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Dynamic simulation and control of teleoperated heavy-duty hydraulic manipulators Sepehri, Nariman 1990

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DYNAMIC SIMULATION AND CONTROL OF T E L E O P E R A T E D H E A V Y - D U T Y HYDRAULIC MANIPULATORS By Nariman Sepehri B. Sc. (Mechanical Engineering) Tehran University of Technology, Iran M. A. Sc. (Mechanical Engineering) University of British Columbia, Canada A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S M E C H A N I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A September 1990 © Nariman Sepehri, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Mechanical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Oof; ly ° f 0 Abstract Some relevant aspects of dynamics and control of heavy-duty hydraulic machines in a teleoperated mode were investigated. These machines, such as excavators and for-est harvesters, are mostly used in primary industries. They have a manipulator-like structure with a nonlinear and coupled actuating system. The aim of the project is to investigate different approaches towards converting Buch machines, with minimum changes, into task-oriented human-supervisory control systems. This provides the op-portunity to use both human supervision and robotic power in hazardous environments and for tasks for which human decision is necessary. A methodology was developed for fast and accurate simulations. Analytical, steady-state and numerical techniques were combined using Large-Scale Systems analysis. The inclusion of nonlinearities in the form of discontinuities (e.g., gear backlash and stick-slip friction) in the model was investigated. Numerical simplifications of the structural dynamics and alternative solutions for the hydraulic part were also studied. The model describing the performance of the machine has been written in ACSL (Advanced Continuous Simulation Language) on a VAX computer system. A modified version of the program is at present running close to real-time on a single processor in conjunction with high speed graphics in a manner similar to a flight simulator used for human interface studies and training. The model also evaluates the performance of the machine in a teleoperated mode and under different control strategies. As a result a velocity control algorithm has been developed which is applied in conjunction with the closed-loop components for teleop-eration of heavy-duty hydraulic machines; it is basically a feedforward compensation ii which uses the measured hydraulic line pressures along with fluid-flow equations as cri-teria to control the joint velocities as well as to uncouple the interconnected actuating system. The control algorithm has been written in C language and is running on an IRONICS computer system, interfaced between the human operator and the machine. The simulation results are supported by the experimental evidence. The experiments were performed on a Caterpillar 215B excavator. Improved operator safety, extension of human capability, job quality and produc-tivity increase are the advantages of a successful implementation of robotic technology to these industrial machines. iii \ Table of Contents Abstract ii List of Tables viii List of Figures ix Nomenclature xiii Graphic Symbols xv Acknowledgements xvii 1 INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 Motivation and General Objective 2 1.3 Scope of the Present Investigation 5 2 A H E A V Y - D U T Y HYDRAULIC MACHINE 9 2.1 Machine Performance 9 2.2 Machine Control 16 2.2.1 Joint-Mode Control 16 2.2.2 Resolved-Mode Control 17 2.3 Hardware Implementation 19 2.3.1 Pilot Servovalves 19 2.3.2 Sensors and Control System 21 iv 3 HYDRAULIC SYSTEMS SIMULATION 24 3.1 Relevant Background 24 3.2 Function-Based Modelling 25 3.2.1 Outline of the Approach 25 3.2.2 Towards Efficient Simulation 31 3.3 Flow-Based Modelling 32 3.3.1 Outline of the Formulation 33 3.3.2 Application to Mobile Hydraulic Systems 36 3.3.3 Simulation Results 38 3.3.4 Discussion 40 3.4 Partitioned Hierarchical Modelling 41 3.4.1 Reconstructing the Model 41 3.4.2 Solution to the Main Valve Hydraulics 46 4 H E A V Y - D U T Y S T R U C T U R A L DYNAMICS 57 4.1 Relevant Background 57 4.2 Outline of the Modelling 58 4.3 Significance Analysis 60 4.3.1 Outline of the Approach 60 4.3.2 Results 64 4.3.3 Numerical Simplification 69 4.4 Discussion 70 5 INCLUSION OF DISCONTINUITIES IN T H E M O D E L 74 5.1 Experimental Observation 74 5.2 Gear Backlash 75 5.3 Friction and Leakage 79 v 5.4 Simulation Result and Discussion 81 6 M A C H I N E CONTROL 83 6.1 Relevant Background 83 6.2 Single-Link Velocity Control 86 6.2.1 Outline of the Approach 86 6.2.2 Choice of Spool Displacement 89 6.2.3 Control Input 93 6.2.4 Results 93 6.2.5 Hysteresis Applied to Control 104 6.3 Multi-link Velocity Control 105 6.3.1 Pump Outputs and Interconnection Constraints 106 6.3.2 Inverse Main Valve Hydraulics 107 6.3.3 Results 108 6.4 Inclusion of Closed-loop Part 114 6.5 Discussion 120 7 CONCLUSIONS 124 7.1 Contributions of this Research 124 7.2 Recommendations for Further Research 126 References 128 Appendices 134 A Hydraulic Power 134 A . l Hydraulic Fluids . 134 A.2 Flow Through Orifices 135 vi A.3 Hydraulic Pumps and Motors 136 A.4 Variable-Displacement Pumps 138 A.5 Hydraulic Valves 140 A. 6 Stroking Force in Hydraulic Valves 143 B Hydraulic Systems for Mobile Machines 144 B. l Constant Flow (CF) System 144 B.2 Constant Pressure (CP) System 145 B.3 Load-Sensing (LS) System 145 C Excavator Dynamics Equations 147 D Solution to a Class of Differential Equations 153 E Curve-Fitting Technique 154 F Excavator Kinematics 156 G Solution to Bucket Linkage 158 H A Frequency-Domain Analysis 160 vii List of Tables 3.1 Computation time (VAX 11/750) for 4sec simulation 38 3.2 Pressure and flow pattern in the main valves (simulation) 51 3.3 Possible configurations for the hydraulic main circuit 53 3.4 Computation time (VAX 11/750) for hydraulic main circuit 55 4.5 Computational efficiency in the simplified model 70 4.6 Torque error in the simplified model 72 viii List of Figures 1.1 Plan of study 8 2.2 Caterpillar 215B excavator 10 2.3 Pumps output flow versus summing pressure 10 2.4 Hydraulic main valve circuit 12 2.5 Fluid-flow pattern in the main valves during boom-up operation . . . . 13 2.6 Variation of orifice areas versus spool displacement (boom) 14 2.7 Pilot system circuit 15 2.8 Excavator machine in a resolved-mode 18 2.9 Control of excavator in a resolved-mode 18 2.10 Pilot servovalve used to retrofit the machine 20 2.11 Hardware structure of sensory and control system 22 3.12 Hydraulic circuit with priority and cross-over action 27 3.13 Single-link actuation with open-center valve 33 3.14 Two-link actuation with priority action 36 3.15 Response to step inputs (two-link) 39 3.16 Subsystems and their input/output arrangement 42 3.17 Response of main valve to a step input (stick) 43 3.18 Response of cross-over valve to a step input (stick) 43 3.19 Main spool displacement versus input voltage (stick) 45 3.20 Cross-over spool displacement versus input voltage (stick) 45 3.21 Schematic diagram of the main valves 47 ix 3.22 Alternative solutions to hydraulic main circuit 54 4.23 Structural model of excavator 59 4.24 Views of different experimental trajectories 61 4.25 Joint angle trajectories over path 1 62 4.26 Joint velocity trajectory over path 1 (link 3) 63 4.27 Joint acceleration trajectory over path 1 (link 3) 63 4.28 Nominal open-loop torque over path 1 63 4.29 Torque composition for link 1 66 4.30 Torque composition for link 2 67 4.31 Torque composition for link 3 68 4.32 Torque error due to dynamic simplification 71 5.33 Gear train arrangement in the Cabin . 75 5.34 Line pressures in swing motion (experiment) 76 5.35 Combined simplified model of gear backlash and hydraulics 76 5.36 Stick-slip friction model 80 5.37 Line pressures in swing motion (simulation) 81 6.38 A heavy-duty hydraulically actuated arm 86 6.39 Different types of interaction with environment 91 6.40 Velocity control flow-chart 92 6.41 Boom up—•down velocity control (experiment) 94 6.42 Actuator force in boom up—•down motion (experiment) 94 6.43 Pressures in boom up—*down motion 95 6.44 Spool displacement in boom up—*down motion (experiment) 96 6.45 Control input in boom up—•down motion (experiment) 96 x 6.46 Stick out—+in velocity control 99 6.47 Variation of Force and Voltage in stick out—nn motion 100 6.48 Pressures in stick out—»in motion 101 6.49 Bucket out—+in velocity control 102 6.50 Pressures in bucket out—>in motion 103 6.51 Hysterisis applied during model switching 104 6.52 Multi-link velocity control flow-chart 109 6.53 Multi-link velocity control (boom and stick) 110 6.54 Outline of the numerical approach to solve case (a) I l l 6.55 Pressures in multi-link motion (boom) 112 6.56 Pressures in multi-link motion (stick) 113 6.57 Decoupled-flow control block diagram 115 6.58 Swing velocity control (experiment) 117 6.59 Swing velocity control (simulation) 117 6.60 Pressures in swing motion (with closed-loop and compensation) . . . . 118 6.61 Control inputs in swing motion (with closed-loop and compensation) . . 119 6.62 Swing joint angle trajectory (with closed-loop and compensation) . . . 119 6.63 Complete block diagram of a teleoperated heavy-duty machine 123 A.64 Flow through an orifice [Merri67] 136 A.65 Positive displacement hydraulic machines [Merri67] 137 A.66 Pressure-flow diagram 139 A.67 Typical hydraulic valves [Merri67] 141 A. 68 Flow gain of different center types [Merri67] 143 B. 69 Principle sketch of different hydraulic systems 146 xi G. 70 Linkage arrangement in bucket 158 H. 71 Typical hydraulically actuated arm 161 H.72 General control block diagram of the system 161 xii N o m e n c l a t u r e P fluid pressure P, supply pressure Pe tank pressure Pi, Pa line pressures Pi = Pi — Pa load pressure Q fluid flow-rate Qi, Q0 flow into and out of the valve Qe flow back to the tank Q3 supply flow Qi ^ ' "j" ^° load flow Qi, Q0 flow into and out of the actuator X piston linear velocity A piston area v control input (voltage) x spool displacement a area of discharge of orifice or restrictor di p u m p to cylinder orifice area ae pump to tank orifice area aa cylinder to tank orifice area a check valve orifice area o u/=~ area gradient x Cd coefficient of discharge of orifice or restrictor xiii V fluid volume on each side of the actuator piston p density of the oil k=Cd> -V P metering coefficient P effective bulk modulus r-Y. hydraulic compliance Dm volumetric displacement of hydraulic motor T applied torque F applied force R resistive force M, m link mass K , K constant (gain) J link inertia e joint angular rotation 9 joint angular velocity 9 joint angular acceleration T time constant xiv Graphic Symbols hydraulic line pilot line lines crossing lines connection reservoir (tank) pressure gauge filter or strainer internal combustion engine ram-type actuator hydraulic motor unidirectional p u m p to I ! A3 I T variable flow pump pilot controlled pump fixed orifice variable flow control orifice check valve -1- — relief valve manually operated (valve) I / | ^ solenoid operated (valve) j spring centered pilot controlled (valve) two-position, two-way control valve mm two-position, three-way control valve two-position, four-way control valve three-position, four-way control valve xvi Acknowledgements I would like to thank my supervisors, Dr. P.D. Lawrence and Dr. F. Sassani for their guidance and encouragement throughout the course of this thesis. Their patience and support are sincerely appreciated. I must express my special gratitude to Real Frenette (Department of Electrical Engineering) whose help and contribution in the implementation of the algorithms has made the completion of this thesis possible. The time and advice given by the members of my thesis committee, Dr. D.B. Cherchas and Dr. G.A.M. Dumont are also gratefully appreciated. Special gratitude is expressed to Allan Hewett (Robotic System International Re-search) and the late Mr. Ken Ryan for the helpful discussions I had with them. Their professional counsel and friendship are very much appreciated. The author also wishes to acknolwledge McMillan Bloedel Research for providing a test bed machine and other field facilities during the experimental studies. xvii Chapter 1 INTRODUCTION 1.1 Preliminary Remarks The science and technology of robotics have developed to a great extent in recent years and many forms of robots have been built and put into use. The application of robotic systems in controlled surroundings is well known. The working conditions are prepared for robots which have limited flexibility. However, some surroundings contain environments which cannot be determined beforehand. Work in these types of conditions has the following characteristics [Shira84]: • the working area is large; • the working environment is complex; • recognition of the environment and decision making is required. Human beings have the ability to adapt and react to changes in these environments, whereas today's fully automated manipulators do not possess such abilities. Many tasks in these surroundings can be automated and the robotic systems are good substitutes for human labour in hazardous or hostile surroundings. Teleoperation provides the opportunity to use both human supervision and robotic power in hazardous areas and for tasks for which human decision is necessary. Examples of teleoperating systems are the spatially-corresponding manipulators used for nuclear material handling. In this system, the human-operated master arm is geometrically 1 Chapter!. INTRODUCTION 2 similar to the operating machine's slave arm; as the operator moves the master arm, each joint of the slave arm follows the movement of the corresponding master arm joint. In a more advanced concept, the master is not an exact copy of the slave. It can even be a keyboard of a micro-computer which is interfaced to the slave. The information such as the positioning of the slave in a base-coordinate system i6 passed from the operator, through the micro-computer. The joint motions and the required position of each joint are then calculated and are passed to the slave actuator servos. This concept can be further extended in such a way that some routine activities such as handling an object from point A to point B be done automatically (a supervisory system). The role of the operator is to give the sub-task specification. This way human intelligence and adaptability is coupled with the machine's strength and immunity to environmental hazards. A similar concept can be used to create intelligent robots. The only difference is that artificial intelligence will be substituted for human intelligence in the control loop. The benefits of the use of teleoperating systems with supervisory control and/or Al in the future have been investigated previously [Hogge85, Sheri86a, Sheri86b]. 1.2 Motivation and General Objective The application of teleoperating systems becomes more challenging when applied to manipulator-like industrial machines. These machines are heavy-duty, hydraulically actuated structures and are used in primary industries. Amongst them, excavators and forest harvesters could be named. The present control of these machines is based on individually actuating each link, controlled by some mechanical means like levers. Most of these machines are mobile and carry their own fuel. The actuation system is complex and coupled; its design is based on acquiring higher efficiency through evolution and Chapter 1. INTRODUCTION 3 innovation and from the stimulus of the competitive market place. The efficiency in gas consumption, speed, smooth operation, etc., depends directly on the operator and his experience with the machine. He is required to learn the operation of the machine which is usually time consuming and costly. Both hands and feet of the operator are involved. Concentration and skill are needed. A successful implementation of robotics technology to these industrial machines could bring about the following advantages [Cherc83]: • productivity increase; • labour cost reduction; • job quality increase; • improved operator safety; • improved operator environment; • extension of human capability; • reduced power consumption. The most important characteristics of these machines can be summarized as follows: • they are designed to work efficiently even with the increased complexity; • they are mobile and thus have infinite work-space; • they are designed to do heavy-duty tasks such as picking and placing or digging. Each characteristic highlights some challenging aspects which have not been investi-gated so far in the literature. Chapter 1. INTRODUCTION 4 As far as the previous work in this area is concerned, the U.S. army started to investigate the application of robotics to rapid excavation [Chave83]. In that work an excavator was modified by replacing the valves with special electrohydraulic valves, in-dependently, to get a nearly linear hydraulic pressure vs current input. The dynamics relations were found empirically by measuring the response of each link to an applied voltage. Karkkainen and Manninen [Karkk83a, Karkk83b] worked on some aspects of the supervisory control for a manipulator aimed at handling timber. The experimen-tal laboratory system was a forest manipulator-like machine controlled by hydraulic valves. Vaha and Halme [Vaha84] applied an adaptive digital controller to control the above mentioned manipulator. They proposed that each hydraulic actuator could be described in state-space form in combination with the mechanical model. Also, Wal-dron, et.al. [Song89] developed a human-controlled, large-scale computer-coordinated legged system to operate in unstructured terrain. Each leg is hydraulically actuated through a variable displacement pump with some unique features [Pery85]. No report was found concerning the interaction or the degree of the complexity of the hydraulic systems. The UBC teleoperation project is investigating conversion of these types of indus-trial machines into task-oriented human-supervisory control systems, with a minimum  change in the original design. The previous experiences in remotely-operated devices such as RSIR, Robotic Systems International Research's remotely controlled hydraulic manipulator or Spar Aerospace's Canada Arm, though valuable, have limited appli-cation to these industrial machines. They require substantial changes in the existing machinery design, which considering the present number of these machines in use, and the existing manufacturing facilities is not feasible and/or desirable. With this background, dynamics and control of hydraulically powered industrial Chapter 1. INTRODUCTION 5 machines are studied in this thesis. Different methods are identified, adapted or devel-oped to model the performance of these machines. The aim is to develop a methodology towards fast simulation and better control of 6uch machines. Additionally, a fast sim-ulator can be embedded into the fault/hazard detection system to improve safety and maintenence. It can be implemented in a manner similar to a flight simulator for human interface studies as well as training. It can also be used to evaluate the performance of the machine in a teleoperated mode and under new control strategies. 