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PAGE 1 1 DESIGN OF AN ADAPTIVE BACKSTEPPING CONTROLLER FOR 2 DOF PARALLEL ROBOT By JING ZOU 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 2014 PAGE 2 2 2014 Jing Zou PAGE 3 3 ACKNOWLEDGMENTS The author expresses his deep gratitude to his advisor, Dr. John K. Schueller, for guiding him throughout his work, and for his support and dedication. The author also expresses his sincere appreciation to his committee Dr. Carl Crane III for his guidance and help. The author also extends his thanks to Dr. Warren Dixon for his support. PAGE 4 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 3 LIST OF TABLES ................................ ................................ ................................ ............ 5 LIST OF FIGURES ................................ ................................ ................................ .......... 6 ABSTRACT ................................ ................................ ................................ ..................... 9 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 11 1.1 Background ................................ ................................ ................................ ....... 11 1.2 Related Work ................................ ................................ ................................ .... 12 2 ADAPTIVE BACKSTEPPING CONTROLL ER FOR PARALLEL ROBOTS ............ 14 2.1 Kinematics and Dynamics Analysis for Parallel Robots ................................ .... 14 2.1.1 Kinematics Analysis ................................ ................................ ................. 14 2.1.2 Dynamics Analysis ................................ ................................ .................. 24 2.2 Adaptive Backstepping Controller for Parallel Robots ................................ ....... 26 2.2.1 Lyapunov Based Design of the Controller ................................ ............... 26 2.2.2 Verification on the Implementation of the Controller ................................ 30 3 ANALY SIS FOR THE 2 DOF PARALLEL ROBOT ................................ ................. 32 3.1 Kinematic and Singularity Analysis of the 2 DOF Parallel Robot ...................... 32 3.2 Accuracy and Effic iency Analysis for the 2 DOF Parallel Robot ....................... 36 3.3 Modeling for the 2 DOF Parallel Robot ................................ ............................. 44 3.3.1 Dynamics Model for the 2 DOF P arallel Robot ................................ ........ 44 3.3.2 Verification of the dynamics model ................................ .......................... 51 3.4 Dynamics Model for the 2 DOF Parallel Robot with Dnamics and Ki nematics Uncertainties ................................ ................................ ................................ ........ 59 4 CONTROL SYSTEM DESIGN FOR 2 DOF PARALLEL ROBOT ........................... 68 4.1 Adaptive Backstepping Controller for the 2 DOF Parallel Robot Model with Uncertainties ................................ ................................ ................................ ........ 68 4.2 Simulation Result and Discussion ................................ ................................ ..... 74 5 CONCLUSION AND FUTURE WORK ................................ ................................ .... 90 REFERENCES ................................ ................................ ................................ .............. 91 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 93 PAGE 5 5 LIST OF TABLES Table : page 3 1 Partial derivative distribution of qa1 to x (rad/dm) ................................ ............... 42 3 2 Partial derivative distribution of qa2 to x (rad/dm) ................................ ............... 42 3 3 Partial derivative distribution of qa1 to y (rad/dm) ................................ ............... 43 3 4 Partial derivative distribution of qa2 to y (rad/dm) ................................ ............... 43 PAGE 6 6 LIST OF FIGURES Fig ure : page 3 1 Symmetrical 2 DOF parallel robot ................................ ................................ ....... 32 3 2 The coordinate system of the 2 DOF parallel robot ................................ ............ 33 3 3 Area definition 1 ................................ ................................ ................................ .. 35 3 4 Area definition 2 ................................ ................................ ................................ .. 35 3 5 Restricted zone for C ................................ ................................ .......................... 36 3 6 Partial derivative of qa1 to x ................................ ................................ ............... 40 3 7 Partial derivative of qa1 to y ................................ ................................ ............... 41 3 8 Partial derivative of qa2 to x ................................ ................................ ............... 41 3 9 Partial derivative of qa2 to y ................................ ................................ ............... 42 3 10 Robot arm coordinate system ................................ ................................ ............. 44 3 11 Revised robot coordinate system ................................ ................................ ....... 45 3 12 Force analysis for bar BC and DC ................................ ................................ ...... 48 3 13 Force analysis for bar AB and ED ................................ ................................ ...... 49 3 14 SimMechanics model for the 2 DOF parallel robot ................................ ............. 52 3 15 Value of qa1 in mathmet ic model and SimMechanics modle ............................. 53 3 16 Value of qa2 in mathematical model and SimMechanics model ......................... 54 3 17 Angular velocity of qa1 in mathematical model and SimMechanics model ......... 54 3 18 Angular velocity of qa2 in mathematical model and SimMechanics model ......... 55 3 19 Input torque at A in mathematical model and SimMechanics model .................. 55 3 20 Input torque at E in mathematical model and SimMechanics model .................. 56 3 21 Error between two models for qa1 ................................ ................................ ...... 56 3 22 Error between two models for qa2 ................................ ................................ ...... 57 3 23 Error between two models in angular velocity of qa1 ................................ ......... 57 PAGE 7 7 3 24 Error between two models in angular velocity of qa2 ................................ ......... 58 3 25 Error between two models for input torque at A ................................ .................. 58 3 26 Error between two models for input torque at E ................................ .................. 59 3 27 2 DOF parallel robot with dynamics and kinematics uncertainties ...................... 60 3 28 Force analysis for bar BC and DC ................................ ................................ ...... 61 3 29 Force analysis for bar AB and ED ................................ ................................ ...... 62 4 1 Simulation Block for Control System ................................ ................................ .. 75 4 2 Destination point and tracking trajectory (ABE) ................................ .................. 76 4 3 Error in x d irection for set point tracking (ABE) ................................ ................... 77 4 4 Error in y direction for set point tracking (ABE) ................................ ................... 77 4 5 Destination point and tracki ng trajectory (BE) ................................ ..................... 78 4 6 Error in x direction for set point tracking (BE) ................................ ..................... 78 4 7 Error in y direction for set point tracking (BE) ................................ ..................... 79 4 8 Destination point and tracking trajectory (ABU) ................................ .................. 79 4 9 Error in x direction for set point tracking (ABU) ................................ ................... 80 4 10 Error in y direction for set point tracking (ABU) ................................ ................... 80 4 11 Destination point and tracking trajectory (BU) ................................ .................... 81 4 12 Error in x direction for set point tracking (BU) ................................ ..................... 81 4 13 Error in y direction for set point tracking (BU) ................................ ..................... 82 4 14 Desired trajectory and tracking trajectory (ABE) ................................ ................. 83 4 15 Error in x direction for trajectory tracking (ABE) ................................ .................. 84 4 16 Error in y direction for trajectory tracking (ABE) ................................ .................. 85 4 17 Desired trajectory and tracking trajectory (BE) ................................ ................... 85 4 18 Error in x direction for trajectory tracking (BE) ................................ .................... 86 4 19 Error in y direction for trajectory tracking (BE) ................................ .................... 86 PAGE 8 8 4 20 Desired trajectory and tracking trajectory (ABU) ................................ ................. 87 4 21 Error in x direction for trajectory tracking (ABU) ................................ ................. 87 4 22 Error in y direction for trajectory tracking (ABU) ................................ ................. 88 4 23 Desired trajectory and tracking trajectory (BU) ................................ ................... 88 4 24 Error in x direction for trajectory tracking (BU) ................................ .................... 89 4 25 Error in y direction for trajectory tracking (BU) ................................ .................... 89 PAGE 9 9 Abstract of Thesis Presented to the Graduate School of the University of Florida i n Partial Fulfillment of the Requirements for the Degree of Master of Science DESIGN OF AN ADAPTIVE BACKSTEPPING CONTROLLER FOR 2 DOF PARALLEL ROBOT By Jing Zou May 2014 Chair: John K. Schueller Major: Mechanical Engineering It is very common in robot tracking control that controllers are designed based on the exact kinematic model of the robot manipulator. However, because of measurement errors and changes of states, the original kinematic model is no longer accurate and will degrade the control result Besides, the structure of the controllers are always much more complicated for robots with the targets expressed in task space, due to the transformation from joint space to task space. In this thesis, a controller is designed for parallel robot systems with kinematics and dynamics uncertainties through backstepping control and adaptive control. Backstepping control is used to simplify the structure of the controller whose target is expressed in the task space and manage the transformation between the err ors in task space and joint space. Adaptive control is utilized to compensate for uncertainties in both dynamics and kinematics. A realization of the proposed controller is achieved based on the Two Degree of Freedom (2 DOF) parallel robot designed in this thesis. The simulation of the control system is carried out using in MATLAB Compared to the simulation result of the system controlled by the backstepping controller, simulation results of the control system indicate that the proposed cont roller has robust performance with regard PAGE 10 10 to dynamics and kinematics uncertainties. The proposed controller gives desired performance to achieve the research goal. PAGE 11 11 CHAPTER 1 INTRODUCTION 1.1 Background Robot manipulators have been widely used in our curr ent society, especially in manufacturing industries. They make their appearance in almost every automatic assembly line. The efficiency and accuracy of the robot manipulators has a great influence on the production and quality of the product. Large number of robot manipulators have been designed over the last half century, and several of these have become standard platforms for R&D efforts [1]. A robot manipulator is a movable chain of links interconnected by joints. One end is fixed to the ground, and a ha nd or end effector that can move freely in space is attached at the other end [2]. Serial robot manipulator, designed as series of links connected by motor actuated joints that extend from a base to an end effector, are the most common industrial robots. H owever, parallel robot manipulator, a mechanical system that uses several computer controlled serial chains to support a single platform or end effector, have the following potential advantages over serial manipulators: better accuracy, higher stiffness an d payload capability, higher velocity, lower moving inertia, and so on. The goal of this thesis is to come up with a new nonlinear control strategy that can enhance robustness of the robot manipulator to both kinematics and dynamics uncertainties through e xplore the kinematic and dynamics characteristics of the parallel robots. PAGE 12 12 1.2 Related Work This work is focused on the kinematics and dynamics analysis of the parallel robot manipulators and the design of a controller to achieve robust performance with reg ard to kinematics uncertainties, dynamics certainties. Robot manipulators are highly nonlinear in their dynamics and kinematics. And even more nonlinearities appear in parallel robot manipulators. In order to have a good tracking performance of parallel