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PAGE 1 1 ADAPTIVE FORCE AND MOTION CONTROL OF ROBOT MANIPULATORS IN CONSTRAINED MOTION WITH DISTURBANCES By YUNG SHENG CHANG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013 PAGE 2 2 2013 Yung Sheng Chang PAGE 3 3 To my wife, Hsiao Chia Hu PAGE 4 4 ACKNOWLEDGMENTS I would like to thank my committee chair, Dr. Carl D. Crane III for his patient guidance and for giving me the great chance to study at the Center for Intelligent Machines and Robotics (CIMAR). I also would like to thank my committee members, Dr. Warren Dixon and Dr. Gloria J. Wiens for their valuable advice to this study. In addition, I would like to thank my companions at CIMAR. From them I learned lot knowledge about robotics. Thanks also go out to my family and wife for their support and love. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 7 ABSTRACT ................................ ................................ ................................ ..................... 9 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 11 2 CONSTRAINED ROBOT MANIPULATOR DYNAMICS ................................ ......... 14 2.1 Euler Lagrange Dynamic Model ................................ ................................ ....... 14 2.2 Model of Contact Surface ................................ ................................ ................. 16 2.3 Dynamic Model of Constrained Robot Manipulator ................................ ........... 17 3 AD APTIVE MOTION AND FORCE CONTROL WITH RISE STRUCTURE ............ 21 3.1 Open Loop Tracking Error System ................................ ................................ ... 21 3.2 Closed Loop Tracking Error System ................................ ................................ 23 3.3 Stability Analysis ................................ ................................ ............................... 24 4 SIMU LATION RESULTS ................................ ................................ ........................ 2 8 4.1 Simulation Environment ................................ ................................ .................... 28 4.2 Simulation Results without Disturbances ................................ .......................... 31 4.3 Simulation Results with Disturbances ................................ ............................... 37 4.4 Discussion ................................ ................................ ................................ ........ 44 5 CONCLUSIONS AND SUGGESTED FUTURE WORKS ................................ ........ 51 5.1 Conclusions ................................ ................................ ................................ ...... 51 5.2 Suggested Future Works ................................ ................................ .................. 52 APPENDIX A PROOF FOR EQUATION 3 21 ................................ ................................ ............... 53 B PROOF FOR EQUATION 3 26 ................................ ................................ ............... 57 C PROOF FOR PROPERTIES 2 4 ................................ ................................ ............ 59 D PROOF FOR PROPERTIES 2 5 ................................ ................................ ............ 60 PAGE 6 6 REFERENCES ................................ ................................ ................................ .............. 61 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 63 PAGE 7 7 LIST OF FIGURES Figure page 4 1 A two degree of freedom robot manipulator moving on a semi circle contact surface. ................................ ................................ ................................ ............... 28 4 2 Desired position trajectory. ................................ ................................ ................. 32 4 3 Desired force trajectory. ................................ ................................ ..................... 32 4 4 Time response of position trajectory in the presence of unknown system parameters but without disturbance. ................................ ................................ ... 33 4 5 The time response of position tracking errors in the presence of unknown system parameters but without disturbance. ................................ ...................... 33 4 6 Enlarged diagram of Figure 4 5. ................................ ................................ ......... 34 4 7 The time response of force trajectory in the presence of unknown system parameters but without disturbance. ................................ ................................ ... 34 4 8 The time response of tracking errors in the presence of unknown system parameters but without disturbance. ................................ ................................ ... 35 4 9 Enlarged diagram of Figure 4 8. ................................ ................................ ......... 35 4 10 The time response of friction force in the presence of unknown system parameters but without disturbance. ................................ ................................ ... 36 4 11 Estimated system parameters in the presence of unknown system parameters but without disturbance. ................................ ................................ ... 