1.3 Scope of the Present Investigation An excavator has been chosen as an application for this 6tudy. It is used extensively in forest and construction industries. This machine incorporates many aspects of a typical robotic system and all aspects of a typical mobile hydraulic system. Thus the analyses and development reported in this thesis can be applied to other similar systems such as feller-bunchers or log-loaders. The candidate machine, a Caterpillar 215B excavator, is first described in Chapter 2. Its performance in the present joint control mode is compared with the proposed resolved mode. It is shown that the hydraulic linkage drive system is complex. The machine is thus considered as a large-scale system. The necessary implementation to retrofit the machine into a teleoperated system is also outlined. Chapter 3 is dedicated to the modelling of the hydraulic system. The equations representing the actuators dynamics are highly nonlinear and interconnected. The initial study, using known techniques, indicates that the state variables of the hydraulic circuit can be divided into two groups; fast-response states and slow-response states. The system is mathematically 6tiff and inefficient in terms of computation and dynamic simplicity. Using integration routines specifically developed for stiff systems could Chapter 1. INTRODUCTION 6 be useful in terms of computational efficiency, however, they are prone to failure or becoming inefficient when they reach a point of discontinuity as occurs quite frequently in the manipulator under investigation. It is also shown that the way the system is modelled has a direct effect on achieving the efficiency. In this regard, the hydraulic dynamic equations are further rearranged in terms of different state variables. The merits and demerits of this alternative arrangement, for the class of machines under investigation, are assessed. This chapter finally concludes with studying the manipulator at the subsystem level rather than the component level. A partitioned hierarchical approach is used. Each subsystem has its own set of dynamic characteristics. The approach ultimately combines analytical, numerical and steady-state solutions for fast simulations. It keeps its simplicity without losing any state information. Chapter 4 studies the dynamics of the structure. The use of Lagrangian approach leads to an explicit and compact mathematical model which allows for further compu-tational arrangement and customization towards more efficient simulations. The model is then examined for inclusion of other nonidealities in the model. Gear backlash and stick-6lip friction are two phenomena which are observed in the perfor-mance of heavy-duty machines. The inclusion of these nonlinearities in the form of discontinuities onto the model is investigated in Chapter 5. Chapter 6 considers the control of the machine in a resolved mode. Most existing hydraulic robots are designed in such a way that each link could be activated inde-pendently and work on a constant main pressure. Excavators use open-center valves which are coupled and are used with a torque-limited pump system. A method is thus proposed which is applied in conjunction with the clo6ed-loop control. It i6 basically a feedforward compensation which uses the hydraulic model to control the joint ve-locities. It also employs the measured line pressures in calculations which contains Chapter 1. INTRODUCTION 7 information related to the structure and the load. Conclusions of this thesis are outlined in Chapter 7. It also includes suggestions for further studies. Emphasis in this thesis is on the development of a system of principles (methodology) as well as validation of the techniques developed. The simulation results are supported by the experimental evidence. The experiments were performed on a Caterpillar 215B excavator on loan to the project from Caterpillar, Peoria. The plan of study is outlined in Figure 1.1. DYNAMICS AND CONTROL OF TELEOPERATED INDUSTRIAL HEAVY-DUTY MACHINES Structure DYNAMICS Hydraulics Sensitivity Analysis T Static v.s. Dynamic Analysis System Simulation Inclusion of more Nonlinearities & Discontinuities Efficient Simulation Cascading the Hydraulics Decoupling the Actuating System . Joint Velocity Control using Hydraulic Fluid-Flow Compensation CONTROL Introducing new Hydraulic State Variables Experimental Evaluation Closed-loop Control Resolved-mode Experimental Evaluation Teleoperated Control Figure 1.1: Plan of study Chapter 2 A H E A V Y - D U T Y HYDRAULIC MACHINE 2.1 Machine Performance To highlight the problems involved, a Caterpillar 215B excavator [Excav], is described. It is a mobile three-degree-of-freedom manipulator with an additional moveable end effector, namely the bucket (Figure 2.2). The bucket is used to dig and carry loads. It could be replaced by an extra accessory for holding and handling objects such as trees. The upper structure of the excavator rotates on the carriage by a "swing" motor through a gear train. "Boom" and "stick" are the other two links which, together with the swing serve for positioning the bucket. Boom, stick and bucket are operated through hydraulic cylinders. The use of hydraulic cylinders, however, restricts the motion of the links due to the added joint angle limitations. These cylinders are activated by means of pressure and flow through the main valves. Modulation of the oil flow in the main valves is controlled by the pilot oil pressure through the manually operated pilot control valves. Joysticks 1 and 2 are used to control these valves. Referring to Figure 2.2, Forward or backward movement or side to side movement of these two levers provide individual control to the link motion. Figure 2.2 also shows the main actuation (hydraulic) circuitjthe output of the engine is used to turn three hydraulic pumps; two axial piston variable displacement pumps and one gear pump (the latter not shown in the diagram). The output flow from the axial 9 Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE Figure 2.2: Caterpillar 215B excavator o to 15000 30000 45000 60000 Summing Pressure (kPa) Figure 2.3: Pumps output flow versus suinming pressure Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 11 piston pumps is used to operate the hydraulic cylinders as well as the swing motor. When the total (summing) of the pressures in the implement circuits become high enough, the axial piston pumps reduce their outputs to prevent engine stall (Figure 2.3). This kind of hydraulic circuit is known as a load-sensing torque-limited circuit; the highest load is sensed and the output flow is changed to meet the maximum power available from the engine. In order to appreciate the complexity of the main hydraulic circuit, the complete diagram for the stationary system is shown in Figure 2.4. Oil from pump 2, for example, goes through swing, stick and boom cross-over valves to the tank. The stick implement, controlled by the stick main valve, cannot be achieved if the swing main valve is fully open. If the latter is partly open, the stick can operate but at a slower rate. The motion of the boom and the stick are coupled by the cross-over valves. This will cause a faster movement of one when the other is at low speed. The configuration of the valves and the action of the cross-over valves, for boom-up, is shown in Figure 2.5. Each valve has its own orifice characteristic with different forms of lapping. As an example, typical variation of the orifice areas, for the boom main valve, versus the 6pool displacement (from manufacturer) is shown in Figure 2.6. The asymmetry in the cylinder to tank orifice area is because of safety considerations and the fact that boom-down motion is mostly controlled by this orifice. The spool displacement is performed through pressure provided by the pilot system. The pilot system circuit has been sketched in Figure 2.7. The oil pressure in the pilot circuit is set by a relief valve. The control valves in the pilot system are shown in the form of a number of orifices with fixed or variable area; when not operated, the flow of the pilot system oil through the valve is stopped and all goes back to the tank through the relief valve. When operated, the oil is sent to either side of the main valves to activate them. In the teleoperated mode, the original pilot valves are bypassed Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 12 Reservoir (tank) Figure 2.4: Hydraulic main valve circuit Figure 2.5: Fluid-flow pattern in the main valves during boom-up operation CO er 2. A HEAVY-DUTY HYDRAULIC MACHINE o--15 -10 -5 0 5 Spool Displacement (mm) 10 15 Figure 2.6: Variation of orifice areas versus spool displacement (boom) Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 15 Gear P u m p _ {bucket d u m p ^ j - f bucket curl Jftf r - t - r -MA Bucket Valve _jj . i to summing valve On-Off Valve Variable Orifice j e w i n g left i swing right ! Swing Valve ^ J Z H * boom up r ± n — boom d o w n ^ i ^ stick out I ! _ _ — _ j Fixed Orifice' stick in B o o m X 1 Figure 2.7: Pilot system circuit Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 16 with specially electrohydraulic servovalves along with a computer to interface with the operator (see Section 2.3.1). The whole machine can move forward or backward on its tracked undercarriage. An additional pair of valves (which should be superimposed onto the diagram shown in the previous figures) control the mobility as well as the steering. This is done by two foot pedals and an additional lever. Due to the mobility, theoretically the manipulator has infinite work-space. The mobility of the machine is not considered in this study. 2.2 Machine Control 2.2.1 Joint-Mode Control The controls found on excavator machines today consist of two two-degree-of-freedom joysticks with a one-to-one mapping between the joysticks motions and the links (Figure 2.2). Each link of the machine is controlled by a specific motion of one of the joysticks; "in/out" or "left/right". A motion of the joystick corresponds to a velocity command. This is called a joint-mode control. This method of control is easy to implement as it does not require any external computation. However, it i6 accompanied by the following drawbacks: • The mapping of the bucket, for example, is somewhat awkward as a left/right motion of the right-hand joystick produces an in/out motion of the bucket. Sim-ilarly, in/out moving the right-hand joystick has the overall result of an up/down motion of the bucket. • This mode requires much coordination on the part of the operator. For instance, suppose the operator desires to bring the bucket towards himself, at constant height and constant angle with respect to the ground such as in scraping or Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 17 flattening the ground. This necessitates the simultaneous coordination of three links, i.e., three simultaneous motions of the joysticks. The efficiency is then directly dependent on the human operator, his judgement and his skill. 2.2.2 Resolved-Mode Control The resolved-mode control addresses the above drawbacks. It is meant to provide a more natural mapping of the joystick motions to the motions of the links. It also diminishes the level of coordination required to move the machine in any way. Intuitively, the most natural way of operating the machine is to be able to move a joystick in the direction that one wants the end-effector to move; for example moving "up/down" (same radius), "in/out" (same height), "left/right", or any combination of these (Figure 2.8). Thus, the position of the end effector can be controlled with a three-degree-of-freedom joystick (joystick 2, Figure 2.8), having an x, y and z axes. The "right/left" motion of the bucket is actually done along an arc with its center at the swing pivot point. The amount of displacement of the joystick indicates the desired linear speed. As far as the orientation of the bucket is concerned, another degree-of-freedom is required. This can be accomplished by either using another joystick (joystick 1, Figure 2.8), or by adding a fourth degree-of-freedom on the first joystick. The angle of orientation of the bucket can refer to two different angles: • Angle o f the bucket with respect to the stick (a l 5 Figure 2.8): changing the position of the bucket would not only affect the bucket but the resulting angle of the bucket with respect to the ground also changes (case of digging). • Angle o f the bucket with respect t o the ground (a2, Figure 2.8): a change in the position of the bucket would also affect the bucket (case of carrying a load Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE Arc L_J 1 T r r r r Height - Radius a: Orientation Figure 2.8: Excavator machine in a resolved-mode J O Y S T I C K S A(flccftua) tec AjHeight) tec £k(Arc) tec ^(Orientation) eec Radius Height Arc Orientation desired PERIOD I INVERSE KINEMATICS Voltage to Servo-Valves CONTROLLER Figure 2.9: Control of excavator in a resolved-mode Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 19 or scrapping). T h e operator can select the type of orientation, depending on the situation Figure 2.9 illustrates different steps involved to control the excavator in mode. First , the joysticks are sampled to get the user commands in terms of -sec A ( H e i g h t ) A(Arc) ^Orientation) , , . , and . Mult iplying these numbers by the sampling sec sec sec period, and adding the result to the previous desired position/orientation of the bucket, the current desired position/orientation is obtained. B y solving the inverse kinematics of the machine, the desired position/orientation in terms of the individual joint angles and joint velocities is determined. Finally, passing these values to a control algorithm, appropriate voltages are sent to the pilot system. 2.3 Hardware Implementation 2.3.1 Pilot Servovalves To retrofit the machine into a teleoperated system, the conventional pilot valves were replaced with a new set of servovalves. The servovalves used, Figure 2.10, are typical commercially available products. They were originally designed to control the flow-rate in proportion to the input current (two-stage electrohydraulic servovalve with direct feedback [Merri67]). A n attachment was built (marked A , Figure 2.10-a) which converts the above flow control servovalves into pressure control ones. This part connects the two outlet ports 1 and 2 to the tank through a fixed orifice, a0. Figure 2.10-b shows the new servovalve orifice arrangement around one of the main valves. A certain given voltage to the servovalve opens orifices oa and o3. Port 2 is then connected to the return tank through two orifices 0 3 and ao which releases the pressure on one side of the main valve spool. The orifice arrangement on the side of port 1 is sketched as in Figure 2.10-c which is similar to the one in the original pilot valves (Figure 2.7). The Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 20 Figure 2.10: Pilot servovalve used to retrofit the machine Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 21 pressure Pi can be shown to be as below: P1 = V " ' ' (2.1) (*+ (£ ) ' ) where o 0 is the fixed orifice area. a x is the variable orifice area which is controlled by the servovalve through an input voltage. Depending on the value of o i , P\ can change from the tank pressure, Pe, to almost the pilot main line pressure, P,. 2.3.2 Sensors and Control System A V M E bus computer system is used to perform the task. As is shown in Figure 2.11, it consists of the following components: 1. a system controller card for the overall control and monitoring of the V M E bus as well as for the user interaction through a terminal port. 2. a C P U board containing a 68020® 16MHz and a 68881 Math Co-Processor with IMbytes memory. A real-time programmable clock is also available. 3. a 12 bit A / D card to sample the joysticks and the pressure transducers. 4. a 12 bit D / A card to send voltages to the servovalves. 5. a 12 bit R / D (Resolver to Digital) card to read the resolvers. The joysticks are three-axi6 units. They provide the means to give velocity com-mands in resolved mode. The output is a differential voltage of magnitude of between ±lvolt. The A / D card (with the gain set to 5) converts the signals for computer use. Pressure transducers (0 —» 5000psi with accuracy of ±0 .5%) are installed to measure the pressures at the cylinders. Also, pressure transducers (0 —• 500pai with accuracy of ±0 .5%) are used to measure the pilot pressures produced by the Bervovalves. Their V M E BUS TT J — \ \ t C P U M E M O R Y 7TT: 7TT GRAPHICS C A R D S Y S T E M C O N T R O L L E R GRAPHICS M O N I T O R 1 T E R M I N A L 7K l I l '1 r S Y S T E M USER Used During Development Phase' JOYSTICKS I I I A / D 1 ± D / A O P E R A T O R ,16 SERVO-VALVES "LZT P R E S S U R E T R A N S D U C E R S H Y D R A U L I C S R/D 7*~ R E S O L V E R S * S T R U C T U R E r • E X C A V A T O R - - f I I I Figure 2.11: Hardware structure of sensory and control system Chapter 2. A HEAVY-DUTY HYDRAULIC MACHINE 23 output is proportional to the pressure, and is between 0 —» 5volts. The A / D card (with the gain set to 1) converts these signals into digital values. In order to keep track of the position of the machine, resolvers (with the resolution of 0.1 deg) are used in conjunction with the R / D card. The resolvers are located at the pivot points of the links and their output gives the angular position of each link. The servovalves provide the interface between the computer and the hydraulics part. They take voltages coming from the D / A card and produce pilot pressures proportional to the voltages. These pressures in turn are applied across the spools of the main valves. A linear potentiometer is also used to measure the main valve spool displacement. The potentiometer ha6 the resolution of 0.025mm. Chapter 3 H Y D R A U L I C S Y S T E M S S I M U L A T I O N 3.1 Relevant Background As systems including fluid power become more complex, the development of accurate models reaches challenging proportions. Normally each system consists of standard components such as valves, lines, actuators and pumps (see Appendix A ) . The principal characteristics of these components have already been modelled [Merri67]. However, different hydraulic system configurations lead to different characteristics with different dynamics (see Appendix B) . As models become larger and more complex, a closed-form solution becomes difficult to achieve. The complexity of the hydraulic system analysis is not only because of the nonlinearity in the dynamics but mostly because of the interaction between the components that makes the system performance analysis difficult. A hydraulic system is generally assumed to be a large-scale system. A large-scale system is one that contains a number of interdependent subsystems which serve par-ticular functions, share resources and are governed by a set of interconnected goals and constraints [Mahmu85]. A n apparent property of a large-scale system is the high degree of interaction between its components, which is the case for hydraulic systems. A systematic approach for analyzing large-scale systems was first introduced by H . M . Paynter. His method, namely Bond-Graph Theory, was later improved [Karno68]. This method was then applied to model [Karno72] and to simulate [Bowns81b] hydraulic 24 Chapter 3. HYDRAULIC SYSTEMS SIMULATION 25 systems. A systematic approach to the analysis of complex hydraulic systems was later developed [Iyeng75]; it is basically the aggregation of component models together with a set of constraints. Study on development of a general purpose hydraulic simulation package has also been conducted [Hull85] 3.2 Function-Based Modelling In this section a method which i6 basically a solution to the large-scale systems is used to model and analyse the machine under investigation. The actuation system is assumed to contain some smaller subsystems and this is continued until a compact and satisfactory solution to each component or subsystem is achieved. Thi6 type of solution, as will be seen later, will lead to the state-space representation and thus time-domain analysis. The equations for each subsystem are usually formed from first principles. 3.2.1 Outline of the Approach A n approach similar to [Iyeng75] is initially used for the hydraulic circuit analysis. For example, in the hydraulic circuit shown in Figure 2.4, each main valve is assumed to be a subsystem which is connected to the other main valves and the actuators by means of components, namely, rigid pipes and flexible hoses. The set of equations for the subsystems may not represent the system completely. A n extra set of relations is thus necessary to indicate the interconnections between the subsystems and the way the external inputs are imposed to the large system. These are called constraint equations [Iyeng75]. Th e constraint equations are usually linear time-invariant for hydraulic systems. T h e general formulation i6 shown in the following: For the subsystem t , the state equations are: Chapter 3. HYDRAULIC SYSTEMS SIMULATION 26 input { U J } — • | subsystem i with {xi} states J — • output {y,} {xi} = fi[{ui},{xi},t] {Vi} = 5i[W.{*i},t] The constraint equations, or compatibility equations, between the subsystems are: f{Vi}\ / Vi \ {y2} {w2} = [*1 + [G\ (3.2) \ K } / V{v»}/ \vm/ where V denotes the external input, n is the number of subsystems which the system is divided into; m is the number of the external inputs. [F] and [G] are matrices consisting of 1 and 0. Hydraulic components which are commonly used in the hydraulic systems are valves, pumps and lines which are equipped with components like check valves and relief valves. Each component has certain characteristics which are mostly nonlinear. The challenge is to use the most convenient mathematical model for each and match them together in such a way that the model best represents the behaviour of the actual system under investigation. Figure 3.12 demonstrates the technique described above. It is a simplified version of the actual system shown in Figure 2.4; it consists of swing and stick main valves as well as a cross-over valve which provides additional fluid to speed up the stick motion. In this demonstration, the hydraulic circuit is divided into three parts: 1. the swing valve and the connecting hoses; Chapter 3. HYDRAULIC SYSTEMS SIMULATION 27 Pi: Pi: — » t c 1 ) ( a to J Q i tA4 T 5 P22 2 Pe 'SW, t o s w i n g _ Psw j ! . Q-SW . Figure 3.12: Hydraulic circuit with priority and cross-over action 2. the stick valve with its cross-over valve and the connecting pipe; 3. the connecting hoses to the stick actuator. The connection between the valves is rigid compared with the connection between the valves and the actuators, i.e., Ch » Ci where Ch is the compliance of the connection between the valves and the actuators, and C\ is the compliance of the connection between the valves themselves. The following equations are written for the first subsystem (notations are shown in Chapter 3. HYDRAULIC SYSTEMS SIMULATION 28 Figure 3.12): ^{Pswi) = ^ j " [kcL>x.wy/P2i - PSwi - Qswi] ~~[t(Pswo) = ^  [Qswo - ku>x,wy/PSWo - Pt (P71) = ^ [Q ~ kux,wJp21 - PSWi - M l - *«u)\/P*i ~ P22 '32 (3.3) (3.4) (3.5) (3.6) where Q is the pump flow. xlw,xlt and xtic are normalized for the sake of simplicity, i . e . 