaced by a 4-DOF joystick (three translational and one rotational degree of freedom). The operator can directly control the motion of the implement in Cartesian-space rather than coordinating the movements of all links to provide desired endpoint (bucket) motion. In Part II of this thesis, issues such as machine instrumentation, modeling of the actuation system, estimation of the static and dynamic parameters of the links, analysis of the actuator friction and joint friction, and estimation of the bucket load, are addressed for this manipulator. 51 Chapter 6 Instrumentation of the Mini Excavator To perform closed loop computer control, several displacement, fluid pressure, and force sensors are required and the pilot stage of the main valves has to be modified. Identification of the manipulator parameters and analysis of friction can be performed using these sensors. 6.1 Structure of the Mini Excavator Figure 6.1 shows the schematic of the mini excavator. The bucket is in fact the movable end-effector of the machine. The upper structure of the machine can rotate on the carriage by a hydraulic reversible swing motor through a reduction gear. The main links "boom" and "stick" together with the swing motion serve to control the position of the bucket. Thus, the major degrees of freedom are as follows: • Bucket in and out (load and dump) • Stick in and out • Boom up and down • Cab swing motion i Figure 6.1 Schematic of the mini excavator. 52 Chapter 6. Instrumentation of the Mini Excavator In the original machine, these links are controlled by pilot-operated main valves whose commands come from two operational levers. Forward-backward or left-right movement of these two levers by a human operator provides individual control to the link motions. Other degrees of freedom in the mini excavator and details of the main valve bank have been explained in Appendix B. The focus of this project will be mainly" on the backhoe links (boom, stick, and bucket). Note that the actuators used for these links are all single-rod (asymmetric) cylinders with limited linear motions. Since the joints are revolute, use of linear actuators results in joint angle limitations. It means that boom, stick, and bucket motions are restricted to specific ranges. Kinematic analysis of the machine can be found in Appendix B. Figure 6.2 shows the four pilot-operated valves in the mini excavator, with the assumption that other degrees of freedom a^re not activated. A Diesel engine is used to turn the three hydraulic pumps; two axial-piston variable-displacement pumps and one gear pump [50]. The two variable-displacement pumps have a common swash plate whose angle is controlled according to the pressure sum (Pi + P2). Consequently, these pumps have identical output flow or equivalently identical displacement coefficient, but different output pressures. The following properties can be obtained by examining the schematic diagram of Figure 6.2: • The boom and bucket actuators have equal priority in receiving and sharing the high pressure fluid provided by pump Pi. • Unlike the Cat-215B machine[61], there is no explicit collaboration between the pumps Pi and P2. However, the two variable-displacement pumps have a common swash plate, therefore, they are not completely independent. 53 Chapter 6. Instrumentation of the Mini Excavator i Figure 6.2 The four pilot-operated main valves in the mini excavator. 54 Chapter 6. Instrumentation of the Mini Excavator Figure 6.3 shows a typical pressure sum versus output flow curve for these pumps. According to the schematic diagram shown in the same figure, the output pressure of the pumps are sampled by two orifices. These pressures are applied to two small pistons that change the angle of the swash plate against two parallel springs. This mechanical feedback system serves to limit the power that is drawn from the Diesel engine, so that the pressure-flow curve is below the power limit curve. Pump P 3 is a fixed-displacement gear pump whose output flow is a function of the engine speed only. As long as the engine spins with constant speed, the output flow of P 3 remains unchanged. Note that the backhoe links have intercoupled actuation system, i.e., operation of the links are not independent, movement of one link might affect the other. This fact must be taken into account in the design of the closed loop computer control. In [61], Sepehri presented a technique for dealing with the coupled actuation system of the CAT-215B machine in the control design. 6.2 Installed Sensors The following sensors have been installed on each of the backhoe actuators: • A linear position sensor that measures relative displacement of the piston. The measurement is performed by a digital resolver. This sensor can be used to measure the displacement at low frequency motions only. max Output Flow, Q Figure 6.3 Variable-displacement pumps in the mini excavator. 55 Chapter 6. Instrumentation of the Mini Excavator • A digital resolver that measures the absolute joint angle. This sensor is installed at the manipulator joint. • Fluid pressure sensors. • A load pin, that measures the reaction force of the cylinder to its hinge. This sensor is sensitive to both tension and compression. In Figure 6.t, the bucket, stick, and boom load pins are indicated by A, B, and C, respectively. Note that these load pins are fixed to the body of the machine. Each load pin has a strain gage bridge inside it. Since the output level of this sensor is very low (in the range of few millivolts), an instrumentation amplifier is required to amplify its output. The operation principle and also characteristics of the motion and force sensors can be found in [16]. A VME-bus based computer system is used to collect the data and issue the control signals. The resolver outputs are directed to an R/D board and the rest of the transducers are connected to an A/D board inside the VME cage. There is also a D/A board inside the cage that is used to send analog control commands to the modified pilot system which is discussed in Section 6.4. 6.3 Joint Torque Measurement The torques applied by the hydraulic actuators about the manipulator joints can be indirectly measured using the installed sensors. In the following, the bucket degree of freedom is discussed in detail. The stick and boom degrees of freedom can be dealt with, in a similar fashion. load pin Figure 6.4 The bucket degree of freedom. 56 Chapter 6. Instrumentation of the Mini Excavator In Figure 6.4: L = fully retracted length of the actuator x = piston displacement 6 = joint angle (see Appendix B) F = force applied by the actuator r = torque applied by the actuator about the bucket joint Note that, by definition, positive torque r is in the direction that acts to increase the joint angle 6. Thus, the positive joint torque is in the direction that is shown in Figure 6.4. This convention applies to all of the backhoe actuators. Neglecting the joint friction, virtual work principle [39, 61] implies that: Td6 = Fdx (6.1) Therefore, the joint torque can be calculated from the following equation: T = FJ(6) (6.2) where the derivative function J(.) is defined as follows: J{9) = dx/d0 (6.3) To calculate the joint torque, one has to measure the applied force F and the coefficient J for the given joint angle. 6.3.1 Force measurement The force applied by the actuator can be measured either by the load pin or by the pressure sensors. Neglecting the fluid compressibility, Newton's third law gives F = Fr (6.4) where, Fr is the reaction force measured by the load pin. On the other hand: F = Fp - Ff (6.5) 57 Chapter 6. Instrumentation of the Mini Excavator where Fp — P\Ai - P2A2 = force measured from pressure readings, Ff = actuator friction (acting between the piston and cylinder). From equations (6.4) and (6.5), one can conclude that the force measured by the load pin is a more accurate representation of the force F, compared to the force measured from pressure readings. By combining equations (6.4) and (6.5), the following equation is obtained for the actuator friction: Ff = Fp- Fr (6.6) Note that pressure transducers usually have considerable offset drift. Therefore, in practice, Ff is caused by both the actuator friction and the error due to the inaccuracy in pressure readings. 6;3.2 Calculation of J(9) Using geometry of the machine, one can obtain the trigonometric mapping x(9), whose derivative is the function under question. This method has been used in the past by Sepehri [61]. A more efficient technique is proposed here, which is very suitable for real-time implementation. The technique is based on polynomial approximation of the function x{9). Coefficients of the polynomial are obtained by minimizing the least squares estimation error. Since both linear and angular position sensors are available on each actuator, there is no need to find the mapping from trigonometric equations. One can easily fully extend and retract each actuator (at low speed) and read the two sensors. Figure 6.5 shows the empirical x{9) curves obtained for the full range of motion of the three degrees of freedom in the machine. These curves are obtained by fully extending and retracting each actuator. 58 Chapter 6. Instrumentation of the Mini Excavator Bucket Stick Boom 9(rad) 6(rad) e(rad) Figure 6.5 The curves x(9) obtained by fully extending and retracting each actuator. As can be seen from Figure 6.5, on flat ground, it is impossible to run the boom actuator over its full range, otherwise, it will hit the ground. It was found that 5th order polynomials are suitable candidates for approximating the x{9) functions. These polynomials are given in Appendix B. The approximation error curves are shown in Figure 6.6. According to the error curves, the maximum approximation error for the boom and bucket is 1mm and for the stick is 2mm, which can be neglected. If needed, piecewise polynomial approximation can be used to achieve higher accuracy. Chapter 6. Instrumentation of the Mini Excavator The derivative functions J(0), calculated using the polynomial approximation, are plotted in Figure 6.7. Bucket Stick Boom •o "D CO -0.2 -0.4 " O CO E, Figure 6.7 The functions J(0) for full angular range of each degree of Horner's algorithm (see [9]) can be employed for recursive computation of the polynomial x(9) and its derivative J(6), for a given angle 9 = 90. For an n-th order polynomial, 2n-l multiplications and 2n-l additions are needed to evaluate the polynomial and its derivative at a given point. Thus for our 5-th order polynomials, 9 multiplications and 9 additions are needed. This is particularly useful in real-time applications. 6.4 Modification of the Pilot Stage As explained in Appendix B, the required pilot supply pressure of the mini excavator is provided by the control valve (sub) unit. The generated pilot pressure, is then modulated through the movements of the mechanical hand levers and is directed to the pilot chambers of the main valves. For computer control, an interface system is needed between the hydraulic main valves and the electronics. Proportional electrohydraulic servovalves are generally used for this purpose, which are expensive and bulky. High speed on/off servovalves can be used instead, as a cost-effective and compact alternative. With proper PWM (Pulse-Width-Modulation) operation, these valves can provide the required pilot signals. To further discuss the idea, consider the schematic of the modified stick valve that is shown in Figure 6.8. 60 Chapter 6. Instrumentation of the Mini Excavator A. Pilot Pressure Supply Unit B. The Manual Hand Lever C. High Speed On/Off Switching Valves (The Modification) D. Stick Main Valve Figure 6.8 The stick actuation system with the modified pilot stage. 6 1 Chapter 6. Instrumentation of the Mini Excavator The modification is devised such that both manual and electronic control are possible: • Manual control is achieved as usual, by leaving the switching valves inactivated. In this mode, with the hand levers in neutral position, the pilot signals are at tank pressure, i.e., P\P = P2P = PT. • For electronic control, the hand levers are pushed down to the end by some holding mechanism. Thus, the pilot supply pressure is passed through the hand levers. With proper PWM excitation of the switching valves, an average pressure difference can be created in the pilot chambers of the main valve. Unlike manual mode, when the switching valves are not activated, the pilot signals are connected to pilot supply pressure, i.e., PLP = P2P = PSP. Muto et al. [51] and Suematsu et al. [66] have pointed out that there are some practical problems to be solved in conventional PWM systems, such as serious nonlinearities (e.g., dead zone) in the valve characteristics. The concept of Differential-Pulse-Width-Modulation (DPWM) has been introduced by Muto et al. [51]. They used this technique to control a hydraulic actuator (double-rod cylinder) by two 3-way solenoid valves and observed that valve nonlinearities can be significantly reduced by using DPWM. Unlike their approach, where DPWM is used to generate the main pressure signals, it is employed here to generate the required pilot signals (see Peussa et al. [55] and Tafazoli et al. [72]). In the conventional PWM method, one applies a binary signal with fixed frequency but variable duty cycle, to create the desired average output, i.e., pilot pressure. Thus, with PWM operation, a high-speed switching device can be used for proportional control. In DPWM, the command signals sent out to the two switching devices are not independent. The technique can be explained using Figure 6.9, which shows ideal DPWM current signals. Note that when the current I is applied to each switching valve, its output is connected to the hydraulic tank, fn the normal condition, identical square waves are sent to both valves with a duty cycle of 50%. To produce an average pressure difference, the duty cycles are varied up and down from 50% (e.g., 35% for one valve and 65% for the other). Instead of generating the DPWM signals with computer software, we designed and built an interface circuit (see Appendix B) that converts a given analog voltage (-5V < u < 5V) to the 62 Chapter 6. Instrumentation of the Mini Excavator corresponding DPWM current signals i\ and i2. The duty cycles of the generated current signals can be computed using the following equations: ' Dx = h/T = 0.5 - O.lu = (50 - 10ti)% (6.7) s D2 = t2/T = 0.5 + 0.1M = (50 + 10u)% Note that there are four degrees of freedom in the machine which require eight switching valves. Therefore, use of the interface circuit considerably reduces software overhead. Peussa et al. [55] found / = 10077^  to be a suitable frequency for DPWM operation in our application. D1=35%, D2=65% 0 T 2T 3T Time Figure 6.9 DPWM current signals for u = 1.5V. 63 Chapter 7 Modeling and Identification of the Actuator Dynamics Hydraulic actuators have significant nonlinear dynamics that affect their performance [50, 76, 24, 78]. Researchers have used several schemes such as self-tuning control [79, 80, 58], adaptive model-following control [84], robust control [25], and feedback linearization [78] to cope with the nonlinear dynamics of the hydraulic actuators. As discussed in the previous chapter, DPWM-operated solenoid valves are used to generate the pilot pressure signals in the mini excavator. A linear model is developed in this chapter for the modified pilot stage [72]. The model is then combined with the main valve and cylinder dynamics to obtain a mathematical model for a single actuator of the mini excavator. Linear identification of the actuator dynamics is addressed next. A brief analysis of the actuator sealing friction is presented at the end of the chapter. This chapter is primarily intended to provide some insight into the performance of the DPWM pilot stage and also the nonlinear dynamic effects that are present in the hydraulic systems. Since our focus in this research is on structural dynamics and frictional effects of the machine, control design using the identified dynamics is not addressed here. The experimental investigations are carried out on the stick actuator. To simplify the system, the stick cylinder is disconnected from the machine (to eliminate the structural dynamic effects), and is used in a horizontal orientation (to eliminate the gravitational effects). 7.1 Nonlinear Dynamic Model of the Actuator In this section, similar to the analysis that was presented in Section 3.2 for the Cartesian manipulator, physical principles are used to model the system. The system under question is a single-rod hydraulic actuator driven by DPWM-operated solenoid valves. A block diagram of the modified pilot stage is shown in Figure 7.1: 64 Chapter 7. Modeling and Identification of the Actuator Dynamics Input Voltage (-5V4u4+5V) Valve Currents Pilot Signals Pilot Pressure Difference u fc. DPWM Interface Circuit & Current Driver 1 » Solenoid Valve #1 1J1P • Solenoid Valve #2 p, Lp Figure 7.1 The new pilot stage with on/off switching valves. For simplicity, we assume that the two solenoid valves have identical switching and hydraulic characteristics. These valves act to change the pressure inside the pilot chambers and their connecting hoses. Thus, the pilot stage can be modeled as a first order linear system [50], i.e., sp -p2p — 1 + T S P x sp 1 + TS (l-nil) (1 - it/I) (7.1) (7.2) where PSp - pilot supply pressure I = amplitude of DPWM current signals T = the physical time constant of the system As explained in Chapter 6, when a solenoid valve is turned on, its output port gets connected to the tank pressure. According to equation (7.1): when i\ = 0 (valve #1 is turned off), the pressure Pip rises to Psp, and when i\ = I (valve #1 is turned on), the pressure P\p drops to zero. Thus, equations (7.1) and (7.2) describe the dynamics of the pilot valves properly. Note that the currents applied to the solenoid valves are assumed to be ideal DPWM signals (see Figure 6.9). Subtract equation (7.2) from equation (7.t), to obtain: 1 + TS IR = A H — 1-2 I (7.3) By subtracting the second line of equation (6.7) from its first line, we observe that the mean value of iR is proportional to the input voltage, i.e., iR = — 0.2u (7.4) 65 Chapter 7. Modeling and Identification of the Actuator Dynamics As a result, the mean value of the pilot pressure difference would be as follows: - 5 - 0.2P s p PLP = u (7.5) 1 + TS Therefore, the mean pilot pressure difference can be controlled by the analog voltage applied to the interface circuit. The generated pilot pressure difference applies the following force on the spool of the main valve: Fv — Pj_,p AV (7.6) where AV is the spool area. A stiff spring mounted on one end of the spool acts as a feedback element. This spring centers the spool against the pressure difference created by the pilot stage. The main valves of the mini excavator have overlapped ports. The nonlinear deadband element which is shown in Figure 7.2 can be used to model the overlap behavior of these valves. Physical Displacement of the Spool Xvd i - D ' k V/ +D Effective Displacement of the Spool Xvd=L(*v) Figure 7.2 Overlap model of the main valve. Now, assuming zero damping length (or equivalently zero transient flow force) for the main spool [50], the valve dynamics can be expressed in the following second order form: where x dx PLPAv = Mv~^- + Bv~jf + K ° x v + Kfxvd Mv = mass of the spool plus the oil inside its chambers By = spool damping coefficient Ks = coefficient of the centering spring Kf = steady-state flow force coefficient (7.7) 66 Chapter 7. Modeling and Identification of the Actuator Dynamics According to equation (7.7), the steady-state flow force and the centering spring have similar effects on the spool. The coefficient Kj can be calculated from the following relation [50]: Kf ~ 0A3w(Ps -PT- sgn{xvd)\PL\) (7.8) where w = valve port area gradient PS = supply pressure PT = tank pressure PL — P\ — P2 = load pressure Note that Kj is a function of the load pressure. The maximum value of w is w = ird, where d is the spool diameter. This value occurs when the ports are full periphery of the spool. For an overlapped valve, the flow equations are as follows [24]: Qi = Kxvd [sVPs - Pi + (1 - s)VPi-PT\ Q2 = Kxvd [s^P2 - PT + (1 - s)VPa ~ P2. (7.9) where, the switching variable s is used to capture the role reversal that occurs in the valve from "extension" to "retraction" of the piston. It is defined as below: '1 xvd > 0 s = < (7.10) .0 xvd <0 The constant K in the flow equations can be computed from the following equation [50]: where K = Cdwx -P Cd = the dimensionless discharge coefficient (Cd ~ 0.61) p - mass density of the fluid (7.11) 67 Chapter 7. Modeling and Identification of the Actuator Dynamics Furthermore, using the continuity principle and taking fluid compressibility into account, one obtains [24, 78]: <2i = Aix + P i (7.12) A2(L-x) + Vh. Q2 — A2x r2 where x = piston displacement L = maximum piston displacement AI,A2 = piston areas j3 = effective bulk modulus of the fluid Vh = volume of the fluid inside the hoses between the main valve and the actuator Applying Newton's second law to the piston gives: P1AL - P2A2 = Mpx + Ff + Fe (7.13) We assume that there is no external force in the system (Fe = 0). Modeling of the frictional force (Ff) was discussed in Chapter 2. A complete block diagram of the system dynamics is shown in Figure 7.3. In this model, we have ignored the hysteresis and deadband behavior of the solenoid valves, ff these effects are important, they can be added to the model. Note that, similar to the analysis that was presented in Section 3.2, all differential equations are presented in the state space form. According to this block diagram, the nonlinear system has an order equal to 7. One should note that several factors affect actuator dynamics, such as the throttle provided by the engine (which can be adjusted by the operator), fluid bulk modulus (which varies with temperature and air entrainment in the hydraulic fluid), and valve wearing. Thus, using the nonlinear model of a hydraulic actuator for control design requires an in-depth knowledge of the system parameters and its operating conditions. 68 Chapter 7. Modeling and Identification of the Actuator Dynamics U Main Valve Dynamics DPWM Interface & Driver The Pilot Stage . _ i P P = — P - i P = P - P 2 j = Xv Xv,=fjxv) K = 0.43w(Ps-PT-sgn(xvd)\PL\) zi = z2 'Z7 = (P A - B z . - X z . - f C . j : ,) 2 ^ 1Lp v v 2 si f vd' Main Valve Flow Equations vd Pi 1 Xvd>0 I 0 Xvd< 0 Qj= Kxjs/P~i3 +(l-s)JTfF~ ] ®2 = K x v d l s / ^ P ~ T + (1 S)/F~7] Piston Dynamics y,=y 2 y4=p2 Figure 7.3 Nonlinear model of a single-rod hydraulic cylinder driven by DPWM-operated pilot valves. 69 Chapter 7. Modeling and Identification of the Actuator Dynamics 7.2 Open Loop Step Response of the Actuator To study the open loop performance of the system, a square wave voltage was applied to the actuator and sensor outputs were recorded with a sampling frequency of fs = 1000Hz. Note that the frequency of DPWM current signals applied to the solenoid valves is 100Hz, thus a high sampling rate is required to capture the behavior of the pilot pressures. The recorded signals are plotted in Figures 7.4, 7.5, and 7.6. A first order low-pass-filtered differentiator was used to estimate the piston velocity (see Appendix C). The time constant of the digital filter was chosen as Tv = 0.05s so that the filter has a faster dynamics than the system. The following conclusions can be drawn from this open loop experiment: • As expected, the pilot stage behaves like a first order system. The pilot pressures are the only signals which contain DPWM frequency component of 100 Hz. This frequency is considerably filtered out in other signals. According to Figure 7.4, DPWM is a suitable way of generating the required pilot pressure. • In Figure 7.4, the pilot pressure difference (PLP) is not symmetric, while the input voltage (u) is a symmetric step signal. This is believed to be due to the difference in the hydraulic characteristics of the on/off solenoid valves, ft might also be caused by the asymmetry (in retraction versus extension regime) of the fluid force on the main valve spool. • Output pressures of the two variable-displacement pumps both stay at a fixed low level until the cylinder is activated. After the. opening of the main valve, the pressure of the corresponding pump (pump 2 for the stick actuator) rises to provide the required flow (Figure 7.5). • The hydraulic force applied by the piston is calculated from the pressure readings, i.e., FP = P\A\ — P2A2. The value of this force when the piston is moving at a constant velocity corresponds to the friction inside the actuator seals (Figure 7.6). 7 0 3 0 CO S= 2 Chapter 7. Modeling and Identification of the Actuator Dynamu Input Voltage, w 1_ I I I Pilot Pressure I, P2 p iwiiniindnminiwiimimmw'ili'mi'iw'i L 3 s Pilot Pressure II, P. 2? ^ I CO CL . mJiimiinmitni>iiniiiiniiintnitit»wii I I Pilot Pressure Difference, P. hp 3 O i^iniwiiiiiiiimmiiiiiiiiiiiiiiiiiiii iimniiiiiiimiiiiiiii "T~ i _i_ 3 s i i _ 4-jmrnmrnmim, r Time Figure 7.4 Open loop experiment: input voltage and pilot pressure signals. 71 o > 10 CO £ 5 10 CO £ 5 Chapter 7. Modeling and Identification of the Actuator Dynamics Input Voltage, u i i i -4--I I 3 s Pump Pressures i / . 1 iiM<7i."(:i?W" i i Pump 2 i i •n'""ra"""*1'^1 4~ ~ r — : jl7ir~)Tnl77ij77..7f*Irl I Pump 1 3 s Cylinder Chamber Pressures 1 1 1 1 ^ P 2 I 1 1 1 1 [•- -\ 1 i 1 E Piston Displacement, x E 0.3 Time Figure 7.5 Open loop experiment: input voltage, pump pressures, load pressures, and piston displacement. 72 Chapter 7. Modeling and Identification of the Actuator Dynamics Input Voltage, u § Or! I I I -4--I I 3 s Piston Velocity, v i 1 i i 1 1 1 r i i i i i • * w T * (I ( I T * I 1 1 T "1 I I 1 1 0 1 2 3 4 5 Applied Force, F p 0 1 2 3 4 5 6 s Time Figure 7.6 Open loop experiment: input voltage, pilot pressure difference, estimated piston velocity, and calculated force. 73 Chapter 7. Modeling and Identification of the Actuator Dynamics Figure 7.7 shows a zoomed plot of the pilot pressure difference. There is a ripple with a frequency of 100772 in this signal, which corresponds to DPWM operation of the solenoid valves. - 0 . 5 1 -j 1 {- -f i I I I i I I I i I - 1 1 ' 1 - 1 • 1 1 1 0.8 0 .9 1 1.1 1.2 1.3 1.4 Time (s) Figure 7.7 Pilot pressure difference (zoomed). 7.3 The System Identification Experiment [32, 44] As discussed in Section 7.1, the electrohydraulic system under consideration is quite nonlinear. In order to ease control design, it is beneficial to obtain an approximate and simple linear model of the system. To cope with the time-varying and nonlinear behavior of the system, the model parameters can be updated recursively and can be used in an adaptive manner. The system identification toolbox in MATLAB™ [43] is used in the next two sections for parametric and non-parametric identification of the stick actuator dynamics. As shown in Figure 7.8, the system under study can be divided into two subsystems: the pilot stage and the main stage. Each subsystem will be dealt with separately. Input Voltage Pilot Pressure Piston Velocity Displacement u • Pilot Stage Main Stage V 1 X • (Main Valve + Piston) ^ s • Figure 7.8 System block diagram. A low-pass-filtered white noise signal with Gaussian distribution was applied as the input voltage to the open loop stick actuator. Pilot pressures and piston displacement were measured at a sampling 74 Chapter 7. Modeling and Identification of the Actuator Dynamics rate of / s = 100Hz, which is about ten times higher than typical bandwidth of the system. The signals are plotted in Figure 7.9. Input Voltage, u 0 1 2 3 4 5 6 7 8 9 10 s 0 1 2 3. 4 5 6 7 8 9 10 s Piston Velocity, v s Piston Displacement, x 0.41 1 1 1 r — 1 1 E 0.35 V 0.3 0 1 2 3 4 5 6 7 8 9 10 s Time Figure 7.9 Open loop system identification experiment using the stick actuator. 