36 4 12 Estimated dry contact surface friction in the presence of unknown system parameters but without disturbance. ................................ ................................ ... 37 4 13 Control torque of the robot manipulator in the presence of unknown system parameters but without disturbance. ................................ ................................ ... 37 4 14 Disturbance used for the simulation. ................................ ................................ .. 38 4 15 Disturbance used for the simulation. ................................ ................................ .. 39 4 16 The time response of position trajectory in the presence of unknown system parameters and disturbance. ................................ ................................ .. 39 4 17 The time response of position tracking errors in the presence of unknown system parameters and disturbance. ................................ ................................ .. 40 PAGE 8 8 4 18 Enlarged diagram of Figure 4 17. ................................ ................................ ....... 40 4 19 The time response of force trajectory in the presence of unknown system parameters and disturbance. ................................ ................................ .............. 41 4 20 The time response of force tracking errors in the presence of unknown system parameters and disturbance. ................................ ................................ .. 41 4 21 Enlarged di agram of Figure 4 20. ................................ ................................ ....... 42 4 22 The time response of friction force in the presence of unknown system parameters and disturbance. ................................ ................................ .............. 42 4 23 Estimated system parameters in the presence of unknown system parameters and disturbance. ................................ ................................ .............. 43 4 24 Estimated dry contact surface friction in the presence of unknown system parameters and disturbance. ................................ ................................ .............. 43 4 25 Control torque of the robot manipulator in the presence of unknown system parameters and disturbance. ................................ ................................ .............. 44 4 26 Position tracking error with disturbance and without disturbance. ...................... 45 4 27 Enlarged diagram of Figure 4 26. ................................ ................................ ....... 46 4 28 Force tracking error with disturbance and without disturbance. .......................... 46 4 29 Enlarged diagram of Figure 4 28. ................................ ................................ ....... 47 4 30 The time response of position tracking errors with different gains. .......... 47 4 31 The time response of force tracking errors with different gains. ............... 48 4 32 The time response of position tracking errors with different gains. ......... 48 4 33 The time response of force tracking errors with different gains. ............. 49 4 34 The time response of control torque with different gains. ............................. 49 4 35 The time response of position tracking errors with ................................ ................................ ................ 50 4 36 The time response of force tracking errors with ................................ ................................ ................ 50 PAGE 9 9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ADAPTIVE FO RCE AND MOTION CONTROL OF ROBOT MANIPULATORS IN CONSTRAINED MOTION WITH DISTURBANCES B y Y ung Sheng Chang May 2013 Chair: Carl D. Crane, III Major: Mechanical Engineering The goal of this study is to design a controller to reject t he disturbances caused by sensor noise and unmodeled effects during hybrid adaptive force and motion control of robot manipulators in constrained motion. A continuous robust integral of the sign of the error (RISE) feedback term is incorporated with adaptive force and motion control to yield asymptotic tracking results in the presence of disturbances and unknown system parameters and dry contact surface friction coefficient The main reason to use the RISE method is that it can enhance disturbance rejection capabilities. It is assumed that the system parameters of the robot manipulator an d the dry friction coefficient of contact surface are unknown. These unknown parameters can be updated by the adaptive update law. The suggested controller can guarantee semi global asymptotic motion and force tracking results which are supported through L yapunov based stability analysis under the condition that the position and velocity of end effector and the normal contact force between end effector and conta ct surface are measurable The contact surface of the environment is modeled by the set of m rig id and mutually independent hypersurfaces. The dynamic model of constrained robot PAGE 10 10 manipulators is developed through an Euler Lagrange formulation. T wo degree of freedom (DOF) robot manipulator s imulation results are given to illustrate the efficacy of the suggested controller. PAGE 11 11 CHAPTER 1 INTRODUCTION Today, more and more robot manipulators are appli ed in many different areas. O nly position control of