Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2004-0410.pdf [ 11.61MB ]
Metadata
JSON: 831-1.0080794.json
JSON-LD: 831-1.0080794-ld.json
RDF/XML (Pretty): 831-1.0080794-rdf.xml
RDF/JSON: 831-1.0080794-rdf.json
Turtle: 831-1.0080794-turtle.txt
N-Triples: 831-1.0080794-rdf-ntriples.txt
Original Record: 831-1.0080794-source.json
Full Text
831-1.0080794-fulltext.txt
Citation
831-1.0080794.ris

Full Text

Equilibrium Point Control of a Programmable Mechanical Compliant Manipulator  by Damien Clapa B.Sc. University of Alberta  A THESIS S U B M I T T E D 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 F O R T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E 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  T H E U N I V E R S I T Y OF B R I T I S H 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)  Title of Thesis:  Degree:  Department of  &^Ov\\WvU.W\  M.A-Sc.  fcv'v'V  lAgcWxv^oJ, E  HleoWxvy C  The University of British Columbia Vancouver, B C  Date (dd/mm/yyyy)  CL \  (E.  Cc^vVr^  C>£  '  \  C a A c C X ^ KKoJa(x.  ^ ^ ^ Y e a r :  v^vAxs\Q  ec \  ^  V\QA —3  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 E P Hypothesis becomes possible. The E P 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 E P of each joint of the manipulator. A benefit of this combined E P 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 E P 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 Table of Contents List of Tables .-. List of Figures Nomenclature Acknowledgements Chapter 1 Introduction  ii iii v vi x xiii 1  1.1 Case for assistive robots and personal care 1.2 What w i l l the robots need to be capable of? 1.3 A i r Muscles 1.4 E P control 1.4.1 Background 1.5 E P Control and P M C Actuators 1.5.1 Interaction Tasks 1.6 Scope and Objective 1.7 Outline o f Thesis ...7.  Chapter 2 Air Muscle Design  1 3 4 4 5 6 8 8 9  10  2.1 Introduction 2.2 System Overview 2.3 A i r Muscle Properties 2.3.1 Observed Limitations 2.3.2 Geometric Models o f A i r Muscles 2.3.3 The empirical Modification to above model 2.4 Symmetric sizing method 2.5 Summary  Chapter 3 Electro-mechanical design and control of a PMC robot 3.1 Introduction 3.2 Manipulator 3.3 Valves 3.3.1 Sizing V a l v e Orifices 3.3.2 Selecting Constant Frequency for operation 3.4 Instrumentation, Drivers and D A Q 3.5 Controller : 3.6 Summary :  Chapter 4 Experimental Methods  10 10 12 13 13 15 17 25  26 26 26 31 32 35 36 36 40  42  4.1 Introduction 4.2 Free-Space Testing 4.2.1 Description o f the test 4.2.2 Experimental measurements 4.3 Transition Testing... ; 4.3.1 Description o f the test  42 44 44 45 45 45 iii  Table o f Contents  iv  4.3.2 Experimental measurements 4.4 Contact Testing 4.4.1 Description o f the test 4.4.2 Experimental measurements 4.5 Summary  46 47 47 50 50  , ;  Chapter 5 Results and Discussion  51  5.1 Introduction 5.2 Free-space task results 5.3 Transition Results 5.4 Contact Results 5.4.1 Velocity, Stiffness and E P Results 5.4.2 Repeatability Results 5.5 Summary  51 ....52 57 62 ...69 72 72  Chapter 6 Conclusions & Recommendations  .  6.1 General recommendations 6.2 Specific Recommendations for this Experimental W o r k  Bibliography Appendix A Air Muscle Equations  .....78 81  A . l A i r Muscle Equations for the Appendix  Appendix B Muscle Construction Appendix C Matlab Optimization Files C.l C.2 C.3 C.4  81  83 88  .'  OptimizeMountLength.m solverbn.m StiffnessLm minimizethis.m  88 89 91 92  Appendix D Manipulator Bill of Materials Appendix E Detailed Machining Drawings Appendix F Assembly Instructions Appendix G Sensors and Calibrations G . l Experimental Equipment Specifications G.l.l Solenoid Valves... G . 1.2 Sensotec Pressure Transducers G . 1.3 AutoTran Pressure Transducers G . 1.4 Force Transducer G . 1.5 Length Encoders G . 2 Calibration o f Equipment G.2.1 Calibration o f Sensotec Pressure Transducers G.2.2 Calibration o f Auto Tran Pressure Transducers G.2.3 Calibration o f Precision Transducers Force Transducer G.2.4 Calibration o f U S Digital Length Encoders  Appendix H H . 1 Free- Space Tests H.2 Contact Tests H.3 Transition Tests  Appendix I Summary of Transition Tests Appendix J Summary of Contact Tests...  74 75 76  93 97 123 126 :  :  126 128 129 130 131 132 133 133 135 ....136 137  138 138 138 139  140 149  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  .'  )  v  96  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 E P 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 - M a x i m u m 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 o f the manipulator  30  Figure 3.5 - Torques for 2 0 N 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  vii  Figure 4.2 - Diagram of the range o f 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 x p versus x for free-space tests  53  Figure 5.3 - Error in x vs. x  54  Figure 5.4 - Error i n k versus x  55  E  a  y  Figure 5.5 - M e a n Absolute Error of k versus v  56  Figure 5.6 - M e a n absolute error XEP versus v  56  y  x  v  Figure 5.7 - Transition test #1 y and yspd with and without the wall versus path  57  a  Figure 5.8 - Transition test #1  yepa and yepa with and without the wall versus path  Figure 5.9 - Transition test #1 k  58  and k d with and without the wall versus path  ya  59  y  Figure 5.10 - Transition test #9 k  and k  ya  yd  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 k versus x  65  Figure 5.15 - Test#10 with bump k versus x  65  y  y  Figure 5.16 - Test#10 force versus x without bump Figure 5.17 - Test#10 force versus x with bump  "..  