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Use of kinematic redundancy for design optimization in robots Wang, Yuyan 1994

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USE OF KINEMATIC REDUNDANCYFOR DESIGN OPTIMIZATION IN ROBOTSByYuyan WangB.Eng., Southeast University, Nanjing, China, 1985A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJanuary 1994© Yuyan Wang, 1994In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of Mechanical EngineeringThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date:Jc /2AbstractFor a manipulator mounted on a dynamic structure, as in space applications, theinteractions with the robot and its supporting structure can play a crucial role. Forexample, stability and accurate control of space station operations are critical; therefore,methods that minimize the dynamic interactions (force and torque ) between the spacestation and robot manipulators are needed. This thesis evaluates two such approaches,and extends these ideas to the design of space robots. General simulations of trajectoryoptimization schemes are conducted based on the redundancy model proposed by de Silva[1], [7], [8]. Furthermore, this optimization theory is extended to cover robot parameterdesign for a redundant manipulator. Two techniques are developed. First method isthe “Joint Angle Manifold Approach”, which uses a path integral to define the globalcost function as the sum of the terms contributed by the reaction forces along the robottrajectory, which are described mathematically as different topological manifolds. Thefinal cost function becomes a polynomial of the design parameters i. After calculating thecost function in each manifold, the one with the minimal value was selected for furtheroptimization to search for the optimal link parameter solution. The second approachuses a global time integral to define the cost function as the sum of two terms; oneterm describing the time integral of the instantaneous base reaction cost function andthe second term describing the peak value of all instantaneous cost function curves.The global cost function is then optimized against the design parameters which are linklengths 1, using a numerical optimization method. Both methodologies are simulated fora three degree-of-freedom manipulator.11Table of ContentsAbstractList of Tables viList of Figures viiAcknowledgment ixNomenclature xi1 Introduction 11.1 Preliminary Remarks 11.2 Literature Review 31.3 Objectives and Contributions of the Research 41.4 Scope of the Thesis 52 Theory of Base Reaction Minimization 72.1 Dynamics of Space Station and Manipulator 72.2 Base Reaction Minimization 112.3 End EfFector Trajectory Generation 142.4 Summary 163 The Simulation Model of the NASA 7 d.o.f. Manipulator 183.1 The Traction Drive Joint 193.2 Kinematics 211113.2.1 Forward Kinematics.3.2.2 Differential Kinematics3.3 Dynamics3.4 Summary4 Simulations of Base Reaction Optimization4.1 The Computer Simulation4.2 Simulation Results4.2.1 Simulation Results for the NASA Seven d.o.f. Robot4.2.2 Simulation Results for the Three d.o.f. Planar Robot4.3 Summary of Results5 Parameter Design Optimization5.1 Background5.2 Global Parameter Optimization5.3 Computer Simulation5.4 Results and Discussions5.4.1 Optimization Results5.4.2 Summary of Results6 The 796.1 796.2 81818283842228364445455252545764646667686872Trajectory Manifold ApproachIntroductionManifold Formulation of Manipulator Kinematics . . .6.2.1 Forword Kinematics6.2.2 Inverse Kinematics6.2.3 Jacobian Relations6.2.4 Manifold Solution for a 3-R Robotiv6.3 Design Parameter Optimization 856.4 Computer Simulation 876.5 Summary of Results 90‘7 Conclusion and Recommendations for Future Work 947.1 Summary of the Thesis 947.2 Major Contributions and Shortcomings of the Research 957.3 Recommendations for Future Work 977.4 Application of the Work 97Bibliography 99A Appendix 102A.1 Kinematics and Dynamics of a Three d.o.f. Planar Robot 102VList of Tables3.1 Denavit-Hartenberg Link Parameters 274.2 Link Length Parameters for the Ten Cases in Figure 4.13 62viList of Figures2.1 A Space Station Based Robot2.2 Nomenclature for Dynamics of a Space Station and Robot3.1 The NASA Seven-Degree-of-Freedom Traction-Drive Manipulator . .3.2 Position Vector Representation in Coordinate Frames3.3 Joint Coordinate Frames for LTM3.4 The Relationship Between the Coordinate Frames of Base, i — 1 th Jointand i th Joint3.5 Schematic Diagram of Link i3.6 Computational Steps for the Recursive Newton-Euler Equations4.1 The Structure of the Software Modules4.2 Flowchart for Trajectory Optimization Using Kinematic Redundancy4.3 Algorithm for Avoiding Singular Configuration Using Kinematic RedundancyEnd Effector TrajectoryBase Reaction Cost FunctionJoint 1 TrajectoryJoint 2 Trajectory.Joint 3 Trajectory.Joint 4 TrajectoryJoint 5 TrajectoryJoint 6 Trajectory• .. 505556575859• . . 60• . . 606181020232638404248494.44.54.64.74.84.94.104.11vii4.12 Joint 7Trajectory 614.13 Optimal Cost Function for Different Link Lengths 624.14 Optimal Cost Function with the First Link Length Decreased 635.1 Algorithm for Design Optimizatiom of Robot Parameters 695.2 A Three-Degree-of-Freedom Planar Robot Mounted on a Space Station 705.3 End Effector Trajectory 735.4 Optimal Local Cost Function Values for 1 = [1 7 7] ( curve 1) and1 = [5 5 5] (curve 2 ) 745.5 Joint 1 Trajectory for 1= [1 77] (curve 1) and 1= [555] (curve 2) 755.6 Joint2Trajectoryforl=[1 7 7](curvel)andl=[5 5 5](curve2) 765.7 Joint 3 Trajectory for 1 [1 7 7] ( curve 1) and 1 [5 5 5] (curve 2) 776.1 A Three-Degree-of-Freedom Planar Robot and a Supporting Space Station 88viiiAcknowledgmentI would like to thank my supervisor, Dr. C. W. de Silva, for providing me theopportunity to carry out this research. His continuous help and advice is gratefully acknowledged.This work has been funded through grants from the Natural Sciences and EngineeringResearch Council of Canada.ix* * * * * * * * * * * *THIS THESIS IS DEDICATED TO MY PARENTS.* * * * * * * * * * * *xNomenclatureRotation matrixA’ D-H transformation matrixForce vector at the i — 1 th joint of the i th linkMoment (torque) vector at the i — 1 th joint of the i th linkVector respresenting the equivalent gravitational force per unit massat the centroid of the i th linkm: Mass of the i th linkMoment of inertial matrix of the i th link about the centroid,and expressed in the body frame of the linkit Length of the i th link.Velocity of the centroid of the i th link with respect to base frameThe linear velocity at the origin of the coordinate frame attached to link i.Angular velocity of the i th link expressed in the body frame of the linkPosition vector of the center of mass of link i from the origin ofthe coordinate frame of the link.Position vector of the center of mass of link i from the origin ofthe base referece frame.Rectilinear velocity of the space station with respect to the inertial frameAngular velocity of the space station with respect to the inertial frameM9 Mass of the space stationInertial matrix of the space stationq The rotation about z_1-axis with respect to i th jointxiin the coordinate system (x1,yjz_1)The rotation aboutz2_1-axis with respect to i th jointin the coordinate system (xj1,y z2_1)Q A set of n joint coordinates.X A set of m end-effector coordinates.xiiChapter 1Introduction1.1 Preliminary RemarksAs the space technology advances, future space missions will progress to encompassmany sophisticated tasks and operations; for example, operating scientific experimentssuch as constructing and repairing space stations, deploying satellites and deliveringmulti-phased payloads. Robots need to provide the flexibility, adaptability, programmability, seif-reconfigurability and intelligence that would be neccessay for autonomous op.eration, and also require to meet engineering specifications such as the end effector speed,load levels and trajectory precisions.Typical tasks of space robots would require extremely precise and accurate motioncontrol. However, manipulator motion control is generally difficult to achieve because,for example, dynamic interaction resulting from the fact that manipulator motions arecoupled with the space station produces unwanted disturbances on the system. Consequently, as the arm moves, it will exert dynamic forces and torques on the space station,causing the latter to move in a perturbed manner by changing its orbit and attitude.Since there is dynamic coupling between the robot and the space station, these disturbances also affect the performance of the manipulator. Hence, it is desirable to minimize,if not eliminate, the dynamic base reaction forces and moments through proper designand control of the robot manipulator.1Chapter 1. Introduction 2There are many considerations in robot system design. Common criteria such asload capacity, load to mass ratio, low parts count and high modularity are important,but even more critical are considerations like large workspace, high dexterity and endeffector dynamics. Preferably it should be a concurrent engineering process; in whichthe design, planning and developmental tasks are carried out in parallel. However, sincemany space missions are multi-tasked, it is difficult to compensate for all unwanted spaceinteractions through physical design of the robot. Then, such disturbances have to beminimized by appropriate control of the robot and the space station. Nevertheless, theneed for interaction compensation through control has to be reduced to a minimum soas to maintain controller simplicity and fuel efficiency.Using the minimization of dynamic interactions as the central criterion for the robotsystem design process, the goal of this research is to develop a unified methodology for optimizing robot parameters for a particular task. Recently, there has been much emphasisin using path planning of redundant manipulators to achieve optimal performance underdisturbances. This is however a limited approach; a general theory and methodologywould be needed to extend the current solutions to include the geometric and physicalparameters of a robot like link length and mass.In order to implement the theory of design optimization, an accurate model of thedynamic interaction of a space robot and path optimization and planning techniques isrequired. Kinematic redundancies may be employed in this optimization. This researchtherefore looks into the aspects of developing a physical model for a robot manipulator inspace operations; develops software programs to simulate robot motions and dynamics;and obtains fast and accurate optimization of the trajectory and design parameters forChapter 1. Introduction 3complex space robot systems such as the NASA seven degree-of-freedom traction drivemanipulator.1.2 Literature ReviewThe earlier work on base reaction minimization was done by Vafa and Dubowsky[3], K.Yamada and K.Tsuchiya [6],and de Silva [1]. Vafa and Dubowsky considered thereduction of altitude variations for satellite bodies by path planning through the modelof disturbance map; which shows the direction of the effect of joint velocities on the basestation disturbance. It was essentially a mapping of the disturbance vector field for manipulator variables. Vafa and Dubowsky attempted to use a virtual manipulator methodfor the analysis of base station dynamic disturbance; this method offers the advantageof decoupling manipulator dynamics from the base station because of the selection ofan appropriate coordinate frame of reference. Algorithms were also developed to usethe disturbance map for trajectory planning in order to achieve minimal base reactioneffects. The search for trajectory in disturbance map approach is an extremely timeconsuming task for high degree-of-freedom manipulators whose kinematics and dynamicsare complex. Yamada and Tsuchyi considered the trajectory planning of a robot manipulator through the application of quantified cost function for the reaction forces ofthe base station. Analytical trajectory is achieved for a simple satellite model. De Silvauses kinematic redundancy to design manipulator trajectories so that base reaction isminimized; In his approach, a local quadratic positive definite cost function is employedto characterize the base disturbance. Unconstrained optimization is used to select theoptimal end effector trajectory of the robot according to motion specifications such asacceleration and jerk limits.Chapter 1. Introduction 41.3 Objectives and Contributions of the ResearchThe objective of this research is to develop a comprehensive methodology for the optimization of design parameters of a robot mounted on a dynamic structure. The researchwill be based on the foundation of the theory developed by de Silva for base reactionminimization through trajectory planning using kinematic redundancy. Computer simulations are developed using this theory for two systems; a three degree-of-freedom planarrobot and a NASA seven-degree-of-freedom traction drive manipulator. The numericalsimulations are used to demonstrate the application of the theory, which are expected toshow significant reduction in the base reactions for both manipulators, through the useof kinematic redundancy. Since redundancy, in a given application only, allows the possibility for optimization of joint trajectories, the optimization model must be extendedto include other geometric and kinematics design variables, such as link parameters,mass, and end efFector velocity of the manipulator. Therefore, an important part ofthis research will be to develop numerical methods for global optimization; including amethod of defining a global cost function, as well as methods for optimizing this complexand multi-valued cost function. The concluding task of the research will be to producenumerical simulations to verify the design parameter optimization methodology for athree-degree-of-freedom planar redundant manipulator.The main contributions of this research are:1. Development of computer simulations for a three-degree-of-freedom planar manipulator and a NASA seven-degree-of-freedom traction drive manipulator based onthe theory developed by de Silva for trajectory planning. Numerical simulationsare performed on a SUN SPARC station computer platform using MATLAB© optimization subroutines. These verify that base reaction minimization is possible forChapter 1. Introduction 5different types of robots, and further predict that the base reaction forces ( costfunction values ) are related to the geometric variables of the robot.2. Development of optimization methodologies for design parameter optimization.Two approaches are developed. They are:(a) Use of the topological manifold method to identify different trajectory flows,and to calculate the global cost function for the manipulator as the pathintegral along each trajectory. The cost function is symbolically estimatedusing first order terms, and the optimal design variables are calculated byoptimization programs in MATLAB©.