ro bot manipulators, people try to compensate for the nonlinearities and use feedback PD control to minimize the tracking error. In [3] and [4], a nonlinear PD controller was proposed by using the nonlinear terms in robot dynamics as nonlinear feedback to can cel those terms and PD feedback to control the tracking error. This controller is very sensitive to uncertainties in the robot model as it needs very accurate knowledge of the robot dynamics to cancel the nonlinear terms in the system. To make the controll er robust to the dynamic uncertainties of the parallel robot manipulator, adaptive control, high gain control and high frequency control methods are introduced. In [5] and [6], an adaptive controller was created with an estimator for the dynamic parameters of the robot to compensate for the uncertainties. And in [7], sliding mode control method is applied to decentralize uncertain dynamic parameters of the robot manipulator to get a more robust performance. Those controllers work well with parallel robots h aving uncertainties in dynamics. However, since there are no estimators to predict the uncertain parameters in kinematic functions and the decentralization method is not applied to uncertain terms appearing in the kinematics, they are not robust to kinemat ic uncertainties. In [8] and [9], adaptive controllers are proposed to make the whole system resistant to uncertainties in both dynamics and kinematics through design of estimator to predict and compensate the uncertain terms in both PAGE 13 13 dynamic and kinematic functions. The controllers give good control results. The researchers produce integrated controllers to compensate for both dynamics and kinematics uncertainties. As the kinematics uncertainties are decoupled from the control input, much more mathematical analysis and structure complexity is required for the controllers. A robust backstepping controller is proposed in [10]. The design needs less effort, but its Lyapunov analysis is based on the slow varying assumption on some parameters, which means the rob ot is not influenced by potentially arbitrarily large and fast external torques, and this is a bad assumption for parallel robot manipulators, where arbitrarily large and fast external torques can appear due to geometric constraints on the bars of the robo t. And in [11], a controller is proposed for system with uncertainties in dynamics, kinematics and actuator, desired armature current model of the actuator is necessary to finish the Lyapunov analysis and controller design for the system. In this thesis, t he mathematical analysis of the Jacobian matrix of a parallel robot helps to conclude that it is linear in physical parameters. And then through the implementation of backstepping control and adaptive control, a controller which is robust to uncertainties in dynamics and kinematics is constructed. With the application of backstepping control, massive mathematical analysis according to the decoupling of control input and kinematics uncertainties is avoided. And the adaptive control has a good performance for the parallel robot with arbitrarily large and fast dynamics caused by geometric constraints. PAGE 14 14 CHAPTER 2 ADAPTIVE BACKSTEPPING CONTROLLER FOR PARALLEL ROBOTS 2.1 Kinematics and Dynamics Analysis for Parallel Robots 2.1.1 Kinematics Analysis Here a kinemati c structure that has a rigid base connected to a rigid end effector by means of n serial kinematic chains in parallel is discussed. Each set of serial degrees of freedom, collected in a vector Let be the total number of joints: and be the vector of all joint angles: The rest of analysis in this section is written with reference to [12]. Not all joints of the parallel structure can be actuated independently; the end effector of the structure has, at each instant in time, a number of degrees of freedom, whic h can never be larger than six. This means that of the joints can be actuated independently (these are called the driving joints), and that their motion completely determines the motion of all other joints (these joints are the driven joints). The relationships between the driving and the driven joints are determined by the so called closure equations. Velocity closure: Given position closure, the velocity of all joints in each leg must be such that the end point of that leg moves with the same spatial velocity as its connection point at the end effector. Mathematically, velocity closure is represented, for example, by the following set of linear equations: PAGE 15 15 ( 2 1 ) with the Jacobian matrix of the th leg. Joint velocity selection matrices: is a t a place corresponding to a driving joint in the vector. A typical looks like ( 2 2) selects the vector of the dr iving joint velocities from the total vector of joint velocities as follows: ( 2 3) Obviously, and is a square with ones on the diagonal at the ind ices of driving joints. Hence ( 2 4) with equal to except that the driven joint velocities are replaced by zeros. is a matrix that differs from the unit matrix in the fact that the rows corresponding to driving joints are eliminated. Hence, it selects the vector of driven joints from the vector: ( 2 5) PAGE 16 16 Similarly as for , is a square with ones on the diagonal at the indices of driven joints. Hence ( 2 6) with equal to except that the driving joint velocities are replaced by zeros. is a in the t h leg: ( 2 7) It selects the joint velocities of the th submanipulator from the vector of all joint velocities: ( 2 8) The following identi ties follow straightforwardly: ( 2 9) and ( 2 10) Dependency matrix: The major point in solving the velocity closure eq between the known driving joint velocities and the unknown driven joint dependency matrix : ( 2 11) PAGE 17 17 With the vector of all joint velocities when the th driving joint is given a unit speed and all other driving joints are kept motionless. Since during the motion generated by the th driving joint. the th column of becomes Hence, the depende ncy matrix can also be written as ( 2 12) Closed form solution: The velocity closure E quation 2 1, together with Eq uation 2 10, give ( 2 13) The matrices and are submatrices of A in Eq uation 2 1 that contain only the columns corresponding to driving and passive joints, respectively. These matrices are fully known once position cl actually is, the vector space spanned by the driving joints must always be a subspace of the vector space spanned by the passive joints: ( 2 14) Equation 2 12 yields an analytical expression of the driven joints as functions of the driving joints: ( 2 15) PAGE 18 18 Since is in general not a square matrix, the normal matrix inverse is not defined, and a Moore Penrose pseudo inverse is required. ( 2 16) Analytical Jacobian matrix: The Jacobian matrix of the