manipulators is not enough to satisfy various tasks. Thus, in order to broaden the tasks of manipulators, it will be necessary to control force exerted between the manipulator end effectors and the contact environment The major tasks requir ing force control are deburring, grinding, contour following and assembly tasks. In order to perform these tasks successfully, it is important to simultaneously control the motion of the end effectors and the force of the contact surfaces. A way to achieve simultaneously motion and force control is hybrid position/force control which controls the position of end effectors in certain directions and controls the force in other directions which is orthogonal to the direction of position [1 3] During the contact with a rigid surface the contact object may impose some constraints on robot manipulators. There are some directions that the end effectors cannot move to. This situation is denoted as constrained motion [ 4 6 ]. It must be noted, however, that the orthogonality conditions used in some of these hybrid contro l algorithms have mixed units. This causes the algorithm to not be invariant to changes i n the choice of units of length [21]. There are many other challenging problems to perform complex tasks well, such as unstructured envir onments, unmodeled effects, and external and force sensor disturbances. Many researches have focused on how to compensate for these effects. Different methods have been propose d to deal with these conditions, such as sliding mode control [ 7 ] and robust con trol [ 8 ]. To address unstructured environment s and unknown parameters of robot manipulators, an adaptive motion and force control method [9 10] is adopted in this study PAGE 12 12 An adaptive law is updated by position and force tracking errors to y ield semi global asymptotic tracking results. To address external and force sensor disturbances, a recently developed continuous Robust Integral of the Sign of the Error (RISE) technique is used [ 11 12 ] The reason to choose RISE technique is that this method can deal wit h the system with sufficiently smooth bounded disturbances and guarantee asymptotic stability results [ 13 14 ] In this study, the dynamic equation of a general rigid link robot manipulator having n degrees of freedom is modeled using the Euler Lagrange formulation. The contact surface of the environment is modeled by the set of m rigid and mutually independent hypersurfaces [4, 5 17] Adaptive motion and force control with RISE feedback term s is used to account for disturbances in the sen sor and unmodeled effects, and the presen ce of uncertain parameters in the robot manipulator and environment Due to the friction force appearing in the contact surface, the dynamic model proposed by [ 4 ] cannot be used A new transformed dynamic model prop osed by [6] is adopted. This model is particularly suitable for the presence of friction force in the contact surface. The uncertain parameters of the robot manipulator and the dry friction coefficient of the contact surface are updated by an adaptive law. The proposed controller can yield semi global asymptotic tracking results. According to the paper s [ 15 16 ] the experimental results propose that the PI type force feedback control yields the best force tracking result. The controller proposed in this study has the same structure suggested in the papers. Simulation results are given to illustrate the suggested controller. PAGE 13 13 This study is organized as follow s The dynamic model of the robot manipulator is developed using the Euler Lagrange formulation in C hapter 2. The filtered tracking errors and t he suggested adaptive motion and force controller with the RISE feedback term is given in Chapter 3. Simulation results with different gains and external disturbances are presented in Chapter 4 and conclusions an d future work s are given in Chapter 5. PAGE 14 14 CHAPTER 2 CONSTRAINED ROBOT MANIPULATOR DYNAMICS 2.1 Euler Lagrange Dynamic Model The nonlinear dynamic system to be controlled is a general rigid link manipulator with n degrees of freedom. This system can be described using the Euler Lagrange formulation: (2 1) where is the generalized inertial matrix, is the generalized Centrifugal and Coriolis force, is the generalized gravitati o nal force is the generalized nonlinear disturbances (e.g., unmodeled effects or force sensor noise), is the applied torque input, and are the generalized joint position, veloc ity and acceleration, respectively. When the robot manipulator make s contact with its environment, the contact force s occur between the end effector an d contact surface. The dynamic Equation 2 1 can be modified to [ 4 ] (2 2) where is the Jacobian matrix is the position and orientation of the end effector in Cartesian space, and is the forces/moments exerted by the end e ffector on the contact surface. T he following dev elopment is based on the assumption that and the normal contact force are measurable, that is nonsingular in a finite work space that the end effector is in contact with the rigid surface at first and the contact forces will keep the end effector on the contact surface that the parameters of are unknown and that is unknown and that the robot manipulator is nonredundant In PAGE 