75 Chapter 7. Modeling and Identification of the Actuator Dynamics In the next sections, we attempt to fit a first order model to each subsystem. The transfer function in the j-domain is as follows: = 777$ = T T ^ ' A > 0 ( 7- 1 4> U(s) s + a which has the following zero-order-hold equivalent (see [6]): Y(z) bz~x E(z) U{z) 1 + az'1 (7.15) a = -e-aT°, - l < o < 0 b= (^1 + a) a Thus, a first order ARX (AutoRegressive with exogenous input) model structure can be used for identification. For a general system with input u and output y, the model is defined by the following difference equation: y(t) + ay(t - 1) = bu(t - 1) + e(t) (7.16) where, t is the discrete time index and {e(t)} is a white noise sequence that represents modeling error. 7.4 Linear Identification of the Pilot Stage Our objective in this section is to estimate the following transfer function for the pilot stage. . . Pilot Pressure PLp(s) H U S ) = Input Voltage = TW ( 7 - 1 7 ) From equation (7.5), one concludes that: Hia{s) = rrfs (7-18) Also, the step response experiment of Section 7.2 verifies that the pilot stage behaves as a first order linear system. Parametric identification is used here, to estimate the coefficients of the following discrete transfer function: = % ^ = (7.19) U(z) 1 + a\z 1 76 Chapter 7. Modeling and Identification of the Actuator Dynamics Result of the parametric identification using an ARX model structure is summarized in Table 7.1. Note that the low standard deviations show high accuracy of the estimated parameters. Table 7.1 Estimated parameters and their standard deviation for H\(z). Parameter Estimated Value Estimated Standard Deviation Relative Standard Deviation -0.8919 0.0027 0.3% h 0.0709 0.00t4 2.0% Using equation (7.f5), the equivalent continuous-time model would be as follows: Hla{s) = (7.20) The parameters Psp and r in the transfer function (7.18) can be calculated from (7.20): Psp = 3 . 2 9 M P a = Allpsi (7.21) T = 0.0885 The following validation tests are carried out on the estimated model. 7.4.1 Test I: Deterministic simulation For a given input voltage, the model response can be compared with the measured output of the system. Figure 7.f0 shows the simulated and measured pilot signal both for the input used in the identification test and also for a different white noise input. According to this figure, the estimated model predicts the system output with a very good accuracy. 77 Chapter 7. Modeling and Identification of the Actuator Dynamics Estimated & Measured Pilot Pressure - The Identification Data 41 i i i i i i i i r \ I i i i i i i i i i I 0 1 2 3 4 5 6 7 8 9 10 S Time Figure 7.10 Measured (solid line) and simulated (dashed line) output (pilot pressure difference) for two different input signals. 7.4.2 Test I I : Residual analysis The residuals represent misfit between data and model. A sequence of residuals which still exhibits some structure would then indicate that either the modeling or the identification is inappropriate. Residual correlations and the 99% confidence interval limits for the uncorrelated residuals are plotted in Figure 7.1 f. According to this figure, the auto-correlation of the residuals resembles that of the white noise, and the cross-correlation with the external input can be neglected. Therefore, the estimated model is a valid one. 78 Chapter 7. Modeling and Identification of the Actuator Dynamics Auto-Correlation Fen. of Residuals Cross-Corr. Between the Input & Residuals . . . 1 0.15, • • • , 0 5 10 15 20 . -20 -10 0 10 20 lag lag Figure 7.11 Residual analysis of the estimated model for the pilot stage. 7.4.3 Test III: Correlation analysis Correlation method is used here, to extract the impulse response (and the step response) of the system from the recorded input-output data. The covariance functions are estimated for M = 80 number of lags with a pre-whitening AR filter of order 80. Figure 7.12, shows the result of this test. The covariance of the pre-whitened input (u) resembles that of white noise (impulse type). Therefore, the pre-whitening filter is a suitable one. The step response obtained using correlation analysis is plotted together with that of the parametric model. The matching between the step responses is good. Chapter 7. Modeling and Identification of the Actuator Dynamics 7.4.4 Test IV: Spectral analysis Frequency response of the system can be extracted from the input-output data using the spectral analysis technique. Good matching is observed between the Bode plot of the system and that of the estimated parametric model, as shown in Figure 7.13. According to this figure, the bandwidth of the pilot stage is around 1.5 Hz. The low-pass-filtered white noise input voltage that was used for identification had a bandwidth of 6 Hz. To better identify the high frequency dynamics, one can repeat the experiment with a different input signal that has wider bandwidth. Magnitude Response of H Phase Response of Ht Figure 7.13 Comparison of the Bode plots for the pilot stage (solid line = from spectral analysis, dashed line = from the estimated model). The validation tests verify the suitability of the estimated linear model of the pilot stage. Since the pilot supply pressure is regulated (see Appendix B) and is almost independent of the engine throttle level, the estimated linear model can be used for a wide range of the operating conditions. 7.5 Linear Identification of the Main Stage Unlike the pilot stage, it is expected that the main stage (combination of the main valve and the cylinder) should be quite nonlinear. In this section, we attempt to fit a simple linear model to the system and test its validity. To avoid the integral term, piston velocity is considered to be the output of the system, instead of piston displacement. Thus, the continuous transfer function under question is Piston Velocity V(s) H2{s) Pilot Pressure (7.22) 80 Chapter 7. Modeling and Identification of the Actuator Dynamics we use the following discrete first-order model for identification purpose: n2{z) = = — — (7.23) PLP{Z) 1 + a2z 1 Table 7.2 summarizes the result of ARX modeling. Note that the standard deviations are relatively higher than what was obtained in Table 7.1 for the pilot stage. We found that increasing the model order or using other model structures does not substantially improve the accuracy. Table 7.2 Estimated parameters and their standard deviation for H^z). Parameter Estimated Value Estimated Standard Deviation Relative Standard Deviation -0.9149 0.0049 0.5% b2 0.0026 0.0001 3.8% The equivalent model in the j-domain is as follows: H2(s) = 0.0026 S-±^£ (7.24) Similar to the pilot stage, validity tests were performed on the estimated linear model of main stage. The results are summarized in the following. 7.5.1 Test I: Deterministic simulation Figure 7.14 shows the simulated and measured output (velocity signal). Note that the matching of the outputs is not as good as the pilot stage. The nonlinear nature of the main-stage dynamics is believed to be the cause for this mismatch. 81 Chapter 7. Modeling and Identification of the Actuator Dynamics Estimated & Measured Velocity - The Identification Data 0.1 i i 1 1 1 1 1 1 1 S Estimated & Measured Velocity - Response to a Different Input 0.1 i 1 1 3 1 1 1 1 1 l i i i i i i i i i I 0 1 2 3 4 5 6 7 8 9 10 S Time Figure 7.14 Measured (solid line) and simulated (dashed line) output (piston velocity) for two different input signals. 7.5.2 Test II: Residual analysis The result of residual analysis for the estimated model is shown in Figure 7.15. The auto-correlation function does not resemble that of the white noise. We believe that this is due to the nonlinear dynamics of the main stage. Chapter 7. Modeling and Identification of the Actuator Dynamics 7.5.3 Test III: Correlation analysis Figure 7.16 shows the result of correlation analysis. The covariance of the pre-whitened input ( P L P ) resembles that of white noise (impulse signal). Therefore, the pre-whitening filter is a suitable one. The step responses are not similar though. Again the matching is not as good as what was obtained for the pilot stage. Figure 7.16 Result of correlation analysis for the main stage. 7.5.4 Test IV: Spectral analysis Figure 7.17 shows the Bode plot of the system together with that of the estimated parametric model. Once again, the comparison shows that the linear time-invariant model is not a good choice for describing the main stage dynamics. As discussed earlier, several algorithms have been reported in the literature, for dealing with this problem in the control design. Thus, it is not considered here. 83 Chapter 7. Modeling and Identification of the Actuator Dynamics Magnitude Response of H2 Phase Response of H2 10 1 10° 101 10"' 10° 101 Hz Hz Figure 7.17 Comparison of the Bode plots for the main stage (solid line = from spectral analysis, dashed line = from the estimated model). 7.6 Analysis of the Actuator Friction To determine the steady-state friction-velocity characteristic of the actuator, the piston has to be driven at several constant velocities, in both directions [2]. The force applied by the actuator in the steady-state regime corresponds to the dynamic friction. A digital PD controller with the following control law is used to close the feedback loop and drive the actuator at the desired velocities: u = -KpAx - KvAv - Kssgn(Ax) (7.25) where Ax = x — x<i is the position error, Av = Ax = v — Vd is the velocity error, Kp = 80 is the position feedback gain, Kv = 5 is the velocity feedback gain. The additional term Kssgn(Ax) with Ks = 1.3 volt is included to compensate for the deadband nonlinearity of the main valve. Figure 7.18 shows the signals obtained from one of such experiments. 84 Chapter 7. Modeling and Identification of the Actuator Dynamics 0.15 0.1 0.05 0 -0.05 -0.1 Desired (solid line) and Actual (dashed line) Velocity I I I I I I I I I I • * 1 - - / - - ) - — -/ . _ • . i r i • 1 ! 1 ,_-| J L » 1 1 1 ^ S . 1 1 - " I I ! I I J- J 1 1 1 1 i 1 1 r , i . J-\±_ r / / 1 1 1 1 1 1 1 1 1 1 1 4000 Force Applied by the Actuator Figure 7.18 Closed loop position control experiment (to study the actuator friction). Observations revealed that actuator friction is higher at negative velocities (retraction of the piston). The possible reason is that the actuator seals have preferential direction and exhibit different frictional behavior in the retraction regime compared to the extension regime. Due to the offset drift of the installed pressure transducers, their measurements are not very dependable. Better pressure transducers have been ordered that will be used in the future to achieve higher accuracy. In the closed loop experiments, we had to re-tune the PD controller for each run. This, once again emphasizes the nonlinear and time-varying dynamics of the actuator. 85 Chapter 8 Estimation of the Gravitational Parameters and Bucket Load Gravity loading dominates manipulator joint torques except for very fast movement of the links. A PD controller with gravity compensation performs almost as well as the standard computed-torque approach where the full inertial terms are taken into account. Accurate values of the gravitational parameters are typically unknown even to the robot manufacturers. Due to the heavy links, gravitational terms play an important role in achieving high performance control of the construction machines, such as the mini excavator. In this chapter, we present a new approach for decoupled estimation of the gravitational parameters (see Tafazoli et al. [71]). The approach employs a least squares method for off-line estimation of the 6 gravitational parameters from the experimental data. To obtain the data (joint angles and torques), static experiments are carried out with the instrumented mini excavator. In the next step, an efficient technique is proposed for on-line estimation of the bucket load in the static condition that uses the estimated gravitational parameters. Experimental investigations show that bucket load estimation can be performed with a 5% accuracy. Analysis of the static friction inside the actuator seals is addressed next. It is investigated that a considerable amount of static friction exists inside the cylinders that cannot be neglected. Therefore, as discussed in Chapter 6, load pins should be used instead of pressure sensors, for accurate measurement of the joint torques. Throughout this chapter, it is assumed that the manipulator is in static condition. Similar issues are addressed in the next chapter for the dynamic condition. 8.1 Related Work In [53], Parker et al. experimentally determined the gravitational parameters of a heavy-duty hydraulic machine. They applied a least squares method to the data obtained statically, in order to identify the gravitational parameters. They used pressure sensors to indirectly measure the joint torques in their application. In their approach, the result of the estimation was found to be poor. The authors pointed out that likely sources of error were the neglected friction component and pressure 86 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load measurement errors. The parameters were needed in order to implement a master-slave force-reflecting resolved-motion control. The usefulness of these parameters in applying force feedback to the machine is twofold: • Gravity compensation in the hand controller; so that the operator does not need to hold the hand controller. In fact, the resolution of the teleoperation system is limited by the ability to compensate for gravity in the hand controller. • Indirect measurement of the endpoint forces; the bucket load estimation technique outlined in this chapter is a simple example. * Zhu [85] and Lawrence et al. [37] also elaborated on the application of force feedback to heavy-duty hydraulic machines. They also used pressure readings to measure the joint torques. Yoshida et al. [83] employed a static test to estimate the gravitational parameters of a PUMA 560 robot. To avoid joint static friction, they increased the motor torque gradually until the motion started. The average value of the applied torque for both directions of the joint was taken as the gravity load and the bias was seen as static friction. The test was proceeded distally to proximally (from joint 6 to joint 2). In a similar approach, Ma and Hollerbach [47] presented a simple, robust, and general procedure to identify the gravitational parameters statically. Their algorithm involves single joint rotation, and fits a sinusoidal curve to the resulting data. They verified their algorithm via simulations and experiments that were performed on the Sarcos Dextrous Arm. In their application, the gravitational parameters were mainly needed for gravity compensation and on-line automatic torque sensor calibration. In [42], Lin proposed a recursive identification algorithm to estimate the torque constants, the joint friction coefficients, and the gravity parameters, at the same time. The procedure was conducted by moving one joint at a time, at constant velocity. Lin tested the algorithm on a PUMA 560 robot, using a stiff controller to hold a constant speed over a large motion range. The estimated parameters were then employed in a PD feedback control law with gravity and friction compensation. The advantage of the above-mentioned techniques to others, such as those requiring simultaneous movement of the joints, is that the general problem is divided into a set of problems of lower complexity, thus achieving good stability and numerical precision. A disadvantage is that identification errors of the distal links will propagate to the proximal links [47]. 