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Adaptive rapid tracking by a 3 DOF robot manipulator Qu, Jing 1991

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ADAPTIVE RAPID TRACKING BY A3 DOF ROBOT MANIPULATORbyQu JingB.S.E., Nanjing Aeronautical Institute, China, 1984A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTERS OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES• (DEPARTMENT OF ELECTRICAL ENGINEERING)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1991© Qu Jing, 1991In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of ^C":18C-Y;c41^‘"4".;",3The University of British ColumbiaVancouver, Canada31DateDE-6 (2/88)Abstract In this thesis, an adaptive rapid tracking control methodology, which allowsa robot manipulator to track an arbitrarily pre-specified short yet maybe complextrajectory quickly and precisely, is simulated. The dynamic equations of motionof the robot manipulator, which has a parallelogram mechanism, are establishedusing Euler-Lagrange method. Recursive least-squares identification is carried outto identify the dynamic parameters of the manipulator system. This is followedby computed torque to approximately linearize the non-linear system using theestimated parameters. Because the approximately linearized plant is time-varyingand has imprecisely-known parameters, and for the purpose of rapid tracking,adaptive control with again the recursive least-squares identification is applied toidentify the parameters of the approximately linearized plant and of the controller,based on which a supplemental tracking control signal is learned and rapid trackingis accomplished by injecting this signal to the input of the approximately linearizedplant. The simulation results show that this tracking methodology allows verygood tracking performance.iiContents Abstract ^List of Figures  ivList of Tables  viAcknowledgments ^  vii1 Introduction  12 Modelling of the 3 DOF RPDDM ^  92.1 Kinematics ^  92.1.1 Introduction ^  92.1.2 Kinematics of the RPDDM ^  122.2 Dynamics ^  182.2.1 Introduction ^  182.2.2 Dynamic equations of motion of the RPDDM ^ 233 Dynamic Parameter Identification ^  303.1 The Estimation Equation set up  303.2 Dynamic parameter identification of the RPDDM ^ 354 Basic Rapid Tracking Control Strategy ^  384.1 Introduction ^  384.2 Linearization  404.3 q—learning algorithm I ^  414.4 Summary ^  455 Adaptive Rapid Tracking Strategy^  475.1 Adaptive Rapid Tracking with General Plant ^ 475.2 Application related issues  505.3 Application of the ARTA to the linearized RPDDM system^525.3.1 Parameter identification for the approximately linearizedplant ^  525.3.2 Controller design and controller parameter identification 536 Simulation Results and Conclusions^  556.1 Simulation Results ^  556.2 Conclusions  67Bibliography ^  68lllList of Figures^Figure 1.1^Block diagram of a MRAC system ^ 5^Figure 1.2^Block diagram of a STR ^  6Figure 1.3^Block diagram of the tracking system ^ 7Figure 2.1^Mechanical configuration of RPDDM ^ 12Figure 3.1^Basic parameter identification structure ^ 31Figure 4.1^Standard feedback control system ^ 39Figure 4.2^Tracking control system structure ^ 40Figure 4.3^Block diagram of computed torque^ 41Figure 5.1^Tracking control system structure ^ 47Figure 5.2^Over-all tracking control system structure ^ 52Figure 6.1^Estimated dynamic parameters ^ 57Figure 6.2^U — ij ^  58Figure 6.3^Estimated controller parameters for joint one .^61Figure 6.4^Estimated controller parameters for joint two .^62Figure 6.5^Estimated controller parameters for joint three^62Figure 6.6^Estimated plant parameters for joint one ^ 63Figure 6.7^Estimated plant parameters for joint two ^ 63Figure 6.8^Estimated plant parameters for joint three ^ 64Figure 6.9^Tracking performance after 50 cycles oflearning—unbounded ^  64ivFigure 6.10^Tracking control for unbounded tracking using truedynamic parameters ^  65Figure 6.11^Tracking performance after 90 cycles oflearning—unbounded ^  65Figure 6.12^Bounded tracking, noise, estimated parameters, 50learning cycles ^  66Figure 6.13^Tracking control for bounded tracking (noise,estimated parameters, 50 cycles) ^ 66VList of TablesTable 6.1Table 6.2Table 6.3True and estimated dynamic parameters ^ 56Initial controller and plant parameters andcovariance matrices for RLS identification ^ 59True and estimated parameters of plant andcontroller ^  61viAcknowledgmentsI wish to thank my supervisor, Dr. Chris C.H. Ma for his advice and helpthroughout my work on this thesis. Without Dr. Ma, this thesis would not havebeen possible. I also thank Dr. P. D. Lawrence from whom I first learnedrobotics. Thanks also due to Dr. T. Salcudean for the helpful discussion on robotkinematics and dynamics.vuChapter 1 IntroductionControl of robot manipulators is an important but difficult subject of study.It has attracted the interests of many researchers because of its high potential forincreasing manufacturing productivity and improving the working conditions forthe human workers. The difficulty of manipulator control lies in the complexity ofthe manipulator system itself, which is characterized by non-linear dynamics withcoupling effects among the links. Good performance of manipulator depends onmany factors, such as high quality manipulator, accurate modelling of manipulatorsystem, appropriate control scheme, etc.The manipulator adopted in this thesis is a rigid, parallelogram direct drivemanipulator (RPDDM). Typical industrial manipulator such as the PUMA hassmall actuators at the joints and use very large gear ratio in order to be ableto exert enough torque onto the links. This arrangement of actuation introduceslarge amount of undesirable nonlinearities such as friction and backlash at thejoints. Because of this, they are in general slow, and can not be used in highspeed applications. In direct drive manipulators, the links are directly coupled tothe motors, the backlash effects are eliminated and the joint frictional effects arereduced immensely. Therefore, the dynamics of the direct drive manipulators canbe modelled accurately by the rigid body dynamics of the links. This characteristicenables them to be controlled by advanced control methodologies for high speedoperations without compromising performance. Since productivity is directlyproportional to the speed of operation, higher speed of operation should yieldhigher productivity.The advantage of the parallelogram manipulators over serial link manipulatorsis obvious. In serial configuration, the motors are located serially at the joints ofthe links, and the weight of the second motor is a load to the first motor downthe serial linkage. Moreover, the reaction torque of the second motor acts onthe first motor. When the second motor accelerates in the clockwise direction,a counter-clockwise torque acts on the first motor, and vice versa. Thus thetwo motors have significant interactions. In parallelogram configuration, on theother hand, two motors are fixed on the base to drive the two input links andcause a two-dimensional motion at the tip of the manipulator. The weight of onemotor is not a load on the other and thus the robot can have high load-to-linkmass ratio. Also, the reaction torque of one motor does not act directly uponthe other, because the motors are fixed on the base. Besides lightweight andlower interaction and nonlinearity, analysis [2] also indicates that parallelogrammanipulators have comparatively lower power consumption.Accurate modelling of manipulator system plays an important role in manip-ulator control. It could occupy as much as 90% of the control designer's efforts.Since the actuators are inherently parts of the links for direct drive manipulators,separate modelling of the mechanical properties of the actuators are unnecessary.The electrical dynamics of the actuators are often orders of magnitude faster thanthe inertial dynamics and it may not be necessary to include them in the robot2model.Usually, the kinematic parameters are known or can be calibrated, as longas the manipulator mechanism and link dimensions are decided. However, theinertial parameters, i.e. the mass, the location of the center of mass, and themoments of inertia of each rigid body link of a robot manipulator are not easilymeasurable. Also, even if the manipulator is modelled accurately, the inertialparameters vary with different payload. Therefore, seeking a good identificationmethodology is essential for the purpose of control, and thus becomes oneimportant part of this thesis.Earlier work in identification of robot dynamics concentrated on