15 15 addition, the following assumption s will be made to facilitate the subsequent Lyapunov based stability analysis. Assumption 2 1: The inertial matrix is symmetric positive definite and satisfies the following inequality : (2 3) where are known positive constant and denotes the standar d Euclidean norm. Assumption 2 2 : The matrix is a skew symmetric matrix satisfied the following relationship (2 4) Assumption 2 3: The desire d trajectory is designed as which exist, and are bounded. Assumption 2 4: The nonlinear disturbance is satisfied such that are bounded by known constants. Assumption 2 5: can be linear ly parameterized in terms of suitably selected p arameters for robot manipulator w here is regression matrix and represents the parameters of the ro bot manipulator. Assumption 2 6: If then and are bounded. In addition, if then the first and second partial derivatives of the elements of and with respect to exist and are boun ded, and the first and PAGE 16 16 second partial derivatives of the elements of with respect to exist and are bounded. 2.2 Model of Contact Surface Suppose that the contact surface of the environment can be des cribed by the following set of m rigid and mutually independent hypersurfaces [ 4 5, 17 ] (2 5) where is assumed to be twice di fferentiable with respect to x. At the point of contact on the surface, the contact force including the normal contact force and friction force can be described by the following equation (2 6) where is a vector of normal contact force components, is normal contact force in Cartesian space, is friction force in Cartesian space and is the direction of is the direction of T he magnitude of is depended on and the friction coefficient and the direction of is the opposite direction of the end effector velocity is differentiable with respect to x except at the point when changes direction. When the robot manipulator is in contact with e nvironment, the end effector of the robot manipulator is constrained to be on that contact surface and only degree of freedom can be control independently. Thus, motion control can be described by the following set of mutually independent c urvilinear coordinates (2 7) PAGE 17 17 where is ass umed to be twice differentiable. Because the position control cannot be done in the m rigid and mutually independent hypersurfaces is independent of and can uniquely determine the configuration of the robot manipulator. 2.3 Dynamic Model of Constrained Robot Manipulator According to the above definition, t he configuration of the robot manipulator can be defined by [ 4 ] (2 8) where D ifferentiati ng Equation 2 8 with respect to time, gives (2 9) where (2 10) Substituting Equation 2 8 and 2 9 into the joint space dynamic model Equation 2 2 and multiplying both sides by gives the operational space dynamic model [1] with the constraints Equation 2 5 and the contact force Equation 2 6 as (2 11) Equation 2 11 can be expressed as PAGE 18 18 (2 12) where In Equation 2 11 the constraint set by the environment can be simply presented by The position of the robot manipulator is described by Note that in operational space dynamic model, normal contact force has a simple structure, i.e., Also, the contact surface friction force only app ears in the second equation of Equation 2 12 Due to this condition, motion and force c ontrol cannot be dealt with separately. In other words, they are coupled. The strategy developed by [ 6 ] is adopted to deal with this condition. According to [ 6 ], is a dded and subtracted to the left hand side of the second equation of Equatio n 2 12 where and is added to the both sides of the first equation of Equation 2 12 where In this way, E quation 2 12 can be reformed as the following PAGE 19 19 (2 13 ) where According to the assum ption 2 1, 2 2 and 2 5, the following properties ca n be obtained for E q uation 2 13 in the Appendix Properties 2 1: The matrix is symmetric positive definite and satisfies the following inequality : (2 14 ) where are known positive constant and denotes the standard Euclidean norm. Properties 2 2 : The matrix is a skew symmetric matrix which satisfies the following relationship (2 15 ) Properties 2 3 : and can be linear parameterized in terms of a suitably selected set of parameters for robot manipulators. PAGE 20 20 Properties 2 4 1 : The matrix is a symmetric positive definite matrix on the assumption that the max imum eigenvalue of is small enough Properties 2 5 2 : The matrix is a skew symmetric matrix which satisfies the following relationship (2 16 ) T he controller proposed in this study is based on Equation 2 13 which design s the control torque to let the position and force follow the desired trajectory. The controller is also incorporated with a RISE feedback structure to reject the disturbance term in Equation 2 13 1 See Appendix for the proof of matrix is a symmetric positive definite matrix 2 See Appendix for the proof of is skew symmetric matrix. PAGE 21 21 CHAPTER 3 AD APTIVE MOTION AND FORCE CONTROL WITH RISE STRUCTURE 3.1 Open Loop Tracking Error System The goal of this study is to design an adaptive motion and force controller with a RISE feedback term to track the desire d time varying motion and force trajectory with disturbance based on the system