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Equilibrium point control of a programmable mechanical compliant manipulator Clapa, Damien 2004

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Equilibrium Point Control of a Programmable Mechanical Compliant Manipulator by Damien Clapa B.Sc. University of Alberta A THESIS SUBMITTED IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF M A S T E R OF A P P L I E D SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard THE U N IVERSITY OF BRITISH jgCJLUMBIA August 2004 © Damien Clapa, 2004 THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES Library Authorization In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Name of Author (please print) Date (dd/mm/yyyy) Title of Thesis: & ^ O v \ \ W v U . W \ fcv'v'V C c ^ v V r ^ C>£ ' \ C a A c C X ^ KKoJa(x. Degree: M . A - S c . lAgcWxv^oJ, E ^ ^ ^ Y e a r : Department of HleoWxvy C C L \ (E. v ^ v A x s \ Q e c \ V\QA The University of British Columbia ^ —3 Vancouver, BC Canada grad.ubc.ca/forms/?formlD=THS page 1 of 1 last updated: 20-Jul-04 Abstract This thesis presents the design and experimental application of the Equilibrium Point Hypothesis as a controller model for a programmable mechanical compliant manipulator. A planar manipulator was designed and constructed with two joints, each powered by a pair of antagonistic McKibben actuators (air muscles). Programmable mechanical compliant manipulators provide increased intrinsic safety and the ability to implement a controller based on the EP Hypothesis becomes possible. The EP Hypothesis presents a model describing how human arm motions may be controlled. A previously developed geometrically derived force model for air muscles was modified leading to the formulation of a linearizing and decoupling compensator. This compensator, in -conjunction with a proportional, integral controller operating on air supplied to the muscles, provided stable control of the stiffness and EP of each joint of the manipulator. A benefit of this combined EP and stiffness control is that a single control strategy can be used both to control the manipulator position in free-space and to provide interaction control for contact tasks. A series of experiments were performed to demonstrate the controller behaviour in free space, in transition from free space to contact, and in contact with the environment. The free space experiments were done mainly to characterize the controller behaviour. The transition task involves moving in free space to contacting a surface at different velocities and contact angles. The contact task is a wiping motion along a surface with a prescribed normal force. The effect on introducing an unexpected "bump" along the surface was examined, as were velocity effects. The stable behavior during transition from free-space to contact is a notable result. Because the manipulator follows an equilibrium-point trajectory with a programmed stiffness, no additional compensation is required when contacting objects in the workspace. Additionally the precise location of the object is not important as the mechanical compliance of the manipulator compensates for small contact position errors. The results of the surface wiping tasks showed that it is possible to generate a wiping EP and stiffness trajectory that results in the predicted normal force while wiping a surface. Additionally, the mechanical compliance of the manipulator allows for stable response to unpredicted disturbances such as the presence of a significant bump on the smooth surface. 11 Table of Contents Abstract ii Table of Contents iii List of Tables .-. v List of Figures vi Nomenclature x Acknowledgements xiii Chapter 1 Introduction 1 1.1 Case for assistive robots and personal care 1 1.2 What wi l l the robots need to be capable of? 3 1.3 A i r Muscles 4 1.4 EP control 4 1.4.1 Background 5 1.5 EP Control and P M C Actuators 6 1.5.1 Interaction Tasks 8 1.6 Scope and Objective 8 1.7 Outline of Thesis ...7. 9 Chapter 2 Air Muscle Design 10 2.1 Introduction 10 2.2 System Overview 10 2.3 A i r Muscle Properties 12 2.3.1 Observed Limitations 13 2.3.2 Geometric Models of A i r Muscles 13 2.3.3 The empirical Modif ication to above model 15 2.4 Symmetric sizing method 17 2.5 Summary 25 Chapter 3 Electro-mechanical design and control of a PMC robot 26 3.1 Introduction 26 3.2 Manipulator 26 3.3 Valves 31 3.3.1 Sizing Valve Orifices 32 3.3.2 Selecting Constant Frequency for operation 35 3.4 Instrumentation, Drivers and D A Q 36 3.5 Controller : 36 3.6 Summary : 40 Chapter 4 Experimental Methods 42 4.1 Introduction 42 4.2 Free-Space Testing 44 4.2.1 Description of the test 44 4.2.2 Experimental measurements 45 4.3 Transition Testing... ; 45 4.3.1 Description of the test 45 i i i Table of Contents iv 4.3.2 Experimental measurements 46 4.4 Contact Testing 47 4.4.1 Description o f the test 47 4.4.2 Experimental measurements , 50 4.5 Summary ; 50 Chapter 5 Results and Discussion 51 5.1 Introduction 51 5.2 Free-space task results ....52 5.3 Transition Results 57 5.4 Contact Results 62 5.4.1 Velocity, Stiffness and EP Results ...69 5.4.2 Repeatability Results 72 5.5 Summary 72 Chapter 6 Conclusions & Recommendations . 74 6.1 General recommendations 75 6.2 Specific Recommendations for this Experimental Work 76 Bibliography .....78 Appendix A Air Muscle Equations 81 A . l A i r Muscle Equations for the Appendix 81 Appendix B Muscle Construction 83 Appendix C Matlab Optimization Files .' 88 C . l Op t im izeMoun tLeng th .m 88 C.2 solverbn.m 89 C.3 Sti f fnessLm 91 C.4 minimizethis.m 92 Appendix D Manipulator Bill of Materials 93 Appendix E Detailed Machining Drawings 97 Appendix F Assembly Instructions 123 Appendix G Sensors and Calibrations 126 G . l Experimental Equipment Specifications : 126 G . l . l Solenoid Valves... 128 G. 1.2 Sensotec Pressure Transducers 129 G. 1.3 AutoTran Pressure Transducers 130 G. 1.4 Force Transducer 131 G. 1.5 Length Encoders 132 G. 2 Calibration of Equipment 133 G.2.1 Calibration of Sensotec Pressure Transducers 133 G.2.2 Calibration of Auto Tran Pressure Transducers 135 G.2.3 Calibration of Precision Transducers Force Transducer : ....136 G.2.4 Calibration of U S Digital Length Encoders 137 Appendix H 138 H. 1 Free- Space Tests 138 H.2 Contact Tests 138 H.3 Transition Tests 139 Appendix I Summary of Transition Tests 140 Appendix J Summary of Contact Tests... 149 List of Tables Table 3.1 - Design Requirements 28 Table 4.1 - Summary of Testing 43 Table 4.2 - List of transition test numbers 46 Table 4.3 - List of all contact test numbers 49 Table 5.1 - Summary of Mean Absolute Error for the transition testing 61 Table 5.2 - Summary of Errors without the bump present 68 Table 5.3 - Summary of Errors with the bump present 69 Table B . l - Air Muscle Supplies 83 Table D . l - B O M - 1 94 Table D.2 - BOM-2 95 Table D.3 - BOM-3 .' 96 ) v List of Figures Figure 1.1 - Rotary Joints Powered by Opposed Pairs of A i r Muscle Actuators 7 Figure 2.1 - Overview of manipulator concept appropriate for demonstrating EP control 11 Figure 2.2 - Additional components required for powering the air muscle 11 Figure 2.3 - Rendering of a section of air muscle (from Shadow Robot Company) 12 Figure 2.4 - Force versus length relationship for an air muscle 12 Figure 2.5 - A i r Muscle Actuator : 13 Figure 2.6 - Manipulator configuration for muscle calibration 15 Figure 2.7 - Measured force and calculated force for an air muscle 16 Figure 2.8 - Absolute error in force between calculation and measured 17 Figure 2.9 - Simple rotary joint powered by a pair of air muscles ....18 Figure 2.10 - Force versus length at different pressures 19 Figure 2.11 - Useful range for a single muscle 19 Figure 2.12 - Working range of each muscle 20 Figure 2.13 - Configuration where maximum and minimum stiffness most constrained 21 Figure 2 . 1 4 - Max imum torque Constraint 21 Figure 2 . 1 5 - Stiffness versus pressure at a constant length 23 Figure 2.16 - Small stiffness constraint 23 Figure 2.17 - Highest stiffness constraint 24 Figure 3.1 - Manipulator with muscles 27 Figure 3.2 - Finalized manipulator design 29 Figure 3.3 - Close-up of back of manipulator 29 Figure 3.4 - Plot of the range of motion of the manipulator 30 Figure 3.5 - Torques for 20N force normal to the wiped surface 31 Figure 3.6 - Partially disassembled Matrix valve with orifice plate 33 Figure 3.7 - Inletorifice sized to allow no more than I N discreet force steps for smallest possible inflation 34 Figure 3.8 - Final inlet and outlet orifice sizes with inflation and deflation times roughly matched 35 Figure 3.9 - Schematic of planar robot controller 37 Figure 4.1 - X and Y axis origin location 43 vi List of Figures v i i Figure 4.2 - Diagram of the range of motion during the free space task 44 Figure 4.3 - Diagram of transition task 46 Figure 4.4 - Diagram of contact test 47 Figure 4.5 - Diagram of E P adjustments • 48 Figure 4.6 - Contact test with a smooth wall 49 Figure 4.7 - Contact task with a bump present 50 Figure 5.1 - Commanded position vs. time for free-space'tests (thick lines are desired and thin lines are measured) 52 Figure 5.2 - Error in xEp versus x for free-space tests 53 Figure 5.3 - Error in xa vs. x 54 Figure 5.4 - Error in ky versus x 55 Figure 5.5 - Mean Absolute Error of ky versus vx 56 Figure 5.6 - Mean absolute error XEP versus v v 56 Figure 5.7 - Transition test #1 ya and yspd with and without the wall versus path 57 Figure 5.8 - Transition test #1 yepa and yepa with and without the wall versus path 58 Figure 5.9 - Transition test #1 kya and kyd with and without the wal l versus path 59 Figure 5.10 - Transition test #9 kya and kyd with and without the wall versus trajectory 60 Figure 5.11 - Transition test #1 actual and predicted force with and without the wall versus trajectory 61 Figure 5.12 - Test #10 without bump (y versus x) 63 Figure 5.13 - Test #10 with bump y versus x 64 Figure 5.14 - Test #10 without bump ky versus x 65 Figure 5.15 - Test#10 with bump ky versus x 65 Figure 5.16 - Test#10 force versus x without bump ".. 66 Figure 5.17 - Test#10 force versus x with bump .' 67 Figure 5.18 - Summary ofy£7> error 70 Figure 5.19 - Summary of ky error 71 Figure 5.20 - Summary of force error 72 Figure B . l - Tools and supplies to make air muscles ,.• 84 Figure B.2 - Soldering the brass inserts 84 Figure B.3 - Putting inserts into the plastic and surgical tubing 85 Figure B.4 - Plastic and surgical tubing connected and plugged with brass inserts 85 Figure B.5 - End loop of the air muscle ...86 List of Figures v i i i Figure B.6 - Exploded view 86 Figure B.7 - Clamping down an O-Clamp 87 Figure B.8 - Completed air muscle mounted to the arm 87 Figure E . l - Drawing 1 98 Figure E.2 - End Effector i 99 Figure E.3 - L ink 2 and sprocket 100 Figure E.4 - Pulley 1 and big gear '. 101 Figure E.5 - Pulley 2 and big gear ...102 Figure E.6 - L ink drive assembly 103 Figure E.7 - Encoder and gear 104 Figure E.8 - Box 105 Figure E.9 - Adaptor 1 106 Figure E. 10 - Adaptor 2 107 Figure E. 11 - Adaptor 3 108 Figure E . 1 2 - L i n k 2 109 Figure E. 13 - Timing belt sprocket 2 110 Figure E . 1 4 - S h a f t 2 I l l Figure E . l5 - Pulley 1 , 112 Figure E . l6 - B i g gear 113 Figure E . l7 - Timing belt sprocket 1 114 Figure E . l 8 - Pulley 2 : 115 Figure E. 19 -Sha f t 1 ! 116 Figure E.20 - L ink 1 117 Figure E.21 - Fitting 118 Figure E.22 - Small gear 119 Figure E.23 - Top 120 Figure E.24 - Side ". 121 Figure E.25 - Base 122 Figure 1.1 - Transition test #1 140 Figure 1.2 - Transition test #2 141 Figure 1.3 - Transition test #3 142 Figure 1.4 - Transition test #4 143 Figure 1.5 - Transition test #5 144 List of Figures ix Figure 1.6 - Transition test #6 145 Figure 1.7 - Transition test #7 : 146 Figure 1.8 - Transition test #8 147 Figure 1.9 - Transition test #9 .' 148 Figure J . l - Contact test #1 150 Figure J.2 - Contact test #2 151 Figure J.3 - Contact test #3 , 152 Figure J.4 - Contact test #4 153 Figure J.5 - Contact test #5 154 Figure J.6 - Contact test #6 155 Figure J.7 - Contact test #7 :..156 Figure J.8 - Contact test #8 157 Figure J.9 - Contact test #9 ...158 Figure J. 10 - Contact test #10 159 Figure J. 11 - Contact test #11 160 Figure J.12 - Contact test #12 161 Figure J. 13 - Contact test #13 162 Figure J.14 - Contact test #14.... 163 Figure J . 15 - Contact test #15 164 Figure J . 16 - Contact test #16 165 Figure J . 17 - Contact test #17 166 Figure J . 18 - Contact test #18 167 Figure J. 19 - Contact test #19 168 Figure J.20 - Contact test #20 169 Figure J.21 - Contact test #21 170 Figure J.22 - Contact test #22 171 Figure J.23 - Contact test #23 .' 172 Nomenclature Ae Area of orifice dlj Inner surface displacement dst Area vector J Jacobian of manipulator Kc Carestian end point stiffness matrix Kj Joint space stiffness matrix L Length of an air muscle Lmount Mounting length of an air muscle Lmax Longest length of muscle at end of range Lmin Shortest length of muscle at end of range Lzero Length at which muscle delivers no axial force P Absolute pressure inside air muscle Patm Atmospheric Pressure (1 bar at sea level) Pg Gauge pressure inside air muscle Pmax Maximum allowable muscle gauge pressure Pmin Minimum allowable muscle gauge pressure R Gas constant T Temperature of air inside muscle Si Inner surface displacement V - Volume inside of muscle X Cartesian end point trajectory Xa Actual Cartesian location of the end point x Nomenclature xi XEP Cartesian end point equilibrium trajectory c A term introduced to account for constant force offset in air muscle b Braid length of an air muscle dV Volume Change / Muscle axial force fmax Maxium available muscle force kt Stiffness of joint i km Stiffness of a single muscle kmax Maximum available joint stiffness kmin Minimum available joint stiffness kx X-axis end point stiffness kxy Cartesian cross stiffness term ky Y-axis end point stiffness m Mass of air inside air muscle n Number of turns in braid of an air muscle p* Critical back pressure p0 Stagnation pressure r Pulley radius vx X-axis velocity of end point xa Actual X-axis position of the end point XEP X-axis endpoint equilibrium position xgpa Actual X-axis endpoint equilibrium position xgpd Desired X-axis endpoint equilibrium position Nomenclature xn ya Actual Y-axis position of the end point yEP Y-ax is endpoint equilibrium position yspa Actual Y-ax is endpoint equilibrium position yEpd Desired Y-ax is endpoint equilibrium position K Muscle Stiffness T Joint torque xmax Max ium available joint torque 6 Joint angle 6a Actual joint angle 6d Desired joint angle 0Ep Equil ibrium joint angle 9max Max ium angular range of motion A D L Activities of daily l iving EP Equil ibrium Position M A E Mean average error P P C Programmable passive compliance P M C c Programmable mechanical compliance P W M Pulse Width Modulation R M S E Root mean squared error Acknowledgements I would like to thank Leanne first and foremost for her unending support and encouragement through this long journey. I would like to thank my supervisors, Dr. Elizabeth Croft and Dr. Antony Hodgson, for their guidance and assistance. I would also l ike to thank the many fellow graduate and undergraduate students at U B C who have helped in so many ways. The assistance of the faculty and staff o f the Mechanical Engineering Department was greatly appreciated. I would also l ike to acknowledge the financial support of the Natural Sciences and Engineering Research Counci l of Canada. v xi i i Chapter 1 Introduction 1.1 Case for assistive robots and personal care In the coming decades, there wi l l be increased demand for nontraditional technologies, such as robotics, for the care of an increasingly dependent elderly population. This increased demand wi l l be driven by a number of factors, including the changing demographics in North America and Europe. While the number of people who can expect to live to advanced age is increasing quickly, the fraction of them who wi l l be disabled wi l l quite likely be no different than it was 25 years ago [1]. The increasing number of disabled elderly people wi l l likely outpace any growth in either formal or informal care sources, and wil l require affordable technologies to assist in tasks of daily living to avoid institutionalization. Affordable in-home robotics is potentially one part of the solution. Elderly people who are experiencing progressive disability are in a precarious situation, particularly i f they are living on low or fixed incomes. As a group, they are likely to face enormous difficulties for three reasons [2]: (i) people have longer life expectancies today than ever before, (ii) disability rates for people over 65 are three times that of those between 35 and 65, and (iii) extended families are shrinking. At present, formal (paid) care is generally insufficient to keep most elderly disabled in their homes. Only those individuals with access to 1 1.1 Case for assistive robots and personal care 2 informal care, usually from family members, are able to live at home [3]. Studies have also shown that the frequency that family members, especially children, tend to visit is inversely proportional to the level of disability [4]. Although modern medicine and improved living conditions have been successful in extending peoples life expectancy, the disability rate of the elderly has not decreased in 20 years [1]. More than 40% of those over the age of 65 are disabled, with the majority reporting disabilities including predominantly either mobility or agility limitations. Based on this fact, there is reason for concern, as the percentage of people over the age of 65 wi l l increase dramatically in the next 30 years [5]. Common sense suggests, and studies [6] have shown, that elderly people would prefer to live in their own homes rather than in an institution. Interestingly, while cognitive disability is an indicator for institutionalization, physical disability is not. Even so, the majority of elderly requiring care suffer from functional disabilities [3]. The best predictor of institutionalization is socio-economic [7]. The five tasks that make it increasingly difficult to live at home with disabilities are: bathing, toilet, transfer, eating and dressing [8]. Collectively these tasks are referred to as the Activities of Daily Living (ADL). The larger the number of these tasks a person requires assistance with, the more difficult it is to continue independent living. There are many diseases and disorders common to the elderly that contribute to the loss of agility and dexterity. Arthritis is the single most reported dexterity related disease of the elderly, affecting 2/3 of those over the age of 65 [9]. Tremor is a symptom of a number of diseases common among elderly people. Something as simple as fastening the buttons of a favorite shirt may be enough to keep someone in their home and away from social interaction. The need for aid with the five A D L ' s leads to a feeling of helplessness and loss of independence and places strain on those who provide informal care [3]. A robotic aid that can assist with the 5 A D L ' s would be enormously helpful to those requiring aid presently. In summary, changing demographics demand a cost effective way of helping functionally disabled people perform simple daily tasks. The ability to perform these tasks without human aid wi l l allow an increasing number of people to live in their own homes with dignity. The work in this thesis is part of an ongoing, worldwide, interest in robotic devices as home assistants. A 1.2 What wi l l the robots need to be capable of? 3 future-developed robotic aid that could physically assist with the five key tasks discussed above would reduce the formal or informal care taking burden. 