7-Degree-Of-Freedom (DOF) Cable-Driven Humanoid Robot Arm. A thesis presented to. the faculty of. In partial fulfillment

Transcription

1 Mechanism Design, Kinematics and Dynamics Analysis of a 7-Degree-Of-Freedom (DOF) Cable-Driven Humanoid Robot Arm A thesis presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Master of Science Jun Ding March Jun Ding. All Rights Reserved.

2 2 This thesis titled Mechanism Design, Kinematics and Dynamics Analysis of a 7-Degree-Of-Freedom (DOF) Cable-Driven Humanoid Robot Arm by JUN DING has been approved for the Department of Mechanical Engineering and the Russ College of Engineering and Technology by Robert L. Williams II Professor of Mechanical Engineering Dennis Irwin Dean, Russ College of Engineering and Technology

3 3 ABSTRACT DING, JUN, M.S., March 2011, Mechanical Engineering Mechanism Design, Kinematics and Dynamics Analysis of a 7-Degree-Of-Freedom (DOF) Cable-Driven Humanoid Robot Arm (118 pp.) Director of Thesis: Robert L. Williams II The purpose of this thesis is to study a 7-DOF humanoid cable-driven robotic arm, implement kinematics and dynamics analysis, present different cable-driven designs and evaluate their merits and drawbacks. Since this is a redundant mechanism, kinematics optimization is used to avoid joint limits, singularities and obstacles. Cable kinematics analysis studies the relations between lengths of cables and pose of the end-effector. This is a design modified from the literature. Several new designs are compared in statics analysis of the whole arm and the most favorable design is suggested in terms of motion range and the consumption of cable tensions. Linear programming is used to optimize cable tensions. The energy consumption of the cable-driven arm is much less than that of the traditional motor-driven arm in dynamics analysis. Cable-driven robots have potential benefits but also some limitations. Approved: Robert L. Williams II Professor of Mechanical Engineering

4 4 ACKNOWLEDGMENTS A sincere and special thanks to my advisor, Dr. Robert L. Williams II. He led me into robotics a complete new field for me and advised me to pick this challenging topic for my thesis. In my research he provided a lot of good suggestions and revised my thesis. I would like to thank Dr. Zhu for giving me an opportunity to see his cable-driven robotic cat and offering some suggestions. I appreciate Dr. Butcher for identifying some practical issues for cable-driven robots. Also, I would like to thank Dr. Pasic for his time and participation. They are my committee members for this thesis. Additionally, I need to thank Elvedin Kljuno for some advice in research. Finally, I want to thank my parents-- Deyan Ding and Chunrong Huang for their support. I love them.

5 5 TABLE OF CONTENTS Page Abstract... 3 Acknowledgments... 4 List of Figures... 7 List of Tables... 9 List of Symbols Introduction Background Literature Review Project Information Thesis Objectives Kinematics Analysis and Kinematics Optimization Introduction to Denavit-Hartenberg (DH) Parameters Arm Model and DH Parameters Forward Pose Kinematics (FPK) Forward Velocity Kinematics Inverse Velocity Kinematics Singularity Analysis of Robotic Arm Model Objective functions Mechanism Design of Cable-Driven Robotic Arm and Statics Analysis Mechanism Design of Cable-Driven Robotic Arm Linear Programming Statics Analysis Motion Range Cable Kinematics Analysis Forward Pose Cable Kinematics (FPCK) Inverse Pose Cable Kinematics (IPCK) Dynamics Analysis Dynamics Analysis of the Cable-Driven Robotic Arm... 63

6 Dynamics Analysis of Traditional Motor-Driven Robotic Arm Comparison of Cable-Driven Robot and Traditional Motor-Driven Robot Conclusions References Appendices Appendix A: Transformation Matrices between Frames Appendix B: Jacobian Matrix and Singularity Analysis Appendix C: Parameters in Dynamics Analysis Appendix D. MATLAB Programs of Simulation... 96

7 7 LIST OF FIGURES Page Figure 1. Elbow joint geometry Figure 2. Schematic diagram of the cable driven robotic arm (Yang et al., 2005) Figure 3. Chen s cable-driven 7-DOF manipulator (Chen et al., 2006) Figure 4: Drawing of Link i-1 Connecting Joint i and Joint i Figure 5. 7-DOF arm model (Venkatayogi, 2007) Figure 6. DH parameters (Venkatayogi, 2007) Figure 7. Simulation of robot position Figure 8. Internal singularity #1 for arm subassembly Figure 9. Internal singularity #2 for arm subassembly Figure 10. Internal singularity #3 for arm subassembly Figure 11.Internal singularity for wrist subassembly Figure 12. Joint angle for without optimization Figure 13. Joint limit avoidance for Figure 14. Joint angle without optimization Figure 15. Singularity avoidance for joint angle Figure 16. Cable-drive mechanism Figure 17. Three designs of a cable-driven spherical joint Figure 18. Four different designs of a cable-driven spherical joint Figure 19. Cable-driven mechanism analysis for shoulder joint (Design 2A) Figure 20. Free body diagram of hand Figure 21. Free body diagram for forearm Figure 22. Free body diagram of upper arm Figure 23. Trajectories of seven joints in the slow motion Figure 24. Trajectories of seven joints in the fast motion Figure 25. The motion path of the robotic arm Figure 26. Cable tensions of cable Figure 27. Cable tensions of cable

8 8 Figure 28. Cable tensions of cable Figure 29. Cable tensions of cable Figure 30. Cable tensions of cable Figure 31. Cable tensions of cable Figure 32. Cable tensions of cable Figure 33. Cable tensions of cable Figure 34. Cable tensions of cable Figure 35. Cable tensions of cable Figure 36. The total cable tensions of 10 cables for six designs Figure 37. Analysis of torques about the z axis for different designs Figure 38. Motion ranges of shoulder joints for different designs Figure DOF cable-driven arm model Figure 40. Shoulder module of cable-driven mechanism Figure 41. Elbow module of cable-driven mechanism Figure 42. Free body diagram of the hand for dynamics analysis Figure 43. Free body diagram of forearm for dynamics analysis Figure 44. Free body diagram of the upper arm for dynamics analysis Figure 45. Optimal cable tensions of Dynamics (slow motion) Figure 46. Opitmal cable tensions of Statics (slow motion) Figure 47. Opitmal cable tensions of Dynamics (fast motion) Figure DOF arm model in real world Figure 49. Energy of dynamics analysis in traditional mechanism in slow motion Figure 50. Energy of dynamics analysis in the traditional mechanism in fast motion Figure 51. Total energy of dynamics needed for different robots in slow motion Figure 52. Total energy of dynamics needed for different robots in fast motion Figure 53. Total instantaneous power of dynamics needed for robots in slow motion Figure 54. Total instantaneous power of dynamics needed for robots in fast motion Figure 55. Relations among different variables... 80