estimatingthe mass of the payload, and this was later extended to estimating also thecenter of mass of the payload and the moment of inertia of the payload. In1985, An, Atkeson, and Hollerbach [1] proposed an approach which uses a wristtorque/force sensor to estimate the inertial characteristics of the payload. Theirapproach has also been extended to estimate the inertial parameters of all thelinks of a robot manipulator. In recent years, more estimation methods havebeen proposed, and most of them can be classified by one of the following twocategories: 1) directly use of the dynamic model and the measurements of the inputactuating torques/forces and output joint positions, velocities, and accelerations,as shown in [4]; 2) Estimation methods in this category are based on the energyconservation theorem [5, 9]. The methods in the first category are actually anextension of the pioneer methodologies in manipulator dynamics estimation. They3are straightforward in nature. Parameters estimated using these methods haveexplicit physical interpretations. However, they need input joint accelerations,which usually are not measurable, in the estimation algorithms. Obviously, thecalculation of joint accelerations increase the computational burden. Besides, thisgives one more source of error. The methods in the second category represent therecent trend in manipulator inertial parameter estimation. The advantages of thesemethods over other methods are the ease of implementation and its applicabilityto both serial link and graph structured manipulators. The only measurementsrequired are the joint positions, velocities and actuator torques. It is known thateach link of a manipulator has ten different inertial parameters. They are themass, the three components of the center of mass vector times the mass, and thesix unique components of the moments of inertia. However, only some of theseparameters are actually present in the manipulator dynamic equations of motion.Some appear in linear combinations (regrouped parameters). The parametersaffecting the dynamic model separately and the regrouped parameters constitutethe base parameters or the set of minimum number of parameters required forcontrol [5], and they can be completely identified by using the energy equationwhich is a linear function of these base parameters. Based on the above analysis,the second approach is employed for the estimation of manipulator dynamicparameters in this thesis.Trajectory tracking control is one of the most common control tasks found inthe industrial robot application. In these tasks, the robot manipulator is required to4move along a given ofter highly complex trajectory. Thus, the goal of the controldesign is to get the control signal which is able to drive the robot manipulatorto follow the desired trajectory.Regarding trajectory tracking, there are many control techniques and method-ologies: independent joint control; multi-variable control; computed torque, etc.However, all these algorithms require one sort of assumption or the other, such asaccurate modelling of the manipulator dynamics, neglecting the coupling effectsbetween the various joints, and so on. If the plant under consideration is linear,time-varying and/or parameter-imprecisely known, then any significant gain inperformance for tracking the desired trajectory closely over the whole range ofmanipulator motion requires the use of adaptive control techniquesIn literature, there are mainly two categories of adaptive control methodsapplied to robotics: Model Reference Adaptive Control (MRAC), and Self-TuningRegulator (STR) based control. The block diagrams of MRAC and STR areillustrated in Figure 1.1 and Figure 1.2.ym --Pi ModelRegulator parameters Adjustmentmechanismr VRegulator u PlantFigure 1.1 Block diagram of a MRAC system5Plant parametersVRegulatorDesignParameter 1.4Estimationr YRegulatortPlantFigure 1.2 Block diagram of a STRBoth MRAC and STR have nice features to cope with parameter uncertainties.However, applying classical feedback adaptive control alone does not seem suit-able enough for tracking tasks where high speed and accurate tracking is requiredalong the entire span of the possibly highly complex yet ofter very short desiredtrajectory, due to the slow asymptotic convergence property. Repetitive controlmethodologies are suitable for tracking short duration trajectories, however, thetrajectories to be tracked are required to be invariant from repetition to repetition.Most recently, Ma [6, 7] proposed a new rapid tracking algorithm, which isiterative in nature, for tracking short complex trajectories precisely, rapidly and toaccommodate the inevitable saturation limits on the actuator. In his algorithm, theentire tracking control sequence q is modified one element at a time, in the orderq0 ,q1 ,...,qN- 1 , where the subscript is the time index. Each time an elementis modified, the resulting tracking error energy for the rest of the trajectory isminimized. So by the time qk is modified, the error element ek in the trackingNerror sequence e = fek l i has been minimized k times. Thus, error sequence e6will be comparatively large at the beginning, then becomes smaller and smalleras time elapses. In order to accommodate the possibly slowly time-varying andimprecisely known dynamics of the plant, an efficient adaptive feedback controlis incorporated with the basic rapid tracking algorithm. Figure 1.3 illustrates thestructure of this adaptive rapid tracking methodology.n •^ Tracking controlgeneratorr +►• - e ^ v^+ u ^^I Controller Plant ^ yFigure 1.3 Block diagram of the tracking systemwhere q is the supplemental tracking control signal and h is the impulseresponse sequence from q to y, which is identified every tracking trial.There are some obvious advantages of this iterative control method overthe existing methods mentioned earlier. Besides the tracking precision andspeed, it allows easy bounding of the actuation signal magnitude with minimumcompromise on the tracking performance and thus mitigates the actuator saturationproblem; the trajectory to be tracked need not be precisely repetitive from onetrial to another; the convergence of the tracking (learning) algorithm has beentheoretically proven; the stability of the closed-loop tracking system is ensured aslong as the reference trajectory to be tracked stays complex as assumed. Anothergood feature of this method is that the supplemental tracking control signal tends7qto converge to very small value (if not zero) before the end of each tracking trial.This is particularly desirable so that the plant needs not be driven to the finalstill position dynamically at the end of each trial. This implies that the trackingtask needs not be precisely repetitive. The idle periods between consecutive trialscan be non-uniform.Though theoretical analysis has given this method much prominence, it hasnever been implemented or simulated on any type of manipulator available, thisthus becomes the main motivation of this thesis.The rest of this thesis are organized as follows: The modelling of the 3 DOFRPDDM is studied in Chapter two. Chapter three deals with the manipulatordynamic parameter identification. Chapters four and five discuss the rapid trackingalgorithm and adaptive rapid tracking methodology. Simulation results andconcluding remarks are given in Chapter six.8Chapter 2 Modelling of the 3 DOF RPDDMThe objective of this chapter is to obtain the dynamic equations of motion ofthe RPDDM system. This includes two steps: The kinematics and the dynamics.Each step will be discussed in detail in a separate section.2.1 KinematicsRobot manipulator kinematics deals with the analytical study of the geometryof motion of a robot manipulator with respect to a fixed reference coordinateframe (base coordinate frame) as a function of time without regard to thetorques/moments that cause the motion. It is essential for deriving the dynamicequations of motion of the manipulator.2.1.1 Introduction The links of a robot manipulator may rotate and/or translatewith respect to a reference coordinate frame, the total spatial displacement of theend-effector is due to the angular rotations and linear translations of the links.To describe this translational and rotational displacement of the end-effector withrespect to the base frame, Denavit and Hartenberg [1955] proposed a matrixmethod of systematically establishing a coordinate system (body-attached frame)to each link of an articulated chain. The Denavit-Hartenberg representation resultsin a 4 x 4 homogeneous transformation matrix representing each link's