Robot. A thesis presented to. the faculty of. In partial fulfillment. of the requirements for the degree. Master of Science. Zachary J.

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1 Uncertainty Analysis throughout the Workspace of a Macro/Micro Cable Suspended Robot 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 Zachary J. Zwahlen April Zachary J. Zwahlen. All Rights Reserved

2 2 This thesis titled Uncertainty Analysis throughout the Workspace of a Macro/Micro Cable Suspended Robot by ZACHARY J. ZWAHLEN has been approved for the Department of Mechanical Engineering and the Russ College of Engineering by Robert L. Williams II Professor of Mechanical Engineering Dennis Irwin Dean, Russ College of Engineering

3 3 Abstract ZWAHLEN, ZACHARY J., M.S., April 2017, Mechanical Engineering Uncertainty Analysis throughout the Workspace of a Macro/Micro Cable Suspended Robot Director of Thesis: Robert L. Williams II Inaccuracy in cable suspended robots has led to research in the addition of a micro system to correct for the inaccuracies of the macro system (cable suspended robot). With the addition of this micro system, the total system inherently becomes kinematically redundant. Due to this, the two systems can be configured in such as to have minimal uncertainty with respect to an end effector pose. For the selection of the minimal uncertainty configuration, a genetic algorithm was used. The genetic algorithm assessed the robotic system with respect to compensability (tracking performance) and minimal configuration change. The reason for these two parameters is to provide an optimal trajectory generation for the control of the macro/micro system to minimize inaccuracy. This controller design dependent on the optimal trajectory generation was compared to the macro/micro system without control to show the improved effect. Finally dependent on these results, recommendations were made with respect to how to improve the genetic algorithm as well as a different trajectory generation method for a macro/micro cable suspended robot.

4 4 Table of Contents Page Abstract... 3 List of Figures Introduction Literature Review Workspace Analysis for Cable Suspend Robot Uncertainty Analysis for Cable Suspended Robot Macro/Micro Systems Cable Suspended Robot Controller Controller Types Sequential Control Method for Macro/Micro Thesis Objectives Methodology Kinematics Forward Pose Kinematics Inverse Pose Kinematics Velocity Kinematics Error Model Static Analysis Statics Task Space Division and Genetic Algorithm Compensability Pseudo Static Control Results Workspace Analysis Uncertainty throughout the Workspace Analysis Sequential Pseudo Static Control Conclusion and Future Work Conclusion... 68

5 4.2 Future Work References

6 6 List of Figures Page Figure 1: Block Diagram of a Macro/Micro Robot with Sequential Control Figure 2: Macro/Micro Cable Suspended Robot Model (Not to scale) Figure 3: Detailed View of Macro/Micro Cable Suspended Robot (Not to Scale) Figure 4: Intersection of Two Circles and Simplified Intersection of Two Circles Model Figure 5: Free Body Diagram of A) Macro B) Micro Platform Figure 6: Macro Only Workspace Figure 7: Micro Only Workspace Figure 8: Micro Workspace with Macro Orientation of A) -30 B) 30 C) -20 D)20 E) - 10 F) 10 G) -5 H) Figure 9: Micro Workspace at Macro Pose x=0 y=-30 θ= Figure 10: Micro Workspace at Macro Pose y= -30 θ= -20 A) x = 8 B) x = 9 C) x = 1044 Figure 11: Macro Workspace with Translational Relationship to Micro of x = A) -.05 B).05 C) -.1 D).1 E)-.2 F) Figure 12: Macro Only Uncertainty Throughout Workspace Figure 13: Macro Only Uncertainty Throughout Workspace Zoomed in Figure 14: Micro Only Uncertainty Throughout Workspace Figure 15: Macro Only Uncertainty Throughout Workspace Zoomed in Figure 16: A) Macro B) Micro Only Uncertainty for Macro Position x=10 y = Figure 17: Total Uncertainty for Macro Position x=10 y = Figure 18: Total Uncertainty Zoomed in for X Uncertainty Figure 19: A) Micro B) Total Uncertainty Zoomed in for Y Uncertainty Figure 20: A) Micro B) Total Uncertainty Zoomed in for θ Uncertainty at Constant Y.. 54 Figure 21: Macro Uncertainty Zoomed in for θ Uncertainty at One Point Figure 22: Total Uncertainty Throughout Macro Workspace for Translational Relationship x = -.1 y = -1.2 θ = Figure 23: Total Uncertainty Throughout Macro Workspace for Translational Relationship x = -.2 y = -.8 θ = Figure 24: Total Uncertainty Throughout Macro Workspace Zoomed in for Translational Relationship A) x = -.1 y = -1.2 θ =5 B) x = -.2 y = -.8 θ = Figure 25: A) Macro B) Micro Motor Orientations (Perfect) Figure 26: A) Macro B) Micro Motor Speed (Perfect) Figure 27: A) Macro B) Micro Motor Torque (Perfect) Figure 28: A) Macro B) Micro Motor Orientation (Error) Figure 29: A) Macro B) Micro Motor Speed (Error) Figure 30: A) Macro B) Micro Motor Torque (Error)... 63

7 Figure 31: Trajectory Uncertainty with and without Controller for Discretized Macro Workspace A) Initial B) Middle C) Final Figure 32: Trajectory Uncertainty with and without Controller for Discretized Macro Workspace (Initial) Zoomed in Figure 33: Trajectory Uncertainty with and without Controller for Discretized Macro Workspace (Final) Zoomed in

8 8 1. Introduction In recent years, cable suspended robotic research has increased due to their advantages over conventional parallel robots. Conventionally, robots are composed of rigid links that can be manipulated to achieve a desired task. In a parallel robot, all links extend from the base to the end effector. This means that the workspace of a parallel robot is heavily dependent on link lengths as well as the link configuration from the base to the end effector. This is why conventional parallel robots have very small workspace to size ratios. Since cables are flexible, there is no hindrance on link length; allowing cable suspended robots to have a huge workspace. One major problem induced by the flexible cables is low accuracy of the system. This is due to the sensing of drum orientation which controls the cable lengths in the system. To confront the inaccuracy of cable suspended robots, some research has been focused on using the cable suspended robot as a macro system with the addition of a micro system for improved accuracy. Accuracy is improved as a result of the addition of a micro system as a result of its higher precision that can compensate for the inaccuracies of the macro cable suspended robot. Currently three large-scale projects are implementing this design are: the Chinese Five-hundred-meter Aperture Spherical Radio Telescope (FAST) [1], NIST two-stage cable robot [2], and the Canadian Large Adaptive Reflector (LAR) [3]. The purpose of this paper is to expand on research in the field of macro/micro cable suspended robots. The specific focus of this paper is to understand the uncertainty of a macro/micro robotic system consisting of a 3-cable-suspended robot for

9 9 each stage, as well as to develop a controller design to minimize this end effector uncertainty. 1.1 Literature Review Workspace Analysis for Cable Suspend Robot In conventional robot workspace analysis, the workspace is defined as all points that the robot end effector can reach. For a cable-suspended robot, it is the same but with the constraint that the robot must be in a configuration corresponding to only positive tension in the cables. From this constraint, a secondary constraint is imposed which is how the particular workspace is classified. The secondary constraint is that the cable suspended robot is either in Static Equilibrium, Dynamic Equilibrium, or Force/Wrench Equilibrium. Static Equilibrium Workspace (SEW) is defined as the set of all poses with positive cable tensions where the end effector is at static equilibrium. For SEW, there is no applied external force or torque and the only agent constraining the robot is gravity [4]. Dynamic Workspace is defined as all the poses where the robot is in a positive tension configuration with an applied dynamic acceleration for the end effector [5]. Controllable Workspace is defined as all poses with positive cable tensions for an end effector in static equilibrium with an applied wrench [6]. A wrench is an external force and/or torque applied to the end effector. Controllable Workspace has a couple of different interpretations of how the workspace is defined. The two different approaches extend from how the magnitude of cable tension is defined. These two approaches are Wrench Feasible Workspace (WFW) [7] and Wrench Closure Workspace (WCW) [8].

10 10 WFW is the workspace defined from all poses of a cable robot where the positive cable tensions are bounded for an applied wrench. WCW is defined as the workspace of the end effector with unbounded positive cable tension and applied wrench. Unbounded positive cable tensions makes WCW reliant on the geometry of the robot as compared to WFW Uncertainty Analysis for Cable Suspended Robot For the uncertainty analysis of the cable suspended robot, all the geometric errors in the system must be considered. These geometric errors can be tolerance errors of the various components like the platform, assembly errors like location of actuators, as well as actuator errors that result in improper cable lengths for the configuration [3]. To relate these geometric errors to the end effector uncertainty the Inverse Pose Kinematics (IPK) Equations are used. Since the IPK equations uses the known pose of the end effector and components (geometric) to solve for the link lengths (geometric). These equations can be used to relate geometric errors to the end effector uncertainty by creating the error model [9] for the system. The error model is a matrix expression that relates the geometric errors of a system to end effector uncertainty. The error model is formed by the partial derivative of the IPK equations with respect to one variable multiplied by its respective uncertainty for each variable independently. The reason for this is that the partial derivative is the sensitivity to change of the respective variable and when multiplied by its uncertainty gives the resulting total effect on the system. One thing to note about this statistical model is that the error model assumes that all variables are independent from one another.