(b) A numerically oriented procedure based on the early version of the path planning optimization is developed. This model uses the instantaneous cost function of the original method as the integrand to calculate the new global costfunction; however, instead of optimization with respect to path parameters,numerical optimization with respect to the geometric parameters is achievedto obtain somewhat global, optimal design variables.1.4 Scope of the ThesisThis section outlines the body of this thesis. Chapter 1 sum.marizes the objectivesof the work, describes the reason for developing robot design optimization, presents themain results, and lists the contribution of the reseach. Chapter 2 introduces the trajectory optimization using kinematic redundancy and its application to base reactionminimization. The theory of de Silva and the techniques of using redundant degrees offreedom to optimize a cost function of the base reactions are reviewed. Chapter 3 is veryimportant for understanding the remaining chapters. Here, the kinematic and dynamicChapter 1. Introduction 6model of the NASA seven-degree-of-freedom robot manipulator is developed. In Chapter4, computer simulations are carried out based on the model derived in Chapter 3. andthe computational algorithm and the methodology of optimization are described. Chapter 5 forms the core of the design parameter optimization. In particarlly, the basis thatleads to a global cost function definition and the methodology of nonlinear constaintedoptimization are outlined. A new approach for incorporating a global cost function inthe optimization is introduced in Chapter 6. Here, the method of topological manifoldsis used to describe the “self motion” of a redundant trajectory, and grouping them intofinite number of submanifolds. This approach allows symbolic computation of the globalcost function, thereby, enabling design optimization of parameters. Finally, the concluding chapter summarizes the design optimization problem and the theory and computermodels developed for resolution of the problem, and provides recommendations for futureresearch.Chapter 2Theory of Base Reaction MinimizationBefore we formulate the theory of design optimization, we shall review the fundamental idea of base reaction minimization through the use of kinematic redundancy fortrajectory planning as developed by de Silva [1], [7], [8]. There are four key steps in theapproach:1. Formulation of space-station manipulator dynamics2. Quantification of the base reaction disturbances in terms of a scalar cost function3. Resolution of the end effector position and orientation in terms of redundant jointangle variables4. Optimization of the redundant joint coordinate to minimize the cost function values.2.1 Dynamics of Space Station and ManipulatorConsider a space station that is orbiting around the earth in a circular orbit ata certain attitude. The open chain robot is rigidly mounted on the base station, asillustrated in Figure 2.1. The links are assumed to be rigid, and joint flexibility, friction,and backlash are assumed to be ideal. The inertial frame is located at the center of theearth. The center of mass of space station is located at rb from the base of the robot.The dynamic equations ( both translational and rotational ) of the rigid space station7Chapter 2. Theory of Base Reaction Minimization 8XeZeInertialReference FrameFigure 2.1: A Space Station Based RobotChapter 2. Theory of Base Reaction Minimization 9are given byf8 + fi, = (mv) (2.1)N3 + Nb + rb 0 fb = + 0 (18w) (2.2)in which f3 is the resultant external force acting at the centre of mass of the spacestation and N8 is the resultant torque.Consider the i th link of a n link manipulator as shown in Figure 2.2. The Newton-Euler equations for this link consist of the force-momentum equations:f2—1 — f: + mg = (mvj) (2.3)and the moment-angular momentum equations about the centroid of the link:N_1— N + rd 0 fj — rj 0 f = (Iriwi) = ‘ci.i + 0 (Iiw) (2.4)Since f, = 0 and N = 0 for an open chain robot, by summing the n equationsgiven by equation (2.3) and (2.4), the bace-reaction forces and moments are obtained asfollows:fb = >mi(ici—gj (2.5)Nb = + 0 (I’.’) + 0 (vc — gj] (2.6)Note that in these summations, the vectors are assumed to have been expressed in acommon reference frame, through suitable coordinate transformations.The required kinematic relations, specifically expressions for w2, and v in termsof the joint velocities q and joint accelerations q are as follows:Chapter 2. Theory of Base Reaction Minimization 10vivciv!-1zsFigure 2.2: Nomenclature for Dynamics of a Space Station and RobotChapter 2. Theory of Base Reaction Minimization 11Angular velocities:= (2.7)Wi = Wi_i + iti=2 ,nAngular Accelerations:(2.8)i=2 nRectilinear velocities:v0=v8+w®r (2.9)= Vj + W 0V = V_1 + W: 0i=l nRectilinear accelerations:0=a.+w3r+CA.’(W) (2.10)= v + 0 r + w ® (w 0a = a_1 + c.’ 0 r_11 + 0 (w, 0 r_1,)i=l,...,n2.2 Base Reaction MinimizationUsing the recursive Newton Euler dynamic formulation, the base reactions obtainedabove can be used to express the cost function. Specifically, the six dimensional vector ofChapter 2. Theory of Base Reaction Minimization 12the base reactions is mapped into a quadratic cost function, using the following measure:CostFunctiom = RTQR (2.11)in which the reaction vector R is given by:R = [f&N&]T—(2.12)Here, the vectors j and N1 are the nominal reactions for a given configuration;and Q is a positive definite weighting matrix, which also takes into consideration of thedimensional differeces in the vector R.Let us assume that the end-effector trajectory is expressed in the normal manner asa sixth order vector containing position and orientation of a body frame attached to theend effector; and expressed in the base frame. The incremental joint motion öq thatwould be needed to produce a trajectory increment of 6y is given as:= J6q (2.13)where J is the Jacobian matrix of the system.The velocity of the end effector is given as:v = Jq (2.14)and the acceleration relationship is:a=Jq+Jq (2.15)The Jacobian matrix can be partitioned into:J = [J, Jr 1 (2.16)Chapter 2. Theory of Base Reaction Minimization 13where J is a nonsigular matrix corresponding to a set of independent joint coordinates q,..1,. is a submatrix corresponding to a particular choice of redundant joint coordinatesq,.Similarly, the vector is partitioned intoq = [q,, q7] (2.17)Thus, the velocity and acceleration equations are represented asv = J4 + J,.47 (2.18)a Jn1n + Jrqr + Jnqn + Jrqr (2.19)The vectors v and a can be expressed in terms of curtate cycloidal motions, whichmay be used to limit both acceleration and jerk.The joint variables and their derivatives are represented as follow:= J’(v—elr4r) (2.20)= J’(a— Jrqr — Jnqn—Jq7) (2.21)Therefore, the cost function is represented in terms of the redundant variables q,which are expressed as a polynomial of time:= qo + c1t + c2t (2.22)Chapter 2. Theory of Base Reaction Minimization 14This in turn means the cost function is depended on the parameters c1 and c2. Theoptimization scheme proceeds as follow: divide the trajectory into sufficiently large number of segments, and find the suitable values for the parametes c1 and c2 to produce alocally optimal solution for all joint coordinates. Such a method is repeated until theend of the task trajectory is reached.2.3 End Effector Trajectory GenerationThe motion for the space robot system is primarily dependent on the end effectortrajectory. The need for engineering specifications on the acceleration and jerk at theend effector of the robot means trajectory kinematics must be appropriately designed.A curtate cycloidal time trajectory is used to meet such criteria [10]. For simplicity reasons in computer simulations, a linear segment is chosen to depict the motion of the endeffector of a manipulator, and also the rotational motion is executed in synchronism forsimulation purposes. An Euclidian rectilinear coordinate y and an angular coodinate 8are chosen to represent the end effector position and orientation respectively.In the curtate cycloidal motion, y and 9 are synchronized, and the velocity v = andangular velocity w = 6 are represented parametrically as:v(p) = b1(1 — cosp) (2.23)w(p) b2(1 — cosp) (2.24)t(p) = a(p— csinp) (2.25)Chapter 2. Theory of Base Reaction Minimization 15a,bi,b2>0,1>c>0,2irpOHere p is the parameter of the motion; corresponding to a scaled time variable. T isthe trajectory duration as given byT = 2ira (2.26)The position and orientation vectors of the trajectory are obtained by integrating theequations (2.23) and (2.24); thus= abi[1 + ()]p — (c + 1) sinp + () sin2p (2.27)S(p) = ab2[1 + ()]p— (c-I- 1)sinp+ ()sin2p (2.28)The final position and orientation at t = T arey(T) = abi[1 + (2.29)6(T) = ab2[1 + ]2ir (2.30)Rectilinear and angular accelerations a and a are obtained by differentiating equations(2.23) and (2.24).b1sinp /ap) = i2.31a(1— ccosp)b2 5flPa(p) = (2.32)a(1 — ccosp)The maximum acceleration values arelamaxi = 1 (2.33)aJ(1_c2)Chapter 2. Theory of Base Reaction Minimization 16b2ImaxI = (2.34)a/(1_c2)Linear and angular jerks a and a are:• b1 (cosp—c)a(p)= -j (2.35)a (1—ccosp)• b2 (cosp—c)cz(p) =— (2.36)a2 (1 — ccosp)These jerks remain finite throughout the trajectory as desired. There are four stationary values for each expression. However, the largest magnitude isb1Iamaxl= a2(1 — c)2 (2.37)b2krnax 2 2 (2.38)a (1—c)These occur at the points with p = 0 and p = 27r.2.4 SummaryIn this chapter, the theoretical framework of the base-reaction optimization of arobot manipulator mounted on a dynamic structure is presented from the point of view ofkinematics and Newton-Euler dynamics. Specifically, the physical description of roboticmotion and reaction forces is presented using Newton-Euler equations, expressed in areference frame fixed to the space station. The fundamental variables are joint motions.Furthermore, the base reactions are defined as a positive-definite functional of the forceand moment vectors. Using the Jacobian relationship between the joint velocities andthe end-effector velocities, the force and moment equations are mapped in terms of thederivatives of the redundant joint varibles q. The optimization of the cost function isChapter 2. Theory of Base Reaction Minimization 17expressed as a stepwise, local, parameter optimization procedure. Finally, a theoreticalmodel for limiting the acceleration and jerk of the end effector is introduced using thecurtate cycloidal motion.Chapter 3The Simulation Model of the NASA 7 d.o.f. ManipulatorAs discussed and shown in Chapter 2, the differential kinematics of a robot manipulator can be described using its Jacobian. Furthermore, the dynamics of the robot and adynamic supporting structure can be formulated using the Newton-Euler equations. Inthis manner, the complete mathematical relationship between the joint angle velocities,accelerations and the dynamic base reactions of F and N can be linked together. Subsequently, the optimization of base reactions (forces and moments) through trajectoryplanning may be carried out as formulated in Chapter 2. In order to carry out the optimization, an analytical model has to be developed in this manner, for the specific robotof interest. In this chapter, such a model is presented for a practical robot.The NASA Langley Research Center sponsored and the Oak Ridge National Laboratory (ORNL) developed a Laboratory Telerobotic Manipulator (LTM) for space applications. This robot will serve as prototype that is used in this chapter for the computersimulation of the optimization scheme. As shown in Figure 3.1, the LTM arm has sevendegrees of freedom that provide one degree of joint-space redundancy over that neededfor the fundamental task of end-effector placement and orientation. The arm is configured from three common pitch/yaw joints which combine to provide shoulder, elbow, andwrist joints. Each of these joints has two degrees of freedom which are provided by adifferential drive mechanism and is termed a traction drive joint, that has two outputs18Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 19providing pitch/yaw motions [11], [12]. This mechanism may be represented by two intersecting revolute joints. A wrist roll mechanism, mounted at the wrist joint, providesthe seventh degree of freedom. Seven degrees of freedom allow the LTM to reorient itselfwithout changing the end-effector position and orientation.The theoretical background that is necessary to carry out a computer simulation ofthe LTM includes kinematics and dynamics. First, a traction drive joint is described.Then, the kinematics of the arm is presented. Finally, the equations of motion are derived.3.1 The Traction Drive JointThe joint mechanisms for conventional robots are usually gear transmissions, timingbelts, sprockets and chain devices. Some others use the technique of harmonic drives.Neither of these mechanisms satisfies the requirement for accurate and repeatable operational specifications for teleoperating robot manipulators in space. In