1.2 What will the robots need to be capable of? Future robotic aids assisting disabled elderly people in their homes wi l l need to safely interact with humans [10,11]. The five key tasks of daily living mentioned above all share one important element: they all require physical interaction with the disabled person. A means by which the designer of an assistive robot can be certain the device wi l l be safe is of great importance. Furthermore, such an assistive robot should emulate human manipulation characteristics. Human muscles are extraordinary actuators. People can vary the force and stiffness of most of their joints independently. One class of actuators, namely, the Programmable Passive Compliant (PPC) actuator, has been identified by other robot designers as being promising for this type of activity [12] in terms of safety and stiffness variability. The goal of this work wi l l be to investigate the potential of this class of actuators for the design and control of safe interactive robots. In this work we wi l l use the term Programmable Mechanical Compliant Actuator or P M C actuator. A P M C actuator is one which is mechanically compliant but whose compliance is variable. This differs from feedback-generated compliance in that the compliance at any instant is a mechanical property of the system, independent of sensors, feedback or control. For interaction with humans, mechanical compliance is intrinsically safer [13]. Electrically- and hydraulically-powered robots are not normally designed to be compliant. Although there are techniques such as impedance control which can make such robots appear to the user as i f they were light and compliant, such techniques are limited by the torque range of the actuators and the bandwidth of the controller system [14]. If the robot becomes un-powered, it wi l l revert to a heavy, stiff state. Thus, relying on control alone to introduce compliance is not an intrinsically safe approach [12,13]. Furthermore, such actuators are generally expensive and therefore unsuited for our intended application. A P M C actuated robot wil l embody the physical elements most suitable for safe human interactions. One of the reasons why humans are good at interacting with an unstructured environment is that our muscles are effectively P M C actuators. There are many tasks where 1.3 A i r Muscles 4 precision is secondary to compliance, for example, shaving. This design approach is hypothesized to reduce the computational load when interacting in unstructured tasks [13]. In this work, a P M C actuated robot is designed and a strategy for control of this robot for a human-interaction type task is developed. 1.3 Air Muscles Of the various P M C actuators that have been developed, one of the most interesting and most developed are air muscle actuators. A i r muscles are simple and inexpensive. They have existed under various names, including: McKibben Muscles, A i r Muscles and Rubbertuators, since their initial development in the 1950's [15]. Most past work has focused on using air muscles as a low cost and lightweight replacement for traditional robotic actuators in high precision positioning tasks. However, while the intrinsic compliance of these actuators makes them unsuited to such tasks, these very properties make them ideal for use in an assistive robotic device. The first proposed use for air muscles in the 1960's was in an orthotic device [15]. Since then at least two companies have attempted to commercialize the actuator. First Bridgestone and later Festo. Neither commercial version has seen significant market penetration. The actuators are nonlinear and have not proven suitable for the types of tasks most researchers have proposed. When implemented in an opposed pair, the resulting joint exhibits similar characteristics to human joints such as the elbow or knee. The compliance is variable and independent of position, and the response to perturbations is also similar to human joint-muscle systems. In this work it is proposed that a neuromotor-science based control model would be appropriate for application to an air muscle actuated robot operating in a human environment, specifically in the context of safe interaction with humans. 1.4 EP control Much of the existing literature on intrinsically compliant actuators is related to the problem of accurate position control of a robot powered by such actuators [16, 17]. However, there is a broad class of tasks where high positional accuracy is secondary to dependable programmable compliance. Many day-to-day tasks carried out by humans do not require fine position control, 1.4 EP control 5 and the inherent compliance of a person's limbs enables appropriate interaction forces to be generated during execution of such tasks [18]. Neuromotor scientists have proposed numerous control schemes to explain how animals control their muscles. One method has been useful in describing a wide range of human motor tasks. Equilibrium Point Control was first proposed in the 1960's by Feldman [19] and can serve as a possible model for a controller for air muscles. E P C is a promising approach for controlling P M C assistive robots. 1.4.1 Background Numerous experiments have been conducted to test the E P C model [ 18-21 ]. There is still controversy regarding how appropriate this model is for understanding actual motor control processes. The controversy is largely irrelevant to our interests. The fact that a large number of tests have shown that this model fits experimental data well suggests that i f this control method is used with P M C actuators, humanlike movement should result. EP control suggests that the brain develops a virtual trajectory for a limb to follow based on what it knows about the environment at the time of the formulation of the trajectory [18]. This virtual trajectory is a set of equilibrium joint positions and stiffnesses. These are two independent trajectories. Because of compliance, the limb wi l l not exactly follow the virtual trajectories but instead wi l l follow one that is governed by interactions between the limb and the environment. Although inertial effects and contact disturbances can cause limbs to deviate from the equilibrium trajectory, the spring-like properties of the peripheral neuromuscular system produce appropriate corrective forces in response to these deviations. With practice, the brain can learn to compensate for the inertial, frictional and contact loads experienced in a particular task and can construct feed-forward EP and compliance trajectories suitable for carrying out very complex motions in space. If the details of the achieved trajectory are important, the subject can compute an inverse model to predict the outcome. One important aspect of compliant control is that for most joints the stiffness can also be chosen. This helps to ensure that the trajectory followed is as planned based on what is known about the environment and possible interactions. When walking, a very compliant posture is maintained by most of the body's joints. When we inadvertently trip over something our body is often able 1.5 EP Control and P M C Actuators 6 to find a new stable posture even before we have a chance to respond. Centrally, we can vary the stiffness of the virtual trajectory as needed for our task. Hitting a tennis ball certainly requires a very stiff forearm, while shaving our face does not. In human arms, controlling the level of coactivation of the muscles and altering reflex gains can vary the stiffness of the elbow joint. From one starting point, a new EP and joint stiffness can be chosen substantially independently. Neuromotor researchers have demonstrated that arm movement has a significant feed-forward component that can be represented as an open loop equilibrium point (EP) trajectory followed by the joints when executing motion tasks [20, 21]. A robot controller based on the EP approach is promising both because of the benefits of the intrinsic safety of this approach and because data collected from observation of humans can serve quite directly as control input to a biomimetic manipulator. This independence of joint stiffness and EP is utilized in the design and control of the experimental manipulator developed in this work. 1.5 EP Control and PMC Actuators Several experimental robots have been constructed utilizing McKibben (air muscle) actuators [16, 17, 22-24]. These actuators behave in many respects similarly to human muscles [33]. By constructing robot joints powered by antagonistic pairs of McKibben air muscles, they, like human joints, exhibit adjustable compliance throughout their range of motion [16, 23]. Figure 1.1 shows a joint constructed with opposed pairs of air muscles. 1.5 EP Control and P M C Actuators 7 A i r Muscles External force Pulley Figure 1 . 1 - Rotary Joints Powered by Opposed Pairs of A i r Muscle Actuators Colbrunn [26] developed a method to independently vary the stiffness and position of a rotary joint powered by air muscles. In a joint as shown above when the forces in both muscles of an opposed pair remain balanced, the joint w i l l not move, but its stiffness wi l l increase. Imbalances in the forces of the two muscles in an opposed pair w i l l cause a change in the equilibrium angle of the joint (9EP), the angle where the joint w i l l move to i f no external joint torque is present. Colbrunn exploited the properties of air muscles to develop a walking robot that remained passive throughout most of its range of motion to conserve air pressure in the tank powering his robot. Colbrunn demonstrated that it is possible to decouple the control of stiffness and desired angular position for a single joint powered by two opposed McK ibben air muscle actuators. The decoupling compensator assumes that the joint stiffness and angular position can be decoupled with a pair of constant gains over the complete workspace of the joint. This simple approximation allowed for acceptable results for the purpose for which the muscles were used. The actual input to the muscles was pressure, which was measured and controlled directly. Stiffness and angular position were calculated. Colbrunn successfully demonstrated independent control of both joint angle and joint stiffness with a set of very simple control laws. Joint angle is directly measured by a rotary encoder and the force in each muscle is measured with a force transducer. Colbrunn reported good success with this method. His measure of success was to have the majority of the motion of the leg happen in the passive phase. That is, he set the equilibrium position and allowed the compliance of the actuators to move the leg into the next position. Tonietti and Bicchi [13, 22] demonstrated an alternative solution for the decoupling of joint stiffness and 64. Their model assumes that the inverse model of stiffness and position to 1.5 EP Control and P M C Actuators 7 A i r Muscles 9. 0 EP External force Pulley Figure 1.1 - Rotary Joints Powered by Opposed Pairs of A i r Muscle Actuators Colbrunn [26] developed a method to independently vary the stiffness and position of a rotary joint powered by air muscles. In a joint as shown above when the forces in both muscles of an opposed pair remain balanced, the joint w i l l not move, but its stiffness wi l l increase. Imbalances in the forces of the two muscles in an opposed pair w i l l cause a change in the equilibrium angle of the joint (OEP), the angle where the joint wi l l move to i f no external joint torque is present. Colbrunn exploited the properties of air muscles to develop a walking robot that remained passive throughout most of its range of motion to conserve air pressure in the tank powering his robot. Colbrunn demonstrated that it is possible to decouple the control of stiffness and desired angular position for a single joint powered by two opposed McKibben air muscle actuators. The decoupling compensator assumes that the joint stiffness and angular position can be decoupled with a pair of constant gains over the complete workspace of the joint. This simple approximation allowed for acceptable results for the purpose for which the muscles were used. The actual input to the muscles was pressure, which was measured and controlled directly. Stiffness and angular position were calculated. Colbrunn successfully demonstrated independent control of both joint angle and joint stiffness with a set of very simple control laws. Joint angle is directly measured by a rotary encoder and the force in each muscle is measured with a force transducer. Colbrunn reported good success with this method. His measure of success was to have the majority of the motion of the leg happen in the passive phase. That is, he set the equilibrium position and allowed the compliance of the actuators to move the leg into the next position. Tonietti and Bicchi [13, 22] demonstrated an alternative solution for the decoupling of joint stiffness and 6^. Their model assumes that the inverse model of stiffness and position to 1.6 Scope and Objective 8 pressures can be found. This method is not appropriate for an error-based controller as it can lead to instability between the muscles. In this work, it is proposed to instead find the map between the differential change in stiffness and position to differential change in mass of air in each muscle over the full operating range. This approach is expected to allow stable, compliant control of multiple P M C actuators. This benefit derives from the fact that the mass of air in a muscle is independent of the length of the muscle. 1.5.1 In teraction Tasks A D L tasks required for assistive l iv ing include free space, transition and contact tasks. In this work we wi l l investigate all three tasks as part of experimental testing of the design and control strategies proposed. In particular, the transition from free space motion to contact is a type of task that poses many difficulties for traditional robotic manipulators. For rigid robots, complex techniques for switching between multiple control strategies [27,28] are used to overcome this difficult type of transition. A wiping task is representative of many A D L tasks and requires free-space, transition and contact motion. Thus in the experimental work of this thesis, a wiping task is used as the exemplar motion. Other researchers have explored the possibility o f using mechanically compliant actuators to create robots that are intrinsically compliant [22- 25]. A manipulator that can use a single control strategy to perform free-space motion, contact interaction, and transition interaction tasks would potentially be very desirable for use in human environments. 1.6 Scope and Objective The objective of the work described here is to demonstrate that a programmable mechanical compliant manipulator can be controlled with a simple control strategy based on EP control. The P M C actuators chosen to use in this demonstration are air muscles. Muscles were designed and built in the lab for this project. A n empirical model was developed to allow for the calculation of muscle force from pressure and length. Because no documented method could be found, an algorithm for sizing a pair of muscles and pulley radius for a joint l ike the one shown in Figure 1.1 was developed. 1.7 Outline of Thesis 9 A manipulator was designed and constructed to perform three tests identified as appropriate for testing the hypothesis. This manipulator has two air muscle actuated links. Valves and other electro-mechanical components were purchased, modified or built to allow for a P C to control the manipulator. A n EP inspired controller was developed and implemented to allow for the desired testing. Three sets of experiments, covering free-space, transition and contact tasks were performed and analyzed. The original contributions of this work are: (i) a method was developed to facilitate the design of rotary air muscle driven joints including proper muscle selection, (ii) a decoupling compensator was developed to map error in joint stiffness and joint EP to error in the mass of air in each muscle, (iii) an EP inspired control algorithm was developed, implemented and tested on the robot. 1.7 Outline of Thesis Chapter 1, Introduction - This chapter discusses the motivation for this work, presents air muscles, EP Hypothesis and the notion of programmable mechanical compliance ( P M C ) and provides a discussion of work that has been done by others in the area of controlling McKibben air muscles. Chapter 2, A i r Muscle Design - This chapter describes the design of the air muscles used in this work. The empirical force model used wi l l be described as well as the symmetric joint sizing method. Chapter 3, Electro-Mechanical Design and Control of a P M W Robot - This chapter details the design of the manipulator, valve selection and development of the EP controller. Chapter 4, Experimental Methods - A description of the three experiments conducted to evaluate the capabilities of the manipulator and controller. The three tests are a free-space motion test, a transition from free-space to contact task and a contact task. Chapter 5, Results and Discussion - A presentation and discussion of the results of the three tests described in Chapter 4. Chapter 6, Conclusions and Recommendations. Chapter 2 Air Muscle Design 2.1 Introduction A i r muscles have unique properties that can be exploited to construct a simple, low cost, P M C robotic device. In this chapter, these properties are investigated with a view to reducing the instrumentation necessary for such a device. A s wel l , in the second part of this chapter, the optimization o f air-muscle properties for a specific robotic design is discussed. 2.2 System Overview A i r muscles, a manipulator and supporting hardware were all required before it would be possible to demonstrate an EP inspired controller of a P M C manipulator. The system envisioned for demonstrating the three experimental tasks chosen is diagramed in Figure 2.1 below. The central disk pictured at the base of the manipulator is actually two concentric pulleys stacked vertically. The first pulley is directly attached to link 1 and the second pulley drives link 2 through a toothed belt. The calibrations can be found to translate the output of the potentiometer shown in the figure to give both the current link angles and the muscles lengths. 