11 11 The error model relates all the geometric errors to the uncertainty of the end effector but yields no significance of each error with respect to the total uncertainty of the end effector. To understand the significance of each error with respect to the uncertainty of the end effector sensitivity analysis [9] is used. Sensitivity analysis looks at the terms in the error model matrix that are related to a specific geometric error. The reason for this is to not include the effects of the amount of error but to look at which geometric errors have a more significant effect on end effector uncertainty. This is why the expression of sensitivity of a geometric error for a point is the Euclidean norm of its respective error model matrix elements. Since this equation is only for a point, it can be integrated over the workspace to understand the sensitivity of the geometric error throughout the workspace. With the sensitivity of a geometric error over the workspace, this can then be related to its effect relative to all geometric error sensitivities over the workspace. This gives an understanding of importance of the geometric errors relative to one another throughout the workspace Macro/Micro Systems Since Macro/Micro systems are kinematically-redundant the method to position both systems is through Task Space Division [10]. Macro/Micro systems are composed of a large robot for coarse motion suspending a smaller robot for fine motion. The addition of the micro system inherently makes the system kinematically-redundant which means that the system has more degrees of freedom than required for the Cartesian space. The extra degrees of freedom over that of Cartesian space means that the robot can have infinite configurations in the Cartesian space. Since there are infinite solutions for the

12 12 macro/micro system, Task Space Division uses the individual robot relative task spaces to create an envelope of possible solutions for the desired end effector pose of the macro/micro robot. From the envelope of possible solutions, a genetic algorithm can be used to select the best task space division system configuration [11]. The selection method used in this genetic algorithm, is dependent on compensability of the micro platform. The compensability looks at the inaccuracy of end effector pose induced by the error in the macro system as well as the manipulability of the micro system. The manipulability of the micro system is defined by the Jacobian for the micro system. The reason for this is that the Jacobian can be thought of as the transformation imposed locally for the system. Since the micro needs to compensate for the error induced by the macro platform the Jacobian times little change in link length provides the necessary amount of compensation. If the error is normalized and the Euclidean norm of all the sets of the error is less than equal to one, resulting in an ellipsoid for the normalized little change in link length sets. For the micro system, this ellipsoid is the compensability ellipsoid. The radius of this ellipsoid corresponds to the amount of compensation required for the micro system. The volume of this ellipsoid, which is the determinate of the expression, represents the difficulty of the micro system to compensate for the end effector pose error. One over the volume is the measure of compensability. To improve tracking performance of the micro robot the relationship taken should have a high manipulability for the micro system as well as small end effector error induced by the macro system.

13 Cable Suspended Robot Controller Controller Types Currently several different types of controllers are being used for the control of macro/micro and or cable suspended robots. The two controllers that will be focused on in this section are the sliding mode controller (SMC) and Proportional Integral Derivative (PID) controllers. The SMC and PID controller were the chosen controllers for the NIST dual stage cable suspended robot [2] and the K.N. Toosi planar cable-driven robot [12] respectively. The reason that the slide mode controller was selected for the NIST project is that it is a variable structure system. Variable structure systems are not sensitive to external disturbances and parameter variations [2]. The design of the NIST robot is for skin to skin transfer of cargo. Since the system will see great variations of loads and disturbances, the sliding mode controller was chosen for this design. Currently PID controllers are being used for positive tension controllers for cable suspended robots [12]. The way that the controller is designed in the K.N. Toosi to provide only positive tension is the use of the null space of the system since it is redundant. This redundancy is a result of having more cables than required for the Cartesian space (planar - 4 cables). Due to this redundancy their system is underdetermined. This is a result of 3 equations and 4 unknown tensions. From this there is a null space for the pseudoinverse solution for tension. This null space can be physically interpreted as internal forces and be used to force the solution of the pseudoinverse to have positive cable tension (pseudoinverse solution of tensions + null space).

14 Sequential Control Method for Macro/Micro For a macro/micro robotic system, sequential control is required since the position of the end effector on the micro robot is dependent on the macro robot position. Sequential control divides the two systems into their own respective closed loops. The two closed loops control the response of the macro or micro respectively. The sequential part of this control method is that positional data from the macro closed loop is used in the positional response for the micro platform s closed loop. This allows for the micro platform to compensate for any error in the position of the macro system versus the desired response of the macro platform. Figure 1 shows sequential control block diagram for a macro/micro robotic system. [6] Figure 1: Block Diagram of a Macro/Micro Robot with Sequential Control From Figure 1, it can be seen how task space division and sequential control are used for controlling a macro/micro robotic system. The task space division is used to

15 15 create two trajectories for the macro and micro platform, respectively. These two trajectories are used as inputs for the control loop of the macro and micro, respectively. Sequential control is then used to relate the positional accuracy of the macro so that the micro can compensate. The ability of the micro platform to compensate for the macro error will control the end effector error of the system. 1.2 Thesis Objectives This thesis aims to expand on existing knowledge of cable-suspended robots, with the uniqueness of the addition of the micro cable suspended robot. The addition of the micro platform is to solve the inherent problem with most cable suspended robots of accuracy due to sensing of shaft orientation. This is a result of the precision of the encoders for the motors on the cable spools. The focus of this paper is to assess uncertainty in both macro- and micro- portions of the system and how control methods can be used to minimize the uncertainty in the overall system. There are four objectives for this thesis: 1) Solve Kinematics and Statics for macro/micro cable suspended robot 2) Analyze workspace and uncertainties in both macro/micro systems independently as well as relate to one another in the total system a. Analysis independent workspaces of each system as well as configuration dependent workspaces of each b. Analysis independent and total uncertainty behavior for the system throughout the workspace 3) Design controller for system to reduce the inaccuracy of the system a. Incorporate task space division and genetic algorithm selection to minimize inaccuracy for a desired position b. Create a controller for sequential control that moves macro then adjust micro position to final desired position c. Simulate controller response for motion

16 4) Simulate uncertainty throughout workspace with and without controller a. Store simulation location uncertainties for each case b. Graph results with relation to one another to see effects of controller design 16

17 17 2. Methodology The purpose of this thesis is to assess the uncertainty throughout the workspace as well as design a controller to minimize end effector uncertainty. To analyze the uncertainty throughout the workspace, kinematic and static analysis are required for Error Model and SEW analysis. To design a controller to minimize end effector uncertainty, task space division and a genetic algorithm are needed to create minimized error trajectory. Kinematics (including velocity) and statics are required for the controller inputs for the system. The model used in this thesis is a two-stage macro/micro cable suspended using two planar 3 cable robots (Figure 2). Figure 3 shows the detailed view for the macro/micro cable suspended robot. Figure 2: Macro/Micro Cable Suspended Robot Model (Not to scale)

18 18 Figure 3: Detailed View of Macro/Micro Cable Suspended Robot (Not to Scale) In Figure 1 the 0 coordinate frame is the origin. The A and P are the coordinate frames of the macro and micro platforms, respectively. The link lengths for the model are expressed as Li. The actuator positions for the macro cable suspended robot are expressed as Bi. The connection points on the macro platform for the various cables are denoted by Ai. It should be noted that the connection points on the macro A3 and A4 are also the actuator points for the micro cable suspended robot. Finally Pi represents the connection points of the cables on the micro platform. 2.1 Kinematics The kinematic analysis of the macro/micro cable suspended robot can be completed two ways: Forward or Inverse Pose Kinematics. Forward Pose Kinematics (FPK) uses the cable lengths of the robot to solve for end-effector pose while Inverse Pose Kinematics (IPK) uses the pose of the end effector to solve for cable lengths of the configuration. Both methods are implemented due to that they are inverse solutions of

19 19 one another and a Circular Check can be used validate both solutions. A circular check is that the outputs of the first solution are inputs of the second solution and the resulting outputs of the second solution are the inputs of the first. The last step in the kinematic analysis will be to differentiate the IPK equations to obtain the velocity expressions for the macro/micro cable suspended robot. Velocity analysis will be required for the inputs for the controller design Forward Pose Kinematics To solve the Forward Pose Kinematics solution, the intersection of two circles method [13] is used. This is due to that the cable robot is planar and the inputs for this solution are cable lengths. The intersection of two circles method uses the two known circle centers (actuator positions/connection points) as well as the radius of the circles (link lengths) to solve for the intersection points (connection point on the platform). This method is analytically difficult due to the coupled equations and can be simplified with a robotic concept of transformation matrices which will be presented later. Figures 4 shows both methods which are Intersection of Two Circles (red) and Simplified Intersection of Two Circles (purple).