87 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load Payload estimation of the robot manipulators in the static condition has been addressed by Paul [54] and Atkeson et al. [8]. Paul presented two techniques; one employs joint torque/forces sensing, and the other uses wrist torque/force sensing. Atkeson et al. also used wrist torque/force sensing to statically estimate the payload mass and center of mass. 8.2 Static Torque Equations When there is no load in the bucket, the actuator torque at each joint acts to compensate the gravitational torque of the backhoe links. Figure 8.1 shows a schematic representation of the backhoe links (see Appendix B). The assigned joint angles for the boom, stick, and bucket degree of freedom are 62,03, and 64, respectively. The cab swing angle is not shown in this figure, as it is not considered in this study. In this figure, cgi is the center of gravity for link i with polar coordinates (r,,a,) in the corresponding link. Figure 8.1 Gravitational forces on the backhoe links. The algebraic link angles with respect to the horizontal plane are: boom : 6-2 stick : 023 = 02 + 03 (8.1) bucket : 0234 = #23 + 84 88 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load and the static torques applied by the hydraulic cylinders about the manipulator joints are as below: T4 = Mbugr4cos(6234 + a4) (8.2) T3 - r 4 + Mbuga3cos823 + Mstgr3cos(623 + a3) r2 = r3+(Mbu + Mst)ga2cos62 + Mbogr2cos(62 + a2) As discussed in Section 6.3, by definition, the actuator torque vector is normal to the plane and its positive direction is upward. Thus, for the manipulator pose shown in Figure 8.1, all the actuator torques are positive. In the next step, we define the gravitational parameter vector (or static parameter vector) $ s as follows: Mbur4cosa4 Mbur4sina4 Mbua3 + Mstr3cosa3 Mstr3sina3 | (Mbu + Mst)a2 + Mbor2cosa2 Mbor2sina2 Intuitively, all the gravitational parameters should be positive. Using definition (8.3), the static torque equations (8.2) can be expressed in the following vector form: (8.3) (8.4) W±g The (3 X 6) regression matrix W = W(62,63, 64) is defined as below: [C234 - 5 2 3 4 0 0 0 0 " C234 - 5 2 3 4 C 2 3 ~S23 0 0 (8.5) LC234 - 5 2 3 4 C23 ~S23 C2 -S2_ where c 2 3 4 = co5t9234, s23 = sin023, and so on. The torque equations can be further converted to the following form: ''r4 ' T34 = W^s (8.6) J 2 3 . 89 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load where r 3 4 = r 3 - r 4 and r 2 3 = r 2 - r 3 are the torque differences between adjacent joints. The new regression matrix W\ is defined as follows: "C234 - « 2 3 4 0 0 0 0 ' 0 0 c 2 3 - s 2 3 0 0 0 0 0 0 C 2 -32. Thus, the static torque equations can also be presented in the following decoupled form: (8.7) TA = #[c234 -5 2 34] 7"34 = #[C23 -S23,} • T23 = g[C2 -S2 Sp2 ¥>6 (8.8) (8.9) (8.10) 8.3 Least Squares Estimation of the Gravitational Parameters The static torque equations derived in Section 8.2 are linear in the gravitational parameters. Therefore, least squares estimation (LSE) can be employed to experimentally determine these parameters. A brief summary of the LSE technique and its statistical properties is given in Appendix D. The required observation data is obtained by recording the sensor outputs for different poses of the manipulator. The joint angles are directly measured by the digital resolvers and the actuator torques are calculated from load pin readings, using the technique that was introduced in Chapter 6. Each pose of the manipulator corresponds to 3 torque equations. Therefore, one has to choose at least 2 different poses, to solve for the 6 unknown gravitational parameters. Due to the undesirable effects such as "minor linkages formed by the cylinders", "measurement errors", and "external disturbances", a much larger set of observations is needed to estimate the parameters with reasonable accuracy. The observation data that is used in this section was obtained by putting the manipulator in iV = 110 different poses. 8.3.1 Coupled Form vs. Decoupled Form of the Equations The gravitational parameters can be estimated by applying LSE to either the coupled form (8.4) or the decoupled form (8.6) of the torque equations. The question is: which form of the equations 90 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load results in a more accurate estimation? As discussed in Appendix D, singular values of the regression matrix can be used to judge the quality of the estimation or even to design a suitable experiment. The N = 110 poses of the manipulator generate (330 x 6) concatenated regression (information) matrices W and W\. Singular values of these two information matrices were calculated and found to be as follows: S(W) = {172.34, 136.75, 83.96, 64.47, 56.47, 24.32} (8.11) S(W1) = {99.52, 89.19, 76.36, 68.80, 51.09, 25.69} (8.12) There are two measures of the information matrix that can be used to judge the quality of estimation (consult Appendix D): • Condition number: according to the singular values K(W) — 7.09 and k{W\) = 3.87. Thus, the decoupled form results in a smaller condition number for the information matrix. Therefore, this form is less sensitive to measurement errors. • Minimum singular value: the two matrices have almost equal minimum singular values. Hence, the decoupled form is more suitable for parameter estimation, fn fact, LSE using the coupled form of the equations resulted in a negative value for <p4. Also, subsequent tests (such as bucket load estimation) revealed that the estimated parameters using the coupled form were not as good as those obtained from the decoupled form of the equations. Another advantage of the decoupled parameter estimation is that the three torque equations (8.8), (8.9), and (8.10) are solved independently and each of them incorporates only 2 of the 6 gravitational parameters. Hence, the original estimation problem is transformed into 3 problems of lower complexity which improves the numerical precision. 8.3.2 Estimated Gravitational Parameters Result of the parameter estimation, using the decoupled torque equations, is given in Table 8.1. As expected, the estimated parameters are all positive. These parameters will be used for our future applications. According to Table 8.1, for the bucket link; ip2 is estimated less accurately compared 91 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load to (fi. Similarly, for the stick link; cp 4 is estimated less accurately compared to <ps, and for the boom link; tpe is estimated less accurately compared to cp5. One can conclude that the parameters (V2j ¥>4> fe) are more sensitive to the variation of the links centers of gravities (see definition (8.3)). Table 8.1. Estimated gravitational parameters using the decoupled static torque equations. Parameter Estimated Value (Kg.m) Estimated Standard Deviation (Kg.m) Relative Standard Deviation 26.75 1.30 4.9% <P2 12.80 1.42 11.1% 113.66 1.56 1.4% <p4 7.44 1.38 18.5% <P5 645.31 3.50 0.5% <P6 17.62 4.69 26.6% 8.4 Estimation of the Actuator Torques for the No-Load Condition Figure 8.2 shows the measured actuator torques and their predicted values using the estimated gravitational parameters. According to this figure, torque estimation for the bucket link is worse than that of the stick and boom. Possible reasons are: • Location of the center of gravity for the bucket link has more variation than the other degrees of freedom. • The static friction for the bucket joint is not negligible. • In the no-load condition, the bucket load pin is measuring a very small force compared to its full-scale range. This means that the accuracy of no-load force measurement for the bucket is poor. 92 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load Bucket Torque Estimation Stick Torque Estimation 20 40 60 80 100 Sample Boom Torque Estimation 20 40 60 80 100 Sample 20 40 60 80 100 Sample Figure 8.2 Measured (solid line) and predicted (dashed line) actuator torques for the static condition when there is no load in the bucket. Standard deviations of the actuator torque estimation errors are listed in Table 8.2. Table 8.2. Standard deviation of the no-load torque estimation errors. Actuator Torque Standard Deviation of the Torque Estimation Error (N.m) Bucket, T4 82.86 Stick, r 3 49.56 Boom, T2 158.28 Using the identified parameters of Table 8.1, the no-load static torques can be estimated from joint angles. This can be employed in the following tasks: • Improving the trajectory tracking performance by gravity compensation. 93 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load • Compensation for gravity loading in the hand controller in force-feedback experiments. • Measurement of the external forces that are applied to the manipulator, fn particular, bucket load estimation is discussed in the next section. 8.5 Bucket Load Estimation 8.5.1 The proposed technique The static torque equations (8.2) can be reformulated as follows: {T2,A)nl - Mbuga3cos623-\- Mstgr3cos(623-\- a3) (8.13) ( r 2 3 W = (Mbu\- Mst)ga2cos62 + Mbogr2cos(02 + a2) Here "NL" corresponds to the "no-load" condition. Now, with a mass M inside the bucket, the torques change as follows: r 3 4 = O 3 4 W + Mga3c23, (8.14) r23=(T23)NL + Mga2c2. Three methods for measuring the external load, based on equations (8.14), are: ga-3C23 M = ^Vt7"23 - (T™)NL\ (8-15) go-2^2 M = an r inn r ^ " ^ N L \ ga2c2 + ga3c23 The denominator of the first two expressions may approach zero for some specific poses of the manipulator. Due to the joint angle limitations, the denominator of the last expression is always positive. This equation also corresponds to the maximum measurable torque which is applied by the external load. Therefore, it will be used for bucket load estimation. Using equations (8.9) and (8.10) for the actuator torques in the no-load condition, one arrives at the following formula: j£ _ T 2 - T 4 - gc23(p3 + gs23(p4 - gc2(p5 + gs2(p6 ga2c2 + ga3c23 94 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load 8.5.2 Experimental result With a known load of M = lOO/fg inside the bucket, the manipulator was put into 10 different poses. Figure 8.3 shows the measured actuator torques and the estimated no-load torques. The following estimated values were obtained for the bucket load, using equation (8.16): M = {100.88, 103.63, 98.07, 98.47, 102.74,102.02,103.37, 95.76, 98.18, 99.07} (8.17) The estimated set has a mean of 100.22I(g with a standard deviation a = 2.68Kg. The main sources of error are as follows: • As discussed earlier, because of the cylinders and their minor linkages, the assumption that the gravitational parameters are fixed is not correct. • Measurement errors in joint angles and torques. Bucket Stick 15000 10000, 5000 4 6 Sample Boom 4 6 Sample 8 10 4 6 8 Sample 10 Figure 8.3 The measured joint torques with load (o) and the estimated no-load torques (*). 95 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load • The load pins are connected to the body of the manipulator instead of the cylinder, therefore, due to the small rotation of the cylinders, they are not measuring the whole reaction force. One can calculate the cylinder angle with respect to the corresponding load pin axis to eliminate this measurement error. To keep the algorithm simple, this issue is not addressed in this research. • Static friction in the manipulator joints. 8.6 Actuator Static Friction As explained in Chapter 6, it is better to use load pin sensors to calculate the actuator torques. Experiments with the mini excavator and the experiments carried out by Parker et al. [53] show that if pressure readings were used for torque measurements, the results would be erroneous. This is due to the significant static friction that exists inside the actuator seals and pressure transducer errors [53]. Figure 8.4a shows the measured force of the bucket actuator using the load pin (FT) and the pressure sensors (Fp). Assuming that these sensors are ideal, the difference of the two readings would be equal to the static friction inside the actuator, i.e., Ff — Fr — Fp. Figure 8.4b shows the calculated static friction. (a) Measured Force (b) Actuator Static Friction 40 60 80 Sample 40 60 80 Sample Figure 8.4 Bucket actuator: (a) forces measured by the load pin and pressure sensors, (b) calculated static Friction. Note that, in practice Ff is partly due to the static friction and partly due to the pressure measurement errors. The mean and standard deviation of the static friction inside the backhoe actuators are listed 96 Chapter 8. Estimation of the Gravitational Parameters and Bucket Load in Table 8.3. According to this table, considerable static friction exists inside the actuators that cannot be neglected. Table 8.3. Static friction of the actuator seals. Actuator Mean of the Static Friction (TV) Standard Deviation of the Static Friction (TV) Bucket 1221.74 601.55 Stick 1226.21 880.97 Boom -2915.59 716.16 An important property of a general rigid-body manipulator dynamics is its linearity in a set of well-defined parameters \P [39, 64]. In this chapter, the static parameter vector 4>s was estimated, which is in fact a subset of the complete parameter vector ^ = [$J $J ]T- 1° the next chapter, we will focus on the estimation of \£, by employing dynamic experiments. 