described by E quation 2 14 Let the position and force tracking error be defined as (3 1) where is the desired robot manipulator trajectory and is the desired constrained force trajectory. To facilitate the following Lyapunov based stability analysis, the filtered tracking errors are defined as (3 2 ) (3 3 ) (3 4) (3 5 ) where denote positive constants The proposed controller is based on the assumption that position, velocity and normal contact force are measurable and acceleration is not measurable, so the filtered tracking error and are not measurable. Multiply ing both side s of Equation 3 5 by and utilizing Equation 2 13 3 1, 3 2 3 3 and 3 4 gives the open loop tracking error system as PAGE 22 22 (3 6 ) By adding and subtracting to Equation 3 6 gives (3 7 ) where is defined as (3 8 ) In Equation 3 8 represents the unknown parameters needed to be updated on line and is the desired regression matrix containing the desired robot manipulator position, velocity, acceleration and force respective ly. In Equation 3 7 the auxiliary function is d efined as (3 9 ) Based on the Equation 3 7 the control torque is designed to be (3 10 ) where is the estimated system parameter vector and is the RISE feedback term. is defined as (3 11 ) where are positive constant control gains. The estimated system parameter vector is generated by the following update law (3 12 ) PAGE 23 23 where is a known, constant, diagonal, positive definite gain matrix. Because Equation 3 12 contains the unknown signal, acceleration and derivati ve of force error can be integrated by parts as follows to get rid of the immeasurable signal (3 13 ) (3 14 ) According to the above equations, the terms in the control torque are all measurable and thus the control torque is implementable. 3.2 Closed Loop Tracking Error System The closed loop tracking error system can be obtained by putting Equation 3 10 into Equation 3 7 as (3 15 ) where denotes the estimated system parameter error vector defined as (3 16 ) According to the subsequent Lyapunov stability analys is, the time derivative of Equation 3 15 is obtained as (3 17 ) where the auxiliary term is defined as (3 18 ) and is defined as (3 19) The time derivative of is obtained as (3 20 ) PAGE 24 2 4 According to [ 18 ], th e Mean Value Theorem can be applied to Equation 3 18 to obtain the upper bound 1 of as (3 21 ) where is defined as (3 22 ) The upper bound of disturbance term and its time derivative can be obtained based on assumption 2 4 as (3 23 ) where are known positive constants. 3.3 Stability Analysis Theorem 3 1: The proposed controller given in Equation 3 10, 3 11 and 3 13 ensures that the signals of the constrained robot manipul ator described by Equation 2 13 are bounded under closed loop operation and the position and force tracking error is regulated as (3 24 ) on the co ndition that gain and are chosen large enough based on the initial conditions of the constrained robot manipulator system, and are chosen according to the subsequent proof as (3 25 ) a nd and are chosen according to the Lemma 2 in the Appendix as (3 26 ) 1 See Lemma 1 of the Appendix for the proof of the up per bound of PAGE 25 25 Proof: A positive definite function is selected as (3 27 ) wher e the auxiliary function are defined as (3 28 ) (3 29 ) and the subscript denotes the th element of the vector. In Equation 3 28 and 3 29 the auxiliary function are defined as (3 30 ) (3 31 ) The derivative of with respect to time can be obtained as (3 32 ) (3 33 ) If the sufficient condition in Equation 3 26 is satisfied, the following inequality can hold 2 (3 34 ) (3 35 ) Thus, are obtained. Taking the time derivative of the following equation can be obtained (3 36) 2 See Lemma 2 of the Appendix for the proof of the inequality in Equation 3 26 PAGE 26 26 (3 37) (3 38) Using Equation 3 22 and the following inequality (3 39) can be upper bounded in terms of as (3 40 ) where and is a positive globally invertible nondecreasing function. Accord ing to Equation 3 27 and 3 40 and Thus, Since and then according to E quation 3 1, 3 2 3 3 and 3 4 Since and is unknown constant vector, then according to Equation 3 16 Since and assumption 2 3 that exist and are bounded, then Since exist and are bounded, then PAGE 27 27 Since then input torque and is implementable. Since then and are uniformly continuous. Moreover, since (3 41 ) applying s lemma, if then and semi globally asymptotically regulates to zero, as PAGE 28 28 CHAPTER 4 SIMULATION RESULTS 4.1 Simulation Environment The simulation environment depicted in Figure 4 1 was used to test the proposed controller. Figure 4 1. A two degree of freedom robot manipulator moving on a semi circle contact surface A two degree of freedom robot manip ulator is chosen for the simulation. The dynamic system matrices forward kinematics and Jacobian matri x are given by PAGE 29 29 where is system parameter calculated by The actual value s of system parameters are and I t is assumed that the value of is unknown in simulation and its initial estimate value is As seen in Figure 4 1, the contact surface between the robot manipulator and the environment is a semi circle surface S with a dry friction coefficient The surface is described as The task space of the robot manipulato r is described