particular, theneed for minimal backlash, high flexibility, high stiffness and smooth operations. TheNASA Oak Ridge National Laboratory developed the traction drive joint mechanism fora seven d.o.f. manipulator. Traction drive is among the most elementary way of implementing speed changing mechanisms. It simply utilizes the application of the frictionwheels of different dimension (diameter) to regulate speed. It can then be construct togive a single, fixed speed ratio, by controlling the rolling surfaces to engage at differentradii. Using lubricant films, this joint poses better power transfer performance characteristics than others mechanisms because of minimal backlash.For the specific case of seven d.o.f. LTM, the traction drives are located at the shoulder, elbow, wrist joints as well as another roll motion at the wrist to provide the extraChapter 3. The Simulation Model of the NASA 7 d.oi Manipulator 20Figure 3.1: The NASA Seven-Degree-of-Freedom Traction-Drive ManipulatorChapter 3. The Simulation Model of the NASA 7 d.oi Manipulator 21degree of redundancy. At each joint, two input rollers are frictionally engaged with twoother wheels of larger diameter dimension, forming a differential mechanism. The two differential wheels are also engaged with a output roller. This whole mechnism will producetwo different independent vector motions pending on the motion of the two differntialwheels. When they are undergoing opposite motions with same speed, a roll motionis produced. When they are undergoing same direction motion at same speed, a pitchmotion is produced about the axis of the output roller. The input rollers are of coursecontrolled by dc servo motors, and any linear combination of pitch and yaw motion canbe generated by varying the motion of input rollers.The dynamic characteristics of the traction drive have been evaluated by de Silvaand Hankins using controllability, observability, stability and response analysis [13]. Using state vector and system matrix analysis, the traction drive joint is found to be wellbehaved with the implementation of a linear quadratic regulator for servo control. Inparticular, satisfactory performance of the traction drive is possible with joint servo usingroll and pitch angle feedback.3.2 KinematicsThe forward-kinematics problem is to find the position and orientation of the endeffector of the maniputor given the joint angles 6. The differential-kinematics problemis to find the corresponding joint volecities that cause the end-effector to move at thedesired velocity. The inverse-kinematics problem is to determine the joint angles whichwill result in a desired end-effector position and orientations, typically given relative toand expressed in a coordinate frame fixed at the base. All of these relations have to beChapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 22determined for the LTM arm.Differential relationships of manipulation are important in several ways. The mostobvious is in the case of motions of accommodation, for instance, when a camera observes the manipulator end-effector position and calculates the differential changes inposition and orientation in order to reach some goal. In this case, we should be able toconvert defferential changes in one coordinate frame ( measurement space ) into changesin another ( the attraction or control space ). Another use of differential relationshipsfollows directly, which is particularly applicable in this work. Given a differential changein the end-effector position and orientation, we will employ differential relationships tofind the corresponding changes in the joint coordinates, since it is the joint velocities andaccelerations that appear in the expressions of the base reactions. The Jacobian matrix,which corresponds to a set of linear equations, that has the differential change in positionand orientation of the end effector as dependent variables, and the differential change inthe joint coordinates as independent variables, as in equation (2.13).3.2.1 Forward KinematicsFirst, a means of relating positions on one link with respect to a fixed frame or toanother moving link is established. In this manner, the position and orientation of theend-effector can be expressed in terms of the joint coordinates. The principal points arecovered here. Details are found in references [15] and [161.Given a point P and two coordinate frames F1 and F as shown in Figure 3.2, theposition of P relative to F1’ is expressed as a vector ‘ in F1,whereas the positionof P relative to Ft is expressed as r in Ft. A superscript denotes which coordinateChapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulatorsystem the vector is referred to.xFigure 3.2: Position Vector Representation in Coordinate FramesThe vectors z and z are related by= d + i’ ‘23(3.1)where R is the matrix of coordinate transformation from Frame i to Frame i— 1 andd is the distance between the origins of the two frames, expressed in Frame i— 1. Notethat=cos — cos a sin 9 sin a2 sin 8cosacos92 —sina2cos61 (3.2)0 sina1 cosajwhere 6 is the joint angle from aj axis to the axis about the z2_1 axis. d is thedistance from the orgin of the i — 1 th coordinate frame to the intersection of the z..1with the axis along the z1 axis. a2 is the offset distance from the intersection ofthe z1 axis with the axis to the origin of the i th frame along the axis. a2 isPzF’d’1YF’1Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 24the offset angle from thez1.. axis to the Z1 axis about the (using the right-hand rule).Note that equation (3.1) represents a rotation and a translation. This transformation between ‘‘ and a’ may be represented by a single matrix through the use ofhomogeneous vectors. An extra unity element is added to the vectors, so that givenx= V (3.3)zthe vector X is definedxX= (3.4)1Note that, the original vectors ‘ and a are augmented by adding a “ 1 “ as thefouth element so that the result is a 4 x 1 vector.Now, the transformations are expressed asX’ = A’ X (3.5)where, the homogeneous transformation matrix is given byd41 (36)0 0 01Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 25By applying equation(3.5) successively, a vector expressed in Frame F’ may begiven by a vector X in Frame F°. Specifically, we haveX = A A ....A’ X (3.7)The combined transformation which is a chain of A matrices is denoted by the Ttransformation matrix, asX = T X1 (3.8)wheren s a dri, s,, a 4n2 s a d0 0 0 1In order to relate positions and velocities at any point on each link of the robotto a fixed base frame, a coordinate frame is attached to each link. Denavit-Hartenbergconvention [16] is used in this work for establishing coordinate frames. For a manipulatorthat has a revolute joint i, the joint variable O is taken to describe the rotation of jointi about the z2_1 axis. The axis is chosen normal to both z_1 and z2 from the crossproduct z1_ ® z. The intersection is the origin. The y axis completes the coordinateaxis triplet according to the right-hand rule, y = 0 The following parametersdescribe the relationship between neighbouring links:• a is the distance between z and z, measured along• d1 is the distance between and measured along z1_• a2 is the angle between z_1 and z2 measured about , using the right hand rule.Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 26z6z5Figure 3.3: Joint Coordinate Frames for LTMThe local coordinate frames of the LTM, established in this manner are shown inFigure 3.3. The Choice of coordinate frames is important, since the whole description ofthe behaviour of the robot, from trajectories to dynamics equations, rests on it. Table3.1 gives the kinematic parameters for the LTM.The next step is to obtain the A2 matrix from the choice of coordinate frames. Using the above definitions, a vector X expressed in link i coordinate frame may be reexpressed in terms of link i — 1 coordinate frame as X’ by performing the followingsequence of sub-transformations:• first, rotate by an angle 9 about z2_1 to bring the axis parallel to the axis• second, translate along z2_1 a distance so that a and z_ coincide• third, translate along x = a distance a to bring the two origins together• finally, rotate by an angle a: about to bring the coordinate frames into coincidence.zizoz2 z7z4Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 27Table 3.1: Denavit-Hartenberg Link Parameters8 d(m) cx2(deg) a1(m) 91(deg)8 0 -90 0 -9092 0 90 ‘2 093 0 90 0 9094 0 -90 13 095 0 90 0 086 0 90 0 9087 d7 0 0 0The result is the general matrix A2 given bycos— cos a1 sin sin a2 sin 82 a1 cos= sin 9 cos aj cos 9 — sin aj cos 9 a1 sin 90 sina1 cosa d10 0 0 1For the LTM, the resulting A matrices, i = 0, 1, ..7 are given below:10011 C1 0 —S100 1 0 0 S1 0 C1 0A=0 0 1 0 0 —1 0 00 0 0 1 •0 0 0 1•C2 0 S2 121.12 C3 0 S3 0S2 0 —C2 l2S S3 0 —C3 0010 0 0100000 1 0001Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 28C4 0 —S4 l4C C5 0 S5 0S4 0 C4 l4S S5 0 — C5 00 —1 0 0 0 1 0 000 0 1 0001C6 0 —S6 0 C7 —S7 0 0S6 0 C6 0 S7 C7 0 0A6=0—1 00 0 0 1d700 01 0 001whereC1 = cos 9, S2 = sin8,C2= cos(61 + 9,), S, = sin(Oj + 9,).and, l is the distance between z0 and z measured along O, 12 is the distance betweenz2 and z1 measured along 2, 13 is the distance between z4 and z3 measured along ,4. is the distance between and z6 measured along Z7.3.2.2 Differential KinematicsFor differential kinematics, the main problem is to obtain the Jacobian matrix. Herewe just briefly discuss some main principles. More details are found in reference [15] and[16].In Figure 3.2, assume that point P is fixed to Frame F’. We have:= d’ + R’ ‘ (3.10)Note that the superscript indicates which coordinate system the vector is referred to.Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 29The expression for the velocity is obtained by differentiating this expression withrespect to time:= Vd1 + + R v (3.11)Note that the point P is fixed with respect to Frame Fl. It can be shown that, forthe present case,= Vd’ + R1 R_1T R’ a’ (3.12)Note that the R’ matrix is given bycos 91 — cos a1 sin 8 sin a sin 6= sin 6 cos a1 cos 61 — sin a1 cos (3.13)o sin a1 cosa1Differentiating R’ with respect to time yields:— sin 6 — cos a1 cos 6 sin a1 coscos 6— cos a1 sin 8 sin a sin 9 Oj (3.14)o o 0Mutiplying R’ and R’ yields:o —1 0R1T= 1 0 0 (3.15)000Hence, we note that R1 R_1PR_1= iz’For a revolute joint that rotates about z axis without translation,= 0 (3.16)So, from equation (3.12), we have=(R’ R_1T) R’ (3.17)Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 30=‘ ói’ [x] (3.18)000If we consider X as the origin point of the end effector frame n, measured withrespect to frame i, equation (3.18) can be rewritten as follows:0 —1 0 Pa,v1 = 1 0 0 8 R p (3.19)0 0 00 —1 0 Pa;= 1 0 0 1 F1,0O_P-F1,Pa,0TiWe multiply the equation (3.19) by the appropriate R transformation matrices toexpress the end effector velocity in Frame F’ with respect to the base coordinate frame.Specifically,i—i-F1,= R R.... R: Fa, 9, (3.20)0nwhere,0R R.... = a1, (3.21)m sa, aChapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 31Hence,0 i—in2, s2, a,2, —P= a, F2, (3.22)ri s a 01—1 nFrom this equation, the translational velocity of the end-effector with respect to thebase frame is found:0 0 0S2,=— fly [Py}’ + y [P2,]’ 9 (3.23)flz 8zi i—i i—iThe joint variable 9 is defined to be a rotation about the z1_.0= 0 8 (3.24)10= 0 6. (3.25)1Then the angular velocity of the end-effector with respect to the base frame is obtainedby multipling the corresponding R trasformation matrices.0 092, ri2, s a,), 0= fly Sy a 0 (3.26)n s a 1I0a2,ayaChapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 32Then the total velocity relation is formed by combining equations (3.23) and (3.27).Specifically,0 0 0th• rD li—i i FD 1:—i— L’ yin 8y Vxinflz •sz0 8. (3.27)aI iiThe velocity relationship between frames can be converted into the differential form,which results in the well-known Jacobian matrix. The Jacobian matrix is a 6 x n matrixconsisting of differential translation and rotation vector elements.d81dxdO2dydO3dz= J(8) (3.28)d8d6d8d8where J is given by the right-hand side matrix of equation(3.27)Each column of the Jacobian matrix consists of the differential translation and rotation vectors corresponding to the differential changes of each joint coordinates. For theLTM, if q is picked as the redundant degree of freedom, the Jacobian is found to be (see equation (2.16) ):J = [J, J7 1 (3.29)where the vectorChapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 33C123456d7—S13C456d7+C1S2456d7—C12S356d7S1C35S6d7 — C1S2C46d7+ C123S46d7— S1S34Cd7—13cC24+13sC4—13CS24—12C ——S1C2S3456d7—S56d7—12456d7+S1C2356d7C13S56d7+S12C46d7—SC3—C1S346d7Jr— 13sc2C4+13cC4+13sS24+12s0001and the 6 x 6 square matrixJ = [Ji J2 J3J4J5J6] (3.30)whereC1S23456d7—C12S456d7—C1S2356d7+C1S2346d7—C1246d7—13CS24+13C2S4— 12CSS12C3456d7—S1C2456d7—12346d7+346d7S1C246d7—13S2C4+13SC24— 12SJnl = C2345S6d7+S24C56d7—C2S356d7+C23S46d7—S2C46d7— 12CC34 — 13S2S4—12S—SiCl0Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 34C12S3456d7+S1C3456d7—S1356d7+C123S56d7S1C346d7+2—13C2S4—13SC4S1C23456d7—C1345S6d7—C13S46d7+256d7S1C2S3S4C6d7+ 13CC34 — 13SC2S3C4Jn2 =—S2C345S6d7+S2C356d7—S234C6d7+13S2C4C’s2S’s2C2C123S456d7— C,S245S6d7— S,S34C56d7— C,C2346d7+S,S3C46d7— C,S2S46d7+ lC,S24+13S,S4S,C2C3456d7— S,S2C456d7+ C,S356d7— S,C2C6d7C,S346d7— S,S246d7+13S,S2C4—13C,SS4+13SC24Jr3 = —S2C3456d7—C245S6d7+S2C346d7—C2S46d7+L3S2C4+L3C24C12s3+ s,c3S,C23— C,C3—s2s3Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 35C1234S56d7+C1235S6d7—S13C456d7-f-C1S2456d7+S1C356d7S1C23456d7—C135S6d7+S1C2356d7-j-C1S3456d7+S12456d7Jn4 =—S2S3C56d7+C2S456d7—S2C3456d7—C1C23s4+s134+c1s24-.