coordinatesystem at the joint with respect to the previous link's coordinate system. Thus,through sequential transformations, the end-effector expressed in the body attached9frame can be transformed and expressed in the base coordinate frame which isthe inertial frame of the dynamic system.For a n DOF manipulator, there will be n+1 coordinate frames: one for eachlink plus the base frame (defined as the 0th coordinate frame). The ith coordinateframe corresponds to joint i+1 and is fixed in link i. When the joint actuatoractivates joint i, link i will move with respect to link i-1. Since the ith coordinatesystem is fixed in link i, it moves together with link i. Thus, the nth coordinateframe moves with the hand (link n).The common rules to establish a coordinate frame are: 1) The zi_ i axis liesalong the axis of motion of the ith joint; 2) The xi axis is normal to the zi_i axis,and pointing away from it; 3) The yi axis completes the right-handed coordinatesystem as required.To determine the transformation relationship between two adjacent links,the following four basic homogeneous matrices are usually used, among whichare three basic homogeneous rotation matrices and one basic homogeneoustranslation matrix1 0^0 0 - cq 0 scb^00 ca^—sa 0 0 1 0^0Tx,a = 0 sa^ca 0 Ty ,4, = —30 0 cq^00 0^0 1_ 0 0 0^1 (2.1)c9 —.50^0 0 1 0 0 dx -sO cO^0 0 0 1 0 dyTz,9 =0 0^1 0 Ttran 0 0 1 dz0 0^0 1_ 0 0 0 110where x, y, z are the three principal axes of the reference coordinate frame,a, 0, 0 are the angles rotated by the body attached frame about x, y, z respectively,and dx, dy, dz are the coordinates of the origin of the body attached frame withrespect to the reference coordinate frame.These basic matrices represent the basic transformation (either rotation ortranslation) of the body attached frame with respect to previous frame. Morecomplex transformation between any two adjacent links can be interpreted as asequence of these basic transformations. Thus, the transformation TiH, whichspecifies the location of the ith coordinate frame with respect to the (i-1)thcoordinate frame, is the product of a sequence of the basic homogeneous rotation-translation matrices.Once all the homogeneous transformation matrices describing the transforma-tion relationships between adjacent links are available, the homogeneous trans-formation matrix Tr , which expresses the end-effector in the base frame, is thechain product of successive coordinate transformation matrices of 71 -1 , and isexpressed asTic). T1T2 • • •^ H 1 -1 for i = 1,2, • • • ,n. (2.2)1 1y4czllink 3y3cY301y0x0I lcy5clink 515c x5clink 4lclink 1Figure 2.1 Mechanical configuration of RPDDM2.1.2 Kinematics of the RPDDM The mechanical configuration of RPDDM isshown in figure 2.1, where the lengths of links one through five are 1 1 , /2 , /3, /4 and15 respectively, and / 1,, -2c, 13 c , /4c , 15c denote the lengths from the origins of thebody-attached frames of the respective links to the origins of their previous frames.According to this configuration and the set up of coordinate frames, we havedifferent homogeneous transformation matrices as follows12TP =^c l^0^s l0^1^0^—sl^0^el0/10 TP, =el^0^s l0^1^0—sl^0^cl100^0^0 1 0^0^0 1c2^—s2^0 12c2 c2^—s2^0 l2c c2= 32^c2^0 1232 T l s2^c2^0 12020^0^1 0 2c 0^0^1 00^0^0 1 0^0^0 1c3^—s3^0 13 c3 c3^—s3^0 13,c3= s3^c3^00^0^113530 T31c =s3^c3^00^0^113,3300^0^0 1 0^0^0 1c32^s32 0^14,c32 - —c32 s32^0 15,c32 -T:c = —s32^c320^00^—14,3321^0 2T5c =—s320—c32^00^14—,33200^0 0^1 0 0^0 1where Ti is the homogeneous transformation matrix which transforms coordinatesin frame i into the ones in frame j, and si = sin 0i, el = cos Oi, s32=sin(03 — 92),c32=cos(03 — 02).Assume that the coordinates of mass centers of the links (one through five)expressed in the corresponding body attached frames, are respectivelyxlc x2c X3c x4c x5cYlc Y2c Y3c Y4c Y5czlc Z2c Z3c Z4c Z5c1^1^1^1^1_Coordinates of the mass centers of the links w.r.t. the base frame Forsimplicity, yi,, x2,, x3,, x4,, x5, can be assumed to be zeros, because the bodyattached frames can always be translated along yi, x2, x3, x4, x5 axes to satisfythis. Then, the coordinates of the mass centers of the links expressed in the base13= TPT1,0Y2cZ2c1-'2c2c0Z2c1Y3cZ3 c10 — 13 ,c1c3 — y3,c1 s3 + z3,31= TPT 1, 13,33 + Y3cc3 +—13 ,31c3 + y3,3133 + z3,c11o3 c0Y3cz30,1(0th) frame can be obtained as follows — 713,x1cc1 + z lcsl1 1c—x1,s1 + zl ccl1Zl c1l2,c1c2 — y2,c1 s2 + z2,3112 cS 2 + Y 2 c C 2 + 11—12,s1c2 + y2,3132 + z2,c11,0 -"4c,0Y4c0Z4c10 — 14cC1C2 — y4 cC132 + 13C1C3 Z4 cS1 -^Y4c^1402 + y4c C2 + 1333 + 11^Z4c^—i4c31C2 + y4cSiS2 — 1331C3 Z4cC11 1= 71 T3712,—15,c1c3 + y5cc1s3 + 12 c1c2 + zusl -—1503 — y5,c3 + 1232 + 111501c3 — y50133 — 1231c2 + z5,c11Velocities of the mass centers of the links w.r.t. the base frame The velocitiesof the mass centers of the links w.r.t. the base coordinate frame can be easilyobtained by taking the first derivatives of the position vectors of these mass centerswith respect to time, and they can be written in the form of the product of theJacobian matrix and the first derivative of the generalized coordinate vector. Thatis- X °5cY5c —0Y5cZ5c1Z 05c114v°x 1 V x° 2Vy i =Jvl 742 =.42e0vzl 0Vz2V z° 3 z° 4 vx°5V °Y3 =434)^[VV y° =44e v °Y5 =J1,5OV °z3 0V z4 0vz5where—xics 1 + z l ccl 0 0Jv1 = 0 0 0— X I ca — zl cs l 0 00=[0i0203Y20 1,52 — 12 cS1C2 Z2eCiJv2 =^0Y2cC 1 S 2 — 12cC1C2 — Z2c ..91Y3c 81 S3 — 13 cS1C3 Z3cCiJv3 7-7^0Y3cC 1 S3 — 13 cCiC3 — Z3 cSi—Y2cc1c2 — 12cc1s2 0—y202 + 12 ce2^0Y2cs1c2 + 120182 00 —y3 c c1c3 — 13 cc1s30^—y303 + 13cc30 y3 cslc3 + 1301s3Y40132 — 14 c S1C2 — 13S1C3 Z4 cCi=^ 04cC1S2 — 14cC1C2 — 13C1C3 — Z4 cSi— Y4cC1 C2 — 14c Ci S2— Y4cS 2 14c C2Y401 c2 + 140 182—/3c1s31/3c3/3s1s3[ 15cs1c3 — y501s3 — 1231C2 + z5 cc1 —12c1s2/2c2/2s1s2Y5c c1 c3 + l5,c1s3Y503 — 15eC3— Y5cS C3 — 150 133.45 =^ 015c C1C3 — y5 cC1S3 — 12CiC2 — Z5 c S1Angular velocities of mass centers w.r.t. the body attached frames To derivethe rotational kinetic energy of the RPDDM, we need to know the angularvelocities of all the mass centers of the links. There are two kinds of framesin which these angular velocities can be calculated: the base frame and the body15attached frame. Here, we choose the latter one. The reason for this will beexplained in the next section.One way to obtain the angular velocities of the mass centers expressed inthe body attached frame is to obtain the angular velocities of the mass centersexpressed in the base frame first, and then transform them to the correspondingbody attached frames respectively by using the homogeneous transformationmatrices.It is clear that the angular velocities of the mass centers of the links of theRPDDM with respect to their previous frames are respectivelywi c001w2c =02, , 1w3c =3W4c0022W5c[03 03Now, transform these angular velocities to the base frame by using propertransformation matrices and we have04c = el[002 .51‘,2c = C44) + TPW1 c = 4, + TPWL = el02 Ci16[ 3.914, = co? + TPwic = (.4c + TPwi c = Oi03 cl(02 + O3) Si0 ^TO( 1 T1 3W4c = W1 + 1 kW3 + 3 W4c^el.(02 + 03) Cl(e2 e3) S 1=^Ti (W2 +nut) =^. el.(02 + 03) ClWhen expressed in their own body attached frames, these angular velocitiesbecome[ o,,,l .Tjcw?,, (Tpc )T4c . a,oW2c2c ncw ric^__nc) Tu4c = (7.1) T (Tio ) T wiL == [O1s2C202W3c3c = TO ^(T3c)T4c = (Tic) T (TP)^ei 33T4c = el C303el S2,,4c^To 0 1 3w4c = 0 W4c^( T3 T4c ) G.) 40^c2e2 e3—0133 -I5c^5c 0w5c = 0 U)5c = (71971 7152cCs) T wrr5c = —BiC392 e317These angular velocities can also be written in the form of the product ofthe Jacobian matrix and the first derivative of the generalized coordinate vector.That islc j 0^,,2c^j eWlc lcwhereW3c =JW3c e^cot =JW4c 0 cot = J,,410 0 s2 0j,,,lc = 1[00 0 JW2c = c2 00100 0 0 0 1 0s3 0 0 s2 0 0 —s3 0Jw3c = c3 0 0 J 4c = c2 0 0 J 5c = —c3 00103c0 0 1 0 1 1 0 1 12.2 DynamicsRobot manipulator dynamics deals with the mathematical formulations ofthe equations of manipulator motion. These equations of motion describe thedynamic behavior of the manipulator. There are many ways of formulating thedynamic equations of motion of a robot manipulator. In this section, we willderive the dynamic equations of motion of the RPDDM system using Lagrange-Euler formulation, and based on the kinematics analysis made in the last section.2.2.1 Introduction Given a conservative n DOF manipulator, its equations ofmotion can be derived from the Lagrange-Euler equations:18d a L aL _dt aqi TiHere L=K-V is the Lagrangian function, where K is the total kinetic energyof the robot manipulator, V is its total potential energy; qi is the generalizedcoordinates of the robot manipulator; Ti is the generalized torque applied at jointi to drive the ith linkFor a rigid body in motion, it is known that the total kinetic energy consists oftwo parts: One is the rotational kinetic energy, and the other is the translationalkinetic energy. That is1^1K = 2—mvT 7), —wT2(2.6)where m is the mass of the rigid body; vc is the velocity of the mass centerof the rigid body with respect to the base frame; w is the angular velocity of themass center of the rigid body; and I is the inertia matrix about the mass center.Obviously, in different frames, w takes different values, and so does I. But thetriple product 27(.0 is the same in all frames [10]. We choose the body attachedframe to compute I, because in this case, I is independent of the motion of thebody. This is why all the angular velocities of the links of the RPDDM werecalculated in the body attached frames in the previous section.