66  .'  67  Figure 5.18 - Summary ofy£7> error  70  Figure 5.19 - Summary of k error  71  Figure 5.20 - Summary of force error  72  y  Figure B . l - Tools and supplies to make air muscles Figure B.2 -  Soldering the brass inserts  ,.•  84 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 - E n d loop of the air muscle  ...86  List of Figures  viii  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 i n k 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 i n k drive assembly  103  Figure E.7 - Encoder and gear  104  Figure E.8 - B o x  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  Ill  Figure E . l 5 - Pulley 1  ,  112  Figure E . l 6 - B i g gear  113  Figure E . l 7 - Timing belt sprocket 1  114  Figure E . l 8 - Pulley 2 Figure E. 1 9 - S h a f t 1  : !  115 116  Figure E.20 - Link 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  :  Figure 1.8 - Transition test #8 Figure 1.9 - Transition test #9  146 147  .'  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 A  Area of orifice  dlj  Inner surface displacement  dst  Area vector  J  Jacobian of manipulator  e  K  Carestian end point stiffness matrix  Kj  Joint space stiffness matrix  L  Length of an air muscle  c  L  Mounting length of an air muscle  mount  Longest length of muscle at end of range  L  max  Li  Shortest length of muscle at end of range  L  Length at which muscle delivers no axial force  P  Absolute pressure inside air muscle  Patm  Atmospheric Pressure (1 bar at sea level)  P  Gauge pressure inside air muscle  m n  zero  g  P  Maximum allowable muscle gauge pressure  P in  Minimum allowable muscle gauge pressure  R  Gas constant  T  Temperature of air inside muscle  Si  Inner surface displacement  max  m  V  X  -  Volume inside of muscle Cartesian end point trajectory  X  a  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  f  Maxium available muscle force  kt  Stiffness of joint i  k  Stiffness of a single muscle  max  m  k  Maximum available joint stiffness  k  Minimum available joint stiffness  k  x  X-axis end point stiffness  k  xy  Cartesian cross stiffness term  max  min  k  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  p  Stagnation pressure  y  0  r  Pulley radius  v  X-axis velocity of end point  x  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  x  a  Nomenclature  xn  y  a  Actual Y - a x i s position of the end point  y  EP  Y - a x i s endpoint equilibrium position  ys  Actual Y - a x i s endpoint equilibrium position  y d  Desired Y - a x i s endpoint equilibrium position  K  Muscle Stiffness  T  Joint torque  pa  Ep  x  M a x i u m available joint torque  6  Joint angle  6  Actual joint angle  6d  Desired joint angle  0p  Equilibrium joint angle  max  a  E  9  M a x i u m angular range of motion  max  ADL  Activities of daily living  EP  Equilibrium Position  MAE  Mean average error  PPC  Programmable passive compliance  PMC  c  Programmable mechanical compliance  PWM  Pulse Width Modulation  RMSE  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 m y supervisors, Dr. Elizabeth Croft and Dr. Antony Hodgson, for their guidance and assistance. I would also like to thank the many fellow graduate and undergraduate students at U B C who have helped i n so many ways. The assistance o f the faculty and staff o f the Mechanical Engineering Department was greatly appreciated. I would also like to acknowledge the financial support o f the Natural Sciences and Engineering Research Council o f Canada.  v  xiii  Chapter 1 Introduction 1.1 Case for assistive robots and personal care In the coming decades, there will be increased demand for nontraditional technologies, such as robotics, for the care of an increasingly dependent elderly population. This increased demand will 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 w i l l be disabled w i l l quite likely be no different than it was 25 years ago [1]. The increasing number of disabled elderly people w i l l likely outpace any growth in either formal or informal care sources, and will 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 if they are living on low or fixed incomes. A s 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. A t 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 will 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 ( A D L ) . 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 will 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 will 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 will 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 will 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 will be to investigate the potential of this class of actuators for the design and control of safe interactive robots. In this work we will 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 will 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 will 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 humaninteraction type task is developed.  