total parallel structure is a matrix; its th column represents the end effector twist that corresponds to a unit speed of the th driving joint and zero speeds for all other driving joints. Hence, the total twist of the end effector is ( 2 17) Th e velocity closure Equations 2 11 can then also be written as ( 2 18) matrix for a serial structure: in that case, the th column depends on the th joint velocity only; irrespective of the velocities of the other joints; in the parallel structure case all other driving joints are explicitly kept motionless. The vector of driving joint speeds that generat es the th column is, by definition, given by the th column of the transpose of the selection matrix Eq uation 2 joints. Equation 2 15 gives the corresponding velocities of the passive joints: ( 2 19) Combining Eq uation s 2 10, 2 15 and 2 17 yields ( 2 20) PAGE 19 19 This equation gives the th column of the dependency matrix in of Eq uation 2 11. Using Eq uation s 2 8 and 2 18, the th column of is then found as ( 2 21) The right the serial subchains from the vector of all joint velocities, and multiplies these subchain joint velocities with the subchain Jacobian matri x to obtain the corresponding end effector twist. All serial subchains are equivalent to calculate a column of the Jacobiann since they all have to follow the same twist of the end effector. Repeating the above mentioned procedure for all driving j oints gives, for all ( 2 22) with the dependency matrix Jacobian matrix property analysis : According to Eq uation 2 20, Eq uation 2 22 ca n be transformed into ( 2 23) where is a matrix. Usually with proper arrangement of in and Then it can be derived that ( 2 24) The conclusion that , , and ar e all linear in physical parameters is obtained from their previous descriptions in this section. PAGE 20 20 The first component of Eq uation 2 24 is linear in a set of physical parameters ( 2 25) where is the regressor matrix. The linearity exploration on the second part of Eq uation 2 24 i.e., requires more mathematical analysis. According the definition of Moore Penrose pseudo inverse, could be transformed as follow ( 2 26) where is linear in a set of physical parameters ( 2 27) where is the regressor matrix. Then the linearity of is relevant with If is scalar linea r with regard to only one physical parameter (or combination of physical parameters), will be linear in a set of physical parameters. For parallel robots of severa between the speed of the end effector and the angular velocities of the joint angles in the ( 2 28) PAGE 21 21 where is the direction vector of angular velocity for th rotation joint, i.e. and is the position vector from th joint to th joint. Given with the direction vector of and the length of and is a vector function of measurable variables and the azimuth angles of joint axis. This leads to ( 2 29) then where corresponds to the length of a bar starting from a passive (driven) j oint in the th leg, , are the first, second and third element of vector function is passive joints. Then, denotes the number of all the passive joints. To get the properties of firstly needs to be calculated. The calculation process and results of is shown below. The component of matrix in row 1, column 1 is The same goes for other components of PAGE 22 22 PAGE 23 23 As can be seen from the listed components of there is no cross product of where and To simplify the expression, unify the sequence number, namely, then will be converted into The analysis on the properties of can be implemented through the knowledge of above. If is a scalar function of measurable variables and the azimuth angles of joint axis. For PAGE 24 24 is also a scal ar function of measurable variables and the azimuth angles of joint axis. Therefore, is linear in a combination of physical parameters and is also l inear in a combination of physical parameters Thus ( 2 30) Bring Eq uation 2 29 back to Eq uation 2 26 ( 2 31) where is the regressor matrix and is the combination of physical parameters in and The second part of Eq uation 2 24, i.e., is linear in a set of physical parameters Substitutes Eq uation s 2 31 and 2 26 into Eq uation 2 24 yields ( 2 32) where is the regressor matrix and is the com bination of physical parameters in and Hence, the kinematics functions (or the kinematic model) of the proposed parallel robot is linear in a set of physical parameters 2.1.2 Dynamics Analysis The dy namic model of a parallel robot with uncertain parameters is: ( 2 33) PAGE 25 25 where and are the angular acceleration and angular velocity of the active joints, is the inertia matrix, is a vector function containing Coriolis and centrifugal forces, is a vector function consisting of gravitational forces. There are several properties for the dynamic equation: Property 1: The inertia matrix is symmetric and uniformly positive definite for all Property 2: The matrix is skew symmetric so that for al l Property 3: The dynamic model as described by (10) is linear in a set of physical parameters as where is called the dynamic regressor matrix. Therefore, for the parallel robot connected by rotational joints and with same sets of physical parameters or sets of combination of physical parameters. Since all uncertain parameters in both dynamics and kinematics are those physical parameters, they can be separated and arranged into uncertain parameters vectors. Uncertain parameters in dynamics are collected in vector and uncertain parameters in kinematics are collected in vector Adaptive control can then be applied to estimate those uncertainties and compensate for them. And the designed controller would have robust performance with regard to uncertain dy namics and kinematics. PAGE 26 26 2.2 Adaptive Backstepping Controller for Parallel Robots This section is focused on designing a controller that gives asymptotical tracking result in task space for the proposed parallel robot. Meanwhile, the controller is robust to kinematic and dynamics uncertainties. 