9 LIST OF TABLES Table 1.The masses of seven motors

10 10 lha lu lf LIST OF SYMBOLS Denavit-Hartenberg parameter Denavit-Hartenberg parameter objective function objective function objective function the length of hand the length of upper arm the length of forearm Moore-Penrose pseudoinverse of Jacobian matrix end-effector velocity density of aluminum Denavit-Hartenberg parameter seven joint rates of arm (frames 1-7) joint angle i joint rate i M frame m external torques rotation matrix of frame m relative to frame n homogeneous transformation matrix of frame m relative to frame n Jacobian matrix expressed in transpose of Jacobian matrix gradient of objective function Hextracting Mathematical notation: det gradient determinant absolute value

11 11 1. INTRODUCTION 1.1. Background An arm model with seven degrees of freedom (DOF) is often proposed for spatial robots. Typically the actuators of the conventional robot are directly installed on the joints. It not only increases the moving weight and inertia of a robot, but also greatly reduces its load capacity. At the same time, this type of design also cannot be used for high-speed motion and rapid response of a robot. However, a cable-driven method introduces a concept to overcome the disadvantages of the conventional robot in motion performance (Chen et al., 2006). The cable-driven structure is a kind of parallel mechanism transmitting forces and motions by cables which are connected to the motors mounted in a fixed base. There are several advantages for it such as lower weight, higher rigidity, higher accuracy and higher load capacity, compared with conventional serial robots. It can also be used in high-speed motion of a robot. Therefore, many researchers have paid attention to this area recently Literature Review For 7 DOF redundant robotic arms, the inverse kinematics is a difficult problem for researchers. Because there are only six equations with seven unknown variables, there are infinite solutions. Williams (1992) specified a certain variable to reduce one DOF and converted this underconstrained problem to a constrained problem, so the solutions are

12 12 unique. However, this method has its own limitations and is not suitable for general cases. Standard linear solution techniques cannot be used in this case too because the Jacobian matrix is not square. Williams (1994) showed the general redundant solution to solve the problem. The Moore-Penrose pseudoinverse of the Jacobian matrix is used in this solution, which solves the problem of the non-square Jacobian matrix. The rotation angle of elbow joint arrangement, the angle is shown in Fig. 1. Because of the specific robot joints can be calculated directly by the cosine law. The elbow joint rate is the time derivative of. Then, a reduced Jacobian solution is calculated. In order to greatly reduce the computation requirement, a partitioned approach for the particular solution was implemented. Figure 1. Elbow joint geometry Because there are infinite solutions for this underconstrained problem, optimization should be used to improve the behavior of the motion of manipulator, in addition to producing the required motion. Liegeois (1977) used an objective function to

13 13 make the manipulator to avoid its joint limits. Williams (1994) showed a function to ensure that the manipulator operates far from singular configurations if the function is maximized. He also showed another function for obstacle avoidance. There are not many papers published about cable-driven mechanism design. To realize the force-closure of the cable-driven mechanism, redundant forces are required because cables can only be pulled unilaterally in tension. In order to generate n DOF motion in one joint, at least n+1 cables are needed as actuating elements (Yang et al., 2005). Yang et al. (2005) proposed to use six cables to drive a spherical joint with 3 DOF shown in Fig. 2. In the figure, the shoulder and wrist joints of the 7 DOF manipulator are driven by six cables each, and the elbow joint is driven by two cables. They also discussed the kinematic relationship between the cable lengths and the poses of endeffector of this humanoid robotic arm. Chen et al. (2006) proposed another design (Fig. 3) to drive the shoulder and wrist joints by four cables each, and the elbow joint by two cables. Figure 2. Schematic diagram of the Yang s cable driven robotic arm (Yang et al., 2005) (figure used by permission)

14 14 Figure 3. Chen s cable-driven 7-DOF manipulator (Chen et al., 2006) (figure used by permission) As mentioned above, cables in cable-driven mechanism can only be pulled unilaterally, so it is critical to always maintain positive cable tensions in the motion. Williams et al. (2006) used a MATLAB function lsqnonneg, which solves the leastsquares problem to maintain all non-negative cable tensions. However, this function is not reliable because in some cases it will still give negative solutions for cable tensions, which is not allowed in a cable-driven mechanism. Additionally, Borgstrom et al. (2009) suggested Linear Programming (LP) as an approach to optimize cable tensions for cabledriven robots.

15 Project Information The subject of this project is Mechanism Design, Kinematics and Dynamics Analysis of a 7-DOF Cable-Driven Humanoid Robot Arm. This project uses a cabledriven approach to replace the traditional motor-driven actuation method. The model is a 7-DOF, kinematically-redundant, human-like arm model (Venkatayogi, 2007). Kinematics analysis includes forward pose kinematics and inverse velocity kinematics. Forward pose kinematics is the computation of the position and orientation of the endeffector of robot given the joint angles and manipulator parameters. Inverse velocity kinematics is to solve the linear equation for the joint rates given the desired Cartesian velocity of end-effector (Williams, 1994). Inverse dynamics analysis is also included based on known motion of the arm. It is the computation of torques needed for each joint given the motion of the arm model. The design of the cable-driven mechanisms of the robotic arm is critical in this project because it affects the efficiency of actuation. A cable-driven mechanism has many advantages compared to traditional actuation mechanism. Comparisons between both mechanisms will be implemented to show the advantages and disadvantages of cabledriven actuation by using numerical examples. Moreover, many simulations are also included in this project.