1.3 Air Muscles O f 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 E P 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 if 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 will not exactly follow the virtual trajectories but instead will 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 E P 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 E P 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 E P 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 E P 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 E P 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 M c K i b b e n (air muscle) actuators [16, 17, 22-24]. These actuators behave in many respects similarly to human muscles [33]. B y 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 E P 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 i n both muscles of an opposed pair remain balanced, the joint w i l l not move, but its stiffness w i l l increase. Imbalances in the forces o f 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 M c K i b b e n 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 o f 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 B i c c h i [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 E P 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 w i l l increase. Imbalances in the forces o f the two muscles i n an opposed pair w i l l cause a change in the equilibrium angle of the joint  (OEP),  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 M c K i b b e n 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 o f 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 B i c c h i [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 o f 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 living include free space, transition and contact tasks. In this work we w i 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 freespace, 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 E P 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 like 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 E P 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 E P to error i n the mass of air in each muscle, (iii) an E P 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, E P 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 M c K i b b e n 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 w i 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 E P 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 well, in the second part o f 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 E P inspired controller o f a P M C manipulator. The system envisioned for demonstrating the three experimental tasks chosen is diagramed i n Figure 2.1 below. The central disk pictured at the base o f 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 o f the potentiometer shown i n the figure to give both the current link angles and the muscles lengths.  10  11  2.2 System Overview  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 Regulator  \  |  Air Muscle  \  Valve  Air Supply Figure 2.2 - Additional components required for powering the air muscle  12  2.3 A i r Muscle Properties  2.3 Air Muscle Properties M c K i b b e n muscles principally consist o f 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 o f a section o f air muscle (from Shadow Robot Company) According to the Shadow Robot Company, a 6mm diameter air muscle has the "strength, speed and fine stroke o f a finger muscle in a human hand" and "an A i r Muscle 30mm i n diameter is capable o f lifting more than 70 K g at a pressure o f only four bar"[29]. The air muscle exerts its maximum force at maximum extension. A s extension decreases, the force that it exerts decreases at a decreasing rate. This means that small changes i n 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  14  2.3 A i r Muscle Properties  the air muscle originally appears i n Chou but was modified by Colbrun [26] to a more useful form. H i s formulation is outlined below. Neglecting the frictional losses, the work done on the system w i l l equal the work extracted from system.  dW = J  (P - P  in  otm  )dl • ds, ={P- P i  alm  J Surface  dl, • ds =P dV  )J  t  J Surface  g  (2.1)  *  Where: P=Absolute  internal gas pressure  Patm Atmospheric =  pressure  P =Gage pressure g  p i n n e r surface displacement dsi=Area vector J/,=Inner surface displacement dV=V o l u m e Change Chou shows that this ultimately yields Equation 2.2 below. The rest o f the formulation can be found i n Appendix B . The force generated by a muscle is a function o f two geometric properties, b and n, and the internal pressure (P ) and the length (L) o f the muscle. The constant g  b is equal to the length o f the nylon strands in the braid i f they were pulled straight. The constant n is equal to the number o f turns i n the helix that makes up the braid.  /=  (2.2)  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:  3L /b = 1 2  2  as derived from Equation 2.2.  (2.3)  15  2.3 A i r Muscle Properties  2.3.3 The empirical Modification to above model The geometric force model for M c K i b b e n 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. 3Z  2  (2-4)  Am  The air muscles chosen for this work had the following 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 o f 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 o f 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 o f lengths and force. This level of error is considered sufficient for our application. U s i n g calculated force in place of force transducers in our system results in only small errors in force.  