(2.5)19For a manipulator having n links, the velocity of each link expressed in thebase frame and the angular velocity of each link expressed in the body attachedframe can be written systematically in the form of the product of the Jacobianmatrix and the first derivative of the generalized coordinate vector q, i.e. q. That isv2, = 4, c (q)4(2.7)iswic =^(q)4Suppose now the mass of link i is rni, and the inertia matrix of link i, evaluatedaround a coordinate frame parallel to frame i but whose origin is at the center ofthe mass of link i, is I. Then, the overall kinetic energy of the manipulator isn^U1^T^+ ± x—•1/4 ( L, 4 ) T 1 iL,.4i=1K = — E mivicvic -i- 7 t2 2 --e 'i=1n1 TT (1 ri j wic (—\];.= —2 q•T \---` [„,„ . TT (,\ j je (q) + J u.,: q /^tc q) q' 16 :'-' vic k`il vi=1This can be re-written in a more compact form as1K = —2 E di ,i (q)qiq—2qi D(q)qNow, consider the potential energy term. In case of a rigid manipulator,the only source of potential energy is gravity. Let g denote the gravity vectorexpressed in the base frame, rti)c denote the three-dimensional coordinate vector(2.8)(2.9)20of the center of mass of link i expressed in base frame. Then the total potentialenergy isV (q) gT E r° m z^ (2.10)i= 1Equations of motion of a general multi-body  Based on the results of theabove analysis, we have the Lagrangian asL = K — V2^j(04i4i — v(q )^ (2.11)andor,dk3 (041aqkd a L= > dki (mi + E Tfidk j(q)4j" jadk •E dki (04i + Ear,^av(q) aqk aqkThus the Lagrange-Euler equations can be written as(2.12)21Edki(04.;+^{adkj ladij }. . avaqi^2 aim qi qi^= TIC^k =1,- • • ,n^(2.13)By interchanging the order of summation and taking advantage of the sym-metric nature of D(q), it can be shown [10] thatE ado l 4i4.^{ado + adki} 4.4.aqi^2^aqiHence^ (2.14)ado^1 adij ) . .aqi^Vgiqi =If we defineadkj + adki^.giq •aqi^aqj^aqk^3i,j1 {ado adk i adii }cijk —^aqi^aqi^aqk(2.15)aV(q) gk —agkthen the Lagrange-Euler equations becomeE dki(mi + E ciik(o4i4i + gk (q) = Tk k = 1,• • • ,n^(2.16)The first term of the left-hand side of the above equation is related to theacceleration of the joint variables. When j = k, dk3 is related to the acceleration22f Odki Odkiaqi^Oqi^aqk(2.18)co =^ciik(q)4i =1= 1 i=1of joint j where the driving torque Ti acts, when j^k, dkj is related to thereaction torque induced by the acceleration of joint k and acting at joint j, orvice versa. The second term consists of centrifugal terms (when i = j) andCoriolis terms (when i j). The third term represents the gravity loading termsdue to the links.Equation (2.16) can be written in matrix form asD(q)q C(q,q)q G(q) = T^ (2.17)where the k, jth element of the matrix C(q, 4) is defined as:2.2.2 Dynamic equations of motion of the RPDDM In section 2.1, we havederived the velocity of each mass center of the links expressed in the base frame,and the angular velocity of each link expressed in its own body attached frame.Suppose that the inertia matrix of link i is /i, and it is evaluated around a coordinateframe parallel to frame i but whose origin is at the center of mass. Noting thatIizz^-hxy --11XzIi =^hyy —hyz^(2.19)Iizzand making use of the Jacobian matrices derived in the last section, then the totalkinetic energy of the RPDDM is235^K -LOT E [migc jvi^,c]c wici=1= 2—1OTD(0)ewhere D(0) is a 3x3 matrix. If we define the inertial parameters as(2.20)pI = [P1 P2 • • • Pz • • • P24] T^(2.21)whereP1 = 2 [m1(xic ZL) rn2 (YL z3c)+ 7113 (Yic zic)+ M4 (Y ,ic zic)+ m5 (ygc zgc)I1 y hxx i3xx i4xx I5xx]P2 = 771 2Y3c —^moic- mvic- rri5 13 -1-2xx I4xx — 1-2yy — Llyy k2P3 = MAC- rn3 13c MAC- 711 5 15c — M4 13 + 13xx I5xx — hyy — 15yy — k3P4 = M2Y2c 12c+ ni4Y4c 14c hxy LlxyP5 = M3Y3c 13c+ M5Y5c 15c I3xy hxyP6 = 77134+ 717,313c + 77l5y5c + in51gc -1- 77143 + i3zz I4zz+ 15zzP7 = n12Y3c+ ni2 11c M4Y4c+ rn4lic+ 712512 1T2zz I4zz+ I5zzP8 = n1 2 12cZ2c+ 771 4 14cZ4c+ rn5 12Z5c I2xz I4xzP9 = in515c Z5 c — 712313 c .Z3 c — M413Z4c hxz — hxzP10 = M2Y2cZ2c+ M4Y4cZ4c hyzP11 = rn5Y5cZ5c — m3y3cz3c hyz hyz24P12 = 771 5 12 15c — 771 4 13 14cP13 = n1 5 12Y5c + M4 13Y4cP14 = M5 12Y5c — m4 13y4cP15 = -1-4xzP16 = hxzP17 = -14yzP18 = I5yzP19 = I4zz + 1-5zzP20 = 7n2 12c + M4 14c + m5 12P21 = m2y2c + rn4Y4cP22 = in3 13c + rn413 — rn515cP23 = M5Y5c — 7113Y3cP24 = k23where k2 , k3 and k23 are three constants and they construct the moment ofinertia of the manipulator about the rotating axis (y o ) of the base frame in the way'lyy = k2 COS2 02 + k3 COS 2 03 + k23 COS 92 COS 03^(2.22)Then, according to equation (2.20), the elements of D(0) will be respectively25P2 cos (202) d11 =^P4 Sill (202) -F(P24 — Pi2) COS (02 + 03)2^ 2P24+P14 sin (02+03) +I —2 — P12) cos (03 — 02)+P13 sin (03 — 02)P sin 20 P3cos (203) + (P1 — P2 — P3) — 5^( 3) 2^^2d12 = d21 = — 1310cos02 — P8sin612 + P18COS03 + P16sin03d13 = d31 = —P1 7cos0 2 — P1 5 sin02 + P11 cos03 + P9 sin03d22 = P7d23 = d32 = —Pi2cos(03 — 02) + Pi3sin(03 — 02) + P19d33 = P6According to definitions (2.15) and (2.18), we have all the elements in C(0, e)matrix as follows:clii = 0c211 = [P2 sin (202) — 2P4 cos (202) + P14 cos (02 + 03) — P13 COS (03 — 02)—^(P24 — P12) Sill (02 + 03) + (P24 — Pi2) sin (03 — 02 )1/22 2c311 = [P14 cos (02 + 03) + P13 COS (03 — 02) — 2P5 COS (203) + P3 sin (203)— (P224^ 2— P12) sin (02 + 03) — ( P24 — Pi2) sin (03 — 02)]/226C121 = C211C221 = P10 sin 02 — P8 cos 02C321 = (PrS11102—pmcose2+P16c003—P18sin03)/2C131 = C311C231 = C321C331 = P9 COS 03 — P11 Sill 03C112 = — C211C212 = 0c312 =(Pm cos02—Pr sin02+Pm cos03—Pi8 sin 03 )/2C122 = 0C222 = 0C322 = 0C132 = C312C232 = 0C332 = P12 sin (03 — 02) + P13 Cos (03 — 02)C113 = — C311C213 = — C312C313 = 027andc(0,e)^c211e2+0311e3= -c2110. 1[ +c31203—C311 01 —0312 020123 = — C312C223 = — c332C323 = 0C133 = 0C233 = 0C333 = 0c2i1e1+C221e2-FC321e30—c312e1 —0332e2C311e1+C321e2+ 0331e3c312 01 +C332 030The total potential energy of the system is5V = gT E mir0 ici=i(2.24)and thus according to equation (2.15), we haveavg1= ao i = °avg2 =^= Fog cos 02 — P21g sin 02ao2av ng3 = ao3 = 1-22g cos 03 + P239 sin 03Finally, with D(0), C(0,0 and G(0) available, and assume that the appliedtorques at the three independent joints are T = [Ti T2 Td then the dynamic28(2.25)equations of motion of the RPDDM areD(0)0 + C (0,00 + G(0) = T^ (2.26)or411 01 d^d13e3 +^C12e2 + C13 e3^=d12151 + d22 O2^23 + C21 e1 + C22 e2 C23^g2^T2^(2.27)d13O1 d23O2 d33O3 + C31 e1 C32O2 + C33 O3 g3 = T3Taking into account the viscous frictions at the three independent joints, thedynamic equations of motion of the RPDDM system becomed1 1 Ol d12 82 d13b3 + C11 O1 + C12 e2 C13e3 µ1e1 + gi =d12b1 + d22O2 + d23e3 + C21 e1 + C22 e2 + C23 03 fij2 g2 T2^(2.28)d13 + d23 e2 d33 O3 C31e1 + C32 e2 + C33 e3 /IA g3 T3where [it1^it3] = [P25 P26 F27] = Piz are the coefficients of viscous frictionat the independent joints.29Chapter 3 Dynamic Parameter IdentificationThe dynamic model of a robot manipulator is always inexact since 1) not allinertial parameters are precisely known; 2) the frictional forces of manipulatorjoints are usually unknown; 3) the inertial parameters of the load are not usuallyknown. These parameters are known as the manipulator dynamic parameters.Inaccuracy in their estimates will degrade the performance of the manipulatorcontroller and thus precise tracking of the desired trajectory can not be achieved.Therefore, a proper method is needed for accurately identifying these parameters.As mentioned in Chapter 1, the identification method used in this thesis isbased on the energy conservation theorem, which states that the work done bythe manipulator actuators minus the energy dissipated by the frictional forcesover any time period is equal to the change in the kinetic and potential energyof the manipulator over