-s1C2c3s4— C1s34+s12C4s2C34+ c24—C1C23456d7+C123S46d7+S13C456d7+C12S356d7—C1S2S4C5Cd7— S1S3S46d7— C1S2C4S6d7+ 51C3S5C6d7—S1C23456d7+C23467—5+2—C1S3456d7—S12C46d7+C1S346d7—C13S56d7Jn5 =S2C3456d7—C2S456d7+S2C346d7—C24S6d7+S235C6d7C1234s5—S13c45+c1C2s35+s1C35s1C2345+C1S345+S1245+S1C235+C135—S2C34S5+c2S45—S23C5Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 36000— C1C2C3C4C556+ SS3C4C5S6 — C1S2S4C556+ C1C2S3S556+Jn6 =s1C356—C123S46+S1s3C46+C1S246—S1C2C3C4C556— C1S3C4C5S6 — S1S2S4C5S6 + S1C2S3S56—C13S56—S1C2346—C1S346+S12C46s2C345S6+C2S456—S2356+S2C346+C2463.3 DynamicsThe Newton-Euler equations of motion are used for deriving the dynamic equations inthis work mainly because of the computational efficiency that results from the associatedrecursive formulation.The necessary mathematical relations are briefly described here. More details arefound in reference [16] and [15].In Figure 3.2, suppose that Frame F is rotating and translating with respect to FrameF. Also suppose that a particle p with mass m is located by vectors Zt and i, withrespect to the origins of the coordinate frames Ft and F’. The origin of Fl is given bya vector d’ with respect to the origin of Frame F’.Then, we have the relation between the position vectors a and z1l:= z + d4 (3.31)Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 37where all vectors are assumed to have been expressed with same coordinate frame.Furthermore, if we define e and v’ as the velocities of the particle p reletive to theframes F’ and F, and if Vd is the velocity of the frame F relative to the frame F1’,the velocity of the particle p with respect to the coordinate system F’ is given by:V’1 = V’ + 1)d (3.32)dr,’ dd’ d) ddV = + dt = cit + ‘4’ ® + di (3.33)where cl*( )/dt is defined as time derivative with respect to the coordinate system F,and is the angular velocity of this frame.Similarly, the acceleration of the particle p with respect to the coordinate frame F’’is found to be:1 d2a &d’a=-;i-+ dt2 =a+ad (3.34)where a’ and & are the accelerations of the moving particle p relative to the coordinateframes F’’ and F’, and ad is the acceleration of Frame F’ relative to Frame F’.Accordingly,d*2& d*a_l dw d2’’= dt2 +2w di + w 0 (w 0 ‘) + 0 a’ + dt2 (3.35)We wish to apply these results to link coordinate systems of the robot, to obtainthe kinematic relationship of the moving links of a robot arm with respect to the basecoordinate system.Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 38Assume that the base coordinate system is (x0, Yo, zo). The coordinate system (x1,yj1,z2_i) with origin O_ is attached to link i — 1 and the coordinate system (x1, y,z) with origin O is attached to link i, as shown in Figure 3.4. Origin O is located bya position vector p1 with respect to the origin 0 and by a position vector p from theorigin O_i with respect to the base coordinate frame.Origin 0_ is located by a positionvector P1—. from origin 0 with respect to base coordinate frame.xo-1i-1 Joint IFigure 3.4: The Relationship Between the Coordinate Frames of Base, i — 1 th Joint andi th JointLet v1_ and w..1 be the linear and angular velocities of the coordinate frame F’(x.4,yjl, z_i) with respect to the base coordinate system. The angular velocity of theframe Ft with respect to the frame F1 is w1.x/z/ZOBase0YoThen, the linear velocity v1 and the angular velocity w1 of the coordinate frame FtChapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 39(xe, y, z) with respect to the base coordinate system are given by:= +w Øp+v2_i = w2 ®p+v1-i (3.36)=‘_i + w (3.37)* i—i i—iHere d ( )/dt denotes time derivative with respect to the frame F , and dt =since the joint is revolute and not prismatic. Also, w1 =The linear and the angular accelerations v and of the coordinate system (xi, yj,z2) with respect to the base frame are:d*p d*p= dt2 + 2w_1 0 + we—i Op + 0 (wi 0 p) + v1_ (3.38)== ‘—i + + w1 (3.39)Here, 2= dW’ ® p + w’ 0 (w_ 0 p) since the joint is revolute.The dynamic equations of a rigid body can be represented by two equations; onedescribing the translational motion of the center of mass, called Newton’s equation ofmotion for a mass particle, while the other describing the rotational motion about thecenter of mass, called Euler’s equation of motion.Let us consider an individual link i as shown in Figure 3.5. All the forces and moments acting on the link are shown in the figure. Let vd be the linear velocity of thecenter of mass of the link with respect to the base frame ( xo, Yo, zo ), w1 be the angularChapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 40-1x0Figure 3.5: Schematic Diagram of Link ivelocity of the link and I be the moment of inertia of the link with respect to center ofmass.The equation of translational motion is obtained by using Newton’s second law,— f1 + m1g = (3.40)and the rotational motion is described by Euler’s equation.— N1 + ?c,,j_1 0 f_1 — rc 0 f = Icài + W: 0 (Icii) (3.41)where w1 is the angular velocity of the body frame of the link.The N-E equations of motion consist of a set of forward and backward recursiveequations. Equations (3.36), (3.37), (3.38), and (3.39) form the forward equations, whileequations (3.40) and (3.41) form the backward equations. For the forward equations,Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 41linear velocity and acceleration, angular velocity and acceleration of each link, are computed from the base reference frame to the end effector. For the backward recursiveequation, the moments and forces acting on each link are computed recursively from theend effector to base reference system. Hence, the forward equations compute the kinematics of each link from the base frame to the end effector, while the backward equationscompute the necessary moments/forces for each joint from the end effector to the basereference frame. The computational algorithm is shown in Figure 3.6.For the LTM, the necessary equations are summarized below:Angular velocities:Wa = W3 (3.42)= W9 + z041W2 Wi + Z1q2= W2 + Z2q3= W3 + z34W4 + Z4q5= W5 + Z5q6W7 W6 + Z6q7Angular Accelerations:=‘.‘ (3.43)“2 = W1 +q2+w1 ®Z1q2Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 42InitializationForward Recursion ISet 1=1Compute Kinematic Variables,c•,1, ye a1:esn0rBackward RecursionSet f1÷, N÷1Figure 3.6: Computational Steps for the Recursive Newton-Euler EquationsChapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 43w3=w2+q®Zqw4=w+q4+®ZW5 =w4+q5+w4ØZqsW6 =4’s+q+wsØZLinear accelerations:(3.44)Vi = Vo + Wi 0 ro,i + Wi 0 (Wi 0r0,i)== V2 + W3 ® r2,3 + W3 0 (w3 0r2,3)V4 = V3V5 = V4 + W5 0r4,5 + 45 0 (w5 0r4,5)= V5V7 = V6 + W7 0 r6,7 + W7 0 (w70)r6,Linear accelerations of the centers of mass:i’d i’i + ‘i 0 1,d1 + l 0 (w1 0r1,) (3.45)Vc2 = VciVc3 = V3 + 43 0 r3,c3 + W3 0 (w3 0 r3,3)Vc4 = Vc3Vc5 = V5 + W5 0 7’5,c5 + W5 0 (‘‘ 0 7’5,c5)Chapter 3. The Simulation Model of the NASA 7 d.o.f. Manipulator 44Vc6 = ‘0c5Vc7 = V + W7 0 V7,c7 + W7 0 (w7 0 r7,7)Note that all vectors here are expressed in the base coordinate frame.The efficient, recursive, Newton-Euler equations of motion, which Luh et al. [14]improved are used. In their approach, all velocities, accelerations, inertial matrices,locations of the centers of mass of the links, and the forces and moments are referencedto the local coordinate systems, thereby improving the computational effeciency.3.4 SummaryIn this chapter, the theoretical framework of kinematics and dynamics of the NASASeven-Degree-of-Freedom traction drive manipulator is presented. The kinematic equations of the robot are built based on the Denevit-Hartenberg convention. The differentialkinematics of the LTM is characterized by Jacobian matrix, which was derived from thefirst principle. The dynamics of the LTM is described using the recursive Newton-Eulerequations. All the dynamic variables derived are expressed in terms of vectors in the basecoordinate frame. This chapter provides the fundamental mechanics for the modelling ofdynamic interaction between a robot and its support structure.Chapter 4Simulations of Base Reaction OptimizationThis chapter will describe the computer simulation process in applying the base reaction optimization technique that has been formulated in the previous chapters. Thedefinition of an end effector path trajectory from the start point to the end point, thedivision of the trajectory into time interval steps, the computation of the joint kinematics at each interval using the inverse Jacobian and computing the instantaneous costfunction at each interval, and the local optimization of the cost function at each intervalusing kinematic redundancy, constitute the main steps of the simulation process. Theassociated software modules and their structure are described. They are utilized to obtain joint trajectories for the seven d.o.f. NASA robot as well as a three d.o.f. planarrevolute robot, that minimize a quadratic function in base reaction. The optimal resultsare compared with non-optimal baseline results. A preliminary design optimization withrespect to robot parameters is also given.4.1 The Computer SimulationBased on the kinematic and dynamic model derived in the previous chapter, numerical software tools have been developed to show the feasibility of using kinematicredundancy for trajectory planning, with the objective of minimizing dynamic interaction of a robot and its support structure. Initially, a simple three degrees of freedom robotfor planar motions is used. Subsequently, the current version of a NASA traction-drive45Chapter 4. Simulations of Base Reaction Optimization 46robot with seven degrees of freedom that is capable of complex operations in the threedimensional space is used. Finally, the three d.o.f. planar robot is used for parametricstudies, aimed at design optimization.The numerical simulation is developed using the MATLAB© software [18] on a SUNSPARC station platform. The structure of the software modules is shown in Figure 4.1.The flowcharts for the simulation procedure and the algorithm for sigularity avoidanceare given in Figure 4.2 and Figure 4.3 respectively. The MATLAB© program consistsof nine main subroutines:• iiitcost.m This routine determines the best initial joint configuration that hasthe minimal cost function value.• optim.m This routine initialize the physical parameters of the program. Constants such as mass, vectors like initial position and orientation of the end effector,and arrays like the cost function are defined in this program. Also, This is themain program that contains the procedures for the optimization algorithm, including generating the end effector trajectory, and optimizing joint angles by callingother subroutines like Kinematic.m, costf.m and other.• costf.m This is the key program that calculates the instantaneous cost functionat any time along the trajectory using the Newton-Euler equations, the Jacobianmatrix, and the local cost function.• Kinematic.m This routine contains the inverse kinematic equations, and it calculates the joint angles using the end effector position and orientation vector as theinput.Chapter 4. Simulations of Base Reaction Optimization 47• grad.m This routine contains the gradient function of the inverse kinematic relationship for the robot manipulator, and it is used for faster and more efficientcomputation in the it Kinematic.m module.• Jnfunc.m This routine contains the functional equations for the normal jointJacobian matrix.• Jrfunc.m This routine contains the functional equations for the redundant jointJacobian matrix.• Jrdotfunc.m This routine contains the partial derivative of the jacobian matrixfor the redundant joint variable.• Jndotfunc.m This routine contains the partial derivative of the jacobian matrixfor the normal joint variable.The initcost.m carries out the initial optimization using an exhaustive search methodto obtain initial confguration for the robot, that minimizes the cost function.The main routine is optim.m and it performs optimization of the trajectory. It divides the end effector trajectory into many time intervals. For each interval, it callsthe MATLAB© OPTIMIZATION TOOL “fminu”to perform an optimization of thetrajectory by selecting a set of parameters c2, c3, which corresponds to selecting thetrajectory segment for the redundant joint, that would give a minimal value for the costfunction.The routine costf.m calculates the value of the cost function for a particular value ofthe redundant coordinate. The six main steps of the computational scheme are:Chapter 4. Simulations of Base Reaction Optimization 48grad.mKinematic.m’ optim.mptime.moptimization • fminsfsolvemoduleof matlab constrmatlabroutinesFigure 4.1: The Structure of the Software ModulesChapter 4. Simulations of Base Reaction Optimization 49InitializationAdjustJoint VariableLCoefficients { C,)Effector TrajectoryDividehito Time StepsCompute Cost Functioncz RQRTKmizedyesNext Step .Set q=c1t+22• Compute q• Compute Jacobian [Jr’ Jn]• Check SingularityCompute Kinematics and Dynamicsof Space Station and Robot•F,NFigure 4.2: Flowchart for Trajectory Optimization Using Kinematic RedundancyChapter 4. Simulations of Base Reaction Optimization 50Figure 4.3: Algorithm for Avoiding Singular Configuration Using Kinematic RedundancyChapter 4. Simulations of Base Reaction Optimization 511. The first joint coordinate q is chosen as the redundant variable q7. It is representedas a polynomial function of in t.qi = C1 + c2t + c3t2The parameters c1, c2 and c3, which are the coefficients of the polynomial, are tobe determined in the optimization process by calling function c= fminu (‘ costf’,[11]’); where [11] is the initial guess of the solution for [c2,c3].2. Given the values for coordinate q, the remaining kinematic variables are constrained by the kinematic equations of the manipulator. Thus, by solving theinverse kinematic problem numerially, we obtain the values for the remaining jointangles. MATLAB© subroutines Kinematic.m and grad.m are used for such computations; Kinematic.m contains the inverse-kinematic equation. Also grad.m isthe gradient matrix of the kinematic equation. MATLAB© solves the equationusing the OPTIMIZATION TOOL “fsolve2 “.3. The Jacobian matrix for a seven-degree-of-freedom manipulator is very complex.The functional form of each of the matrix elements has been derived from first principles. They are listed in the programs Jnfunc.m and Jrfunc.rn. Their corresponding derivatives are