20 20 Figure 4: Intersection of Two Circles and Simplified Intersection of Two Circles Model The difference between the two solutions is what coordinate frame the intersection points are expressed in. In the Intersection of Two Circles solution the results are expressed in the 0 frame where as for the Simplified Intersection of two circles the solution is expressed in the 1 frame. In both methods, the solution to the problem is the Pythagorean theorem with respect to the two radii expressed as the hypotenuse for the right triangle. The resulting legs lengths for the respective triangles are just the difference in the x and y values between the intersection point and corresponding circle center. In the case of the Intersection of Two Circles, since the solution is expressed in the 0 frame, this forces the solution to be coupled and neither x or y can be solved for explicitly. The intersection of two circles can be simplified by using a transformation matrix. Transformation matrices resolve vectors in one coordinate frame to another. The use of the transformation matrix in the Simplified Intersection of Two Circle method allows one to express the coordinate frame (1) with respect to one circle point at the next

21 21 circle center. This decouples the equation and x can be explicitly solved by the subtraction the two equations from one another which removes y from the expression. The way that the FPK solution is implemented for the macro/micro design is the same for both systems. The first intersection point solved for uses the two cables that attach at the same connection point. The second intersection point is solved for using the last cable as well as the platform itself as the second cable. One thing to note about FPK solution is that the macro has to be solved for first since the micro solution is controlled by the pose of the macro system. Also the value of y is double valued as seen in Figure 1, however due to the system only being constrained by gravity, only the negative solution exist Inverse Pose Kinematics For the Inverse Pose Kinematic solution, the link lengths can be resolved into a vector that extends from its respective connection point to its actuator point. Since the pose of the robot is known, consequently the connection points are known in the 0 frame using a transformation matrix. Vector subtraction between the two points results in the cable length vector. The reason for looking at the vector from the connection point to actuator point is a result of positive cable tension direction. Since the macro/micro cable suspended robot is kinematically redundant, there are infinite solutions. This can be seen in the Inverse Pose Kinematics solution with respect to that it only defines the end effector pose as an input while there are 6 unknown cable lengths (3 equations 6 unknowns). This means that the macro platform could be anywhere in the useable workspace. To acquire the last 3 equations, Task Space division

22 22 is used to define the pose of the macro system. Task Space division will be discussed later in its respective section of this thesis. Equation 1 shows the IPK solution for link length and Equation 3 shows the transformation matrix used in the solution to solve for connection or actuation points expressed in the 0 frame. 0 0 L # = B, A # A, P # (1) [ A,P 0 A,P T] { A, P 1,2,3,4 } = [ A,P 0 A,P 0 R]{ A, P 1,2,3,4 } + { A, P} = [ [ A,P 0 R 0 A,P ] { A, P} ] { A, P 1,2,3, { A, P 1,2,3,4} = [ A,P 0 T] { 1 A,P } (2) A, P 1,2,3,4 } (3) 1 [ A,P O T] is the respective transformation matrices that transforms vectors that are expressed in either the A or P frame to the 0 frame. [ coordinate frame A or P expressed in the 0 frame. { A,P 0 R] is the rotation matrix for the 0 A, P} is the vector for the position of the A,P macro or micro platform respectively. { A, P 1,2,3,4 } is the connection or actuator positions expressed in the A or P frame. Equation 2 shows the formation of the transformation matrix. The transformation matrix is a convenient matrix that maps the translational and orientation relationship between two coordinate frames. This is why both the rotation matrix as well as the position of the coordinate in the 0 frame are included. The inclusion of the 1 under the vector is to account for the addition of the coordinate frame position in the matrix multiplication. Due to the augmented 1 in the vector, the zeros and 1 are a result to provide a vector solution that is the same as how the original vector was augmented. To apply the IPK solution, the connection or actuator positions for the respective platform can be resolved into the 0 coordinate frame using the transformation matrix (Equation 2).

23 23 With the connection and actuator positions expressed in the 0 frame Equation 1 can be used to solve for the link length Velocity Kinematics For the velocity analysis of the macro/micro cable suspended robot, the IPK equations presented earlier are differentiated with respect to time. The differentiation with respect to time for the 3 equations for each platform will result in the Velocity Jacobian for each platform respectively. Due to the macro/micro design there is another Velocity Jacobian that appears that maps the macro system positional change with respect to time to the micro systems positional change with respect to time. It should be noted that there are moment arm expressions in the Velocity Jacobian expressions due to the known relationship between the Static and Velocity Jacobians for cable suspended robots [13]. These are shown in the Velocity expressions so the Static Jacobians can be validated in their respective section. Also, the expressions for these moment arms appear in the Static section of this thesis. Equation 4 shows the differentiation with respect to time and Equations 5 and 6, 7 and 8, and 9 through 12 show the Velocity Jacobian expression for macro, micro, and total system respectively. d 0 0 L dt # = d B, A dt # A, P # (4) {L } = [J M ]{X } (5) L L 1 1y { L 2} = [ L 2x L 3 L 3x L 1y L 2y L 3y L 1x r A1y L 1y r A1x x A L 2x r A1y L 2y r A1x ] { y A } (6) r A2y L 3y r A2x θ A L 3x {l } = [J Mm ]{X } + [J m ]{x } (7)

24 24 L L 4 4 x { L 5} = [ L 5 x L 6 L 6 x L 4 y L 5 y L 6 y (r A 4 x ( L 4 y ) r A 4 y (L 4 x )) x A (r A 4 x (L 5 y ) r A 4 y (L 5 x )) ] { y A (r A 3 x (L 6 y ) r A 3 y (L 6 x )) θ A } + [ L 4y L 5x L 6x L 4y L 5y L 6y L 4x r P1y L 4y r P1x x P L 5x r P2y L 5y r P2x ] { y P r P2y L 6y r P2x θ P L 6x } (8) [J T ] = [ J M 0 J Mm J m ] (9) {L T } = [{L }; {l }] (10) {X T } = [{X }; {x }] (11) {L T } = [J T ]{X T } (12) L, l and X, x are the cable length and positional rate of change with respect to time. J T, J M, J Mm, J m are the respective Velocity Jacobains for the system. L # x or y is a short hand for the vector component either x or y over its total magnitude. r A or P #x or y are the respective moment arm short hands. As previously stated there is a known relationship between the Velocity Jacobian and Static Jacobian. By inspection it can be seen that the first two columns of the Velocity Jacobian are the components of the cable vector over its total magnitude. While the third column is the cross product of the cable vector with its respective moment arm. This relates to the Static Jacobian. In the first two columns are x and y components of the tension and the third column corresponds to the moment induced by the tension at some distance away. To validate the Velocity Jacobian, the Resolved Rate method was used [14]. The method for Resolved Rate is that given constant Cartesian rate inputs and an initial pose for the robot find the resulting link lengths for the next configuration and the resulting pose of the robot. The steps for Resolved Rate are as follows. Given the initial configuration of the robot and constant Cartesian rates for position, solve for the required length rates of change for this step. The reason that the initial configuration is required, is that gives a configuration to start from as well as provides the necessary components of

25 25 the Velocity Jacobian since it is link vector based. With the known link length rates, using Equation 13 which is numeric integration for link length rates, solve for the new link lengths. The new link lengths will be the input for the FPK method and will result in the new pose for cable suspended robot. The steps are repeated for each, consequently following iterations of Resolved Rate. Since the Cartesian rate inputs are constant, this means that the integral of the Cartesian rates which is position has to be linear. One thing to note since Equation 13 is the numeric integration of cable length rates, the step used must be small enough so that this numeric approach does not affect the solution of FPK. L new = L old + L t (13) L new is the solved for new cable lengths for the next iteration of FPK. L old is the cable lengths for the current iteration to solve FPK. t is differential time Error Model The last step in the kinematic analysis of the macro/micro cable suspended robot is the derivation of the Error Model. The Error Model uses the IPK equations presented earlier due to relates the geometric parameters of the model to end effector pose. The Error Model will be used to look at the uncertainty in each system independently as well as total uncertainty for the system. To complete the error models, the IPK equations are rewritten as one function set equal to zero (Equation 14). The summation of the partial derivatives of this function multiplied by its respective uncertainty (Equation 15) creates one of the 3 equations for the Error model for each system respectively. It should be stated that only link one s expression are shown in Equation 14 and 15 for brevity. The