10 2.2 System Overview 11 Figure 2.1 - Overview of manipulator concept appropriate for demonstrating E P control In addition to the manipulator and muscles, a valve and pressure transducer for each muscle are required. The overall system design for the hardware shown in Figure 2.1 and Figure 2.2 is discussed in Chapter 3. The above-diagramed system has the properties required such that each joint stiffness (k/ and ki) as well as equilibrium angles for each link (OEPI and 0EP2) can each be independently controlled. The relationships between these parameters are discussed in the remainder of this chapter and in Chapter 3. Pressure A i r Regulator Muscle \ \ | Valve A i r Supply Figure 2.2 - Additional components required for powering the air muscle 2.3 A i r Muscle Properties 12 2.3 Air Muscle Properties McKibben muscles principally consist of a nylon braid encasing a latex rubber tube. The nylon braiding can be purchased from electrical supply stores and the rubber tubing was standard surgical natural latex tubing available from medical supply stores. The construction method is described in Appendix A[29]. Plastic net Figure 2.3 - Rendering of a section of air muscle (from Shadow Robot Company) According to the Shadow Robot Company, a 6mm diameter air muscle has the "strength, speed and fine stroke of a finger muscle in a human hand" and "an A i r Muscle 30mm in diameter is capable of lifting more than 70 K g at a pressure of only four bar"[29]. The air muscle exerts its maximum force at maximum extension. As extension decreases, the force that it exerts decreases at a decreasing rate. This means that small changes in force can be achieved by using a larger muscle at an extension below its maximum. The sketch graph below shows the relationship between force and extension for a constant pressure. Force Length Figure 2.4 - Force versus length relationship for an air muscle 2.3 A i r Muscle Properties 13 The air muscle takes advantage of the geometry of its outer shell to generate a contracting force when inflated. The muscle has two main components: an outer shell and an inner bladder. The outer shell is typically made of nylon and the inner bladder of latex or synthetic rubber. One of the air muscles used for this work is shown in Figure 2.5 below. The bladder is required to contain the gas used to power the actuator. The nylon braid converts the pressure in the actuator to tension in the braid, which exerts force in the axial direction. Figure 2.5 - A i r Muscle Actuator The theoretical rest length of an air muscle is equal to the length at which its volume is maximized. In reality, due to end effects the rest length of an unloaded actuator is not quite at the point of maximum volume. For the purposes of this work, it is desirable to reduce the amount of instrumentation required. One way to accomplish this is to avoid the use of force transducers for each muscle. Instead, the pressure and length of each air muscle is used to calculate the force and stiffness of each muscle. The empirical equation that is fit to each muscle is presented in the following sections. 2.3.1 Observed Limitations There is a maximum and minimum force achievable for the air muscles. The maximum force and pressure are physical design limitations particular to the way the muscles are constructed. The minimum pressure is required to keep the bladder inflated and the minimum force is required to avoid large hysteresis. These mechanical limits are: maximum pressure of 7 bar, minimum pressure of 1 bar and a maximum force of 100 N . 2.3.2 Geometric Models of Air Muscles Various groups have modeled air muscles in different ways. A geometric model suggested by Chou and Hannaford [31 ] is the principal model used in this research. The geometric model of 2.3 A i r Muscle Properties 14 the air muscle originally appears in Chou but was modified by Colbrun [26] to a more useful form. His formulation is outlined below. Neglecting the frictional losses, the work done on the system wi l l equal the work extracted from system. P=Absolute internal gas pressure Patm=Atmospheric pressure P g =Gage pressure p i n n e r surface displacement dsi=Area vector J/,=Inner surface displacement dV= Volume Change Chou shows that this ultimately yields Equation 2.2 below. The rest of the formulation can be found in Appendix B. The force generated by a muscle is a function of two geometric properties, b and n, and the internal pressure (Pg) and the length (L) of the muscle. The constant b is equal to the length of the nylon strands in the braid i f they were pulled straight. The constant n is equal to the number o f turns in the helix that makes up the braid. In theory the muscle should have maximum force at its most extreme length (where the maximum possible length is equal to b) and generate no force at the position where the maximum volume is achieved, which can be shown to be when: dWin = J (P - Potm )dli • ds, ={P- Palm ) J dl, • dst =PgdV J Surface J Surface * (2.1) Where: / = (2.2) 3L2/b2 = 1 (2.3) as derived from Equation 2.2. 2.3 A i r Muscle Properties 15 2.3.3 The empirical Modification to above model The geometric force model for McKibben actuators was used to solve for force in each muscle as a function of pressure and current length as shown by [30]. This theoretical model was found to be unsatisfactory for this purpose. A n offset (c) was subtracted to account for end-effects. This near constant offset has been reported by others [26]. Rather than trying to measure the geometric properties b and n, instead the terms b,n,c were empirically fit to data collected for each actuator throughout the pressure, length and force ranges of interest. A least squares fit was used to solve for the values. Am 3Z2 (2-4) The air muscles chosen for this work had the fol lowing physical characteristics: Vi inch nylon braid, !4 inch latex tubing, b = 480mm, and n = 6.8 turns. The calibrated values are approximately in agreement with the geometric values. The values for one of the muscle calibrations were: Z?=501mm, n=5.6782 and c=28.1193. A n example plot of a single muscle calibration is shown below in Figure 2.7. The absolute force error over the range of motion for the wiping task is shown in Figure 2.8. This plot was generated from data collected from the completed manipulator with the stiffness controller, described in Chapter 3, implemented. To calibrate all of the muscles each joint was in turn cycled through the full required range of motion for the experiments (6EP minimum 6EP maximum) with three constant joint stiffnesses (k,) of 15Nm/rad, 22.5Nm/rad and 30Nm/rad. Figure 2.6 below shows the setup used to calibrate the muscles. Force Figure 2.6 - Manipulator configuration for muscle calibration 2.3 A i r Muscle Properties 16 Figure 2.7 - Measured force and calculated force for an air muscle The error in the calculated force exhibits hysteresis as shown in Figure 2.8. The error does not increase linearly with the magnitude of the force. The absolute error is generally less than 2 N throughout the entire muscle operating range of lengths and force. This level of error is considered sufficient for our application. Using calculated force in place of force transducers in our system results in only small errors in force. 2.4 Symmetric sizing method 17 0.34 0.345 0.35 0.355 0.36 0.365 0.37 0.375 Muscle Length (m) Figure 2.8 - Absolute error in force between calculation and measured 2.4 Symmetric sizing method Although quite a few people have built manipulators from air muscle actuators, there is no published description of how one might select the most appropriate muscles for a given task. A s this research is primarily intended to show the benefits of using P M C joints to perform interaction tasks a method for properly choosing joint parameters to satisfy constraints derived from a desired task was developed. Given a joint such as the one shown in Figure 2.9 below, the parameters Lmount, r, b and n can be chosen to yield different available ranges of k and 6EP as well as joint torque (r). Because of the properties of the actuator, the true range of available k and T w i l l vary with the actual angle of the joint (9n). The goal of the method described below is to ensure between the desired limits of 6EP the joint wi l l possess the ability to achieve a prescribed range of k and r. 2.4 Symmetric sizing method 18 Figure 2.9 - Simple rotary joint powered by a pair of air muscles For the hypothetical joint task a required maximum joint torque is known and is defined as x m a x . Also, for simplicity, it is assumed the range of motion is symmetric and known and defined as ± dmax as shown in the above figure. In addition the maximum and minimum joint stiffnesses are symmetric and defined as kmax and kmin There are also several constraints that are relevant: maximum axial force in muscle is a constant across all muscle sizes and defined asfmax. After building several muscles and exposing them to sufficient axial force to initiate failure, this was found to be primarily a limitation on the end fitting. The nylon braiding and tubing can shear i f exposed to excess clamping force. This presented a limit to how much axial force the end fittings could take before coming apart. For different designs this may change but the premise that some maximum force is achievable still holds although the l imiting factor may change. The maximum inflation pressure is assumed to be constant across all muscle sizes and is defined as Pmax. The minimum inflation pressure is a constant across all muscle sizes and defined as Pmin. A lso , the ratio of n/b is bounded above and below based on available braid sizes. Although commercially available nylon braid is only available in discreet steps of n/b, any size between could be custom built in theory. Practically choosing the closest n/b wi l l suffice. It is helpful to see what the force output of an air muscle is versus length for a given pressure. 2.4 Symmetric sizing method 19 Force Pressure Length Figure 2.10 - Force versus length at different pressures Addit ional assumptions for this method include the following. The mounting length o f both muscles is equal when 8 is equal to zero; for example, Lmountj=Lmomt2. The working range of the manipulator is defined by the application and is symmetric about 6=0. The working range is equal to ± 6. Muscle length can never exceed b as this is the length o f a single strand of the braid. Muscle length can never be less than the length where volume is maximum and axial force is zero defined as Lzero. Length b Figure 2 . 1 1 - Useful range for a single muscle The mounting length of the muscle must fall within this region and the working region for the muscle is defined as shown below in Figure 2.12. The width of the working region is equal to Omaxr to allow for the range of motion defined by 6max. 2.4 Symmetric sizing method 20 Lmount Figure 2.12 - Working range of each muscle N o w the maximum and minimum working lengths of each muscle can be defined: A™ =Ln,o«m -A<*" Two constraints must hold at this point: (2.5) (2.6) (2.7) mm — zero (2.8) Examining the torque constraint, one can note that it is most difficult to satisfy this when the joint is configured as shown in Figure 2.13. 2.4 Symmetric sizing method 21 Figure 2.13 - Configuration where maximum and minimum stiffness most constrained In this configuration muscle 1 is at length Lmin and muscle 2 is at length Lmax. Superimposing both muscles onto one force length plot as shown in Figure 2.14 below is helpful to visualize the impact of the torque constraint. Muscle 1 is at point A and muscle 2 at point B. f B L l Lmin L 2 - L m a x Figure 2.14 - Max imum torque Constraint In this configuration the maximum torque that can be generated is defined by: 7 m a x act if A fB ) r Where fA is equal to either (2.9) 2.4 Symmetric sizing method 22 / , = / ( £ m i n , ^ ) i f (2-10) / ( Z m i n ; J P r a M ) < / m a x or (2.11) fA=fma (2-12) and fB=f(Lw,PmiB)- (2.13) T m a x must satisfy the constraint: T >T • (2.14) max _ act — max V / To investigate the joint stiffness constraints, the stiffness relations of the air muscles are established. Rearranging the Equation 2.2, yields: f = Pg3AL2-BPg (2.15) 1 b2 where, A=—-, and B = . Am Ami2 Each muscle volume[30] is calculated as: V = BL-AL3. (2.16) Thus, the change in volume with respect to length is <p = ~ = B-3AL2 (2.17) aL Using equations 2.15-17 yields the solution of muscle stiffness, K , as the change in force with respect to length where mass is held constant as, K = — dL T mR-^-T-ip1 +P 6LA (2.18) V2 where, gauge pressure is related to air mass and volume by, This equation shows that for a given length, the stiffness varies linearly with pressure. 2.4 Symmetric sizing method 23 Stiffness Figure 2.15 - Stiffness versus pressure at a constant length The most difficult configuration to generate a small joint stiffness occurs when attempting to generate a large clockwise torque in the configuration shown in Figure 2.13 above. Figure 2.16 below shows the points on the force length plot for each muscle to achieve minimum joint stiffness, k m i n Muscle 1 is at point C and Muscle 2 at point B. B M L m i n L 2 - L m a x Figure 2.16 - Small stiffness constraint To obtain the lowest possible stiffness at this configuration while applying the largest required torque, both muscles would need to generate the least force possible to apply the maximum torque. 2.4 Symmetric sizing method 24 fc ^ m i n act ~ ^ ( K c + K B ) (2.20) (2.21) k < k min act — min (2.22) The most difficult configuration to generate a large joint stiffness occurs when attempting to generate a large counter-clockwise torque in the configuration shown in Figure 2.13 above. L i L m i n L 2 - L m a x Figure 2.17 - Highest stiffness constraint To obtain the highest possible stiffness at this configuration while applying the largest required counter-clockwise torque, both muscles would need to generate the most force possible to apply the maximum force. This is shown in Figure 2.17 above. /D — fA ^max act ~ ^ ( K A + KD) k > k max act max (2.23) (2.24) (2.25) This above method for sizing air-muscles was programmed into Matlab. The function ' fmincon' , which is an optimization routine that accepts nonlinear constraints, was used to minimize the mounting length subject to the above nonlinear constraints given in Equations 2.5 to 2.25. The MatLab files can be found in Appendix C. For the case where ^ m ^=30Nm/rad, 2.5 Summary 25 &m/„=15Nm/rad, TO T a x=3.1Nm and Omax= n/\6 the fol lowing values were found: 6=514mm, n=5.74, r=90mm and L m o w „ ( =393mm. Unfortunately, there was no material available with an n/b ratio as suggested by the optimization. Muscles with £>=480mm, n=6.S and L m o u „ ,=360mm are predicted by the same equations used in the optimization to yield a joint with: kmin=\S.O, kmax=l>2.% Nm/rad and rmax = 6.15N when used with a r=90mm pulley over the same 6max.. In reality, the working stiffness range was slightly greater than this prediction and was in fact satisfactory for the experiments. 2.5 Summary It was shown that using a simple empirical model of the air muscle force relationship to length and pressure, an accurate force calculation can be made. A l l four muscles used in testing were calibrated using the method outlined in Section 2.3. The errors in using a calculated force rather than measured force are small (typically less than 2N). Addit ionally, a method for solving optimal air muscle parameters for desired P M C joint characteristics was discussed. The relevant constraints that are important when performing this optimization were presented along with a description of the logic behind their importance. Matlab code was developed to allow for choosing muscle parameters to yield minimum mounting length of air muscle in a P M C to minimize the space of the device. Chapter 3 Electro-mechanical design and control of a PMC robot 3.1 Introduction In this chapter, the design of an air muscle actuated two-link planar manipulator is discussed 1. A n air source and suitable valves to inflate and deflate the muscles were required along with a suite of electronics including sensors, valve drivers and the D A Q and computer to implement the EP-controller in an experimental setup. Addit ionally, the controller development is detailed. This chapter wi l l describe the above requirements and the chosen solutions leading to the complete electro-mechanical system. 3.2 Manipulator A two link planar manipulator was chosen as the platform for testing for several reasons. The two-link manipulator is sufficient to allow for control of the stiffness of the endpoint in the This work was done with the assistance of two undergraduate students as part of their fourth year project 26 3.2 Manipulator 27 direction of the surface to be wiped. For the proposed simple wiping task, the 2-degree of freedom manipulator met the necessary requirements for a test platform. Figure 3.1 below shows the final system installed in the Industrial Automation Laboratory at U B C . Figure 3.1 - Manipulator with muscles Table 3.1 lists the specifications and constraints for the design of the manipulator. 3.2 Manipulator 28 Table 3.1 - Design Requirements Dimensions Link length 227mm Pulley diameter =180 mm Forces Maximum Y-axis force - 2 0 N Maximum force on a pulley -200 N Configuration - Angle of the second joint independent of the first - Design must include two encoders - Easy to install on the lab table - The motion must be in the horizontal plan Material A l l custom parts in aluminum or steel The sizing of the manipulator and the choice of appropriate air muscles were inherently linked. The final sizes chosen were eventually derived from a few simple constraints imposed at the beginning of the design process. The manipulator was sized based on the desired workspace and forces. The finalized planar manipulator design is shown in Figure 3.2. The bi l l of materials for the assembled manipulator can be found in Appendix D. As shown in Figure 3.3, the two links are driven from the base of the manipulator. The distal joint is driven from the base through a timing belt and pair of sprockets. This allows the air muscles to be longer than the links and also reduces link mass and complexity of the manipulator. Each joint has a pulley mounted at the base and a pair o f antagonistic P M C actuators. 3.2 Manipulator 29 Gears Figure 3.2 - Finalized manipulator design Figure 3.3 - Close-up of back of manipulator 3.2 Manipulator 30 A l l components were designed in Pro/Engineer. The production drawings are in Appendix E and the assembly procedure is in Appendix F. Figure 3.4 shows a sweep of postures of the manipulator wiping the surface at the prescribed distance of 0.4m. Figure 3.4 - Plot of the range of motion of the manipulator The expected torques at each joint are shown in Figure 3.5 for an end point force of 20N in the Y-ax is direction. 3.3 Valves 31 § 0 O - 2 - 3 - Joint 1 Joint 2 •150 -100 - 5 0 0 x (mm) 50 100 150 Figure 3.5 - Torques for 20N force normal to the wiped surface Encoders mounted on the joints were used for all early development work. For the final experiments the encoders were replaced with single turn potentiometers configured to vary between 0-5 Volts each turn, to integrate with the final (Labview) control platform. 3.3 Valves There are a number of ways that the state of the P M C ' s can be varied. The two basic methods are pressure control and mass flow control. There are several valve choices that could be considered: proportional pressure control valves, proportional mass flow control valves, and solenoid valves. Mass flow control is the preferred method for operating air muscles. Because of their low cost and controllability, solenoid valves were chosen. Unfortunately, solenoid valves only offer one steady state mass flow (on/off). However, advances in solenoid valves have led to very fast solenoid opening times. Therefore, a P W M strategy can be used to vary the average mass flow rate through the valve. A s wi l l be discussed in Section 3.5, a control strategy that varies mass 3.3 Valves 32 flow rate to cause the joints to fol low EP and K trajectories wi l l be developed. The output of this controller is a duty cycle to the valves. Matr ix valves[34] were selected based on price and speed. They produce a 3 position, 3 way solenoid with opening time around 2ms. This allows for a single valve per P M C . Each valve has 3 ports and 3 positions, meaning they can be open to supply, open to vent or closed. In the manipulator setup, the four Matrix solenoid valves operated on the P W M signal. They have a maximum frequency of 200 H z and the minimum time to open of 2 ms. They have three different positions to allow for: an inlet from an air supply to the actuator, an outlet from the actuator to the atmosphere and a closed position where no air is exchanged. For further specifications refer to Appendix B. The following sub-section discusses the sizing of orifice plates for both the inlet and outlet of the valves for effective P W M control of the valves. As wel l , the selection of the operating frequency is described. 