97 Chapter 9 Estimation of the Full Dynamic Parameters and Friction Coefficients 9.1 Background The dynamics of a rigid-link robot manipulator is generally expressed in the following vector form [14, 39]: M(q)q + V(q,q) + G(q) = r (9.1) where, q is the joint-variable vector, M(q) is the symmetric inertia matrix, V(q, q) is the vector containing Coriolis and centrifugal terms, G(q) is the gravity torque vector, and r is the joint torque vector. Thus, the structural dynamics of a robot is, in general, in the form of second-order coupled nonlinear differential equations. The dynamic equation of a robot enjoys an important property that is of great use in the process of identification and control design [39, 63]. Namely, it is linear in a suitably defined set of parameters. In other words, one can write equation (9.1) as r = W(q,q,q)* (9.2) with ^ the dynamic parameter vector and W(q, q, q) a matrix of nonlinear functions depending on joint variables, joint velocities, and joint accelerations. One can compute the matrix W for any given robot assuming that the kinematic parameters are known a priori. The structural parameter vector * is a function of the link inertia parameters (mass, center of mass, and moments of inertia). It is sometimes referred to as the vector of "dynamic parameters". In reality, motion of the manipulator links are affected by friction, which is a local effect in the joints. Therefore, by assuming a friction model that is linear in parameters, one can incorporate frictional terms in the dynamics and still keep the property of being linear in parameters [39]. Least squares estimation of the structural parameters and joint friction coefficients of the backhoe links is studied in this chapter. When the links are at rest, the effective structural parameters are the static (gravitational) parameters that were estimated in the previous chapter. In other words, the gravity vector G(q) was statically identified in the previous chapter. Dynamic experiments are carried out in this chapter in order to identify the full dynamic and frictional effects. The gravitational parameters are once again estimated here as a subset of the full parameters. 98 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients 9.2 Related Work Experimental identification of the robot dynamics has been studied by many researchers for the past 20 years. Khosla and Kanade reported one of the earliest work in [36]. They outlined fundamental properties of the Newton-Euler formulation of the rigid-body dynamics from the view point of parameter identification. They showed that the model can be transformed to a form that is linear in the dynamic parameters and classified the dynamic parameters in three categories: fully identifiable, identifiable in linear combinations, and unidentifiable. Obviously, only those parameters are important that affect the manipulator dynamics and in fact those are the ones that can be identified from the experimental observations. Khosla and Kanade proposed the strategy of estimating off-line the dynamic parameters and then estimating on-line the inertial characteristics of the payload and the dynamic friction coefficients. In [8], Atkeson et al. studied estimation of the inertial parameters of the manipulator loads and links. Each link of a robot has in general 10 inertial parameters (mass, center of mass, and inertia tensor). Atkeson et al. showed that manipulator dynamics can be expressed in terms of a reduced set of the inertial parameters that are independently identifiable and that allow the application of a straight linear least squares estimation. The reduced set can be generated by examination of the closed form dynamic equations for linear combinations of the parameters. We have also used this intuitive regrouping approach in this chapter, to find a suitable set of the identifiable parameters for the backhoe links. According to Atkeson et al, the major sources of the estimation error are: sensor errors, kinematic errors and unmodeled dynamics such as link flexibility. Similar sources of error exist in our system that affect the estimation accuracy. Lin proposed an off-line identification method in [40], to estimate the minimal knowledge of the inertia parameters that determines the manipulator dynamics. In his algorithm, the parameters were recursively estimated by moving one joint at a time. In a more recent work [41], he dealt with the minimal parameters of a manipulator in the least squares sense. He introduced the terminology of "minimal linear combinations" or MLCs of the system parameters that define a set of linear combinations of the inertial parameters which uniquely determines the system dynamics. He also 99 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients presented a systematic approach to finding a set of MLCs for a general manipulator. In fact, the reduced set of the inertial parameters introduced by Atkeson in [8] is a set of MLCs for the manipulator. fn [34], Khoshzaban et al. presented an experimental work on identification of the manipulator dynamics for a Caterpillar 215B excavator. They considered the manipulator as a hybrid open-and-closed-chain mechanism, where the main links form the open-chain and the prismatic actuators form a number of closed kinematic chains with the main links. They used the Newton-Euler formulation to obtain a comprehensive model of the system which is linear in the unknown dynamic parameters. Pressure readings were used in their approach for computing the joint torques, and joint velocity and acceleration were estimated from joint position. The authors found that some parameters were not identifiable and some were identifiable only in linear combinations. They used least squares estimation and singular value decomposition to estimate the parameters that were identifiable. Only 26 out of the total 82 parameters were found to be identifiable for the combination of the cab swing, boom, and stick degrees of freedom. Their algorithm resulted in negative values for some of the parameters that were supposed to be positive. In this chapter, we will obtain a dynamic model of the backhoe links by considering the manipulator as a simple open kinematic structure. An explicit closed form model that incorporates the minimal number of the dynamic parameters is then obtained and the unknown parameters are estimated by employing least squares estimation. In [45], Lu et al. reported an experimental work on the estimation of the robot parameters including the friction coefficients. They also presented a technique to eliminate acceleration measurement. A similar approach is adopted in this chapter for identification of the dynamic parameters and friction coefficients of the backhoe links. 9.3 Dynamic Torque Equations Our objective in this section is to obtain a dynamical model for the mini excavator that relates the actuator torques to the joint angles. Either the Euler-Lagrange or Newton-Euler formulation can be used to determine the dynamics of the manipulator [14, 64, 39]. The Euler-Lagrange formulation which is an "energy-based" approach to the dynamics will be employed here. Assuming that the 100 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients system is conservative, the so-called Lagrange equation of motion (for link i) can be expressed as follows: d_ dL_ _ dL_ _ d~t dft~ dfi - t% (9-3) In this equation: L = the Lagrangian function 0i = joint angle of the link i T{ = actuator torque about the corresponding joint The scalar Lagrangian function L is defined as the difference between the kinetic energy K and the potential energy U of the manipulator; i.e., L = K — U (9.4) Thus, the kinetic and potential energy of the manipulator should be derived first. Kinetic energy is a function of the joint angles and velocities while potential energy is a function of the joint angles only. Using the convention of Appendix B, the boom, stick, and bucket links are respectively labeled as links 2, 3, and 4. 9.3.1 Kinetic energy of the manipulator Assuming that the cab is not moving, the kinetic energy of the backhoe links would be: K = k2 + k3 + k4 (9.5) where, ki is the kinetic energy of link i. Now, define the coordinate frame C; with the origin at the link center of mass cgi and axes parallel to the link frame specified in Appendix B. Then, k{ can be computed from the following equation [14]: h = V T g i V c g i + ^ Jflwi (9.6) where .Mi = mass of the link i 101 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients V c g l = velocity of the eg, w.r.t the base frame U\ = angular velocity of the attached frame C; w.r.t the base frame X\ = fnertia tensor of the link i w.r.t the attached frame C; The linear velocity V c g . can be easily obtained by differentiating the position vector of c<?;. The backhoe links are symmetric with respect to the XY plane of the corresponding attached frames. Therefore, according to [14], the link inertia tensor has the following form: IxX{ IxYi 0 (9.7) 0 0 Izz. On the other hand, the angular velocity vector of link i has only one component; that is, 0 U\ = I 0 .<*>»• For the sake of simplicity, the rotational kinetic energy of link / will be presented as (9.8) (9.9) One can verify that ui2 = #2, u>3 = 023, and to3 = t9234. Using equation (9.6), the following expressions were obtained for the kinetic energies of the links: k2 = ^Mbor22022 + -IbJl (9.10) k3 = -Mst api + 46l23 + 2a2r362923cos(63 + a3) + olstO 23 (9.11) k4 = -Mbu[al6l + a236223 + r\8\34 + 2a2a362623c3 + 2a2r4626234cos(634 + a4) + 2a3r4e239234cos(d4 + a4)] + ^hJl34 (9.12) where 023 = j^623, c3 = cos93, and so on. 102 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients 9.3.2 Potential energy of the manipulator The potential energy of the manipulator is the sum of the individual potential energies of the links; i.e., U = u2 + u3 + u4. The individual potential energies of the links are as follows: u2 = Mbogr2sin(82 + a2) (9.13) u3 = Mstg[a2s2 + r3sin(923 + a3)} (9.14) u4 = Mbug[a2s2 + a3s23 + r4sin(0234 + a4)} (9.15) 9.3.3 The dynamic equations of the manipulator From the expressions for the kinetic and potential energy, one can derive the Lagrangian. The following differential equations were obtained by using equation (9.3) : r4 =(lbu + Mburl)e234 + Mbua2r4 02cos(634 + a4) + 8lsin(634 + a4) + Mbua3r4 023cos(94 + a4) + 8l3sin(94 + a4) + Mbugr4cos(0234 + a4) + Tf4 (9.16) r 3 =T4 + (Ist + Mstrl + Mbual)823 + Mbua2a3(02c3 + 8ls3^j + Mbua3r4 0234cos(04 + a4) - 0234sin(04 + a4) + Msta2r3 92cos(03 + a3) + 0lsin(03 + a3) + Mbuga3c23 + Mstgr3cos(023 + a3) + rf3 (9.17) r2 = r3+ [lbo + Mbor\ + (Mst + Mbu)a22} 02 + Msta2r3 823cos(63 + a3) - 9j3sin(93 + a3) + Mbua2a3[023c3 - 0l3s3^ + Mbua2r4 0234cos(034 + a4) - 0l34sin(034 + a4) + (Mbu + Mst)ga2c2 + Mbogr2cos(02 + a2) + r / 2 (9.18) 103 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients In the above equations 1 7 4 , 7 7 3 , and 7 7 2 are the frictional torques of the bucket, stick, and boom joints, respectively. 9.4 Modeling of the Joint Friction For linear least squares estimation, a friction model is needed that is linear in the coefficients. Various models have been used in the literature for precise modeling of friction in the robot joints. In [45], Lu et al. employed an asymmetric Coulomb plus viscous friction model. They further used this model for least squares estimation of the friction coefficients. In [12], Canudas de Wit et al. proposed a symmetric friction model which is linear in its 3 parameters (see Chapters 2 and 3). Their model is specifically suitable for capturing the Stribeck effect at low velocities. A very interesting and instructive work on robot friction modeling has been reported by Phillips and Ballou in [57], where the authors identified the components of the joint frictions for a 3-dof industrial robot in a step-by-step approach. They initially used an asymmetric Coulomb plus viscous friction model and detected other significant components such as gravity load-dependent and position-dependent frictions in the next steps. For our application, we will assume a simple viscous friction model, that is: rf3 = Bj3 (9.19) T/2 = Bboe2 where Biu, Bst, and Bb0 are the viscous friction coefficients of the joints. The experimental investigations on the mini excavator show that the joint frictional components are not significant and a simple viscous model can sufficiently describe their behavior. More complicated models (such as a position-dependent friction model) can also be used. Our emphasis here is more on the structural dynamics rather than the frictional behavior. 9.5 Parameterization of the Dynamic Equations The manipulator parameter vector which consists of 9 structural parameters and 3 friction coefficients, is defined in equation (9.20). This parameter vector consists of 3 parts: 104 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients • The dynamic part <I>d that has 3 elements and is only effective when the links are moving. • The static (gravitational) parameter vector <l>s that has 6 elements. This vector was estimated in the previous chapter by static experiments. • The friction parameter vector <frf which contains the viscous friction coefficients of the 3 joints. hu + Mburj Ist + Mstr\ + Mbua\ Iho + Mhor\ + (Mst + Mbu)a\ Mhuficosa^ Mbur4sina4 Mbua3 + Mstr3cosa3 Mstr3sina3 (Mbu + Mst)a2 + Mbor2cosa2 Mbor2sina2 bu B St B bo (9.20) Using the definition (9.20), the dynamic torque equations can be presented in the following vector form: Ar = (9.21) 105 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients The torque difference vector A T and the regression matrix W\ are defined as below: >4 " A r = - T3 - n ,T23. J2 - T3. 'ton 0 0 w 1 4 w15 0 0 0 0 0 4 0 0 0 W22 0 w 2 4 w2s w26 w 2 7 0 0 — 0 4 #3 0 ( 9 . 