as where is orthogonal to the curvilinear coordinate of According to Equation 2 6 the interaction force between the contact surface and the end effector are given by PAGE 30 30 where normal contact force component, can be measured by the force sensor Based on the above equations, Operational space dynamics can be derived and the dynamics Equation 2 13 controlled by the proposed controller can be obtained T he following simulation is based on the assumption that the position and velocity of the end effector and the normal contact force are measurable, that is nonsingular in a finite work space that the end effector is in contact with the rigid surface with dry friction force at first and the end effector will not leave the contact surface The contr ol torque is given by Equation 3 10 3 11 and 3 13 The parameter estimated vector is defin ed as (4 1) The parameter estimated vector Equation 4 1 is generated by the updated law Equation 3 13 The regression matrix can be obtained by Equation 3 8 Parameter values of the controller are and The desired position and force trajectory are given by Sampling time is 0.001 sec PAGE 31 31 4.2 Simulation Results without Disturbance s The desired posit ion trajectory is shown in F igure 4 2 and desire d force trajectory is shown in F igure 4 3. The time response of position trajectory and its position tracking errors in the presence of unknown system parameters but without disturbance are shown i n Figure 4 4 and Figure 4 5, respectively. Figure 4 6 is the enlarged diagram of Figure 4 5. The time response of force trajectory and its force tracking errors in the presence of unknown system parameters but without disturbance are shown in Figu re 4 7 and Figure 4 8 respectively. Figure 4 9 is the enlarged diagram of Figure 4 8. The time response of friction force is shown in Figure 4 10 The estimated system parameters and dry contact surface friction are shown in Figure 4 11 and Figure 4 12 respecti vely. Figure 4 13 shows the control torque of the robot manipulator. As shown in the Figure 4 5 and Figure 4 6 t he position tracking error falls between 0.001 m and 0.001 m The proposed controller has a good position tracking abi lity. According to Figure 4 8 and Figure 4 9, the proposed controller also has good force tracking ability. The force tracking error falls between 1 N and 1 N In Figure 4 11 and Figure 4 12 the estimated parameters do not converge to their actual value. T his phenomenon may be addressed by using composite adaptive update law which may be studied in the future work. PAGE 32 32 Figure 4 2. Desired position trajectory. Figur e 4 3. Desired force trajectory. PAGE 33 33 Figure 4 4. Time response of position trajectory in the presence of unknown system parameters but without disturbance Figure 4 5 The time response of position tracking errors in the presence of unknown system parameters but without disturbance PAGE 34 34 Figure 4 6. Enlarged diagram of Figure 4 5 Figure 4 7 The time response of force trajectory in the presence of unknown system parameters but without disturbance PAGE 35 35 Figure 4 8 The time response of tracking errors in the presence of unknown system parameters but without disturbance Figu re 4 9. Enlarged diagram of Figure 4 8 PAGE 36 36 Figure 4 10. The time response of friction force in the presence of unknown system parameters but without disturbance. Figure 4 11 Estimated system parameters in the presence of unknown system parameters but without disturbance. PAGE 37 37 Figure 4 1 2 Estimated dry contact surface friction in the presence of unknown system parameters but without disturbance. Figure 4 13 C ontrol torque of the robot manipulator in the presence of unknown system parameters but without disturbance. 4.3 Simulation Results wit h Disturbances The desired position and force trajectory is the same as shown in Figure 4 2 and F igure 4 3, respectively. The disturbance is shown in Figure 4 14 and Figure 4 15 As per assumpti on 2 4, the nonlinear disturbance is bounded by known constants. The time response of the position trajectory and its position tracking errors in the PAGE 38 38 presence of unknown system parameters and dis turbance are shown in Figure 4 16 and Figure 4 17, respectively. Figure 4 18 is th e enlarged diagram of Figure 4 17 The time response of force trajectory and its force tracking errors in the presence of unknown system parameters and dis turbance are shown in Figure 4 19 and Figure 4 20, respectively. Figure 4 21 is th e enlarged diagram of Figure 4 20 The time response of friction force is shown in F igure 4 22 The estimated system parameters and dry contact surface friction are shown in Figure 4 23 and Figure 4 24 respe ctively. Figure 4 25 shows the control torque of the robot manipulator. As shown in the Figure 4 17 and Figure 4 18 the position tracking error falls between 0.001 m and 0.001 m like Figure 4 8 and Figure 4 9 The proposed controller has good position dist urbance rejection ability. According to Figure 4 20 and Figure 4 21 the proposed controller also has good force disturbance rejection ability. The force tracking error falls between 1 N and 1 N Like the estimated system parameter vector in Figure 4 11 and Figure 4 12, the estimated parameters of system with disturbances