3.3.1 Sizing Valve Orifices When dealing with a large pressure drop from the supply to muscle, compressible flow must be considered. This introduces choked flow through the orifice that graduates into subsonic f low as the back pressure increases past the critical values. These relations are useful in sizing the orifice and theoretical mapping of the mass f low rate. The pressure drop over the orifice governs whether the f low is choked. The critical back pressure to stagnant pressure ratio is: This ratio value is specific for air. Here, p* is the critical back pressure at which the f low becomes sonic. The stagnation pressure , p 0 , is the pressure of the air with no velocity. For any back pressure lower than the critical pressure, the flow through the orifice is choked. Under these conditions, the mass f low rate is independent of the back pressure. ^ - = 0.5283 Po (3.1) Wl m a x — 0 .6847 P o A e (RT0f2 (3.2) 3.3 Valves 3 3 It is assumed that the supply air is at room temperature and is stagnant. A e is the area of the orifice and R is the gas constant. When the back pressure has increased such that the flow is subsonic, the calculations are more complex. At subsonic conditions, the back pressure is equal to the pressure in the orifice. Now the mass flow rate is dependent on the back pressure as illustrated in the equation below, For deflation of the muscles, the same theory applies where it is assumed that the air in the muscle is stagnant and and the back pressure is atmospheric pressure. Instead of having a changing back pressure, the supply pressure is changing. (3 .3) Inlet Solenoid Orifice Plate Manifold Exhaust To muscle Supply Figure 3 .6 - Partially disassembled Matrix valve with orifice plate The Matrix valves were not able to deliver exactly the performance required without modification. A n orifice plate was added to both the inlet and outlet side of the valves to lower 3.3 Valves 34 the maximum flow rate through the valve. The testing method for selecting the orifice is described in the Appendix E. The results of these tests show that the best inflation orifice hole diameter is 0.508mm while the best deflation orifice hole diameter is 0.787mm. 25 0 2 4 6 8 10 12 Time (s) Figure 3.7 - Inlet orifice sized to allow no more than I N discreet force steps for smallest possible inflation With the chosen inlet orifice size it is shown in the above figure that the maximum change in normal force to the surface is 1 N per injection of gas into the muscle. The graph in Figure 3.7 was generated from data with the manipulator in the orientation where the force normal to the surface is most sensitive to actuator changes. The force steps show small ringing due to the manipulator joints being underdamped. The outlet was also tested to ensure that the f low in and out of the actuator was roughly balanced. The f i l l and deflate time are 1.8s and 1.75s respectively. This balance is considered 3.3 Valves 35 satisfactory, given in-house machining capabilities, and precision machining of the orifices was not considered necessary. Figure 3.8 shows a plot of the inflation and deflation pressure versus time of a muscle with the modified valves installed. 6 OH 1 1 1 1 1 1 1 0 2 4 6 8 10 12 14 Time (s) Figure 3.8 - Final inlet and outlet orifice sizes with inflation and deflation times roughly matched 3.3.2 Selecting Constant Frequency for operation It was desired to have the air f low into the muscles appear as close to infinitely variable as possible. However, this results in a trade off between frequency and range of useful duty cycles available. The valves had a minimum time to open of approximately 2 ms. The controller is designed in such a way that any commanded duty cycle that results in an open command of less 3.4 Instrumentation, Drivers and D A Q 36 than 2 ms is held for 2 ms regardless. This is the minimum command time for this valve. A frequency of 500 H z would result in two flows being allowable, either closed or open for the minimum pulse. In reality the valves have a maximum recommended operating frequency of 200 Hz . Instead at least a 10:1 turndown ratio for the valve was selected. The quantity of gas released with a 3ms pulse is roughly 10 times less than the full open value. The operating frequency was set at 30 H z to give a 10:1 ratio between maximum and minimum continuous flow. The flow is variable in very small increments between these limits. The counter/timer chip driving the P W M signal is capable of 0.4 microsecond steps. 3.4 Instrumentation, Drivers and DAQ ORTS[31], a U B C developed real-time operating system, was used to run all early testing. National Instruments hardware and Labview software was used instead for the final experiments due to the added flexibil ity o f that package and the wealth of examples and support available. A l l of the sensors were calibrated for their expected operating range before proceeding from this point. The sensor information and calibrations are listed in Appendix D. The values from the calibrations were entered into National Instruments Measurement Explorer for use in all Labview code used in this research. The sensors were recalibrated as required throughout the experiments. 3.5 Controller Using the force model given in Equation 2.4, a controller was developed to allow for the simultaneous control of both joint stiffness and equilibrium position of each joint in the robot. The controller used for the manipulator is shown in Figure 3.8. In the experiments that follow, the Cartesian space trajectories are pre-computed and converted to joint space trajectories before motion begins. The controller operates at 30 H z and each control decision is based on digitally filtered data collected at 300 Hz . 3.5 Controller 37 EPd Cartesian to joint space stiffness Equation 3.6 -K, Inverse Kinematics Equations 3.4 -9. epd -K AO. ep -Pressure, 0 -Decouple Equation -Am*- PI - D C * - Plant 3.8 Computations Equations: 3.11 &3.12 Figure 3.9 - Schematic of planar robot controller The desired Cartesian equilibrium trajectory, Xgpd, is first converted to a joint space trajectory using the inverse kinematics of the manipulator^ 5]. 0'2 = tan - 1 (±V l - J D 2 ) Where D is given by (3.4a) 2 , 2 2 2 ^ x +y -a , —a, D = cos0 2 = ^ ! 2-2axa2 Where aj is the link length. Yie ld ing: (3.4b) 3= t a n " 1 t a n a2 sin02 ax + a2 cos# 2 (3.4c) and 02 — 6X + 02 (3.4d) Next, the stiffness is transformed from Cartesian to joint space. The Cartesian stiffness matrix, Kc is defined as K. (3.5) Cartesian stiffness and position equilibrium point trajectories are generated and converted to joint space with inverse kinematics. ky is prescribed and kx is solved to satisfy ki=k2. This 3.5 Controller 38 constraint minimizes the amount of gas used over the prescribed task. kxy=kyx is solved such that the cross terms in the joint space stiffness matrix are zero, reflecting the physical nature o f the system. K = kx 0 0 fc, = JTKCJ [32] (3.6) Where the manipulator Jacobian, J , is: J - a{ sin(0,) - a2 sin(0 2) a, cos(0!) a2 cos(0 2) (3.7) The singular positions of the Jacobian are outside the task workspace. A s shown in Figure 3.9, the errors in stiffness, K, and equilibrium position, 9ep, along with the most recent observation of the angular position and pressures is fed into a decoupling block. The decoupler uses the partial derivatives of stiffness and theta with respect to mass to transform from stiffness and EP to error in mass for each muscle. Am, A m , dk dk dm, dm2 dd„„ de eq eq dm, dm-. Ak A 0 EP In order to obtain these derivatives, one can note that torque in each joint is given by: T = k(0-9EP), and can also be represented by, T = r{fl-fl) The joint stiffness is: k = r2(Kx-K2), (3.8) (3.9) (3.10) (3.11) Solving Equation 3.9 for 0 £ / > and substituting from Equations 3.10 and 3.11 yields, 3.5 Controller 39 a —a (/i fi) V E P - 0 —7—;—\ • r [KX+K2) (3.12) The partial derivatives required for the decoupler are then: dk dm. • = r (3.13) fd_K^ (3.14) de EP ML dm. (fx-A) 3*. dmx r(icx + K2) r(Kx + K2 )2 dmx ' (3.15) dd dm. if-fi) dm2 r(Kx + K2) r(Kx + K2 f dm2 (3.16) Equation (2.4) yields: f = Pv3AL2-BPV-c. (3.17) The equation derived for joint stiffness, X " i n Section 2.4 is still val id even with the constant c in the above equation. K = ^- =mR^(b2 +P6LA. dL V1 (3.18) For each muscle the change in muscle stiffness with respect to mass is * K _ R T 3A2L*+B7 dm ~ L2(-B + AL2)2 and the change in force with respect to mass is (3.19) df ^(-B + 7>AL2) — = -RT- '-jm L(- B + AL2)' (3.20) 3.6 Summary 40 The muscle mass errors for each muscle then enter the PID block and a resulting duty cycle input to the valves is generated. The PI controller was tuned using a Ziegler-Nichols technique on the actual hardware. The manipulator was given constant stiffness trajectory for each joint o f 15Nm/rad and a step input for desired joint equilibrium position, dspd. The integral term was set to zero and the proportional gain was increased until continuous oscillations were observed. The gains were then solved according to the Ziegler-Nichols method. The gains were set to P=0.25 and 7=0.05. The duty cycle (DC) in the valve controller is then updated and the airflow in and out of the valves varies accordingly. A positive output from the PI controller demands in airflow into the valves and a negative output from the PI controller demands exhaust of air from the actuator. Sensors measure the angular positions and muscle pressures. The sensor data is fed back to the decoupler and forward into the calculation block. The calculation block solves the current actual stiffness of each joint and the current actual EP of each joint using Equations 3.11 and 3.12. 3.6 Summary In this section the steps required to ready all electrical and mechanical hardware for our experiments was presented. A manipulator was designed and built to satisfy some general design constraints introduced to ensure the final manipulator would be appropriate for desired testing. The manipulator designed was a planar 2 link robot powered by two pairs of antagonistically mounted air muscles. Solenoid valves were chosen to control air f low to the air muscles. The valves were chosen for their speed and suitability for use with P W M control. Orif ice plates were sized and added to the modified valves to reduce the maximum flow rate through the valves. A n operating frequency of 30 H z was chosen to run the P W M controller. A set of equations to allow for the decoupling o f joint stiffness and joint equilibrium angle were developed. This decoupler converts errors in these variables to error values for the quantity of mass of air in each air muscle. 3.6 Summary 41 A n EP controller was developed and implemented in Labview. A simple PI control loop was used to control the mass of air in and out of the air muscles. The gains were tuned using the Ziegler-Nichols method. Chapter 4 Experimental Methods 4.1 Introduction > The experimentation described in this chapter was designed to show the strengths and weaknesses of the EP controller, coupled with the air-muscle actuated robot, in free-space, contact and transition tasks. Three sets of experiments were performed, one for each type of task. The experiments are summarized in Table 4.1. The desired trajectory in Cartesian space for X - Y position and stiffness in the Y-direction used for each test was calculated offline. The joint space stiffnesses and equilibrium positions were then calculated and stored in a binary file that was loaded as required for the actual experiments. 42 r 4.1 Introduction 43 Table 4.1 - Summary of Testing Test Type Number of Tests Stiffness Speeds Other Variants Free-Space 10 1200N/m 10 speeds: 15-150 mm/s None Contact 46 800N/m l lOON/m 1400N/m 15 mm/s 30 mm/s 75 mm/s Bump/No Bump yEP = 405,410,415mm (nominal) Transition 9 1000N/m 2 mm/s 5 mm/s 10 mm/s 3 approach angles: 30,60,90° A complete list of all testing performed is provided in Appendix H. Figure 4.1 below shows the location of the X and Y-axis on the manipulator. A l l measurements given in this and subsequent chapters are referenced from this origin. The arrows show the positive directions of these two axes. Figure 4.1 - X and Y axis origin location For each experiment, data acquisition was performed using the same 16-bit D A Q card as used for the controller and streamed to a binary file at 30 Hz . Data was collected from all pressure sensors as well as a force sensor mounted to the wall used in two of the sets of tests. The complete data sets are listed in Appendix H and the analysis of the data is in Chapter 5. 4.2 Free-Space Testing 44 4.2 Free-Space Testing The free-space tests were performed to evaluate the ability of the controller to fol low prescribed non-contact trajectories throughout a range of velocities. The response of the manipulator to increased operating velocity was used to determine the velocities used in subsequent test modes. In this experiment, the equilibrium trajectory of the end effector for a non-contact task is expected to match the actual trajectory with error increasing with velocity due to inertial effects that are unaccounted for in the open loop EP trajectory. 4.2.1 Description of the test The manipulator was run back and forth along a 150mm, straight-line end point trajectory with a constant Y-axis position of 400mm. The starting X-axis position was at x=75mm and with a turnaround point at x=-75mm. A constant velocity trajectory with instantaneous start/stop and instantaneous change in turnaround velocity was commanded to present a worst-case scenario for each velocity profile. Ten profiles between 15mm/second up to 150mm/ second were tested. Figure 4.2 shows the free-space trajectory. The stiffness along the Y-ax is , ky< was set to 1200 N/m for all velocities. / 150 mm Figure 4.2 - Diagram of the range of motion during the free space task 4.3 Transition Testing 45 4.2.2 Experimental measurements The purpose of this test was to determine the range of useful operating velocities for the manipulator. This range is limited by the size of the muscles and the speed at which the valves can f i l l them. A t some commanded velocity the manipulator wi l l cease to be able to converge to the trajectory that it was ordered to follow. The highest velocity the manipulator can follow and still converge to the desired EP trajectory in the 150mm straight-line motion was established by this test and documented in Section 5.2. 4.3 Transition Testing The second set o f experiments was used to observe the system response when transitioning from free space to contact. The same surface from the contact tests (shown later) was also used for this set of experiments. When transitioning from free-space motion to contact motion, industrial robots typically require a change in controller. Making this switch requires sensing the moment of contact and stable methods to switch smoothly from one controller to another. EP control should require no switching of controllers. The transition should be smooth due to the compliance of the manipulator and the nature of the control scheme. 4.3.1 Description of the test A single Y-ax is stiffness value of 1000 N/m was.chosen for all of the tests; this value is in the middle of the manipulator's available stiffness range in the test configuration. Straight-line path velocities of 2, 5, 10, 20 and 40mm/s were evaluated. These speeds were chosen based on observations o f the behavior from the free-space tests. Three different angles of attack into the surface were tested: 30°, 60° and 90°. The wall was placed 400mm away from the origin along the Y-axis . A s shown below in Figure 4.3. Each path is 20mm in length for all velocities and angles with 10mm of travel before contacting the wall and 10mm after making contact. The point of contact for all tests was at the point where the X-axis crosses the surface. 4.3 Transition Testing 46 x = Omm 90<* 60° Surface \ r, , i y = 400mm \ Figure 4.3 - Diagram of transition task Table 4.2 below is a list of the test numbers for the different combinations of velocity and approach angles investigated in this set of tests. Table 4.2 - List of transition test numbers Velocity Angle 2mm/s 5mm/s lOmm/s 30° 1 4 7 60° 2 5 8 90° 3 6 9 4.3.2 Experimental measurements The behavior during the transition from free-space to contact should be stable and the manipulator should remain controllable. The forces generated should agree within some percentage of the expected forces based on commanded end-point stiffness and E P . The results of this experiment are discussed in Section 5.3. 4.4 Contact Testing 47 4.4 Contact Testing These experiments were designed to evaluate the forces generated normal to a surface while wiping with a prescribed stiffness and equilibrium position. The manipulator end effector was in contact with a surface throughout the task duration. The experiments were performed with and without an unpredicted "bump" disturbance along the surface. The forces generated due to the contact with the surface should be predictable from the trajectory and the location of the surface. It is expected that the normal force in the surface wi l l be bounded and the behavior of the controller stable and predictable for the contact testing. 4.4.1 Description of the test The surface was placed 400mm in front of the manipulator. A force transducer was used to record the force normal to the surface. The wiped length had the same position and length as the free space trajectory. A l l tests followed a left to right motion and back again. There was a 3 second pause after data acquisition began at the beginning and at the turnaround point of the wipe. The tests were performed with three different Y-axis (ky) stiffness levels: 800N/m, 1 lOON/m and 1400N/m. The equilibrium path was chosen to maintain a constant force in the absence of a disturbance, The EP path was calculated to compensate for expected deflection in the X-axis and the resulting effect this has on the normal force in the Y-axis . Early tests showed the surface deflected slightly due to its Figure 4.4 - Diagram of contact test 4.4 Contact Testing 48 compliance. This was modeled as a varying contact stiffness across the surface and the EP trajectory (nominally labeled &syEp= 405, 410 and 415 mm) was adjusted accordingly (stiffer On the side where the surface was attached to the force transducer and less stiff moving in the positive x direction). Three velocities were chosen based on results from the free-space tests as listed in Table 4.3. Figure 4.5 below shows the logic behind the adjustments made to the XEP trajectories to account for both the shift from the contact and from the compliance of the wal l . The dashed line indicates the nominal ygp; this is the EP trajectory that would yield the desired force i f the wal l was infinitely stiff and the principle directions of the Cartesian stiffness matrix of the manipulator were perfectly aligned (normal and perpendicular) to the wal l (k^ =0). In fact at the one point in the trajectory, jc=0mm, the cross coupling term (kxy) is zero, and then the nominal trajectory yields the desired force with a stiff wal l . Points A and C show how the EP trajectory has to be varied to achieve the same force over the surface. A t point A , the cross term shifts the end-point farther left and less force is generated than expected. The point AEP represents the direction in which the trajectory must be corrected to counter this effect. The point A'EP goes further to show how the EP trajectory must be adjusted deeper into the surface to achieve the desired force due to the compliance of the wall . Point B and C show the result of this method at the middle and positive end of the trajectory. Figure 4.5 - Diagram of EP adjustments 4.4 Contact Testing 49 The tests were repeated with a 50mm long, 4mm "bump" present in the center of the wiped path. The results with the bump were also captured. To assess the repeatability of the measurements and testing several points, namely, the 30 mm/s - 410mmy£'p points for each stiffness were repeated several times as listed in the second data row of Table 4.3. This table shows the numeric designations of all of the contact tests performed. Figure 4.6 below is a photograph of the manipulator contacting the smooth wiped surface and Figure 4.7 is a photo of the same test with the bump present. Table 4.3 - List of all contact test numbers 15 mm/s 30 mm/s 75 mm/s y E P 800 N/m 1400 N/m 800 N/m 1100 N/m 1400 N/m 800 N/m 1400 N/m 405 mm 410 mm 415 mm 1 3 2 4 5 10 15 6,7,8 11,12,13 16,17,18 9 14 19 20 22 21 23 - v.. IS -: f^jffljS! Figure 4.6 - Contact test with a smooth wall 4.5 Summary 50 Figure 4.7 - Contact task with a bump present 4.4.2 Experimental measurements The deviation of the measured forces generated due to contact with the surface from the expected forces based on the commanded end-point stiffness and EP are obtained from this experiment and the results are discussed in Section 5.4. 