26 26 Error Model for each system independently can be seen in Equation 16 and 18 for the macro and micro systems respectively. The Error Model Jacobians for the two systems can be seen in Equations 17 and 19 for the macro and micro respectively. The total Error Model for the system can be seen in Equation 20. f(x A, y A, θ A, L 1, x B1, y B1, x A1, y A1 ) = 0 (14) f x A dx A + f y A dy A + f θ A dθ A = f L 1 dl 1 f x B1 dx B1 f y B1 dy B1 f x A1 dx A1 f dy y A1 (15) A1 L 1x L 1y L 1x r A1y L 1y r A1x dx A [ L 2x L 2y L 2x r A1y L 2y r A1x ] { dy A } = L 3x L 3y L 3x r A2y L 3y r A2x dθ A [ L L L 3 L 1x 0 0 L 1y L 2x L 3x 0 L 2y L 3y (L 1x ( c A ) + L 1y ( s A )) (L 2x ( c A ) + L 2y ( s A )) 0 (L 1x (s A ) + L 1y ( c A )) (L 2x (s A ) + L 2y ( c A )) (L 3x ( c A ) + L 3y ( s A )) 0 0 (L 3x (s A ) + L 3y ( c A )) ] T dl 1 dl 2 dl 3 dx B1 dy B1 dx B2 dy B2 dx A1 dy A1 dx A2 { dx A2 } (16) 1 L 1x L 1y L 1x r A1y L 1y r A1x [J AE ] = [ L 2x L 2y L 2x r A1y L 2y r A1x ] L 3x L 3y L 3x r A2y L 3y r A2x L L L 3 L 1x 0 0 L 1y L 2x L 3x 0 L 2y L 3y (L 1x ( c A ) + L 1y ( s A )) (L 2x ( c A ) + L 2y ( s A )) 0 (L 1x (s A ) + L 1y ( c A )) (L 2x (s A ) + L 2y ( c A )) (L 3x ( c A ) + L 3y ( s A )) T (17) [ 0 0 (L 3x (s A ) + L 3y ( c A )) ] L 4x L 4y L 4x r P1y L 4y r P1x dx P [ L 5x L 5y L 5x r P2y L 5y r P2x ] { dy P } = L 6x L 6y L 6x r P2y L 6y r P2x dθ P L L L (L 6x (c A ) + L 6y (s A ) 0 0 (L 6x ( s A ) + L 6y (c A )) (L 4x (c A ) + L 4y (s A )) (L 5x (c A ) + L 5y (s A )) 0 (L 4x ( s A ) + L 4y (c A )) (L 5x ( s A ) + L 5y (c A )) (L 4x ( c P ) + L 4y ( s P )) 0 0 (L 4x (s P ) + L 4y ( c P )) 0 (L 5x ( c P ) + L 5y ( s P ) 0 0 (L 6x ( c P ) + L 6y ( s P )) 0 L 4x L 5x L 6x L 4y L 5y L 6y dy A [ ( L 4x r A4y + L 4y r A4x ) ( L 5x r A4y + L 5y r A4x ) ( L 6x r A3y + L 6y r A3x ) ] { dθ A } T dl 4 dl 5 dl 6 dx A3 dy A3 dx A4 dy A4 dx P1 dy P1 dx P2 dx P2 dx A (18)

27 27 1 L 4x L 4y L 4x r P1y L 4y r P1x [J PE ] = [ L 5x L 5y L 5x r P2y L 5y r P2x ] L 6x L 6y L 6x r P2y L 6y r P2x L L L (L 6x (c A ) + L 6y (s A ) 0 0 (L 6x ( s A ) + L 6y (c A )) (L 4x (c A ) + L 4y (s A )) (L 5x (c A ) + L 5y (s A )) 0 (L 4x ( s A ) + L 4y (c A )) (L 5x ( s A ) + L 5y (c A )) (L 4x ( c P ) + L 4y ( s P )) 0 0 (L 4x (s P ) + L 4y ( c P )) 0 (L 5x ( c P ) + L 5y ( s P ) 0 0 (L 6x ( c P ) + L 6y ( s P )) 0 L 4x L 5x L 6x L 4y L 5y L 6y [ ( L 4x r A4y + L 4y r A4x ) ( L 5x r A4y + L 5y r A4x ) ( L 6x r A3y + L 6y r A3x ) ] T (19) { P} = [J PE ] { [J AE ] dl 4 dl 5 dl 6 dx A3 dy A3 dx A4 dy A4 dx P1 dy P1 dx P2 dx P2 dl 1 dl 2 dl 3 dx B1 dy B1 dx B2 dy B2 dx A1 dy A1 dx A2 { dx A2 }} (20) s A or P and c A or P are short hand for sine and cosine of the orientation angle of the platforms respectively. [J AE ] and [J PE ] are the Error Model Jacobains for the macro and micro systems respectively. { P} is the uncertainty in end effector pose. d denotes the uncertainty with respect to a specific variable. The error model will be used in the uncertainty analysis for this thesis due to the error model relates the geometric tolerance errors in the system to end effector pose error. Like previously stated, uncertainties will be solved for each system independently as well as total for a specific configuration. For the macro uncertainty analysis only, the macro error model will be used with all macro geometric tolerance errors. In the case of the

28 28 micro uncertainty analysis only, micro geometric tolerance errors without macro pose uncertainties will be used. The reason for this is to only look at the uncertainty in the micro due to its configuration and geometric tolerance errors. For the total uncertainty analysis all geometric errors in the system will be used. The error results for each system will be used to create error bars for the respective uncertainty in x and y as well as a triangular patch to show the plus minus effect of orientation uncertainty. It should be noted like stated in the Literature Review that the Error Model assumes that all variable are independent of one another. Since the model used has small errors and actuation is independent for cable lengths, this is a valid method for uncertainty analysis for this model. To validate the derived expressions, these results were compared to the results produced using symbolic MATLAB differentiation. Both methods agree on the expressions for the Error Models for their respective system. 2.2 Static Analysis The static analysis for this thesis is used to define the Static Equilibrium Workspace with respect to each system independently, as well as, how the respective workspaces develop due to the configuration between the two platforms. The Static Equilibrium Workspace is divided into 4 categories which are as followed: macro workspace only, micro workspace only, macro based on micro workspace, and micro based on macro workspace. Macro Workspace only assumes that there is no micro system attached to the macro system. Micro Workspace only assumes that the macro platform is the ground link. Macro based on micro workspace is the workspace of the

29 29 macro due to a defined pose for the micro system. Finally, micro based on macro workspace is the workspace of the micro due to a defined pose of the macro. The reason for the 4 categories is to have the unaltered workspaces of each platform independently so that these workspaces can be compared against how the workspace is changed with respect to configuration relationships. To implement these solutions, the relative area with respect to each system is discretized. At each discretized point, two constraints are imposed for a useable solution. These are cable length limits with respect to the respective system and that all cable tensions are positive. With these points the workspace envelope can be created to show the workspace for a particular configuration Statics To implement SEW, the Static equations that govern both systems must be derived. To start the static analysis the Free Body Diagrams (FBDs) are required to expose the unknown cable tension. The two FBDs for each platform can be seen in Figures 5 A and B for the macro and micro system respectively. With the FBDs, 6 equations can be derived to solve for the 6 unknown cable tensions expressed in the 0 frame. It should be noted that there is a known relationship from literature between the Static and Velocity Jacobian for cable suspended robots [13]. Due to known relationship, the static approach used in this thesis is sequential which solves for micro tension followed by solving the macro tensions. The reason for this is to validate the Static Jacobian expressions using the previously validated Velocity Jacobians expressions. To derive the 6 static equations, Equations 21, 22, and 23 are applied to each of the FBDs.

30 30 With these equations, the matrix expression for the system can be written which are the Static Jacobian for the respective systems. Equations 24 25, and 26 is the first matrix expression to solve for micro tensions and the Static Jacobian expression for the micro platform. Equations 26, 27, 28, and 29 is the second matrix expression to solve for the macro tensions and the two Static Jacobian expression for the macro and macro on micro respectively. Equations 30 and 31 show the moment arm expressions that are used in the Static Jacobians. A) B) Figure 5: Free Body Diagram of A) Macro B) Micro Platform ΣF x = 0 (21) ΣF y = 0 (22) ΣM z (A,P) = 0 (23) [J SP ]{T 2 } = {b} (24)

31 L 4 x [ L 4 y (r P 1 x L 4 y r P 1 y L 4 x ) L 5 x L 5 y (r P 2 x L 5 y r P 2 y L 5 x ) L 6 x L 6 y (r P 2 x L 6 y r P 2 y L 6 x ) T 4 ] { F ex T 5 } = { F ey m micro g T 6 31 M z } (25) [J SA ]{T 1 } + [J SAP ]{T 2 } = {b} (26) T 1 [J SA ] { T 2 } + [J SAP ] { T 3 T 4 T 5 F ex } = { F ey m macro g} (27) T 6 M z L 1 x [J SA ] = [ L 1 y (r A 1 x L 1 y r A 1 y L 1 x ) L 2 x L 2 y (r A 1 x L 2 y r A 1 y L 2 x ) L 3 x L 3 y (r A 2 x L 3 y r A 2 y L 3 x ) ] (28) L 4 x [J SAP ] = [ L 4 y (r A 4 x ( L 4 y ) r ( L A 4 y 4 x )) L 5 x L 5 y (r A 4 x ( L 5 y ) r ( L A 4 y 5 x )) L 6 x L 6 y (r A 3 x ( L 6 y ) r ( L A 3 y 6 x )) ] (29) O {r A,P 1,2,3,4 } = [ cos (θ A,P) sin (θ A,P ) sin (θ A,P ) cos (θ A,P ) ] { A,P A, P 1,2,3,4} (30) O {r A,P 1,2,3,4 } = { r A,P 1,2,3,4 x r A,P 1,2,3,4 y } (31) To validate the expressions 24 and 26, the known relationship from literature is the Velocity Jacobian is the negative transpose of the Static Jacobian [13]. Just like with the Velocity Jacobian, the Static Jacobian has the secondary Jacobian that relates the macro to the micro system. For all the Jacobians the relationship holds validating the expressions presented. Equations 32, 33, and 34 show the relationships between the 3 Jacobians for the system, either being macro, macro on micro, or micro respectively. [J M ] = [J SA ] (32) [J Mm ] = [J SAP ] (33)