do not c onverge to their actual value. Figure 4 14 Disturbance used for the simulation PAGE 39 39 Figure 4 15 Disturbance used for the simulation Figure 4 16 The time response of position trajectory in the presence of unknown system parameters and disturbance. PAGE 40 40 Figure 4 17 The time response of position tracking errors in the presence of unknown system parameters and disturbance. Figure 4 18 Enl arged diagram of Figure 4 17. PAGE 41 41 Figure 4 1 9 The time response of force trajectory in the presence of unknown system parameters and disturbance Figure 4 20 The time response of force tracking errors in the presence of unknown system paramete rs and disturbance PAGE 42 42 4 Figure 4 2 1 Enlarged diagram of Figure 4 20. Figure 4 22. The time response of friction force in the presence of unknown system parameters and disturbance PAGE 43 43 Figure 4 23 Estimated system parameters in the presence of unknown system parameters and disturbance Figure 4 2 4 Estimated dry contact surface friction in the presence of unknown system parameters and disturbance PAGE 44 44 Figure 4 25 Control torque of the robot manipulator in the presence of unknown system parameters and disturbance 4.4 Discussion Figure s 4 26 and 4 27 show the position tracking error without disturbance and with disturbance in the same figure. F rom the s e figure s the position tracking error with disturbance is almost the same as the position tracking error without disturbance. It shows that, due to the RISE feedback term, the disturbance rejection ability is good Figure s 4 28 and 4 29 show the force tracking error without disturbance and with disturbance in the same figure. Fr om these figure s the force tracking error with disturbance is a little bigger than the force tracking error without disturbance. Despite this situation the proposed controller also has a good force disturbance rejection ability. Noted that in these figures, there are abrupt changes at Because the direction of the friction force depends on the direction of end effector velocity, the abrupt direction change of the end effector velocity can be seen in Figure s 4 4 and 4 16 To make co mparison, the simulation is run with the same parameters as before except to show how the different gain will affect the result in Figure 4 30 and 4 31 PAGE 45 45 The position tracking errors look alike, but the force tracking errors have much difference. It agrees with the Properties 2 4 that has to be small enough. T he simulation is also run with the same parameters as before except in Figure s 4 32, 4 33 and 4 34. The transient position tracking error improves a lot but the ameliorate in this condition and the joint torque become bigger. In the last simulation the system is run to test the validity of Theorem 3 1. Figure s 4 35 and 4 36 are converge to zero under the condition that These figures show that if do not satisfy Theorem 3 1 con ditio ns, the system would not be stab le. Figure 4 26 Position tracking error with disturbance and without disturbance PAGE 46 46 Figure 4 27 Enlarged diagram of Figure 4 26. Figure 4 28 Force tracking error with disturbance and without disturbance PAGE 47 47 Figure 4 29 Enlarged diagram of Figure 4 28. Figure 4 30. The time response of position tracking errors with different gains. PAGE 48 48 Figure 4 31 The time response of force tracking errors with different gains. Figure 4 32 The time response of position tracking errors with different gains. PAGE 49 49 Figure 4 33 The time response of force tracking errors with different gains. Figure 4 34 The time response of control torque with different gains. PAGE 50 50 Figure 4 35 The time response of position tracking errors with Figure 4 36 The time response of force tracking errors with PAGE 51 51 CHAPTER 5 CONCLUSIONS AND SUGGESTED FUTURE WORKS 5.1 Conclusions This work focus ed on designing an adaptive controller for constrained robot manipulator with unknown parameters both in the robot manipulator and the contact surface with disturbance caused by force sensor noise and unmodeled effects. First, the dynamic equation of a gene ral rigid link robot manipulator having n degrees of freedom modeled by Euler Lagrange formulation is used in this study. Because of the friction force occurring between contact surface and end effector, the position and force control cannot be controlled separately. The Euler Lagrange dynamic model is modified to suit this condition. Second, the contact surface of the environment is modeled by the set of m rigid and mutually independent hypersurfaces. When the robot manipulator is in contact with environme nt, the end effector of robot manipulator is constrained to be on that contact surface. Thus, only (n m) position coordinates and m force coordinates need to be controlled. Third t he proposed controller is an adaptive hybrid position/force controller with RISE feedback structure. By using the Lyapunov stability analysis, the suggested controller can guarantee semi global asymptotic position and force tracking result even if in the presence of the disturbance. The only assumption about the disturbance is th at it has to be bounded by known constants. In conclusion, the suggested force feedback term in the controller is in accordance with PI type Intensive