16 Thesis Objectives This project includes kinematics and dynamics analysis for the 7-DOF cabledriven humanoid robot arm model. Forward Pose Kinematics (FPK) and Inverse Velocity Kinematics (IVK) will be implemented in the kinematics analysis. FPK is straightforward. The general kinematically-redundant solution is used for solving IVK. In this solution, objective functions are used to optimize the inverse velocity kinematics and attempt to avoid joint limits, singularities and obstacles during the motion of the arm. For Inverse Dynamics, the classical Newton-Euler recursion inverse dynamics algorithm is used, and torques for each joint in a known motion of the arm model are calculated. In addition, this project also includes the design of a cable-driven mechanism and the calculation of the forces for each cable with known motion of the arm. As mentioned in the introduction, cable-driven actuation has certain potential advantages over traditional motor-driven actuation. Comparison of the input efforts needed for both mechanisms to perform the same motion will be analyzed. Finally, many simulation results will be shown. The specific thesis objectives are listed as follows: Kinematics Analysis for the 7-DOF arm model including: Forward Pose Kinematics, Inverse Velocity Kinematics and Kinematics Optimization. Inverse Dynamics for the 7-DOF arm model using the recursive Newton- Euler dynamics algorithm. Design cable-driven actuation mechanisms for the arm model and calculate forces of each cable for the certain motion.

17 17 Compare energy and power needed for the motor-driven and cable-driven actuation mechanisms to perform the same simulated motions. Compare energy and power of high speed motion and low speed motion for both cable-driven and motor-driven actuation mechanisms. Perform the following simulations using MATLAB. i. Simulate FPK 3D numeric results. ii. Simulate and compare the results before and after kinematics optimization. iii. Simulate the energy and power needed for each joint in a time period with changing inputs for both cable-driven and motordriven actuation mechanisms. iv. Simulate the changes of forces needed for each cable of the cabledriven robot over time.

18 18 2. KINEMATICS ANALYSIS AND KINEMATICS OPTIMIZATION 2.1. Introduction to Denavit-Hartenberg (DH) Parameters (Section 2.1 except Fig. 4 is all derived from Williams (2008).) DH Parameters were first proposed by Denavit and Hartenberg (1955). They are a standard description of the geometric pose of joints and links in a robotic system. The version used in this thesis is Craig s (2005) interpretation. Generally, there are four parameters used to describe the complete pose relationship between consecutive frames {i} with respect to {i-1}. Link {i-1} connects joints {i-1} and {i}. Cartesian coordinate frame {i-1} is fixed to link {i-1} and moves with it. Four parameters stand for two rotations and two translations. Those parameters represent the changes from frames {i-1} to {i}. Frame Attachment Rules are listed as follows: 1) The direction of Z axis is always same as the rotation direction for revolute joints and the translation direction for prismatic joints. 2) The X axis is always pointed along the common normal between consecutive Z axes. If consecutive Z axes intersect, X axis still needs to be along the perpendicular direction of both axes. If consecutive Z axes are directed along the same line, there is considerable freedom in defining X axis and the only requirement is that it must be perpendicular to this line. 3) Y axis is found by right hand rule with known Z and X axes. Since the Y axis is already decided if Z and X axes are known, usually it is not necessary to show Y axis in a figure.

19 For the first link, it is common to choose { } along { } and ensure frames {0} 19 and {1} are identical when the first parameter is 0. Figure 4 shows consecutive frames {i} and {i-1} and DH parameters between the two frames. Those DH Parameters are defined by: : the angle between { } to { } measured about { } the distance from { } to { } measured along { } the distance from { } to { } measured along { } the angle between { } to { } measured about { } Figure 4: Drawing of Link i-1 Connecting Joint i and Joint i-1

20 Arm Model and DH Parameters A 7-DOF arm model and its DH Parameters (Venkatayogi, 2007) are used in this project. The model shown in Fig. 5 includes 3 DOF for shoulder, 1 DOF for elbow and 3 DOF for wrist. Fingers are not included in this model. The DH parameters are shown in Fig. 6. Figure 5. 7-DOF arm model (Venkatayogi, 2007) (figure used by permission) Figure 6. DH parameters (Venkatayogi, 2007) (figure used by permission)

21 Forward Pose Kinematics (FPK) Forward pose kinematics is the calculation of end-effector pose of robot with known rotational angles in each frame. The homogeneous transformation matrix between the frames i-1 and i is (Craig, 2005): (1) where,, and., and are the DH parameters. This matrix is a special matrix. The first three components of last column of the matrix are the coordinates of the origin of the frame i based on the frame i-1. So the total transformation matrix from frame 0 to end-effector is given by (2) The full expressions of all components in Eq. 2 are listed in Appendix A. The rotational matrix between the frames i-1 and i is the submatrix of (Craig, 2005) (3) For example, given the following joint angles and link parameters,,,,,,, cm, cm, cm, the FPK result is:

22 22 The simulation result of robot position for this example is shown in Fig. 7: 40 Left View 40 Front View z(cm) 20 z(cm) y(cm) Space View x(cm) Top View 0 z(cm) y(cm) x(cm) y(cm) x(cm) Figure 7. Simulation of robot position This is the case where the base is [0, 0, 0] and the pose of end-effector is [ , -30.0, ]. Use FPK analysis, the location of end-effector is easily calculated by multiplying all the transformation matrices from base to the end-effector.