0< Xtw , Using the notations for the general large scale systems as described before, Q I P21 \ Pswi \PSW0J { « l } = {yi} = QsWo P,2 \ Pe ) Subsystem 2 consists of only two flexible hoses. The following equations are written: Pswi V PsWo ) Jt(PSTi) = £ T \kvXltJPHc - PSTi - 4STi) ^{PsTo) = ^ QsTo ~ ku)X,t\JPSTo ~ Pe (3.7) (3.8) w here PsTi STo, ( *« \ e QsTi {u2} = QSTO , {yz} = P.tc \ Pe J PsTi STo , Chapter 3. HYDRAULIC SYSTEMS SIMULATION 29 The stick valve with its corresponding cross-over and the connecting pipes are consid-ered as the third subsystem: where Ci 1_ Cx (3.9) (3.10) dt Qn - M l " *«)y/Pn ~ Pe ~ kayJP72 - P, - [kay/Pu - Pttc - kcuxHy/PHc - PSTi + kasfP22 - Pttc] (3.11) P*2 \P«c) {u3} = / x'tc \ Q Q21 PSTi \ P.'/ {y*} = P32 \P*cJ The external input vector is then, {V} = ( Xtw X.t X.tc Q QsWi QsWo QsTi QST0 Pe ) For the above arrangement, matrices [F] and [G], Equation (3.2), are as follows; [F? 0 0 0 0 0 0 0 \ o °\ 0 0 0 0 0 0 0 0/ and Chapter 3. HYDRAULIC SYSTEMS SIMULATION 30 [G]T = Z 1 0 0 0 0 0 0 0 \ o 0 ' 0 0 0 0 0 0 0 l / As is seen from these matrices, any input to a subsystem is related to either an external input or an output from other subsystems. This compatibility which is directly dependent on the way the subsystems are modelled and their boundaries are defined is important to ensure that the set of equations derived for each subsystem could together describe the dynamics of the whole system. This method is desirable, since it enables one to start with components with known dynamics to describe the whole system. The complexity of the actuation system is not a major problem and it is easy to trace back the performance of a subsystem to the inputs and the parameters responsible for it [Iyeng75]. In this example, the actuators flow rates (Qswi, Qsw0, •••)> spool displacements (xtw,xtt,...) and pump outputs (Q) were considered a6 external inputs. This is be-cause only the main valve dynamics were of interest. However, these variables are related to the structure, pilot servosystem and the engine and pump dynamics, respec-tively. If the dynamics of the structure for example, is included, some of these variables will then become outputs from the structure subsystem rather than being as external inputs. Also, in the above formulation, the flow across the cross-over check valves had to be modelled for the sake of compatibility. They however serve only to direct the flow in one direction; their effect can be included differently as will be shown in Section 3.4.2. Chapter 3. HYDRAULIC SYSTEMS SIMULATION 31 The concept which was exemplified here has been applied to the entire machine. The general equations describing the complete machine dynamics are listed in Appendix C. 3.2.2 Towards Efficient Simulation Once the equations are derived, they could be directly used in the simulation. The pri-mary study showed that the state variables of the hydraulic circuit could be divided into two groups; fast-response states and slow-response states. Example of a slow-response state is the pressure in the flexible hoses connecting the main valves to the actuators which have a considerable effect on the dynamics and the vibrational behaviour of the structure. An element involving a restriction linking two rigid pipes (e.g., in the main valve system) is a typical example of a component with a high-frequency response. The existence of fast-respon6e states results in the loss of computational efficiency. Using integration routines specifically developed for stiff systems may useful in terms of computational efficiency, however, they are prone to failure [Bowns81a] or becoming inefficient when they reach a point of discontinuity as this occurs quite frequently in the machine under investigation. On the other hand, explicit integration methods could handle these discontinuities. They are basically stable and require just one starting value for each state and are simple to program, but they cannot handle stiff systems as the smallest integration time interval should be used for the whole system. In the process towards more efficient simulations, the system of equations were grouped and studied with respect to different criteria. The main stiffness was found to originate from the connection between the valves and is localized in one part. This can be shown, for example, by rearranging the previously derived differential equations (3.3) to (3.11) as follows: h (3.12) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 32 Jt{PSWo) = ^ 5<p-"> = ch Jt(PsTo) = (3.13) (3.14) (3.15) and 5 ™ -1_ Q - kay/Pn-Ptte - fcw(l - **c)JPn ~ Pe] ] dl(p'tc) ~ cl (3.16) (3.17) (3.18) (3.19) The last four equations, (3.16) to (3.19), determine the explicit integration time-interval for the simulation. It is also seen that, the coefficients of the stiff variables are constant, permitting division which allows them to be rewritten in a different form. Equation (3.16), for example, is written as below (note that Ci is a small value compared with Ch): Cijt(pn) « 0 = [Q ~ ka^Pn - P.tc - kw(l - x.te)y/Pn - Pe] The differential equation responsible for the stiffness has this way been converted to a steady-state equation which allows a larger integration step size [Hrons73]. Also the number of states to be integrated has been reduced. However, some iterations are required. 3.3 Flow-Based Modelling This section considers the possibility of rearranging the hydraulic equations in terms of different state-spaces. It is shown that the way a hydraulic system i6 modelled has a Chapter 3. HYDRAULIC SYSTEMS SIMULATION 33 direct effect on the acquired computational efficiency. An alternative modelling tech-nique is thus investigated. Fluid-flow rate, rather than the conventional fluid pressure, is chosen as a state-space variable. This concept, which was originally developed by [Bown675], is investigated and exemplified with the basic hydraulics used for the class of heavy-duty machines [Sepeh90b]. The merits and demerits of the new approach are then outlined. 3.3.1 Outline of the Formulation Figure 3.13 shows a single actuated system, similar to subsystem 1 in Figure 3.12. The pump flow, Q, is distributed between the actuator and the tank. Pressures across the line as well as flow are shown in the figure. Orifices a;, o„ and a e control the flow distribution. Given certain values for orifice areas and assuming a constant pump (C) r . p. (Co) Figure 3.13: Single-link actuation with open-center valve flow for the time being, equations describing the hydraulic dynamics can be written as follows: (To the actuator): Qi = katy/P - Pi (3.20) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 34 Qe = kaey/p - Pe (3.21) Q = (Qi + Qe) (3.22) f = ^ Q i - d i ) (3-23) (Out of actuator): Qa = ka0y/P0 - Pe (3.24) dP dt = ( 3 - 2 5 ) (§i and Q0 are related to the actuator piston linear velocity. C{ and C0 are the compliance of the hoses connecting the valve to the actuator. The inclusion of Equation (3.22) requires some iterations during the simulations (see Section 3.2.2). Equations (3.20) to (3.23) can be rearranged in terms of different states in a similar manner as in [Bowns75]. Equations (3.20), (3.21) and (3.22) are combined and rewritten as follows: ( £ , ) ' = (p-Pi) ( 3 - 2 6 > when differentiated with respect to time, Hfcoi)^  dt ~ dt dt V'z*> /2(Q - Qi)\dQj _ dP V (fcae)2 > dt ~ dt K6-™> The tank pressure, P e , is usually constant. Substituting from (3.28) and (3.29) into (3.23) a new differential equation in term6 of Qi is derived: dQi (4i-Qi) dt 7C ( _L (3.30) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 35 which is used to update Qi during the simulation whose value can be used to determine other flow and pressure states recursively, as follows: Q, = Q-Qi (3.31) P = ( £ ) ' + i>, (3.32) fl = P (3.33) Equations (3.30) to (3.33) can replace equations (3.20) to (3.23). In this new arrange-ment, iteration is no longer required as well as the number of differential equations to be integrated still remains the same. Equation (3.30) can be further studied. Assuming Qi is constant during the time-interval At, it has the following form: dQi _ aQj + b dt ~ cQi + d where a = — (fca;ae)2 b = (kaidefQi c = 2Cia\-2Cia\ d = 2Cia,iQ when integrated will become (see Appendix D): At = t - t , = Z ( Q i - Q » ) + f i - $ l n ( ! ^ ) (3.34) o v a a2' KaQi<b + b' where is the flow rate at time Qi, the flow-rate at time t, is determined iteratively. The updating rate, At, is decided according to the rate of change of Qi. This equation can be substituted for the differential Equation (3.30). Chapter 3. HYDRAULIC SYSTEMS SIMULATION 36 3.3.2 Application to Mobile Hydraulic Systems Figure 3.14 shows a hydraulic actuation system. Two open-center valves are connected in series, a case similar to an excavator's hydraulic system. In this configuration, one actuator has priority over the other. The connection between the two valves has a low i i (C) IP Pe Oil TV— ( C O 1 ) PO1 Qi2 { C A ) "i2 >o2 Figure 3.14: Two-link actuation with priority action compliance. Flexible hoses deliver fluid to the actuators. Referring to Figure 3.14, the following equations are applied to the pump side in their conventional form: Q Q* = kaay/P-Pu = kaelyJP-P = Qi7 + Qe7 (3.35) (3.36) (3.37) (3.38) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 37 Qi2 = kai2^P-Pi2 (3.39) Qe2 = kae2y/p-Pe (3.40) = h Q i ' ~ ^ ] ( 3 - 4 1 ) dPi7 1 ~dt = c72iQii~Qi2) (3-42) As an alternative solution to the above equations, fluid flows Qtl and Qi2 are taken as state variables. Following the same approach as before, Equations (3.35) to (3.40) are differentiated with respect to time. After some manipulations and substitutions, the following equations are derived: dQn a dQi2 ,„ . . . a i - d f + ^~dT = 7 1 <3-43) a*-dT + ^-dT = 7 2 <3-44> where 2Qn 2(Q-Qn) 2 { Q - Q i l - Q i 2 ) ( K ) J (kaeiy + (kae2y C*2 Pi 2(Q - QH - Qn) (kae2y 2(Q - Qn - Qi2) (kae2y 2 (kai2)2 (kae2)2 * - - ( ^ ) which could be easily solved for and . Other states are then calculated at at recursively as follows: Qei = Q-Qn (3.45) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 38 Qe2 = P = Qel ~ Qi2 (3.46) (3.47) P = Pit = Pil = (3.48) (3.49) (3.50) 3.3.3 Simulation Results Figure 3.15 shows the simulation results for a two-link manipulation. Both conventional and alternative modelling techniques were used and the results were compared. The Fourth Order Runge Kutta integration routine with the same integration time interval was used for both models. It i6 seen that both models provide similar results which i6 expected. Table 3.1 compares the computation time required for both models during Asec simulation when one or both links are activated. It is seen that using fluid-flow as an alternative state-space variable can reduce the computation time. However, the computation reduction due to the flow-based modelling technique decreases as the system complexity increases, which is not desirable. Table 3.1: Computation time (VAX 11/750) for Asec simulation First Method Second Method computation reduction Single Link 9.51aec 8.34sec 14% Two Link 10.50sec 9.82sec 7% Chapter 3. HYDRAULIC SYSTEMS SIMULATION 39 e Time (s) Time (s) Figure 3.15: Response to step inputs (two-link) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 40 3.3.4 Discussion As will be discussed in the following , the new arrangement, in spite of its merits, does not contains the strength of the original approach for the class of hydraulic systems un-der investigation. Compared with the conventional method which considers "pressure" as the cause and "flow" as the result, great care should be taken during the program-ming when the flow-based modelling technique is used. Equations describing pressure as a function of flow are inversely related to the orifices areas. These equations fail to characterize the pressure properly when the orifices areas approach zero. However, this is not a problem in the conventional method where flow is calculated based on pressure. The differential equations describing the fluid-flow may change for different orifice arrangements, or, when some orifices are closed (cases when the valve is fully open, ae = 0, or when it passes through its dead-band, a; & aa = 0). This requires redefining the equations during the executions. The inclusion of check valve or relief valve action, which is trivial for the conventional method, reaches challenging proportions when the new technique is applied. Finally, the cross-over action, the torque-limited pump control and pilot system dynamics were not included in the example demonstrated in this section. The derivation of flow-based equations for such a complex system will become difficult. Also, further changes may not be easily performed once the equations are derived. As a result, this approach will not be further utilized for the main simulation in this research. o Chapter 3. HYDRAULIC SYSTEMS SIMULATION 41 3.4 Partitioned Hierarchical Modelling 3.4.1 Reconstructing the Model The findings based on initial studies, outlined in the previous sections, led to a new arrangement in the model, for the class of machines under investigation. The manip-ulator is studied at a different subsystem level [Sepeh89a]. Figure 3.16 shows the new subsystems and their input/output arrangement. The structure is now considered as one subsystem. The hydraulic part is considered to be composed of three subsystems. Referring to Figure 3.16, the main valve system (level 2) with its corresponding cross-overs and the connections is considered as one subsystem which is activated by the pilot subsystem (level 1), and connected to the links (level 4) by means of flexible hoses (level 3). Each subsystem has its own set of dynamic characteristics. This arrangement is the result of the study based on initial formulation, outlined in Section 3.2.1, as well as experimental observations. The subsystems are outlined in the following: Subsystem one: The experimental study (Figures 3.17 and 3.18) showed a first-order relation be-tween the input and the output in the pilot system (level 1, Figure 3.16). The input to the pilot system is the voltage applied to the servovalves. Figures 3.17 and 3.18 illustrate the experimental results showing the displacement of stick main valve and its corresponding cross-over valve given a step input. A given step voltage input to the stick servovalve activates both main valve and cross-over valve spools. This is per-formed by the pilot valve's application of a certain pressure at the end of the main spool rod, Figure 3.17. The same pilot pressure is applied to the stick cross-over valve spool, Figure 3.18. It is also noted that the cross-over valve spool moves only in one direction. This is because the pilot pressure from both sides of the main valve spool is Chapter 3. HYDRAULIC SYSTEMS SIMULATION 42 Figure 3.16: Subsystems and their input/output arrangement applied to the same side of the cross-over valve spool. The relationship between the input voltage and the spool displacement can be mod-elled as a first-order differential equation. The servovalves operate independently and their inputs are external. Thus, an analytical solution could be written, for each valve separately, in the following form: Xi{t) = A < [ l - e - £ ] (3.51) Xi = fi{vi(t)} (3.52) where Xi(t) is the current value of the spool displacement given a voltage, vit to the pilot valve. T; is the time constant. er 3. HYDRAULIC SYSTEMS SIMULATION o o o in o «-i O VO CS v -CD ~ »> O u S-0- ^ o s (Experiment) Volt-Pressure-Spool Displ.-0.0 Time (ms) 15000.0 Figure 3.17: Response of main valve to a step input (stick) o © o iri o «-J ,-C O l O es C us OH g « — E £ > — f O & ^ > 5 -s 1 s a C/3 es (Exper it) Volt-Pressure-Spool Displ. -0.0 Time (ms) 15000.0 Figure 3.18: Response of cross-over valve to a step input (stick) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 44 Figures 3.19 and 3.20 present typical experimental results showing the steady-state relations between the input voltage and the spool displacement, Equation (3.52). The valves do not necessarily behave symmetrically. A curve-fitting technique was applied to find this relationship. The curve-fitting technique finds the best polynomial function with the lowest order of power and an acceptable variance (see Appendix E). Subsystem two: All fast-response states are localized in the main valve subsystem (level 2, Figure 3.16). The variables were redefined and the equations were rearranged to form a set of steady-state equations (see Section 3.2.2). The solution for this part needs some iterations. The numerical solution for this subsystem will be described in the next section. Subsystems three and four: Lumped parameter theory [Watto87] for the connecting hoses (level 3, Figure 3.16) and Lagrangian approach [Paul83a] for the structure (level 4, Figure 3.16) are used to model these two subsystems. The resulting differential equations contain only states with similar response time which determine the time-interval for the numerical inte-gration. As is noticed, the approach combines analytical, numerical and steady-state solu-tions all together. The number of the state spaces to be integrated is confined to only slow-response elements. The approach keeps its simplicity without losing any state information. One characteristic of this modelling is that all fast-response elements are confined into one subsystem which is possible for the class of systems under investiga-tion. er 3. HYDRAULIC SYSTEMS SIMULATION B B e CD CD O © ee CO "o O (X IT: o I Stick Main Valve £ 0 ° o 0 0 ° , _ , /  D V - / a - O O O O O O O O G o o o o o p L1.5 -1 -0.5 0 0.5 1 1.5 Input Voltage (v) Figure 3.19: Main spool displacement versus input voltage (stick) E E 5-6 ft. O o Cfl , Stick X-Over Valve o (Experiment) 0 ° 0 0 \ o o °o 0 ° o o o o o o 1 1 , — . — , I I -1.5 -1 -0.5 0 0.5' 1 1.5 Input Voltage (v) Figure 3.20: Cross-over spool displacement versus input voltage (stick) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 46 3.4.2 Solution to the Main Valve Hydraulics The part of the hydraulic circuit which drives the actuators (i.e., main valves, subsystem 2, Figure 3.16) has been modelled as in Figure 3.21. It is connected to the pumps at one side and distributes the flow to different actuators. As was indicated earlier, due to the low compliance in the connections in the main valve system, steady-state equations could simulate the dynamics of this level accurately. These equations can be derived using fluid flow continuity and orifice flow equations as follows (refer to Figue 3.21): (Branch 1): Q Qn + QBUX Q u = Q\2 + QBOm Ql2 Q\S + QsTc Qn = kaebu\/Pu — P\2 Ql2 = kaebo\f P12 — P13 C?13 kaettcy/pl3 - Pe QBUI = kaibuy/Pn - PBL QBOm = kd.yJPlt - Pboc QsTc kaJ P 1 3 — Pttc (Branch 2): Q = Qn + Qswi Q i \ — Q22 + QsTm Q22 = Q23 + QBOc QIX — kaetw yP21 — P22 Q22 = kaeH y P j 2 — P23 Chapter 3. HYDRAULIC SYSTEMS SIMULATION 47 Pump 1 (Branch 1) Q Pump I (Branch 2) Q i = input e — exit m = main e = cross — over bo{BO) = boom bu{BV) = bucket rt(ST) = stick *w(SW) = swing Figure 3.21: Schematic diagram of the main valves Chapter 3. HYDRA ULIC SYSTEMS SIMULATION 48 Q23 = kaeboeyJP23 - Pe Qswi = kaitwy/P2i — Pswi Q s T m = ka,\jp22 — Ptte QBOc = kcLy/P23 - Pkoc (Interconnection between the branches): QsTi = QsTc + Q s T m QBOi = QBOc + Q,BOm Qsn = kaitty/Pttc - Psn QBOI = kauyyij Pboc — PBO{ Referring to Figure 3.21, pressures at the actuator sides along with the orifice areas are known in the above equations which are to be solved for the flow distribution (direct solution). An attempt to solve these nonlinear equations, using the general iterative techniques and regardless of the physics of the problem, was unsuccessful due to poor convergence. In the following two customized techniques, suitable for this configuration, will