the same time period. The advantages of this methodare the ease of implementation, applicability to both serial-linkage manipulatorsand graph structured manipulators and computational efficiency (because it doesnot require the measurement or computation of joint accelerations as some of theother methods do).3.1 The Estimation Equation set upFigure 3.1 is the block diagram of dynamic parameter identification. Thelower rectangle contains a set of identification equations which will be derivedbelow based on energy conservation theorem.30Robot Dynamics 0, Dynamic ParameterIdentificationFigure 3.1 Basic parameter identification structureTo derive the identification equations, let's first define the total energy of ann—link manipulator system asH(q,4)= K(q,q)+V(q)^ (3.1)where q^[q i q2 • • • qn ]T is the generalized coordinate vector containingall independent joint variables, and K, V are the manipulator kinetic energy andpotential energy, respectively. It is shown in [8] thatdH Vdt = 2_, ri4iwhere^is the non-conservative force at joint i, and it can be modelled asra = Ti = Ti - ILA^(3.3)where Ti and fi are the input joint torque and frictional force at joint i respectively.ft, is the constant frictional coefficient of joint i. If we integrate equation (3.2)from to to t, it becomes(3.2)i= 131f dtdH—dt =Ef ri4i dt1=1to^toorH (t) — H (t o ) = E I ri4idti=iSince the total energy H is linear in the inertial parameters [5], it can bewritten asH(q,4, 52) = ET (q,4)12^ (3.6)where E(q, 4) is a non-linear function of the joint positions and velocities.St E R113xn is the inertial parameter vector (IPV), which contains all the basicinertial parameters of the manipulator, i.e.Q = [ Q1 Q2 •^f2n ]T^(3.7)and= [m, mix, mo i m,z, 1,xx Ityy Itzz Iixy izyz Itxz ]T (3.8)Usually, only some of the parameters in equation (3.8) will appear in thedynamic equations of motion (this depends on if or not (a9 ipc, ) 2 +^) 2 = 0 ,(3.4)(3.5)32where K and V are the total kinetic and potential energy of the system. BecauseK and V are both linear in inertial parameters, i.e. K =^MPi, V = Etherefore if ( apR: ) 2 -Li av \ 2 = 0, then the change in parameter Pi can not cause anychange in K, nor in V. Thus , Pi has no effect on the dynamic equation of motion= alfq = 1 • • • n), and some appear in the form of linearcombination (regrouped parameters) of the basic parameters. The parametersaffecting the dynamic model separately and the regrouped parameters constitutethe base parameters required for control, and they are the only parameters neededto be identified. To this end, we rewrite equation (3.6) in terms of the baseparameter vector PI asH (q, 4, PI) = FT (q, OPT^(3.9)If we substitute equation (3.3) and (3.9) into equation (3.5) and defineF(ti) = F(q(ti), (t i )) , we will obtain the following linear equation of theunknown base parameter vector Pi and frictional coefficient vector Pm[F(t) — F(to)1T^E I dt = E I Ti 4idti=1^ to(3.10)By defining function W(to, t) and Ji(to, t) as33F(t)— F(to)14)(to,t)=[^J(to,t)o =(3.15)W(to, t) = E f Ti4idti=i to(3.11)Ji(to,t)= I 4Nttoequation (3.10) becomes[F(t) — F(to)FPI+J(to,t) T13,=W(to,t)^(3.12)whereJ(to, t) = [ (to , t) J2(to , t) • • • AN, OF= [P1^it2^• • • lin ] TFinally, we obtain the equation used for identification as4) (to , oTo = w (to , t)where(3.13)(3.14)Equation (3.14) is a linear function of the unknown 0 (DPV—dynamic pa-rameter vector) which can be identified by applying a recursive identificationalgorithm, such as RLS.343.2 Dynamic parameter identification of the RPDDMFor RPDDM, the generalized coordinates are 0 = [0 1 02 031T . Accordingto the definitions made in the last section, the total energy of the RPDDM canbe written asH (0, O, PI) = FT (0, O PI^ (3.16)where vector Pi have been previously defined asPI = [ P1 P2 P3 P4 P5 P6 P7 P8P9 P10 P11 P12 P13 P14 P15 P16^(3.17)P17 P18 P19 P20 P21 P22 P23 PA TFT is a row vector and for RPDDM with given configuration, its elements aree2F(1) = , -.-iF(2) —^1e2 COS 2 (02) 2F(3) —^1e2c os2 (03)29i ^(202) F(4) =^1 292 sin (203 )F(5) =^1 2 92F(6) = 13592F(7) . -12F(8) = — e1e2 sin 02F(9) = 0103 sin 03F(10) = — OA COS 02F(11) = 01e3 COS 03F(12) = Of cos (03 — 02)F(13) = e? sin (03 — 02) +F(14) = elf sin (03 + 02) 2F(15) = —01O3 sin 02F(16) = 0102 sin 03F(17) = —e1e3 COS 02F(18) = e102 cos 03F(19) := 0203F(20) = g sin 02F(21) = g cos 02F(22) = g sin 03F(23) = —g cos 03F(24)= iq, cos 02 cos 032+ q COS (03 + 02) + 202O3 cos (03 — 02)220203 sin (03 — 02)236Also according to earlier definitions, we have= [11 1 112 [1 3] T^(3.19)3^t2W(11,12) , E TAdt^(3.20)i=1Ati,t2) = ft2 qdt ft2 qdt It2 eidtti^ti^tiiT[ Finally, the equation for identification of dynamic parameters can be obtained asW(ti, t2) = (Ht2 - HO + JT (tl, t2)Pit= [F(12) — F(ti)JT PI + JT (ti,t2)Pit^(3.22)= (DT (ti,t2)Pwhere(DT (1 1 , 12) = [[F(12) — F(ii)}T^JT (ti, 12)]Pi (3.23)P =Note that in equation (3.22), the term JT(ti, t2)131, represents the energy absorbedby the viscous fricions at the three indenpendent joints.By applying the RLS identification to equation (3.22), parameters P can becompletely identified.(3.21)37Chapter 4 Basic Rapid TrackingControl StrategyIn the previous chapters, we have studied the issues of modelling and dynamicparameter identification. In the following two chapters, we will discuss the rapidtracking control methodology and adaptive rapid tracking control strategy.4.1 IntroductionIndustrial robot manipulators are often required to follow a given trajectoryclosely, such as picking, placing and pelletizing, etc. For some of these problems,only the final position and/or orientation of the end-effector is of interest and thein-motion tracking performance is of little value. For these problems, the timingof the motion of the manipulator can be traded off for better tracking accuracyif necessary. However, if a task requires the manipulator to track the referencetrajectory precisely over the entire most likely short trajectory , and also the timingover the tracking task is important, then the ability to track the reference trajectoryrapidly and precisely by the manipulator end-effector becomes highly desirable.Given a linear time-invariant plant P and an arbitrary reference trajectory rwith transfer function R, the standard feedback control system structure is shownin Figure 4.138 RFigure 4.1 Standard feedback control systemIt is known [3] that for such a control system, unless the open loop transferfunction, PC, is divisible by the largest invariant factor of the denominator of R,it can not track reference trajectory r, not even asymptotically. If the referencetrajectory is complex so that R is of high order, the controller C would have tobe of high order as well in order to achieve just asymptotic tracking. Therefore,in order to track a possibly complex trajectory rapidly and precisely, a differentcontrol methodology must be applied.Most recently, Ma [6, 7] proposed a new tracking algorithm, namely theq-learning algorithm. The control system structure is shown in 4.2, where qrepresents the tracking control signal, and is to be generated through q-learningalgorithm, C is the controller which stabilizes the linear, time-invariant plant Pthat may not be stable. In his method, the supplemental tracking control signal qis modified one element at a time, each time minimizing the tracking error energyfor the remainder of the short-duration and complex trajectory. In this way,the resulting tracking error energy is minimized each time when an individualelement in q is modified. When all elements in q have been modified, the next`learning' cycle can be started again from the very beginning and so on untiltolerable tracking error is obtained. This results in a supplemental tracking control39generation (q-learning) methodology which would normally allow a particularvariation to take place on an element of q only if such variation will lead to themaximum possible decrease in the resulting tracking error energy. Should suchvariation result in the particular control value exceeding a possible bound set onq, the variation can be simply limited to meet the bound, resulting in suboptimalminimization of the tracking error energy. Then the learning process can beallowed to continue, if more reduction in the error energy is desired, until nextactual tracking task must begin.e, q, h^iPiFigure 4.2 Tracking control system structure4.2 LinearizationBefore the rapid tracking algorithm can be applied, the non-linear manipulatormodel must be linearized. For this purpose, computed torque technique is used.The idea is to equal the desired torques