23 Forward Velocity Kinematics (The entire Sections are derived from Williams, To maximize compatibility, continuity, and understanding, the symbols, terms, and definitions used in this reference have been faithfully retained in the text, whenever possible, because this approach represents the single best method for solving problems of this sort. Any apparent similarity in content is superficial.) The forward velocity kinematics calculates the Cartesian velocity with known joint rates. (4) where is the Jacobian matrix based on the frame m. The order of is for this 7 DOF arm model and the full form of is attached in Appendix B Inverse Velocity Kinematics Inverse velocity kinematics is a problem of solving the linear equation for the joint rates given Cartesian velocity. In this case, standard linear solution techniques cannot be used because the Jacobian matrix is not square. The general redundant solution is used for solving this problem. The formula is listed as follows: (5) The first term is the particular solution. The second term is the homogeneous solution causing zero motion of the end-effector. The matrix is the Moore-Penrose pseudoinverse of Jacobian matrix and. The vector is

24 24 the gradient of an objective function of joint angles. The gain k is positive to maximize H and negative to minimize H. In Fig. 1 in Section 1.2, use cosine law and get the following equation. (6) where P X : X component of coordinate values of end-effector with respect to frame 0 P Y : Y component of coordinate values of end-effector with respect to frame 0 P Z : Z component of coordinate values of end-effector with respect to frame 0 Take time derivative of Eq. 6 and get directly: (7) where : Time derivative of P X with respect to frame 0 : Time derivative of P Y with respect to frame 0 : Time derivative of P Z with respect to frame 0 Since is known, the reduced Jacobian solution is calculated. The particular solution for the remaining joint rates is (8) where is the Cartesian velocity command and is the fourth column of the Jacobian matrix; both of them have row 1 removed. The psueudoinverse of the reduced Jacobian matrix is. The homogeneous solution for local redundancy optimization is (9)

25 25 The total solution for the remaining joints is the sum of the particular and homogeneous solutions: (10) In order to reduce the computation requirement, a partitioned approach for particular solution of Eq. 4 wrist partitioning is used. For a robotic arm with a spherical wrist, Eq. 4 can be written in the following partitioned form: (11) where the vectors v and are the translational and rotational Cartesian velocity commands. The Jacobian matrix is partitioned into upper-left, lower-left and lower-right submatrices, where and are submatrices and is a submatrix. The vector represents translational (arm) joint rates 1 to 4 and is the rotational (wrist) joint rates 5 to 7. The next step of wrist partitioning is also using is Eq.6 and Eq. 7 to get. The particular solution for the remaining arm joint rates is as follows: (12) Where is with row 1 removed and is column 4 of with row 1 removed. The reduced matrix is the pseudoinverse of without column 4 and row 1. The vector contains the translational joint rates without. The homogeneous solution for arm joint rates excluding is given as follows (13)

26 The particular solution for the elbow joint rate is given in Eq. 7 and the 26 homogeneous solution for is zero. The solution of the remaining arm joints is the sum of the particular and homogeneous solutions: (14) Because is a square matrix, the inverse matrix of is easy to get directly and solution can be obtained by the following standard linear techniques as: (15) where is the inverse matrix of Singularity Analysis of Robotic Arm Model Singularities are the certain configurations that robots can not or are not allowed to reach in the motion because they lose one or more degrees of freedom in those configurations. Singularity conditions for redundant manipulators appear when where det in the equation denotes the determinant operation. The calculation of in Eq. 7 fails when the denominator is zero. That means. The singularity condition for remaining translational joints will be obtained with the Eq. 16. (16) (The full expression for Eq. 16 is attached in Appendix B.) The term can be zero in following three ways:

27 27 1. (See Fig. 8). In this case, joints 3 and 4 can move the wrist tangentially to the link between elbow and wrist. Joint 2 can move the wrist perpendicularly to the plane of the front view. However, the freedom to translate radially along this link between elbow and wrist is lost. The singular direction is shown in Fig. 8. Figure 8. Internal singularity #1 for arm subassembly 2. (See Fig. 9). In this case, joints 3 and 4 can move the wrist tangentially to the link between elbow and wrist. Joints 1 and 2 can move the wrist perpendicularly to the plane of the front view. However, the freedom to translate radially along this link between elbow and wrist is lost. The singular direction is shown in Fig. 9.

28 28 Figure 9. Internal singularity #2 for arm subassembly 3. (See Fig. 10). In this case, joint 2 can move the wrist tangentially to the link between elbow and wrist. Joints 1, 3 and 4 can move the wrist perpendicularly to the plane of the front view. However, the freedom to translate radially along this link between elbow and wrist is lost. The singular direction is shown in Fig. 10. Figure 10. Internal singularity #3 for arm subassembly

29 29 The singularity condition for remaining wrist joints is calculated by the equation: (17) which shows that is the singularity condition for the wrist joint. As shown in Fig. 11, joints 5 and 7 provide roll and joint 6 provides yaw, but the freedom to pitch is lost. The symbol EE in Fig. 11 represents the end-effector. Figure 11. Internal singularity for wrist subassembly 2.7. Objective functions The first objective function was proposed by Liegeois (1977) to make the manipulator to avoid joint limits. (18) where is the current angle for joint i, is the center of motion range for joint i and is half the motion range for joint i. This function is minimized for joint limit avoidance.

30 30 Example 1: In a certain task, the end-effector of robot arm should maintain a velocity--, the first three components of are the translational Cartesian velocity and their units are cm/s. The rest three components are rotational Cartesian velocity and their units are rad/s. The initial angles for 7 frames are. Use the general redundant solution from Section 2.5 (Eq. 5) to get the rates of rotational angles for each frame. In this case, in the equation is the gradient of the objective function. When, there is no optimization involved in this solution. The result shows that is beyond the joint limit (See Fig.12) when t is around 1.8 seconds. Here we assume joint limits of are k=0 joint limit (degree) Time( sec) Figure 12. Joint angle for without optimization The objective function to avoid joint limits mentioned in Eq. 18 should be used in this case. The center of range for joint 6 is equal to and half the range of joint 6 is Then, use Eq. 5 and choose to avoid joint limits for. Figure 13

31 shows that the objective function improves result significantly and helps joint 6 to avoid its limit. This function can also avoid joint limits for different joints simultaneously k= -2 joint limit (degree) Time (sec) Figure 13. Joint limit avoidance for Williams (1994) showed a function to ensure that manipulator operates far from singular configurations if the function is maximized. (19) Here Example 1 in Page 28 will be used to show how the objective function works for singularity avoidance. To simplify the problem, only singularity conditions for the wrist joint are considered here. It is known from Section 2.6 that the singularity conditions for wrist joint are. It can be seen from Fig. 14 that also reaches one of the singularities in the motion. In this case, the objective function for singularity avoidance is

32 32 (20) where the symbol denotes absolute value. This function is maximized to avoid singularity, so k in Eq. 5 in Section 2.5 should be positive to maximize the function. Plug Eq. 20 back to Eq. 5 and choose to avoid singularity ( ). It is shown in Fig. 15 that the objective function keeps far away from singularity. This function can also avoid different singularities for a robot arm simultaneously k=0 singularity condition -60 6(degree) Time (sec) Figure 14. Joint angle without optimization

33 k=1.5 singularity condition -60 6(degree) Time (sec) Figure 15. Singularity avoidance for joint angle Besides, Williams (1994) also showed another function for obstacle avoidance when the function is minimized. (21) where is a single constant configuration that is good for avoiding collisions with an obstacle and W is a diagonal matrix with positive gains. Three objective functions, and can be combined in a certain way so that different optimizations can be implemented simultaneously.