be discussed: Method one: The hydraulic circuit is viewed as consisting of three subsystems connected together at nodes 1, 1, 2 and 2, with the following input/output arrangement: (Branch 1): {yi}T = ({yi}T,Q.BOm,QsTc) Chapter 3. HYDRAULIC SYSTEMS SIMULATION 49 (Branch 2): {V2}T = ({y2}T,QsTm,Q.BOc) (Interconnection): {U3}T = ({V3}T,QBOm,QsTm,QBOc,QsTc) {y3}T = {{y3}T,P.u,Pboc) where, { y i } T = ( Q B U U P I U P ^ P I Z ) { y z } r = (Qswi, P21, P22, P23) {y3}T = [ Q S T U Q B O ) The external inputs are: { V } T = ( {V 1 } r , {V,} r ) {V 3 } r ) where, {Vx} T = (Q,PC,PBUU aibu, &ebu,<lebo, < » e » t c ) {^}T = {Q,Pe,PsWi,a-iMW,ae»w,<Le*t,aebo^) These notations suggest an iterative technique which has to be performed at two levels; given the current values of the cross-over line pressures (Pboc and P,tc), subsys-tems 1 and 2 are solved iteratively in parallel, and their outputs are then used for the cross-over subsystem to update the corresponding line pressures. An incremental search Chapter 3. HYDRAULIC SYSTEMS SIMULATION 50 technique and linear interpolations, starting with the state pressures downstream, is used for branches 1 and 2. The solution is robust and is natural for the hydraulic sys-tems. Iterations are performed at two levels and the final results determine the correct interconnections in the cross-over part. The check valves in the cross-over connection are still to be modelled as orifices in this method. Table 3.2 illustrates the simulated flow pattern for the hydraulic main circuit for different spool displacement arrangements. Actuator pressures, Pswi, PBUU PBOI a n d PsTi a r e assumed to be known in this illustration. The pressure pattern in the main lines is shown in terms of the ratio of their values to the ones in the corresponding actuators, P Pn i.e., — , — . Spool displacements and flow are shown as a percentage of full PBUI Pswi stroke and pump flow, respectively. Table 3.2(Case 1) 6hows the case where both stick and boom valves are open, however, the pressures in the main lines are not high enough to provide any flow to the actuators. As these valves open further, the pressures rise enough to provide flow [Table 3.2(Case 2)]. In Case 2 there is no flow to swing or bucket because their corresponding valves are not yet activated. Also, there is no cross-over flow. Table 3.2(Case 3) shows the case where the boom valve is fully open while the stick valve remains the same. It is seen that the pressure at the boom cross-over side is not high enough to provide additional flow to speed up the boom motion. However, when the stick valve is moved back to 35% of its full stroke [Table 3.2(Case 4)] the boom cross-over becomes active 6uch that the flow provided to the boom becomes 1.77 times greater than the last case. Tables 3.2(Case 5) and 3.2(Case 6) show the flow distribution and interconnection between the main lines when bucket, boom and stick are operating simultaneously. As is seen, when the bucket valve is fully open, the only flow available for the boom is through the cross-over valve. Chapter 3. HYDRAULIC SYSTEMS SIMULATION 51 c .2 f—( a 2 1 c "3 B V c h i O « e a! v u OS B V tl OH CN ci 5 o II 9 o II e m o II © o II to «N O © II II © II o o II to CM o o II al 9 to cs II « _ 1 o «"9 O II o © II 1 LJL CO US ft* - M 00 to rt II P5 IO es 2 cj| r -2.2 h io ro rr o ro o o © IO 3 53 i $ H H IO es o II ro 0 o II II © rt II 'IS o o o © II a? ro IO Chapter 3. HYDRAULIC SYSTEMS SIMULATION 52 s .£ c c O CN CO V Z5 1 •o 9 ) s i a. o CS II II o II 1* CO cs II E w 2 r-II o i l o| P i o II Ic o o II E o eo eo I o o o II op »o CO U u Hi i o cs I C cs II 3 s o II CO II II tO CO CO II o 8 N H CS II IN *» CO r-II i loi o o o II Sol PI • a. 2 to CO © II 8 •a CS II t « e II CO o o t .e N H CS II o x CN eo CS r-II II 6! u OF o II El So-CO o o o II o 2 to CO S o H u I i Chapter 3. HYDRAULIC SYSTEMS SIMULATION 53 It is observed that depending on the orifice arrangement and the actuator pressures, different hydraulic configurations with different flow patterns can be obtained. The algorithm, based on method one, is capable of finding the suitable configuration. Method two: Alternatively, the circuit in Figure 3.21 can be studied beforehand to decide on possible configurations. Depending on which one of the four check valves in the cross-over lines are functioning, nine different hydraulic configurations can be achieved, which are tabulated in Table 3.3. Table 3.3: Possible configurations for the hydraulic main circuit Boom main & cross-over no cross-over Boomcross-over only Pa > Pis Pn > Pn Pa > Pn Pli = PiS Pn > - P a s Pn < Pa / Stick cross-over only / Pn <yPis Pn < Pn Pa </Pn Pl/=PiZ Pu > Pa PitA Pa / Stick main & cross-over / Pa y/Pis Pa — Pn Pa =/Pii Pir^ Pit Pn> Pa Piy< Pa It can be shown that four out of nine configurations are not physically possible. For example, P J J < P13 & P J J = PJS are not possible simultaneously because, assuming Pa < Pn then, since Pn < Pn and PXz < Pu thus, Pas < P I A . Therefore, the solution to the hydraulic circuit shown in Figure 3.21 is within five alternatives as shown in Figure 3.22. Referring to this figure, Case (a) is when the cross-over valves are not functioning. Each main line is capable of providing the amount of flow needed to the corresponding actuator. The condition to be met to accept this as Chapter 3. HYDRAULIC SYSTEMS SIMULATION 54 Figure 3.22: Alternative solutions to hydraulic main circuit Chapter 3. HYDRAULIC SYSTEMS SIMULATION 55 a solution is given in terms of the pressure distribution along the lines. Cases (b) and (c) are two more possible solutions; the two main lines could still be studied separately. The last two cases are the more common ones. Case (d) shows the stick in/out motion which speeds up through getting more fluid from the other main line. Case (e) shows the resulting circuit where boom-up motion is speeded up. Either boom cro6s-over or stick cross-over can function at a time, but never both. The previously demonstrated example, Table 3.2, ultimately presented Case6 (a), (b) and (e). The search for the solution amongst the five alternatives is performed through an iterative search strategy. This is done by examining the five alternative circuits in par-allel. The one which satisfies the condition for the pressure distibution is the solution. In this method only one set of iterations is required. Also, the cross-over check valves are not required to be modelled. Both methods were tested through simulation and were found to be robust. How-ever, the second method has proven to be computationally more efficient. This is shown in Table 3.4. Table 3.4: Computation time (VAX 11/750) for hydraulic main circuit Case 1 Case £ Case S Case 4 Case 5 Case 6 First Method 0.04*ec 0A3sec Q.Zbsec 2.38«ec 1.82aec 0.56sec Alternative Method 0.044*ec 0.049aec 0.051«ec 0.062«ec 0.050*ec 0.049sec As is seen from Table 3.4, the computation time required for the first method to provide Table 3.2 varied from 0.04aee to 2.38jec, whereas for the new algorithm this range is as small as 0.044«ec to 0.062aec which shows an improvement. Further simplifications can still be performed on the second method to ease the Chapter 3. HYDRAULIC SYSTEMS SIMULATION 56 search for the solution. For example, it is noted that when 6tick or boom is not activated, Cases (d) or (e) (Figure 3.22) are not amongst the possible solutions and thus are not considered. Also, Case (a) is the only solution when neither boom nor stick is activated. Chapter 4 H E A V Y - D U T Y S T R U C T U R A L D Y N A M I C S 4.1 Relevant Background The methods used in robot structure modelling could be classified with respect to the laws of mechanics on the basis of which motion equations are formed. Two well-known methods are Lagrangian and Newtonian. The first results in the class of Lagrange's methods was reported by Uicker [Uicke64]. It was originally used for a certain class of closed-chain mechanisms. Kahn [Kahn69] extended the algorithm for open-kinematic chains. A complete formulation of open-kinematic chain manipulators based on the Lagrange approach has been derived in [Paul83a]. One major advantage of the La-grangian approach is that it allows us to recognize the nature of coupling between the linkages in term6 of inertia, centripetal force, gravity and Coriolis force. Application of the Newton-Euler dynamics to modelling of joint interconnected rigid bodies was first introduced by Kane [Kane62]. A special application of the Newton-Euler method for open-kinematic chain manipulators has been derived [Luh80]. Both Lagrangian and Newtonian approaches were shown to be equivalent [Silve81]. Manipulators with cylindrical hydraulic actuators, form a class of closed-kinematic chain robots. This type of geometry offers higher loading capacities and more rigidity [Arday84]. There have been many attempts to develop a general purpose program for gen-erating equations for manipulator dynamics [Burdi86, Somoy84, Izagu86, Walke82]. 57 Chapter 4. HEAVY-D UTY STRUCTURAL DYNAMICS 58 Different computer/computing architectures have been proposed for fast simulation [Lathr85, Lee86, Gu85]. However, this large amount of theoretical work done has re-vealed that, for general arms, there exists no complete formulation with enough reduc-tion of calculational complexity to be used for real-time simulation purposes [Aldon84]. For the PUMA robot, the simplified symbolic equations, suitable for on-line computa-tion have been developed [Paul83b, Leahy86, Izagu85, Armst86]. 4.2 Outline of the Modelling The machine is composed of two large subsystems; the structure and the actuation system. The dynamics of the actuation system were described in Chapter 3. The dynamics of the structure could be determined by partitioning it into links (the Newton-Euler approach) or by considering it as a system (the Lagrangian approach). The Lagrangian approach is used for the dynamics of the structure. The use of the Lagrangian approach will lead to an explicit and compact mathematical model which allows for any further computational arrangement and customization for fast simula-tions. Although the configuration of hydraulic manipulators, due to the attachment of the cylindrical actuators, implies that they belong to the class of closed-kinematic chain manipulators, however, compared with the main structure, the actuator linkage dynamics is not significant and thus the simpler serial link dynamic analysis is used here. A complete formulation of manipulators based on Lagrange's approach has been described in [Paul83a]. It has the following form: Ti = £ +jt.il D i i J A + Di (4-53) j=i j=i fc=i where, • n is the number of degrees of freedom; Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 59 • Ti is the torque applied to link t through joint t; • Du represent the effective inertia at joint i; • Dij represent the coupling inertia between joint i and joint j; • Dijj represent the centripetal forces at joint i due to velocity at joint j ; • Dijk represent the Coriolis forces at joint i due to velocities at joints j and Jfe; 9 Di represents the gravity loading at joint i. The structure model is shown in Figure 4.23 along with the individual link parameters (link 1 is the swing, link 2 is the boom and so on). The Denavit-Hartenberg coordinate frames [Paul83a] assigned to each link is also shown (see Appendix F). There is an Link Variable Q a d 1 e, 90° 0] 2 e, 0° «: 0 3 e3 0C 0 4 e« 0° 0 0 Figure 4.23: Structural model of excavator offset between the first link (swing) and the others. The derivations were performed symbolically, using the symbolic software package "REDUCE". The equations of mo-tions are in closed-form and are in terms of the centripetal, gravity, Coriolis and inertia coefficients. Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 60 Due to the nature of the actuation, each link excluding the first one, has a limited joint angle. The principle of virtual work was used to determine on the relationship between the acting force on the actuators and the resulting torque at the joints (see Appendix G). The gravity vector in Lagrangian equations as well as the pseudo inertia matrix for the bucket have been defined in a more general form such that the final equations could both include the machine orientation as well a6 to simulate a log-loader. 4.3 Significance Analysis In this section, the exact model is used along with the experimental data to determine the significance of various terms involved in the linkage dynamic equations. Inertia, gravity loading, Coriolis and centrifugal effects are evaluated in terms of their absolute contribution to the absolute total torque at each link. The study is performed under different loading, speed and machine orientations. The less significant terms or elements are identified. This helps in identification of the nature of the linkage interaction in heavy-duty manipulation, which is useful for design, control and simulation studies [Sepeh90a]. 4.3.1 Outline of the Approach Different experimental pick and place tasks were performed. They were designed to include all possible realistic motions involving all the links and covering the whole workspace. These paths are shown in Figure 4.24. The joint angles were measured during each path. Figure 4.25 shows these angles over a typical path trajectory. Joint velocities and accelerations were then calculated, using a piece-wise least-squares linear curve-fitting technique. Figures 4.26 and 4.27 illustrates the results for a link. The Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 61 77 (Experiment) **1 T ^ I X Y-X Plane i Figure 4.24: Views of different experimental trajectories Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 62 be S- " v IK < c ° o 1 e I Link 1 (Swing) (Experiment) Time (s) Figure 4.25: Joint angle trajectories over path 1 Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 63 Time (s) Figure 4.26: Joint velocity trajectory over path 1 (link 3) » 1 1 i I i " 0 2 4 6 8 Time (s) Figure 4.27: Joint acceleration trajectory over path 1 (link 3) Figure 4.28: Nominal open-loop torque over path 1 Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 64 speed can artificially be scaled by changing the time required to complete a task. New velocity and acceleration profiles are then to be determined for the new path. These data were used along with the exact model to determine the nominal open-loop torques generated over the trajectories. Figure 4.28 shows a typical result. A method similar to [Leahy86] and [Fu87] is used for the significance analysis. The torque equations are broken into different components in a hierarchical manner; they are first broken into inertia, gravity, centripetal and Coriolis components. The contribution of each component is evaluated in terms of its absolute value with respect to the total of absolute values. The approach is then carried out at a lower level of the hierarchy; inertia, Cori-olis and centripetal torques are each broken into their contributing components. For example, the coupling inertia and the joint inertia are separated and are evaluated independently. Coriolis and centripetal terms are divided into the contributing terms due to swing motion or the motion of the other links. 4.3.2 Results The results of significance tests are shown for a typical path trajectory in Figures 4.29, 4.30 and 4.31. Figure 4.29 shows the percentage of the contributing terms forming the open-loop torque for link one (swing). As is seen, torques due to inertia and Coriolis are the dominant ones (4.29-a). Their relative contribution, however, remains constant even for slower motions (same deduction as in [Brady82]); the same results were achieved when the trajectory speed for the same path was reduced to one half. Contribution of the centripetal torque has been found to be negligible (4.29-a). The gravity component showed no contribution; however, for cases when the machine is on a uphill or downhill slope, the gravity effect becomes significant (4.29-b). The inertia torque breakdown shows that the swing motion is not heavily coupled Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 65 with other links (4.29-c). Also the breakdown of Coriolis term6 reveals that the greatest contribution is the Coriolis effect due to the inclusion of 6wing motion (4.29-d). Gravity and inertia terms are important for link two torque calculation (Figure 4.30-a). Coupling inertia due to the swing motion can be ignored (4.30-b); it i6 noted that, at times, the swing coupling contribution seems to be noticeable, however, those are instances when the total absolute inertia torque itself is very 6mall (4.30-c). There has not been found any difference between the centripetal effect on link two due to the swing, or due to link three motions (4.30-d). However, the contribution of total centripetal torque is less than 10% (4.30-a). Figure 4.31 shows the results for link three. It is seen that the centripetal torque becomes noticeable (4.31-a). The swing coupling inertia shows no significant contri-bution on link three motion (4.31-b). However, swing motion is shown to be effective on link three centripetal torque calculation (4.31-c). Based on the above and other similar tests, performed over different trajectories, speed, loading and machine orientation (uphill/downhill), the following results can be summarized: Inertia terms: Coupling inertia terms have a small effect on the 6wing motion. For the other two links, coupling inertia only due to the inclusion of swing motion is negligible. Coriolis terms: Coriolis torque due to the coupled motions between the swing and other two links (boom and stick) is noticeable for swing torque calculation. Contribu-tion of Coriolis terms on the other two links is insignificant. Centripetal terms: Centripetal terms in the swing torque calculation can be ignored. However, they become significant for link three (stick) torque calculation. Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 66 3 I i -< 5 O T o Link 1, Path 1 Inertia . ( a ) . Centripetal-^J \\ ft Coriolii ravity s cr in o Link 1, PathJ, Loaded, Uphill \ | \ /Gravity . • (b) L i n k l Path CI -S cr t -*" m e . C o . < c K o ^Jomt Inertia ' n i Coupling Inertia 1 , |, A'Li *> -3 cr e ' Link J_. Path II Swing motion -(d) Time (s) Figure 4.29: Torque composition for link 1 Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 67 s •5 < o M O _ Gravity Inertia Link 2, Path 1 Centripetal Coriolis e e C K e Link •(b). 