17  2.4 Symmetric sizing method  0.34  0.345  0.35  0.355 0.36 0.365 Muscle Length (m)  0.37  0.375  Figure 2.8 - Absolute error i n 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 o f how one might select the most appropriate muscles for a given task. A s this research is primarily intended to show the benefits o f 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 L , mount  r, b and n can be  chosen to yield different available ranges o f k and 6EP as well as joint torque (r). Because o f the properties o f the actuator, the true range o f available k and T w i l l vary with the actual angle o f the joint (9n). The goal o f the method described below is to ensure between the desired limits o f 6EP the joint w i l l possess the ability to achieve a prescribed range o f 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.  A l s o , 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 k  max  and k  min  There are also several constraints that are relevant: maximum axial force in muscle is a constant across all muscle sizes and defined asf . After building several muscles and exposing them to max  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 limiting factor may change. The maximum inflation pressure is assumed to be constant across all muscle sizes and is defined as P . The minimum inflation pressure is a max  constant across all muscle sizes and defined as P . min  A l s o , 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 w i 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  Pressure  Force  Length Figure 2.10 - Force versus length at different pressures Additional assumptions for this method include the following. The mounting length o f both muscles is equal when 8 is equal to zero; for example, L  j=L  2.  momt  mount  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  L . zero  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  6  .  max  2.4 Symmetric sizing method  20  Lmount  Figure 2.12 - Working range o f each muscle N o w the maximum and minimum working lengths o f each muscle can be defined: (2.5)  A™  = n,o«m L  - <*" A  (2.6)  Two constraints must hold at this point: (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.  21  2.4 Symmetric sizing method  Figure 2.13 - Configuration where maximum and minimum stiffness most constrained In this configuration muscle 1 is at length L  min  and muscle 2 is at length L . max  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  L2-L  m a x  Figure 2.14 - M a x i m u m torque Constraint In this configuration the maximum torque that can be generated is defined by:  7  max  if  act  Where f  A  A  fB  )  r  is equal to either  (2.9)  22  2.4 Symmetric sizing method  /, =/ ( £ /(Z  P  m i n ; J  ,^)  m i n  )</  r a M  (2-10)  i f  m a x  or  (2.11) (2-12)  f =f A  ma  and (2.13)  f =f(L ,P )B  T ax m  w  miB  must satisfy the constraint:  T  >T  max _ act —  •  (2.14) V  max  /  To investigate the joint stiffness constraints, the stiffness relations of the air muscles are established. Rearranging the Equation 2.2, yields: (2.15)  f = P 3AL -BP 2  g  g  1 where, A=—-,  b  2  and B =  Am  .  Ami  2  Each muscle volume[30] is calculated as: (2.16)  V = BL-AL . 3  Thus, the change in volume with respect to length is <p = ~  aL  (2.17)  = B-3AL  2  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-ip +P 6LA 1  V  2  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.  (2.18)  23  2.4 Symmetric sizing method  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 i n 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 i m  n  L -L 2  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  (2.20)  fc  ^min  k  min  act  ~  act  —  ^  <k  ( c K  +  K  B  (2.21)  )  (2.22) min  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 in m  L -L 2  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  —  ^max  k  max  (2.23)  fA  act  ~  ^( A  >k act  K  +  (2.24)  D)  K  (2.25) max  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 ^ ^=30Nm/rad, m  2.5 Summary  & /„=15Nm/rad, T m  25  OTax  =3.1Nm and  n=5.74, r=90mm and L  mow  O = n/\6 the following values were found: 6=514mm, max  „ =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  mou  „ , = 3 6 0 m m are predicted by the same equations used in  the optimization to yield a joint with: k =\S.O, k =l>2.% Nm/rad and r min  with a r=90mm pulley over the same  6  max  max  = 6.15N when used  .. In reality, the working stiffness range was slightly  max  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). Additionally, 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 o f 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 o f electronics including sensors, valve drivers and the D A Q and computer to implement the EP-controller i n an experimental setup. Additionally, the controller development is detailed. This chapter w i 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 o f the stiffness o f 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.  28  3.2 Manipulator  Table 3.1 - Design Requirements  Dimensions L i n k length Pulley diameter  227mm =180 m m  Forces M a x i m u m Y - a x i s force M a x i m u m force on a pulley  Configuration  Material  -20 N -200 N - Angle o f the second joint independent o f the first - Design must include two encoders - Easy to install on the lab table - The motion must be i n the horizontal plan A l l custom parts i n aluminum or steel  The sizing o f the manipulator and the choice o f appropriate air muscles were inherently linked. The final sizes chosen were eventually derived from a few simple constraints imposed at the beginning o f 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 b i l l o f materials for the assembled manipulator can be found in Appendix D. A s shown i n Figure 3.3, the two links are driven from the base o f the manipulator. The distal joint is driven from the base through a timing belt and pair o f sprockets. This allows the air muscles to be longer than the links and also reduces link mass and complexity o f 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 o f 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 o f 2 0 N in the Y - a x i s direction.  