calculated from the real model, whichis D(0)9 + C (0,0)0 + G(0) = T, to the computed torques calculated from theestimated model, which is b(0)u+0 (0, 0O+ = T, where u is the commandinput. and represents the identification. In this way, if the parameter estimatesare close enough to the true ones, by equaling the above two equations, we willr^e  ^V1 TrackingSignalGenerator40_ilt Dynamic parameteridentificationDynamic parameterse, 6^1,1Computed torqueARobot dynamicste, 6, 6end up withu P::: ã^ (4.1)which approximately is a double integrator, and thus the non-linear manipulatorsystem is approximately linearized. Figure 4.3 shows the block diagram of thecomputed torque and the resulting approximately linearized plant.approximately linearized plant -.:=double integratoruFigure 4.3 Block diagram of computed torque4.3 q—learning algorithm IThe q-learning algorithm is derived as follows. Suppose 1 cycles of learninghave been made on the supplemental tracking control signal q, and we are justabout to modify the kth element of q in the (1+1)th learning cycle to minimizethe tracking error further. For the purpose of formulating an optimal modificationalgorithm, let's first define the following notations:41q l,k-1 : the latest computed desired supplemental tracking control signal, i.e.the control signal sequence obtained after the (k-1)th element of q in the (l+1)thlearning cycle has been modified;,k-i.qk^. the kth element of the vector q l,k-1 ;A q l,k^y l,kqki ' k : the kth element of q l,k , i.e. the modification to be made on the kthelement of q during the (/+1)th learning cycle;e l,k-1 : the tracking error which would be obtained if q1,k-1 were appliedto the tracking system;e l,k^e l,k-1 —With the notations defined, one can obviously draw thatqk = qk^aqk1,k^1,k-1. 1,k^ (4.2)Referring to Figure 4.2, and assuming that the pulse transfer functions fromq to y and from r to y are H and Hr respectively, then, in terms of transferfunctions, the output y isy = Hq + lir r^ (4.3)while in the time domain, equation (4.3) can be written in the convolution formas (if q1,1c-1 has been applied as the tracking control)y l,k-1 = h 0 47 1,k-1 + hr 0 r^ (4.4)42where h = [h i , h2, • • • ,11A] T is the impulse response sequence at output y due to aunit pulse applied to input q at time k=0. N is the duration of a trial in a normallyrepetitive tracking process. The tracking error corresponding to equation (4.4) islk-1e '^= r- y l ' ic-1(4.5)= r-h 0 q l,k-1 ___ hr orNow, if the modification on the kth element of q has been made, then wewill haveq l,k = q l,k-1 + A y l,k^ (4.6)and if q l,k were applied to the control system instead of q 1 A -1 , we would gety l,k _ _ _ h 0q1,k + hr orh 0 (71,k-1 + Aq i,k) + hr 0 r^(4.7)=h0 ,71,k-i + h 0 Aq l,k + hr or= y 1,k-1 + h 0 Aq l,kwhich yieldsSincee l,k = r _ y l,k= r _ y l,k-1 — h 0 A q l,k=e1,k-1 _ h 0 Aq i,kAq l,k = q l,k _ q l,k-1=L:;,2, Aqk ' k , fyL,..v_juu•••,O Tk^N—k-1(4.8)(4.9)43SOh 0 Aq l,k^0,- ,O, hi Aqk,k h20qk,k^h N qk,ka^o,o,•••oTftEN1•1■11■11011■1=11■,k^N N—k-1 (4.10)=k^N^N—k-1Assuming that the plant will be stopped from responding after a tracking trialuntil the next trial begins (i.e. by applying brakes or turning off the motors,etc.), then only the first N elements in equation (4.10) are of interest to us, thush Aql ,k can be expressed ash Aq l,k 0, • • ,O, hi, h2, • • • ,hN_kjTk^N—k^ (4.11)Aqk,kh kwhere h k =^hi, h 2 , • , hN_k] T. Therefore equation (4.8) can be writtenk^N—kas1 k^1 k-1^i,khke^e^qk (4.12)If Aqki ' k were to minimize the error energy ji,k =< e l,k ,61,k >, it must satisfydjl,kik = 0 = _ 2h k^_ Aq 1k,k hk)dAqk'This yields the optimal k-th modification(4.13)1 k^< hk ,e1,k-1 >Aqk < hk , hk > k = 0,1,•••,N— 1^(4.14)Note that the unique solution in equation (4.14) is a minimum (rather than amaximum) point. This can be easily checked by taking the second derivative of44J1,k with respect to Aqk' k . Equations (4.2), (4.12) and (4.14) constitute the basicrapid tracking control algorithm, which from here on is referred to as q-learningalgorithm I. With q l,k and ei , k now available, the process can go on to calculatethe next optimal modification Aqk'±+/ / (to yield q1,k+1 and e l,k+1\) until all Nelements of q have been modified. Then a new learning cycle can be started bycalculating qi+1,1 , • • • and so on.The above learning algorithm has been proven to converge [6] with J-1, kapproaching zero and q l,k approaching the corresponding optimal control q°Pt , ifthe learning cycle is allowed to go on forever.4.4 SummaryIn this chapter, the basic rapid tracking algorithm, namely q-learning algorithmI was studied. The algorithm, which is made up by equations (4.2), (4.12) and(4.14), is iterative in nature and theoretically converge with el , k approaching zeroand q l,k approaching q°Pt . Since the error element ek in the tracking error sequencee = fekl i gets minimized k times each modification cycle, the resulting valueof ek tends to be larger when k is small than when k is large. Hence the trackingperformance will be comparatively the worst at the beginning, then becomes betterand better as time elapses during an actual tracking trial.Because the algorithm allows easy bounding of the actuation signal magnitudewith minimum compromise on the tracking performance, it thus mitigates theactuator saturation problem. Another good feature of the algorithm is that the45tracking control signals begins with zero values and also ends with zeros (or verysmall) values at the end of each tracking trial, so that the manipulator does notneed to be driven to the original still position dynamically. This implies that thereference trajectory to be tracked needs not be truly repetitive.46Chapter 5 Adaptive Rapid Tracking StrategyIn the last chapter, we have discussed the basic rapid tracking algorithm, whichis applicable only to linear time-invariant parameter-known systems. However,in practice, all industrial processes are at least slowly time-varying, parameterunknown or imprecisely known. In such case, adaptation should be applied.5.1 Adaptive Rapid Tracking with General PlantConsider the system in the following figure where plant P is a linear plant.srGeneratoru yVeP-I 1.0I^+Figure 5.1 Tracking control system structureLet AB -1 and Y -1X denote the polynomial coprime factorizations of P andcontroller C respectively. ThenPC^PY = 1+ PC r+ 1 +POAX^AY= AX + BY r + AX + BO(5.1)IfAX+BY=1^ (5.2)TrackingSignal47theny = AXr AYq (5.3)The fact that AX and AY are both polynomials with finite orders impliesthat the system is stable. Further, because AY is a polynomial with finite orderand hence a finite impulse response from q to y, the convergence of q-learningalgorithm can be relatively faster. In another word, the controller designed willnot only stabilize the plant AB -1 if it is open-loop unstable, but also allow afinite impulse response from q to y and thus guarantee a fast convergence rateof the q-learning algorithm.Practically, as any physical plant operates, its components wear out, theoperating environment changes and the dynamics of the plant changes (most likelyslowly) with time. Thus, to maintain top performance under such situation, h,the impulse response sequence from q to y, the plant and controller parametersmust be adaptively identified.To develop an adaptive control algorithm for the q-learning control system,let 6, V denote the degrees (in z -1 ) of B, A respectively, and U be a Hurwitzpolynomial of degree v = v — 1, then there exist X1, Y1 [11] of orders— 1, v — 1 respectively such thatAXid- BY1 = U^ (5.4)48Now, if we choose X1, Y1 as the controller, then according to equation (5.1),AX1 BY1 = U becomes the closed-loop characteristic polynomial.Note that U here is known, its coefficients can be determined by the pre-specified performance requirements, such as on rise-time, overshoot, etc. Multi-plying the right side of equation (5.4) by AX + BY yieldsAX1 BY1 = U(AX + BY)^(5.5)To identify the controller, an error equation must be set up, the convergenceof which to zero would lead to the controller being identified. Towards this end,let y = AB —l u = Ax, then u = Bx. Multiplying equation (5.5) by x yieldsXly + Y1 u = U(Xy + Yu)^ (5.6)Since U is known and u, y can be measured, equation (5.6) is an equationlinear in the unknown X, Y, X1, Y1. Therefore, X, Y, X1, Y1 can be identified byapplying any reliable linear recursive identification algorithm after defining theerror E to beE XUy kUu — Xiy — friu^(5.7)where ^ denotes estimates.Before applying the rapid tracking algorithm, it is still necessary to know h,the impulse