34 34 3. MECHANISM DESIGN OF CABLE-DRIVEN ROBOTIC ARM AND STATICS ANALYSIS 3.1. Mechanism Design of Cable-Driven Robotic Arm As mentioned in literature review (Section 1.2), a spherical joint with four cables is proposed by Chen et al. (2006). The spherical joint contains the minimum number of cables required for a cable-driven mechanism (See Fig. 16). However, the paper did not specify the parameters of the design of the spherical joint nor did it compare the design with other designs. Additionally, the statics and dynamics of the arm were not discussed in the paper either. Therefore more complete analysis of the arm will be done in the following sections. Figure 16. Cable-drive mechanism In order to understand cable-drive mechanism of the spherical joint better, its structure and parameters needed to be clarified. The main surface of platform P 1 P 2 P 3 P 4 is a square with side length equal to 2lc. The thickness of the platform is small and

35 35 negligible. The platform and a supporting bar are connected by a ball joint. The platform can rotate about the ball joint by pulling the four cables connected to it. Those cables go through holes B 1, B 2, B 3, B 4 in a fixed beam and connect to motors in a fixed base. Actually, B 1 and B 2 are the same hole so are B 3 and B 4. The distance between holes B 1 and B 3 is 2lb. In the original design (Chen et al. 2006), it doesn t specify the lengths of lb and lc. Based on this design, some designs come up: First, there are three designs Design 1A, 1B and 1C (Fig. 17(i), (ii) and (iii) respectively) by specifying the relation between the lengths of lb and lc. (i) Design la (lb<lc) (ii) Design lb (lb=lc) (iii) Design 1C (lb>lc) Figure 17. Three designs of a cable-driven spherical joint

36 Second, there are three more designs Design 2A, 2B and 2C by rotating the 36 beam by shown in Fig. 18(i)-(iii), respectively. (i) Design 2A (lb<lc) (ii) Design 2B (lb=lc) (iii) Design 2C (lb>lc) (iv) Design 3A Figure 18. Four different designs of a cable-driven spherical joint

37 37 Third, there is one more design (Design 3A) by changing the surface of the base to a square instead of a rectangle as shown in Fig. 18(iv). However, in this design the cables can not provide torques about Z axis (perpendicular to the moving plane P 1 P 2 P 3 P 4 ) when the planes B 1 B 2 B 3 B 4 and P 1 P 2 P 3 P 4 are parallel to each other because the torques about Z axis produced by every cable have the same magnitude but opposite signs. This design has more limitations than others so it will not be considered in the following analysis. Statics analysis and motion ranges will be discussed in the following sections to show which one is the best design out of those designs Linear Programming Linear Programming (LP) is a mathematical approach to obtain the best output in a given mathematical model with linear equations and inequations. Usually, it is a technique used to optimize a linear objective function, subject to linear equality and inequality constraints. Those problems can be expressed in the following canonical form: Maximize or Minimize: Subject to: Ax b. Where x is the vector of unknown variables, c and b represent vectors of known coefficients and A represents a known coefficient matrix. The objective function c T x should be maximized or minimized according to the requirement. (The content above in this section is derived from "Linear programming," 2010, para. 1-4)

38 38 (The following content of this section is derived from Help document of MATLAB 2009.) In MATLAB software there is a function linprog used to solve linear programming problems. This function is used to find the minimum value of an objective function subject to: where f, b, beq, lb, and ub are known vectors; A, Aeq are known matrices; lb and ub are the lower and upper bounds of the variables x; are the inequality constraints and are the equality constraints. Only the vector x is unknown. The function will find the optimal solution for vector x so that the objective function is minimized Statics Analysis Before statics analysis, an approach that calculates the coordinates of points in the moving platform of a spherical joint with known rotational angles of three frames (Chen et al., 2006) is introduced and shown here. To maximize compatibility, continuity, and understanding, the symbols, terms, and definitions used in this reference have been faithfully retained in the text, whenever possible, because this approach represents the single best method for solving problems of this sort. Any apparent similarity in content is superficial. The Design 2A from Section 3.1 is used to exemplify this approach. As shown in Fig. 19, a fixed frame B is set the same as frame 0 in the arm model (Fig. 5) and

39 frame P is set the same as frame 3 in the model. The frame P is fixed to a moving platform P 1 P 2 P 3 P 4. The origins of both frames overlap in the point O that is the center of 39 square P 1 P 2 P 3 P 4. If the pose of this spherical joint first rotates by an angle around the Z axis of frame 1, then rotates by an angle around the Z axis of frame 2, finally rotates by an angle around the Z axis of frame 3, the rotational matrix of the shoulder joint with respect to frame B is (Chen et al., 2006): (22) The platform P 1 P 2 P 3 P 4 is a square with side length equal to 2lc. The length of the supporting bar is lh and the length between holes B 1 and B 3 in the bottom beam is 2lb. The coordinate values of B 1, B 2, B 3 and B 4 expressed in frame B are:, The coordinate values of P 1, P 2, P 3 and P 4 expressed in frame P are:,,,. Then the coordinate values of P i in frame B is expressed by (Chen et al., 2006) (23)

40 40 Figure 19. Cable-driven mechanism analysis for shoulder joint (Design 2A) The lengths of cables are calculated by (24) The vectors along cable tensions expressed in the base frame are (25) In Fig. 19, 1, 2, 3 and 4 are the cable tensions provided by the four motors mounted to the fixed base. To maintain a balance of torques about point O, the equation can be expressed by (26) where is the sum of external torques, is the vector starting from point O to the point at which the cable force exerts, is the vector of cable force, is a unit vector along the direction of the cable force, and magnitude of the cable force. So the unit vector along is a scalar representing the in Eq. 25 can be calculated by