3.3 Valves  31  - Joint 1 Joint 2  §  0  O  -2  -3 •150  -100  -50  0 x (mm)  50  100  150  Figure 3.5 - Torques for 2 0 N 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 o f ways that the state o f 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 o f 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 w i l l be discussed in Section 3.5, a control strategy that varies mass  32  3.3 Valves  flow rate to cause the joints to follow EP and K trajectories w i l l be developed. The output of this controller is a duty cycle to the valves. Matrix 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 o f 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. A s well, 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 flow as the back pressure increases past the critical values. These relations are useful in sizing the orifice and theoretical mapping of the mass flow rate. The pressure drop over the orifice governs whether the flow is choked. The critical back pressure to stagnant pressure ratio is: ^ - = 0.5283  (3.1)  Po  This ratio value is specific for air. Here, p* is the critical back pressure at which the flow becomes sonic. The stagnation pressure , p , is the pressure of the air with no velocity. For any 0  back pressure lower than the critical pressure, the flow through the orifice is choked. Under these conditions, the mass flow rate is independent of the back pressure. 0.6847 A Po  Wl m a x  —  (RT f  2  0  e  (3.2)  3.3 Valves  3 3  It is assumed that the supply air is at room temperature and is stagnant. A is the area o f the e  orifice and R is the gas constant. When the back pressure has increased such that the flow is subsonic, the calculations are more complex. A t subsonic conditions, the back pressure is equal to the pressure in the orifice. N o w the mass flow rate is dependent on the back pressure as illustrated in the equation below,  (3.3)  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.  Inlet Solenoid Orifice Plate  Manifold Exhaust  Supply  To muscle  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 W i t h 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 flow 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  35  3.3 Valves  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 0  1  1  1  1  1  1  2  4  6  8  10  12  1 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 flow 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 o f 500 H z would result i n two flows being allowable, either closed or open for the minimum pulse. In reality the valves have a maximum recommended operating frequency o f 200 H z . Instead at least a 10:1 turndown ratio for the valve was selected. The quantity o f 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 o f 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 flexibility o f that package and the wealth o f examples and support available. A l l o f 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 i n all Labview code used i n this research. The sensors were recalibrated as required throughout the experiments.  3.5 Controller Using the force model given i n Equation 2.4, a controller was developed to allow for the simultaneous control o f both joint stiffness and equilibrium position o f 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 H z .  37  3.5 Controller  -K  Cartesian to joint space stiffness Equation  -K,  3.6  AO. ep  Inverse Kinematics Equations  EPd  Decouple Equation 3.8  -Am*-  PI  -DC*-  Plant  Computations Equations: 3.11 &3.12  -Pressure, 0 -  -9.epd  3.4  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' = t a n - ( ± V l - D ) 1  (3.4a)  2  2  J  Where D is given by 2  ^  ,  2  2  D = cos0 = 2  ^  2  - a , —a,  x +y  2a a x  !  2  (3.4b)  -  2  Where aj is the link length. Y i e l d i n g :  3=  a  sin0  2  t a n "  2  (3.4c)  t a n  1  a + a cos# x  2  2  and 0  2  (3.4d)  — 6 + 0 X  2  Next, the stiffness is transformed from Cartesian to joint space. The Cartesian stiffness matrix, K is defined as c  K.  (3.5)  Cartesian stiffness and position equilibrium point trajectories are generated and converted to joint space with inverse kinematics. k is prescribed and k is solved to satisfy ki=k2. This y  x  3.5 Controller  38  constraint minimizes the amount of gas used over the prescribed task. k =k xy  yx  is solved such that  the cross terms in the joint space stiffness matrix are zero, reflecting the physical nature o f the system.  K =  k  x  0  0  fc,  [32]  = JKJ T  C  (3.6)  Where the manipulator Jacobian, J , is: - a sin(0,) {  J  a, cos(0!)  - a  sin(0 ) 2  2  (3.7)  a cos(0 ) 2  2  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, 9 , along with the most recent observation of ep  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, Am,  dk  dk  dm, dd„„eq  dm  Ak  deeq  A 0 EP  dm,  dm-.  