32 32 [J m ] = [J SP ] (34) 2.3 Task Space Division and Genetic Algorithm Since the system is kinematically redundant, it has infinite possible solutions for a given end effector pose. Task Space division is used to solve for these infinite possible solutions. To implement Task Space Division, the workspace above the micro platforms end effector pose is discretized for possible solutions for macro pose. Task Space Division checks two different types of constraints at each of these points. The constraints are with respect to length limits of the cables for the respective platform and if the configuration has only positive cable tension (static equilibrium). If the defined pose for the macro meets both constraints defined by the system characteristics then it is an acceptable solution to be used in the Genetic Algorithm to select the optimized solution. The genetic algorithm used in this thesis restricts this possible solution set from Task Space Division to a smaller set dependent on the compensability values for the possible configuration solutions. Compensability assesses the amount of error induced by the macro system and the manipulability of the micro system to correct for this error. This will be discussed later in the Compensability section of this thesis. With the refined solution set based on compensability, the genetic algorithm then looks at the relationship change between the two platforms at the start and stop of the trajectory. The genetic algorithm first checks for a minimal change in the translational relationship between the two platforms for the two configurations (start and stop). This selection is to force the solution that has the most minimal translational change between the two platforms to minimize micro movement for the motion. It should be noted there is

33 33 a benefit due to how the workspace is discretized with respect to end effector pose. The discretized poses for the macro extend away from the micro platform position vertically. As will be seen in Workspace Analysis section of this thesis, micro workspace in relation to macro pose is more dependent in y than as compared to x. This means that the genetic algorithm inherently chooses points with minimal cable lengths. After this selection if there are still multiple solutions there is one more check. The secondary check forces the macro orientation change between the configurations to be minimal. There are two reasons for this, first is to have smallest orientation change for macro platform. The secondary benefit is that the micro will need minimal motor orientation change to correct translationally for this macro orientation change. With this selection process the optimized solution can be used with respect to trajectory generation to solve for the pseudo static controller inputs throughout the trajectory Compensability Compensability is the measure of the ability micro system to compensate for the end effector error due to the micro system. The end effector error with respect to the macro system is defined by the Error Model which was previously defined in this thesis. For the compensating of the micro, this is the Jacobain matrix ([J m ]) that relates to micro motion only. The reason for this is the Jacobian maps little change in link length to little positional change to correct for the error induced by the macro system. Equations 35 and 36 show the expression for manipulability and end effector error due to the macro system. Equation 37 equates these two expression to one another. Equation 38 shows the expression for compensability for the given configuration of the system.

34 { P} = [J m ] 1 { L} (35) 34 dx A { P} = [J PE ] (1:3),(12:14) [ dy A ] (36) dθ A dx A [J m ] 1 { L} = [J PE ] (1:3),(12:14) [ dy A ] (37) dθ A I c = 1 dx A 0 0 det ([J m ][J PE ] (1:3),(12:14) [ 0 dy A 0 ]) 0 0 dθ A (38) I c is the compensability value for the configuration of the system. The function det is the determinate call for the matrix. (1: 3), (12: 14) denotes the row and column values for the Error Model Matrix [J PE ]. From the compensability expression, it can be observed how the value of compensability evaluates the compensating ability of the micro to correct for macro induced end effector error. In the expression, the error in the macro system is the diagonal which is a result of how the error is normalized; so that the determinate of the matrix relates to a volume of an ellipsoid for normalized link length change. The volume of the ellipsoid can be equated to the tracking performance of the micro system. One over this ellipsoid is the compensability value which if larger than the tracking error seen in the micro system is minimized. The reason that only part of the Error Model is used, instead of the total assuming no error in the micro is to remove unnecessary zeros in the matrix that effect the value of compensability. The removing of these extra zeros does not change the mathematical result of the solution. Removing these zeros, only results in that the singularity effects of

35 35 the zeros are not included in the determinate value. This is the reason that only part of the Error Model is used in the Compensability analysis for the macro/micro cable suspended robot. 2.4 Pseudo Static Control For the Pseudo Static Control simulation, the trajectory of the motion as well as the system behavior throughout this trajectory is required. To create the trajectories for the motion, 5 th order and 6 th order with via point polynomial functions [14] were used. The reason for this selection is to have position (orientation) and velocity (angular velocity) profile for the trajectory that start and stop with zero velocity and acceleration. Position and velocity profiles are required for the controller inputs for the system. Another reason for the 5 th order and 6 th order with via point polynomial functions is to simulate the controller response for both the desired response as well as the response for sensed error of the macro platform respectively. The via point provides another point in the trajectory that can be controlled to simulate an error induced trajectory for the macro system. Due to the macro/micro design trajectories need to be defined for each system independently. The position and velocity equations for the 5 th order and 6 th order with via points can be seen in Equations 39-40, and respectively. It should be noted the x is shown as a one of the pose elements for the trajectory for brevity. x(t) = a 5 t 5 + a 4 t 4 + a 3 t 3 + a 2 t 2 + a 1 t + a 0 (39) x (t) = 5a 5 t 4 + 4a 4 t 3 + 3a 3 t 2 + 2a 2 + a 1 (40) x(t) = a 6 t 6 + a 5 t 5 + a 4 t 4 + a 3 t 3 + a 2 t 2 + a 1 t + a 0 (41) x (t) = 6a 6 t 5 + 5a 5 t 4 + 4a 4 t 3 + 3a 3 t 2 + 2a 2 t + a 1 (42)

36 36 x(t) is position with respect to time. x (t) is the velocity with respect to time. a i is order respective constant. t is time. With the trajectories for the motion, the controller inputs required for this motion can be obtained. Since the trajectories will define the kinematic position and velocity at each step, these can be used in the IPK and Velocity Analysis to solve for the required link lengths and link length rates of change respectively. Also since the trajectories define the configuration that the robot is in at every step, this can be used to calculate the resulting tensions in the cables using the Static Analysis. These inputs must be resolved to their actual controller inputs for the motors used in the design. To do this, the link lengths, link length rates of change, and tensions must be converted into motor orientation, motor speed, and torque of the motor respectively. Equations 43, 44, and 45 show the relationship between these inputs for motor orientation, motor speed, and torque respectively. θ i = L i L 0i r i (43) θ i = L i r i (44) τ i = T i r i (45) θ i is the motor orientation with respect to initial configuration for motor i, L 0i is initial length of cable for the starting configuration for motor i. r is radius of the cable spool for motor i. θ i is motor speed for motor i. τ i is motor torque for motor i. To simulate the Pseudo Static Controller response for the system the macro/micro correct trajectories as well as the macro error induced trajectory are used. The reason for the error induced macro platform is to simulate the sequential control for the micro

37 37 system. Correct macro/micro trajectories are used to show the desired response of the system. The error induced macro trajectory is to show the compensating of the micro for macro pose error due to sequential control.

38 38 3. Results The results presented for this thesis will be in regards to: workspace for the system, uncertainty throughout the workspace for the system, and pseudo-static control inputs as well as compare with and without controller uncertainty throughout the trajectory. Each of these three categories will be explained in more detail in their respective sub-sections with respect to this section of the thesis. The workspace analysis will assess the workspace of each platform independently as well as how they relate for total system workspace. For the uncertainty analysis, like workspace each system will be evaluated independently as well as how they relate in the total system. Finally the trajectory defined by the genetic algorithm will simulate the pseudo-static control inputs as well as compare the uncertainty throughout the trajectory with and without the controller. 3.1 Workspace Analysis To start the Workspace Analysis for the macro/micro cable suspended robots, the independent workspaces for each system were used as baselines to compare to configuration dependent workspaces for the macro/micro design. The independent workspaces for the macro and micro platforms respectively can be seen in Figures 6 and 7. Figure 6 shows the workspace of the macro platform without an attached micro system. Figure 7 shows the micro workspace assuming the macro platform as a ground link.

39 39 Figure 6: Macro Only Workspace Figure 7: Micro Only Workspace

40 40 From Figure 6 and 7 it can be seen that due to the reversed middle link, the workspace size due to orientation of the platform is reversed. The meaning of this is that for the macro platform, the workspace extends to the left for more negative orientations where as for the micro platform the workspace extends to the right for more positive platform orientations. These two plots as well as the reversed middle link relationship will be used to understand how the workspace develops for particular configurations for the system. To start the configuration relationship analysis with respect to workspace, the effects of macro pose on micro workspace is assessed first. This is due to, the micro configuration relationship (counterweight) to the macro which will govern the workspace size and possible orientations for the macro system. This will be discussed later in this section of the thesis. For the micro workspace response to macro pose, first the development of the workspace with respect to macro orientation is assessed. The reason for this is to understand which macro/micro configurations are useable throughout the macro orientation dependent workspace. It should be noted that all these plots (Figures 8 A-F) appear at the same point, to not include the effects of macro orientation workspace limits on the development of the micro workspace. This orientation dependent workspace limits of the macro system on the micro workspace will appear later on in this thesis. The point used in all these plots is for x and y equal 10 m and -30 m respectively.