2^Peth 1 Swing motion • W tl C i s e o • e < Link 2 Path 1 -(d) nn _ 1 ' 0 Time (s) Figure 4.30: Torque composition for link 2 Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 68 Time (s) Figure 4.31: Torque composition for link 3 Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 69 Gravity terms: Gravity terms in general can not be ignored. Depending upon the path, loading and/or machine orientation, their contribution may vary. 4.3.3 Numerical Simplification The above analyses indicate that inertia, Coriolis, centripetal and gravity terms some-how all contribute to the open-loop torque calculations. The centripetal and Coriolis forces were usually assumed to be important only when the manipulator is working at high speed. This assumption was used to justify dynamics simplification [Paul83a]. In this chapter, it was demonstrated that this assumption is false for an excavator. Basically, two facts are involved-the velocity terms (i.e., Coriolis and centripetal) have similar magnitudes relative to the inertia terms for all speeds of movements, and, there are some circumstances in which the velocity terms are near a maximum whereas the inertial terms are zero, e.g., near the points at which the link motion moves from ac-celeration to deceleration [Brady82]. For the class of machines under investigation, the terms involved in the structural dynamics can be grouped into three: • Terms which have no contribution; their values are always zero and are shown as • Terms with negligible contribution; these terms are neglected and are shown as "0". "0". • Significant terms which should be kept in the matrices. The simplified matrices are thus shown as below: Inertia matrix: (D 0 0 \ 0 D 22 D 23 V 0 D, '32 # 3 3 / Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 70 Coriolis matrix: (Dm Dn3 0\ 0 0 0 V o o o / Centripetal matrix: I 0 0 0\ 0 0 0 VP311 0 0/ The gravity vector is not simplified. Table 4.5 shows the numerical improvement, in terms of computation, due to thi6 simplification. The error in torque calculation i6 shown in Figure 4.32 which can be compared with the nominal torque trajectory shown in Figure 4.28. Table 4.6 shows the percentage of error due to this modification. Table 4.5: Computational efficiency in the simplified model % of Numerical Calculation Improvement Link 1 Link 2 Link 3 Overall Multiplications 11.7 % 14.3 % 1.1 % 27.1 % Additions 12.6 % 14.8 % 1.6 % 29 % 4.4 Discussion The simplified model contains all the necessary information and includes the major dynamic terms, and thus can be used in the simulation program for control or design studies. For example, the program is currently being used to evaluate the application of teleoperated control on the excavator machine performance. On the other hand, the simplified model may not be sufficient to accurately simulate the open-loop machine performance (i.e., given a prescribed input torque trajectory) Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 71 Figure 4.32: Torque error due to dynamic simplification Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 72 Table 4.6: Torque error in the simplified model (a) Mean % of Torque Error Path Link 1 Link 2 Link 3 Path 1 0.9 % 2.2 % 0.6 % Path 2 1.2 % 2.0 % 0.6 % Path S 1.0 % 0.6 % 1.2% Path 4 1.3 % 1.7 % 1.3 % (b) Maximum % of Tore ue Error Path Link 1 Link 2 Link 3 Path 1 3.4 % 6.3 % 3.7 % Path 2 3.9 % 5.2 % 2.5 % Path S 5.7 % 2.5 % 5.6 % Path 4 10.2 % 6.7 % 5.0 % over a long period of running time. The simplified model, although close to the machine dynamics, presents in principle a different model. An error in the acceleration calcula-tion, at each integration time-interval may result in a different joint velocity and angle, which will be used to decide on the next value of acceleration, thus accumulating an error. This is different from the numerical round-off error in the integration routines. The inaccuracies in velocity also affect the torque calculation in a hydraulically actuated system as well. The equations describing the actuator pressures are, in their simplest form, first order differential equations in terms of flow-rates and joint velocities. Any inaccuracy in the velocity calculation affects the pressures, which are used to calculate torque. For these cases, in which accuracy is necessary, the less significant terms should be kept in the model. They can be updated at a slower rate than the dominant terms (e.g., two times slower for the excavator model). This technique proved to be accurate in the open-loop simulation of of typical pick and place tasks; the path trajectory from Chapter 4. HEAVY-DUTY STRUCTURAL DYNAMICS 73 simulation results were similar to the experimental ones within the accuracy of the numerical integration. Chapter 5 INCLUSION OF DISCONTINUITIES IN T H E M O D E L Gear backlash and stick-slip friction are two nonideal phenomena which are often ob-served in the performance of heavy-duty machines. The swing motion of the operator's cabin in an excavator is a typical example. In this chapter the inclusion of these two nonlinearities in the form of discontinuities in the model is investigated. The inclusion of the conventional model of backlash in the simulation, as will be seen, is unnecessary in terms of computation time and the final results. An algorithm which combines the fluid-flow dynamics and the gear train is thus developed [Sepeh89b]. The contact and non-contact cases are analyzed which brings about proper 6ets of static and dynamic equations to simulate this phenomenon. 5.1 Experimental Observation The torque generated by the hydraulic actuator is transmitted through a gear train to rotate the upper structure (referred to here as the Cabin). There is some slack between the gear teeth which results in a non-rigid connection in the transmission (Figure 5.33). Figure 5.34 shows the experimental variation in the line pressures in the swing motion in response to a step in spool displacement. The pattern is changed into an oscillatory form once the valve is closed (starts at t*, Figure 5.34). In this case, the line pressures follow in the opposite directions; there are some periods where they remain constant as is shown in the circle. The amplitudes as well as the peak values decrease with time. Thi6 typical pattern in the swing motion has been found to be different 74 Chapter 5. INCLUSION OF DISCONTINUITIES IN THE MODEL 75 Figure 5.33: Gear train arrangement in the Cabin from those of the other links. Nonidealitie6 such as gear backlash, friction and leakage are contributors to this effect. 5.2 Gear Backlash Figure 5.35 shows the combined hydraulic system and the gear train. The cabin has been modeled as a rotating disk with high inertia, IM- The gear train has been simplified as a single pair of gears (ratio n : 1) with an equivalant backlash, 6. Backlash has been shown in its conventional form inside the circle, Figure 5.35. However, its magnitude is given in terms of angular rotation reflected on the motor shaft. The motor shaft, which drives the cabin, is assumed to have inertia 7m. The following equations represent the dynamics of the above system: Chapter 5. INCLUSION OF DISCONTINUITIES IN THE MODEL 76 Figure 5.35: Combined simplified model of gear backlash and hydraulics Chapter 5. INCLUSION OF DISCONTINUITIES IN THE MODEL 77 Hydraulic circuit: Pi = (Qi-Dmem)/c Po = {Dm0m-Qo)/C Qi = kuz^y/P-Pi Q0 = fewi,wyJPD - Pe where C is the hydraulic compliance. P and Pe are the pump and tank pressures, respectively. The above equations do not include the changes in the pump pressure due to the torque-limited system for the sake of simplicity in the argument. However, thi6 effect has been included in the simulation program. Motor shaft: L = [(Pi-P0)Dm-Te}/Im L = J Ldt Om = J Ldt Cabin shaft: B M = (nTe)/IM 0M = J 0Afdt 6M = J 6Mdt Te is the torque acting on the gear teeth due to elastic collision. It is formulated as below: Te = Ke.fgb{-6,6,(6m-neM)} Depending on whether there is a contact or not, fgb is defined as follows: fgb = 0 when -8 <{8m- n9M) < 6 U « {Om - n$M) when \{6m - n9M)\ > 8 Chapter 5. INCLUSION OF DISCONTINUITIES IN THE MODEL 78 Ke is related to the elasticity of the gear teeth and is very high. The above sets of equations simulate the swing motion with gear backlash. The motor shaft dynamics is included, although, it could be ignored when it is geared with the cabin dynamics. Also, the system of equations contains high frequency phenomena such as the collision in the gear teeth. The above properties are further used to simplify the model. In the process of simplifying the model, the dynamics is divided into two parts; "contact" and "non-contact". When the gear6 are in contact, the dominant part which characterizes the motion is the Cabin. The total torque generated by the hydraulic motor is used to move the Cabin and the motor shaft together. The following equations are thus used to characterize Cabin and motor shafts performance during the contact period: Te = (Pi-P0)Dm $M = (nTE)/IM 8M = J ^Afdt 9M = J 0Mdt 9m = n9M 6m = JIQM It is noted that IM is ignored compared with IM. When there is no contact, both the motor shaft and the cabin follow their own dynamics. The collision torque, T e , is set to zero. Further simplification could be performed by combining the hydraulics and the motor shaft dynamics. The study showed that in the non-contact case, the response of the motor shaft can be simulated Chapter 5. INCLUSION OF DISCONTINUITIES IN THE MODEL 79 by the following equations: Pi = Po Qi Qc Dm D, m 0m = J 9 mdt Iteration may be required to find the appropriate flow-rates. The original equations for the cabin are still valid and are used during the non-contact period. A logic system has been set up which recognizes the two modes and decides on the models to be used [Sepeh89b]. It i6 based on constantly monitoring the relative angular displacement between the motor shaft and the cabin as well as the hydraulic line pressures. 5.3 Friction and Leakage The rotation of the upper stucture of the excavator on its tracks is accompanied with a significant resistance in the form of dry friction. The weight of the upper structure is the most contributing factor. Dry friction is also called stick-slip friction [Cheok88]. Referring to Figure 5.36, dry friction is divided into two components [Shear83]; in the absence of relative rotation, the frictional torque is variable and is equal and opposite to the total sum of external torques applied (Text). This opposing torque increases with the increase of the external torque until it reaches to saturation limit namely stick friction (T,t). Once the rotation starts, the friction decreases almost instantly and remains constant afterwards as slip friction (T,;). The switchover from the stick (static) friction to the slip (dynamic) friction causes some instability in the numerical simulation. Therefore, the switchover is assumed Chapter 5. INCLUSION OF DISCONTINUITIES IN THE MODEL 80 Tsi e Figure 5.36: Stick-slip friction model to take place at certain velocity, 60 [Shear83]. It i6 small enough to be considered negligible but large enough to prevent excessive stiffness in the numerical integration process. Dry friction, Te, is then described mathematically as follows: Tc = T„t Tc = T.tsgn{T„t) if \Temt\>T.t k \B\ < 90 Tc = Ttl»gn{6) if \0\ > 0o Two types of leakage can exist; cross-port (internal) leakage between the lines, and, external leakage to the tank. Leakage can be the result of defective parts such as in relief valves or lack of proper sealing. Leakage flows are usually laminar and therefore proportional to the first power of pressure difference [Merri67]. In the excavator, there is an additional cross-port flow on the swing motion which is due to an element called "soft-swing". Soft-swing balances the pressure difference in the line6 to rapidly damp the undesirable oscillations when the motion i6 suddenly stopped. It i6 modelled as an orifice. Chapter 5. INCLUSION OF DISCONTINUITIES IN THE MODEL 81 Time (s) Figure 5.37: Line pressures in swing motion (simulation) 5.4 Simulation Result and Discussion Figure 5.37 shows the simulation result when gear backlash, dry friction and leakage are included in the model. Dry friction has been included on both the cabin and the motor shaft. The simulation result revealed that the decrease in the differential pressure during the oscillation is because of the dry friction in the cabin as well as the cross-port flow. However, the mean value of the line pressures should remain constant. The inclusion of the leakage to the reservior causes this mean value to decrease. Chapter 5. INCLUSION OF DISCONTINUITIES IN THE MODEL 82 There are a number of short intervals where the line pressures stay relatively con-stant (shown inside the circle in Figure 5.37). These are circumstances when the gear backlash is in effect and there is no connection between the cabin and the motor. The cabin keeps moving in the same direction because of the inertia, whereas, the motor stops until the backlash is taken up. The pressure difference in the lines during this period is due to the inclusion of the friction in the motor shaft and its corresponding gear train. This study shows that dry friction and leakage (cross-port or external) are more significant than gear backlash in determining the vibrational behaviour of the swing motion in heavy-duty manipulations when the Cabin is suddenly brought to 6top. Chapter 6 MACHINE CONTROL A technique for motion control of the machine in a teleoperated mode is presented. It is a closed-loop control with feedforward compensation. The feedforward part which is based on the hydraulic model, developed in Chapter 3, uses fluid-flow equations to control the joint velocities. It also employs the measured hydraulic pressures as information related to the structure, load and interaction with the environment. The closed-loop part then compensates for the modelling errors. 6.1 Relevant Background Given the motion equations of a manipulator, the purpose of robot arm control is to maintain a prescribed motion for the arm along a desired arm trajectory by applying corrective compensation torque [Lee86]. There exists a large number of papers ded-icated to the problem of manipulator control. However, very few control algorithms have been offered which have been actually implemented in the control of industrial manipulators. One class of control is the joint servo mechanism. Although a robot arm is a multi-variable nonlinear system, current industrial practice for the control of manipulators is to use independent servo-loops for each joint [Lee86]. In order to be able to eliminate the effects of nonlinear dynamic coupling between the links as well as changes in the effective loading, these effects should be predicted and cancelled by feedforward com-pensation. The question which always arises is to what extent it is necessary to take 83 Chapter 6. MACHINE CONTROL 84 dynamics into account for the purpose of compensation. A type of compensation can be achieved through computed-torque technique in which the dynamics compensation is included in the feedback loop [Craig86]. Another class of joint control is the sliding mode control based on the theory of variable structure systems [Young78]. In some cases a control which commands the manipulator end-effector to move in a desired Cartesian direction may be more appro-priate. A control strategy based on this idea, namely resolved-motion control, has been developed [Whitn69]. In this control all the joints must run simultaneously at different rates to achieve a desired hand motion along the world coordinates. Adaptive control is another class of control; it is a generic control technique which can be used for both joint motion and resolved motion control. It stems from having a system with good performance in tracking the desired trajectory over a wide range of manipulator motions and payloads. Compared to classical control, in adaptive control there is no need to have a priori knowledge of the parameters and the disturbances. One method is the model reference adaptive control [Dubow79]. Another approach is the self-tuning control which uses a model to fit the input/output data from the manip-ulator [Koivo83, Leini84]. There have also been some studies on hybrid position/force control which are suitable for complex handling and assembly tasks [Raibe81]. A number of studies relevant to the closed-loop control of manipulators were re-viewed in which the emphases were put on the structure. Nonlinearities due to the linkage coupling were discussed and different algorithms were introduced in order to take them into account. In all these studies, it was assumed that each link is actu-ated individually and the actuation system is simple. However, this is not true for hydraulically actuated heavy-duty machines. The efficiency considerations of these mobile machines lead to a complicated series of interconnections between the actuators which makes the task of the control rather challenging. Chapter 6. MACHINE CONTROL 85 As far as hydraulic systems are concerned, there have been few attempts to apply modern control theory to some types of industrial hydraulic equipments. One is the adaptive control for an ideal electrohydraulic position-servo [Kulka84]. Also, a multi-variable adaptive technique was proposed for the control of hydraulic manipulators with individual joint control [Halme85]. The dangers of using linearization theory have been mentioned [Davie81]. Some experimental work on hydraulic manipulators was also per-formed [HanafSO]. Nonlinearity inherent in hydraulic systems were discussed [Cox86]. A method was also developed which is based on cascading the actuation component with the structural part [Sepeh89c]. Different control strategies were used for each part; self-tuning control was applied to the structure and classical control was applied to the hydraulics. The two controllers then communicate during the manipulations by interchanging their respective outputs. In this chapter, a method similar to [Sepeh89c], in concept, is presented for teleoper-ated control of heavy-duty hydraulic manipulators with coupled actuating system. The approach is basically a simple closed-loop with feedforward compensation. The feed-forward part, namely a decoupled-flow compensator, employs the measured hydraulic line pressures. This information is used in the cascaded hydraulic part along with the fluid-flow equations, developed in Chapter 3, to control the joint velocities. A prior knowledge of some hydraulic parameters is the only requirement in this algorithm. In the following sections, the decoupled-flow compensator is first introduced through single-link and multi-link velocity control. Inclusion of the closed-loop part to the above algorithm will then be discussed. The entire algorithm, entitled a decoupled-flow controller, has some features which will be outlined when concluding this chapter. Chapter 6. MACHINE CONTROL 86 6.2 Single-Link Velocity Control 6.2.1 Outline of the Approach Figure 6.38 shows a schematic of a hydraulically actuated link (similar to the stick of an excavator). The three orifices in the main valve are controlled by a single 6pool dis-m X Figure 6.38: A heavy-duty hydraulically actuated arm placement and thus are interdependent. The pump flow, Q, is assumed to be constant and known for the time being. The link experiences variable loading due to the grav-ity, interaction with the environment, inertia, etc. There is a kinematic relationship between the joint angular velocity, 8, and the actuator piston linear velocity, X (for example, see Appendix G). Chapter 6. MACHINE CONTROL 87 The torque, T, required to provide a desired joint trajectory can be calculated based on the equation of motion for the link as follows: T = M6 + Q + T (6.54) where T oc F = (PiAi - P0A0) (6.55) Ai, Q and T denote inertia, gravity loading and friction/disturbing torques, respec-tively. F is the force provided by the hydraulic actuator. Line pressures Pi and P0 are shown in Figure 6.38; they are related to the flow-rates as follows: Q = Qi-rQe (6.56) Qe = kae^P - Pe (6.57) Qi = k a i s j p ~ Pi (6.58) Qa = ka0s/P0-Pe (6.59) The flow is controlled through changing the orifice areas o^ , ae and a„. Qe is the pump flow to the tank. The flow into the actuator, Qi, and out of the actuator, QD, are related to the piston linear velocity as follows: Qi = AiX + PiCi (6.60) Q0 = A0X-P0C0 (6.61) Given a desired motion (including desired joint acceleration), these equations can de-termine the desired spool displacement (inverse problem), providing that a complete knowledge about the hydraulic system, structural parameters and load is available. However, this work assumes to have no knowledge about the structure or load. It thus Chapter 6. MACHINE CONTROL 88 uses the information contained in the hydraulic pressures which is, in a Bense, a reflec-tion of the dynamics of the structure, load or interaction and thus using it is in fact a form of load compensation [Viers80]. Therefore, knowing that the joysticks operate in the velocity mode, the measured values of Pi and P0 are used and the spool displacement is determined such that the fluid-flow provided by the hydraulic system, aims at bringing the link to the desired velocity, i.e., Qi = AiXd (6.62) Q0 = A0Xd (6.63) This way, only the above two equations along with Equations (6.56) to (6.59) are re-quired to be solved for the required spool displacement to achieve the desired velocity, Xd- However, since the three orifices are related to a single spool displacement, de-pending on whether the pump to cylinder flow Equations (6.56) to (6.58)] or cylinder to tank flow [Equation (6.59)] is intended to be controlled, two different solutions may be obtained for the spool displacement. This is because the number of equations is more than the number of unknowns, due to the simplified algorithm, and thus a unique solution is not guaranteed. However, both approaches finally converge to a unique solution during steady-state motion. Some criteria are thus required during the choice of the appropriate solution. Before describing these criteria, the methods used to calculate the spool displacement are first outlined. Model No. l(Flow Control from Pump to Actuator): In this case, it is assumed that the flow from pump to cylinder controls the motion. Chapter 6. MACHINE CONTROL 89 Equations (6.56) to (6.58) are rearranged as below: Q-Qi = kaey/P - Pe (6.64) Qi = kaiyjp - Pi (6.65) where Qi is the desired flow into the actuator. The unknowns are P, oe and a;. The latter two are dependent and related to the spool displacement. These two equations should be solved simultaneously in order to find the right spool displacement. They are nonlinear and an iterative method is applied to solve them. Pressures Pi and Pe are known through measurement. Pe is the tank pressure and is usually constant. Model No. 2 (Flow Control from Actuator to Tank): Equation (6.59) is used as follows to determine the cylinder to tank orifice area: where Qa is the desired flow to the tank, calculated based on the desired velocity. P0 and Pe are the current return line and tank perssures, measured through pressure transducers. Once aa is calculated, the corresponding spool displacement is determined. The calculation is simple and does not require any iteration. 