2  (3.8)  In order to obtain these derivatives, one can note that torque in each joint is given by:  T=  k(0-9 ),  (3.9)  EP  and can also be represented by,  T=  (3.10)  r{f -f ) l  l  The joint stiffness is:  k =  r (K -K ),  (3.11)  2  x  2  Solving Equation 3.9 for 0 a n d substituting from Equations 3.10 and 3.11 yields, £/>  39  3.5 Controller  a  —a  V E P -  0  fi)  (/i  (3.12)  —7—;—\ • [K +K )  r  X  2  The partial derivatives required for the decoupler are then:  dk dm.• = r  (3.13)  d_K^  f  (3.14)  ML  deEP  dm.  dm  (fx-A)  r(ic + K )  x  x  2  2  r  x  r(K + K )  dm  x  2  2  (K  r  2  + K f  x  2  (3.15)  x  if-fi)  dm.  dd  3*.  (K + K ) dm '  (3.16)  dm  2  Equation (2.4) yields:  f=  P 3AL -BP -c.  (3.17)  2  v  V  The equation derived for joint stiffness, X"in Section 2.4 is still valid even with the constant c i n the above equation.  K = ^dL  =mR^(b V  +P6LA.  2  1  (3.18)  For each muscle the change in muscle stiffness with respect to mass is  * K _  dm  3A L*+B 2  R  T  7  (3.19)  L (-B~ + AL ) 2  2  2  and the change in force with respect to mass is  df ^(-B + 7>AL ) — = -RT'L(- B + AL )' jm 2  2  (3.20)  3.6 Summary  40  The muscle mass errors for each muscle then enter the P I D 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 o f each joint and the current actual E P o f 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 flow to the air muscles. The valves were chosen for their speed and suitability for use with P W M control. Orifice 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  A n E P 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.  41  Chapter 4 Experimental Methods 4.1 Introduction >  The experimentation described i n this chapter was designed to show the strengths and weaknesses o f the E P controller, coupled with the air-muscle actuated robot, i n free-space, contact and transition tasks. Three sets o f experiments were performed, one for each type o f task. The experiments are summarized i n Table 4.1. The desired trajectory i n 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 i n 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  46  800N/m llOON/m 1400N/m  15 mm/s 30 mm/s 75 mm/s  Bump/No Bump y = 405,410,415mm (nominal)  1000N/m  2 mm/s 5 mm/s 10 mm/s  3 approach angles: 30,60,90°  Contact  Transition  9  EP  A complete list of all testing performed is provided in Appendix H . Figure 4.1 below shows the location of the X and Y - a x i s 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 H z . 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 follow 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 E P 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 - a x i s 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 - a x i s , k was set to 1200 y<  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 o f useful operating velocities for the manipulator. This range is limited by the size of the muscles and the speed at which the valves can fill them. A t some commanded velocity the manipulator w i 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 E P 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. M a k i n g this switch requires sensing the moment of contact and stable methods to switch smoothly from one controller to another. E P 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 - a x i s 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 60° 90<*  \  Surface  r, y = 400mm  ,  i  \  Figure 4.3 - Diagram of transition task Table 4.2 below is a list o f the test numbers for the different combinations o f velocity and approach angles investigated in this set o f tests. Table 4.2 - List o f 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 o f 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 " b u m p " 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 w i 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.  Figure 4.4 - Diagram o f contact test 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 - a x i s (k ) stiffness levels: 800N/m, 1 lOON/m and 1400N/m. The equilibrium y  path was chosen to maintain a constant force in the absence of a disturbance, The E P 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  48  4.4 Contact Testing  compliance. This was modeled as a varying contact stiffness across the surface and the E P 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 o f the wall. The dashed line indicates the nominal ygp; this is the E P trajectory that would yield the desired force i f the wall was infinitely stiff and the principle directions o f the Cartesian stiffness matrix o f the manipulator were perfectly aligned (normal and perpendicular) to the wall (k^ =0). In fact at the one point in the trajectory, jc=0mm, the cross coupling term (k ) is zero, and then the nominal xy  trajectory yields the desired force with a stiff wall. Points A and C show how the E P 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 E P trajectory must be adjusted deeper into the surface to achieve the desired force due to the compliance o f the wall. Point B and C show the result of this method at the middle and positive end o f the trajectory.  Figure 4.5 - Diagram o f E P 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 yEP  405 m m 410 mm 415 mm  30 mm/s  800 N/m 1  1400 N / m 3  2  4  75 mm/s  800 N / m 1100 N/m 1400 N/m 5 10 15 6,7,8 11,12,13 16,17,18 9 14 19  800 N/m 20  1400 N/m 22  21  23  -  IS  -^fjffljS! :  Figure 4.6 - Contact test with a smooth wall  v..  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 E P 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 E P 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 ( M A E ) is used as a measure of the deviation from the desired value whenever error for a data set is discussed.  \measured -predicted \+\measured -predicted^... + \measured„ -predicted^ l  [  1  ^ ^  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  52  5.2 Free-space task results  5.2 Free-space task results Figure 5.1 shows the x component o f the desired equilibrium point trajectory (xspd) for different x-direction Cartesian velocities (v ). The y component (y>EPd ) is equal to a constant value of x  400mm for the entire trajectory. Since there is no surface for the manipulator to interact with, in this test the actual position o f the end-point, X„ (where X =[x a  a  y ]) should be close to the n  commanded XEP (where XEP=[XEP ysp]) for slow movements and diverge as dynamic effects create joint torques.  -80  1  1  0  5  1  1  10 Time (s)  15  1  20  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 - a x i s equilibrium position, XEP, is small and increases as the velocity o f the manipulator endpoint X - a x i s velocity ,v  X}  increases. The change i n 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, i n fact, have the largest errors. For v = 15mm/s and 30mm/s x  trajectories, the error converges as the manipulator has time to correct. The v^=50mm/s case  53  5.2 Free-space task results  takes nearly then entire length of the surface to approach zero error. The 150mm/s case does not converge.  15 10  I  x a  -5 r  \ v.. —  g  v = 15mm/s X  -10  v =30mm/s X  -15  _ . v =50mm/s _ . v =150mm/s  -20 -80  -60  -40  -20  0  20  40  60  80  x (mm) Figure 5.2 - Error in x p versus x for free-space tests E  If there were no (or small) dynamic effects, (i.e. for the low velocity experiments) the X-axis equilibrium position, x p, and X-axis actual position, x , position would be expected to be near E  a  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 i n 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 i n computed torques in the controller result in XEP varying from the x over the a  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  25 20 15 10  x  5  l  0  x  -5  o  -10 — v =15mm/s  -15  X  — v =30mm/s X  -20  . v =50mm/s X  -25 -80  _ v =150mm/s  -60  -40  -20  0 x (mm)  20  40  60  80  Figure 5.3 - Error in x vs. x a  The set point for Y - a x i s stiffness, k , i n Figure 5.4 below was 1200 N / m throughout the range o f y  the motion. The error in k increases as the velocity of the endpoint increases. The trend is y  similar to the error in X - a x i s equilibrium position, XEP- A g a i n for the case where X - a x i s velocity, v is 150mm/s, the value for Y - a x i s end-point stiffness, ky, does not converge to the set point x  over the duration of the test.  5.2 Free-space task results  55  100  Figure 5.4 - Error in k versus x y  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 limiting 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.  56  5.2 Free-space task results  140 120  §1  100[  o  80  & 3  60  <  40  c  a  20 0 0  50  100  Velocity (mm/s)  150  Figure 5.5 - Mean Absolute Error of k versus v y  O  16  x  x  MAE  C  14 12 o fc wcu 10 •i—»  _3  "3  8  <  6  x> cu  O  c>  4  +  2 0  h  4H++*  '  +  50  Velocity (mm/s)  100  Figure 5.6 - Mean absolute error x p versus v E  150  x  57  5.3 Transition Results  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 k stiffness was set y  to 1000 N/m. Without a wall present the commanded Cartesian end-point trajectory, y g , and the P  actual Cartesian end-point >> (measured) trajectories overlap very well. When the wall was put a  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  408406-  Path (mm)  Figure 5.7 - Transition test #1 y  a  and  y pd E  with and without the wall versus path  58  5.3 Transition Results Figure 5.8 shows the values of Y-axis equilibrium point trajectory, y p, both with and without E  the wall, as well as the desired Y-axis equilibrium point trajectory, y pd- Both agree very well E  with the desired trajectory. The presence of the wall does not interfere with following the y d Ep  Y-axis equilibrium point trajectory. The distinction between y d and y allows the end-point to Ep  a  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, k , must also be y  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, k , behaves somewhat differently than expected. Because the y  Cartesian end point stiffness, K is translated into joint space stiffness, Kj before the task begins, c  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 k was due to a y  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  Actual no wall Actual wall Desired  115011001050-  ?  1000' 950900850800 L 0  10  15  20  Path (mm)  Figure 5.9 - Transition test #1 k  ya  The k  ya  and k d with and without the wall versus path y  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 k errors. y  5.3 Transition Results  60  Actual no wall Actual wall Desired  10  20  Path (mm)  Figure 5.10 - Transition test #9 k  ya  and k d with and without the wall versus trajectory y  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, k , for these tests because the manipulator is very close to the Y x  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 - a x i s equilibrium position 5mm into the surface and Y - a x i s end-point stiffness of 1000 N / m .  