4.5 Summary Three sets of tests were devised to test the performance of the manipulator in different tasks. Free-space, contact and transition tests were created to assess the capabilities of the manipulator in each of these three modes of operation. The effect of velocity, stiffness and EP as observed in these experiments are discussed in the following chapter. Chapter 5 Results and Discussion 5.1 Introduction The follow sections present the results of the three sets of tests outlined in Sections 4.2-4.4 The key results from each set of tests are presented. Summary data is presented where relevant. The statistic Mean Absolute Error (MAE) is used as a measure of the deviation from the desired value whenever error for a data set is discussed. \measuredl -predicted[\+\measured1-predicted^... + \measured„ -predicted^ ^ ^ n Mean absolute error is the average of the difference between predicted and actual value in all test cases; it is the average prediction error. This statistic is appropriate for data that is not normally distributed as in this case, unlike Root Mean Squared Error (RMSE) which magnifies the effect of outlying data. 51 5.2 Free-space task results 52 5.2 Free-space task results Figure 5.1 shows the x component of the desired equilibrium point trajectory (xspd) for different x-direction Cartesian velocities (vx). The y component (y>EPd ) is equal to a constant value of 400mm for the entire trajectory. Since there is no surface for the manipulator to interact with, in this test the actual position of the end-point, X„ (where Xa=[xa yn]) should be close to the commanded XEP (where XEP=[XEP ysp]) for slow movements and diverge as dynamic effects create joint torques. -80 1 1 1 1 1 0 5 10 15 20 Time (s) Figure 5.1 - Commanded position vs. time for free-space tests (thick lines are desired and thin lines are measured) For slower motions it can be observed in Figure 5.2 that the error in X-axis equilibrium position, XEP, is small and increases as the velocity of the manipulator endpoint X-axis velocity ,vX} increases. The change in commanded trajectory direction requires an instantaneous change in velocity from positive to negative, and would be expected to generate controller error. The start and turn-around points do, in fact, have the largest errors. For vx = 15mm/s and 30mm/s trajectories, the error converges as the manipulator has time to correct. The v^=50mm/s case 5.2 Free-space task results 53 takes nearly then entire length of the surface to approach zero error. The 150mm/s case does not converge. I 15 10 - 5 x a g - 1 0 -15 - 2 0 r \ v.. -— v = 15mm/s X v =30mm/s X _ . v =50mm/s _ . v =150mm/s -80 - 6 0 - 4 0 - 2 0 0 20 40 60 80 x (mm) Figure 5.2 - Error in xEp versus x for free-space tests If there were no (or small) dynamic effects, (i.e. for the low velocity experiments) the X-axis equilibrium position, xEp, and X-axis actual position, xa, position would be expected to be near coincidental. The error in x in Figure 5.3 shown below, indicates that the results do not follow exactly as predicted. The error in x does increase as the velocity increases but is greater than the XEP error in all cases. There is also a hysteretic effect evident in the figure. Non-zero errors in torque are computed by the system over the motion cycle as the muscles switch from inflation to deflation and the friction force between the muscle braiding and the tubing switches direction. These small errors in computed torques in the controller result in XEP varying from the xa over the trajectory. Implementation of the controller using force transducers would be expected to remove the hysteresis. However, since the proposed tasks for this robot are not position precision sensitive (stiffness/force behavior is considered primary), once identified, this small hysteresis was not considered problematic. 5.2 Free-space task results 54 x l x o 25 20 15 10 5 0 - 5 -10 -15 -20 -25 — v =15mm/s X — v =30mm/s X . v =50mm/s X _ v =150mm/s -80 - 6 0 - 4 0 - 2 0 0 20 x (mm) 40 60 80 Figure 5.3 - Error in xa vs. x The set point for Y-ax is stiffness, ky, in Figure 5.4 below was 1200 N/m throughout the range of the motion. The error in k y increases as the velocity of the endpoint increases. The trend is similar to the error in X-ax is equilibrium position, XEP- Again for the case where X-axis velocity, vx is 150mm/s, the value for Y-ax is end-point stiffness, ky, does not converge to the set point over the duration of the test. 5.2 Free-space task results 55 100 Figure 5.4 - Error in ky versus x The summary data presented in Figure 5.5 and Figure 5.6 shows that the error increases as the velocity increases. The speed of the valves is the principal l imiting factor at higher speeds. Were the time to open and close the valve faster, the orifice size could be chosen such that the manipulator was faster i f this were required. The commanded duty cycle to the valves becomes fully saturated with a commanded path velocity of 150mm/s. 5.2 Free-space task results 56 140 120 §1 100[ o & 3 < c a 80 60 40 20 0 0 50 100 Velocity (mm/s) 150 Figure 5.5 - Mean Absolute Error of ky versus vx o fc w cu •i—» _ 3 "3 x> < cu 16 14 12 10 8 6 4 2 0 O x M A E C O c > + 4H++* + h ' 50 100 Velocity (mm/s) 150 Figure 5.6 - Mean absolute error xEp versus v x 5.3 Transition Results 57 5.3 Transition Results The second set of tests demonstrated the transition of the end-effector from free space to contact task. For the range of velocities and angles of approach tested there was little difference in the behavior of the system. A l l cases behaved as expected. Appendix I contains all test results. In this section only the two most extreme cases are discussed as most of the test results are quite similar. Test #1 was chosen as the case used to present comprehensive example results. Figure 5.7 shows the result of following a 30° angle of approach trajectory through the center of the workspace. The commanded velocity was 10 mm/s along the path. The ky stiffness was set to 1000 N/m. Without a wall present the commanded Cartesian end-point trajectory, y g P , and the actual Cartesian end-point >>a (measured) trajectories overlap very well. When the wall was put in place and the manipulator transitions from free-space to contact the results diverge as expected. The 1mm drift into the surface is a result of the non-infinite stiffness of the wall and manipulator. 410: . . 1 1 408-406-Path (mm) Figure 5.7 - Transition test #1 ya and y E p d with and without the wall versus path 5.3 Transition Results 58 Figure 5.8 shows the values of Y-axis equilibrium point trajectory, yEp, both with and without the wall, as well as the desired Y-axis equilibrium point trajectory, yEpd- Both agree very well with the desired trajectory. The presence of the wall does not interfere with following the yEpd Y-axis equilibrium point trajectory. The distinction between yEpd and ya allows the end-point to smoothly transition between free-space and contact with no alteration to the controller. Ultimately, to control the interaction forces, the Y-axis end-point stiffness, ky, must also be programmed to a desired value. 410: • • 408 -406 - -Path (mm) Figure 5.8 - Transition test #1 yepa and ynpa with and without the wall versus path The Y-axis end-point stiffness, ky, behaves somewhat differently than expected. Because the Cartesian end point stiffness, Kc is translated into joint space stiffness, Kj before the task begins, there is no opportunity to adjust for the change in the configuration of the manipulator. The transform from Cartesian to Joint stiffness uses the manipulator Jacobian, which is configuration dependant. The effect of the manipulator not truly being on the EP configuration leads to the Cartesian stiffness at the end point being different than desired. In fact the errors observed in the individual joint stiffnesses, k/ and were very small, showing the error in ky was due to a difference in configuration from planning to execution. For the contact testing experiments 5.3 Transition Results 59 discussed in the following section, this difference was compensated as explained in Section 4.4. Figure 5.9 below shows this resulting shift from interaction with the wall in test #1. 1200r 1150-1100-1050-? 1000 ' 950-900-850-800 L 0 Actual no wall Actual wall Desired 10 Path (mm) 15 20 Figure 5.9 - Transition test #1 kya and kyd with and without the wall versus path The kya trajectory varied slightly between tests. It was most different in test #9 (2mm/s, 30°) as shown below in Figure 5.10. The small errors in position coupled with increased velocity created the largest ky errors. 5.3 Transition Results 60 Actual no wall Actual wall Desired 10 Path (mm) 20 Figure 5.10 - Transition test #9 kya and kyd with and without the wall versus trajectory The observed force normal to the wall follows the predicted value within I N based on the commanded endpoint stiffness and commanded end point position. There is little effect from the X-axis end-point stiffness term, kx, for these tests because the manipulator is very close to the Y -axis for the entire trajectory. The resulting force into the surface is shown below in Figure 5.11. The expected normal force is 5 N when the manipulator is resting on the surface with an Y-ax is equilibrium position 5mm into the surface and Y-ax is end-point stiffness of 1000 N/m. 5.3 Transition Results 61 CD O S-l O 0 10 Path (mm) Figure 5.11 - Transition test #1 actual and predicted force with and without the wal l versus trajectory Table 5.1 below shows a complete summary of the Mean Absolute Error for the different variables presented in the above plots for all of the transition tests. Table 5.1 - Summary of Mean Absolute Error for the transition testing " ^ ^ ^ Test 1 2 3 4 5 6 7 8 9 v=2 v=2 v=2 v=5 v=5 v=5 v=10 v=10 v=10 Error Type ^ ^ ^ ^ 0=30 0=60 0=90 0=30 0=60 0=90 0=30 0=60 0=90 ysp M A E no wall (m) 0.10 0.11 0.15 0.13 4.14 0.18 0.20 0.20 0.35 kv M A E no wall (N/m) 7.09 9.01 9.70 7.75 17.64 11.48 7.33 12.50 26.90 ysp M A E wall (m) 0.10 0.12 0.13 0.15 0.17 0.15 0.18 0.19 0.20 kv M A E wall (N/m) 18.60 35.15 41.60 19.77 35.19 41.46 18.17 32.33 52.84 F M A E wall (N) 0.56 0.80 0.93 0.56 0.91 0.95 0.58 0.89 0.94 Testing was performed using three different approach angles. No significant differences in the response of the manipulator were seen for the different approach angles to the wall . The manipulator performed the transition task well for all velocities tested. The highest velocity of 5.4 Contact Results 62 lOmm/s did result in larger deviations from the predicted force but these were not considered to be significant in comparison to the predicted force value (i.e. less than <10%). 5.4 Contact Results The third set of tests executed was a contact task. The manipulator was commanded to wipe a surface with a variety of position E P and stiffness trajectories. The trajectories were generated such that the ky and the force into the surface should remain equal over the full surface. A s discussed in Chapter 4, forces resulting from displacement in X were countered by adjusting the position EP trajectory to compensate. A l l trajectories were run against a surface mounted to a force transducer. The resulting force into the surface was measured. The tests were run once against the unmodified flat smooth surface and then again with a 4mm smooth bump in the middle of the surface. The test conditions in Table 4.3 are reproduced below for convenience. Test #10 (vx =30mm/s, ygp =405mm, ky =1 lOON/m) was chosen for example results. Detailed results for all of the 23 test points with and without the bump are presented in Appendix J . The Y-axis equilibrium position, yEp, is the more important component of the equilibrium position vector, XEP, for ensuring forces due to unexpected position disturbances are as predicted by the EP and stiffness trajectories. Figure 5.12 shows good agreement between the actual and desired Y-axis end-point position, yEpa and yEpd- The value of Y-ax is actual position, ya, is offset as expected, due to the presence of the wal l between the manipulator, from the Y-ax is equilibrium position, yEp. Table 4.3 - List of all contact test numbers 15 mm/s 30 mm/s 75 mm/s yep 800 N/m 1100 N/m 1400 N/m 800 N/m 1100 N/m 1400 N/m 800 N/m 1100 N/m 1400 N/m 405 mm 410 mm 415 mm 1 3 2 4 5 10 15 6,7,8 11,12,13 16,17,18 9 14 19 20 22 21 23 5.4 Contact Results 63 Figure 5.12 - Test #10 without bump (y versus x) Again adding a bump does not make the task of tracking the Y-ax is equilibrium position, yEp particularly more difficult as shown in Figure 5.13, only the Y-ax is actual position, ya, deviates due to the presence of the wall . The 4mm bump is well within the capabilities of the controller's capability for rejection. 5.4 Contact Results 64 100 Figure 5.13 - Test #10 with bump y versus x It is also important that the Y-ax is end-point stiffness, ky, tracks close to the desired value for the forces from the contact to result close to the desired values. Figure 5.14 shows that the ky values drift around somewhat, but are generally close to the desired value. Figure 5.15 shows that this remains true when the bump is present as well . 5.4 Contact Results 65 1600 1500 1400 1300 B 1200 w * 1100 1000 r 900 800 700 -100 Actual Desired - 5 0 0 x (m) 50 100 Figure 5.14 - Test #10 without bump ky versus x 1600r 1500-1400-1300-1200-1100 -1000-900-800-700--100 Figure 5.15 - Test#10 with bump ky versus x 5.4 Contact Results 66 Figure 5.16 and Figure 5.17 show the force response from contact with the wall and without the bump present. These results are again for the case where Y-axis end-point stiffness, ky, was set to 1100 N/m and Y-ax is equilibrium position, ygp, to 405mm, but are representative of the general behavior for all cases tested. The behavior generally followed closely to the predicted response and did not cause instabilities in the system. 30 — Actual - Predicted 25 20 10 5 0 1 --100 -50 0 x (mm) 50 100 Figure 5.16 - Test#10 force versus x without bump 5.4 Contact Results 67 Figure 5.17 - Test#10 force versus x with bump Table 5.2 and Table 5.3 detail the M A E values for all of the test cases with and without the bump present. The error is given as both the M A E and also the M A E relative the expected value. In the case of Y-ax is equilibrium position the error is given relative to the requested depth of contact into the surface. 5.4 Contact Results 68 Table 5.2 - Summary of Errors without the bump present yEP M A E kv MAE F M A E Test# (mm) rel .% (N/m) rel. % (N) rel.% 1 0.12 2.44% 18.49 2.31% - 0.50 12.41% 2 0.14 0.93% 25.21 3.15% 1.72 14.31% 3 0.09 1.85% 97.88 6.99% 1.35 19.34% 4 0.10 0.69% 87.95 6.28% 0.67 3.18% 5 0.14 2.78% 18.41 2.30% 0.59 14.78% 6 0.16 1.63% 17.49 2.19% 0.89 11.14% 7 0.16 1.63% 16.93 2.12% 0.97 12.18% 8 0.17 1.67% 17.40 2.18% 0.97 12.07% 9 0.27 1.78% 22.38 2.80% 1.72 14.36% 10 0.14 2.77% 23.78 2.16% 0.37 6.67% 11 0.17 1.70% 26.91 2.45% 0.77 6.97% 12 0.16 1.57% 27.45 2.50% 0.88 8.04% 13 0.16 1.60% 27.02 2.46% 0.87 7.94% 14 0.21 1.41% 39.44 3.59% 1.46 8.87% 15 0.12 2.46% 35.66 3.24% 0.55 10.09% 16 0.17 1.69% 40.56 2.90% 0.55 3.96% 17 0.16 1.58% 40.72 2.91% 0.68 4.88% 18 0.16 1.64% 41.62 2.97% 0.68 4.86% 19 0.21 1.40% 67.31 4.81% 1.10 5.24% 20 0.25 5.02% 19.70 2.46% 0.64 16.05% 21 0.42 2.79% 25.23 3.15% 1.49 12.42% 22 0.22 4.36% 43.92 3.14% 0.69 9.91% 23 0.39 2.62% 66.18 4.73% 1.15 5.50% Without the bump present, the errors seen during the contact task are very small. The error in Y -axis equilibrium position, tends to be less than 5% regardless o f the speed, depth or stiffness. The relative error in Y-ax is stiffness is also generally less than 5%. The error in force is the largest relative to the expected value. This error was seen to be as large as 16.05%. Test case #20 {yEp=A05, ft/=800N/m and vx=75mm/s) generated the largest error in force. 5.4 Contact Results 69 Table 5.3 - Summary of Errors with the bump present yEP M A E kv M A E F M A E Test# (mm) rel .% (N/m) rel. % (N) rel .% 1 0.16 3.23% 15.75 1.97% 0.50 12.54% 2 0.14 0.95% 19.05 2.38% 1.15 9.62% 3 0.09 1.87% 20.03 1.43% 1.00 14.28% 4 0.09 0.61% 82.52 5.89% 1.52 7.22% 5 0.14 2.89% 15.55 1.94% 0.60 14.91% 6 0.22 2.21% 14.19 1.77% 0.83 10.40% 7 0.18 1.83% 14.40 1.80% 0.81 10.11% 8 0.17 1.70% 14.37 1.80% 0.84 10.55% 9 0.22 1.44% 18.93 2.37% 1.32 11.03% 10 0.13 2.62% 20.11 1.83% 0.60 10.93% 11 0.21 2.07% 23.74 2.16% 0.80 7.28% 12 0.18 1.78% 23.27 2.12% 0.90 8.16% 13 0.17 1.65% 22.81 2.07% 0.91 8.27% 14 0.26 1.71% 36.09 3.28% 1.38 8.35% 15 0.12 2.46% 31.30 2.85% 0.97 17.57% 16 0.17 1.68% 35.15 2.51% 0.93 6.65% 17 0.16 1.56% 35.29 2.52% 1.07 7.68% 18 0.16 1.56% 35.97 2.57% 1.09 7.78% 19 0.21 1.38% 61.69 4.41% 1.45 6.92% 20 0.25 4.96% 20.61 2.58% 0.70 17.58% 21 0.43 2.89% 25.53 3.19% 1.36 11.30% 22 0.22 4.34% 50.98 3.64% 0.97 13.79% 23 0.39 2.60% 63.77 4.56% 1.31 6.23% The M A E results with the bump present are almost identical to those without. Test case #20 was still the most challenging, showing the largest relative errors. Test case #20 represents the lowest Y-axis end-point stiffness and Y-ax is equilibrium position commanded at the highest velocity. The combination of low predicted force and the errors introduced by the high velocity caused the largest observed errors to occur during this test case. 5.4.1 Velocity, Stiffness and EP Results The following three plots show the effect of changing X-ax is velocity (vx), Y-ax is end-point stiffness (ky) and Y-axis equilibrium position (ysp) had on the tests run without the bump present. 5.4 Contact Results 70 The tests chosen for these plots were four sets of ky and yEp that were run at each of the three values. Figure 5.18 is a summary plot of the Y-axis equilibrium position, yEp, M A E for four sets of cases. This plot shows that M A E of yEp increases as the X-axis velocity, vx, increases. It also increases as the yEp increases. The yEp M A E decreases as the Y-ax is end-point stiffness ky increases for the data shown below. 