2.2.1 Lyapunov Based Design of the Controller Let , denote the tracking error of the end effector, the position of the end effector and the destination position of the end effector. And for simplicity, replace with Then ( 2 34) Taking the time d erivative on both sides of Eq uation 2 34 and substituting Eq uation 2 32 into it ( 2 35) here backstepping control is introduced through plus and subtract an d on the left side of Eq uation 2 35. is the estimate Jacobian matrix of the parallel robot, where all uncertain elements of in the Jacobian matrix are replaced by corresponding elements in which are the estimators of those uncertain elements in . is a value which can be designed to achieve specified goals. And Equation 2 35 is transformed into ( 2 36) PAGE 27 27 where is the error between the set o f physical parameters and the estimator of the same set of physical parameters ; and take the derivative of result in Now can be designed as ( 2 37) Substitute and into the dynamics function of the parallel robot, i.e., Eq uation 2 33 ( 2 38) Applying Property 3 in Equation 2 38 ( 2 39) Eq uation 2 39 can be reformulated as ( 2 40) Defining the error between the set of physical parameters and the estimator of the same set of physical parameters Select the Lyapunov candidate as ( 2 41) The derivative of the Lyapunov candidate is ( 2 42) Substitute Eq uaiton s 2 3 7 and 2 39 into Eq uation 2 42 and apply Property 2 PAGE 28 28 ( 2 43) Design the input controller as ( 2 44) Substitute Eq uation 2 44 into Eq uation 2 43 ( 2 45) For simplicity, replace with Now we propose the adaptation laws for and as follows ( 2 46) ( 2 47) where and are designed positive numbers. Substitute Eq uation s 2 46, 2 47 into Eq uation 2 4 5 ( 2 48) where is a negative semi definite funct ion. PAGE 29 29 If a scalar function is such that is lower bounded by zero and is uniformly continuous in time Then as I t has already been proven that is a negative semi definite function along the trajectories of and is a positive definite function, which mean is decreasing and the value of is always bigger than 0. Therefore, could be lower bounded by Moreover ( 2 49) Under the reasonable assumption that it can be drawn from Equation 2 49 that ( 2 50) Eq uation 2 48 could then be rewritten as where Take the derivative of and substitute Eq uation 2 16 into the derivative According to the results in Eq uation 2 50 PAGE 30 30 Consequently, 1) is lower bo unded by ; 2) ; and 3) and is uniformly continuous. All the conditions in Corollary are satisfied. Applying to the Lyapunov candidate in Equation 2 42 l eads to the conclusion as i.e., as The designed controller could achieve asymptotical tracking for the proposed parallel robot. 2.2.2 Verification on the Implementation of the Controller is a given desired value and and are the sets of some constant uncertain physical parameters and would never expand to infinity, thus, Singularities in kinematics and dynamics could be avoided by the selection of working ar ea, which guarantees Apply the conclusions from Equation 2 29 to Eq uation s 2 25 and 2 26, , and gives the following results Introduce the above results in the equation PAGE 31 31 Thence, all those designed and measured values have been proven to be bounded. could be measured and are made of measurable par ameters, from Eq uaiton 2 26, is achievable. Through integration of is obtained. The value of can be acquired from the equation is a given desired value and known, can be calculated through and is calculated by taking the derivative of is measurable, then is available from Using Eq uation 2 25, the value of is accessible, and take integration of gives Consequently, all elements of the control input can be constructed. All designed and measured values are bounded and the control input is implementable. Therefore, the controller is implementable. PAGE 32 32 CHAPTER 3 ANALYSIS FOR THE 2 DOF PARALLEL ROBOT 3.1 Kinematic and Singularity Analysis of the 2 DOF Parallel Robot Kinematic and singularity analysis is given to the kind of 2 DOF parallel robot presented in Figure 3 1. The length of th e linkages are set to be: , According to [13], kinematic analysis for the 2 DOF parallel robot consists of forward and inverse kinematics analysis. The forward kinematics problem for the parallel robot is to obtain the coordinates of the end effector from a set of given joint angles Figure 3 1 Symmetrical 2 DOF parallel robot The coordinate system of the parallel robot is set as exhibited in Figure 3 2. Let denotes the coordinates of the end effector. All othe r parameters of the parallel robot are represented by the characters shown in Figure 3 1. The following equations are driven from the geometric relationship. PAGE 33 33 Figure 3 2 The coordinate system of the 2 DOF parallel robot There are four joint variables , with only two independ ent variables while the rest joints and are functions of and .Assume that PAGE 34 34 Two solutions exist for the forward kinematics. Solution 1 (up configuration): Solution 2 (down configuration): Inverse kinematics: The inverse kinematics problem for the parallel robot is to obtain a set of joint angles from given coordinates of the end effector. Assume that ( 3 1) ( 3 2) ( 3 3) ( 3 4) then, ( 3 5) ( 3 6) And those are the so lutions for the inverse kinematic analysis. Singularity: Due to the analysis in [13], singularity happens under three cases. PAGE 35 35 Type I singularity happens when or Type II singularity happens when and Type III singular ity happens when The definition of and can be found in Figure 3 3 and Figure 3 4. To avoid the happening of singularity which would cause a change in the solution number of the kinemat ics, we should make sure that neither A, B, C nor E, D, C should be on the same line. Therefore, the shadowed area in Figure 3 4 should be assigned as the restricted zone for the location of end effector point C. Figure 3 3 Area definition 1 Figure 3 4 Area definition 2 PAGE 36 36 Figure 3 5 Restricted zone for C The restricted zone is bonded by a circle centered at point A with the radius of a circle centered at point E with the radius of and the line cross the points A, E. 3.2 Accuracy and Efficiency Analysis for the 2 DOF Parallel Robot The changes of the coordinates of the end effector C, denoted as with reg ard to the changes of and denoted as should not be too small to ensure the accuracy and efficiency of the parallel robot. With the ratio too big, the accuracy of the position of the end effector cannot be guarant eed as small error in the active joint angles could lead to big difference in the position of the end effector. If the ratio is too small, the efficiency of the parallel robot cannot be assured as it takes much greater change in the active j oint angles to achieve the same amount of variation in the position of the end effector, making it hard to achieve fast motion control. Meanwhile, the work range of the end effector is also limited. In this section, a briefly mathematical analysis on the r elationship between the ratio and the values of and the structure parameters of the parallel robot is PAGE 37 37 performed. This could forge a general understanding about their interactions and support with some guidelines on the selections of those according to the specific requirements. First, the derivatives of and with regard to x and y value of the end effector are calculated. Recall the inverse kinematic analysis result in section 3.1. By choosing the mechanical structure, Eq uations 3 5 and 3 6 can be narrowed to ( 3 7) ( 3 8) Assume that Take the partial derivatives of