derived and listed in Jndotfunc.m and Jrdotfunc.m respectively.4. The dynamic variables of 4, q, , and 4 are then calculated using the Jacobianequations along with the values of the end effector velocity and acceleration, andJocobian matrix J and Jr. Next the determinent of J, is checked. If it is zero, 4’is replaced with another coordinate in q, and return to step 1. Else the steps arecontinued.Chapter 4. Simulations of Base Reaction Optimization 525. The joint variables e1, e, w and w, and their time derivatives are calculatedusing the dynamic equations.6. The force and torque at the base point are calculated using the Newton-Eulerequations.7. Finally for the given weighting matrix Q the Cost Function is calculated.cost = RT(QR)4.2 Simulation Results4.2.1 Simulation Results for the NASA Seven d.o.f. RobotFirst consider the seven-degree-of-freedom NASA-ORNL traction-drive robot. Inthis simulation, the space station is assumed to be orbiting the earth at geosynchronousperiod of 24 hours; this means that the circular orbital radius is approximately .1? =42330 km. The tangential velocity of the space station is calculated using the conservationof kinetic and potential energy.1 2 GmMmv3=(4.1)/2GM (4.2)where Me is the mass of the earth, and G is the universal gravitational constant.Also, from simple kinematics,Wa = (4.3)The space station parameters values used are:Moment of inertia about the axis of rotation I, = 19.7 x 106 kg.m2Mass M3 = 18.0 x iO kgChapter 4. Simulations of Base Reaction Optimization 53The link parameters values used are assumed identical links:Link length i = 5.0 mLink mass m = 20.0 kgLink moment of inertia about the joint axis I 200.0 kg.rn2The reference frame is located at the centre of mass of the space station. The end effector is assumed to move along a straight line from the point (7.5,6.5, 6.5) to (7.2,5.5, 6.2),and it is assumed to be rotated to the appropriate orientation at the end point of the trajectory. The distance travelled is y = /(7.5 — 7.2)2 + (6.5 — 5.5)2 + (6.5 — 6.2)2 = 1.09m. Assume that the time duration of motion T ( equation (2.26) ) is 2.0 seconds, theacceleration limit a ( equation (2.33) ) is 10 rn/s2 and the angular acceleration limit (equation (2.34) ) is 10 rad/s2. Then the curtate cycloidal parameters a, b1, b2 and c ( seeequations (2.23) through (2.25) ) are obtained for this specific end effector trajectory, asa = 0.31831, b1 = 0.3597, b2 = 0.0297 and c = 0.9934. The linear jerk limit ( equation(2.37) ) and the angular jerk limit ( equation (2.38) ) are calculated as a = 814.98 m/s3and a = 672.93 rad/s3,using equations (2.37) and (2.38) respectively.The end effector trajectory is divided into 200 intervals. During the initial step, theoptimal joint configuration of the robot is determined by carryiny out an optimization (an exhaustive search method ) as, q = —2.2605 rad, q = 1.1250 rad, q3 = 1.5507 rad,q4 = 2.0675 rad, q5 = —0.6290 rad, q = 1.2293 rad, and q7 = —0.1662 rad. Next, theunconstrained optimization is carried out for each successive interval of the trajectory toobtain the optimal joint trajectory.A nonoptimal case is also simulated for the same end effector trajectory, by selecting an arbitrary trajectory for qr by assigning the values for c1 = 1 and c2 = 1 for theChapter 4. Simulations of Base Reaction Optimization 54ploynomial coefficients; and then obtaining qn and other variables to satisfy the specifiedend effector trajectory.The results are shown in Figures 4.4 through 4.12. These figures compare the optimalcase with the nonoptimal case.Figure 4.4 shows the end effector trajectory for both the optimal and the nonoptimal cases, which are constrained to be identical. The time interval is divided into smallenough steps so that there would be no deviation present due to numerical errors.Figure 4.5 shows the optimal and the nonoptimal cost functions as functions withrespect to trajectory time. The relative magnitude of the optimal cost function is muchlower in comparision to that of the nonoptimal one, thereby indicating the potentialreduction of the dynamic interaction through trajectory optimization using the presentedapproach. The corresponding joint trajectories (q to q7) are shown in Figures 5.5 through4.12.4.2.2 Simulation Results for the Three d.o.f. Planar RobotThe parametric studies are carried out using a three-degree-of-freedom planar robotwith revolute joints. Its kinematic and dynamic models are given in Appendix A. Theparameters of the space station remained the same as in last simulation while the parameters of the robot are listing in Table 4.2.2. Noted that the mass of each link remainedconstant, while the link lengths and the moment of inertia for each link have been changedaccordingly. Specifically, the moment of inertia changes proportionaly to the square ofthe link length.Chapter 4. Simulations of Base Reaction Optimization 5565—645EQ:.:6350c 6.3 . . .(N 6 25626462 7456..• 7.4.• 735.. 7.35.6 7.25Y- axis Position (m) X - axis Position (m)Figure 4.4: End Effector TrajectoryThe end effector is assumed to move along a straight line, from point (7.5,4.0) to(8.0,4.5), through a distance y = /(7.5 — 8.0)2 + (4.0 — 4.5)2 = 0.71 rn. Assume thetrajectory duration T ( equation (2.26) ) as 0.7 s, the acceleration limit a ( equation (2.33)) as 10 m/s2. Then the curtate cycloidal parameters a, b1, and c of the end-effector (equations (2.23), (2.24), and (2.25) ) are obtained for this trajectory, as a = 0.11141,= 0.73406 and c = 0.75224. The linear jerk limit is computed using equation (2.37)as a = 963.43 rn/s3.The end effector trajectory is divided into 200 intervals. The initial optimized jointconfiguration is found to be q = 1.48 rad, q = —0.42 rad, q3 = —1.44 rad. Simulations are carried out with different link lengths for the parametric studies. The resultsare shown in Figures 4.13 and 4.14.Chapter 4. Simulations of Base Reaction Optimization 56101110:0CD 109C0 8 :I4o6oI841I21iiI82Time (s)Figure 4.5: Base Reaction Cost FunctionFigure 4.13 shows the optimal cost functions for different link lengths. The results ofcurve 7 and curve 8 show clearly to be better than that of other curves, indicating anenhanced optimal case. Curve 7 indicates the situation of having the length of the firstlink decreased, and curve 8 indicates the situation of the length of the first link decreasedwhile the length of the third link is increased.Figure 4.14 shows the curves 7 and 8 in Figure 4.13. Note that the link lengths forthe case of smaller cost function ( curve 1 ) are = 2.5 m, 12 = 5 m and 13 = 7.5 m,and for the larger one ( curve 2 ) they are = 2.5 m, 12 = 5 m and 13 = 5 m.It shows that the cost function is highly depended on the parameter values of linklengths. The functional dependance is complex, but can be numerically simulated andoptimized. In particular, reducing the length of the first link for the manipulator seemsChapter 4. Simulations of Base Reaction Optimization 57to have the biggest impact on minimizing the cost function values. This phenamenonis in agreement with physical interpretation because the first link produces the biggestdirect torque contribution to the base station, which is directly proportional to the lengthof the first link. Therefore, It implies there are opportunities for design optimization.4.3 Summary of ResultsUsing the framework of trajectory optimization, that was presented in the previouschapter, several simulation studies were carried out. The method utilizes the kinematically redundant degrees of freedom of a robot as means for instantaneous local optimization of a quadratic cost function of the disturbance forces and moments at thesupport structure of the robot. In this manner the best joint angle trajectories may be0.5 1Time (s)Figure 4.6: Joint 1 TrajectoryChapter 4. Simulations of Base Reaction Optimization 581.81.7nQnoptimalc’Jo1.5.5Coptimal1.21.10 0.5 1 1.5 2Time (s)Figure 4.7: Joint 2 Trajectoryplanned. The computer simulation has been developed using MATLAB© software ona SUN SPARC station platform. The software is structured into a modular high levelform. The processing time of simulation is excessive because of the complexity of theoptimization. The simulation algorithm is however robust, with a built in capability toavoid singular configuration of the robot and options to select a level of accuracy for theoptimization. Simulations were carried out based on the model of optimizing the jointangle trajectory to satisfy a specific straight-line trajectory for the end effector. Thedifference between optimal and non-optimal cost function values ranges from 2 dB to 3dB. The individual joint angle motions of the optimal trajectory are also found to besmoother than those of the non-optimal trajectory. Figure 4.13 illustrates how the costfunction changes with the link length of the manipulator. This plot shows that a locallyoptimized cost function is still functionally dependent on the physical parameters suchas link lengths of the robot. This strongly suggest that the optimization using kinematicChapter 4. Simulations of Base Reaction Optimization 591.651.6 floflopt ai1.55c 1.5optimaI1.45 I< 1.31.251.2o 0.5 1 1.5 2Time (s)Figure 4.8: Joint 3 Trajectoryredundancy may be further utilized to do design optimization. However, a method ofdefining a global cost function is required, and a technique to optimize such a cost function must be developed and implemented. This aspect is addressed in the next chapter.Chapter 4. Simulations of Base Reaction Optimization 602..2*1.8optimalo-‘ti.6Co&1.4 ---nono imaLD0,1.21 0 0.5 1 1.5 2Time (s)Figure 4.9: Joint 4 Trajectory4I..to3: nonoptimalC2oCOoptimal0—1 .____________________________________________0 0.5 1 1.5 2Time (s)Figure 4.10: Joint 5 TrajectoryChapter 4. Simulations of Base Reaction Optimization 61—- nonoptimal2.82.6co2.40‘22optimalc. .D<1.61.40:5 1 1.5 2Time ( s)Figure 4.11: Joint 6 Trajectory65o-) 3nonoptimal2ooptimalo—1 .i______________________ _____________________0 0.5 1 1.5 2Time (s)Figure 4.12: Joint 7 Trajectory-I.CD I-’ C 0 O I-. 0 CDCD CDC) z 0 0 0-I 3 cD 0)I.-. I CD CD 00 p,C00—0C.nCD z 0 —c— t.3rrrr0000.EL t.3©00©0000©. ELz0000.ELChapter 4. Simulations of Base Reaction Optimization 635x 10____________ ___________________________:3.51———3C-)1.5 •10.5o . .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time (s)Figure 4.14: Optimal Cost Function with the First Link Length DecreasedChapter 5Parameter Design Optimization5.1 BackgroundDesign engineering for space robots covers many aspect of the space system; desirableparameters include high capacity/mass ratio, low parts count, modularity in configuration, large work envelope for the activity environment, dexerterity in the end effector andalso low dynamic interaction with the supporting structure which is typically tIie spacestation.Reduction of dynamic interactions in a space robotic system by path planning formsthe basis of present thesis. This will minimize the required control effort, and thus reducecontrol fuel in the space operations. This idea may be extended to the design of a spacerobot system. Specifically, minimization of dynamic interactions between a robot and thespace station is used as the criterion for parameter design optimization that is describedin this chapter. The dynamic interaction is quantified by a mapping of reaction forcesand torques into a scalar functional. Such a mapping is expressed as a cost function,which has to be minimized in the design procedure. Both local and global disturbanceeffects have to be studied using an appropriate cost function, in the design optimization.As discussed in chapters 2 and 3, in addition to its use in dexterity improvement andsigularity avoidance, kinematic redundancy can be successfully applied to minimize the64Chapter 5. Parameter Design Optimization 65base reactions through appropriate planning of the joint trajectory, for a specified endeffector trajectory. This is possible because the extra and redundant degrees of jointangle freedom can be utilized to locally minimize a cost function in terms of reactionalforces and torques at the support structure, throughout the trajectory. In particular, it isattempted in this manner, to maintain the dynamic forces and moments at the locationof interaction with the support structure ( space station ) to be the same as referencevalues set in the cost functional formulation.A unified approach for robot trajectory optimization, design and control has beenproposed by de Silva [24]. Here the design of the robot system is linked to the problemof trajectory planning. This approach integrates the latest technology in each field in thespirit of concurrent engineering. The main steps of the procedure of the unified designoptimization are:1. Define the robotic tasks.2. Group the tasks into appropriate, general subtasks, using a group technology appropriate.3. Design the end effector trajectory for each subtask.4. Select robotic parameters for individual groups.5. Select a common set of parameters for all groups.6. Optimize trajectory of each subtask using kinematic redundancy built into therobot.7. Shape the optimal cost function by parameter redesign if needed.Chapter 5. Parameter Design Optimization 668. Design the controller to achieve the optimized joint trajectories.An effort to apply this general approach for a simple 3-R robot is given in the following section.5.2 Global Parameter OptimizationOptimization of the joint angle trajectories of a redundant manipulator to minimizethe reaction forces and torques generated at the base of the manipulator, according toa cost function defined locally along the trajectory, is the fundamental problem in thepresent work. This locally optimized cost function may be reshaped globally throughparameter changes, and could constitute a design approach for the robot system.In a unified parameter design optimization approach, it follows that, a global optimization can be quite useful. Unlike joint variables, geometric parameters such as linklengths and