41 41 A) B) C) D)

42 42 E) F) G) H) Figure 8: Micro Workspace with Macro Orientation of A) -30 B) 30 C) -20 D)20 E) - 10 F) 10 G) -5 H) 5 From Figures 8 A through F, the development of micro workspace on macro orientation can be seen. One finding from these figures is that the workspace extends forward with respect to positive macro orientation and consequently extends backwards for negative macro orientation. This can be seen if one compares the arbitrary zero (x = 10 m) with 0 orientation line for micro orientation in the respective figure. This can be

43 43 understood by accessing the middle link at this edge of the workspace since this is where it becomes slack. Secondly that the workspace width diminishes with respect to increased orientation. This is a result of minimized platform width in the x direction, with larger orientation, this results in a smaller x distance between the actuator reducing the useable workspace of the micro system. Lastly that the orientation dependent size of the workspace holds true throughout the orientation changes of the macro. The meaning of this is that as the micro platform orientation goes positive the workspace is extending to the right. One thing to note about these plots is that dependent on opposite orientations of macro the shape of the workspace is flipped but this relationship diminishes with respect to larger orientation changes. This is a result of the middle point on the 0 macro orientation (Figure 8) which is cable length limit induced is altered by the movement of the actuators with respect to orientation of the macro. The reason that these plots (Figures 8 A-F) are useful is that, as long as the macro is in the macro workspace defined by its orientation these micro orientation dependent workspaces hold true as seen in Figure 9. If the macro platform is near the limit of the macro orientation dependent workspace then the workspace obtainable is minimized to keep the system in static equilibrium. From Figures 10 A through C, it can be seen how the workspace is reduced as the macro nears the limit of the orientation defined workspace for the macro system. In the Figures 10 A through C, the workspace is collapsing to the left as the macro approaches the orientation dependent workspace limit. This shows the influence of the micro system as a counterweight in the analysis of the macro system since the micro can only be positioned in such a way as to keep the

44 44 resulting configuration in static equilibrium. What is meant by the word counterweight with regards to this thesis is depending on the configuration between the two platforms, the micro platform effect on the macro can be modeled as a weight that forces an unequal distribution of platform weight for the macro system. Figure 9: Micro Workspace at Macro Pose x=0 y=-30 θ=-30 A) B) C) Figure 10: Micro Workspace at Macro Pose y= -30 θ= -20 A) x = 8 B) x = 9 C) x = 10 For the analysis of the macro configuration dependent workspace, the concept of the micro counterweight presented earlier will be used to assess how the macro workspaces changes due to the configuration relationship of the two platforms. To

45 45 reiterate, the counterweight concept is that the micro can be positioned in such a manner to force the macro platform to be weighted to one side. This weighting property will either extend or reduce the workspace size due to which side is weighted more heavily, left and right respectively. To show this behavior, the orientation of the micro used in the configuration relationship was 30 because it has the largest workspace for all orientation sets of the macro. X configuration relationship is varied from ±.2 m to show the forced reduction or extension of the workspace respectively and Y configuration relationship is held constant -1.2 m. From Figures 11 A through F, the counterweighting property of the micro system can be seen. A) B)

46 46 C) D) E) F) Figure 11: Macro Workspace with Translational Relationship to Micro of x = A) -.05 B).05 C) -.1 D).1 E)-.2 F).2 From the plots (Figures 11 A through F), one can see how the offset micro platform extends or contract the macro workspace. Looking at the magenta line with respect to 0 (except in the case of x =.2 m (blue)) one can see how the workspace contracts and extends with respect to the induced counterweight of the micro being either positive or negative with respect to macro X configuration relationship. It should be noted that the ±.2 m plots start and stop at -10 and 10 respectively. This is a result of

47 47 the micro workspace static equilibrium limits due to macro orientation. From Figures 11 E and F, it can be seen that these are the last maximum angles (discretizing angle with respect to 10 ) with respect to macro pose that have an obtainable workspace for ±.2 m. 3.2 Uncertainty throughout the Workspace Analysis For the uncertainty analysis throughout the workspace, the independent uncertainties as well as the total uncertainty in the system will be assessed. The uncertainty analysis is applied to the poses of the platform(s) that are defined by SEW and cable length constraints. The results of the uncertainty analysis will be used to create error bars with respect to x, y, and θ for each of these poses inside the defined workspace. Like in the case of the workspace analysis, the macro and micro systems will be evaluated independently as a baseline to compare against configuration effects of macro/micro design. Figures 12 and 13 show the uncertainty development for the macro system only throughout its workspace. It should be noted that the error bars are magnified to show the effect throughout the workspace.

48 48 Figure 12: Macro Only Uncertainty Throughout Workspace Figure 13: Macro Only Uncertainty Throughout Workspace Zoomed in

49 49 From Figures 12 and 13 general trends of uncertainty development with respect to x, y, and θ for the macro platform can be drawn. With respect to the uncertainty in x, towards the center of the workspace (x 0) the uncertainty is minimal. This is a result of the symmetry of links 1 and 3. Secondly extending away from 0 for the workspace (holding y constant) x uncertainty increases. This is a result of the cables not being symmetric to one another. It should be noted that this is not a symmetric relationship with respect to x uncertainty. This means that if the same value of ±x is assessed the negative x value will have a larger uncertainty. These two trends results being offset from 0 are a result of the middle link. Lastly is that the x uncertainty decreases the further down the workspace the macro is positioned. This is a result of cable vectors becoming more vertical (y) reducing effect for x displacement. With respect to y uncertainty development, the uncertainty is constant for most of the workspace, except near the upper limit of the workspace where the uncertainty increases exponentially. This is a result of cable length uncertainty and small angle approximation effect near the top of the workspace. Since cable length uncertainty is the largest uncertainty in the system this results in the largest uncertainty to be in y with vastly less effect on x and θ throughout most of the workspace. Only near the top of the workspace does the cable vectors become horizontal enough that x uncertainty becomes comparable to y. This is why y uncertainty reduces near the ends (maximum x). Also this exact behavior that was discussed earlier for x holding y constant, is why y is maximum near 0 (symmetric cables) as seen in Figure 13.

50 50 With respect to the exponentially increasing y uncertainty, this is a result of the small angle approximation. The small angle approximation is that if there is little orientation change with respect to a radius (x), there is negligible change in x but the change in y is in relationship to the little orientation change. Due to this at the top of the workspace since the cable length is increasing (the x radius with respect to the small angle approximation) this results in a large change in the y uncertainty near the top of the workspace. As a result of the compounding effect of the increased radius and small angle change approximation. With regards to the micro platform, the only difference that is seen is due to the reversed middle link. All previous general trends hold true except for the offset to the right. Since the middle link is reverse for the platform, the minimal x uncertainty is offset to the left for the micro system. From Figure 14, it can be seen that the general trends developed for the macro platform hold true for the micro platform with the exception to the offset x minimum uncertainty to the right. Also, from Figure 14 it can be seen the further up the workspace the minimal uncertainty moves towards x=0. This trend can also be seen in the macro platform which can be seen in Figure 15. It should be noted that this trend has not been presented until now due to it is visually difficult to see this behavior in the macro system. The reason for this trend is that as the cables become more horizontal the effects of the outside link on the uncertainty in x become larger. This results in a diminished effect of only the middle link on the uncertainty in x. This is why it can be seen that approaching the top of the workspace the minimal x uncertainty moves towards 0.