s imulation results were given to show the validity of the proposed controller. PAGE 52 52 5.2 Suggested Future Wor ks The contact surface model in this study is assumed to be a rigid contact surface, but in most application areas, the contact surface may not be rigid. The stiffness of the contact surface needs to be considered in the contact model [19] In addition, t he robot manipulator is assumed to be non redundant in this study. The future work can focus on how to extend the controller proposed in this study to the redundant robot manipulator which has some advantages. For example, a problem occu r s when the Jacobian matrix becomes linearly dependent. In this case, a non redundant robot manipulator is at a singular configuration and loses the degree of freedom in some direction [1]. Redundant robot manipulator can deal with this situation. Another suggested future work is to improve the estimated system parameters and dry contact surface friction coefficient. In this study, the estimated vector converge to their actual values. According to [20], the composite adaptive control skil l may be incorporated with the controller proposed in this study. The main idea of composite adaptive control skill is that it includes the prediction error of the system parameters estimation in the adaptive updated law. The prediction error is defined as the difference between the estimated system parameters and actual system parameters. Thus, including prediction error in the Lyapunov stability analysis can prove that estimated system parameters converge to their actual values as time goes to infinity. PAGE 53 53 APPENDIX A PROOF FOR EQUATION 3 21 Lemma 1. The Mean Value Theorem can be applied to prove the upper bound of in Equation 3 21 where is defined as Proof: can be written in the following form PAGE 54 54 The Mean Value Theorem can be applied to and rewritten as where Thus, can be upper bounded as Noted that the partial derivatives can be upper bou nded as PAGE 55 55 Using the following inequality and can be further upper bounded as Thus, PAGE 56 56 where is some positive globally invertible nondecreasing function. PAGE 57 57 APPENDIX B PROOF FOR EQUATION 3 26 Lemma 2 The function is defined as If the following sufficient conditions is satisfied Then Proof: In tegrating both sides of Equation 3 30 (A 1) Using integrat ion by part, the first term of Equation A 1 can be expressed as Using the following property, Equation A 1 can be rewritten as PAGE 58 58 T hus, according to Equation 3 23 if then is positive definite function. In the similar way as above proof i n tegrating both sides of Equation 3 31 (A 2 ) Using integration by part, the first term of Equation A 2 can be expressed as Using the following property, Equation A 2 can be rewritten as Thus, according to E qua tion 3 23 if then is positive definite function PAGE 59 59 APPENDIX C PROOF FOR PROPERTIES 2 4 Properties 2 4 The matrix is a symmetric positive definite matrix on the assumption that the maximum eigenvalue of is small enough Proof: can be rewritten as (A 3) If the maximum eigenvalue of then the minimum eigenvalue of According to assumption 2 1, is a symmetric positive semi definite matrix. Thus, is a symmetric positive definite matrix on the assumption that the maximum eigenvalue of is small enough PAGE 60 60 APPENDIX D PROOF FOR PROPERTIES 2 5 Properties 2 5 The matrix is a skew symmetric matrix satisfied the following relationship Proof: According to Equatio n A 3 can be rewritten as the following From assumption 2 2 and the above equation lead to the properties 2 5. PAGE 61 61 REFERENCES [1] O. Khatib "A Unified Approach for Motion and Force Control of Robot Manipulators the Operational Space Formulation," I EEE Journal of Robotics and Automation, vol. 3, pp. 43 53, Feb 1987. [2] M. H. Raibert and J. J. Craig, "Hybrid Position Force Control of Manipulato rs," Journal of Dynamic Systems Measurement and Control Transactions of the A SME vol. 103, pp. 126 133, 1981. [3] D. E. Whitney, "Historical Perspective and State of the Art in Robot Force Control," International Journal of Robotics Research, vol. 6, pp. 3 14 Spr 1987. [4] N. H. Mcclamroch and D. W. 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[19] R. Volpe and P. Khosla, "An experimental evaluation and comparison of explicit force control strategies for robotic manipulators," in Robotics and Automation, 1992. Proceedings., 1992 IEEE International Conference on 1992, pp. 1387 1393 vol.2. [20] P. M. Patre, W. MacKunis, M. Johnson, and W. E. Dixon, "Composite adaptive control for Euler Lagrange systems with additive disturbances," Automatica, vol. 46, pp. 140 147, Jan 2010. [21] Journal of Robotic Systems, vol. 7, pp. 139 144, 1990. PAGE 63 63 BIOGRAPHICAL SKETCH Yung Sheng Chang was born in Taipei, t he capital city of Taiwan. He received his degree in the Department of Electrical and Control Engineering from the National Chiao Tung University in June 2009. He then began working in Ministry of Nation Defense in August 2009. After that, he co ntinued his education for pursuing degree in the Department of Mechanical Engineering from the University of Florida and joined the Center for Intelligent Machines and Robotics under the advisement of Dr. Carl Crane in August 2011 