and with regard to and PAGE 38 38 Take the partial derivatives of , and from Eq uation 3 1 to Equation 3 4 with regard to and Then based on Eq uation s 3 7 and 3 8, the partial derivatives of and with regard to and are ( 3 9) PAGE 39 39 ( 3 1 0) ( 3 1 1) ( 3 1 2) The restriction on the working zone is set as follow to simplify the analysis It is meaningless if or as well as setting to be much smaller than that of and Because for those cases, the working range for the parall el robot will be too small, making it incapable for any applications. The value of is fixed as to reduce the amount of calculation. Calculate those derivations from Eq uation 3 9 to Equation 3 1 2 returns the evolution of the ratios due t o the change of and Responding to different requirements on the parallel robot, the value range for , and varies. Here as an example, the ratio ranges are set as and Since this could satisfy the requirement for the robot to move fast and have precise positioning ability. Assume the size of the working zone is F rom the calculation result, the robot structure parameters are Then at the working area PAGE 40 40 the ratio range requirements are met. The corresponding angle range is: The value distribution of in its working area is shown in Figure 3 6. Some values are recorded in Table 3 1. The results of , are shown in Figure 3 7, Figure 3 8 and Figure 3 9. And some values of them are recorded in Table 3 2, Table 3 3 and Table 3 4. B y observing the partial derivatives value distribution, relationship between the partial derivatives and locations of the end effector can be revealed. Figur e 3 6 Partial derivative of qa1 to x PAGE 41 41 Figure 3 7 Partial derivative of qa1 to y Figure 3 8 Partial derivative of qa2 to x PAGE 42 42 Figure 3 9 Partial derivative of qa2 to y Table 3 1. Partial derivative distribution of qa1 to x (rad/dm) x, y values (dm) 0.6 0.4 0.2 0 0.2 0.4 0.6 2.6 0.42618 0.44063 0.45163 0.46011 0.46712 0.47388 0.48175 2.8 0.4 0.41649 0.43037 0.44244 0.45368 0.46532 0.47889 3.0 0.37804 0.39662 0.41337 0.4291 0.44486 0.46209 0.48285 3.2 0.36011 0.38119 0.4013 0.42141 0.44291 0.46791 0.5 Table 3 2. Partial derivative distribution of qa 2 to x (rad/dm) x, y values (dm) 0.6 0.4 0.2 0 0.2 0.4 0.6 2.6 0.48175 0.47388 0.46712 0.46011 0.45163 0.44063 0.42618 2.8 0.47889 0.46532 0.45368 0 .44244 0.43037 0.41649 0.4 3.0 0.48285 0.46209 0.44486 0.4291 0.41337 0.39662 0.37804 3.2 0.5 0.46791 0.44291 0.42141 0.4013 0.38119 0.36011 PAGE 43 43 Table 3 3. Partial derivative distribution of qa 1 to y (rad/dm) x, y values (dm) 0.6 0.4 0. 2 0 0.2 0.4 0.6 2.6 0.2702 0.24271 0.21781 0.19628 0.17876 0.16577 0.15784 2.8 0.3 0.27694 0.25629 0.23882 0.22526 0.21635 0.21305 3.0 0.33534 0.31644 0.30015 0.2873 0.27882 0.27591 0.28034 3.2 0.38085 0.36636 0.35522 0.34852 0.34775 0.35522 0.375 Table 3 4. Partial derivative distribution of qa2 to y (rad/dm) x, y values (dm) 0.6 0.4 0.2 0 0.2 0.4 0.6 2.6 0.15784 0.16577 0.17876 0.19628 0.21781 0.24271 0.2702 2.8 0.21305 0.21635 0.22526 0.23882 0.25629 0 .27694 0.3 3.0 0.28034 0.27591 0.27882 0.2873 0.30015 0.31644 0.33534 3.2 0.375 0.35522 0.34775 0.34852 0.35522 0.36636 0.38085 From the derivatives distribution tables and the 3D derivatives distribution plots, it can be seen that for 2 DOF parallel robots with four bars similar in length, values of all partial derivatives reach extremely large value at the edge of the restricted zone for C in Figure 3 5 and A, E points together with the area close to them. Meanwhile, for the rest area, the values of all four partial derivatives set in four sets of relative stable range and the mean values of those range is decided by the selection of and For the cases where the mean values of these range get bigger with and taking smaller value. As a result, for specific desired values of partial derivatives, the optimal value of and can be decid ed with the mean value mentioned above to be tuned the same as or close to the desired values of partial derivatives. And the working area can be finally determined by set up the sets of value range for the partial derivatives. PAGE 44 44 3.3 Modeling for the 2 DOF P arallel Robot 3.3.1 Dynamics Model for the 2 DOF Parallel Robot Based on the analysis and calculation on section 3.2, choose the diameter of the robot arm to be The mass of the four robot arms are identical, which is: The length of all four robot arms is i.e., Treat the robot arm as thin rod, with a coordinate system attached to the rod as shown on Figure 3 10. Cal culate moment of inertia for the rod returns The restrictions for the joint angles are: Figure 3 10 Robot arm coordinate system For convenience, reset the parallel robot coordinate system as Figure 3 11. The relationship between the angles can be drawn through the close loop ABCDEA: PAGE 45 45 Namely, ( 3 1 3) ( 3 1 4) Take first and second derivatives of Eq uations 3 1 3 and 3 1 4 gives ( 3 1 5) (3 1 6) ( 3 1 7 ) ( 3 1 8 ) Figure 3 11 Revised robot coordinate system Here New ton Euler method is applied to set up the relationship between the input torque at the two active joints and the states of all the joints. Let denote the position vector from A to B ,B to C, D to C and E to D PAGE 46 46 denote the interacting force at A, B, C, D and E. denote the gravity force of bar AB, BC, CD and DE. denote the angular velocities of bar AB, BC, CD and DE. denote the angular accelerations of bar AB,BC,CD and DE. PAGE 47 47 and denote the torque applied at the active joint A and E. , and denote the acceleration of the mass centers of bar AB, BC, CD and DE. Newton Eule r method is applied to different bars and the whole robot system as follow: Apply Newton's second law of motion and Euler's laws of motion to bar BC, with all forces defined in Figure 3 12: PAGE 48 48 Figure 3 12 Force analysis for bar BC and DC Apply Newto n's second law of motion and Euler's laws of motion to bar BC, with all forces defined in Figure 3 12: i.e., ( 3 19 ) Apply Newton's second law of motion and Euler's laws of motion to bar CD, with all forces defined i n Figure 3 12: i.e., ( 3 20 ) Apply Newton's second law of motion and Euler's laws of motion to bar AB, with all forces defined in Figure 3 13: PAGE 49 49 i.e., ( 3 2 1 ) F igure 3 13 Force analysis for bar AB and ED Apply Newton's second law of motion and Euler's laws of motion to bar DE, with all forces defined in Figure 3 13: i.e., ( 3 2 2 ) Solve E quations 3 19 3 2 0 3 2 1 a nd 3 2 2 gives PAGE 50 50 ( 3 2 3 ) ( 3 2 4 ) where The relationship between the input torques and the angular states is therefore shown in Eq uation s 3 2 3 and 3 2 4 Combined Eq uation s 3 23 3 2 4 with Eq uation s 3 1 7 3 1 8 returns the dynamic equat ion of the 2 DOF parallel robot. PAGE 51 51 where 3.3.2 Verification of the dynamics model To verify the