mass are considered global parameters. These parameters affect the dynamicforces and interactions because link lengths and inertia are directly present in the dynamic equations. For example, the link length is a global parameter in the sense thatvariations in it affects the workspace of the robot manipulator irrespective of joint angleconfigurations. Therefore, the end effector path imposes a constraint on the link lengthsand vice versa. Note that link parameters do not have the same level of freedom as thelocal variables such as joint angles.To optimize the choice of the design parameters i for a particular task, a global costfunction F2 is defined as:F2=Fdt + /3Fmax (5.1)Chapter 5. Parameter Design Optimization 67Fmax = Max(F) (5.2)where a, /3 are positive constants, t1 is the final time of the trajectory, F is the instantaneous cost function value at the i th interval, as defined in the previous chapter, andFmax is the peak value of the entire cost function along the end effector trajectory. Inother words, the global cost function is the sum of the time integral of the local costfunction and a weighted measure of the peak value of that cost function. The purposeof this cost function is to limit the peak values of the previous cost function, and also torestrain the global time average of dynamic reaction of the manipulator. In other words,in the global optimization process the local cost function would be reshaped through theproper selection of the design parameters.The parameters of optimization here are 4, which may represent, for example, linklengths and link inertia values. In what follows, we assume that the design parameters 1,are link length alone. Then, the link inertia values could be adjusted depending on thelink lengths according to same design strategy.5.3 Computer SimulationBased on the global cost function presented in the previous section, numerical simulations are carried out in the MATLAB© environment. The flowchart of the simulationsis shown in Figure 5.1. The method of optimization used is the nonuniform convergenceof a feasible set using a gradient line search algorithm.The procedure of global optimization of the base reactions using link lengths i as theChapter 5. Parameter Design Optimization 68optimization parameters, for a given end effector trajectory, is as follows:1. Assume an initial best guess for i.2. Use joint angle redundancy to minimize the instantaneous forces and moments atthe supporting structure using a quadratic cost function, with the design parametersi being kept unchanged.3. Integrate the instantaneous cost function, and identify the overall peak value ofthe cost function. Calculate the global cost function for the end effector trajectoryusing this information.4. The global cost function Fg is used to globally optimize the system using linkparameters 4 subject to physical constraints. During each search step, goto Step 2to obtain a best fit joint trajectory for a specific end effector trajectory.The flowchart of computer simulation of this design procedure is shown in Figure 5.15.4 Results and Discussions5.4.1 Optimization ResultsA 3-R planar robot manipulator in 9?2, as shown in Figure 5.2, serves as the model forthe computer simulation, for parameter optimization. Assume that the reference frameis located at the centre of mass of the space station, with the x— y plane coincidingwith the motion plane. The link length is denoted by 4. The joint angle q is the anglebetween link 1 and x-axis. The joint angle q2 is the angle between link 2 and link 1, andq3 is the angle between the links 3 and 2.Chapter 5. Parameter Design Optimization 69InitializationSelect Link Lengths [.4AdjustJoint VariableCoefficients { C! }Divide End Effector TrajectoryintoTime Steps12Set q=c1t’c2t• Compute q• Compute Jacobian [Jr’ Jn I• Check SingularityCompute Kinematics and Dynamicsof Space Station and Robota Wi,’VCI,• F, NCompute Local Cost FunctionfeRQRTnoCompute Global Cost Functiontffgaffcdt+13fcyesnoFigure 5.1: Algorithm for Design Optimizatiom of Robot ParametersChapter 5. Parameter Design Optimization 70XbYsxsFigure 5.2: A Three-Degree-of-Freedom Planar Robot Mounted on a Space StationChapter 5. Parameter Design Optimization 71The space station is assumed to be orbiting the earth at a geosynchronous period of24 hours. The following parameters values are used for the space station:Moment of inertia about the axis of rotation I,, = 19.7 x 106 kg.m2.Mass M3 = 18.0 x iO kg.The link mass values used areLink mass m1 = 20.0 kg.Therefore, the moment of inertia will also be changing as the square of the link length,which is to be optimized.An initial guess of link length is set at 1 = [5 5 5]. In each optimization searchstep, for every change in link length, program calls the local cost function optimizationroutines to obtain the best set of joint trajectories.The optimization program is set to search for the best numerical solution that satisfya given physical constraint [20].ll+l2+1315 (5.3)l3120.512 13The constraint equations set the limits of the workspace of the manipulator as well asthe ranges of the values for link 2 and link 3 based on physical considerations.During local optimization period, the end effector is assumed to move along thesame straight line, from point (7.5,4.0) to point (8.0,4.5), traveling the distanceChapter 5. Parameter Design Optimization 72y = J(7.5 — 8.0)2 + (4.0 — 4.5)2 = 0.71 m. Assume that the time of the trajectory motion ( T in equation (2.26) ) to be 0.7 second, and the acceleration limit ( a in equation(2.33) ) to be 10 rn/s2. Then the curtate cycloidal parameters a, b1, and c are obtainedfor this specific end effector trajectory as, a = 0.11141, b1 = 0.73406 and c 0.75224.The end effector trajectory is divided into 200 time intervals. The starting joint configurations are obtained through optimization within the first step, and was found to beqi = 1.48 rad, q = —0.42 rad, q3 = —1.44 rad, This remained unchanged in eachglobal optimization.The globally optimal set of link variables was found to be 1 = [1 7 7] with the globalcost function value of f9 = 1.19 x iO. Figures 5.3 and 5.7 present the results. Figure5.3 gives the end-effector trajectory used in the simulation. In Figure 5.4 the locallyoptimized cost function for link lengths 1 = [5 5 5] is shown as the reference ( curve 2) with which the global optimization began. In the same figure, the instantaneous costfunction for the final step of global optimization, which corresponds to 1 = [1 7 7], isalso shown. As expected, the instantaneous cost function for 1 = [1 7 7] is found to besubstantially smaller than that of 1 = [5 5 5] throughout the end effector trajectory.This shows the potential of using the dynamic reactions at the support structure as theperformance index.5.4.2 Summary of ResultsBased on the simulation model for trajectory planning using kinematic redundancy,a global cost function is defined as the time integral of the local cost function with anadditional term given by a linear function of the overall maximum value of that localcost function. The optimization algorithm uses a recursive technique to optimize theChapter 5. Parameter Design Optimization 73Figure 5.3: End Effector TrajectoryEC0Co00Cl)>-.5 7.6 7.7 7.8X-axis Position (m)7.9 8Chapter 5. Parameter Design Optimization 7487Time (s)Figure 5.4: Optimal Local Cost Function Values for 1=[1 7 7] (curve 1) and I = [5 5 5j( curve 2 )Chapter 5. Parameter Design Optimization 752.32.2—‘ 1:21.8O : :o : :c. . . . .D : : :• •.C •<1.61.514I I I I I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time(s)Figure 5.5: Joint 1 Trajectory for 1= [1. 7 7] (curve 1) and 1= [5 5 5] (curve 2)Chapter 5. Parameter Design Optimization 76-0.4-0.5---1—-0.7.cj : .C . . .---2. . .<-1.2-1.3-14I I I I I I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7lime(s)Figure 5.6: Joint 2 Trajectory for 1= [1 7 7 (curve 1) and 1= [5 5 5] (curve 2)Chapter 5. Parameter Design Optimization 77-0.2-0.4•D :Co.: :Co : :C.—,-u.o‘I-.oC.o--—2-. .o : :0- :Co.D.0)• : : :—1 6 I I I I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time(s)Figure 5.7: Joint 3 Trajectory for 1 = [1 7 7] (curve 1) and 1 = [5 5 5] (curve 2)Chapter 5. Parameter Design Optimization 78link length parameters globally and the joint trajectory locally, in succession. In thismanner, the link lengths of the robot are optimized as well as the joint trajectories. Thelink lengths are optimized with respect to a set of physical constraints while the jointtrajectories are considered unconstrained. The computer simulation were carried out fora three-degree-of-freedom, revolute, planar manipulator. This method of optimizationtakes many hours of CPU time because of the high level of computational complexityof the global optimization. However, the results indicate that the algorithm and thenumerical model are robust, with the cost function of the final optimal design being 1dBto 2dB below that of the initially, locally optimized case. As shown in the computersimulation results, a set of optimized design parameter values is obtained as 1 = [1 7 7].It is expected that the optimal link length values will always be depend ent on the definition of the global cost function and the physical link constraints. This is a brute forcemethod of doing design optimization. The methodology is essentially a local analysisapproach because the global cost function is still directly related to the locally optimizedjoint trajectory. This functional relationship is embeded in the cost function definition,and the looped optimization procedure. A more global approach to exploit the fundamental relationship between link parameters and the cost function would be desirable.In the next chapter, a relatively global technique for analysing of inverse kinematics of aredundant manipulator from a topological point of view will be used for the parameterdesign optimization of the manipulator.Chapter 6The Trajectory Manifold Approach6.1 IntroductionIn the investigation described in the previous chapter, the trajectory optimizationof robots has been based on the instantaneous kinematics approach. This approach hasbeen applied to both local trajectory optimization and global design optimization withthe cost function that expressed in terms of the dynamic forcs and torques at the support structure. Burdick [27] has, however, look at the global application of redundantkinematics of manipulators from a topological point of view. Here, the infinite numberof trajectory solutions, which is possible in view of the kinematic redundacy, is groupedinto a finite and bounded set of disjoint and continuous manifolds, known as self motionmanifold. This concept of self motion manifold for a redundant robot manipulator is extended in the present chapter into a systematic and convienient method for redundancyresolution and parameter design optimization.Smooth surfaces may be generalized as manifolds in an abstract, multidimensionalhyperspace. In manifold theory, a smooth n dimensional surface may be characterizeby a set of flat Euclidian surfaces in the n dimensional Eucidian space. In such aframework, useful mathematical concepts like differentiation become meaningful in theneighborhood of any locally defined coordinates. The concept of manifold has usefulapplications in robotics because the trajectories of joint angle coordinates for a general n79Chapter 6. The Trajectory Manifold Approach 80degree-of-freedom manipulator can be treated as an ri dimensional manifold flow, and themathematical tools used in manifold theory can be applied to solve robotics problems.This can provide new insights into understanding robotic dynamics through a new geometric perspective. In particular, for kinematically redundant manipulators, manifoldtheory allows for the resolution of infinitely many possible trajectories into a finite groupof submanifolds with similar characteristics. Furthermore, this approach allows for theimplementation of a global cost function through a natural definition using the conceptof a path integral along a family of submanifold.The forward kinematic function of a manipulator is a nonlinear vector function thatmaps joint angle vectors into the end effector position and orientation vectors. For aredundant manipulator, there exist infinitely many joint angle configurations that can bemapped into a single end effector position and orientation. However, this set of infinitepoints can be grouped into a finite set of manifolds, each with a set of distinct globalgeometric characteristics. These submanifolds are known as “self-motion” manifolds. Inother words, the motion in joint angle configuration in a self-motion manifold leaves theend effector motionless.From the foregoing discussion it should be clear that a specific manipulator task mapsout a specific end effector trajectory, which in turn maps out a family of flow submanifold, or “self-motion” submanifold. By defining the global cost function, which representsdynamic interactions between the robot and its support structure, along each submanifold, the design parameters of the robot system can be optimized by computing the costfunction. Here the concept of Path Integrals is introduced. Since the full descriptionof reaction forces and torques is complex for symbolic computation, only the first orderapproximations of F and N are considered. In other words, only higher order terms inChapter 6. The Trajectory Manifold Approach 81F and N containing link variables l are considered in evaluating the path integral inthe global cost function calculation.6.2 Manifold Formulation of Manipulator KinematicsIn order to analyse the global characteristics of redundant manipulators, one needsto understand the kinematics of the robot in terms of manifold mapping instead of localkinematic analysis [26], [27].6.2.1 Forword KinematicsLet Q = {q} be the set of joint angle coordinates for the manipulator. Here, {q2} isof dimension n. Let X= {} be the set of end effector coordinates in vector spaceof dimension m. Note that, for kinematic redundancy of a manipulator we must havenm+1The forward kinematics of a manipulator may be expressed as a nonlinear vector function which relates a set of Ti joint coordinates, Q, to a set of m end effector coordinates:X= f(Q) (6.1)The forward kinematic function in equation (6.1) maps a given joint configuration,Q, to a unique end effector location, X. The set of joint configurations forms the “configuration space”, C, and the end effector locations form the “workspace”, W. Therefore,the forward kinematic function can be understood as a mapping of points from the configuration space to the workspace.f(Q) : C = W (6.2)Chapter 6. The Trajectory Manifold Approach 82where, C = configuration space; W = workspace. The configuration space has an ndimesional torus manifold. The workspace has a sheet liked structure.6.2.2 Inverse KinematicsThe inverse-kinematics relation is given by:Q = f(X) (6.3)For a redundant manipulator, and a given end-effector position and orientation, thereexists an infinite number of solutions in {q2} to satisfy the f operation. In other words,f is a one-to-many mapping, that is, for a fixed X = {z1} then Q = {qj is not acompact set. However, a non-redundant manipulator would have a bounded set of discrete and finite points in the configuration space for a given end effector position andorientation in the workspace.For a redundant manipulator, let r = n — m be the degree of redundancy. Theinverse-kinematic solution is represented as a group of one or more disjoint r-dimensionalmanifolds M, such that,f_i(X)= UM() (6.4)andM2()flM3b)= 0 V i j (6.5)Note that M2 is the i th r-dimensional submanifold, with i = 1, 2, . . s, where, s isthe number of self motion in the inverse kinematics solution. Furthermore, is a vectorof r independent parameters, b =...,/i,.