6.2.2 Choice of Spool Displacement As described earlier, the desired angular velocity is related to the desired fluid flow-rate. The relationship between the flow-rate and the pressure could be established in the form of a set of nonlinear equations as was described before. These equations are then solved to determine the magnitude of the spool displacement repeatedly within an appropriate time interval during the motion. A knowledge about the relationship between the orifice Qo (6.66) Chapter 6. MACHINE CONTROL 90 areas and the spool displacement for each valve is therefore necessary. It was also shown that in general there could be two different solutions to each case. Figure 6.39 demonstrates how the appropriate solution is selected. Referring to Fig-ure 6.39, the side of the actuator which is connected to the pump is named pump(power) side, and the side which is connected to the tank is called drain side. Figure 6.39-a shows the case when the load is moving with gravity. The direction of the gravity force acting on the link, R, and the direction of the net force acting on the actuator piston, F, and the direction of the desired ram velocity, Xj, are shown in the figure. As is seen, the force F is in the opposite direction to the desired motion. It is also understood that in this case, the drain side is controlling the load and therefore the flow from the cylinder to the tank needs to be regulated (model no. 2). Figure 6.39-b shows the case when the velocity and the force are in the same direc-tion. The orifices on the pump side control the motion. The pressure at the pump side is thus utilized to decide on the spool displacement (model no. 1). Figures 6.39-c and 6.39-d show two other cases in which the manipulator is in interaction with the environment in a closed-loop fashion. In both cases the direction of the net force, F, and the desired velocity, Xj, are the same and the pump side drives the link (model no. 1). The algorithm, based on the above discussion, has been summarized in the form of flow-chart shown in Figure 6.40. The algorithm just introduced is not the only way of selecting an appropriate so-lution. There are other ways to decide on the solution; some were considered during this study. However, they did not provide enough physical insight and thus their ap-plications were not as successful as the one described. This will be discussed later. er6. MACHINE CONTROL DESIRED ANGULAR VELOCITY 6 DESIRED ACTUATOR'S LINEAR VELOCITY . 9 < ? » LINE PRESSURES ACTING FORCE ON ACTUATOR'S PISTON MODEL # 1 DESIRED FLOW INTO ACTUATOR Q, ORIFICE SIZE (PUMP SIDE) SPOOL STROKE MODEL # 2 1 DESIRED FLOW OUT OF ACTUATOR Qc ORIFICE SIZE (DRAIN SIDE) SPOOL STROKE PILOT SERVO-VALVE VOLTAGE To the Machine Figure 6.40: Velocity control flow-chart Chapter 6. MACHINE CONTROL 93 6.2.3 Control Input i Once the desired spool displacement has been decided, the input voltage (control input) to the corresponding pilot servovalve is determined. The steady-state relationship between the input voltage and the spool displacement in the pilot servosystem is used to determine the voltage necessary to be applied. To find this relationship, different step voltages were applied and the corresponding spool displacement was measured. A curve-fitting technique (see Appendix E) was used to find the analytical relationship between the voltage and the spool displacement for each valve (see also Section 3.4.1). 6.2.4 Results Some results from experiments that were performed on a real-world excavator are pre-sented. Simulation results are also included along with experimental ones to validate the machine model and the simulation program developed. Figure 6.41 shows the velocity control of the boom when it moves up—*down with a constant velocity towards its joint limits. As is seen, after a certain delay, the boom reaches the desired up velocity. The delay is due to the time required for the pressure to be built up in the pilot system to move the main valve spool. Also, the spool has to travel a bit (valve dead-band) before any orifice can open. The same delay is also seen when the boom changes its direction. The valve spool has to move to the other direction and has to pass through its dead-band. Figure 6.42 shows the variation of the net force on the actuator piston, during the experiment; it is positive at all times, mostly to compensate for gravity. Thus, for boom-up in which the desired velocity is positive, the "pump side" should control the motion (model no. 1), whereas for boom-down the drain orifice controls the motion (model no. 2). Intuitively, it is well understood that for boom-up motion, the pump Chapters. MACHINE CONTROL 94 p be v u O c "5 p in Desired •• Actual (Experiment) i i i • i i i 0.0 Time (ms) 32580.0 Figure 6.41: Boom up—•down velocity control (experiment) o o C-5 U I-iS o (Experiment) 0.0 Time (ms) 32580.0 Figure 6.42: Actuator force in boom up-*down motion (experiment) Chapter 6. MACHINE CONTROL 95 (experiment)^Pump 1 Time (ms) 32580.0 16 Time (s) 24 Figure 6.43: Pressures in boom up—rdown motion Chapter 6. MACHINE CONTROL © 0 c a (experiment) from Model # 1 Desired Actual V from Model # 2 0.0 Time (ms) 32580.0 Figure 6.44: Spool displacement in boom up—»down motion (experiment) ( experiment) 1 1 1 1 1 1 i 1 1 1 1 1 1 0.0 Time (ms) 32580.0 Figure 6.45: Control input in boom up—>down motion (experiment) Chapter 6. MACHINE CONTROL 97 should push fluid into the actuator cylinder in order to make the motion happen, whereas the boom could move under it's own weight by letting the fluid to drain from the actuator to the tank. Figure 6.43-a shows how the line pressures (shown as Pi and P„) change during the motion. Pump 1 is connected to the boom main valve and is thus active; it senses one of the line pressures. During boom-up motion, the line with the pressure shown as Pi is connected to the pump but it is connected to the reservoir in boom-down operation and thus is noted as P0. The simulation results are also shown in Figure 6.43-b. As is seen, there is a good agreement between the simulation results and the experimental measurements. Figure 6.44 shows the desired (calculated) and the measured (actual) spool displace-ment during boom up—»down motion. The measured value of Pi along with "model no. 1" is used to determine the amount of spool displacement needed to provide the boom-up motion. Referring to Figure 6.43, the pressure Pi rises as the motion starts. This pressure is constantly used to decide on the new spool displacement. During boom-down operation, the line with pressure Pi is connected to the return tank and is noted as P0. "Model no. 2" is used along with the pressure P0 to decide on the orifice area to allow a certain flow (proportional to the desired velocity) to be drained to the tank. The input voltage which causes this displacement is shown in Figure 6.45. As is seen it follows the same pattern a6 the desired (calculated) spool displacement. The next example illustrates the results obtained during the stick velocity control tests. The stick is commanded to move out—+in at a constant velocity over a joint angle range of about 80°. The desired velocity profile along with the actual velocity are shown in Figure 6.46-a. The results from the simulation are also shown in Figure 6.46-b. The oscillations during the change in velocity direction, observed in the simulation results, are believed to be the result of early/rapid closure of the cylinder to tank orifice area Chapter 6. MACHINE CONTROL 98 when the spool moves to its neutral position. This suggests a difference between the orifice model used in the simulation and the actual one. Figure 6.47-a shows the actuator force and the voltage to the stick servovalve. the actuator force changes the sign during the motion. The algorithm automatically decides which model to use and then determines the appropriate spool displacement and sends voltage to provide this displacement. The oscillation in the control input (shown inside circles, Figure 7.47-a) is due to the model switching when the acting force changes direction. This oscillation can however be prevented which will be discussed next. The fact that the input voltage does not change drastically when the model switching occurs, indicates that the two models are converging to a unique solution while the steady-state velocity is achieved which is expected. Figure 6.47-b illustrates the simulation results. The model switching during stick out—>in operation is also shown in the simulation. The variation of the system pressures from both experiment and simulation is pre-sented in Figure 6.48. In this example, both lines with pressures P; and Pa were used, one at a time, in the control algorithm. Pump 2 pressure is calculated automatically which will be further used to update the pump outflow. To conclude this section, the results of the velocity control when applied to the bucket are also included. However, normally the bucket is in either dump (out) or curl (in) position and does not require velocity control. Figure 6.49 shows the application of the algorithm during the velocity control of the bucket dump—>curl; both experimental and simulation results are presented. The system pressures during the bucket operation are also shown in Figure 6.50-a. As is seen Pump 2 does not contribute during the bucket operation. The simulation results are also included in Figure 6.50-b which can be compared with the experimental ones shown in Figure 6.50-a. The velocity control algorithm outlined here, has been tested for different loading Chapter 6. MACHINE CONTROL 99 '—- \ u £ >> c u o f < J> c 0 > o f"1 © in Slick (simulation) (b) o " 0 •<r*Jotn< Angle 10 15 20 Time (s) 25 30 Figure 6.46: Stick out—»in velocity control Chapter 6. MACHINE CONTROL 100 Figure 6.47: Variation of Force and Voltage in stick out—fin motion Chapter 6. MACHINE CONTROL 101 <= L • . — • i — . 0.0. Time (ms) 34480.0 Figure 6.48: Pressures in stick out—• in motion Chapter 6. MACHINE CONTROL 102 Desired —• Actual — — Time (ms) 30000.0 o. o o 5 T -Bucket (simulation) ' 0 16 Time (s) 24 32 Figure 6.49: Bucket out—•in velocity control Chapter 6. MACHINE CONTROL 103 ( experiment) (a) / N J J s V •» v. Pump £ 0.0. Time (ms) 30000.0 16 Time (s) Figure 6.50: Pressures in bucket out—fin motion Chapter 6. MACHINE CONTROL 104 and speed. The results were satisfactory within the range of speed and accuracy ex-pected for such machines. The control frequency (i.e., number of control signals per second) during these experiments was 50Hz. 6.2.5 Hysteresis Applied to Control As was mentioned in the previous section, due to measurement noise, the actuator force calculated based on the pressure readings, contains high frequency oscillations. Unless these oscillations are filtered (through a hardware or software), the control algorithm tends to switch back and forth between the two models when the net force changes sign (shown in circles, Figure 6.47-a). To prevent this, a simple method has been applied which, in concept, is similar to hysteresis effect. Referring to Figure 6.51, the algorithm keeps its current model until the transition period for the force sign change is passed. The distance H (Figure 6.51) is determined according to the actuator areas and the noise level. 6 z; © S Model # / Model # 2 H Force Figure 6.51: Hysterisis applied during model switching Chapter 6. MACHINE CONTROL 105 6.3 Multi-link Velocity Control The single-link velocity control algorithm becomes challenging when more than one link is in motion. The principle is the same; i.e., given the desired end-effector path trajectory, the control inputs are selected such that the flow distribution amongst the cylinders always be proportional to the desired joint velocities. Loading, coupled actuation and pumps limitation are issues that have to be addressed in multi-link velocity control scheme. The equations representing the linkage dynamics are in general coupled. Also, the machine experiences different loads. The hydraulic circuit consists of two main lines (see Figure 2.2). Each line consists of main valves with priority action. The two lines are coupled, i.e., they share their outputs to speed up the motion. The two pumps which support the flow, change their outputs according to the sum of the line pressures (see Figure 2.3). The maximum flow-rate that each pump can provide is limited. The flow distribution to the actuators should satisfy these conditions. For example, the maximum joint velocity for swing is limited to the pump flow which itself is variable and load dependent. The maximum velocity that stick could achieve, on the other hand, depends on the remaining flow that has not been consumed by the swing. The stick cro6s-over valve could, however, provide additional flow to the stick from the other main line. The boom cross-over valve doe6 the same thing as the stick cross-over to 6peed up the boom-up motion. The load effect on the actuation is removed by applying the measured line pressures into the control algorithm. The limitations due to the pump outputs or interconnections are taken into account by satisfying a set of constraints (Section 6.3.1). Finally, the hydraulic actuation system is decoupled by solving the inverse fluid-flow equations (Section 6.3.2). Chapter 6. MACHINE CONTROL 106 6.3.1 Pump Outputs and Interconnection Constraints The relative angular velocities of the links provide a certain endpoint trajectory. The speed in which the path is followed could be changed by scaling these velocities up or down. Each joint velocity is related to the flow directed to its corresponding actuator. It is thus necessary to assure that the required flow could be provided by the hydraulic circuit. The necessary changes in the desired flow should be performed in order to have close to an optimum 6peed for the manipulator when following a path. In the following, this approach is outlined: step 1: For each link, the desired flow-rate from the main line to the corresponding actuator is first calculated based on the desired joint velocity from the joysticks. This flow should primarily satisfy the following constraints namely "The Maximum Availability Constraints". QBU i c t i T l / C u m j . < Q (6.67) Q s w i L c f t l R i g k t < Q (6.68) QBOiVj, < 2Q (6.69) QBOiDown < Q (6.70) Q STiln/0ut < 2Q (6.71) where Q in the output flow from each pump. QBUU Qswi, QBOI and Qsn are the desired flow into the bucket, swing, boom and stick, respectively. 2Q is the maximum flow provided from both pumps when croBS-over valves are active. Note that the boom cross-over valve is not active during boom-down motion. Any violation from the above constraints should be modified before proceeding to Chapter 6. MACHINE CONTROL 107 the next step. This is done by scaling down all the flow-rates. The proportional rela-tionship between the joint angular velocities and the corresponding flow-rate suggests that the desired path is still obtained but at a lower speed which is actually the highest possible one. step 2: The updated flow-rates should now satisfy the following constraints namely 11 The Interconnection Constraints": Q - QBUiD„mr/Curl > QBOiDown (6.72) 2<? - QBUiDump/CnT, - Q s w i L c t t l R i g h t > Q s T i I n / 0 u t + QBOiUp/Down (6.73) Modifications are performed, if necessary, by introducing a new scaling factor. For example, referring to the last inequality the scaling factor is calculated as below: K = —— W 11Right QSTiIn/0ut ~\~ QBOiup/Down) If K is found to be less than unity, then the flow-rates appearing in the denominator should be modified by multiplying them by K. The updated flow-rates are then used for the control purpose. If K is greater than one, it indicates that the output from the pumps could handle the required flow-rates and thus there is no need for any correction. The algorithm outlined here provides a satisfying solution by checking the constraints and modifying the desired joint velocities. 6.3.2 Inverse Main Valve Hydraulics The decoupling of the actuation system i6 actually performed by solving the inverse fluid-flow equations, i.e., based on the current pressure status of the system, the amount of the valve spool displacements are determined such that the desired fluid-flow pattern could be obtained. Chapter 6. MACHINE CONTROL 108 The schematic diagram of the hydraulic main lines and their interconnection has already been shown in Figure 3.21. As was mentioned earlier, this circuit could be represented by five possible alternatives (Figure 3.22). It was also noted that one of the solutions is always active. The search for the best solution amongst the five alternatives, [cases (a) to (e), Figure 3.22 is performed through an iterative search strategy. This is done by ex-amining the newly modified flow distribution from Section 6.3.1. The most probable circuit i6 chosen first; the appropriate spool displacement for the main valves as well as the pressure distribution along the main lines are then determined. The conditions in which the circuit could be the solution are then examined next. If these conditions are not satisfied, the algorithm considers the next possible solution and so on. Figure 6.52 shows the general flow-chart. The algorithm first considers the case where the main lines are not interconnected and each one is able to supply the right amount of flow-rates, i.e., case (a) in Figure 3.22. However, if this case is the solution, the conditions of the circuit must be satisfied, otherwise the next possible solution is examined. The way the algorithm has been set up, makes it easy to locate the solution promptly. All five possible solutions could also be evaluated simultaneously. The solution to each case is a numerical problem; one of which is exemplified with an experiment in the following Bection. 6.3.3 Results Figure 6.53-a shows the experimental results when two links, boom and stick, are simultaneously moving at a constant velocity throughout their joint limits. It is seen that the desired values of velocity can be ultimately obtained using only decoupled-flow compensator and no closed-loop control. Figure 6.53-b shows the simulation results. Referring to Figure 6.53-b, there is a Chapter 6. MACHINE CONTROL • Figure 6.52: Multi-link velocity control flow-chart er 6. MACHINE CONTROL Figure 6.53: Multi-link velocity control (boom and stick) M O D E L # I DECIDE ON B O O M M O D E L DECIDE ON STICK M O D E L M O D E L # 2 DETERMINE THE BOOM VALVE SPOOL DISPLACEMENT •Given cylinder to tank flow, decide on C - T orifice area. 'Calculate P - C and P - T orifice*. Calculate C - T eroaa-over orifice * USE THE CURRENT VALUE POR BOOM SPOOL DISPLACEMENT • DETERMINE THE STICK VALVE SPOOL DISPLACEMENT DETERMINE THE STICK VALVE SPOOL DISPLACEMENT * UPDATE THE BOOM VALVE SPOOL DISPLACEMENT ' DECIDE ON ANOTHER ITERATION DETERMINE THE BOOM VALVE SPOOL DISPLACEMENT DECIDE O N STICK M O D E L DETERMINE THE STICK VALVE SPOOL DISPLACEMENT 'Knowing flow pattern in the main line, find P - C and P-T oriflcei area. •Calculate C - T orifice area. •Calculate P -T crow-over orifice. DETERMINE THE STICK VALVE SPOOL DISPLACEMENT DECIDE ON S W I N G / B U C K E T M O D E L DETERMINE THE SWINCA»UCKET VALVE SPOOL DISPLACEMENT DETERMINE THE SWINC/BUCKET VALVE SPOOL DISPLACEMENT CALCULATE THE PRESSURE DISTRIBUTION ALON<; THE MAIN LINES Figure 6.54: Outline of the numerical approach to solve case (a) Chapter 5. MACHINE CONTROL 112 Time (s) Figure 6.55: Pressures in multi-link motion (boom) Chapter 6. MACHINE CONTROL 113 o o o •O STICK ( experiment) (a) Time (ms) 13440.0 o o o in , o es © S t i c k ( atmu/ahon ) (b) r^._ ^PsTo j ys~Pump2 a> h. S on V) u CU E a> 03 Time (s) 12 Figure 6.56: Pressures in multi-link motion (stick) Chapter 6. MACHINE CONTROL 114 drift in both boom and stick velocity profiles. The time-delay in the pilot system, when the load seen by the actuator changes, is believed to be the cause; however, the maximum error observed, due to the drift, is less than %4. The cross-over valves were enabled in this experiment to provide the interconnection. The algorithm based on case (a), Figure 3.22, was performed during this experiment. Figure 6.54 shows the flow-chart in which this case was solved; the circuit is decomposed in smaller modules and numerical techniques are performed at these levels. Other possible cases have similar flow-charts, however, more computations must be performed. Figures 6.55 and 6.56 show both experimental and simulated values of line and pump pressures during the simultaneous motion of boom and stick. As is seen both pumps were active in this experiment. Due to computational load, the control frequency was 2bHz in this experiment. The oscillations in the pump pressures, which were observed during this experiment, are believed to be originated from the swash plate control system. 