0.5 0.45 0.4 0.35 0.3 J 3 0 .25 S-i O fcl W 0.2 0 .15 0.1 0 .05 0 0 o + Test #1,5,20 (kyd=800N/m, yEpd=405mm) " E P ( f 4 0 5 m m ) Test #2,9,21 (k >800N/m, y c =415mm) yd J EPd Test #4,19,23 (kyd=1400N/m, yEpd=415mm) Test #3,15,25 (kyd=1400N/m, yp X + 20 + o x + O 40 60 v (mm/s) 80 1 0 0 Figure 5.18 - Summary of yEp error Figure 5.19 is a summary plot of the Y-axis end-point stiffness, k y, M A E for four sets of cases. This plot shows that M A E of ky does not appear to be correlated with X-ax is velocity, vx, or Y -axis equilibrium position, yEp, The observed ky M A E increases as the commanded ky increases as shown below. 5.4 Contact Results 71 100 r 90 -80 -7 0 -B 6 0 -50 -a 4 0 -30 -2 0 -10-0 -0 o x + 20 + Test # 1,5,20 (kyd=800N/m, yEpd=405mm) O Test #3,15,25 (k =1400N/m, y c =405mm) ^ yd J EPd x Test #2,9,21 (kyd=800N/m, yEpd=415mm) ^ Test #4,19,23 (lyd=1400N/m, yEpd=415mm) O x + 40 60 (mm/s) o X + 80 100 Figure 5 . 1 9 - Summary of ky error Figure 5.20 is a summary plot of the force M A E for four sets of cases. This plot shows that M A E of force does not appear to be correlated with X-axis velocity, vx, or Y-ax is end-point stiffness, ky. The Force M A E increases as the Y-ax is equilibrium position, yEp, increases for the data shown below. 5.5 Summary 72 2.51 i-i o b 1.5 o i i i + Test #1,5,20 (k =800N/m, yCDJ=405mm) yd J EPd O Test #3,15,25 (kyd=1400N/m, yEpd=405mm) x Test #2,9,21 (kJ=800N/m, yEpd=415mm) * Test #4,19,23 (I =1400N/m, yEpd=415mm) 0.5 20 40 60 80 100 (mm/s) Figure 5.20 - Summary of force error 5.4.2 Repeatability Results Three test points were repeated three times each. The repeated test data lines up very wel l . The manipulator delivers near identical behavior on each wipe of the surface when the trajectory is repeated. Detailed results presented in the Appendix and the data from the two above tables supports this assertion. Tests 6,7,8 were a group of three repeated identical trajectories, as were 11,12,13 and 16,17,18. A s seen in the above tables, the error for these batches of tests did not vary substantially. 5.5 Summary The results gathered from these three experiments are very encouraging regarding the usefulness of air muscle actuated P M C manipulators. A l l three experiments resulted in data in line with expectations. The free space tests were very useful in determining the appropriate speeds to run the two subsequent tests. Whi le velocities up to 150mm/s were tested it was found to be impractical to 5.5 Summary 73 command velocities faster than 75mm/s. Although the end point ceased to follow its commanded trajectory at higher velocities, no system instabilities resulted from executing these trajectories. The valve was simply not able to keep up with the required f low to achieve high velocities. This is due to the sizing of the orifice when the valves were initially calibrated. The orifice sizing, although limiting in terms of velocity, allowed for very smooth motion by keeping the force pulses small. The transition tests demonstrated the ability of this type of manipulator to transition between free space and contact without either planning for the contact or a change of control strategy. In the case of the tests that were performed in this work, the transition was unplanned. The end point was simply commanded to interfere with the surface. None of the velocities or angles of approach tested caused any instabilities or other unexpected behavior. The forces generated from the unexpected contact with the surface were as expected. The end point stiffness in fact becomes increasingly reduced, as the depth of contact grows larger. This was a result of the particular posture the manipulator was in when making contact and is not a general result. The contact task demonstrated the capabilities of the controller and manipulator for a wiping task. Force errors remained under 20% from predicted and interestingly did not vary greatly with the presence of an unexpected bump on the surface. The 4mm bump caused the end point to produce greater normal force into the surface, but the actual forces produced matched as well as those without the bump. The larger force is due to the increased deviation from yEp due to the presence of the bmp. It was observed that at higher speeds the error in force normal to the surface increases most appreciably for test cases where the predicted interaction force is low. Chapter 6 Conclusions & Recommendations The objectives of this work were to demonstrate the capabilities of a P M C manipulator controlled with an equilibrium point hypothesis inspired controller. Particular interest was taken in the ability of such a device to perform tasks that share characteristics with the activities of daily l iving. In specific, it was important to demonstrate that a simple P M C manipulator could perform basic free space, transition and contact tasks using a simple and stable controller. In this thesis a P M C robot with a simple linear PI controller based on the E P hypothesis was presented. A novel method for sizing a P M C joint using air muscles was presented, and an E P controller for the robot was designed and implemented. In a series of experiments it was shown that at low to moderate speeds (given the limitations of the valves of the air muscles) the controller tracks a demanding commanded trajectory, with some hysteresis induced in the computation (rather than direct measurement) of the actuator forces. The manipulator was designed to be large enough to carry out a wiping task. The muscles for this project were constructed in the lab as suitable muscles were not available for purchase. Solenoid valves were chosen for metering air in and out of the air muscles. Because the behavior of these valves was not well suited to the size of the muscles, orifice plates were designed and fit to each of the valves to reduce the available maximum flow rate in and out of each valve. 74 6.1 General recommendations 75 The equilibrium point controller was quite simple to program and worked very well . Independent control of joint stiffness and equilibrium position was possible. Close tracking o f both of these variables was achievable with simple PI control o f the mass flow in and out of each muscle. The results of the surface wiping tasks showed that it is possible to generate a wiping EP and stiffness trajectory that results in the predicted normal force while wiping the surface. Addit ionally the mechanical compliance of the manipulator allows for stable response to unpredicted disturbances such as the presence o f a significant bump on the smooth surface. Finally, stable behavior during transition from free-space to contact is a notable result. Because the manipulator follows an equilibrium-point trajectory with a programmed stiffness, no additional compensation is required when contacting objects in the workspace. In addition, knowledge of the precise location of the contact object is not important as the mechanical compliance of the manipulator compensates for small contact position errors. The particularly low cost of implementation of the technologies used in this work is a promising factor in the development of affordable assistive robotic devices for in home use. The assumption that programmable mechanical compliance adds intrinsic safety to a robot that may interact with people has been supported through the demonstration of transition and contact tasks. This result holds where significant unmodeled disturbances are present, being easily handled by a robot of this type. The biggest trade-off is that it is not possible to generate superhuman stiffnesses with this manipulator were they desired. The usefulness of this approach is supported by the fact that the EP hypothesis fits a broad range of human motion tasks. The controller demonstrated in this work shows that it is possible to closely control the EP and joint stiffness values of a manipulator. The success of this controller was independently verified by the external force measurement that showed the actual behavior of the manipulator matched the expected behavior. 6.1 General recommendations With the successful demonstration of a stiffness/EP controller on a planar manipulator now carried out, it is possible for this work to be expanded. There is a wealth of information available from neuromotor control studies regarding the stiffness and EP trajectories that humans 6.2 Specific Recommendations for this Experimental Work 76 follow when carrying out tasks. Results from the observation of humans can be directly implemented on this system that shares the capability o f simultaneously varying EP and stiffness. A variety of assistive devices using this technology can be imagined. Research can be carried out to help better understand the right characteristics assistive manipulator should possess. Topics for investigation include: (i) the combination of mass, stiffness and size most practical for a reaching manipulator to possess, (ii) methods of user activation, (iii) a practical method for learning from human task examples. The method detailing the sizing of air muscle parameters could also benefit from further investigation. The method used in this work was only developed to allow for minimization of the mounting length of the air muscles. Relatively simple modifications could allow for other features to be minimized. O f particular interest for mobile applications would be to alter the method to allow for the minimization of the difference in the mass of air in maximum activation versus minimum activation, allowing for increased "fuel efficiency." 6.2 Specific Recommendations for this Experimental Work The compliance o f the surface that was used presented some difficulties and the means by which this compliance was compensated for creates some confusion when interpreting the results. For future testing it would be preferable to use a surface with either uniform stiffness or very high stiffness so that deflections of the wall are negligible. In this case they were neither. A i r muscle models including friction exist. One of these models could be implemented to improve the force prediction capabilities of the model. If an even greater increase in force accuracy is required a transducer on each link could be used. Another approach would be to instrument the end effector with a 3-axis force transducer. This would allow the control loop around end effector force to be closed completely. The added benefit of measuring end point force directly is redundancy and increased safety. Wi th the current configurations, additional tasks could be attempted. It would be interesting to determine the effect of changing the mass of the endpoint during a motion to simulate picking up an object partway through a motion. Another variant on the tests performed in this work would be to try the wiping task with a higher compliance surface. 6.2 Specific Recommendations for this Experimental Work 77 Increasing the manipulator to a 3-degree of freedom device would allow for more realistic assistive task demonstrations. 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M . , Hannaford, B., "McK ibben Art i f ic ial A i r Muscles: Pneumatic Actuators with Biomechanical Intelligence," Proc. of IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 221-226 [34] http:// www.matrix.to.it/ - Matr ix Valve Website [35] Spong, M.W. , Vidyasagar, M . , 1989 Robot Dynamics and Control, Wi ley, New York Appendix A Air Muscle Equations A. l Air Muscle Equations for the Appendix From Chou dWin = f {P-P0)dl • ds, ={P-P0)\ dl, • ds, =P'dV Where: P=Absolute internal gas pressure Po=Atmospheric pressure P'=Gage pressure Si=Inner surface dispacment Dsi=Area vector Dli=Inner surface displacement dV=Volume Change Equation A . 1 Where: dW0U, = -fdL and, Equation A.2 dW0Ut = dWin The force in the muscle can be written as: f = -P dV dL where the length of the muscle can be represented by Equation A.3 Equation A.4 L = b- cos(<9) Equation A . 5 81 Appendix A 82 D = b • sin(0) Ml The volume in the muscle is given by: Equation A .6 1 h3 V = —uD2L - - ^ s i n 2 (0)cos(0) Am Equation A .7 f _ p,dV _ /Vde _ Fl>(2cos2(<9)-sin2(fl)) dL dL/ Ann1 / dtt Equation A.8 P'fr 2(3cos 2(fl)-l) 4;Z?J 2 cos 2 (0) = ^ -v ' Z>2 Ultimately yielding Z 2 W ( 3 - = — 1 ) 4;zw2 Equation A .9 Equation A . 10 Equation A . 11 Appendix B Muscle Construction Collect all the needed materials for an air muscle and size to correct lengths (see Table 1 and Figure 1 below). A i r Muscle Supplies Description Dimensions Quantity Supplier Product # Make Muscle Braiding 1/2" dia 18 1/2" length Radar Inc. (Seattle) 625300113 N /A Surgical Tubing 3/16" O.D., 1/32" thick 8" length Lancaster Medical Supplies N / A N /A Plastic Tubing 4mm O.D., 0.75mm thick 25cm length Festo 152 584 Festo Large Brass Insert 1/4" O . D . - 3 / 1 6 " l .D. 2 Columbia Valve & Fitting B-405-3 Swagelo k Small Brass Insert 1/4" O.D. - 1/8" l.D. 1 Columbia Valve & Fitting B-405-2 Swagelo k Aircraft Cable 1/16" dia 2 loops Steveston Marine & Hardware N / A N /A Aluminum Sleeves 1/16" dia 4 Steveston Marine & Hardware N / A N /A O-Clamps 1/4" nominal dia 3 Acklands & Grainger F A R HC9-4 Fairview Fittings Muscle Braiding 2 l /4"dia 2 X 3 3/4" Length Radar Inc. (Seattle) 624900113 N / A 0 . D. - Outer Diameter 1. D. - Inner Diameter Note: Muscle Braiding 2 was added A u g 13, 2002 because 1/4" braiding was used to make the loops of muscles X I and X 2 instead o f Aircraft Cable Table B . l - A i r Muscle Supplies 83 Appendix B 84 Figure B . l - Tools and supplies to make air muscles Solder large and small brass inserts together and plug the other large insert by filling it with solder (see Figure B.2). Figure B.2 - Soldering the brass inserts Push 4mm O.D. plastic tubing over small insert and surgical tubing over the large insert (see Figure B.3). Appendix B 85 Figure B.4 - Plastic and surgical tubing connected and plugged with brass inserts Slightly melt both ends of the mesh braiding so they don't fray apart. Insert the 4mm O.D. plastic tubing into mesh and push it through until it exits the other end. Now pull the plastic tubing until only the plug at the end of the surgical tubing is showing. Slide an O-Clamp over the brass plug and also over the braiding. Pul l or push on the plastic tubing until the end of the brass insert is flush with the end of the meshing. Make a loop with one of the smaller pieces o f meshing and slide its 2 ends underneath the O-Clamp. Slide the O-Clamp back over the brass plug snuggly with the two ends of the loop sandwiched between the O-Clamp and the larger braiding (see Figure B.5). Appendix B 86 Figure B.5 - End loop of the air muscle Alternate clamping down either side of the O-Clamp until it's snug and the loop can't be pulled out. Whi le holding the large mesh braiding pull the plastic tube until the junction of the two tubes comes out. Now repeat steps 9-12 to the other end of the muscle (see Figure B.6) Slide another O-Clamp onto the plastic tube and over the braiding and clamp it down just l ike the others (see Figure B.7). Figure B.6 - Exploded view Note: Top layout is exploded view of the bottom layout except that the mesh has been removed Appendix B 87 Figure B.8 - Completed air muscle mounted to the arm Appendix C Matlab Optimization Files C l OptimizeMountLength.m X=fmincon('minimizethis',[6.8 .4 .11 .36],[ ],[ ],[ ],[ ],[5 .4 .02 .32],[6.5 .6 .2 .4],'solverbn') n=X( l ) ; b=X(2); r=X(3); Lmount=X(4); %this script solves the following constraints %The maximum obtainable stiffness is greater then Kstiffmax %The minimum obtainable stiffness is less then Kst i f fmin %Kstiffmin>0 %The Max imum obtainable torque is greater than TorqueMax %The working range o f the robot is greater than Deltheta %minimum ratio of n/b>8 %max ratio of n/b<21 % C is a vector that the solver tries to set <=0 %Lmount < Kstiffmax = 30; Kst i f fmin = 15; TorqueMax = 3.1; DelTheta = pi/16; P(l)=15; %min pressure is 15 psi P(2) = 100 ;% max pressure is 100 psi Fmax=100; %max force is 100N Lmin=b*cos(54.73561/l 80*pi); LPmaxCrossFmax = l /1050/P(2)*210 A( l /2)*(P(2)*(1750*P(2)*b A2+(Fmax+0)*pi*n A2)) A( l /2); LPminCrossFmax = l /1050/P( l )*210 A ( l /2)*(P( l )* (1750*P( l )*b A 2+(Fmax+0)*pi*n A 2)) A ( l /2) ; FPmaxCrossb = P(2)*7000*b A 2*(3*b A 2/b A 2- l ) / (4*p i*n A 2)-0; • FPminCrossb = P( l ) *7000*b A 2*(3*b A 2/b A 2- l ) / (4*p i *n A 2) -0 ; i f LPmaxCrossFmax >b Lmax=b; else Lmax=LPmaxCrossFmax; end i f LPminCrossFmax <b L m i n l = LPminCrossFmax; 88 Appendix C 89 Fmaxl=Fmax; else L m i n l = b; Fmax 1 =FPminCrossb; end Lmaxs=[Lmin:.001 :Lmax]; Lmins=[Lmin:.001 :Lmin l ] ; Fmaxs=P(2)*7000*b A 2.*(3.*Lmaxs. A 2./b A 2- l ) . / (4*pi*n A 2)-0; Fmins=P( l )*7000*b A 2.*(3.*Lmins. A 2. /b A 2- l ) . / (4*pi*n A 2)-0; TmaxDes=TorqueMax; L l =Lmount+r*DelTheta; L2=Lmount-r*DelTheta; DeltaF=TmaxDes/r F A = P( l ) *7000*b A 2* (3* (L l ) A 2/b A 2- l ) / (4 *p i *n A 2) -0 ; F B = P(2)*7000*b A 2*(3*(L2) A 2/b A 2- l ) / (4*p i*n A 2)-0; i f F B > F m a x FB=Fmax End F C = F A + DeltaF F D = F B - D e l t a F K A = st i f fnessl(n,b,Ll,P(l)*7000) K C = st i f fnessl(n,b,L2,(FC+0)/(b A2*(3*(L2) A2/b A2-l)/(4*pi*n A2))) Kmin=r A 2 * ( K A + K C ) K B = stiffnessl(n,b,L2,P(2)*7000) K D = st i f fhessl(n,b,Ll , (FD+0)/(b A 2*(3*(Ll) A 2/b A 2- l ) / (4*pi*n A 2))) Kmax=r A 2 * (KB+KD) MaxTorqueAct = ( F B - F A ) * r figure plot(Lmins,Fmins,'r') hold plot(Lmaxs,Fmaxs); line([ b b],[Fmaxl Fmax]); l ine([LPmaxCrossFmax b],[Fmax Fmax]); l ine([Ll L1 ] , [FAFC] ) ; line([L2 L2],[FB FD]) ; line([Lmount Lmount],[0 100]) C.2 solverbn.m Appendix C 90 function[C,Ceq]=solverbn(X) n=X( l ) ; b=X(2); r=X(3); Lmount=X(4); %this function solves the fol lowing constraints %The maximum obtainable stiffness is greater then Kstiffmax %The minimum obtainable stiffness is less then Ksti f fmin %Kstiffmin>0 %The Max imum obtainable torque is greater than TorqueMax %The working range of the robot is greater than Deltheta %minimum ratio of b/n>.05 %max ratio o f b/n<.l % C is a vector that the solver tries to set <=0 %Lmount < b Kstiffmax = 30; Kst i f fmin =15; TorqueMax = 3.1; DelTheta = pi/16; P(l)=20; %min pressure is 20 psi P(2) = 100 ;% max pressure is 100 psi Fmax=100; %max force is 100N Lmin=b*cos(54.73561/l 80*pi); LPmaxCrossFmax = l /1050/P(2)*210 A( l /2)*(P(2)*(1750*P(2)*b A2+(Fmax+0)*pi*n A2)) A( l /2); LPminCrossFmax = l /1050/P( l )*210 A ( l /2)*(P( l )* (1750*P( l )*b A 2+(Fmax+0)*pi*n A 2)) A ( l /2) ; FPmaxCrossb = P(2)*7000*b A 2*(3*b A 2/b A 2- l ) / (4*p i*n A 2)-0; FPminCrossb = P( l ) *7000*b A 2*(3*b A 2/b A 2- l ) / (4*p i *n A 2) -0 ; i f LPmaxCrossFmax >b Lmax=b; else Lmax=LPmaxCrossFmax; end i f LPminCrossFmax <b L m i n l = LPminCrossFmax; Fmax l=Fmax; else L m i n l = b; Appendix C 91 Fmax 1 =FPminCrossb; end Lmaxs=[Lmin:.001 :Lmax]; Lmins=[Lmin:.001 :Lmin l ] ; Fmaxs=P(2)*7000*b A 2.*(3.*Lmaxs. A 2./b A 2-l) . / (4*pi*n A 2)-0; Fmins=P(l)*7000*b A 2.*(3.*Lmins A 2. /b A 2- l ) . / (4*p i*n A 2)-0; TmaxDes=TorqueMax; L1 =Lmount+r*DelTheta; L2=Lmount-r*DelTheta; DeltaF=TmaxDes/r F A = P( l ) *7000*b A 2* (3* (L l ) A 2/b A 2- l ) / (4*p i *n A 2) -0 ; F B = P(2)*7000*b A 2*(3*(L2) A 2/b A 2- l ) / (4*p i*n A 2)-0; i f F B > Fmax FB=Fmax end F C = F A + DeltaF F D = F B - DeltaF K A = sti f fnessl(n,b,Ll,P(l)*7000) K C = st i f fnessl(n,b,L2,(FC+0)/(b A2*(3*(L2) A2/b A2-l)/(4*pi*n A2))) Kmin=r A 2 * ( K A + K C ) K B = stiffnessl(n,b,L2,P(2)*7000) K D = st i f fnessl(n,b,Ll , (FD+0)/(b A 2*(3*(Ll) A 2/b A 2-l) / (4*pi*n A 2))) Kmax=r A 2 * (KB+KD) MaxTorqueAct = (FB-FA) * r C ( l ) = Kstiffmax - Kmax; %ensure the maximum stiffness is possible C(2) = K m i n - Kst i f fmin; %ensure the minimum stiffness is possible C(3) = -Kstiffmin %ensure the minimum stiffness is positive C(4) = TorqueMax - MaxTorqueAct; %ensure the max torque is achievable C(5) = 0; C(7) = n/b-21; %check the b/n ratio C(6) = 8 - n/b; %check it on the other side C(8) = DelTheta*r + Lmount -b ; %check the theta range on the right side Ceq=[] C.3 Stiffnessl.m Appendix C 92 function stiffnessl=stiffhessl(n,b,L,P) A= l / (4*p i *n A 2) ; B=b A 2/(4*p i*n A 2); P h i = B - 3 * A * L A 2 ; V o l = B * L - A * ( L A 3 ) ; st i f fnessl=(P+101000)A^ol*Phi A2+P*6*L*A; C.4 minimizethis.m function valuetomin = objfun(X) n=X( l ) ; b=X(2); r=X(3); Lmount=X(4); valuetomin=Lmount; Appendix D Manipulator Bill of Materials 93 Table D.l - BOM-1 3 o 1 « S 3 < f 5 5 - a1 2 ou 13 .2 S ffl E 2 3 ," °P>A • S o % ° ca c*t < c D U g o > T o 0 3 ^ t- B h S 2 c o * i l ,1 .a >» -^ t oo on o Si 0 0 D o g 0 0 ~ § D - > ! 