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 1
2014 Jing Zou 2
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. 3
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 CONTROLLER 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 ANALYSIS FOR THE 2 DOF PARALLEL ROBOT... 32 3.1 Kinematic and Singularity Analysis of the 2 DOF Parallel Robot... 32 3.2 Accuracy and Efficiency 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 Parallel Robot... 44 3.3.2 Verification of the dynamics model... 51 3.4 Dynamics Model for the 2 DOF Parallel Robot with Dnamics and Kinematics 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 4
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 5
LIST OF FIGURES Figure: 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 mathmetic 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 6
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 direction for set point tracking (ABE)... 77 4-4 Error in y direction for set point tracking (ABE)... 77 4-5 Destination point and tracking 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 7
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 8
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 DESIGN OF AN ADAPTIVE BACKSTEPPING CONTROLLER FOR 2 DOF PARALLEL ROBOT Chair: John K. Schueller Major: Mechanical Engineering By Jing Zou May 2014 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 errors 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 SimMechanics 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 controller has robust performance with regard 9
to dynamics and kinematics uncertainties. The proposed controller gives desired performance to achieve the research goal. 10
CHAPTER 1 INTRODUCTION 1.1 Background Robot manipulators have been widely used in our current 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 hand 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 motoractuated joints that extend from a base to an end-effector, are the most common industrial robots. However, 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 and 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 explore the kinematic and dynamics characteristics of the parallel robots. 11
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 regard 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 robot 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 cancel 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 controller 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 having 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 kinematic 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 12
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 robot 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 robot. 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, the 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. 13
CHAPTER 2 ADAPTIVE BACKSTEPPING CONTROLLER FOR PARALLEL ROBOTS 2.1 Kinematics and Dynamics Analysis for Parallel Robots 2.1.1 Kinematics Analysis Here a kinematic 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 kinematic chain is defined as a leg. The th leg has 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,, which 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: 14
[ ] [ ] (2-1) with the Jacobian matrix of the th leg. Joint velocity selection matrices: is a matrix with one 1 in each row, at a place corresponding to a driving joint in the vector. A typical looks like [ ] (2-2) selects the vector ( ) of the driving 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 indices 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) 15
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 matrix with one 1 in each row, at a place corresponding to a joint in the th leg: [ ] (2-7) It selects the joint velocities of the th submanipulator from the vector of all joint velocities: (2-8) The following identities follow straightforwardly: (2-9) and (2-10) Dependency matrix: The major point in solving the velocity closure equations is to find a relationship between the known driving joint velocities and the unknown driven joint velocities. Let s formally capture this relationship in the following definition of the dependency matrix : ( ) (2-11) 16
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 dependency matrix can also be written as (2-12) Closed-form solution: The velocity closure Equation 2-1, together with Equation 2-10, give ( ) (2-13) The matrices and are submatrices of A in Equation 2-1 that contain only the columns corresponding to driving and passive joints, respectively. These matrices are fully known once position closure is achieved. By definition of what a driving joint actually is, the vector space subspace of the vector space spanned by the driving joints must always be a 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) 17
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) The velocity closure Equations 2-11 can then also be written as (2-18) Note the important difference with the definition of a column of the Jacobian 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 generates the th column is, by definition, given by, the th column of the transpose of the selection matrix, Equation 2-2, that has a 1 on the place of the th driving joint, and 0 for all other joints. Equation 2-15 gives the corresponding velocities of the passive joints: (2-19) Combining Equations 2-10, 2-15 and 2-17 yields (2-20) 18
This equation gives the th column of the dependency matrix in of Equation 2-11. Using Equations 2-8 and 2-18, the th column of is then found as (2-21) The right-hand sides of these equations first select the joint velocities of one of the serial subchains from the vector of all joint velocities, and multiplies these subchain joint velocities with the subchain Jacobian matrix 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 abovementioned procedure for all driving joints gives, for all, ( ) (2-22) with the dependency matrix as defined in (11). Jacobian matrix property analysis: According to Equation 2-20, Equation 2-22 can 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 are all linear in physical parameters is obtained from their previous descriptions in this section. 19
The first component of Equation 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 Equation 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 linear 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 several legs connected by rotation joints, the relationship between the speed of the end effector and the angular velocities of the joint angles in the th leg could be treated like a serial robot with the same number of linkage, then (2-28) 20