}, which are the generalized coordinates forthe self motions.Chapter 6. The Trajectory Manifold Approach 83Each of these submanifolds is thought of as a “self-motion” manifold, which is a setof continuous joint motions that has no effect on the end effector motion.6.2.3 Jacobian RelationsDifferential relationship between the joint displacement and the end effector positionis given by:ax = J6Q (6.6)where J is the m x n Jacobian matrix of the redundant manipulator. The incrementalself motion which corresponds to the incremental joint motion 6Q, is given by thedifferential relationship.Si/i— J, SQ (6.7)where J is the r x n Jocabian matrix, and r = n — m.Let us consider the end effector coordinates and the self motion variables as an augmented task vector {X, i/i } . The differential relationship corresponding to this new setof generalized coordinates is:ax= JA SQ (6.8)where JA is an n x n augmented Jacobian matrix, which is given byax,aqJA = (6.9)8b/ãQNote that the augmented Jacobian JA will be singular whenever J loses rank and/orwhen J is not full rank. Assuming that this is not the case, the joint velocities areChapter 6. The Trajectory Manifold Approach 84obtained from equation (6.8), asxQ = (6.10)1’and the joint accelerations are obtained by differentiating equation (6.8), asxQ = .. — JC’JAQ (6.11)7,1’6.2.4 Manifold Solution for a 3-R RobotConsider the three-degree-of-freedom planar robot that has been discussed in theprevious chapters. There are two possible sets of joint configurations for this 3-R manipulator.1. 62a, 83a} = {(cz + 77), (,‘ — 7r), ( — a —77 —7 + ir)} (6.12)2.{81b,2b,O} = {(a—77),(1r—7),(’’—a+77+7—r)} (6.13)where,a = tan’( Ye — 13 siflb) (6.14)Xe — 13 C05cos’(21112(6.15)12 12 R277 co&(211122) (6.16)(6.17)Chapter 6. The Trajectory Manifold Approach 85Here, {Oia, °2o 6} is a set of joint motions in manifold a. {Olb, 62b, 8} is a set of jointmotions in manifold b. {e, ye} is the position vector of end-effector. Also, b is a redundant independent variable which is defined as the orientation of the third link relativeto the base coordinate frame, and 11, 12, 13 are lengths of links 1, 2, and 3 respectively.As b changes in its feasible range [7r, —7rj, equations ((6.12) and (6.13) ) will define two1-dimensional manifolds in the configuration space. These manifolds are disjoint for allvalues of tb unless the Jacobian submatrix formed by links 1 and 2 is not full rank.6.3 Design Parameter OptimizationAs in the previous chapter, the objective of the parameter design problem for a spacerobot is considered as the optimal selection of the most suitable set of link length {l}for the manipulator such that the dynamic interactions with the support structure areminimized in a global manner. In particular, we wish to select a set of suitable designparameters such as link lengths to find an appropriate trajectory £, such that,jFdt = mm [jFdt} (6.18)subject to the following constraints:11 + 12 + 13 = 15 (6.19)l3120.512 13where F is the global cost function. F is path dependent, and hence depends on thefundamental joint motion variable such as q, j, and . F is also dependent on the designChapter 6. The Trajectory Manifold Approach 86parameters {l}The cost function used for parameter optimization must be global and representativedescription of the dynamic objective ( e.g. minimization of the dynamic interaction ).Also, it must explicitly depend on the design parameters. However, for the simplificationof numerical simulations, the partial functional dependance is considered to include themost significant contributing terms. Suppose that the functional form of the global costfunction isFc=+ 4qkllk)dt (6.20)The first term of the integrand is representive of the direct inertia forces while the secondterm is representative of the cross coupling effects between joints.The procedures used in the design parameter optimization using the monifold methodare as following:1. Initialize the self motion parameter 1’ and the design parameters (link length i ).2. Generate the end effector trajectory. Divide the trajectory into a suitable numberof time intevals.3. Compute the multiple set of self-motion manifolds.4. Compute the augmented Jacobian matrix JA, and also JA’, JA, along with theend-effector velocity e and acceleration a. Compute the joint velocities Q andaccelerations Q.5. Repeat Steps 2, 3 and 4 for the complete trajectory.Chapter 6. The Trajectory Manifold Approach 876. Compute the global cost function for each self motion monifold. Select the manifoldwith the smallest cost function value, for use in the optimization.7. Increment the link lengths so as to minimize the cost function, subject to thephysical constraints of these parameters.8. If the minimum cost is not reached, go to Step 2. Else stop.6.4 Computer SimulationAs before, let us consider a 3-R planar robot manipulator in J2 as the model forthe computer simulation. This robot and supporting space station are schematicallyshown in Figure 6.1. Assume that the reference frame is located at the centre of massof the base, with the x— y plane coinciding with the motion plane. The link lengths aredenoted by l. The joint angle q1 is the angle between link 1 and the x-axis. The joint angle q2 is the angle between links 2 and link 1, and q3 is the angle between the links 3 and 2.Let the self motion parameter /‘ be defined as the orientation of the third link relativeto the x— y — z inertial reference frame. For a given end effector position (se, ye), thereare two possible sets of joint configurations.Suppose that the global cost function is defined by:F=(6.21)In order to compute the global cost function, let us assume that the initial positionof the end effector is set at {a,,y0}. Also the end effector velocity v and acceleration aare obtained from cycloid motion assumptions as described in Chapter 2.Chapter 6. The Trajectory Manifold Approach 88t2.XbYsxsFigure 6.1: A Three-Degree-of-Freedom Planar Robot and a Supporting Space StationChapter 6. The Trajectory Manifold Approach 89The self motion parameter b, which is defined as the angle between the third link ofthe robot and the inertial reference frame, is given by:‘‘=q1+q2+q3 (6.22)The Jacobian matrix J of the 3-R planar manipulator is given by:J= us1 —— 13s23 —12s12— 13s23 —13s123(6.23)lid— 12Ci — 13C23 —l2Ci—l3Ci23 —l3Ci23and the Jacobian matrix J corresponding to incremental self motion S’çb and the incremental joint angle SQ is given by:= [] = [111] (6.24)the augmented Jacobian matrix JA, is given by:us1 — 12512 — 13S23 —12S12 — 13S23 —l3S123JJA = = u1c— 12 C12 — 13c23 —i2c12 — u3c123 —i3c123 (6.25)Jb1 1 1Now, the joint velocities can be obtained from equation (6.10), and the joint accelerations from equation (6.11).The space station model is assumed to be the same as in the previous simulations.The robot parameters are as follows:Link length 1: = 5.0 m.Link mass m1 = 20.0 kg.Link moment of inertia about the joint axis I = 200.0 kg.m2.It is assumed that the end effector moves along the straight line, from point (7.5,4.0)to (8.0,4.5), as before through a distance y = /(7.5 — 8.0)2 + (4.0 — 4.5)2 = 0.71 m.Chapter 6. The Trajectory Manifold Approach 90Assume that the time duration of the trajectory ( T in equation (2.26) ) is 0.7 s, thetrajectory acceleration limit ( a in equation (2.33) ) is 10 rn/s2. Then the curtate cycloidalparameters a, b1, and c are obtained for this specific end effector trajectory, as a =0.11141, b1 = 0.73406 and c = 0.75224. The linear jerk limit ( as given by equation(2.37) ) is obtained as a 963.43 rn/s3. The end effector trajectory is divided into200 time intervals. The global cost function is calculated for the two manifolds, and thesmaller is chosen for the cost function optimizatiom. The optimal set of link lengths isfound to be 1 = [5.4 6.4 3.2 J with the minimum value of the global cost function asfg = 4.78 X iO.Table 6.1 presents a set of global cost function values and the corresponding linklengths. The optimal result is given by the shaded row.6.5 Summary of ResultsIn this chapter, the idea of flow manifolds, the separation of inverse-kinematic solutions of a robot into self motion manifolds, and the application of the concept of selfmotion for design optimization have been developed. Base on the definition of the globalcost function as a path integral of design parameters on individual self motion manifolds,the cost function was computed along all self motion manifolds using the augmented Jacobian matrix, and the one with minimum value was set to optimize for the best designparameters. The global cost function is defined through a heuristic interpretations ofthe dependance of the base dynamic interaction on the link parameters, such that only afew quadratic terms are used to approximate the functional relationships. This methodwas used to simulate the parameter design optimization for a 3-R planar robot, andsimulation results shown on Table 6.1 indicated that the method is capable for designoptimization. This is another alternative to the brute force design optimization processChapter 6. The Trajectory Manifold Approach 91Table 6.1 Cost Function Values versus Link LengthsLink Lengths Cost FunctionlI 12 132.0 7.0 6.0 7.16x10e33.6 4.6 6.8 7.80x10e33.8 6.4 4.8 5.62x10e34.0 7.6 3.4 4.90x10e34.4 4.0 5.6 5.94x10e34.8 5.2 5.0 5.77x10e354 64 32 { 478x1OeS5.0 6.6 3.4 4.82x10e35.2 6.0 3.8 4.99x10e36.0 5.4 3.6 5.04x10e37.8 3.2 4.0 5.41x10e37.6 4.0 3.4 5.61x10e38.0 4.0 3.0 5.66x10e3Chapter 6. The Trajectory Manifold Approach 92as described in the previous chapter. This method provides a more transparent designoptimization because of the simplified cost function. Also, as a result of the computationsbeing able to be done symbolically rather than numerically, the computed global costfunction becomes a simple polynomial ( quadratic ) function of the design parameters.This method has its limitations as well. One is the difficulty to resolve the redundant,inverse kinematics into self motion manifolds for a manipulator with large number ofdegrees of freedom. Another is the error introduced as a result of using a simplified costfunction.In comparision to the results obtained by the method of the previous simulations,the optimal set of link variables obtained by the manifold method, in this Chapter, isdifferent for the following reasons:1. The global cost functions used are different. In the manifold method, the symboliccost function consists only of a few terms expressed as a quadratic form of l,whereas the cost function in the original simulation in Chapter 5 uses the completeexpression for the base reactions.2. The approach in Chapter 5 utilizes the joint angle redundancy to locally minimizethe instantaneous forces and moments along each interval of the end effector trajectory, whereas in the manifold method, the joint angles are computed using themanifold equations.Topology is the study of global geometric characteristics of mathematical objects,such as the curvature of surfaces, which is invariant under continuous transformations.For deisgn optimization, this approach is suitable and convienient because of many simiplications that can be used. These advantages include the ability to an group the infinitenumber of inverse-kinematic solutions into a finite number of classes which depend onChapter 6. The Trajectory Manifold Approach 93the link parameters. This provides a more heuristic and direct definition of a g1obaI costfunction with respect to design parameters, even though not exact. Also, a simplifiedunderstanding of the functional relationship between the design parameters and the costfunction is possible through such a cost function. Thereby, the optimization complexityis reducded which is an advantege in numerical simulations.Chapter 7Conclusion and Recommendations for Future Work7.1 Summary of the ThesisAdvances in the space technology and the development of robotics will lead to moreand more sophisticated robotic designs and advanced control techniques. Unwanteddynamic disturbances can adversely affect the performance of a space robot system.The objective of this research has been to explore and develop a methodolgy for theoptimization of robot design