51 51 Figure 14: Micro Only Uncertainty Throughout Workspace Figure 15: Macro Only Uncertainty Throughout Workspace Zoomed in 2

52 52 To start the macro/micro uncertainty analysis throughout the workspace, a constant pose for the macro will be given. The reason for this is two see how the macro uncertainty (constant) and micro relate to total uncertainty in the system. This provides insight into how the micro can be positioned with respect to the macro to provide a configuration with the minimal uncertainty for the system. Figures 16 through 21 will be used to provide general trends for macro/micro configuration to total end effector uncertainty. Figure 16 A shows the constant uncertainty for macro platform for each of the possible end effector locations. Figure 16 B shows the uncertainty for the micro platform only to relate to the total end effector uncertainty. Figures 17 and 18 show the total end effector uncertainty and the zoomed in uncertainty due to a behavior that is seen as the micro moves further forward in its respective workspace. Figures 20 and 21 show the additive and subtractive effects of the macro and micro uncertainty in total uncertainty for y and θ respectively. A) B) Figure 16: A) Macro B) Micro Only Uncertainty for Macro Position x=10 y =-30

53 53 Figure 17: Total Uncertainty for Macro Position x=10 y =-30 Figure 18: Total Uncertainty Zoomed in for X Uncertainty

54 54 A) B) Figure 19: A) Micro B) Total Uncertainty Zoomed in for Y Uncertainty A) B) Figure 20: A) Micro B) Total Uncertainty Zoomed in for θ Uncertainty at Constant Y A) B) C) Figure 21: Macro Uncertainty Zoomed in for θ Uncertainty at One Point

55 55 To start the general trends for macro/micro configuration to total end effector uncertainty, y and θ are assessed first because they are additive and subtractive respectively in regards to total uncertainty. Also, both of these total uncertainties are independent of other uncertainties unlike x. The reason that the uncertainty in y is additive is due to it is heavily dependent on cable length uncertainty which are directionally the same in both systems as seen in Figures 16, 17, and 19. Figures 19 A and B show the additive property of the macro/micro system to provide the same y uncertainty development at the ends for both micro and total end effector uncertainty in y (constant macro). The reason that θ uncertainty is subtractive is due to the reversal between the two middle links as seen in Figures 20 and 21. Due to the reversed middle links, the total system behaves counterproductively as compared to the two uncertainties in the independent systems. Figures 20 A and B, show how the counterproductive behavior develops for constant y and Figure 21 A through C show subtractive effect at one point. The total uncertainty in x is a function of both the respective uncertainties in x (induced by cable lengths) as well as orientation uncertainty in the macro platform. The reason that the uncertainty in x increases on the left side towards the bottom of the workspace is orientation uncertainty of the macro platform. Further down the workspace the cables are longer and provide a greater effect for the uncertainty in x caused by orientation uncertainty of the macro platform. The reason that the x uncertainty reduces to the left is a result of the total system transition from productive to counterproductive nature. This transition is a result of how the uncertainty develops with respect to how the

56 56 systems are placed (configured) with respect to one another. The last trend to be discussed can be seen in Figure 18. From Figure 18, it can be seen that at some point the orientation induced trend reverses. This is a result of the micro uncertainty (productive with configuration relationship to micro) in x is having a greater effect on total end effector uncertainty than macro orientation. With the analysis of macro/micro configuration to total uncertainty, these trends effects can be seen throughout the total workspace for the system. The figures that will be shown are with respect to θ = 10 and zoomed in around x = 10 m and y = 30 m. The reason for this is that they will align with the responses previously shown. One thing to note is since total uncertainty is dependent on both macro and micro uncertainties; the total uncertainty development throughout the workspace varies with respect to the pose of the macro and configuration relationship with respect to the micro. What is meant by this is that given the macro pose uncertainty the resulting total minimum uncertainty is dependent on micro uncertainty to mitigate this macro uncertainty. This is a result of the previously defined counterproductive nature of the system with respect to some uncertainties. It should be noted that only in the proximity of the derived relationship trends for uncertainty (zoomed in), will these hold true due to minimal macro uncertainty magnitude change. Figures 24 through 26 will be used to understand these general trends and behaviors.

57 57 Figure 22: Total Uncertainty Throughout Macro Workspace for Translational Relationship x = -.1 y = -1.2 θ = 5 Figure 23: Total Uncertainty Throughout Macro Workspace for Translational Relationship x = -.2 y = -.8 θ = 5

58 58 A) B) Figure 24: Total Uncertainty Throughout Macro Workspace Zoomed in for Translational Relationship A) x = -.1 y = -1.2 θ =5 B) x = -.2 y = -.8 θ =5 Like previously stated, position of the macro and configuration relationship with respect to the micro cause different behaviors for total uncertainty throughout the workspace. Due to variable macro uncertainty dependent on pose. From Figure 17, it can be seen that (-.1 m,-1.2 m) and (-.2 m, -.8 m) have more and less total orientation and x uncertainty respectively. It should be noted that these points are not defined, but the response of total uncertainty for these points will be relative to how the total uncertainty develops near them in the micro workspace. If the configuration response would hold true for all macro workspace poses, this behavior would be seen. This is not the case as previously defined since the macro uncertainty varies with pose. From Figures 24 and 25, it can be seen that the last row of the workspace (-.1 m,-1.2 m) has less total x uncertainty than (-.2 m,-.8 m) which is the opposite response as compared to behavior at macro pose (10 m,-30 m). Also, this can be seen in the second row of the workspace from the top where (-.2 m,-.8 m) has minimal total orientation uncertainty as compared to that of (-.1 m,-1.2 m).

59 59 The second comment stated was that the relationship will hold true with respect to poses near the pose that the relationships were defined. This is a result of minimal uncertainty change with respect to the positional change of the macro. From Figures 26 A and B, it can be seen that the previous total uncertainty relationship with respect to configuration for minimal x and θ hold true for configuration relationships (-.2 m, -.8 m) and (-.1 m,-1.2 m) respectively. From the analysis of configuration relationship between the two platforms, the minimal total uncertainty in the system can be understood for a given end effector location. Also this analysis can be used to provide the minimal total uncertainty with respect to x, y, and θ or a weighted combination of these for a given end effector pose. 3.3 Sequential Pseudo Static Control The results for the Sequential Pseudo Static Control simulation are in regards to controller inputs as well as uncertainty with and without the controller throughout the trajectory. For the sequential control, θ is shown as loop inputs for both systems as well as for micro system to correct for the macro pose error. The controller inputs for the motor are θ and τ. It should be noted even though the trajectory generation provides a dynamic response, the motion is slow enough that dynamic effects are negligible (Pseudo Static) and Static Equilibrium will define the motor torques throughout the trajectory. With regards to the uncertainty throughout the trajectory with and without control, total end effector uncertainty will be compared to micro only uncertainty (perfect sensing of macro) at each step of the trajectory. To begin the results for the Sequential Pseudo Static

60 Control simulation, the controller inputs for the system will be assessed first. Figures 25 through 27 show the controller inputs assuming no error in the macro system. 60 A) B) Figure 25: A) Macro B) Micro Motor Orientations (Perfect) A) B) Figure 26: A) Macro B) Micro Motor Speed (Perfect)

61 61 A) B) Figure 27: A) Macro B) Micro Motor Torque (Perfect) From Figures 25 and 26 (same as Figures 28 and 29 but error response), the validation of the motor speed response can be seen with respect to derivative relationships. Also from these figures the response defined by the trajectory generation is correct since start and stop with zero angular velocity. From Figures 27 (same as Figure 30 but error response), it can be seen that task space division, genetic algorithm, and trajectory generation are providing configurations that provide positive cable tensions throughout the trajectory. With the results for a perfect macro trajectory, these can be compared to error induced trajectory to access the compensating ability of the micro system in sequential control. The error induced controller inputs can be seen in Figures 28 through 30. From Figures 28 and 29 the compensating effect of the micro system can be seen to correct for macro pose error. This shows the effect of the sequential control to correct for sensed macro pose error.

62 62 A) B) Figure 28: A) Macro B) Micro Motor Orientation (Error) A) B) Figure 29: A) Macro B) Micro Motor Speed (Error)

63 63 A) B) Figure 30: A) Macro B) Micro Motor Torque (Error) The last part of the Sequential Pseudo Static Control results is with respect to uncertainty throughout the trajectory with and without the controller. Figures 31 A through C are used to assess the genetic algorithm configuration selections for the trajectory. Figures 31 A through C show the influence of the selection method of the genetic algorithm on minimal micro movement throughout the trajectory. Looking at Figures 31 A and C the micro position is (-2 m,-35 m) and (5 m, -30 m) respectively. It can be seen that there is little movement in x between the two configurations but relatively large movement with respect to y. This a result of compensability, since the value of compensability was not assessed with respect to configuration relationship change between the two configurations. It should be noted that compensability was arbitrarily constrained to keep solutions within a working precision. From these results it can be seen that more work needs to be accomplished to relate these two principles to one another. This will result in a more efficient genetic algorithm for the configuration selection for the system. It should be noted that the workspaces shown are with respect to

64 64 end effector location, this is why the discretized workspace shown is macro dependent with respect to the current orientation of the macro. Also the red and green are the uncertainties with and without control respectively. A) B) C) Figure 31: Trajectory Uncertainty with and without Controller for Discretized Macro Workspace A) Initial B) Middle C) Final The final assessment for the genetic algorithm selection is with regards to the uncertainty in the controlled system as compared to the possible configuration uncertainty without control in the discretized workspace of the macro system. This behavior can be seen in Figures 32 and 33. As seen in Figure 32, the controller uncertainty with respect to x and θ is not the minimal uncertainty throughout the discretized macro workspace as compared to without the controller. This is not the case for Figure 33. It should be noted that y uncertainty is not assessed due to sensed macro platform in the controller. This is a result of that compensability does not account for total end effector uncertainty. It should be reiterated that this compares micro (with control) and total (without control) uncertainty to one another. What is meant by this is that the total points due to the counterproductive nature of the system shows that without