response from q to y. This can be obtained by identifying the plant49P = AB- 1 first, which can be further done by applying another identificationprocess to the error equation7,b = y — Au^ (5.8)then h can be calculated from the transfer function AY^- AY1 AX1+BY1 U •There should be no stability problem with this adaptive control system becausethe assumption of complexity of the reference trajectory will lead to persistentexcitation in the signal pair (u,y) which in turn lead to parameter convergenceand hence stability.5.2 Application related issuesBefore applying the adaptive rapid tracking algorithm (ARTA), the initial qand e or q°, e° must be available. For a parameter known time-invariant plant,this implies that one tracking trial must normally be attempted first. During thistrial, q°, e° can be actually acquired to allow the learning algorithm to start, thenrapid tracking algorithm can be invoked so that ever-more precise tracking canbe attained in the second and future trials.For a parameter imprecisely known or unknown plant, however, one more trialis normally needed before q° , e° can be obtained for use by the rapid trackingalgorithm. During the very first trial, the controller adapts to the desired one, andthe plant gets identified, but the tracking error generated would not completelyreflect that of the controlled system due to the adaptation at the start of the trial.50Hence a second trial is needed to generate the q° , e° required by the rapid trackingalgorithm for precise and rapid tracking in the third and future trials.In the time-varying case, h, the impulse response from q to y, must be eitherrecursively identified during a tracking trial or in one-shot after the trial, then beused for generating the new tracking control q for the next trial. The identificationcan be carried out exactly as pointed out earlier. Since the reference trajectory tobe tracked is assumed to be highly complex, the data acquired for identificationwill be sufficiently rich. The identification process will be uniformly stable.Therefore, the identified impulse response will be uniformly bounded as long asC continues to stabilize the plant P and the true impulse response of the slowlyvarying system is uniformly bounded. Since a uniform bound can be easily seton the tracking control q, q can also be assumed to be uniformly bounded. Hencethe entire adaptive tracking system will be uniformly stable.510uq,e,h^Rapid trackingalgorithmIv Robot dynamics(non-linear)r + e ^---11-0^►1 ControllerA AControllerdesignuLinearized plant...4.1Dynamics parameterkidentification Parameters^I-IdentificationyAController parameters ^►Identificationi■ ^ue eiComputed ttorque•■5.3 Application of the ARTA to the linearized RPDDM systemParameters of linearized plantFigure 5.2 Over-all tracking control system structureFigure 5.2 illustrates the over-all configuration of the control system whenthe adaptive rapid tracking algorithm is applied to the approximately linearizedRPDDM. The dashed box contains the supposedly linearized plant, whose param-eters are imprecisely known and need to be further identified.5.3.1 Parameter identification for the approximately linearized plant Sincethe approximately linearized plant P is supposed to be a double integrator, that is______ITz 1)u = or y =^in discrete time domain, where T is the sampling period,(i -z-i it is reasonable to assume that P takes the form ofA^bpiz-1P=—=B 1 + apiz -1 + ap2z -2(5.9)where bp1, api and ap2 are coefficients and need to be identified.52Hence we can set up the error equation (5.8) as= (1 + apiz -1 + ap2z -2 )y —which givesapl(5.10)y(k)=[—y(k —1)^—y(k —2)^u(k —1)] ap2[bpi+ 0 (5.11),f,T nBy applying RLS, parameter Op can be completely identified.5.3.2 Controller design and controller parameter identification Accordingto section 6.2 and equation (5.9), given a Hurwitz polynomial U of degreev = S + v —1 = 2 + 1 — 1 = 2, there would exist X1, Y1 of degrees S —1 =1and v — 1 = 0 respectively, such thatAX1 + BY1 = U^ (5.12)Equation (5.12) is the characteristic polynomial of the closed-loop system, whichcan be specified by specifying U given the performance specifications. Now, if atracking rise-time of 0.1 sec is required, this together with the consideration thatthe end-effector should have no overshoot, which implies a critical damping ratioof = 1, will determine U asU = fo + fiz-1 + fez-2(5.13)= 1 — 1.921578877z -1 + 0.923116346z-2Now, considering equation (5.2) and the degree assignments for X1, Y1, we canassume thatXi = be 0 +{^ (5.14)= 153and similarlyX = bo + biz -1{^(5.15)Y = 1Substituting X ,Y, X1, Y1 into equation (5.7), we have6 = Uu — (bo + biZ -1 )Uy — u — (bc0 + bciZly^(5.16)ore = (Uu — u) + (bo + biz -1 )Uy — (boo + bcizly^(5.17)By definingUu(k) — u(k) = u u (k){^(5.18)Uy(k) = yu (k)equation (5.17) becomesuu (k) = [y(k) y(k — 1) —yu (k) —yu (k — 1)]bababob1+ 6 (5.19) _ 4:1 7: Oc 4. eAgain, by applying RLS, the controller parameters b oo, bo l can be identified.To make sure that the plant's and controller's parameters are correctly iden-tified, we can check to see if the equationsAX + BY = 1^ (5.20)andAX1 + 13f71 = U^ (5.21)hold, using the identified parameters.54Chapter 6 Simulation Resultsand Conclusions6.1 Simulation ResultsFor the purpose of demonstrating the identification of the dynamic parametersof RPDDM, the true (hypothetical) dynamic parameters to be used to simulate therobot dynamics are assumed to be given as in table 6.1. Also, the initial states ofthe manipulator are chosen to be 0 = [0 27r/5 57r/6] T and 0 = [0 0 0]T .Identification of dynamic parameters are carried out using the actual workdone by the torques applied at the three independent joints. The measurements ofjoint angles, angular velocities and applied torques at the joints are incorporatedwith normally distributed random noises. The variances of the noises incorpo-rated with the applied torques and joint angles and joint angular velocities areassumed to be 4% and 1% of the peak values of the corresponding measure-ments, (i.e.[0.012;0.008;0.001] for the torques, [0.00035;0.002;0.0025] for 0 and[0.00015;0.00075;0.001] for 0). The initial dynamic parameters are assumed to bezeros. The covariance matrix for RLS identification is set to 10 71. Since the workdone by applied torques is obtained through integration, the smaller the samplingperiod, the more accurately it will be calculated. On the other hand, consideringthe practical realizability, a sampling rate of 0.001 sec. is selected.55Figure 6.1 shows the 24 dynamic parameters identified under the conditionsspecified above. Their final values are also listed in table 6.1.•^Parameters Tme value Initial value Estimated valueP(1) 0.8698 0 0.8687P(2) -0.2503 0 -0.2380P(3) -0.0226 0 -0.0254P(4) 0.0789 0 0.0792P(5) 0.0315 0 0.0334P(6) 0.0163 0 0.0164P(7) 0.1349 0 0.1321P(8) 0.0660 0 0.0649P(9) -0.0628 0 -0.0627P(10) 0.0335 0 0.0334P(11) 0.0048 0 0.0044P(12) 0.0022 0 0.0021P(13) 0.0092 0 0.0082P(14) 0.0035 0 0.0046P(15) 0.0100 0 0.0102P(16) 0.0150 0 0.0146P(17) 0.0200 0 0.0200P(18) 0.0250 0 0.0242P(19) 0.0069 0 0.0068P(20) 0.4348 0 0.4254P(21) 0.0334 0 0.0327P(22) 0.0408 0 0.0408P(23) 0.0129 0 0.0129P(24) -0.0183 0 -0.0225P(25) 0.0400 0 0.0403P(26) 0.037 0 0.0353P(27) 0.0410 0 0.0416Table 6.1 True and estimated dynamic parameters56160 180^20014010.