parameters, by exploiting kinematic redundancy in therobot. The scope of this research includes the analytical formulation and computersimulation of the optimization of trajectory planning using kinematic redundancy, andthe development of approaches and software tools for optimal parameter design of arobot. Initially, a robot manipulator with redundant degree of freedom is consideredfor minimization of dynamic interaction between the robot and its support structure.Using this as the preformance measure as expressed by a quadratic cost function, localoptimization along the end effector trajectory, for a specific task, is formulated using theredundant joint motion as the means of optimization. On this basis, numerial modelshave been built using the MATLAB© software language. Computer simulations for both3-R planar robot and the NASA seven d.o.f. traction drive manipulator have shownthat the base reactions can be effectively minimized through the optimization process.These software tools are subsequently extended for the analysis of the parameter designproblem for a robot. Two approaches have been developed for this purpose. The first is a94Chapter 7. Conclusion and Recommendations for Future Work 95brute force method that numerically describes the global cost function for the parameterdesign problem as the sum of a time integral of the previously explored instantaneous costfunction and the scaled maximum of the instantaneous cost function. The second methoduses the mathematical concept of topological manifolds to separate the trajectory flowsof redundant manipulators into different but finite number of “self motion” submanifoldsthat have similar geometric characteristics. In each submanifold, the end effector ismotionless, and hence an optimization may be carried out with a proper definition ofglobal cost function for base reaction minimization. This function is typically a simplequadratic of the design parameters and can be easily computed and eventually optimized.Even though there are computational advantages, because the cost function is simplified,the design results become less accurate. Both methods have been verified through thecomputer simulation of a 3-R planar robot manipulator, and have shown to performeffectively.7.2 Major Contributions and Shortcomings of the ResearchThe key accomplishments of this research are:1. Formulation of the optimization problem for minimizing dynamic interactions between a robot and its support structure, in terms of kinematic redundacy.2. The MATLAB® modelling of the associated trajectory optimization problem.3. The development of global techniques for design parameter optimization. Thisincludes:(a) Definition of a global cost function.(b) Development of computer simulation algorithms for optimizing the global costfunction.Chapter 7. Conclusion and Recommendations for Future Work 96(c) The application of the manifold flow concept to redundant robots for computing the multivalued global cost function as a path integral along the redundantd.o.f. trajectory.(d) Development of computer simulation for carrying out the design optimizationprocess.(e) Carrying out several computer simulations to illustrate the techniques.Even though an approach for design optimization has been developed, which wasfound to be effective, there are still some issues that need to be addressed, and techniquesthat need to be further developed. For example:1. The overall optimization process is computationally inefficient.2. The results of the design optimization depond on the specific definition of theglobal cost function. The numerical stability of the optimization methods, resultsto various parameters of the deisgn problem have not been investigated.3. The manifold flow method, in this stage, serves only as illustrative purposes. Inparticular, only a quadratic function in the design parameters is used to computethe path integral which forms the global cost function.4. In the design simulations, only the link length have been used as the optimizationparameters. The explicit use of other parameters such as mass, moment of inertia,and strength, in the design has not been addressed.Chapter 7. Conclusion and Recommendations for Future Work 977.3 Recommendations for Future WorkThis thesis has made contributions towards the use of kinematic redundancy forminimizing dynamic interactions between a robot and its support structure, when carrying out specific classes of trajectory following tasks by an end effector. This research hasalso made progress towards the development of a robot parameter design methodology,on that basis. In order to enhance the theory and to realistically apply this work, thereare several reconmmendations that can be made for future research:1. Development of efficient numerical algorithms for constrained optimization of anonlinear global cost function.2. Extension of the concepts developed here to include other robot parameters andstructures.3. Development of control techniques that will guarantee the execution of the designed,optimal trajectories.4. Enhancement of the design optimization to cover several robotic tasks, throughfor example, a group technology approach.5. Application of the techniques to a realistic design, and evaluation of the design.7.4 Application of the WorkSince the software modules developed here are general, they may be applied tosimulate any robot. Specific application considered here is in space robotics. For example,the methodology developed can be used to design a robot manipulator for space stationoperation. Consider a robotic task such as deployment of a scientific instrument like theHubble Telescope using a redundant Canadian Arm on a NASA space shuttle mission.Chapter 7. Conclusion and Recommendations for Future Work 98For the purpose of saving fuel and minimizing the dynamic disturbance on the spacestation, the design parameters of the Canada Arm can be optimized according to thetask specifications of the mission. First, kinematic and dynamic requirement of the endeffector of the arm are used to calculate the kinematic parameters in the curtate cycloidalmotion planning for the manipulator. Secondly, physical considerations on dynamicdisturbance can be translate into an appropriate weigthing matrix Q for the descriptionof the cost function at the base station. The robot arm trajectory, either a straight lineor more complex motion, is used to help generate the appropriate joint trajectory for therobot arm using inverse kinematics in the kinematic.m software module. The link lengthoptimization for the Canada arm can be done using the brute force method described inChapter 5. In this case, the Jacobian matrix for the Canada arm need to be derived, thephysical parameters of mass, moment of inertia, and the parameters in the description ofthe global cost function all need to be realistically estimated according to the engineeringspecification of the space mission, but the methodology of design optimization is identicalto the procedures outlined in the thesis. After the optimal selection of link parameters,the redundant trajectory planning software module can also be applied to generate thebest joint angle path for the Canada Arm to deploy the instrument from the start pointto the end point according to the specified end effector trajectory, either a straight line ora curve. However, there is of course the need to design controllers to regulate the jointsof the robot accordingly. But nevertheless, the Canada Arm can now deploy a payloadswiftly and smoothly.Bibliography[1] De Silva, C.W., “1987 NASA-ASEE Summer Fellowship at the Lewis Research Center”, Final report of the NASA-ASEE Case-lewis summer faculty fellowship program,pp. D-47-54, September 1987.[2] De Silva, C.W., Ching, C.L., and Lawrence, C., “Base Reaction Optimization ofRobotic Manipulators for Space Applications”, proc. of the 19th International Symposium on Industrial Robots, Springer-Verlag, New York, Nov. 1988, pp. 829-852.[3] Vafa, Z. and Dubowski, S., “Minimization of Spacecraft Disturbances in SpaceRobotic Systems”, Proc. 11th AAS Guidance and Control Conf. Keystone, 00,1988.[4] Papadoponlos, E. and Dubowsky, S., “ On the Nature of Control Algorithms forSpace Manipulators” proc. IEEE International Conf. on Robotics and Automation,pp. 1102-1108, 1989.[5] Smith,J.H., Estus, J., Heneghan, C., and Nainan, C., “The Space Station SreedomEvolution-Phase: Crew-EVA Demand for Robotic substitution by Task Primitive”,proc. IEEE International Conf. on Robotics and Automation, Scottsdale, AZ, pp.1472-1477, 1989.[6] Yamada, K. and Tsuchiya, K., “Trajectory Planning for a Space Manipulator”, Proc.AAS/AIAA Astrodynamics Specialist Conference, Stowe, VM, Paper No. AAS89-440, August 1989.[7] De Silva, C.W., “Trajectory Design for Robotic Manipulator in Space Applications”.J. Guidance, Control and Dynamics, Vol.14, No.3, pp. 670-674, 1991.[8] De Silva, C.W., “Use of Kinematic Redundancy for Minimizing Dynamic Interactionin Space-Station Robots”, Modeling and simulation, Instrument Society of AmericaPublication, Vol. 24,Part 4,pp. 2231-2238, April 1992.[9] Wee, L.B. and Walker, M.W.,”Space Based Robot Manipulators: Dynamics of Contact and Trajectory Planning for Impact Minimization”, Proc. American ControlConference, Chicago,IL ,pp. 771-775, June 1992.[10] Shigley, J.E., Theory of Machines and Mechanisms, McGraw Hill Book Co., NewYork, 1980.99Bibliography 100[11] Herndon, J.N., Hamel, W.R., and Kuban, D.P., “Traction-Drive Telerobot forSpace Manipulation”, proc. IEEE International Conf. on Robotics and Automation,Raleigh, NC, 1987.[12] Kuban, D. P., and Hamel, W.R., “Application of A Traction - Drive Seven - Degreesof Freedom Telerobot to Space Manipulation”, proc. American Astronautical Society10th Annual Guidance and Control Conference, Keystone, Colorado, 1987.[13] De Silva, C.W., and Hankins, W.W., “ Dynamic Evaluation of the NASA-ORNLtraction-Drive Joint”, Journal of Guidance, Control, and Dynamics, vol. 14, no. 3,pp. 688-692, May-June 1991.[14] Luh, J.Y.S., Walker, M.W., and Paul, R.P., “On-Line Computational Scheme forMechanical Manipulators”, J. Dynamic systems, Measurement, and Control, Trans.ASME, vol. 120, pp. 69-76, 1980.[15] Asada, H. and Slotine,J.J.E., Robot Analysis and Control, Wiley, New York, 1986.[16] Fu, K.S., Gonzalez, R.C., and Lee, C.S.G., Robotics: Control, Sensing, Vision, andIntelligence, McGraw-Hill, New York, 1987.[17] Ranky, P.G., and Ho, C.Y., Robot Modelling Control and Applications with Software,Springer-Verlag, New York, 1985.[18] MATLAB User’s Guide, Published by The MathWorks, Inc. 1990.[19] Rao, S.S., Optimization Theory and Applizations, John Wiley and Sons, New York,1985.[20] Mayorga, R.V., Ressa, B., and Wong, A.K.C., “A Kinematic Criterion for the DesignOptimization of Robot Manipulators”, proc. IEEE International Conf. on Roboticsand Automation, Sacramento, CA, pp. 578-583, 1991.[21] Wolfe, M.A., Numerical Methods for Uncontrained Optimization, Van NostrandReinhold Company, New York, 1978.[22] Gill, P.E., and Murray, W., Numerical Methods for Constrained Optimization, Academic Press, London,1974.[23] Stadler, W., Multicriteria Optimization in Engineering and in the Sciences, PlenumPress, New York, 1988.[24] De Silva, C.W., and Wang,Y., “Use of Kinematic Redundancy for Design Optimization in Space-Robot Systems”, proc. 1993 American Control Conference, SanFrancisco, CA, vol. 2, pp. 1806-1810, June 1993.Bibliography 101[25] Ighal, H. Dubey, R.V., and Euler, J.A., “Efficient Gradient Projection Optimizationfor Manipulator with Multiple Degree of Redundancy”, , proc. Internationa Conf.on Robotics and Automation, pplOO6-lOll, 1990.[26] Burdick,J.W. and Sereji, H. “Charecterization and Control of Self Motions in Redundant Manipulators”, NASA Conference on Space Telerobotics, Pasadena, Calif.Vol. 2, pp.3-14, 1989.[27] Burdick,J.W., “ On the Inverse Kinematics of Redundant Manipulator: Characterization of the Self-Motion Manifolds”, proc. International Conf. on Robotics andAutomation, Scottsdale, AZ, Vol.1, pp264-2’TO, 1989.[28] Gottlieb, D.H., “Robots and Topology”, proc. International Conf. on Robotics andAutomation, San Francisco, CA, pp. 1689-1691, 1987.[29] Guillemin, V., and Pollack, A., Differential Topology, Prentice-Hall, Inc., EnglewoodCliff, New Jersey, 1974.[30] Massey, W.S., Algebraic Topology: An Introduction, Springer-verlag, New York,1967.Appendix AAppendixA.1 Kinematics and Dynamics of a Three d.o.f. Planar RobotFor a 3 d.o.f. planar robot with rotary joints, the Jacobian matrix is given by:J= —us1 — 12s2 — 13s23 —12s12 —13s123(A.1)—lid— 12C2 — 13C23 —l2Ci —l3Ci23The positions of the centers of mass of the links with respect to the base referenceframe are:(r1) lCCr1 = = (A.2)(r1) —lcisi(r2) l2C1 +42C12== (A.3)(r) l2S1 +l2S12(r) 42C1 +l2C12 +143C23rC3 = = (A.4)(r3) 1c25 + lc2Sl +l3S123whereC1 = cos 8, S1 = sin 8, C1 = cos(81 + 63), S, = sin(62 + 6,),C, = cos(61 + 63 + 6k), St3 = sin(61 + 0, + Ok)102Appendix A. Appendix 103The velocities of the centers of mass of the links are obtained by differentiating equations (A.2) through (A.4):— id 1S1= (A.5)id 1C1—iC2(qL +2)S12= (A.6)lciqiCi + 1c2(ql +2)C12—l1qSi — 1d2(ql +2)S12 — lC3(1 + 2 +3)S123= (A.7)liqiCi + 1c2(ql +2)C12 + 1d3(ql + 2 +43)C123The accelerations of the centers of mass of the links are obtained by differentiatingequations (A.5) through (A.7):1 2’ IciqL’i — ‘ciq1’i= (A.8)— id ?si + C1—l1qC — liij1S— l2( +)2C12 — Ld2(l +2)S12= (A.9)—l1qS + liq1Ci —l2(q +)2S12 + 1c2(ql +.2)C12— L1jS — l2(q +)2C12— i2(ql + 1j2)S2.—ld3(q + + j)2C123 — lC3(ql + ‘2 + )S123Vc3 =—l1qS—i1jC—i2(q + 4)2S12—l2(1 +2)C12—lC3(q + 4 +)2C123 — id3(ql + q2 + )S123The angular accelerations of the three links are:= (A.10)w2=q1+ (A.11)Appendix A. Appendix 104w3=ql+q2+ (A.12)The base reaction forces and moment are given by:= mi(ii) +m2(i)+m3(i’) (A.13)= mi(ii) +m2(i)+m3(i) (A.14)mi(r1)(i’i) — (A.15)+ m2(rC)(vC2)Y—m2(rC)Y(vC2)+ m3(rC)(vC3) —+ Ic14) + 1c2W + IC3W3

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