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) joint in the th leg,,, are the first, second and third element of vector function. is the number of passive joint in one leg and assume all legs have the same number of 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 The component of matrix is shown below. in row 1, column 1 is The same goes for other components of 21
( ) 22
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 If above., which applies for parallel robots with no less than three legs, then is a scalar function of measurable variables and the azimuth angles of joint axis. For, which applies for parallel robots with two legs, then 23
is also a scalar function of measurable variables and the azimuth angles of joint axis. Therefore, is linear in a combination of physical parameters, and is also linear in a combination of physical parameters. Thus ( ) Bring Equation 2-29 back to Equation 2-26 (2-30) ( ) ( ) (2-31) where ( ) is the regressor matrix and is the combination of physical parameters in and. The second part of Equation 2-24, i.e., is linear in a set of physical parameters Substitutes Equations 2-31 and 2-26 into Equation 2-24 yields (2-32) where is the regressor matrix and is the combination 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 dynamic model of a parallel robot with uncertain parameters is: (2-33) 24
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 all. 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 number of linkage in each leg, both their kinematic and dynamic models are linear in 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 dynamics and kinematics. 25
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 endeffector and the destination position of the end-effector. And for simplicity, replace ( ) with. Then (2-34) Taking the time derivative on both sides of Equation 2-34 and substituting Equation 2-32 into it (2-35) here backstepping control is introduced through plus and subtract and on the left side of Equation 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) 26
where is the error between the set of 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., Equation 2-33 ( ) ( ) ( )( ) ( ) ( ) (2-38) Applying Property 3 in Equation 2-38 ( ) ( ) (2-39) Equation 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 Equaitons 2-37 and 2-39 into Equation 2-42 and apply Property 2 ( ) ( ( ) ( ) ) 27
( ( ) ) ( ( ) ) ( ( ) ) (2-43) Design the input controller as Substitute Equation 2-44 into Equation 2-43 ( ) (2-44) ( ( ) ( ) ) ( ( ) ) ( ) ( ) (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 Equations 2-46, 2-47 into Equation 2-45 (2-48) where is a negative semi-definite function. 28
Barbalate s Lemma Corollary: If a scalar function, is such that is lower bounded by zero and is uniformly continuous in time Then, as. It 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) Equation 2-48 could then be rewritten as ( ) where. Take the derivative of and substitute Equation 2-16 into the derivative ( ) According to the results in Equation 2-50 29
( ) Consequently, 1) is lower bounded by ; 2) ; and 3) and is uniformly continuous. All the conditions in Barbalate s Lemma Corollary are satisfied. Applying Barbalate s Lemma Corollary to the Lyapunov candidate in Equation 2-42 leads 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 area, which guarantees,. Apply the conclusions from Equation 2-29 to Equations 2-25 and 2-26,,, ( ), and gives the following results { { Introduce the above results in the equation 30
bounded. Thence, all those designed and measured values have been proven to be could be measured and are made of measurable parameters, from Equaiton 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 Equation 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. 31
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 the 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 other parameters of the parallel robot are represented by the characters shown in Figure 3-1. The following equations are driven from the geometric relationship. 32
Figure 3-2. The coordinate system of the 2 DOF parallel robot There are four joint variables,,, with only two independent variables,. while the rest joints and are functions of and.assume that 33
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 solutions for the inverse kinematic analysis. Singularity: Due to the analysis in [13], singularity happens under three cases. 34
Type I singularity happens when or, Type II singularity happens when, and Type III singularity 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 kinematics, 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 35
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 regard 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 guaranteed 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 joint 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 relationship between the ratio and the values of and the structure parameters of the parallel robot is 36
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 endeffector are calculated. Recall the inverse kinematic analysis result in section 3.1. By choosing the mechanical structure, Equations 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 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 37
Take the partial derivatives of,, and from Equation 3-1 to Equation 3-4 with regard to and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Then based on Equations 3-7 and 3-8, the partial derivatives of and with regard to and are ( ) ( ) (3-9) 38
( ) ( ) (3-10) ( ) ( ) (3-11) ( ) ( ) (3-12) 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 parallel 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 Equation 3-9 to Equation 3-12 returns the evolution of the ratios due to 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. From the calculation result, the robot structure parameters are. Then at the working area 39
, 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. By observing the partial derivatives value distribution, relationship between the partial derivatives and locations of the end-effector can be revealed. Figure 3-6. Partial derivative of qa1 to x 40
Figure 3-7. Partial derivative of qa1 to y Figure 3-8. Partial derivative of qa2 to x 41
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 qa2 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 42
Table 3-3. Partial derivative distribution of qa1 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 decided 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. 43
3.3 Modeling for the 2 DOF Parallel 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. Calculate 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: 44
[ ] [ ] [ ] Namely, (3-13) (3-14) Take first and second derivatives of Equations 3-13 and 3-14 gives (3-15) (3-16) (3-17) (3-18) Figure 3-11. Revised robot coordinate system Here Newton-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 45