^40^60^80^100^120-0.2-0.40-1‘27 estimated dynamic parameterstime (sec)Figure 6.1 Estimated dynamic parametersThe simulation of the computed torque is one part of the simulation of thewhole system. It is carried out using the estimated dynamic parameters of themanipulator. Because of the limitation in choosing the sampling rate, the workdone by applied torques can not be accurately calculated so that it equals to thetotal energy change of the manipulator system, thus resulting in some deviationof the estimated dynamic parameters away from the true ones. As a result, thenon-linear manipulator system can only be approximately linearized. Fortunately,this approximate linearization does not degrade the tracking performance verymuch, according to the simulation on tracking.Figure 6.2 shows the differences between the inputs and the second derivativeof output angles of the approximately linearized plant, which depict the extend57of linearization. These differences may seem large. However, the correspondingrelative differences are much smaller. This can be seen by refering to Figure 6.10.U-THETA_ddtime (sec)Figure 6.2 U — eBecause in practice, the approximately linearized plant is still parameterimprecisely-known, therefore, RLS is applied again to identify the parameters.At the same time the parameters of the controller are also identified. The initialcontroller and plant parameters and the covariance matrices for RLS identificationare shown in table 6.2. The measurements for identification are the joint angles,which are the output of the approximately linearized plant, and u, the commandinput of the same plant. Both are assumed to be incorporated with normallydistributed random noises with variances of 3% and 2% of the peak values of58the respective measurement. The covariance matrices are reset at the beginningof each trial.It should be mentioned that for the purpose of identifying the controller andplant parameters, a second-order linear model is used. Of course, higher-ordermodel can also be adopted for this purpose. When a second-order model or ahigher-order one should be used depends on how well the non-linear manipulatorsystem is linearized.Controller PlantJoint Initial parameters[bc0; bcl; b0; bl]Initial covariance matrix Initial parameters[apl; apt; bpl]Initial covariance matrix1 [1;-0.6;3;2] 1.Oe5eye(4) [0;0;0] 1.0e5eye(3)2 [0.5;-1;1.2;0.7] 1.0e5eye(4) [0;0;0] 1.0e6eye(3)3 [4.1;0.3;2.3;2] 1.0e4eye(4) [0;0;0] 1.0e7eye(3)Table 6.2 Initial controller and plant parametersand covariance matrices for RLS identificationBefore applying the adaptive rapid tracking algorithm, two trials should beattempted. In the first trial, the controller is allowed to adapt to the desired one,but the tracking error generated does not yet fully reflect that of the controlledsystem due to the adaptation of the controller at the beginning of the trial. Thus,a second trial is needed to generate the q°, e° required by the rapid trackingalgorithm for precise tracking in the third and future trials. Figures 6.3 to 6.8show the controller and the plant parameter estimates for the first two trials.59Figure 6.9 shows the unbounded tracking performance of tracking two suc-cessive tasks after 50 cycles of learning. Its control signals are shown in Figure6.10. Comparing with Figure 6.11, which shows the tracking performance underthe same conditions as that of Figure 6.9 except that it is obtained after 90 learningcycles, it can be known that the more cycles learnt, the better the tracking will be.Since in practice, saturation always exists, and for reality, the actual boundson the DC motors are calculated and converted to the corresponding bounds onthe control signals U. This can be done by using the dynamic equations of motionand assuming the worst case of actuation. Also, to make it realistic, normallydistributed random noises are introduced into the identifier which identifys thecontroller and plant parameters. Figure 6.12 shows the tracking performancewith the actual bound set on the control signals, and noise introduced to theidentifier. The corresponding control signals are shown in Figure 6.13. The actualbounds on the control signals are found to be [109.7; 83.7; 225.3] for joints 1,2,3respectively. The values of control signals for exact tracking are approximately[116; 108; 280]. Comparing Figure 6.12 and Figure 6.9, we can see that even whenthe tracking control signals are bounded at actual bounds, and the measurementsfor identification are noisy, the plant output can still track the input referencefast with fairly good precision. This is very important because it mitigates thepractical problem of saturation of actuator while still allows satisfactory trackingperformance.It is noted from Figures 6.10 and 6.13 that before the end of each tracking60150100500-50-100task, the command signals tend to be very small. This is important in practice,because it implies that the manipulator does not need to be driven dynamicallyto its original still position after each tracking trial.plant controllerapl ap2 bp 1 bc0 bc1joint 1 -2.0031 0.9990 0.0101 66.136 -56.788joint 2 -2.001 1.0001 0.0107 66.359 -55.637joint 3 -2.007 1.0005 0.0103 81.203 -67.012Table 6.3 True and estimated parameters of plant and controllercontroller parameters for joint one250200300-1500 0.5 1.5 2time (sec)2.5 3 35Figure 6.3 Estimated controller parameters for joint one61-300-4000-200-10020030010000.5 1 1.5^2time (sec)2.5 3 35250200150100500-50-100-150-2000 0.5 1 1.5 2time (sec)2.5 3 35controller parameters for joint twoFigure 6.4 Estimated controller parameters for joint twocontroller parameters for joint threeFigure 6.5 Estimated controller parameters for joint three62estimated plant parameters for joint oneFigure 6.6 Estimated plant parameters for joint oneestimated plant parameters for joint twoFigure 6.7 Estimated plant parameters for joint two631.510.50-0.5-1-1.5-22.50 0.5 1 1.5^2time (sec)2.5 3 3532.521.50.53-0.5 ^0 05 352.51.5^2estimated plant parameters for joint threeFigure 6.8 Estimated plant parameters for joint threeunbounded tracking after 50 cycles of learningtime (sec)Figure 6.9 Tracking performance after 50 cycles of learning—unbounded64unbounded command signals U after 50 cycles of learningtime (sec)Figure 6.10 Tracking control for unboundedtracking using true dynamic parametersunbounded tracking after 90 cycles of learning2.5 -,, ..... I 1.5 -0.5-0.5 ^0 1.5^2time (sec)OS^1 2.5^3^35Figure 6.11 Tracking performance after 90 cycles of learning—unbounded651 2.5 3515^2 3250200 -150 -100 -50-50-100-150-200-2500bounded tracking after 50 cycles of learning, noise introducedtime (sec)Figure 6.12 Bounded tracking, noise, estimated parameters, 50 learning cyclesbounded command signals after 50 cycles of learning,noise introducedtime (sec)Figure 6.13 Tracking control for boundedtracking (noise, estimated parameters, 50 cycles)666.2 ConclusionsIn this thesis, an adaptive rapid tracking control methodology is applied to asimulated 3 DOF direct drive rigid manipulator with parallelogram mechanism.The non-linear, coupled dynamics of the manipulator was approximately linearizedand decoupled via computed torque technique and by using the identified dynamicparameters. The identification was based on the Energy Conservation Theorem.Because of the limitation in choosing sampling rate and the presence of noisein the measurements, the identified dynamic parameters are some distance awayfrom the true (hypothetical) ones.Though the linearization is approximate due to the imperfect identification ofthe dynamic parameters, it does not degrade the tracking performance very much.Since the parameters of the approximately linearized plant are still imprecisely-known, RLS is used to identify the plant and controller parameters adaptively,using the data collected from the input of the linearization block and the sys-tem's output, both of which are assumed noisy. Based on the identified controllerand plant parameters, the supplemental tracking control signal is generated. Sim-ulation results show that this rapid tracking methodology gives good trackingperformance, even when the actual bound is set on the tracking control signal.This is particularly important in practice, since every actuator saturates.67Bibliography[1] C.H. An, C.G. Atkeson, and J.M. Hollerbach. "estimation of inertial param-eters of rigid body links of manipulators". Proceedings of the 24th CDC, pp990-995, 1985.[2] Haruhiko Asada and Kamal Youcef-Toumi. "analysis and design of a direct-drive arm with a five-bar-link parallel drive mechanism". Journal of DynamicSystems, Measurements and Control, Vol. 106, pp 225-230, September, 1984.[3] B.A. Francis and M. Vidyasagar. "algebraic and topological aspects of theregulator problem for lumped linear systems". Automatica, Vol. 19, No. 1,pp. 87-90, 1983.[4] M. Gautier. "identification of robots dynamics". Proc. IFAC, Symp. on Theoryof Robots, pp 125-130, Vienne, 1986.[5] M. Gautier and W. Khalil. "a direct determination of minimum inertialparameters of robots". IEEE Conf. on Robotics and Automation, pp 1682-1687, Philadelphia, April, 1988.[6] C.C.H. Ma. "rapid tracking of complex trajectory in short-duration processes".Automatica, The Journal of IFAC, 1990.[7] C.C.H. Ma. "direct adaptive rapid tracking of complex trajectories". ASMEJournal of Dynamics, Measurement and Control, submitted in 1991.[8] E.N. Moore. Theoretical Mechanics. John Wiley & Sons, 1983.68[9] S.Y. Sheu and M.W. Walker. "estimating the essential parameter space ofrobot manipulator dynamics". Proc. of the 28th CDC, pp 2135-2140, Tamps,Florida, December, 1989.[10]Mark W. Spong and M. Vidyasagar. Robot Dynamics and Control. John Wileyand Sons, Inc., 1989.[MM. Vidyasagar. Control System Synthesis: A Factorization Approach. TheMIT Press, 1985.69


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