validity of the dynamics model, some specific values of the parameters are given to the dynamics model. Simulation result from SimMechanics and result deducted directly from mathematical equations in MATLAB are compared to each other. Substitute into the dynamics model. PAGE 52 52 As verification for the accuracy of the mathematical model, the dynamic characteristics of mathematical model will b e compared to the results from the SimMechanics model in Figure 3 14. Figure 3 14 SimMechanics model for the 2 DOF parallel robot PAGE 53 53 The simulation results of the mathematical model and the SimMechanics model would be greatly affected by the appearance o f singularity. PD control is applied to these two models to avoid the singularity. The PD control output is And the initial parameters are set as Simulation time is set as The simulation results shown from Figure 3 15 to Figure 3 20 indicate that the curves of angles, angular velocities, and control inputs for both models are well matched or closed to each other. The errors for those six parameters between the two models are presented from Figure 3 21 to Figure 3 26. And the error percentages for those six parameters between the models are always smaller than 0.1%. From the results shown in those figures, the conc lusion can be made that the input output characteristic and dynamic behavior of these two models are close to each other, i.e., the dynamics model built in section 3.3.1 is valid. Figure 3 15 Value of qa1 in mathmetic model and SimMechanics modle PAGE 54 54 Figure 3 16 Value of qa2 in mathematical model and SimMechanics model Figure 3 17 Angular velocity of qa1 in mathematical model and SimMechanics model PAGE 55 55 Figure 3 18 Angular velocity of qa2 in mathematical model and SimMechanics model Figure 3 19 Input torque at A in mathematical model and SimMechanics model PAGE 56 56 Figure 3 20 Input torque at E in mathematical model and SimMechanics model Figure 3 21 Error between two models for qa1 PAGE 57 57 Figure 3 22 Error between two models for qa2 Figur e 3 23 Error between two models in angular velocity of qa1 PAGE 58 58 Figure 3 24 Error between two models in angular velocity of qa2 Figure 3 25 Error between two models for input torque at A PAGE 59 59 Figure 3 26 Error between two models for input torque at E 3.4 Dynamics Model for the 2 DOF Parallel Robot with Dnamics and Kinematics Uncertainties The 2 DOF parallel robot with parameters is shown in Figure 3 27. Modeling for the 2 DOF parallel robot with dynamics and kinematics uncertainties has the same proc edures as modeling for the 2 DOF parallel robot with no uncertainties, except that and are not necessarily identical due to the dynamics uncertainties and and are not necessarily identical due to the kinematics uncertainties. Restrictions for the joint angles: The relationship between the angles is driven through the close loop ABCDEA: Take first and second derivatives of combined with gives the equati ons: PAGE 60 60 ( 3 2 5 ) ( 3 2 6 ) Figure 3 27 2 DOF parallel robot wi th dynamics and kinematics uncertainties Follow the steps in section 3.3.1, using the Newton Euler method and rewrite the equations with those new characters: PAGE 61 61 Figure 3 28 Force analysis for bar BC and DC For bar BC: For bar CD: PAGE 62 62 Figure 3 29 Force analysis for bar AB and ED For bar AB: For bar ED: Rewrite the equations as: ( 3 2 7 ) PAGE 63 63 ( 3 2 8 ) where Combine Eq uations 3 2 7 and 3 2 8 gives ( 3 29 ) where PAGE 64 64 Substitute and with and in Eq uation 3 29 where with PAGE 65 65 where PAGE 66 66 PAGE 67 67 PAGE 68 68 CHAPTER 4 CONTROL S YSTEM DESIGN FOR 2 DOF PARALLEL ROBOT 4.1 Adaptive Backstepping Controller for the 2 DOF Parallel Robot Model with Uncertainties According to the geometric relationship, the Jacobian matrix mapping from the angular velocities of the active joints A, E to t he velocity of the end effector is where Separate the kinematic uncertain phy sical parameters and collect them in the vector where where PAGE 69 69 Estimate of where From the analysis in section 3.4, the dynamic model for 2DOF Parallel Robot ignoring friction would be According to Property3 PAGE 70 70 where with PAGE 71 71 PAGE 72 72 PAGE 73 73 The actual structure parameters of the 2 DOF parallel robot in the simulation are designed as (a) For the implementation, the parameter estimates are initialized as (b) The desired the point i s designed as PAGE 74 74 The desired trajectory is designed as Bring those values into expressions of the dynamics m odel. With the assistance of MATLAB SimMechanics the simulation model of the 2 DOF parallel robot with structure parameters in (a) controlled by the adaptive backstepping controller using initial estimate parameters in (b) is built up and shown in Figure 4 1. 4.2 Simulation Result and Discussion In this section, two types of tracking control, set point tracking control and trajectory tracking control, are implemented. For the set point tracking control, the assignment for the controller is to adjust the in put torque on the active joints so that the end effector could eventually reach the destination point. For the trajectory tracking control, the controller manages the input torque on the active joints with the goal to follow the desired trajectory. As a co ntrast, set point control and trajectory control using the only backsteppping controller carried out on the same 2 DOF parallel robot. Set Point Tracking For the set point tracking control, the desired point is given as The destination poin t and tracking trajectory, tracking errors between the end effector and the destination point in X direction and Y direction during this p rocess are displayed in figures. PAGE 75 75 Figure 4 1. Simulation Block for Control System PAGE 76 76 Trajectory Tracking For the t rajectory tracking control, the desired trajectory is set as in meters. The desired trajectory and tracking trajectory, corresponding tracking errors between the tracking trajectory of end effector and the desired traje ctory in X direction and Y direction during this process are shown in figures. Simulation r esult of set point tracking for: 1) the adaptive backstepping controller proposed in this thesis for the 2 DOF parallel robot with exact model knowledge (ABE). From Fig ure 4 2 to Figure 4 4 ; 2) the backstepping controller for the 2 DOF parallel robot with exact model knowledge (BE). From Figure 4 5 to Figure 4 7 ; 3) the adaptive backstepping controller proposed in this thesis for the 2 DOF parallel robot model with dynamics an d kinematics uncertainties (ABU). From Figure 4 8 to Figure 4 10; 4) the backstepping controller for the 2 DOF parallel robot model with dynamics and kinematics uncertainties (BU). From Figure 4 11 to Figure 4 13. Figure 4 2 Destination point and trackin g