2 2 ) ( 9 . 2 3 ) 0 0 w 3 3 w 3 4 w 3 5 w 3 6 w 3 7 w 3 8 w 3 9 0 -0 3 02 J The regressors W{j, which are nonlinear functions of the joint angles, velocities, and accelerations, are: wn = 8234 w14 = a2^ 02cos034 + Q\sin034J + a3(§23cos04 + 6l3sin64^j + gcos6234 wis = a2(-82sin934 + 6lcos634^ + a3(^—623sin04 + 0\zcos94^j - gsin0234 ( 9 . 2 4 ) w 2 2 — t>23 w24 = a3(o234cos64 - 6\34sin04 W25 = a3(-6234sin64 - 0 3^4co504) ( 9 . 2 5 ) W26 = a2 (§2cos63 + Ojsindz^ + gcos623 w2j = a2 (—02sin63 + Q^cosO^ — gsin623 W33 = #2 W34 = a2 (6234C0S634 - 02346771034) ^35 = a2 (-6234sin634 - 0l34cos934^j w36 = a2 (o23cos63 - 6l3sin63^ ( 9 . 2 6 ) W37 = a2(-023sinQ3 - 6l3cos03 w38 = gcos02 W39 = -gsin62 1 0 6 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients The dynamic equations of motion of the backhoe links have the following properties: • Unlike the static torque equations of the previous chapter, one cannot completely decouple these equations. • As expected, the equations reduce to the static torque equations (derived in the previous chapter) by setting the velocity and acceleration terms equal to zero. Using the chain rule, the regressor functions can be expressed in the following form: W 1 5 = "77 ™14 = ~n « 2 ^ 2 C 3 4 + a3^23C4j + 02^2^234^34 + «3^23^234-54 + #C 2 3 4 — « 2 ^ 2 ^ 3 4 — dsO^S^) + ^2^2^234^34 + &3^23^234C4 ~ <?s234 (9.27) (9.28) (9.29) ^38 = gc2 w39 = -gs2 107 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients This formulation of the regressors will be used in the next section, to eliminate the need for acceleration measurement. 9.6 Least Squares Estimation of the Parameters Similar to the static case, least squares estimation (LSE) can be employed for experimental determination of the parameters. Basics of the LSE are summarized in Appendix D 9.6.1 Eliminating acceleration measurement Only position sensors have been installed on the backhoe joints of the mini excavator. Joint velocities are obtained by numerical differentiation of the position signals. There is no acceleration sensor in the system. Numerical calculation of the acceleration signals from the joint angles significantly amplifies the noise and is not recommended, as pointed out by Atkeson et al. in [8]. The specific form of the rigid body dynamics can be used to eliminate the need for acceleration measurement, as suggested by Hsu et al. [31] and Lu et al. [45]. To discuss the technique, consider the dynamics of the backhoe links: Ar(t) = Wi(t)* (9.30) Define the low-pass filter L and the high-pass filter H according to Figure 9.1. H( S ) 1 +T„s l+Ts Figure 9.1 First-order high-pass and low-pass filters. The two filters are related to each other according to the following equation: H(s)=-(1-L(s)) •l-V (9.31) Note that Tv is the time constant of these filters. Since the parameter vector \& is constant, low-pass filtering of the equation (9.30) gives: (Ar ) 7 j = ( W 1 ) L * 108 (9.32) Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients Equation (9.32) will be used for least squares estimation of the parameters. As shown in the following, the matrix (W\)L can be computed without knowing the accelerations. By examining the regressor coefficients defined in (9.27), (9.28), and (9.29), one observes that a general regressor w can be expressed as follows: w(t) = jtxx(t) + x2{t) (9.33) where x\ and x2 are functions of the joint positions and velocities. For example, for the regressor W24, we have: %i = a30234cos04 , • x2 = -a30234623sin04 (9.34) The regressors that reflect the joint friction are functions of the joint velocities only. Laplace transformation of (9.33) yields: ™(s) = szxO) - z i O ) | i = 0 + x2(s) (9.35) Assuming that the manipulator was initially at rest, we have Xi(t)\t=0 = 0. Therefore: w(s) = 5Xx(s) + x2(s) (9.36) Multiplying both sides of (9.36) by L(s) gives: 1 s 1 —w(s) = = — + x2(s) (9.37) 1 + Tvs y J 1 + Tvs u ; 1 + Tvs A 1 y ' Which is equivalent to the following equation: W L = ( * i ) f f + ( ^ ) i (9-38) This means that the matrix {W\)L can be computed from the position and velocity information by using the two filters. Digital implementation of these filters have been discussed in Appendix C. 109 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients 9.6.2 Experimental results To obtain the required joint angle and torque data, a dynamic experiment was carried out on the backhoe links. A 3-dof joystick was used to generate the inputs of the open loop system (voltage signals sent to the valves). In [3], Armstrong discussed the issue of finding "exciting" trajectories for identification of the manipulator dynamics. He used calculus of variations in order to find an optimal trajectory for the identification purpose. For the mini excavator, we are mainly interested in the dynamic effects that contribute to the typical operation of the machine in a field. Thus, the experiment was performed as if the manipulator was used for a standard job. The joint resolvers and actuator load pins were sampled at a frequency of fs — fOOHz for a time duration of 60 seconds. Therefore, each recorded signal is comprised of 6000 points. The high-order delay-free low-pass-filtered differentiator discussed in Appendix C was used for off-line estimation of the joint velocities from the joint angles. Figure 9.2 shows the measured joint angles and the estimated joint velocities. The low-pass-filtered regression matrix and torque vectors were obtained with a time constant of Tv = 5TS = 0.053 for the first-order filters. The singular values of the composite (18000 X 12) matrix {W\)L were calculated and found to be as follows: 5i.= {731.30, 715.25, 671.86, 374.84, 260.53, 203.58, (9.39) 41.11, 20.11, 18.48, 14.25, 9.19, 4.74} which gives a condition number of 154.38 for (W\)L. If absolute torques were used instead of the torque differences, the singular values would be: S = {1548.36, 686.38, 582.52, 402.40, 281.25, 179.03, (9.40) 71.62, 24.95, 13.33, 10.08, 8.60, 4.65} which gives a condition number of 333.25. Obviously, the difference form of the torque equations, given by equation (9.30), results in better numerical accuracy. Therefore, equation (9.30) was used for LSE of the parameters. The result of the LSE is summarized in Table 9.1. The following can be observed from this table: • As expected, all the parameters are positive. •As expected, both inertia related parameters (tpi,^,^) and friction coefficients (Bbu,Bst,Bbo) increase as we go from the distal link (bucket) to the proximal link (boom). 110 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients • The estimated gravitational parameters ipi to cp6 to some extent match the result of the static experiments reported in Chapter 8. As expected, the parameters that were obtained with high accuracy (ipi,<ps,(fs) from the static experiments have better matching with the result of the dynamic experiment. The low-pass-filtered joint torques can be estimated using the estimated parameters and joint angle measurements. Standard deviations of the torque estimation errors are listed in Table 9.2. Figure 9.3 shows the low-pass-filtered actuator torques along with their estimation using the identified parameters listed in Table 9.1. Also, figure 9.4 shows the result of cross validation with a different input. According to these figures, the estimated joint torques match the measurement with a good accuracy. Once again we see that bucket torque estimation contains more relative error than the stick and boom. A list of possible reasons for this discrepancy was given in Chapter 8. Ill Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients Bucket Joint Angle, 94 Bucket Joint Velocity, 94 T3 co -0.5 Stick Joint Angle, 93 Stick Joint Velocity, 93 Boom Joint Angle, 92 Boom Joint Velocity, 92 Figure 9.2 Open loop dynamic experiment: joint angles and velocities. 112 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients Table 9.1. Estimated structural parameters and viscous friction coefficients. Parameter Estimated Value Estimated Standard Deviation Relative Standard Deviation i>i = hu + Mhur\ 14.04 Kg.m? 3.20 Kg.m2 22.8% fa = ht + Mstrl + Mbua2 87.68 Kg.m2 6.58 Kg.m2 7.5% V>3 = ho + Mbor\ + (Mst + Mbu)a2 1142.65 Kg.m2 14.30 Kg.m2 1.3% (fi = Mbur4coscx4 29.44 Kg.m 0.36 Kg.m 1.2% ip2 - Mbur4sina4 15.82 Kg.m 0.20 Kg.m 1.3% ¥>3 = Mbua3 + Mstr3cosa3 119.65 Kg.m 0.41 Kg.m 0.3% <p4 = Mstr3sina3 18.39 Kg.m 0.38 Kg.m 2.1% ¥ 5 = (Mbu + Mst)a2 + Mbor2coscx2 648.09 Kg.m 0.41 Kg.m 0.1% ipe = Mbor2sina2 8.65 Kg.m 0.53 Kg.m 6.1% Bbu 111.68 N/(rad/s) 7.11 N/(rad/s) 6.4% Bst 132.90 N/(rad/s) 9.20 N/(rad/s) 6.9% Bbo 584.55 N/(rad/s) 27.75 N/(rad/s) 4.7% Table 9.2. Standard deviation of the dynamic torque estimation errors. Actuator Torque Standard Deviation of the Torque Estimation Error (N.m) Bucket, (T4)L 80.12 Stick, (r 3)L 63.18 Boom, (r 2)L 221.04 113 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients Bucket 600 400 200 0 -200 -400 -i I* • i 1 1 r * - -iv H i J ll --\ 1 / 1 1 -10 20 30 s 40 50 60 Stick 2000 1000 -1000 Boom Figure 9.3 Low-pass-filtered joint torques (solid lines) and their estimation (dashed lines) using the identified parameters. 114 Chapter 9. Estimation of the Full Dynamic Parameters and Friction Coefficients Bucket Boom Time Figure 9.4 Low-pass-filtered joint torques (solid lines) and their estimation (dashed lines) using the identified parameters (cross validation with a different input). 115 Chapter 10 Conclusions 10.1 Contributions of This Research Identification of frictional effects and structural parameters for improved control of hydraulic manipulators were considered in this thesis. These issues were experimentally investigated on two industrial prototypes. Novel approaches were presented for friction estimation and compensation, and also identification of the structural parameters for these systems. The major contributions of this thesis are as follows: (1) The nonlinear observer proposed by Friedland-Park was employed for friction esti-mation in the Cartesian electrohydraulic manipulator. As an important contribution of this thesis, the convergence rate of the observer was analyzed so that one can choose its parameters to achieve the desired speed of convergence. This analysis is needed for proper use of the Friedland-Park friction observer in general applications. (2) A novel approach was presented for friction compensation in position tracking control. The approach combined friction estimation with acceleration feedback control. Experimental investigations showed that the tracking performance of the manipulator can be considerably improved by employing the proposed controller. In our particular application, the RMS position tracking error was reduced by a factor of 1.65 to 1.9 in comparison to PD control. The proposed friction compensation technique is quite general and can be used for the control of any manipulator that has significant actuator dynamics. (3) An efficient method was introduced for fast and accurate measurement of the joint torques in excavator-type manipulators. The technique employs polynomial approximation to calculate joint torques from force readings. It was also found that actuator friction is not negligible. As a result, using load pins (instead of pressure sensors) leads to "more accurate measurement of the joint torques. 116 Chapter 10. Conclusions (4) A new technique was introduced for modification of the pilot stage of the mini excavator. The technique involved the use of two on/off solenoid valves for each actuator. Experiments showed that DPWM operation of these valves results in a well-behaved pilot system. A linear model was developed for the proposed pilot stage and system identification results validated the model. The proposed technique is a cost-effective and compact alternative to the conventional electrohydraulic proportional servovalves, which are both expensive and bulky. (5) For the first time, the gravitational parameters were successfully estimated from static experiments that were performed on the backhoe links. It was verified that the decoupled form of the torque equations results in better estimation of the parameters. (6) An efficient technique was presented for real-time estimation of the bucket load when the manipulator is at rest. The technique employed the estimated gravitational parameters to predict the joint torques in the no-load condition. Experiments showed that bucket load estimation can be carried out with a 5% accuracy. (7) The Euler-Lagrange formulation was used to derive the dynamic equations governing motion of the backhoe links. Using intuitive grouping of the equations, they were converted to a form that is linear in parameters. Both dynamic parameters and joint friction coefficients were determined without the need for acceleration measurement. Experimental results showed that torque estimation using the identified parameters can be performed with a good accuracy. Successful identification of the manipulator dynamics in this thesis establishes the groundwork for further research on implementation of the advanced control techniques on the mini excavator. 10.2 Suggestions for Further Work There are many open venues in which this research can be further continued. Some highlights are as follows: (1) Recursive version of the least-squares estimation approach can be used for real-time estimation of the parameters. For example, if the parameters of the linear friction 117 Chapter 10. Conclusions model of the Cartesian manipulator are estimated recursively, the resulting control strategy will be adaptive and can cope with the time-varying nature of friction. (2) Bucket load estimation in the dynamic condition (when the links are in motion) is a natural extension of the static version that was developed in this thesis. It is particularly useful for load estimation when there is no time to stop the machine links from moving and the tasks are required to be carried out fast. (3) Optimal selection of the static poses of the excavator in the system identification experiment is an interesting problem. For instance, one can attempt to minimize the condition number of the corresponding regression matrix. This would result in more accurate estimation of the unknown parameters. (4) Implementation of the advanced model-based trajectory control techniques on the mini excavator is an important task. 