65 65 control for some points have less uncertainty as compared to sensed macro pose with control. If the points are compared one to one either being total or sensed (micro uncertainty analysis), it can be seen for with respect to both cases that the chosen points do not always provide the minimal uncertainty configuration for the system. The chosen points are with respect to positional and orientation induced end effector uncertainty and manipulability of the macro platform. This is why Figure 32 has more orientation uncertainty and a smaller configuration spacing as compared to Figure 33. Since Figure 32 has a larger orientation uncertainty the spacing between the platforms is small to minimize uncertainty. This minimal uncertainty can be handled by the manipulability at this spacing relationship. Since Figure 33 has smaller orientation, the uncertainty at the end effector can be corrected by the manipulability of the micro system when fully extended (maximum manipulability). It should be noted that compensability is a value to access the configuration relationship to tracking performance. This means that even though it appears that in Figure 33 the spacing should be minimal, the chosen point is selected due to manipulability and consequently in Figure 32 macro orientation induced uncertainty. From this discussion, it can be seen that compensability is independent of end effector uncertainty. This is why compensability also needs to be related to total end effector uncertainty (micro). To have a genetic algorithm that will choose best minimal uncertainty configurations for end effector uncertainty with minimal tracking error of the micro platform. Even though with and without control does not directly compare what the end effector uncertainty would be at the different positions in the discretized macro

66 66 workspace. With the known micro uncertainty workspace analysis, this can be used to access the chosen points. The points chosen are both offset to the left for minimal x uncertainty but consequently have maximum θ uncertainty. Since end effector uncertainty is dependent on the purpose of the robot. These chosen points will have to be a function of the desired end effector uncertainty of the system with respect to x, y, and θ. These results were only for the purpose of showing the improved accuracy of the system with the addition of the controller. Figure 32: Trajectory Uncertainty with and without Controller for Discretized Macro Workspace (Initial) Zoomed in

67 Figure 33: Trajectory Uncertainty with and without Controller for Discretized Macro Workspace (Final) Zoomed in 67

68 68 4. Conclusion and Future Work 4.1 Conclusion From the simulations that were conducted, conclusions with respect to: workspace, uncertainty throughout the workspace, and genetic algorithm configuration selection for pseudo static control can be drawn. With respect to workspace, the adaptation of each system s respective workspaces to the others has been developed. For the uncertainty throughout the workspace, relationships have been drawn on uncertainty development with respect to each system as well as how they relate to total uncertainty in the system. Finally for the genetic algorithm configuration selection, compensability has been assessed with respect to configuration changes as well as resulting total end effector uncertainty. The conclusions for workspace are with respect to the independent as well as total system. The independent workspaces have been developed to access how each system s workspace adapts with respect to the other. In the case of the macro platform it was seen that the micro system can be positioned in such a way that acts as a counterweight on the system. This counterweight can be used to extend or contract the macro platforms workspace (total). With respect to the micro system, the orientation dependent as well as the workspace limits of the macro platform were accessed. From the orientation dependent workspaces, it was seen how the shape and size of the workspace adapted with respect to macro platform orientation. With respect to workspace limits of the macro platform, it was seen how this reduces the useable workspace of the micro platform. One thing to note is that due to the reversed middle link, the platform s respective workspaces

69 69 have more useable workspace in the opposite direction of one another. This results in a total system that can have a larger workspace with more possible configuration than if the link was the same as the macro platform. The conclusions drawn with respect to the uncertainty throughout the workspace are as followed: reasons for uncertainty development throughout workspace for each system independently as well as how the uncertainty in each system relates to the total uncertainty in the system. For the development of uncertainty throughout the workspace, it was seen that due to the reversed middle link, the development for the system was the opposite with respect to one another. The reasons for how the uncertainty developed through the workspace in relation to x, y, and θ respectively was assessed with respect to the cable uncertainty. Like previously stated, the uncertainty in the independent systems, is the opposite of one another. This resulted in the total uncertainty in the system to behave productively or counterproductively. This behavior, as well as macro orientation uncertainty were used to assess how the total uncertainty in the system developed. Since total uncertainty is with respect to the independent uncertainties (productive/counterproductive) as well as chained uncertainty (orientation). The minimal uncertainty with respect to x, y, θ or a combination of them for a specific end effector pose will be a result of configuration relationship between the two platforms due to their respective uncertainties. The last conclusions to be drawn are with respect to genetic algorithms configuration selections for pseudo static control simulation. The genetic algorithm does provide a response that is feasible since all cable tension remain positive throughout the

70 70 trajectory as a result of task space division. The genetic algorithm does provide optimal configurations with respect to tracking performance due to compensability. The only problem with compensability is since it was not the focus of this thesis, the value of compensability for a specific configuration was not related to configuration relationship change and end effector uncertainty. If relationships can be drawn with respect to these three parameters the genetic algorithm could provide a more overall optimal choice for the two configures used for trajectory generation. 4.2 Future Work From the conclusion section of this thesis, it can be seen that more work needs to be done with respect to compensability to have a more optimal genetic algorithm to choose the two configurations for a trajectory. To do this the compensability value needs to be assessed with respect to its tracking performance. With this information, either the number of possible solutions can increase or this can be related to configuration change and total end effector uncertainty. What is meant by number of possible solutions can increase is that with a minimal desired tracking performance there will be more possible solutions for a given end effector pose. This will result in a more optimal genetic algorithm that will be able to account for the relationship configuration change as well as total end effector uncertainty. If the desired tracking performance is not providing a substantial increase in possible solutions, then it can be related (weighted) to configuration change and total end effector uncertainty. In this genetic algorithm, the solution found will be a weighted function of all three, for an optimal configuration dependent on the required trajectory and specifications set for the system.

71 71 The last idea for future work on this project is with respect to not using standard trajectory generation techniques. What is meant by this is that, instead of fitting a function for movement with respect to start and stop. The motion could be discretized with respect to steps, these steps would have configuration solution sets defined by the genetic algorithm. Using all these possible solutions at each step of the trajectory, this could provide an optimal solution with regards to tracking performance, configuration change, and total end effector uncertainty for the desired trajectory. The way that the trajectory would be developed is that all steps would be in correlation to their future as well as past configurations throughout the trajectory. This would require a secondary genetic algorithm that would assess the correlation between all of these points and solution sets to achieve the optimal trajectory for the system, for the desired motion. This form of the solution, would only be useful with respect to line following and would provide no added advantage for pick and place.

72 72 References [1] X. Tang, X. Chai, L. Tang, and Z. Shao, Accuracy synthesis of a multi-level hybrid positioning mechanism for the feed support system in FAST, Robotics and Computer- Integrated Manufacturing, vol. 30, no. 5, pp , Oct [2] S.-R. Oh, K. Mankala, S. K. Agrawal, and J. S. Albus, Robust Control of Dual-Stage Cable Suspended Robots With Input Constraints for Cargo Handling, 2004, vol. 2004, pp [3] H. D. Taghirad and M. Nahon, Kinematic Analysis of a Macro Micro Redundantly Actuated Parallel Manipulator, Advanced Robotics, vol. 22, no. 6 7, pp , Jan [4] A. Ghasemi, M. Eghtesad, and M. Farid, Workspace Analysis for Planar and Spatial Redundant Cable Robots, Journal of Mechanisms and Robotics, vol. 1, no. 4, p , [5] G. Barrette and C. M. Gosselin, Determination of the Dynamic Workspace of Cable- Driven Planar Parallel Mechanisms, Journal of Mechanical Design, vol. 127, no. 2, p. 242, [6] R. Verhoeven and M. Hiller, Estimating the Controllable Workspace of Tendon- Based Stewart Platforms, in Advances in Robot Kinematics, J. Lenarčič and M. M. Stanišić, Eds. Dordrecht: Springer Netherlands, 2000, pp [7] P. Bosscher, A. T. Riechel, and I. Ebert-Uphoff, Wrench-feasible workspace generation for cable-driven robots, IEEE Transactions on Robotics, vol. 22, no. 5, pp , Oct [8] M. Gouttefarde and C. M. Gosselin, Analysis of the wrench-closure workspace of planar parallel cable-driven mechanisms, IEEE Transactions on Robotics, vol. 22, no. 3, pp , Jun [9] B. Zi, H. Ding, X. Wu, and A. Kecskeméthy, Error modeling and sensitivity analysis of a hybrid-driven based cable parallel manipulator, Precision Engineering, vol. 38, no. 1, pp , Jan [10] T. Yoshikawa, K. Hosoda, T. Doi, and H. Murakami, Quasi-static trajectory tracking control of flexible manipulator by macro-micro manipulator system, 1993, pp

73 [11] L. Luo, Y. Zhang, and Z. Sun, Task space division and trajectory planning for a flexible macro-micro manipulator system, Tsinghua Science and Technology, vol. 12, no. 5, pp , Oct [12] M. A. Khosravi, H. D. Taghirad, and R. Oftadeh, A positive tensions PID controller for a planar cable robot: An experimental study, 2013, pp [13] R. Williams II, Three-Cable-Suspended Planar Robot for Macro-/Micro-Robot Simulations, October Unpublished internal document, Ohio University Robotics Lab. [14] R. Williams II. (2016). Robotic Mechanics [Online]. Available: 73

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