6.4 Inclusion of Closed-loop Part The decoupled-flow compensator block diagram together with the closed-loop part is shown in Figure 6.57 and is entitled decoupled-Flow Controller. The blocks shown as uAn, "Jf - 1" and " P _ 1 " are related to the decoupled-flow compensator; their roles have been described in Sections 6.3.1, 6.3.2 and 6.2.3, repectively. Referring to Figure 6.57, the control law consists of two parts. The first part (decoupled-flow compensator) is based on the hydraulic model; it observes the current status of the machine through pressure measurements, and provides appropriate inputs to the actuators to accomplish the task. The effects of load and coupled hydraulics have this way been eliminated, to a great extent, which makes the design of the second Chapter 6. MACHINE CONTROL 115 Operator T dt J K-1 A H-1 p - i M K - l L7J H - l Joystick Inverse kinematic Algorithm to modify joint velocities Inverse main valves hydraulics Inverse steady-state pilot system Machine dynamics Proportional gain Derivative gain Av Kv dt Desired velocity (Base coordinate) 84 Desired velocity (Joint coordinate) 6™ Modified desired velocity xd Desired spool displacement Vi Desired voltage Av Added voltage Vc Current voltage (Control input) p Line pressures vf Voltage due to position error vv Voltage due to velocity error Figure 6.57: Decoupled-fiow control block diagram Chapter 6. MACHINE CONTROL 116 part of the controller (i.e., the closed-loop part) easier. Appendix H highlights the benefits of this technique with an example in a frequency-domain analysis; it is shown that the inclusion of the hydraulic model-based part causes a reduced sensitivity of the hydraulic actuation to the load, and an improved response. The inclusion of the closed-loop part is to compensate for the modelling error, unmodelled dynamics, lack of knowledge of the parameters and/or other uncertainties. For example, it played an important role in the control of swing motion in the presence of dry friction and valve parameters uncertainty. Figure 6.58-(curve a) shows a typical experimental result of swing velocity control with no closed-loop. It is seen that after a long delay and an overshoot, the velocity does not approach to the desired value. Dry friction in the form of stick-slip as well as modelling error are responsible for this behaviour. Dry friction can be seen as an external force like gravity loading. However, compared to the gravity loading, which is a position dependent, friction is a nonlinear function of velocity [Tsai88]; the overshoot is partly known to be the result of the change in the order of the system's differential equations, depending on whether there i6 a motion or not [Tsai88]. The modelling error is responsible for the low tendency of the velocity to converge to the desired one. The swing velocity control in the presence of modelling error and dry friction, was applied in the simulation. The result is shown in Figure 6.59-(curve a). The uncertainty was placed in the valve spool displacement which gave a similar result as the one from experimental observation. The delay can be reduced by introducing a secondary compensation; when there is no motion (starting point), dry friction cannot be noticed until a certain pressure is applied. Therefore in the absence of any motion or when changing the swing direction, the "calculated" line pressure is used rather than the "measured" one. The calculated Chapter 6. MACHINE CONTROL 117 'O.O Time(ms) 12000.0 Figure 6.58: Swing velocity control (experiment) ' 0 2 4 6 6 10 12 Time (s) Figure 6.59: Swing velocity control (simulation) Chapter 6. MACHINE CONTROL 118 0.0. Time (ms) 19520.0 o o o o f I I • I 0 5 10 15 20 Time (s) Figure 6.60: Pressures in swing motion (with closed-loop and compensation) Chapter 6. MACHINE CONTROL 119 o © © ' 00 o ( experiment) " ~* ~ K •» Volt (Vel.) Volt (Ang.) Volt(feedforward)' 0.0 Time (ms) 12000.0 Figure 6.61: Control inputs in swing motion (with closed-loop and compensation) ( experiment) Desired Actual Time (ms) 12000.0 19520.0 Figure 6.62: Swing joint angle trajectory (with closed-loop and compensation) Chapter 6. MACHINE CONTROL 120 pressure is the one which exceeds dry friction to start. The value of stick friction is thus to be estimated. Once the motion starts, the control begins to use the measured value of pressures. Figure 6.59-(curve b) shows the result of the simulation when the improved method, including the closed-loop part with low gains, was applied to the same system as in Figure 6.59-(curve a). The closed-loop part was selected as a PD control. The error between the desired angle, 9m, and the actual angle, 9a, is multiplied by a constant Kp\ the same is done for the velocity, using constant Kv. This provides an added voltage to the control input from the compensator as below: A t ; = Kp(&X-9a) + Kv(0X-9a) (6.74) Figure 6.58-(curve b) showB the experimental observation. The measured line pres-sures are shown in Figure 6.60-a. The simulation results for the same motion are also included in Figure 6.60-b. As is seen, the line pressure Pi rises at the beginning as a result of overcoming the stick friction. Once the motion starts, friction decreases abruptly resulting in an increased velocity and pressure drop. The control inputs from the feedforward part,u<f, position feedback,vp, and velocity feedback,v„, loops are shown in Figure 6.61. It is seen that the inputs from the closed-loop part only serve to stabilize the velocity at the beginning of the motion. The resulting joint angle trajectory is also shown in Figure 6.62. 6.5 Discussion The complexity of teleoperated control of heavy-duty hydraulic machines can be re-duced by cascading the hydraulic actuation from the structure and applying a feedfor-ward compensation technique to the hydraulic part. The control loop i6 then closed, using simple control components. The entire control system is 6hown in Figure 6.63 Chapter 6. MACHINE CONTROL 121 along with the machine dynamics. Experiments were performed on a real-world exca-vator. The emphasis was put on the effectiveness of the decoupled-flow compensator based on the model developed in Chapter 3. The following discussion is beneficial to conclude this chapter: 1. The pressures in the hydraulic lines are measured and are used in the control to determine the control input. Other information such as the pump pressures are automatically determined by the algorithm which are subsequently used to update the pump outputs limitations. 2. The choice of spool displacement follows the same rationale as the original design of the orifices. This approach has been found to be more effective than other methods tried; two of which are briefly mentioned herewith: • In the first method, the angular velocity is chosen as the criterion. If the actual velocity is greater than the desired velocity, a smaller calculated spool displacement will be selected. • In the second method, the desired orifice areas are first determined based on "model no 2". The calculated spool displacement is then used to calculate the flow-rate from the pump to the cylinder. If this flow is less than the desired incoming flow, then the spool displacement based on "model no. 2" is used; otherwise, the calculation should be carried out using "model no. 1". 3. The use of "model no. 1" requires numerical iterations. An alternative algorithm of similar principle was considered to reduce the computational load; it used both measured line pressures and pump pressures to decide on the spool dispalcement. However, its application in practice did not show any satisfactory result. Chapter 6. MACHINE CONTROL 122 4. A knowledge of hydraulic parameters is necessary. Different methods of direct and indirect measurements were performed to indentify the characteristics of the hydraulic part: • direct measurement; • experimental measurement; • from the data given by the manufacturer; • by comparing the response or behaviour of the actual system and the model. For example, experimental measurements were applied to determine the relation-ship between the main valve spool displacement and the input voltage to the servovalve system. Nonlinearities such as dead-bands, time-delay and dry friction were indirectly studied by comparing the simulation results with the experimental observations. 5. With the existing computing facilities, it was only possible to control simple tasks involving a single link or two links in real-time. Study on a typical duty-cycle of the excavator shows that most of the time two links, out of four links, are in motion simultaneously. This knowledge would help to simplify the algorithm for faster implementation of the whole algorithm, especially with the inclusion of closed-loop part, in the future. 6. The delay in the pilot system as well as the second order effect (i.e., the time derivative) of the pressures were not considered in the decoupled-flow compen-sator. Also the potentials/limitations of the control in closed-loop interaction with the environment (e.g., digging) and for more complex tasks was not experi-mented and is subject to future study. Operator JOYSTICK Desired Motion (Base Coordinate^ INVERSE KINEMATICS Desired Motion (Joint Coordinate) ALGORITHM TO DETERMINE APPROPRIATE FLUID-FLOW RATES Desired Flow Rale Modified Desired Motion (Joint Coordinate) CLOSED-LOOP CONTROL INVERSE Desired Spool Displacement INVERSE STEADY-STATE MAIN VALVES HYDRAULICS PILOT SYSTEM • Desired Voltage Added Voltage Current Voltage M A C H I N E DYNAMICS PILOT SERVO SYSTEM Pilot Pressure MAIN VALVES HYDRAULIC SYSTEM Fluid Flow (to Actuators) C O N N E C T I N G HOSES Pressure (on Cylinders) » w LINKAGES Current Motion I Figure 6.63: Complete block diagram of a teleoperated heavy-duty machine Chapter 7 C O N C L U S I O N S 7.1 Contributions of this Research A large number of heavy-duty manipulator-like machines are used in primary industries such as excavators in construction and feller bunchers in forestry. The environments in which these machines operate are not very well defined and are potentially hazardous. Consequently, the operation and control of these machines are very much operator dependent and require significant visual feedback, judgement and skill. Application of robotic technology to this class of machines would bring about enhanced operator safety and increased productivity. The objectives of the research reported in this thesis were to develop the means for converting these machines into task-oriented human supervisory control systems. In order to accomplish this, a study comprising experimental, theoretical, mathematical, mechanical and simulation components was conducted. An intended characteristic of this study is its direct application to a real-world ma-chine. Mechanical examination of the candidate machine, a Caterpillar 215B excavator, revealed that it was suitable for supervisory control providing that some changes were made in the pilot system. The machine was then used as a test bed to validate the methods and approaches developed during the course of this research. The contributions of this research can be divided into two areas; Modelling/Simulation and Control: 124 Chapter 7. CONCLUSIONS 125 Modelling and Simulation: A simulation of an excavator has been developed with several unique features. The mathematically stiff hydraulic system was simplified by appropriate partitioning of the differential equations. The high frequency components were partitioned from the low frequency components and were located in one subsystem through a specially devel-oped subsystem arrangement. It was then shown that efficient simulation was possible by combining the best properties of the transient and steady-state solutions in the modelling phase. In an effort to improve the computational efficiency, a flow-based modelling tech-nique was investigated. It was found that this can reduce the computation time; how-ever, flow-based equations for complex systems were found to be difficult to derive and did not show any advantage to the original method for the class of machines under investigation. The coupled and interconnected main valve system was further studied. It was shown that the complexity of the computations could be reduced through representing the main valve circuit by possible alternatives with simpler configurations. A logic system was then developed which finds the applicable solution amongst these possible alternatives. Inherent to any heavy-duty hydraulic machine with a large number of interconnected components are nonidealities such as gear backlash, friction and leakage. Thus, it was necessary to conduct analyses which took these effects into consideration. As a result, an algorithm was developed to efficiently simulate the gear backlash for the class of heavy-duty machines. Experimental and simulation studies in the structural dynamics were also performed which provided invaluable data in terms of the interactions between linkages and the Chapter 7. CONCLUSIONS 126 overall characteristics of the machine in heavy-duty manipulation. It was found that inertia, Coriolis, centripetal and gravity terms contribute in the dynamics; contribution of the centripetal forces to the overal torque, for example, can become as high as 40%. Based on this analysis, a numerical simplification was performed which reduced the computation effort by 27%. Control: A decoupled-flow controller has been developed for resolved-mode teleoperation of heavy-duty hydraulic machines having coupled actuation systems. In this approach the hydraulic part senses loads in the structural part. The control law is partitioned into two parts. The first part is based on the hydraulic model; it employs the measured line pressures along with fluid-flow equations to control the joint velocities. The load sensitivity, pump limitations and coupled hydraulics are considered in this portion of the control law. The algorithm was tested on the machine and was shown to successfully compensate for the effects of load variation in velocity control of heavy-duty machines within the range of their operating speed and accuracy. The second part of the control law is a simple closed-loop control consisting of PD components and static friction compensator. Its effectiveness was demonstrated experimentally in compensating for modelling errors and unmodelled dynamics. 7.2 Recommendations for Further Research The scope for further research is open and valuable contributions can be made. The followings are the highlights: er 7. CONCLUSIONS 127 Real-Time Simulation: The model can be further studied through implement-ing a suitablly developed parrallel computer architecture with more efficient nu-merical schemes and interactive features for real-time simulation. A real-time simulator can be used in conjunction with A l in a fault/hazard diagnosis system. It can also be used with computer graphic displays similar to a flight simulator for human interfce studies. Machine Mobility and Stability: Mobile manipulators have almost infinite workspace; e.g., to reach a point it is possible to define a domain in which the base can maneuver and the access to that point is possible. However the machine needs to maintain its balance. The stable workspace is thus dependent on the dynamic states of the machine structure. T h e simulation model can be further extended to include the machine mobility as well. Motion and Force Control: The decoupled-flow control technique needs fur-ther consideration to assess its potentials and limitations when fully applied to the machine specially when the machine is in closed-loop interaction with the environment. Also, the fundamentals of the proposed control can be applied for static force control which is important for machines such as feller-bunchers. Hydraulic Parameter Identification: Parameter identification of a hydraulic component of a complex system, based on input/output measurement of the system is a difficult and complex task and falls into the category of nonlinear identification. The simple identification techniques used during the course of this study, can be substituted with a more thorough and systematic approach which could further be imbedded into a failure analysis and detection system. References [Aldon84] Aldon M . J . and Liegeois A., "Real-Time Aspects in Robot Dynamics Modelling" (A. Danthine and M. Geradin eds.), Elservier Science Publishers B.V., North-Holland, 1984. [Arday84] Ardayfio D.D. and Qiao D., "The Kinematics of Industrial Robots Having Closed-Kinematic Chains", Robotics and Automation '84 (M.H. 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References 131 [Lambe83] Lambeck R.P. , "Hydraulic Pumps and Motors", Marcell Dekker Inc., New-York, 1983. [Lathr85] Lathrop R . H . , "Parallelism in Manipulator Dynamics", Proc. IEEE Inter-national Conference on Robotics and Automation, St. Louis, Missouri, March 1985, pp. 772-778. [Leahy86] Leahy M . B . , et.al., "Efficient Dynamics for a PUMA 600", Proc. IEEE International Conferene on Robotics and Automation^ San-francisco, Cali-fornia, April 1986, pp. 519-524. [Lee86] Lee C . S . G . and Chang P.R., "Efficient Parallel Algorithm for Robot In-verse Dynamics Computation", Proc. IEEE International Conference on Robotics and Automation, San-Francisco, California, April 1986, pp. 851-857. [Leini84] Leininger G . G . , "Self-Tuning Adaptive of Manipulators", Advanced Soft-ware in Robotics (A. Danthine and M. Geradin eds.), Elsevier Science Pub-lisher B.V., North-Holland, 1984. [Luh80] Luh J .Y .S . , Walker M . 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D . , "Controller Design for a Manipulator Using Theory of Variable Structure Systems", IEEE Transactions on Systems, Man and Cy-bernetics, Vol. SMC-8, No. 2, February 1978, pp. 101-109. Appendix A Hydraulic Power Any hydraulic device contains some standard components which are common; amongst them valves, connecting lines, pumps, check valves and relief valves could be named. In this appendix the basics of fluid power and the most common components are described. A . l Hydraulic Fluids Hydraulic systems use liquid for transferring the energy. The characteristics of a hy-draulic fluid are known by its states. The states of the hydraulic fluids are density, specific heat and viscosity. These states are temperature and pressure dependent, the variation of which in the presence of heat and/or pressure is outlined in [Merri67, pages 6-12]. One important characteristic of the hydraulic fluids is their compressibility. The compressibility of a fluid is defined as follows compressibility = ~ where /? is called the bulk modulus of the liquid and is equal to 0 = -V(?