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ASME Journal of Dynamic Systems, Measurement, and Control, 115:95-102, March 1993. [83] K. Yoshida, N. Ikeda, and H. Mayeda. Experimental study of the identification methods for an industrial robot manipulator. In Proc. of the IEEE/RSJ International Conf. on Intelligent Robots and Systems, pages 263-270, Raleigh, NC, July 1992. [84] J.S. Yun and H.S. Cho. Application of adaptive model following control technique to a hydraulic servo system subjected to unknown disturbances. ASME Journal of Dynamic Systems, Measurement, and Control, 113:479-486, September 1991. [85] M. Zhu. Master-slave force-reflecting resolved motion control of hydraulic mobile machines. Master's thesis, Department of Electrical Engineering, University of British Columbia, April 1994. 125 Appendix A Specifications of the Cartesian Manipulator A.l The System Configuration Figure A.l shows schematic diagram of the basic electrohydraulic manipulator that is used in the automated fish processing machine. The X and Y degrees of freedom are actuated by identical single-rod cylinders. Position Transducer Pneumatic On-Off Actuator in Z Direction Plus Fish Head Cutter Blade o -M tn Position Transducer Figure A.l Schematic diagram of the basic planar manipulator. 126 Physical parameters of the prototype machine are as follows: • Moving mass in the X direction = Mx — 32.7 Kg • Moving mass in the Y direction = My = 55.7 Kg • Maximum stroke of the pistons = L = 2 in. = 5.08 X 10 - 2 m • Volume of the hoses connecting the valve to the X cylinder = VQX = 8.9 X 10 _ 5m 3 • Volume of the hoses connecting the valves to the Y cylinder = Voy = 1.14 x 10 - 4 m3 • Piston bore diameter = 7J(, o r e = 1.5 in. = 3.81 X IO - 2 m • Piston rod diameter = Drod = 1 in. = 2.54 x 10 - 2 m The following values were computed from the piston diameters: • Piston head-side area = Ai = 1.140 x 10 - 3 m2 • Piston rod-side area = A2 = 6.333 X 10~4 m2 • Area ratio = 7 = A\jA2 = 1.8 As shown in Figure A.l, the high pressure fluid is provided by a pressure-compensated, variable displacement, axial piston pump (Sauer-Sundstrand model L23-RBKY-FPx3xx-xx). This pump has the following specifications: • Maximum pressure = 3000 psi • Displacement = 1.41 in.3/rev = 23 cc/rev • Input speed (max) = 3200 rpm • Input Speed (min) = 500 rpm • Theoretical delivery at maximum speed = 19.5 gpm = 73.8 Ipm In the experiments that are reported in this thesis, the pump pressure was set to Ps = 852 psi. A.2 The Interface Components A.2.1 Flow Control Servovalves (Sauer-Sundstrand Co.) Two electrohydraulic servovalves serve as interfaces between the electric section and hydraulic cylinders. The flow control servovalves (FCS) used in this machine are two-stage valves that provide an output flow rate proportional to the input current signal. The FCS consists of a closed loop 127 pressure control pilot valve (double jet flapper valve) and a spring-centered second stage that features a dual spool arrangement. Dynamic modeling of a similar valve has been discussed in [74]. The main valve (model MCV113A6t09) has the following specifications: • Supply pressure = 500 to 3000 psi • Return pressure = atmospheric to 300 psi • Flow rating = 10 gpm (38 lit/min) based on tOOO psi pressure drop and rated input signal • Null pressure = 50% of the supply pressure • Hysteresis = 5% of the rated input signal • Frequency response = 18 Hz (-3 dB bandwidth) • Filtration = 10 microns in-line pressure filter The pilot valve (model MCVlOfAf4f2) has the following specifications: • Nominal (rated) full flow current = 42 mA • 4 pin MS connector, two of which (A & B) are used • Single 106 fi coil (measurement = 100 Cl) A.2.2 Voltage to Current Converter Current signals (in the range of -42 to 42 mA) are required for the servovalves, while the output signal provided by PC (D2A output) is a voltage in the range of 0 to 5V. Therefore, an electronic interface circuit is needed in order to activate the FCS by PC. The circuit shown in Figure A.2 was designed for this purpose. A.3 Sensor Specifications A.3.1 Position transducers (MTS Systems Co.) The Temposonics™ If linear displacement transducers precisely sense the position of an external magnet to measure piston position with a high degree of resolution. This sensor measures the time interval between an interrogation pulse and a return pulse. The interrogation pulse is transmitted through the transducer waveguide, and the return pulse is generated by the movable permanent 128 470 Q -15V • WAA— 5.1V +15 -2.5V 1 0 K i o . l r r F Ref=-2.5V +15 10K 10K i—vw—T—WA-i Ref 10K -AAAA 10K -AW—1 i = ( V^n- 2.5) / 0.0459 2N2222 V.-2.5 V_ fb 741 82 Q I - 5 — 1 — -2N4126 :82 Q FCS Coil '(HOQ) 47 Q Figure A.2 The interface circuit, magnet representing the displacement to be measured. Specifications of the installed sensors (model no. TTRCU0020) are as follows: • Stroke = 2.0 in. • Current draw = 100 mA (max) and 25 mA (min) for +/-15 Vdc • Continuous operation pressure = up to 3000 psi Since the original transducers have PWM digital output, two APM (Analog Personality Module) were installed to obtain DC output voltage. The APM is an internally mounted digital module that processes digital data into an analog output via a digital to analog converter. The digital programming of the APM makes the full-scale adjustment of the starting and ending of the stroke possible. The output range of the sensors were set to -5 to +5 V. A.3.2 Pressure transducers (Sensotec Inc.) Six pressure transducers (model LV/2359-12) have been installed on the machine. These sensors 129 have amplified DC output (voltage) with the following specifications : • Pressure capacity = 3000 psig • Accuracy (min) = +1-0.5% FSO • Output = (0 to +5 Vdc) • Supply voltage = +/-15 Vdc • Overload safe = 50% over capacity A.4 The Data Acquisition Board (Advantech Co.) A high-performance full-sized add-on board for IBM PC (PC-LabCard model PCL-812PG) used for data acquisition . The key features of this board are as follows: • 16 single-ended analog input channels (12 bit) • Software programmable analog input ranges. Bipolar : +/-5V, +/-2.5V, +/-1.25V, +/-0.625V, +/-0.3125V • The ability to transfer A/D converted data by interrupt handler routine • An Intel 8253-5 programmable Timer/Counter that can be used for periodic sampling and timing purposes. • Two 12 bit D/A output channels. An output range (0 to +5V) or (0 to +10V) can be produced. • 16 digital input and 16 digital output channels (TTL compatible). 130 Appendix B Specifications of the Mini Excavator B.l The Prototype Machine The machine used in this project is the mini excavator model TB035 from Takeuchi company [73]. It is a new urban type full hydraulic compact excavator. A sketch of the machine was shown in Figure 5.1. In the original machine, there are 4 joint motions that are controlled by two joysticks through modulation of the oil by the pilot operated main valves. These motions are • Bucket in and out (load and dump) • Stick in and out • Boom up and down • Cab swing motion (slew motion) The other 5 degrees of freedom which are controlled by manually operated hydraulic valves are • Right travel (traction) • Left travel (traction) • Dozer blade motion • Boom swing motion • Auxiliary motion (optional attachment such as breaker for the end-effector) Note that there are two types of swing motion in this machine, i.e., cab swing and boom swing. The first one is controlled by a pilot operated valve while the second one is controlled by a manually operated valve. Whenever the word "swing" is used alone, it refers to the cab swing (or slew) motion. Three degrees of freedom in the machine are actuated by reversible hydraulic motors: right travel, left travel and cab swing. All other actuators are single-rod cylinders with limited linear motion. In this work, group of the motions that are controlled by the pilot operated hydraulic valves are of main interest. 131 B.2 The Main Valve System Schematic diagram of the main valve system is shown in Figure B.l. Takeuchi designers have been very conservative in their design. They have installed several pressure relief and check valves wherever it might be useful for protection. According to this diagram, the circuits which provide oil to the right and left travel motors have priority for the supply of oil from pumps Pt and P2. On the other hand, the hydraulic circuit is also configured so that when a travel lever is operated, the solenoid valve and the logic valve are activated, causing hydraulic oil from pump P3 to flow to the attachment (backhoe actuators). That is, during travel, operation of the attachment is possible. Load check valve In Figure B.f, one can see that a load check valve is included in each section except the travel sections. This valve prevents oil from flowing backward due to the load pressure from the actuator port during switching of the spool. Main relief valve A main relief valve is mounted between the pump circuit and tank circuit of each main valve inlet housing and serves to maintain the circuit pressure below the corresponding set value. Port relief valve Valves of this type are incorporated into the boom, stick, bucket and the dozer blade sections. These valves are located between each actuator port and the tank circuit. If there is abnormal pressure due to shock or external pressure generated (when the actuator port is suddenly sectioned by the spool, or when an overload occurs), they act to protect the actuator. Anti-cavitation valve Some anti-cavitation valves which are essentially check valves are also incorporated, for example, between the cylinder port of the boom swing section and the tank circuit. When a cylinder is operated at high speed and supply of hydraulic oil can't keep up with it, causing a vacuum to form at the cylinder port, this valve supplies oil from the tank, preventing cavitation from occurring. Solenoid and logic valve When the travel levers are pushed, the solenoid valve is excited, causing the spool to operate and the logic valve reacts to it, acting as a switching valve to cause oil from pump P3 to flow to the Trasm section of the control valve. 132 Main Relief Valve V T K Right AlJ^O Travel I Activated by a limit switch through^ I movement of travel levers Figure B.l Schematic diagram of the main valve bank in the mini excavator. 133 Figure B.2 also shows the main valve system with less details. According to this figure, there are some couplings in the main valve system. Engine Control Valve (sub) Left Travel ^ Dozer Blade ^ Cab Swing _L T 1 L. Right Travel \ , X. Stick Boom Swing ^ Trasm ^ Valve Boom Bucket Auxiliary ^ Solenoid Valve and Logic Valve LU Activated by a limit switch through movement of travel leverS| Figure B.2 Simplified diagram of the main valve system of the mini excavator. 134 Depending on the traction, we have two different cases as below : • When travel levers are in neutral position, the fluid flow coming from pump P3, first passes through dozer blade and then goes back to the tank through the cab swing valve. This means that, the flow received by cab swing valve depends on whether dozer blade is working or not and vice versa. These valves have equal priority in getting fluid from pump P3. In any condition, except for auxiliary valve operation (which is not considered in this research), there is no collaboration between pumps PI and P2. This is particularly different from Caterpillar excavator discussed in [61]. In the Takeuchi machine, the boom and bucket have equal priority in receiving fluid from pump Pf. Similarly, the stick and boom swing valves have equal priority in receiving fluid from pump P2. • When travel levers are pushed, the solenoid and logic valve activate, blocking the fluid path from pump P3 to the tank. If both dozer blade and cab swing valves are in neutral position, flow of pump P3 enters the Trasm circuit and can supply oil for the backhoe attachments, wherever it is needed. It is worth noting that movement of right or left travel levers, blocks flow of Pf or P2 respectively from going to the attachment. The reason for using the Trasm valve is to keep the constant flow of pump P3 to tank if none of the backhoe links are activated. The orifice inside this valve prevents the pressure in Trasm circuit to fall to zero, so that fairly high pressure from P3 will be present whenever any of the backhoe actuators need it. Note that there will be no fluid in the Trasm circuit if any of the two actuators (dozer blade or cab swing) are activated. Thus dozer blade and cab swing have higher priority (than the backhoe actuators) in receiving fluid from pump P3, which is reasonable. 135 B.3 The Pilot System The required pilot pressure is provided by the control valve (sub) block in the main valve diagram. As shown in Figure B.3, the control valve (sub) section consists of a pressure relief valve, a pressure reducing valve and an electric on/off valve. The pressure reducing valve is used to obtain the low pressure pilot supply from the high pressure output of the pump P3. The pressure relief valve is activated if the pilot pressure is lower than P3 and provides the pressure required for the main control valves. The on/off control valve activates the pilot pressure. The pilot supply pressure generated, is then modulated through the movements of the mechanical joysticks and is used to activate the pilot operated main valves. Engine )!