2 ! oo s; 'S 1 3 ' O ° M-> <^  d- — f - IX, W •I « 55 55 Table D.2 - BOM -2 °3 •a « ° : o cu f— C J ^> C/3 O f l O OH 5 " -k-' S CJ a, CJ ; — CN O C J CZ) O M O x> o — • • "2 73 u * 8 8 31 73 3 a iuate iuate iuate iuate ca i— 60 She gra< She w» uate CJ uate c uate c uate •a -a cd UBC o UBC O o CJ CJ a .1 •a I l< l3 op 5 CJ o 73 6 1 2 Appendix D 96 Table D.3 - B O M - 3 c l l |3 3 M 13 CJ T 3 JO Z B O O N g -o " s S3 eg i *c3 « . cd ca • U CU P GO ON cd cd 1 6b i-a c 3 Cu cd 2 CO Q £ f c 2 u c u O-o o . Z Cfl OS 1) 55 55 3 l a M o o t-a . t/3 ca 0 0 c Z I H Appendix E Detailed Machining Drawings 9 7 Appendix E 98 Figure E . l - Drawing 1 Appendix E 99 Figure E.2 - End Effector Appendix E 100 Figure E.3 - L ink 2 and sprocket Appendix E 101 Figure E.4 - Pulley 1 and big gear Appendix E 102 Figure E.5 - Pulley 2 and big gear Appendix E 103 Figure E . 6 - L ink drive assembly Appendix E 104 DRAW 1NG US DIGITAL C V J SHOP CQ O , CJ O ^ - - -o • — — — 1  i — O O 1 — r o Q_ — CC ' — =tt= < C C O CC L_i_ CJ CC CC CP 1 1 1 oo L U L_i_ cCj <J O O OO CJ CJ ( z : 2: LJ-J LU 1—1—1 00 ITEM o c j 00 CC >— ro co CC > 0 0 ce C V J 1 — <=c <=t 1 1 1 -1 C J C N J > — Cl_ + > — — CC ce 0 L U -•••! LLJ CD CQ : • ' 0 < J L U L U C J ::>• 1 CC O C J L U L U 00 no : • 1 1 1 CC >_ — -=E CQ 1— -: L U ce <c 1 «=C CC <c Q_ LO =H= I— m 1— C J - c ^ c j LiJ m > C J O 1 1 1 CC •< Z- Q_ CC cm 0 — CJ>_ — 0 c > — CO Li J UJ Z O cz> cz> CZ > <=> -H-H-H-H CC LiJ LiJ Li J 0 0 1 TT. X X o u s X X X X ' — UJ 1 r n X X X X 1 -z. 1— -< -- O Figure E.7 - Encoder and gear Appendix E 105 X L J L . L Z J Q_ Cl_ CC O o CC ro O zc zc L O L _ u C V I = C V J O O C V J <z> ro i cv L O ro -< 1 X O O — CQ -::: O =H= = 3 — oo 1— co CC Q _ i s : u _ CC CC 1 i ! Z D -=c Z D ^ oo L_i_l O 1 C O oo 1— C _ J La_ Q _ X X X X oo o -< 1— cvj C V J oo CQ — C V J ro ^3" L O L O CJ>. C C Z D cz> C C ::> <z> O C V J :— -—^  X C V J Q _ > — C C O LaJ C Q C Q z ~ X LaJ X > o ICC o -. C Q LaJ oo Z D i i i C C >_ — C Q l — - z LaJ C C <c 1 C C <c Q _ Q co =H= J L O 1 L O | C J D <c Li_l o -U_l C C > _ I S Cl_ C C r-i — i — o oo OO CTi — o> c > — CO u L U _ J L U LJ ^ O <o o <_• > o - H - H - H - H -< CC OO L U L U la J C/3 X I ZH x x 0 O O 3: X X X X LU I n_ X X >< : x 1 ^ 1 <c — - O 0 Figure E.8 - Box Appendix E 106 CO CO CO C J oo CC O C J oo CC LO LO I C J C J O U J c c CL_ c o CO CVJ oo c e O I— Q _ <=c o OQ C J CC CQ o C J o c c Q_ CC Q C_j =t=t= C O O O O c=> - H - H - H - H x x 0 X X X X x x x x Figure E.9 - Adaptor 1 Appendix E 107 Figure E. 10 - Adaptor 2 Appendix E 108 Figure E. l 1 - Adaptor 3 Appendix E 109 Figure E.12 - L ink 2 Appendix E 110 Figure E . l3 - Timing belt sprocket 2 Appendix E 111 Figure E .14 -Sha f t 2 Appendix E 112 Figure E . l5 - Pulley 1 Appendix E 113 Figure E.16 - B i g gear Appendix E 114 Figure E.l7 - Timing belt sprocket 1 Appendix E 115 Figure E . l 8 - Pulley 2 Appendix E 116 <z> CD <r> CD CD CD CD o oo z^: <c CC I— zz> CC > — O CO-I— CD <=c CVJ 1 — ) — CL ::::> L O - — O <_ - : CO - S 1— _Q u _ O LU <c — I zc CC e j oo UJ o OO —i - S 1 , 1 t CC >-— CQ z<: :— -: UJ CC «=C 1— <c CC -<c o_ L Z J L Z J cn <L> 1 L O =8= <L> |— L O 1 UJ OO :> zc > c_> O UJ CC > -"— o oo — < -OO C2t — o o 0 OO L U LU C J ^ O o> cz> <z: o ^ -H-H-H-H OO LU LU L i J CO X i nz — x x 0 o <_> s X X X X L U I x 1 ^ 1— <c — - O Figure E .19 -Sha f t 1 Appendix E 117 Figure E.20 - L ink 1 Appendix E 118 Figure E.21 - Fitting Appendix E 119 Figure E.22 - Small gear Figure E.23 - Top Appendix E 121 O O o CJ o C J O O L O u~> 3^" C J o C C Q _ C C o oo o C Q C J C C C C Li_l C Q o C C C C LH J =1* C J C C M O — <0 <0 . <=3 O o -H-H-H x x 0 X X X X X X CO Q CO UJ uu l_l_t I— <_> z : O Figure E.24 - Side Appendix E 122 cc O L O CVJ CJ oo C C L O CO oo L O L O I -=r CJ CJ O U J C C C C 00 C Q o C Q CJ C C C O CVJ I C C L-lJ C O O C C C l _ L O CVJ CO C C C C Q C_j o O O O O -H-H-H-H X X XQ X X X X X X X X or 00 U J Figure E.25 - Base Appendix F Assembly Instructions Step 1: End Effector (see drawing # 2) • First attach the back adapter to one side of the force transducer. Use three M 3 screws and tight it gently. The adapter can be mounted on any of the two sides and in any o f the three different orientations. It would probably be better to use the. same configuration every time you want to record data. • Put the side adapter around the force transducer on the opposite side of the back adapter. • Insert the front adapter inside the side adapter and fix it with three M 3 . The back and front adapters must have the same orientation. Once again, tight gently to avoid damaging the expensive force transducer. • Finally, screw the bearing and its shaft on the front adapter. Step 2: Link 2 + sprocket (see drawing # 3) • Insert the timing belt sprocket in link 2. (Note: the hole for shaft 2 is drilled after these two parts are assembled together. Because of that, it is preferable not to take this assembly apart) • Use four screws and two dowel pins to fasten the link and the sprocket together. • Don't insert the shaft right now like shown on the drawing. Step 3: Pulley one + big gear (see drawing # 4) 123 Appendix F 124 • First position one of the big gears on pulley 1 using two dowel pins 3/32. Make sure you place it on the right side. If you are not sure which side to choose, you better go trough Step 4 first. • Fixe the pulley on place with two countersunk screws 8-32. Step 4: Pulley 2 + sprocket + big gear (see drawing # 5) • Attach the other big gear on pulley 2 the same way as Step 3. • Insert the timing belt sprocket in the pulley and fixe it with four 4-40 A l lan screws. (Note, you have to make sure that the hole is perpendicular to the pulley) Step 5: Link 1 drive assembly (see drawing # 6) Note: before you start this step make sure you have a vice plus a vice-grip. A lso make sure all the previous steps are done correctly. Believe me, you don't want to this twice. • First introduce link 2 in the shorter slot of l ink 1. • Insert shaft 2 inside link 2 and the sprocket. The shaft is larger than link 1 and both his tips should be out of the holes. L ink 2 and the sprocket are slightly smaller than the slot and should not touch the sides. • Insert both bearings on shaft 2 inside link 1. • Put on the belt • Introduce pulley 2 inside the belt and place it in the larger slot. Both timing belt sprockets should be aligned. • Orient link 2 correctly according to the big gear on pulley 2. This is to make sure that the gears wi l l not disengage themselves when operating the robot. • Insert shaft 1 by the upper side. The shoulder of shaft 1 should mate the pulley. • Insert the metric bearings. This is where you need a vice and some vice-grips. A hammer and your imagination might be useful. • Final ly bolt pulley 1 on link 1, using four A l lan screws. • Y o u can also add 6 set screws: 1 for each bearing and two for link 2. Step 6: Encoder (see drawing # 7) • Insert the fitting on the encoder shaft. • Insert the small gear on the fitting and tight it there with a set screw. If you tight too much you wi l l deform the plastic gear which is not good for the alignment. (Note: this should be done only once the encoder is placed on the top. See step 8) Step 7: The Box (see drawing # 8) • First screw the two sides on the base using 4 countersunk screws 10-32. The bearing housing should be on the bottom side. • Insert shaft 1 (and this includes the two links attached to it) in the base. Appendix F 125 • Insert the bearing in its housing. • Put on the top and screw it with 4 countersunk screws 10-32. The housing should face up. • Insert the last bearing in the top housing. • Screw the base on the lab table using 4 T-nuts. Step 8: Tota l assembly • Put the two encoders at their place on the top of the box. • Put the small gear on (see step 6) • Bolt the end effector on link 2. It is preferable to use one block washer on each side to avoid slippage. We suggest orienting the end effector with an angle of 20 degrees with link 2. • Here we do not explain how to mount the air muscles since they might be constructed differently. Appendix G Sensors and Calibrations G.l Experimental Equipment Specifications Pneumatic Actuators Each pneumatic actuator is approximately 33 cm in length. Inside, each consists of rubber surgical tubing (3-mm diameter) and is covered by a tough plastic weave. When the surgical tubing is being inflated, this provides a radial force and the weave contracts, resulting in a decreased length. In addition, the actuators need to be held taut when initially inflated or the tubing inflates non-uniformly against the mesh. These actuators were made in the laboratory with no rigid specifications. Valves Solenoid valves: The four Matrix solenoid valves used operate on a pulse width modulation signal. They have a maximum frequency of 200-Hz and their minimum open time is 2-ms. They have three different positions: one to allow for air to be supplied to the actuator, a second to serve as an outlet for air from the actuator to the atmosphere, and a third closed position where no air is exchanged. Force Transducer 126 Appendix G 127 The Precision Transducers force transducer used has a capacity of 50 kg. It has tapped holes on either end, which were used to attach it to a flat plate on which the applied force was impressed, and to the base. Pressure Transducers There were two each of two types of pressure transducers. The transducers by Sensotec have a range of 150 psig and came with calibration papers. AutoTran transducers have a minimum range of 100 psi and a 1% accuracy. Length Encoders The U S Digital length optical encoders measure real-time shaft angle. In the initial set-up they were attached directly to the shaft of the pulley. In the final set-up they were attached to the shafts of toothed gears that meshed with gears on the respective pulleys. Coupled with the gears, they provide % degree of precision on the link position. Manipulator The manipulator is a two-link arm that is controlled by two pulleys mounted on a single shaft. The rotation of one pulley translates to the rotation of one link in the same plane of motion. The inner link is directly attached to its pulley and the second link is attached at the end of the first l ink with freedom to rotate. A high-torque timing belt transmits the force and motion from the second pulley to the second link. B y rotating the pulleys, the linkage assembly performs simulated wiping motions. The manipulator was constructed from aluminum. The pulleys are 18 cm in diameter and have a thickness of 0.64 cm. The link directly attached to the pulley is 22.5 cm long and the second link is 18 cm. The links are also 0.64 cm in thickness. Voltmeter The Fluke 801 OA digital multimeter was used to read out the voltage while doing calibrations. Digital Scale Weighing o f all of the components used in the calibration of the force transducer was done by a Toledo S M - F digital scale. The accuracy was one tenth of a gram. Appendix G 128 G.I.J Solenoid Valves Matrix Solenoid Valve Model 821 3/3 N C , Identification code G N K 8 2 1 2 0 3 C 3 K K 3 Port, 3 Way High Frequency Valve www.matrix.to.it/pd009.htm Description - The Pneumatic Solenoid Valves 820 Series The research about materials and new technological solutions allowed the realization of a shutter solenoid valve with an extremely simple operation principle and with avant-garde dynamic characteristics. The mass of the moving elements has been reduced to the minimum and every inner friction has been eliminated: in this way, we obtained response times of milliseconds and an operation life over 500 mi l l ion cycles. Due to the possibility of controls of speed-up type, their dynamic characteristics are even more improved. Standard solenoid valves with 24 V D C control have a response time lower than 5 ms in opening and 2 ms in closing, with a maximum operation frequency of 200 Hz . On the contrary, solenoid valves with speed-up control have a response time lower than 1 ms, both in opening and in closing, with a maximum operation frequency of 500 Hz . Besides high-speed characteristics, solenoid valves 820 Series offer flow rate values up to 180 dm3/min (ANR) , with feeding pressure from 0 to 8 bar. Controll ing the valve through either P W M or P F M techniques, it is possible to vary the passing flow rate and to obtain, in this way, a solenoid valve having a proportional flow rate. General Characteristics { P R I V A T E "TYPE=PICT;ALT=Pneumat ic scheme"}Control Direct - P F M -P W M Type and function 3/3 N C Dimensions (mm) 24.2 x 37 x 48.5 Fluid Non-lubricated dry air, neutral gases (-10 +50 °C) Filtration rating M i n 40 micron Temperature -10 +50 °C (standard version) Response time in opening 24 < 6ms X X / K K < 3 ms Response time in closing 24 < 2 ms X X / K K < 1 ms Maximum frequency 100 H z 200 H z Weight 130 g Product life expectancy > 500 M i s cycles Flow rate (at 6 bar) 90 N l /min - Control tension X X / K K No. Outlets 1 Outlet No . Electrical controls 2 Controls Port connection Integrated cables IP 62 L = 500mm / 100mm Control tension Speed-up in tension (24VDC) 0.8 W Operating pressure 0 - 8 bar Materials Body in PPS, Flanges in A l , Seals in N B Appendix G 129 G.1.2 Sensotec Pressure Transducers Sensotec Model L M 150, serial numbers 70258, 702583 150psig range www.sensotec.com/pdf/lm.pdf Sensotec offers the Model L M pressure transducer as a low cost alternative with good performance for high volume applications. Each unit is constructed of welded stainless steel for durability in dry rugged environments. Both gas and l iquid pressure overloads of up to 50% over capacity are safely accepted. Performance Pressure Range 150psig Output Resolution Accuracy (min.) 1.8mV/V(nom) infinite ±0.5% F.S. Environmental Temperature, Operating Temperature, Compensated 60°F to 160°F Temperature Effect* -Zero (max.) 0.01%F.S./°F -Span (max) 0.02% Rdg/°F -65°F to 250°F Electrical Input 10VDC Bridge Resistance 350 ohms** Electrical Termination (std.) Cable 3 ft. Mechanical Media Gas, Liquid 50% over capacity V4-I8NPT female Stainless steel Gage Stainless steel Overload-Safe Pressure Port Wetted Parts Material Type Case Material * Consult Sensotec on units below 150psi * * 5000 ohm below 150psi Appendix G 130 G.1.3 AutoTran Pressure Transducers Autotran Model 250G, serial numbers 8-B6107213 and 8-B6107156 1 OOpsi range www.autotraninc.com/specs/250g.html The series 250G is machined from a solid piece of stainless steel and employs a micromachined piezoresistive strain gage fused with high temperature glass to a stainless steel diaphragm. This design provides an exceptionally stable sensor ideal for use in a wide variety o f applications. There are no welds, no O-rings, and no silicone oi l to leak and cause potential problems. This is a truly tough and compact pressure transducer that comes in a 2-wire, 4-20mA version, or a 3-wire, 1-5V version. Oto 100 PSI (0 to 7 Bar) < l % o f F S +/- 0.25% FS typical <+ / -2%ofFS 30 to 130 degrees F (0 to 55 degrees C) -4 to 185 degrees F (-20 to 85 degrees C) Any media wet or dry compatible with 17-4 P H stainless steel 10-30 V D C 10mA maximum (for voltage output) 5 K ohm 1.IK ohm 1 to 5 V D C , 4 to 20mA two wire +/- 2% 24" 3-wire cable (1-5V), 24" 2-wire cable (4-20mA) Solid one piece 17-4 P H stainless steel !4" N P T 2.2" L x 7/8" D ia (54.8 m m L x 21.4 mmDia) Specifications {PRIVATE} Pressure Range: Accuracy: Stability: Thermal Effects: Compensated Range: Operating Temperature: Media Compatibility: Input Supply: Supply Current: Load Resistance (Voltage Output): Load Resistance (Current Output): Output Signal: Zero Offset: Electrical Connection: Housing: Connections: Dimensions: Appendix G 131 G.1.4 Force Transducer PT (Precision Transducers) Model ST 5, serial number 65266 50kg capacity www.precisiontransducers.com/pdf/product/ST SERIES.pdf Features - tension and compression universal loading - tool steel design for high accuracy -compact, lightweight, and easy to handle - N .S .C. approved models -temperature compensation, both zero and span -electroless nickel plated -compatible with international standard fixings - moisture protected -can be used for multi-point weighing or scale conversion -full range of mounting accessories (refer over) Specifications Nominal capacity 50kg Nominal output at capacity 2 m V / V ± 0 . 1 % Factory calibration mode compression Linearity error 0.017% Repeatability 0.01% Zero return, creep (30mins) 0.015% Temp, effect span/10 deg. C 0.01% Temp, effect zero/10 deg. C 0.015% Insulation resistance - brg. to gnd >5000 M ohms Insulation resistance - cbl. to gnd >1000Mohms Compensated temp, range -10 to 50 deg. C Output resistance 352.2 ohms Input resistance 410 ohms nominal Service load 100% of capacity Safe load 150% of capacity Mechanical failure >300% of capacity Recommended excitation 5V to 15V ac/dc Maximum excitation 15V dc Environmental protection IP65 Appendix G 132 G.1.5 Length Encoders US Digital Corp. SI series Model S1-360-IB www.usdigital.com/products/s 1 s2 Features -2-channel quadrature, T T L square wave outputs -Small size - -40 to +100°C operating temperature - Low cost -Tracks from 0 to 100,000 cycles/sec - Single +5V supply -Bal l bearing option tracks to 10,000 R P M -3rd channel index option Description The SI and S2 series optical shaft encoders are non-contacting rotary to digital converters. Useful for position feedback or manual interface, the encoders convert real-time shaft angle, speed, and direction into TTL-compatible quadrature outputs with or without index. The encoders utilize an unbreakable mylar disk, metal shaft and bushing, L E D light source, and monolithic electronics. They may operate from a single +5VDC supply. The SI and S2 encoders are available with ball bearings for motion control applications or torque-loaded to feel l ike a potentiometer for front-panel manual interface. Mechanical Notes Bal l Bearing: Sleeve Bushing j {PRIVATE} Acceleratio jn 10,000 rad/sec2 j 10,000 rad/sec 2 [ j Vibration j 20 g. 5 to 2 K H z |20 g. 5 to 2Khz ! j Shaft Speed 10,000 R P M max. continuous j 100 R P M max. continuous j Shaft Rotation N/A j Continuous & reversible {Acceleration ! 50K rad/sec2 |N/A ' | Shaft Torque 0.05 in. oz. max. J0.5 ±0.2 in. oz. | Shaft Loading 1 lb. max. |2 lbs. max. dynamic ; |20 lbs. max. static j j Bearing Life (40/P) = Li fe in mil l ions of revs, where P = Radial load in pounds. JN/A j (Weight 0.7 oz. J0.7 oz. | j Shaft Runout \ 0.0015 T.I.R. max. jo.0015 T.I.R. max. j Materials & Mounting: {PRIVATE} Shaft Brass or stainless i Bushing \ Brass Connector Gold plated Hole Diameter J 0.375 in. +0.005 - 0 Panel Thickness 0.125 in. max. : Panel Nut Max. Torque 20 in.