41 41 (k=1, 2, 3, 4) (27) In order to maintain a balance of forces exerted on the platform, the equation can be expressed by (28) where is the sum of external forces. Figure 20. Free body diagram of hand In order to make the problem easier to solve, statics of the whole arm will be analyzed part by part separately. First, the hand is analyzed; its free body diagram is shown in Fig. 20. The weight of the moving platform Q 7 Q 8 Q 9 Q 10 is G 4 and its side length is 2lq. The length of R 7 R 9 in the beam is 2lp. The length of the supporting bar is lg. A ball joint is used to connect the

42 42 supporting bar with the platform Q 7 Q 8 Q 9 Q 10. The platform is fixed to the hand. The weight of the hand is G 5. The force exerted by the supporting bar on the platform is. The scalars t 7, t 8, t 9 and t 10 are the magnitude of cable tensions for the four cables. To maintain static equilibrium of forces exerted on the hand and platform, the equation can be expressed by (29) To maintain static equilibrium of torques about point I, the equation is (30) All vectors in Equations 29 and 30 are expressed in Frame 0 (a fixed frame). There are seven unknowns in those equations including t 7, t 8, t 9, t 10 and the x, y, z components of. There are three equations in each of those equations so there are six equations in total. This is an underconstrained problem because there are more unknowns than equations. Therefore, it is possible to minimize the sum of the cable tensions by linear programming function in MATLAB. In this case the objective function is min and the lower bound of cable tensions lb should be bigger than zero because cables can only provide positive tensions in a cable-driven mechanism. The upper bound ub is obtained by the limit of cable tension divided by a safety factor. The limits of cable tension of different materials are different. The optimal solution of cable tensions and the solution of are obtained by this method.

43 43 Figure 21. Free body diagram for forearm Second, consider the forearm. Its free body diagram is shown in Fig. 21. In the figure, is the force exerted on the forearm by the hand. It is the reaction force of in Fig. 20. The cables 7, 8, 9 and 10 in Fig. 20 that control the motion of the hand go through the holes R 7 and R 9 in the beam fixed in the forearm and connect with four motors in a fixed base. and are forces applied on the beam by the tensions of the four cables (j=7, 8, 9, 10). Those cable tensions are obtained from statics analysis of the hand. Frictions between cables and holes are not considered here. The weight of the beam is negligible. Point G is the center of gravity of the forearm and G 3 is the weight of the forearm. The beam P 5 P 6 is perpendicular and fixed to forearm EH. The beam is driven by two cables P 5 Q 5 and P 6 Q 6 and it rotates with the forearm around a revolute joint E. The two cables go across the holes Q 5, Q 6 and connect to the motors in the fixed base. The scalars t 5 and t 6 are magintudes of these cable tensions. The force is exerted on the forearm by the upper arm. In order to maintain static equilibrium of forces exerted on the forearm, the equation can be expressed by (31)

44 In order to maintain static equilibrium of torques about joint E, the equation can be expressed by 44 (32) Because the revolute joint E can only rotate along the z axis in Fig. 21, it can only balance torques in the z direction. That is why Eq. 32 only contains equation of torques in the z direction. Since is known from the analysis of the hand, there are five unknowns and four equations in Equations 31 and 32. The five unknowns are t 5, t 6 and three components of. This is still an underconstrained problem. Linear programming is used for optimizing the solutions of this problem, so the optimal solution of t5, t6 and solution of can be obtained from those equations. The torques in the x and y axes will transfer to the upper arm by this revolute joint. They can be calculated by (33) where is a vector. The first two variables of are the torques in the x and y axes of frame 3 of the arm model. The third variable is zero because torques in the z axis can be balanced by the revolute joint.

45 45 Figure 22. Free body diagram of upper arm Third, the statics of the upper arm is analyzed here. Its free body diagram is shown in Fig. 22. In the figure, are the torques transferred by the revolute joint in elbow. And is the force exerted by the forearm on the upper arm. The cables 5 and 6 in Fig. 21 that control the motion of the forearm go through the holes Q 5 and Q 6 in the beam fixed in the upper arm and connect with two motors in a fixed base. and are forces applied on the beam by the two cable tensions (i=5, 6). The cable tensions are obtained from statics analysis of the forearm. Frictions between cables and holes are not considered here and the weight of the beam is negligible. The vector G 2 is the weight of the upper arm and point C is the center of gravity of upper arm BD. The vector G 1 is the weight of the moving platform P 1 P 2 P 3 P 4 and point B is its center of gravity. The side length of platform is 2lc. The force is exerted by the supporting bar on the platform. The length of the supporting bar is lh and the length of the upper arm is lu. The length of B 1 B 3 is 2lb. The platform is driven by four cables going through holes B 1 and B 3 and connecting to motors in the fixed base. To maintain static equilibrium of forces exerted on the upper arm and platform, the equation is

46 46 (34) To maintain static equilibrium of torques about point B, the equation is (35) There are seven unknowns and six equations here. The seven unknowns are t1, t2, t3, t4 and three components of. This is also an underconstrained problem and the optimal solutions can be obtained by linear programming. The complete optimal solutions of cable tensions for the whole arm are available after statics analysis of the hand, the forearm and the upper arm. Next, a numerical example will be used here to test which design cosumes the least cable tensions for the same pseudostatic motion (slow enough that acceleration of the motion is negligible) of the robotic arm. Linear programming is used to minimize the sum of cable tensions for all the designs. Example 2: The robotic arm will perform a pseudostatic motion with rotations about 7 joints. In each joint, the angle changes from one limit to the other in the following ranges:,,,,,,. The cycloidal function (Wiliams, 2009) is used to form an array of angles changing smoothly from one limit to the other. (36) where i is from 1 to 7; and is the first and second component of the motion range above, respectively; t is the time variable and t F is the final time. Two motions (a slow motion and a fast motion) are used in this thesis. The trajectories of seven joints in those