( • -L-I I I I Control Valve (sub) 1/4 of the Pilot Valve • I- -fcxd sp ->— Handle Pilot Supply pressure ,_|__ Output Port Pressure Supply for the Main Valve Figure B.3 Schematic diagram of the original pilot system of the mini excavator. B.4 The Interface Circuit for DPWM-Operated Solenoid Valves Experiments have verified that a DPWM frequency of JDPWM = 100 Hz with a center duty cycle of 50% is a suitable choice for our system. In the very first experiments the digital (TTL compatible) DPWM signals were generated by computer. In order to reduce software overhead, a hardware circuit has been designed that converts an analog voltage (—5V < u < 5V") to a pair of 136 DPWM digital signals whose duty cycles depend on the analog voltage. The circuit is shown in Figure B.5. It is based on the comparison of the analog voltage (Vi = u) with a symmetric triangular signal. The DPWM frequency is IDPWM = AR'C (B.l) and can be adjusted by the 10 turn potentiometer R'. Values of C = 0.047/J,F and R' = 53.2 K would result in JDPWM = 100 Hz. Figure B.4 shows the signals for u — 2V. The characteristic equations of the circuit are as below f Di = (50 - lOtt) [ D2 = (50 + 10tt)% (B.2) where D\ and D2 are the positive-duty cycles of the two DPWM outputs. The two outputs of the convenor circuit are then connected to the current amplifier stage that drives the solenoid valves. 10 VR & Vi -5 -10 I 10 10 20 ms DPWM (1) 30 10 ms DPWM (2) 10 20 ms 30 10 20 ms 30 Figure B.4 Typical signals of the analog to DPWM convenor circuit. 137 10nF D2A Output (u) •-R -AAA/v-R R •AMA-R -AAAA-1/4 LM324 1/4LM324 R=100K, 1% i I I Lowpass Filter j | and Inverter 1/4 LM324 7nr 56Q iv*; r— -12V -AAAA-R 100 K, 10 turn 0.047 uF I + 777/ J z 1/4 LM324 R=100K, 1% I Triangular I Wave Generator D P W M (1) D P W M (2) Figure B.5 Analog to DPWM convenor circuit. 138 B.5 Sensor Specifications B.5.1 Pressure transducers (GP:50 New York Ltd.) The main pressure sensors (model 211-D-RV-2-CA/JJ/GE) have a capacity of 5000 psig and the pilot pressure sensors (model 211-D-RH-2-CA/JJ/GE) have a capacity of 500 psig. These sensors feature amplified DC output (voltage) with the following specifications: • Accuracy = 0.1% FSO • Output = (0 to +5 Vdc) • Supply voltage = 10.5-32 Vdc • Proof pressure = 2 times full-scale range • Boost pressure = 5 times full-scale range • Response time = 0.5 ms B.5.2 Load pins (Sensor Developments Inc.) The 1-DOF load pins (model 20006) have been custom-designed for the mini excavator. They measure the reaction force of the backhoe cylinders to their hinges by using strain-gage Wheatstone bridge. These sensors have the following specifications: • Capacity (bucket load pin) = 20000 Lbf = 88.960 KN • Capacity (stick & boom load pins) = 30000 Lbf = 133.44 KN • Nominal output = 2 mVN • Bridge resistance = 700 fl • Hysteresis & nonlinearity = 0.25% F.S.. • Compensated temperature range = +21 to +77 °C • Usable temperature range = -54 to +121 °C Each load pin output is connected to a self-contained, AC powered, signal conditioning module (model 90131). This module contains a precision differential instrumentation amplifier with filtered output and a regulated adjustable bridge excitation source. This encapsulated module has the following specifications: • Dynamic response (DC to -3 dB two pole Bessel filter) = 2 kHz 139 • Gain range = 40-250 • Gain temperature coefficient = 200 ppmfC • Common Mode Rejection Ratio (CMRR) = 90-t00 dB The following adjustments were made for our application • Gain = 250 • Filter bandwidth = 100 Hz • Bridge excitation voltage = fO V Using the raw calibration data provided by the manufacturing company, the following equations were obtained for force measurement: Boom : y = ^  f (17230.37 + 318.28a:)a; x > 0 (17505.54 - 325.64z)a; x < 0 (-18044.19 - 469.40z)a: x > 0 Stick : y = i (B.3) (-18801.13+ 467.77z)a: x < 0 . (-17267- 608.13x)x x^O Bucket : y (-20123.60 - 154.23a;)a; x < 0 where, x is the amplified output of the load pin (in Volts) and y is the calculated reaction force (in Newtons). B.6 Manipulator Kinematics The mini excavator can be considered as a serial manipulator with rigid links, fn this section, the forward and inverse kinematic equations are derived for the mini excavator. The kinematic analysis of the Cat-215B excavator can be found in [85, 61]. There are two major differences in the kinematic structure of the Cat-2f5B and the mini excavator: • There is no offset distance between the backhoe links and the cab joint in the mini excavator. 140 • An additional degree of freedom exists in the mini excavator which is the "boom swing motion". Since the corresponding actuator is not pilot-operated, it will be ignored in the following analysis. Figure B.6 shows the schematics of the mini excavator, as a 4-DOF manipulator, along with the assigned link frames. Denavit-Hartenberg (D-H) convention is used to assign the coordinate frames [64]. The link parameters are listed in Table B.l. i Figure B.6 Mini excavator along with the assigned D-H coordinate frames Link Link Length (m) a. i d. 1 Cab Swing (1) a1 = 0.760 0 Boom (2) a2 = 2.500 0 0 62 Stick (3) a3 = 1.290 0 0 63 Bucket (4) a4 = 0.716 0 0 64 Table B.l Link parameters for the mini excavator. B.6.1 Approximating polynomials for the x(0) mappings As discussed in Section 6.3.2, fifth-order polynomials are used to approximate the mappings Xi{0i), i = 2,3,4, for the backhoe links. Note that 0; is the joint angle and 0 < Xi < Li is the linear 141 displacement of the corresponding piston. Table B.2 shows both the angular and linear limits of the motion for backhoe links along with the calculated approximating polynomials. Table B.2 The approximating polynomials for the x(6) functions. Boom -1.085 rad < 62 < 1.2075 rad , 0 < x2 < 0.575 m x2 = 2.0654 x 10~4^ + 8.5559 x 10" 36»| - 5.0482 x 10_ 26»| + 3.7527 x lO - 30f + 3.1380 x 10_ x62 + 2.60 55 x 10"1 Stick -2.6927rad < 03 < -0.6102 rad , 0 < x3 < 0.615 m x3 = -3.9231 x 10"36»! - 2.1809 x 10 - 2 0 | + 2.1045 x 10_26>| + 2.5698 x 10_1(9| + 8.1529 x IO - 2 63 - 2.7975 x 10"2 Bucket -2.6843 rad < 64 < 0.5182 rad , 0 < x4 < 0.490 m x4 = 4.7953 x 10_ 56»| + 6.2684 x 10" 36i + 3.8262 x 10 - 2 ^ + 3.8016 X 10 - 2 ^ - 2.0399 x 10~ ^4 + 8.9810 x IO"2 B.6.2 Forward kinematics Problem: Given the joint angles 62, 93,04), calculate position and orientation of the bucket w.r.t the base frame. For simplicity, we first assume that the endpoint is the bucket joint. It is beneficial (see [85]), to express the endpoint position in cylindrical coordinates (re,0e,ze) . From Table B.l, one obtains the following A matrices: 142 c i 0 s i S i 0 —c\ 0 1 0 0 0 0 O l C l 0 ,A 2 c 2 - 5 2 0 a 2 c 2 s 2 c 2 0 a 2 s 2 0 0 1 0 0 0 0 1 ,A3 c3 - 5 3 0 a3c3 5 3 c3 0 a 3 5 3 0 0 1 0 0 0 0 1 (B.7) Thus, the arm T matrix becomes: T = A i A 2 A 3 = C i c 2 3 - c i 5 2 3 5 i c i ( a i + a 2 c 2 + a 3 c 2 3 ) 5 i c 2 3 - 5 i 5 2 3 -c i 5i(ai + a2c2 + a3c23) •S23 C 2 3 0 a 2 5 2 + a 3 5 2 3 0 0 0 1 (B.8) where c 23 = cos(02 + 63), and so on. The following forward kinematic equations can be obtained from the last column of T. re = a i + a2c2 + a3c23 0e = #1 (B.9) ze = a2s2 + a3s23 It can be easily verified that for the endpoint defined as the tip of the bucket (rt,0t,zt), the forward kinematic equations would be as follows: rt = ax+ a2c2 + a3c23 + a4c234 0t = 0i (B.10) zt = a2s2 + a3s23 + a4s234 a = 0234 where a is the angle between the bucket link and horizontal plane and 0 2 3 4 = 02 + O3 + 04. B.6.3 Inverse kinematics Problem: Given the bucket tip position (rt,0t,zt) and its orientation (a), find the joint angles (Oi,O2,03,O4). Figure B.7 shows the projection of the manipulator links onto the vertical plane formed by the (xi, yi) axes. This geometric diagram can be used to obtain the inverse kinematic equations [64], as follows: 143 Figure B.7 Projection of the links onto the vertical plane 61 = 0e h = \J(n - a\f + *f, ix = tan" I = ^Jl\ + a2 — 2l\a4cos(i\ — a), £2 = sin _x(a\ + a\-l2 zt .\( zt — a^sina ^3 = —7T + COS~ 2a2a3 (B.ll) \ 2aol 04 = " - 623 One should note that, due to the joint angle constraints, "elbow up" is the only possible configuration. Therefore, the problem of inverse kinematics has a unique solution provided that the desired position and orientation of the bucket is within the workspace of the machine. 144 Appendix C Differentiation and Low-Pass Filtering C.l Numerical Differentiation It is sometimes required to differentiate a signal in real-time, for example to estimate velocity from position signal. The following first order transfer function represents a basic low-pass filtered differentiator in the s-domain: H(s) (C.l) 1 + Tvs where Tv > 0 is the time constant of the filter. Bilinear transformation is used next to discretize the filter: 2 1 - z'1 s = (C.2) Ts 1 + 2-1-where Ts is the sampling time of the digital filter. Using this transformation, one obtains the following discrete approximation: a ( l _ ^ - i ) H e " ( z ) = 1 - /3*-i with the coefficients a and (3 defined as below: ' a = 2/(27; + Ta) < . (5 = (2TV - TS)/(2TV + T„) For implementation, consider the following filter: (C.3) (C.4) X H (z) eq V ^ Figure C.l Digital differentiator. The difference equation equivalent to the discrete transfer function (C.3) can be expressed in the following state-space form: ' zv = x + /3zvoid (C.5) s v = a(zv - zvoid) 145 here zv is the state variable. In an off-line application, one can use a non-causal high-order filter for delay-free differentiation. An order f 6 digital Finite-Impulse-Response (FIR) filter (see [52]), with the typical amplitude response shown in Figure C.2 can be employed for this purpose. * / s / 5 0) TS 3 ° nfs/5 (rad/s) * / S Frequency Figure C.2 Amplitude response of a typical off-line low-pass filtered differentiator. C.2 Numerical Low-Pass Filtering It is sometimes required to low-pass filter a signal in real-time, for example, to remove the noise or for anti-aliasing. A first order low-pass filter can be represented by the following transfer function in the ^ -domain: i (C.6) 1 + Tvs where Tv > 0 is the time constant of the filter. Using bilinear transformation, the approximation in the z-domain would be: with the coefficients 7 and (3 are defined as below: ' 7 = TS/(2TV + Ta) < . (3 = (2TV - TS)/(2TV + Ts) (C.7) (C.8) 146 For implementation, consider the following filter: X L J Z > eq y Figure C.3 Digital low-pass filter. Defining the state-variable zy, one arrives at the following description in the state-space: Zy X -\- fiZygld . V = l{zv + zyold) (C.9) 147 Appendix D Linear Least Squares Estimation Basics of the linear least squares estimation and corresponding properties are briefly reviewed in this appendix. More details can be found in [7, 32, 29]. To start, consider the general estimation model for linear regression: where Y = the measured system output, (nxl) X = the unknown parameter vector, (p x 1) A — the measured regression matrix, (n X p) E = [a] — the residual vector, (n x 1) The observation Y and the regressor A are usually obtained from an experiment. The least squares criterion aims to minimize the following quadratic loss function: fn other words, the objective is to make the predicted output Y = AX as close as possible (in a least squares sense) to the measured output. The parameter vector X which minimizes V is such that Equation (D.3) is called the normal equation. If the matrix ATA is nonsingular (invertible), the optimal solution is unique and is given by Y = AX + E (D.l) (D.2) ATAX = ATY. (D.3) (D.4) A necessary and sufficient condition for the matrix ATA to be invertible is that the columns of A be linearly independent [9]. This condition is called the excitation condition. 148 For statistical interpretation, assume that the real process can be defined by the following relation: Y = AX° + E (D.5) where X° is the true parameter vector and the modeling error E = {e,-, i = 1, ...,n} is a zero mean white noise sequence which is uncorrelated with the regressors and has a standard deviation of ae. It can be shown that the following statistical properties hold: I. X is an unbiased estimate of the true parameter vector X°, i.e., E X = X° where E[.] is the expectation function. II. The covariance of X is as below: cov{x^ = a2e(ATA)~1 (D.6) III. An unbiased estimate of a2 is ' < D - 7 > One can directly compute the estimated covariance matrix o2(ATA)~1 from the experimental data. According to the properties II and III, diagonal elements of this covariance matrix are the estimated parameter variances. The quality of estimation depends on the regression matrix A which is also called the information matrix. This matrix is usually a function of some independent variables in the system (joint angles of a robot, for example). Singular values of the information matrix A can be used to judge the level of excitation. The concept of Singular Value Decomposition (SVD) can be found in the mathematical texts such as [29]. Singular values of a general matrix A are in fact square roots of the eigenvalues of ATA. Note that AT A is a symmetric positive semi-definite matrix, therefore, all of its eigenvalues are non-negative. For a full-rank matrix A, the singular values can be sorted as follows: S(A) = {01, <T2, o-p} (D.8) The 2-norm condition number of A is calculated as below: 149 thus K(A) > 1. A large condition number shows that the problem is ill-conditioned and the matrix A is close to rank deficiency, while, a small condition number shows that the problem is well-conditioned. In [29], Golub and Van Loan have shown that in nonzero residual problems, K2(A) measures the sensitivity of the solution X to errors in Y and A. In contrast, residual sensitivity depends only linearly on K(A). On the other hand, in a least squares estimation problem, according to property II, it is desirable that ATA be as large as possible to obtain high accuracy in the estimation. One can equivalently say that, the minimum singular value of A should be large. Thus, there are two measures for quality of excitation: • The condition number of A A small condition number is desirable. The best condition number (K(A) = 1) is obtained if A is an orthogonal matrix, i.e., ATA = Ip, where Ip is the (p X p) identity matrix. • The minimum singular value of A A large value is desired for <rmj„(A) to achieve high accuracy. Note that ermiN(A) grows with the size of the equations (n). Using the above measures, one can design a suitable identification experiment that corresponds to a high level of excitation. 150 

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