-lbs. j Appendix G 133 G.2 Calibration of Equipment G.2.1 Calibration of Sensotec Pressure Transducers Sensotec model: LM/2345-03 Serial Number: 702583 Certificate of Calibration Calibrated at 150 psig Excited voltage = 10V Shunt Resistor = 59 ohms Calibration = 1.7557 m V / V Shunt Calibration = 1.4851 m V / V Data taken with Fluke Digital Multimeter (03/13/02): offset = -2.52 V shunt = 4.01 V current shunt resistance = 87.325 ohms excitation voltage = 8 V Calculations: Current shunt calibration = (59 ohms/ 87.325 ohms)*(1.4851 m V / V ) =1.003388 m V / V Current shunt pressure = (1.003388 m V / V / 1.7557 mV/V)* (150 psig) = 85.725 psig Calibration Equation: V = offset + P*cal ibrat ion*8*G/150 psig => G is factor to be determined 4.01 V = -2.52 +85.725*1.7.557*8*G/150 =>G = 0.813498 Therefore inverted calibration curve with gain and offset for O R T S : Pressure (psig) = 13.12787*Voltage (V) + 33.0822 Appendix G 134 Sensotec model: LM/2345-03 Serial Number: 702581 Certificate of Calibration Calibrated at 150 psig Excited voltage = 10 V Shunt Resistor = 59 ohms Calibration = 1.6228 m V / V Shunt Calibration = 1.4843 m V / V Data taken with Fluke Digital Multimeter (03/19/02): offset = -2.29 V shunt = 4.40 V current shunt resistance = 87.325 ohms excitation voltage = 8 V Calculations: Current shunt calibration = (59 ohms/ 87.325 ohms)*(1.4843 m V / V ) =1.002274 m V / V Current shunt pressure = (1.002274 m V / V / 1.6228 mV/V)*(150 psig) = 92.643 psig Calibration Equation: V = offset + P*calibration*8*G/150 psig => G is factor to be determined 4.01 V = -2.52 +85.725*1.6228*8*G/150 => G = 0.83435 Therefore inverted calibration curve with gain and offset for O R T S : Pressure (psig)•= 13.84804*Voltage (V) + 31.712 Appendix G 135 G.2.2 Calibration of Auto Tran Pressure Transducers Voltage #2: Auto Tran Inc. S N : 8-B6107213 Voltage #3: Auto Tran Inc. S N : 8-B6107156 Voltage #4: Sensotec S N : 702581 Pressure #4: Pressure calculated using the above calibration equation for Sensotec S N : 702581 Pressure Voltage(#4) Pressure(#4) Voltage(#2) Voltage(#3) 0 -2.29 0 -1.13 -1.1 10 -1.52 10.66905545 -1.58 -1.54 20 -0.84 20.09107844 -2.01 -1.97 30 -0.14 29.79021976 -2.37 -2.35 40 0.56 39.48936107 -2.87 -2.84 50 1.41 51.26688982 -3.4 -3.38 60 2.08 60.55035365 -3.82 -3.8 70 2.94 72.46644155 -4.36 -4.35 80 3.52 80.50287293 -4.71 -4.7 | Results: (V, psi) j Pressure(#2) = -22.314*Voltage(V) - 24.548 Pressure(#3) = -22.145*Voltage(V) - 23.511 The resultant calibration equations are linear best fits done automatically in Excel . Note that the pressure is measured in psig. Appendix G 136 G.2.3 Calibration of Precision Transducers Force Transducer The variety of weights used to calibrate in compression: Part# Part Mass (g) Part weight(N) #1 2709.7 26.582157 #4 3059.5 30.013695 #6 1653.2 16.217892 #11 1198.8 11.760228 #12 525.7 5.157117 03/19/02 part# Weight(N) voltmeter(V) w/ horizontal shift (V) none 0 2.47 2.51 12 5.157117 2.31 2.35 11 11.760228 2.1 2.14 6 16.217892 1.96 2 12+6 21.375009 1.79 1.83 1 26.582157 1.63 1.67 4 30.013695 1.52 1.56 4+12 35.170812 1.35 1.39 4+11 41.773923 1.14 1.18 The orientation of the transducer is horizontal but it was vertical for the measurements o f compression under the given weight. Therefore, the right column accounts for a shift in the voltages. The final calibration equation is: Force (N) = -31.374*Voltage (V) + 78.867 Appendix G 137 G.2.4 Calibration of US Digital Length Encoders These are the measured lengths of each actuator at the starting position (end pt. l ) and at the end position (end pt. 2) muscle length (m) end pt. 1 (offset) end pt. 2 M1 0.302 0.388 M2 0.363 0.33 M3 0.366 0.39 M4 0.412 0.326 The offset is the length at the starting position. The gain was determined by comparison with previous calibration data. Previous calibration data | muscle length(m) end pt. 1 (offset) slope M1 0.536254 0.0002638 M2 0.3304 -0.0001 M3 0.374916 0.0001 M4 0.3235 -0.0001 Slope = previous slope * (actual difference in length)/(perceived difference in length) Sample Calcu la t ion: Slope of M l = 0.0002638*(0.388-0.302)/(0.536254-0.302) = 9.685E-05 These results are tabulated below: muscle length(m) end pt. 1 (offset) slope M1 0.302 9.685E-05 M2 0.363 -1.012E-04 M3 0.366 2.692E-04 M4 0.412 -9.718E-05 M l & M 4 are a paired muscle group. They were attached around the same pulley so they same encoder information. The same goes for M 2 & M 3 . The offset is the measurement of the length of the muscles at the starting position for calibration. Appendix H H . l Free-Space Tests Each of the below trajectories was followed this the manipulator in free-space. N o wall was present for any of the tests. 1_1200 K 1.5cms Jan 30 2004 2_1200 K 1.667cms Jan 30 2004 3_1200 K 1.875cms Jan 30 2004 4_1200 K 2.143cms Jan 30 2004 5_1200 K 2.5cms Jan 30 2004 6_1200 K 3cms Jan 30 2004 7_1200 K 3.75cms Jan 30 2004 8_1200 K 5cms Jan 30 2004 9_1200 K 7.5cms Jan 30 2004 10_1200 K 15cms Jan 30 2004 H.2 Contact Tests Each of the following points were with and without a bump present on the instrumented contact surface. 11.5cms_800K_40.5_comp 2_1.5cms_800K_41.5_comp 3_1.5cms_1400K_40.5_comp 4_1.5 cms_l 400K_41.5_comp 5_3 cms_800K_40.5_comp 6 3 cms_800K_41 _comp 7_3cms_800K_41_comp 8_3 cms_800K_41 _comp 9_3cms_800K_41.5_comp 10_3cms_l 100K_40.5_comp 1 l_3cms_l 100K_41_comp 12_3cms_l 100K_41_comp 13_3cms_l 100K_41_comp 14_3 cms_l 100K41.5_comp 15_3 cms_l 400K_40.5_comp 1 6 3 cms_l 400K_41 _comp 1 7 3 cms_l 400K_41 _comp 18_3cms_1400K_41_comp 19_3 cms_l 400K41 .5_comp 20_7.5cms_800K_40.5_comp 21_7.5cms_800K_41.5_comp 138 Appendix H 139 22_7.5cms_1400K_40.5_comp 23_7.5cms_1400K_41.5_comp H.3 Transition Tests A l l of these tests were performed once with the wall present. 1_1000K_0.2cm_s_30_deg 21000K_0.2cm_s_60_deg 3_1000K_0.2cm_s_90_deg 4_1 OOOK_0.5cm_s_30_deg 5_1000K_0.5cm_s_60_deg 6_1000K_0.5cm_s_90_deg 7_1000K_1 cm_s_30_deg 8 _ 1 0 0 0 K J cm_s_60_deg 9_1000K_1 cm_s_90_deg 10 1000K_2cm_s_90_deg 11_1000K_4cm_s_90_deg Appendix I Summary of Transition Tests y versus Path 410 405 6 E 400 395 390 y no wall J a y with wall • y, EPd 5 10 15 Path (mm) 20 410 y__ versus Path 5 10 Path (mm) 1200 1150 1100 1050 1000 950 900 850 800 k versus Path y Force versus Path Actual no wall Actual wall Desired 0 5 10 15 Path (mm) 20 5 10 15 Path (mm) 20 Test #1 - 30 degrees, v=0.2cm/s Figure 1.1 - Transition test #1 140 Appendix I 141 y versus Path y__ versus Path ^EP 410 405 6 6 400 395 390 y no wall — - y with wall a • " yEPd 5 10 15 Path (mm) 20 410 5 10 15 Path (mm) 20 k versus Path y Force versus Path 1200 1150 1100 1050 c Z 1000 950 900 850 800 Actual no wall Actual wall Desired 5 10 Path (mm) 5 20 0 Test #2 - 60 degrees, v=0.2cm/s 5 10 15 Path (mm) Figure 1.2 - Transition test #2 Appendix I 142 y versus Path y__ versus Path 410 405 E E 400 395 390 - M ^  - "" y-'t /[•? y -j - — y no wall — - y with wall J a yEPd 5 10 15 Path (mm) 20 410 405 £ 400 395 390 5 10 15 Path (mm) 20 k versus Path y Force versus Path 1200 1150 1100 B 1050 z 1000 950 900 850 800 Actual no wall — - Actual wall Desired 5 10 15 Path (mm) 20 0 Test #3 - 90 degrees, v=0.2cm/s 5 10 15 Path (mm) 20 Figure 1.3 - Transition test #3 Appendix I 143 y versus Path J a y^,, versus Path . • \y -~ y~ ^  — y no wall J a _ - y with wall yEPd-10 Path (mm) 15 20 410 5 10 15 Path (mm) 1200 1150 1100 1050 c z; 1000 950 900 850 800 k versus Path y Force versus Path Actual no wall Actual wall Desired 5 10 15 Path (mm) o PL, -10 20 0 Test #4 - 30 degrees, v=0.5cm/s 5 10 Path (mm) Figure 1.4 - Transition test #4 Appendix I 144 y versus Path y__ versus Path ^EP 410 405 E 400 395 390 y no wall J a — - y with wall y__, •'EPd 5 10 15 Path (mm) 20 410 405 E 400 395 390 5 10 15 Path (mm) 20 1200 1150 1100 1050 950 900 850 800 k versus Path y Actual no wall — - Actual wall Desired 0 5 10 15 20 Path (mm) 15 10 o & 0 -10 Test #5 - 60 degrees, v=0.5cm/s Force versus Path Actual Predicted 5 10 15 Path (mm) 20 Figure 1.5 - Transition test #5 Appendix I 145 y versus Path y ^ versus Path ^EP 410 405 395 390 /~ jr / S / /• r rz-r y no wall J a /•'r — - y with wall J a _A y E P d 10 15 Path (mm) 410 10 15 Path (mm) k versus Path y Force versus Path 1200 1150 1100 1050 c Z 1000 950 900 850 800 - — Actual no wall — - Actual wall Desired 5 10 15 Path (mm) 20 0 Test #6 - 90 degrees, v=0.5cm/s 5 10 15 Path (mm) Figure 1.6 - Transition test #6 Appendix I 146 y versus Path Ja y no wall J a — y with wall Ja yEPd 5 10 15 Path (mm) 20 410 V™ versus Path E^P 5 10 Path (mm) k versus Path y Force versus Path 1200 1150 1100 c 1050 c z 1000 950 900 850 800 Actual no wall Actual wall Desired 5 10 15 Path (mm) o 0* 20 0 Test #7 - 30 degrees, v=lcm/s 5 10 15 Path (mm) Figure 1.7 - Transition test #7 Appendix I 147 y versus Path J a y versus Path / yy y no wall J a — - y with wall yEPd 5 10 15 Path (mm) 20 5 10 15 Path (mm) 20 k versus Path y Force versus Path 1200 1150 1100 1050 z 1000 950 900 850 800 — Actual no wall - Actual wall Desired 5 10 15 20 Path (mm) Test #8 - 60 degrees, v=lcm/s 5 10 15 Path (mm) 20 Figure 1.8 - Transition test #8 Appendix I 148 y versus Path y__ versus Path 410 405 6 S 400 395 390 r -0/ y no wall a _ - y with wall J a yEPd 10 15 Path (mm) 20 410 5 10 15 Path (mm) 20 k versus Path y Force versus Path 1200 1150 1100 r—\ 6 1050 Z 1000 950 900 850 800 Actual no wall — - Actual wall -Desired — "X \ \ -\ \ 20 15 10 o -10 0 5 10 15 Path (mm) Test #9 - 90 degrees, v=lcm/s — Actual Predicted 5 10 15 20 Path (mm) Figure 1.9 - Transition test #9 Appendix J Summary of Contact Tests 149 Appendix J 150 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 ? 1200 z 1100 1000 900 800 700 -100 versus x without bump -50 0 X(m) Actual — Desired 50 100 S z 1600 1500 1400 1300 1200 1100 1000 900 800 700 -100 versus x-position with bump -50 0 x (mm) Actual — Desired 50 100 Force versus x-position without bump Force versus x-position with bump -50 0 x (mm) Test #1 - v =15mm/s, Y =405mm, k =800N/m x ep y Figure J . l - Contact test #1 Appendix J 151 y-position versus x without bump y-position versus x with bump 100 1600 1500 1400 1300 E 1200 Z w 1100 1000 900 800 700 -100 k versus x without bump -50 0 X(m) Actual Desired 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 k versus x-position with bump -50 0 x (mm) Actual Desired 50 100 30 25 20 Z g 15 o 10 5 Force versus x-position without bump Force versus x-position with bump 0 1— -100 Actual Predicted -50 0 x (mm) 50 100 0 50 x (mm) Test #2 - v =15mm/s, Y =415mm, k =800N/m x ep y Figure J.2 - Contact test #2 Appendix J 152 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 versus x without bump -50 0 X(m) Actual Desired 50 100 1600 1500 1400 1300 B 1200 z > 1100 1000 900 800 7001— -100 k^  versus x-position with bump -50 0 x (mm) Actual Desired 50 100 Force versus x-position without bump Force versus x-position with bump Test #3 - v =15mm/s, Y =405mm, k =1400N/m x ep y Figure J.3 - Contact test #3 Appendix J 153 y-position versus x without bump y-position versus x with bump 100 -100 k versus x without bump -50 0 X(m) Actual Desired 50 100 1600 1500 1400 1300 e 1200 z 1100 1000 900 800 700 ^ versus x-position with bump Actual — Desired -100 -50 0 50 x (mm) 100 Force versus x-position without bump Force versus x-position with bump 0 50 x (mm) Test #4 v =15mm/s, Y =415mm, k =1400N/m x ep y Figure J.4 - Contact test #4 Appendix J 154 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 B 1200 Z 1100 1000 900 800 700 -100 ky versus x without bump Actual — Desired -50 0 X(m) 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 k versus x-position with bump Actual — Desired -50 0 x (mm) 50 100 Force versus x-position without bump 30 25 20 Z | 15 o 10 5 Force versus x-position with bump 0'— -100 Actual — Predicted -50 0 x (mm) 50 100 Test #5 - v =30mm/s, Y =405mm, k =800N/m x ep y Figure J.5 - Contact test #5 Appendix J 155 3901— -100 y-position versus x without bump yEPd — y. E^Pa -50 0 x (mm) 50 100 430 425 420 415 6" B 410 >, 405 400 395 3901— -100 y-position versus x with bump I — -50 0 x (mm) 50 100 -100 ky versus x without bump -50 0 X(m) Actual Desired 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ^ versus x-position with bump -50 0 x (mm) Actual Desired 50 100 Force versus x-position without bump Force versus x-position with bump -100 0 50 x (mm) Test #6 - v =30mm/s, Y =410mm, k =800N/m x ep y Figure J.6 - Contact test #6 Appendix J 156 y-position versus x without bump y-position versus x with bump S z 1600 1500 1400 1300 1200 1100 1000 900 800 700'— -100 ky versus x without bump -50 0 X(m) Actual Desired 50 100 1600 1500 1400 1300 1200 >, 1100 1000 900 800 700 -100 ^ versus x-position with bump -50 0 x (mm) Actual — Desired 50 100 Force versus x-position without bump Force versus x-position with bump Test #7 - v =30mm/s, Y =410mm, k =800N/m x ep y Figure J.7 - Contact test #7 Appendix J 157 y-position versus x without bump y-position versus x with bump 100 100 1600 1500 1400 1300 e 1200 z >. 1100 1000 900 800 700 -100 ky versus x without bump Actual Desired -50 0 X(m) 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 700 — —100 ^ versus x-position with bump Actual Desired -50 0 x (mm) 50 100 Force versus x-position without bump Force versus x-position with bump -50 0 x (mm) Test #8 - v =30mm/s, Y =410mm, k =800N/m x ep y Figure J.8 - Contact test #8 Appendix J 158 y-position versus x without bump y-position versus x with bump 100 Z -100 ky versus x without bump 1600 Actual 1600 1500 — Desired 1500 1400 1400 1300 1300 1200 e 1200 z 1100 1100 1000 1000 900 900 800 800 700 700 -50 0 X(m) 50 100 -100 ^ versus x-position with bump -50 0 x (mm) Actual — Desired 50 100 Force versus x-position without bump Force versus x-position with bump Test #9 - v =30mm/s, Y =415mm, k =800N/m x ep y Figure J.9 - Contact test #9 Appendix J 159 y-position versus x without bump y-position versus x with bump 700'— —100 k versus x without bump Actual — Desired -50 0 X(m) 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 700 — —100 k versus x-position with bump -50 0 x (mm) Actual Desired 50 100 Force versus x-position without bump Force versus x-position with bump -100 0 50 x (mm) Test #10 - v =30mm/s, Y =405mm, k =1 lOON/m x ep y Figure J. 10 - Contact test #10 Appendix J 160 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 1200 '» 1100 1000 900 800 7001— -100 ky versus x without bump Actual Desired -50 0 X(m) 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ^ versus x-position with bump Actual Desired -50 0 x (mm) 50 100 Force versus x-position without bump Force versus x-position with bump Test #11 - v =30mm/s, Y =410mm, k =1 lOON/m x ep y Figure J . l 1 - Contact test #11 Appendix J 161 y-position versus x without bump y-position versus x with bump 100 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ^ versus x without bump —— Actual — Desired -50 0 X(m) 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ^ versus x-position with bump Actual — Desired -50 0 x (mm) 50 100 Force versus x-position without bump Force versus x-position with bump Test #12 - v =30mm/s, Y =410mm, k =1 lOON/m x ep y Figure J.12 - Contact test #12 Appendix J 162 y-position versus x without bump y-position versus x with bump 7001— -100 ^ versus x without bump -50 0 X(m) Actual Desired 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ^ versus x-position with bump Actual — Desired -50 0 x (mm) 50 100 Force versus x-position without bump Force versus x-position with bump -100 Test #13 - v =30mm/s, Y =410mm, k =1 lOON/m x ep y Figure J.13 - Contact test #13 Appendix J 163 y-position versus x without bump y-position versus x with bump 100. -100 ky versus x without bump -50 0 X(m) Actual — Desired 50 100 1600 1500 1400 1300 ? 1200 z 1100 1000 900 800 700 -100 ky versus x-position with bump -50 0 x (mm) Actual Desired 50 100 -100 Force versus x-position without bump -50 • Actual — Predicted 0 50 x (mm) 100 Force versus x-position with bump Test #14 - v =30mm/s, Y =415mm, k =1 lOON/m x ep y Figure J.14 - Contact test #14 Appendix J 164 y-position versus x without bump y-position versus x with bump 7001— -100 ky versus x without bump Actual Desired -50 0 X(m) 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ^ versus x-position with bump -50 0 x (mm) Actual Desired 50 100 Force versus x-position without bump Force versus x-position with bump -50 0 x (mm) Test #15 - v =30mm/s, Y =405mm, k =1400N/m x ep y Figure J . 15 - Contact test #15 Appendix J 165 y-position versus x without bump y-position versus x with bump 100 1600 1500 1400 1300 1200 1100 1000 900 800 700'— —100 ky versus x without bump Actual Desired -50 0 X(m) 50 100 1600 1500 1400 1300 1200 1100 1000 900 800 700 -100 ^ versus x-position with bump -50 0 x (mm) Actual — Desired 50 100 Force versus x-position without bump Force versus x-position with bump -50 0 x (mm) Test #16 - v =30mm/s, Y =410mm, k =1400N/m x ep y Figure J.16 - Contact test #16 Appendix J 166 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 E 1200 Z 1100 1000 900 800 700 -100 k versus x without bump -50 0 X(m) Actual Desired 50 100 S z 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ^ versus x-position with bump -50 0 x (mm) Actual — Desired 50 100 Force versus x-position without bump Force versus x-position with bump -50 0 x (mm) Test #17 - v =30mm/s, Y =410mm, k =1400N/m x ep y Figure J.17 - Contact test #17 Appendix J 167 y-position versus x without bump y-position versus x with bump -100 ^ versus x without bump 1600 1600 1500 ^^_^_ = _ = ^_____^ 1500 1400 1400 1300 1300 1200 B 1200 Z 1100 > 1100 1000 1000 900 900 800 Actual 800 700 — Desired 700 -50 0 X(m) 50 100 -100 ^ versus x-position with bump -50 0 x (mm) Actual — Desired 50 100 Force versus x-position without bump Force versus x-position with bump -50 0 x (mm) Test #18 - v =30mm/s, Y =410mm, k =1400N/m x ep y Figure J. 18 - Contact test #18 Appendix J 168 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ^ versus x without bump -50 0 X(m) Actual — Desired 50 100 1600 1500 1400 1300 £ 1200 Z >> 1100 1000 900 800 700 -100 ky versus x-position with bump -50 0 x (mm) Actual Desired 50 100 30 25 20 Z 1 15 o It 10 5 Force versus x-position without bump Force versus x-position with bump 0 1— -100 -50 Actual Predicted 0 50 x (mm) 100 Test #19 - v =30mm/s, Y =415mm, k =1400N/m x ep y Figure J.19 - Contact test #19 Appendix J 169 y-position versus x without bump y-position versus x with bump -100 ky versus x without bump -50 Actual Desired 0 X(m) 50 100 1600 1500 1400 1300 s 1200 z 1100 1000 900 800 700 -100 ky versus x-position with bump Actual Desired -50 0 x (mm) 50 100 Force versus x-position without bump Force versus x-position with bump Test #20 - v =75mm/s, Y =405mm, k =800N/m x ep y Figure J.20 - Contact test #20 Appendix J 170 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 B 1200 Z 1100 1000 900 800 700 -100 ky versus x without bump Actual Desired -50 0 X(m) 50 100 1600 1500 1400 1300 f 1200 % 1100 1000 900 800 7001— -100 ^ versus x-position with bump -50 0 x (mm) Actual — Desired 50 100 Force versus x-position without bump Force versus x-position with bump -50 0 x (mm) Test #21 - v =75mm/s, Y . =415mm, k =800N/m x ep y Figure J.21 - Contact test #21 Appendix J 171 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 a 1200 z 1100 1000 900 800 700 -100 ky versus x without bump -50 0 X(m) Actual Desired 50 100 1600 1500 1400 1300 £ 1200 Z w ^ 1100 M 1000 900 800 700 -100 k versus x-position with bump Actual Desired -50 0 x (mm) 50 100 Force versus x-position without bump Force versus x-position with bump -100 100 Test #22 - v =75mm/s, Y =405mm, k =1400N/m x ep y Figure J.22 - Contact test #22 Appendix J 172 y-position versus x without bump y-position versus x with bump 1600 1500 1400 1300 1200 1100 1000 900 800 700 — —100 ky versus x without bump Actual Desired -50 0 50 100 X(m) 1600 1500 1400 1300 1200 1100 1000 900 800 7001— -100 ky versus x-position with bump -50 0 x (mm) Actual Desired 50 100 Force versus x-position without bump Force versus x-position with bump 0 50 x (mm) Test #23 - v =75mm/s, Y =415mm, k =1400N/m x ep y Figure J.23 - Contact test #23 

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