47 47 two motions in shown in Fig. 23 and Fig. 24. The motion path of the robotic arm is shown in Fig. 25. In the slow motion, t F =10 seconds and in the fast motion t F =0.3 second. Angles (deg) theta 1 theta 2 theta 3 theta 4 theta 5 theta 6 theta Time (sec) Figure 23. Trajectories of seven joints in the slow motion Angles (deg) theta 1 theta 2 theta 3 theta 4 theta 5 theta 6 theta Time (sec) Figure 24. Trajectories of seven joints in the fast motion

48 48 Left View Front View z (m) 0.2 z (m) z (m) y (m) y (m) Space View x (m) y (m) x (m) Top View x (m) Figure 25. The motion path of the robotic arm In statics analysis t changes from 0 to 10 seconds, so this motion is slow and the acceleration of the motion is negligible. The data for this example are listed as follows: Density of material of mechanism:, lower bound cable tension: 5 N. In upper arm: lh=0.075 m, lc=0.05 m, lu=0.315m, radius of upper arm bar r=0.01 m, thickness of plaform d 1 =0.01 m. In forearm: EF=0.005 m, RE=0.04 m, P 5 F=P 6 F=0.03 m, lf=0.287 m, radius of forearm bar r=0.01 m. In hand: lg=0.04m, lq=0.025m, lha=0.105m, radius of hand bar r=0.01m, thickness of the platform d 2 =0.005m. For Design 1A, lb=0.015m<lc, Q 5 R=0.015m<P 5 F, lp=0.01m<lq. For Design 1B, lb=0.05m=lc, Q 5 R=0.03m=P 5 F, lp=0.025m=lq. For Design 1C, lb=0.06m>lc, Q 5 R=0.04m>P 5 F, lp=0.03m>lq. For Design 2A, lb=0.015m<lc, Q 5 R=0.015m<P 5 F,

49 49 lp=0.01m<lq. For Design 2B, lb=0.05m=lc, Q 5 R=0.03m=P 5 F, lp=0.025m=lq. For Design 2C, lb=0.06m>lc, Q 5 R=0.04m>P 5 F, lp=0.03m>lq. The optimal cable tensions of those six desgins are shown in Fig and the total optimal cable tensions of those designs are shown in Fig. 36. Cable tensions for cable 1 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (sec) Figure 26. Cable tensions of cable 1 for six designs Cable tensions for cable 2 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 27. Cable tensions of cable 2 for six designs

50 50 Cable tensions for cable 3 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 28. Cable tensions of cable 3 for six designs Cable tensions for cable 4 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 29. Cable tensions of cable 4 for six designs Cable tensions for cable 5 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 30. Cable tensions of cable 5 for six designs

51 51 Cable tensions for cable 6 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 31. Cable tensions of cable 6 for six designs Cable tensions for cable 7 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 32. Cable tensions of cable 7 for six designs Cable tensions for cable 8 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 33. Cable tensions of cable 8 for six designs

52 52 Cable tensions for cable 9 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 34. Cable tensions of cable 9 for six designs Cable tensions for cable 10 (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 35. Cable tensions of cable 10 for six designs Total cable tensions for 10 cables (N) Design 1A Design 1B Design 1C Design 2A Design 2B Design 2C Time (Sec) Figure 36. The total cable tensions of 10 cables for six designs

53 53 The MATLAB program for pseudostatics analysis of the whole arm optimized by linear programming is shown in Appendix D (1). In Fig. 36, it proves that with length increase of lb, Q 5 R and lp, the total cable tensions needed for Design 1A, 1B and 1C decrease and Design 1C uses the least cable tensions to perform the same motion in those three designs. It is similar that with length increase of lb, Q 5 R and lp, the total cable tensions needed for Design 2A, 2B and 2C decrease and Design 2C uses the least cable tensions to perform the same motion in those three designs. Comparing those six designs in Fig. 36, it proves that Design 1C uses the least cable tensions to perform the same motion. Actually, the trend of the decrease of cable tensions with length increase of lb, Q 5 R and lp is not a coincidence. Because of the special characteristics of this 3 DOF spherical joint, it is much harder to create torque in z axis than torques in x or y axes (see Fig. 37). Therefore, the design can produce the same torque in the z axis with less cable tensions for the motion of the whole arm is better. One cable tension in three different designs is analyzed in Fig. 37. It is disassembled in the x, y and z axes. In Design 1A (lb<lc), the component forces that affect torque in the z axis are T x and T y. The directions of torques provided by T x and T y are opposite, so the two torques will partially cancel out. In Design 1B (lb=lc), there is no component force in the x direction so the torque in z axis is only provided by component force T y. In Design 1C (lb>lc), the direction of its component force T x is opposite to that of component force T x in Design 1A. The torques provided by component forces T x and T y are in the same direction so the total torque is the sum of two torques. In order to provide the same torque about the z axis, the cable

54 54 tension in Design 1C is smallest. And producing torque in z axis is the hardest part of the motion, so that is why Design 1C uses the smallest cable tensions to perform the same motion. Figure 37. Analysis of torques about the z axis for different designs 3.4. Motion Range It is known that Design 1C is the best design for the using the least cable tensions from the previous section. It is uncertain if its motion range is bigger than the other designs or not. It is vital to always maintain a positive tension for every cable in a cabledriven mechansim, which can be used to find the practical motion range of this mechanism (Chen et al., 2006). Simplifing Eq. 26 from Section 3.3: (37) where A=[ and, the cable tensions can be expressed as (Chen et al., 2006):

55 55 (38) where is the pseudoinverse matrix of matrix A, is an arbitrary scalar and is a null vector of A. According to the Cramer s rule (Pham et al., 2004), the null vector N can be expressed as: (39) where is the determinant of the matrix formed by extracting i th column of matrix A. Because the cable tensions should be positive all the time, the following constraints are required (Pham et al., 2004): (40) The motion ranges of shoulder joint are calculated using those constraints in Eq. 40 by MATLAB and shown in Fig. 38. The approximate volumes of motion ranges of shoulder joints of Designs 1A, 1B and 1C are 15, 8 and 7, repectively. It is also shown in the figure that the motion range of shoulder joint becomes smaller when the length of lb increases. Because the structures of shoulder joint and wrist joint are simliar, the motion range of wrist joint of Design 1A is also bigger than that of Design 1C. Therefore Design la has the biggest motion range out of three designs and Design 1C has the smallest one.