~)>O (A.75) Bulk modulus is important in determining the dynamic performance of the hydraulic systems [Merri67]. Although the theoretical value for the bulk modulus is high, in practice, however, such values are rarely achieved because they decrease sharply with a small amount of air entrained in the fluid. When the hydraulic fluid is carried through 134 Appendix A. Hydraulic Power 135 flexible containers, such as connecting hoses, the bulk modulus is reduced. The effective or total bulk modulus, /3e, is then defined by the following equation irbbvSi) <A76> where 0C is the mechanical compliance of the container, Pi is the bulk modulus of the liquid, (3g is the bulk modulus of the entrapped air, Vg is the volume of the air entrapped, Vt \s the total volume of the container. The major source of mechanical compliance is the hydraulic hoses connecting valves and pumps to the actuators. The effect of air entrapped in the hydraulic fluid becomes more at lower pressures. Also, there are some cases where the hydraulic circuits are not capable of keeping the fluid temperature down. The increased temperatur reduces the bulk modulus. Equation (A.75) is the basis for describing pressure changes in transmission lines [Watto87]. A review on the effect of the transmission lines on the dynamic performance of hydraulic systems ha6 been reported in [Bower72], in which an ideal valve/motor circuit was considered and the effect of the length of the transmission lines was inves-tigated. A .2 Flow Through Orifices Referring to Figure A.64, the flow rate through an orifice is given by Q = d*fyPi ~ Pi) (A-77) Appendix A. Hydraulic Power 136 cj is the discharge coefficient; its value is approximated by 0.60 (for more details see [Merri67, page 42]). I I r 1 2 3 Figure A.64: Flow through an orifice [Merri67] Sometimes fixed orifices by length are used. The above equation could still be used, however, the discharge coefficient for short tubes should be determined according to the length, radius of the tube and the velocity of the fluid. A.3 Hydraul ic Pumps and Motors Hydraulic pumps and motors convert mechanical energy into hydraulic energy and vice versa, respectively. The most common type of these machines, used in the fluid power hydraulics, is classified as positive-displacement machines [Merri67]. Positive-displacement machines are quite efficient and find extensive uses in hydraulic systems. Figure A.65 shows three types of positive-displacement hydraulic machines; Figure A.65-a shows a limited-travel type; whereas A.65-b and A.65-C show continuous-travel devices. The ideal governing equations for pumps and motors are simple. They are written as follows: Q T D m 9 m DmPi (A.78) (A.79) Appendix A. Hydraulic Power 137 1 1 J h — — * r ( a ) Double acting actuator V c / Axial piston type pump Figure A.65: Positive displacement hydraulic machines [Merri67] Appendix A. Hydraulic Power 138 where Q and T are the motor/pump flow and torque, respectively. In practice, these equations may not be exact due to two sources of losses; leakage flow and internal friction. However, hydraulic machines are known to be quite efficient and system design is often based on ideal machines [Merri67]. A.4 Variable-Displacement Pumps Future generation of mobile equipments will require energy efficiency and more con-trollable hydraulic systems. The objective of the system efficiency is to avoid the unnecessary pressure drops and pressure-flow losses [Merri67]. One way to achieve a greater efficiency is by means of the use of variable-displacement pumps. The efficiency through variable-displacement pumps could be graphically repre-sented by pressure/flow diagrams as shown in Figure A.66. Point a is the maximum flow available from the pump. It is controlled by changing the displacement of the pump pistons or the operating speed. Point b specifies the maximum pressure avail-able from the pump. The value of which is determined by the system relief valves or through the pressure compensation. Point c is called the corner power and indicates the maximum possible power output. Point / is a typical working point and represents the pressure required to move a load and the flow needed to impart the desired velocity. The shaded area shows the actual power required. The area Inbkl is the power loss if pressure-compensated pump is used. The area amlha is the power loss if a fixed-displacement pump is used. A more efficient way is to use a pump which adjusts its pressure output to the preset differential above the load pressure. In this case the power loss is limited to the area of ll'k'kl. In general the maximum input power to the pump is constant and is provided through a power source such as an internal combustion engine. The maximum power Appendix A. Hydraulic Power 139 Figure A.66: Pressure-flow diagram limits the working area to the area of amm'n'boa. To increase the productivity [Lambe83], however, it is desirable to be able to have a pump control to performe with a given level of installed power. This is specially important where loads are highly variable. The torque-limiting pump enables to operate with the constant power curve. As the pres-sure increases, the stroke of the pump is reduced to prevent the pump from exceeding the desired maximum torque or power limit, whereas for light loads high flow becomes available for rapid machine cycle-time. In some machines, such as excavators, the prime mover drives two large variable pumps. If each pump is provided with an individual torque-limiting control, it does not permit the optimum use of the available power. The full power is made available with optimum productivity if torque-summation concept is used. In this case each pump not only senses its own torque position, but also that of the opposite pump. So when Appendix A. Hydraulic Power 140 one pump is operating at low power, the second automatically raises its own torque limit to correspond. In this type of control, the pumps as a pair, still operate with the power limit of the engine, but each is automatically free to make use of any increment of power that is not being used by the other. A . 5 Hydraulic Valves Hydraulic valves are devices that control the fluid power via the use of mechanical motion. The most widely used valve is the sliding valve with spool type construction. Spool valves are classified by a number of ways flow enter or leave the valve, the number of lands and the type of center when the valve spool is in neutral position. Figure A.67 shows typical spool valves. The general equations for the valve could be written using the continuity equa-tions for the orifices. These equations are nonlinear. For the very special case of critical-center valve with matched and symmetrical orifices, Figure A.67-a, the follow-ing equations hold (Pe = 0) [Merri67]: Qi = Qi - QA = <?3 - Qi Wl = Cd<l\\ Cdd4\ Cd<i3\ cda2\ IP. ~ Pi P. + Pi cda P   Pi fj+Jl I A ori\ Cef tx .y— d 2J—-— (A.80) I2(P. - Pi) P I2(P. - Po) 2{Pi) - cda3\l — '- + cda4\ — Appendix A. Hydraulic Power 141 Pi = Pi ~ Pc (a) Three-land-four-way spool valve to load (b) Four-land-four-way open-center spool valve Figure A.67: Typical hydraulic valves [Merri67] Appendix A. Hydraulic Power 142 /P,-Pi , j P. + Pt , A B 1 , = c ^ i y — - — + c d a 2 J — - — (A.81) where Pt = Pi — Pa , Pt = Pi + Pa , a x = o3 and o2 = a 4. In making a dynamic analysis, sometimes, it may be useful that the nonlinear equations be linearized. Using the taylor series about the operating point, then the equation for Qi becomes: where x denotes spool displacement. Defining K.x = ——=flow gam ox K,p = — -\~Qp~) ^flow-pressure gam the linearized equation then becomes: AC?, = KtAx - 2!CpAPl (A.82) The flow gain characteristic of valves is directly related to the types of the valve center. Figure A.68 shows the shape of the flow gain for three types of center. A.6 Stroking Force in Hydraulic Valves Forces acting on the valve spool during the motion of the spool are FS = FD + FE 4- FR (A.83) FJJ is the D'Alembert force and is equal to = "*> FE is the external force from the spring with the stiffness K T , FE = K.x Appendix A. Hydraulic Power 143 o Open Center Underlap Region / / Critical Center Closed Center PS \ ^ \ Spool Displacement Overlap Region Figure A.68: Flow gain of different center types [Merri67] and FR is the flow-induced forces acting on the spool due to the motion of flow (for more details, see [Merri67, pages 92-105]). It is defined as (referring to Figure A.67-a): FR oc 0.43u>(P, - Pt)x + 0.60u;(L2 - Ljjp{P. - P , ) ^ where the first statement on the right hand side, acts as a centering spring, and the Becond statement, acts as viscous damping. The contribution of each force depends on the type of the valve used and its appli-cation. Sometimes the magnitude of forces needed are so high that it is necessary to use a two-stage servovalve in which the first stage provides an adequate hydraulic force to stroke the second stage spool valve. Appendix B Hydraulic Systems for Mobile Machines Competitive push to improve efficiency and productivity and to reduce the system cost, leads to the design and the implementation of new hydraulic systems with different in-teraction. There are initially three types of systems need to be considered; constant flow , constant pressure and load sensing [Sulli72, Taken74]. They differ from each other in design and interaction of their components. They also give different operating economy and energy consumption depending on the working-cycles and the relation-ships between functions being operated simultaneously. These three systems are briefly described here. B . l Constant Flow (CF) System In this system, Figure B.69-a, the flow at a certain engine speed is constant while the pressure is adapted to meet the requirements. Depending on the type of direc-tional valves being used, the CF system could be CFO (open-centre valve) or CFC (closed-cent re valve). The CFO system is less complicated and is less sensitive to con-tamination. The pump output pressure is determined by the heaviest load. So, the simultaneously actuated functions should have more or less the same pressure require-ments, or be sub-divided between several circuits. 144 Appendix B. Hydraulic Systems for Mobile Machines 145 B.2 Constant Pressure (CP) System In a constant pressure system, Figure B.69-b, the pressure is constant and the flow varies as required. The pump is type of pressure-compensated variable-displacement pump. Since the constant pressure system keeps the pressure constant, when the ma-chine is not in operation, then small leakage in a valve results in creeping movements. This system is desirable in a sense that the operating functions are independent and do not influence each other. Therefore, when control accuracy and interference free simultaneous operation of functions are required, this type of system is recommended. B .3 Load-Sensing (LS) System In a load sensing system, Figure B.69-C, both pressure and flow are adjusted as required. The pump displacement is regulated such that there is always a constant pressure difference between the pump and the signal line (usually the heaviest load). In LS, as in CF systems, simultaneously operated functions should have the same pressure requirements or should be sub-divided into several circuits for better operating economy. The main deficiency of this system is that if the mechanical constructions is not rigid enough, the load sensing system interprets the oscillations as being varying pressure requirements and this may cause the whole system to oscillate. A more efficient but complex in design for load-sensing systems is the one which provides a pressure signal to the variable-displacement pump. The highest pressure directs the control signal to stroke the variable-displacement pump. The individually compensated valve sections then keep the pressure difference, belonging to the other functions constant. In this system, though looking interconnected, the functions are independent and still the efficiency is held. Appendix B. Hydraulic Systems for Mobile Machines Appendix C Excavator Dynamics Equations In this appendix, the general equations describing the excavator dynamics are shown. The equations are grouped as in Figures 3.16. The same notations as in Figures 3.16 and 3.21 are used. Level 1; Pilot Valves Xbo l/noc *boc x,w i.t x.t 1/r.e *.t \x.tJ 11/r.tc ) Xbu = Xbu{vbu{t)} Xbo = ^6C.{UCK,(*)} Xboc = Xboc{vbo(t)} where r is the time constant. Function X denotes the steady-state relationship between the input voltage and the spool displacement (see Figures 3.19 and 3.21). Level 2: Main Valves and Pumps  Pump Side: Q = Q{Pii,P*i,Q) 147 Appendix C. Excavator Dynamics Equations Qn Q12 Ql3 Q21 Q22 <?23 QBOX QBUX QsTi Qswi QsTm QsTc QBOm. QBOc Pl7 JPl2- Pis }/pl»- Pe y/Pn- P22 \JP22 — P23 \Z-P23 - Pe /Pboc — PBOI PBUX jP.tc - PSTi P22 ~~ Pttc Pis " Pete P12 -Pboc P23 -Pboc (Pn\ 1 P12 Pis P21 1 P22 c, P23 Pboc \P*tc) V Q l l — Ql2 — QBOm Ql2 — Ql3 — QsTc Q — Q21 — Qswi Q 2 I — Q22 — QsTm Q22 ~ Q23 — QBOC QBOm — QBOc — QBOx QsTm — QsTc - QsTi / Appendix C. Excavator Dynamics Equations 149 Tank Side: QBVO — kaobusjPBUO — Pe QBOo — kaoboyfpB~oo~—~Pe QsWo = kaotwyJ PsWo — Pe QsTo = kaoat\JPsTo - Pe Q denotes the pump output dynamics; its steady-state relation has been shown in Figure 2.3. a denotes the orifice area and is a function of 6pool displacement (see also Figure 2.6): aitm ~ Abti{*bu} O-obu = A>feu{z{m} O-ebu — A s b u { « 6 u } Level 3: Connecting Hoses PBVX = rP~ [QBUI — QBUI) PBUO = ^rjj {QBUO — QBUO) PBOi = (QBOi — QBOiJ PBOO — ($Buo — QBUO} Pswi = yf— (Qswi — Qswi) iwi PsWo = TT^~ ($SWo — Q-SWo) 'two PsTi = {QsTi ~ QsTi) 'Hi PSTO = -rr~ ($STo — QsTo) Vtto Appendix C. Excavator Dynamics Equations 150 = •A-buiQbuFbu. Q B U O — Aouo9\)UF}ni QBOi = $ B O O = A-booGboFbo = nDm8,w QsWo = nDm6,w QsTi = QsTo A,t09ltFlt Dm is the hydraulic motor constant with the gear ratio n. V is the fluid volume trapped on each side of the actuator piston. F is the ratio of actuator piston linear velocity with respect to its corresponding joint velocity (see Appendix G). Both V and F are joint angle dependent, i.e., Vhu = v(ebu) VLo = v(9bu) Level 4: Actuators and Structure TBU = (PBUiAbui — PBUoAbuo) Fbu TBO — (PBOiA-boi — PBOoAboo^J F{,o TST — (PsTiA,« — PsToAtto) Ftt Tsw = n^Pswi - PsWo) Dm Appendix C. Excavator Dynamics Equations 151 9bo / Tj3V — fdhu^bu — fCbu \ TBO — fdboffbo — fCbo TST — fd,t6,i — fc,t V Tsw — fd,w8aw — fc,w J ( fa \ 2Qbu8bo 20bu8it 20bu8tw Ds ijk Ml 2Obo0,w ft 26,t8$w \ 82 ) /c and / d are the columb friction and damping, respectively. Dy, D^ and D{ are ma-trices describing the coefficients related to the inertia, Coriolis, centripetal and gravity terms, respectively (see Chapter 4); they include the dynamic and kinematic parame-ters of the machine. The input/output arrangements are as follows (see Section 3.2.1 for notations): External Inputs: Subsystem Level 1: Subsystem Level 2: {u2} {V2}T {V}T = {tV,*>fco, = {v} {Vl}T = {*bu, Xbo, Xboe, * » u ; , * « « ) * # « c } Uv.}/ = {QBUU Q B U O , QBOi, Q B O O , QsWi, QsWo, QsTi, QsTo} Appendix C. Excavator Dynamics Equations 152 Subsystem Level 3: {yz}T — {PBUI, PBUO, PBOU PBOO, Pswi, Pswo, Psn, PSTO} Subsystem Level 4'-" M = {y*} fcu) 8bo, 9iw,9tt, Obo, 6iw, 6»t] The compatibility between the inputs and the outputs is evident in this arrangement. Appendix D Solution to a Class of Differential Equations Given the following differential equation, dx ax + b dt cx + d it can be written in the following form, dt = ——~dx ax + o or t* , f' cx , r* d I dt = I -dx + / zdx Jto Jxo ax + b Jxo ax + b where tx cx c. . be ax + b I -dx = -(x - x0) -Ln -J x 0 ax -j- o a a axo + o and L x d d ax + b f -dx = — Ln -r 0 ax 4" b a axo -\- o The solution to the above differential equation is thus: /d bc\ ax + b c. t-t0={ -)Ln — + - ( * - x 0) vo or' ax0 + b a 153 Appendix E Curve-Fitting Technique The purpose of curve-fitting is to find an analytical form for describing a set of discrete data points. Given a set of experimentally measured data points, (zi,yi), (x2,y2), ••• and (xm,ym), the objective is to find a continuous function y = /(*) which provides the most reseanable representation of this set. The unknown parameters of the fitting relationship must be chosen such that the sum of the square of differences, i.e., S = $ ^ [ y t — /(£»)] i s minimized. The continuous function is assumed to be a polynomial «=i form: f(x) = a i 4- a2x + a3x2 + ... + anxn~l which is to be fitted in the least square sense to m discrete data points, let, S = ~ + a*Xi + a*Xi + - + a n * " ~ 1 ) ] «=1 note that 5 = S(di,a2, ...,an). S is minimum with repect to alt a2, a n , when, 9S__dS__ _dS__ _ 55 _ Q dai da2 da,j dan It can be shown that, dS m o - = £ 2 [ y i - (ai + + °»a!.? + - + ^ r1)]!-1?"1) = o t=i or d > r > i + ( £ * 0 a > + ( £ * ? ' + > 3 +... + ( E ^ ' + n " > n = E ^ ' " 1 i=l t = l t'=l t = l »=1 154 Appendix E. Curve-Fitting Technique 155 Thus the least-squares analysis leads to n linear equations which can be written in matrix form as follow: m m m m E« • E*r* N i = i i = l m m m m E-f • • E-r-t=i t=i m m m m t=i t'=i E«? • .=i • E*r+1 «=i i 7 s > \ 1=1 m E v » * * i = l m t=i VE^r1/ VE-r1 E*r E*r+1 ••• E*?(B_1V an i = l i = l i = l t=l t=l The set can now be solved for 01, 02, a„. Gauss elimination method is used to solve the above equations. It is a direct method which consecuitively uses each equation as pivot to eliminate the unknowns from the other remaining equations. This will cause the last equation to contain only one unknown. Other unknowns are then calculated by back substitution. Appendix F Excavator Kinematics The homogeneous transformations matrix [Paul83a] related the coordinate frames of link n to the one belong to link (n — 1), Figure 4.23, for each link is as follows: Link 1: Ai = Rot(6l,zo)Tran(a1,-di,0)Rot(90°,x0) Link 2: Link 3: (Cot(6i) -Stn(Sj) 0 0 \ / l 5«n(«i) Cot{9t) 0 0 0 0 1 0 0 0 /Co»(«!) Sin(9i) 0 1 / 0 0 1 / Sin(9i) aiCoi(9i) + diSin{6i)\ - Coi(9i ) oi Sin(9i) - d\ Cot{9x) I / I 0 0 \ o °\ 0 0 1 / Aj = Rot(92, zi)Tran(a.7,0,0) = A3 = Rot(93,ti)Tran(a3,0,0) = (Coa(9i) -Sin(93) 0 Sin(92) Cos(92) 0 0 0 1 \ 0 0 0 /Cos(93) -Sin(93) 0 Sin(93) Coi(93) 0 0 0 1 \ 0 0 0 a2Cot{97) \ 0 1 / a3Cos(93)\ a3Sin(93) 0 J Link 4: i * 4 =Rot(e 4,» 3) = (Co*{9i) -Sin(9t) 0 0 \ Sm(9 4) Coi(94) 0 0 0 0 1 0 \ 0 0 0 l / 156 Appendix F. Excavator Kinematics 157 The position of the end effector, in cartesian coordinate is the last column of a matrix resulting from multiplying matrices A\, A2 and A3: X = Co«(fli) [ai + a3Cot(83) + a3Co»{83 + 83) Y = Stn(0,) a i + a3Cot{83 ) + a3Cot(83 + 83 ) + diStn(fi,) - di Co§(e1) Z = at Sin(82 ) + 03 Sin(83 + 83 ) In a cylindrical mode, Figure 2.8, this position is easily defined as: r = a3Cos(83) + a3Coi{83 + 83) 6 = 8! Z = aj Sin(8j) + o 3 Sm(0 j + 83 ) Given the position of the end effector in cylindrical coordinate, the joint angles are calculated through geometric relations as below: e3 1 = 8 = tan" = Cot' = L 20303 J Appendix G Solution to Bucket Linkage The torque, T, applied to the bucket (Figure G.70) given the actuator force, F, can be decided using the principle of virtual work: Figure G.70: Linkage arrangement in bucket where dl dl dO d&4 d6 d&4 From rectangle ABC, I2 = L2 + r l - 2r4LCos{6 - a) 158 Appendix G. Solution to Bucket Linkage 159 taking derivative with respect to 0, From four bar linkage BCDE [Mabie86], where taJ) = r-c2-^d-c Ci = 2rjr4 — 2r2r4Cos(04) C2 = —2^^5171(04) c3 = r \ + r \ - r l + r\-2r1riCoa(94t) taking derivative with respect to 04, $ 2 / 0 \ ( c 3 ~ ci)[-c2 - 0.5(2clCl + 2c2c2 - 2c s c 3 )-°- 5 ] + (c3 - ci)[c2 + Jc\ + c\ - cj — — 2G 03 I T ) ; — : where ca = 2r2r45m(04) c2 = —2r2r4Coa(04) c3 = 2r!r25tn(04) Appendix H A Frequency-Domain Analysis Figure H.71 shows the schematic diagram of a simple hydraulically actuated arm. Each link is activated by a hydraulic motor which is connected to a servovalve through expandable hoses. Components such as check vlaves and relief valves are for machine safety. The servovalve monitors the flow to and from the motor. To know the direct relationship between the input (spool displacement, a;) and the output (joint angular position, 6) in the frequency domain, the model is linearized. The following equations are then written in Laplace transform: motor and link dynamics: T(s) = (Pi - P0)Dm = J(s2Q) + fd(sQ) where fd denotes damping. valve dynamics: Qi = KaX(s)-2KpPi Q0 = KmX(s) + 2KPP0 or Qi = KmX(a) - KPP, where p, = ^-P. * 2 160 Appendix H. A Frequency-Domain Analysis Servovalvc Rotary Shock Relief Figure H.71: Typical hydraulically actuated arm 9i D m S Figure H.72: General control block diagram of the system Appendix H. A Frequency-Domain Analysis 162 Km and JCP are flow gain and flow-pressure coefficients, respectively (see Appendix A). pipe dynamics: C(sPi) = Qi-{ae)Dn C(sP0) = (sQ)Dm-Q0 or C{aPi) = 2Ql-2(se)Dm In this analysis the servovalve dynamics response was assumed to be fast enough and its effect was thus neglected. The above equations are shown in the form of block diagram (solid lines) in Figure H.72. The following relationship can then be written: _ UC*DmX(s) - 2KpDmPl e { 6 ) ~ "7c /^7cl^ T2^7 ( H-8 4 ) As is seen, the output (0) is a function of both the control input (x) and the load (Pi) seen by the hydraulic system. This is due to the fact that the flow provided by the hydraulic valve depends on both spool displacement and the load. Assuming x is the output from a PD controller, e.g., X(s) = Kp(Qd - 0) + Kvs(®d ~ ©) the error in the system, E(s) = 0<*(a) — ©(«)], with a disturbing torque Tj(s) added will be: _ [JCS3 + (2JKP + fdC)s2 + (2fdKp + 2Dl)s]©d(s) - [Cs 4- 2Kp\Td(s)  E ^ ~ [JCs3 4- (2JKP 4- fdC)s* 4- (2fdKp + 2Dm + 2KmDmKv)s 4- 2KxDmKp] T The steady-state error, 9e, of this system corresponding a 6tep input Td(s) = —, and a s ed desired constant velocity ©d(s) = —, is s2 0e = UmsE(s) »-»o Appendix H. A Frequency-Domain Analysis 163 h = 2fdKp + 2Dm.a 2KP T 2KxDmKp 2KwDmKp The above block digram can be improved by introducing a compensation; based on the measured value of load pressure (Fi m) a value for spool displacement (xd) is decided such that the flow to the system be proportional to the desired velocity (84), i.e., Qt = DmsQd(s) = K ^ s ) - KpPr or Xd(s) = ^sQd(s) + ^ p r This is included (dashed lines) to the previously shown block diagram in Figure H.72. The transfer funtion of the new system then becomes: 0 ( S ) = JCs* + fdCs> + 2Dms ( H - 8 5 ) which can be compared with Equation (H.84). It is seen that the effect of load on fluid-flow (i.e., Kp) has this way been eliminated. The error in the system is: [JCS3 + fdCs2]ed(s) - [Cs)Td(s) E ^ ~ [jCs* -f fdCs* + (2Dm + 2KmDmKv)s + 2KmDmKp] The steady-state error corresponding to a disturbing torque and a desired constant velocity then becomes: 8e = 0 

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