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. [ ] [ ] 46
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-Euler 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: 47
Figure 3-12. Force analysis for bar BC and DC Apply Newton'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 in 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: 48
{ i.e., { (3-21) Figure 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., { (3-22) Solve Equations 3-19, 3-20, 3-21 and 3-22 gives 49
(3-23) (3-24) where ( ) ( ) The relationship between the input torques and the angular states is therefore shown in Equations 3-23 and 3-24. Combined Equations 3-23, 3-24 with Equations 3-17, 3-18 returns the dynamic equation of the 2 DOF parallel robot. ( ) 50
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. [ ] 51
( ) [ [ ] ( ) ( ) ] As verification for the accuracy of the mathematical model, the dynamic characteristics of mathematical model will be compared to the results from the SimMechanics model in Figure 3-14. Figure 3-14. SimMechanics model for the 2 DOF parallel robot 52
The simulation results of the mathematical model and the SimMechanics model would be greatly affected by the appearance of 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 wellmatched 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 conclusion 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 53
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 54
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 55
Figure 3-20. Input torque at E in mathematical model and SimMechanics model Figure 3-21. Error between two models for qa1 56
Figure 3-22. Error between two models for qa2 Figure 3-23. Error between two models in angular velocity of qa1 57
Figure 3-24. Error between two models in angular velocity of qa2 Figure 3-25. Error between two models for input torque at A 58
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 procedures as modeling for the 2 DOF parallel robot with no uncertainties, except that and and are not necessarily identical due to the dynamics uncertainties 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: [ ] [ ] [ ] equations: Take first and second derivatives of, combined with gives the 59
( ) (3-25) ( ) (3-26) Figure 3-27. 2 DOF parallel robot with 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: 60
[ ] [ ] [ ] [ ] Figure 3-28. Force analysis for bar BC and DC For bar BC: { For bar CD: 61
{ Figure 3-29. Force analysis for bar AB and ED For bar AB: { For bar ED: { Rewrite the equations as: (3-27) 62
(3-28) where ( ) ( ) Combine Equations 3-27 and 3-28 gives [ ] [ ] [ ] [ ] (3-29) where 63
[ ] [ ] [ ] ( ) [ ( ) ] Substitute and with and in Equation 3-29 [ ] [ ] where [ [ ] ] ( ) [ ( ) ] with ( ) 64
( ) ( ) ( ) ( ) ( ) where ( ) ( ( ) ) ( ) ( ) ( ) ( ) 65
( ) ( ) ( ) ( ) ( ( ( ) ) ) ( ) 66
( ) ( ( ) ) 67
CHAPTER 4 CONTROL SYSTEM 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 the velocity of the end-effector is [ ] where Separate the kinematic uncertain physical parameters and collect them in the vector ( ) where [ ] ( ) [ ] where 68
( ( ( ( ) ) ) ) Estimate of [ ] where From the analysis in section 3.4, the dynamic model for 2DOF Parallel Robot ignoring friction would be [ ] [ ] According to Property3 ( ) ( ) 69
where [ ] ( ) [ ] with ( ) ( ) ( ) ( ) 70
( ) ( ) ( ) ( ) ( ) ( ) ( ) 71
( ) ( ) ( ( ) ) ( ) 72
( ( ) ) ( ) ( ( ) ) designed as The actual structure parameters of the 2-DOF parallel robot in the simulation are For the implementation, the parameter estimates are initialized as (a) The desired the point is designed as (b) 73
[ ] The desired trajectory is designed as [ ] Bring those values into expressions of the dynamics model. 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 input 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 contrast, 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 point and tracking trajectory, tracking errors between the end-effector and the destination point in X direction and Y direction during this process are displayed in figures. 74
Figure 4-1. Simulation Block for Control System 75
Trajectory Tracking. For the trajectory 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 trajectory in X direction and Y direction during this process are shown in figures. Simulation result 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 Figure 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 and 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 tracking trajectory (ABE) 76
Figure 4-3. Error in x direction for set point tracking (ABE) Figure 4-4. Error in y direction for set point tracking (ABE) 77
Figure 4-5. Destination point and tracking trajectory (BE) Figure 4-6. Error in x direction for set point tracking (BE) 78
Figure 4-7. Error in y direction for set point tracking (BE) Figure 4-8. Destination point and tracking trajectory (ABU) 79
Figure 4-9. Error in x direction for set point tracking (ABU) Figure 4-10. Error in y direction for set point tracking (ABU) 80
Figure 4-11. Destination point and tracking trajectory (BU) Figure 4-12. Error in x direction for set point tracking (BU) 81
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 backstepping controller 82
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 faster 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 but never disappear. Figure 4-14. Desired trajectory and tracking trajectory (ABE) 83
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 estimators. For exact model system, those controllers without the estimator could 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) 84
Figure 4-16. Error in y direction for trajectory tracking (ABE) Figure 4-17. Desired trajectory and tracking trajectory (BE) 85
Figure 4-18. Error in x direction for trajectory tracking (BE) Figure 4-19. Error in y direction for trajectory tracking (BE) 86
Figure 4-20. Desired trajectory and tracking trajectory (ABU) Figure 4-21. Error in x direction for trajectory tracking (ABU) 87
Figure 4-22. Error in y direction for trajectory tracking (ABU) Figure 4-23. Desired trajectory and tracking trajectory (BU) 88
Figure 4-24. Error in x direction for trajectory tracking (BU) Figure 4-25. Error in y direction for trajectory tracking (BU) 89
CHAPTER 5 CONCLUSION AND FUTURE WORK The proposed 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 realize 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 the 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 result 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 get 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. Similar analysis can be applied on other sorts of parallel robot. 90
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