56 56 (i) Design 1A, lb=0.015 m (ii) Design 1B, lb=0.05m (iii)design 1C, lb=0.06m Figure 38. Motion ranges of shoulder joints for different designs

57 57 4. CABLE KINEMATICS ANALYSIS 4.1. Forward Pose Cable Kinematics (FPCK) Cable kinematics represents the relationship between cable lengths and the endeffector pose. Forward pose cable kinematics is used to get the end-effector pose given the lengths of the driving cables. The 7-DOF cable-driven robotic arm is shown in Fig. 39. The shoulder and wrist joints are the spherical joints (Design 1B) mentioned in Section 3.1. The elbow joint is a revolute joint driven by two cables. Figure DOF cable-driven arm model A method (Yang et al., 2005) is used here to implement forward pose cable kinematics of this model. To maximize compatibility, continuity, and understanding, the symbols, terms, and definitions used in this reference have been faithfully retained in the text, whenever possible, because this approach represents the single best method for solving problems of this sort. Any apparent similarity in content is superficial. According to forward pose kinematics in Section 2.3, the end-effector pose can be expressed by

58 58 (41) where because there is no rotation between Frames 7 and 8; ; lha is the length of the hand;, and represent the pose of Frame 3 with respect to Frame 0 (a fixed frame), Frame 4 with respect to Frame 3, and Frame 7 with resepct to Frame 4, respectively;, where is the rotational matrix of Frame 3 with respect to Frame 0. Because the origins of Frames 3 and 0 are the same point, the pose of Frame 3 expressed in Frame 0 is ;, where and are the rotational matrix and pose of Frame 4 with respect to Frame 3; and lu is the length of the upper arm;, where and are the rotational matrix and pose of Frame 7 with respect to Frame 4; and lf is the length of the forearm. In Eq. 41 only, and are unknown. They are relative to rotational angles in seven frames, however those angles are also unknown so obtaining, and directly by those angles doesn t work. Finding the direct relationships between the cable lengths and rotational matrices is an appropriate approach (Yang et al., 2005).

59 59 Figure 40. Shoulder module of cable-driven mechanism There are two steps in Forward Pose Cable Kinematics of the whole arm. First, the relation between cable lengths and the end-effector pose in the shoulder joint is analyzed here. As shown in Fig. 40, the shoulder module is redundantly driven by four cables to create 3-DOF rotational motions about the ball joint B. In this case, the lengths of four cables-- and are known. The origins of Frames 0 and 3 are the same point B. Suppose that the coordinates of P i and B i (i=1, 2, 3, 4) with respect to Frame 0 are given by and, respectively. Then the following equations are obtained by analysis: (42) (43) (44) Because P 3 and P 1 are symmetric about point B and the coordinates of B with respect to Frame 0 is, the coordinates of P 1 are expressed by

60 60 (45) Plug Eq. 45 back into Eq. 44 to replace the coordinates of P 1. Then for Equations 42-44, the only unknowns are the coordinates of P 3 (x p3, y p3, z p3 ). At most two sets of solutions can be found by solving those three equations. Similarly, two sets of solutions for the coordinates of P 2 (x p2, y p2, z p2 ) can be obtained. The distance between P 2 and P 3 -- is known, which can been used to check the solutions for those two points. Normally, only one set of solutions for P 2 and one set of solutions for P 3 are valid. Suppose that the known coordinates of P i with respect to Frame 3 are. The connections between and are (Yang et al., 2005): So can be obtained by (Yang et al., 2005) where is the rotation matrix between Frame 0 and Frame 3, so in Eq. 41 is known. Figure 41. Elbow module of cable-driven mechanism

61 61 Second, the relationship between cable lengths and end-effector pose in the elbow joint is analyzed here. As shown in Fig. 41, the elbow module is redundantly driven by two cables to generate 1-DOF rotational motion. Because the revolute joint on the elbow can only rotate about z axis, so the coordinates of all points in the figure in z axis are the same. In order to simplify the problem, the z coordinates of all points will not be shown in the analysis. Suppose that the coordinates of C 6 are with respect to Frame 3. The known coordinates of C i (i=1, 2, 3, 5) with respect to Frame 3 are. For this mechanism, the equations are obtained by analysis: At most two sets of solutions of coordinates for C 6 can be obtained by solving two equations above. Only one set of solutions is valid by checking the length of the other cable, so is obtained with known coordinates of C 6. For the triangle C 2 C 5 C 6, can be calculated by the cosine law: Since is known, the transformation matrix can be calculated by Eq. 1 in section 2.3. Because the shoulder and wrist joints are similar spherical joints, the transformation matrix for the wrist joint can be obtained by a similar method. With known and, can be obtained by Eq. 41 and the pose of end-effector can be obtained. So forward pose cable kinematics of the whole arm is finished.

62 Inverse Pose Cable Kinematics (IPCK) Inverse pose cable kinematics is the calculation of required cable lengths given the end-effector pose. Since this process is difficult, it is divided into two steps. First, use inverse pose kinematics to find optimal rotational angles in each frame with the known pose of the end-effector. Inverse pose kinematics for the 7-DOF redundant arm model has been published by Tarokh et al. (2010), so the details of that method will not be discussed here. Second, the method in Section 3.3 is used to calculate cable lengths with known optimal rotational angles in each frame. With those two steps, IPCK can be accomplished.

63 63 5. DYNAMICS ANALYSIS 5.1. Dynamics Analysis of the Cable-Driven Robotic Arm Dynamics analysis considers not only forces and torques applied on the system like statics analysis, but also the effects of motion to the system including translational and rotational accelerations of the system. The dynamics analysis of the whole robotic arm is also done part by part. First, the hand is analyzed; its free body diagram is shown in Fig. 42. There are only two new variables in the figure: a 1 and α 1, where a 1 is the acceleration of the hand and α 1 is the angular acceleration of the hand about a ball joint I. Both of them are calculated with respect to the fixed Frame 0. The rest of variables are the same as in statics analysis of the hand in Section 3.3. Figure 42. Free body diagram of the hand for dynamics analysis The equation of forces exerted on the hand and platform can be expressed by