trajectory (ABE) PAGE 77 77 Figure 4 3 Error in x direction for set point tracking (ABE) Figure 4 4 Error in y direction for set point tracking (ABE) PAGE 78 78 Figure 4 5 Destination point and tracking trajectory (BE) Figure 4 6 Error in x direction for s et point tracking (BE) PAGE 79 79 Figure 4 7 Error in y direction for set point tracking (BE) Figure 4 8 Destination point and tracking trajectory (ABU) PAGE 80 80 Figure 4 9 Error in x direction for set point tracking (ABU) Figure 4 10 Error in y direction for set point tracking (ABU) PAGE 81 81 Figure 4 11 Destination point and tracking trajectory (BU) Figure 4 12 Error in x direction for set point tracking (BU) PAGE 82 82 Figure 4 13 Error in y direction for set point tracking (BU) Simulation result of trajectory tracking for : 1) the adaptive backstepping controller proposed in this thesis for the 2 DOF parallel robot with exact model knowledge (ABE). From Figure 4 14 to Figure 4 16; 2) the backstepping controller for the 2 DOF parallel robot with exact model knowledge (BE). From Figure 4 17 to Figure 4 19; 3) the adaptive backstepping controller proposed in this thesis for the 2 DOF parallel robot model with dynamics and kinematics uncertainties (ABU). From Figure 4 20 to Figure 4 22; 4) the backstepping controller for the 2 DOF parallel robot model with dynamics and kinematics uncertainties (BU). From Figure 4 23 to Figure 4 25. From the simulation results, it is can be seen that the performance of the adaptive backstepping controller is comparable with that of the backstep ping controller PAGE 83 83 for systems with exact model knowledge. Compare the simulation results for two controllers from Figure 4 2 to Figure 4 7 and from Figure 4 14 to Figure 4 19, it is clear that the backstepping controller even makes the errors converge to 0 f aster than the adaptive backstepping controller. Compare the simulation results for two controllers from Figure 4 8 to Figure 4 13 and from Figure 4 20 to Figure 4 25, it is clear that the proposed adaptive backstepping controller gives a robust result for the 2 DOF parallel robot system with both dynamics and kinematics uncertainties while the backstepping controller cannot even give asymptotic result as the tracking errors for the system controlled by the backstepping controller oscillate around 0 over bu t never disappear. Figure 4 14. Desired trajectory and tracking trajectory (ABE) PAGE 84 84 Those results fit well with the corresponding theory. Because the backstepping controller used as a contrast is exact model based controller, it does not involve the estim instantly give the required or desired control output on the system to reach the destination in a more efficient and faster way. And that is one of the advantages for the exact model knowledge based controller. Though giving inspiring control result for exact model system, those controllers act badly when no exact model knowledge about the system is available. Therefore, for the uncertain model cases, the backstepping controller cannot give asymptotic tracking results. As for the proposed adaptive backstepping controller, the estimator in the controller can adjust to the dynamics and kinematics uncertainties through self adaptive process, thus drive the errors to 0. Figure 4 15 Error in x direction for trajectory tracking (ABE) PAGE 85 85 Figure 4 16 Error in y direction for trajectory tracking (ABE) Figure 4 17 Desired trajectory and tracking trajectory (BE) PAGE 86 86 Figure 4 18 Error in x direction for trajectory tracking (BE) Figure 4 19 Error in y direction for trajectory tracking (BE) PAGE 87 87 Figure 4 20 Desired trajectory and tracking trajectory (ABU) Figure 4 21 Error in x direction for trajectory tracking (ABU) PAGE 88 88 Figure 4 22 Error in y direction for trajectory trac king (ABU) Figure 4 23 Desired trajectory and tracking trajectory (BU) PAGE 89 89 Figure 4 24. Error in x direction for trajectory tracking (BU) Figure 4 25 Error in y direction for trajectory tracking (BU) PAGE 90 90 CHAPTER 5 CONCLUSION AND FUTURE WORK The pro posed adaptive backstepping controller could well address the issue of precise position control or tracking control for the 2 DOF parallel robots with dynamics and kinematics uncertainties. Actually, from the analysis in section 2, it is not hard to realiz e that the proposed adaptive backstepping controller could achieve precise position control or tracking control for all parallel robot connected only by revolute joints. This research opens up to new challenges. One of them is to explore the influence of t he coefficients on the control result. The influence of on the control result is quite different. From some sample tests, it appears has a greater influence on the control r esult than the other coefficients. Further work could be done to reveal a relatively more complete relationship between those coefficients and the control result. This is a very useful aspect as it provides guidelines on how to tune those coefficients to g et different types of desired results. In addition, the coverage of the proposed adaptive controller can be extended. The kinematics analysis in section 2 is limited to parallel robot connected only by revolute joints. But this restriction is not final. Si milar analysis can be applied on other sorts of parallel robot. PAGE 91 91 REFERENCES Magazine. 16, 75 83 (2009). 2. Carl D. Crane, III and Joseph Duffy, Kinematic Analysis of Robot Manipulators (Cambridge University Press, New York, US, 1998). Nonlinear PD Controller for a Redundan Advanced Robotics 23, 1725 1742 (2009). 4. Hui Cheng, Yiu 491 (2003). 5. Jin Qinglong th Chinese Control Conference (2011) pp. 2440 2445. Machine Theory. 45, 80 90 (2010). 7. Jinkun Liu and Xinhua Wang, Advanced Sliding Mode Control for Mechanical Systems (Tsinghua University Press, Beijing, China, 2011). robots Automatic Control. 51, 1024 1029 (2006). onal Confannee on Robtics &Automation. 3, 3075 3080 (2004). Nonlinear Control of Robot Manipulator With Uncertainties in Kinematics Dynamics and al Journal of Innovative Computing, Information and Control. 8, 5487 5498 (2012). Interna tional Conference on Control, Instrumentation and Automation (2011) pp. 911 916. and Automation, Albuquerque. 4, 2961 2966( 1997). PAGE 92 92 13. Tien Dung Le, Hee Jun Kang and Young Strategic Technology (2012) pp. 1 4. 14. W. 3939. 15. Wenbin Deng, Jae Won Lee and Hyuk of a New 2 D OF Parallel Mechanism Based on Matlab / SimMechanics International Colloquium on Computing, Communication, Control, and Management. 3,233 236 (2009). PAGE 93 93 BIOGRAPHICAL SKETCH Jing Zou was born in Nanchang, China, in the year 1989. He received a ba Mechanical and Aerospace Engineering Department at the University of Florida in August 20 12 and received his MS degree in May 2014. 