61  5.3 Transition Results  CD O  S-l  O  0  10 Path (mm)  Figure 5.11 - Transition test #1 actual and predicted force with and without the wall versus trajectory Table 5.1 below shows a complete summary of the M e a n 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 1 v=2  2 v=2  ^ ^ ^ ^  0=30  ysp M A E no wall (m) k M A E no wall (N/m) ysp M A E wall (m) k M A E wall (N/m) F M A E wall (N)  0.10 7.09 0.10 18.60 0.56  " ^ ^ ^ Error Type  v  v  Test  0=60  3 v=2 0=90  0.11 9.01 0.12 35.15 0.80  0.15 9.70 0.13 41.60 0.93  0=30  5 v=5 0=60  6 v=5 0=90  7 v=10 0=30  8 v=10 0=60  9 v=10 0=90  0.13 7.75 0.15 19.77 0.56  4.14 17.64 0.17 35.19 0.91  0.18 11.48 0.15 41.46 0.95  0.20 7.33 0.18 18.17 0.58  0.20 12.50 0.19 32.33 0.89  0.35 26.90 0.20 52.84 0.94  4 v=5  Testing was performed using three different approach angles. N o 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  62  5.4 Contact Results  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 k and the force into the surface should remain equal over the full surface. A s y  discussed in Chapter 4, forces resulting from displacement in X were countered by adjusting the position E P 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 (v =30mm/s, ygp =405mm, k =1 lOON/m) was chosen for example results. Detailed x  y  results for all of the 23 test points with and without the bump are presented in Appendix J . The Y - a x i s equilibrium position, y p, is the more important component of the equilibrium E  position vector, XEP, for ensuring forces due to unexpected position disturbances are as predicted by the E P and stiffness trajectories. Figure 5.12 shows good agreement between the actual and desired Y - a x i s end-point position, y p and y pd- The value of Y - a x i s actual position, y , is offset E  a  E  a  as expected, due to the presence o f the wall between the manipulator, from the Y - a x i s equilibrium position, y p. E  Table 4.3 - List of all contact test numbers 15 mm/s  30 mm/s  75 mm/s  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 yep 20 22 405 m m 1 3 5 10 15 410 m m 6,7,8 11,12,13 16,17,18 2 4 21 23 415 m m 9 14 19  63  5.4 Contact Results  Figure 5.12 - Test #10 without bump (y versus x) A g a i n adding a bump does not make the task of tracking the Y - a x i s equilibrium position, y p E  particularly more difficult as shown in Figure 5.13, only the Y - a x i s actual position, y , deviates a  due to the presence of the wall. The 4mm bump is well within the capabilities of the controller's capability for rejection.  64  5.4 Contact Results  100  Figure 5.13 - Test #10 with bump y versus x It is also important that the Y - a x i s end-point stiffness, k , tracks close to the desired value for the y  forces from the contact to result close to the desired values. Figure 5.14 shows that the k values y  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 Actual Desired  1500 1400 1300  B  1200  *  1100  w  1000  r  900 800 700 -100  -50  0  50  x (m)  Figure 5.14 - Test #10 without bump k versus x y  1600r 15001400130012001100 1000900800700-100  Figure 5.15 - Test#10 with bump k versus x y  100  66  5.4 Contact Results  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 - a x i s end-point stiffness, k , was set y  to 1100 N / m and Y - a x i s 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 -100 1  -50  0 x (mm)  50  Figure 5.16 - Test#10 force versus x without bump  100  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 o f 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 o f Y - a x i s equilibrium position the error is given relative to the requested depth o f contact into the surface.  5.4 Contact Results  68  Table 5.2 - Summary of Errors without the bump present yEP  k MAE  MAE  Test#  (mm)  rel.%  (N/m)  rel. %  1  0.12  2.44%  18.49  2.31%  2  0.14  3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  0.09 0.10 0.14  0.93% 1.85% 0.69% 2.78% 1.63% 1.63% 1.67% 1.78% 2.77% 1.70% 1.57% 1.60% 1.41% 2.46% 1.69% 1.58% 1.64% 1.40% 5.02%  25.21 97.88 87.95 18.41 17.49 16.93 17.40 22.38 23.78 26.91 27.45 27.02 39.44 35.66 40.56 40.72 41.62 67.31 19.70 25.23 43.92 66.18  3.15% 6.99% 6.28% 2.30% 2.19% 2.12% 2.18% 2.80% 2.16% 2.45% 2.50% 2.46% 3.59% 3.24% 2.90% 2.91% 2.97% 4.81% 2.46% 3.15% 3.14% 4.73%  0.16 0.16 0.17 0.27 0.14 0.17 0.16 0.16 0.21 0.12 0.17 0.16 0.16 0.21 0.25 0.42 0.22 0.39  2.79% 4.36% 2.62%  FMAE  v  -  (N) 0.50 1.72 1.35 0.67 0.59 0.89 0.97 0.97 1.72 0.37 0.77 0.88 0.87 1.46 0.55 0.55 0.68 0.68 1.10 0.64 1.49 0.69 1.15  rel.% 12.41% 14.31% 19.34% 3.18% 14.78% 11.14% 12.18% 12.07% 14.36% 6.67% 6.97% 8.04% 7.94% 8.87% 10.09% 3.96% 4.88% 4.86% 5.24% 16.05% 12.42% 9.91% 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 - a x i s stiffness is also generally less than 5%. The error i n force is the largest relative to the expected value. This error was seen to be as large as 16.05%. Test case #20 {y p=A05, ft/=800N/m and v =75mm/s) generated the largest error in force. E  x  69  5.4 Contact Results  Table 5.3 - Summary of Errors with the bump present yEP  MAE  k  v  FMAE  MAE  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 4  0.09 0.09 0.14  1.87% 0.61%  20.03 82.52  1.00 1.52  14.28% 7.22%  2.89% 2.21%  15.55 14.19  1.43% 5.89% 1.94%  0.60  14.91%  1.83% 1.70% 1.44% 2.62%  10.40% 10.11%  1.80% 2.37% 1.83%  2.07%  14.40 14.37 18.93 20.11 23.74  0.83 0.81 0.84 1.32 0.60  0.18 0.17  1.78% 1.65%  23.27 22.81  2.16% 2.12% 2.07%  0.80 0.90 0.91  7.28% 8.16% 8.27%  0.26 0.12  1.71% 2.46%  36.09 31.30  3.28% 2.85%  16  0.17 0.16 0.16 0.21 0.25  1.56% 1.38% 4.96%  35.15 35.29 35.97 61.69 20.61  2.51% 2.52% 2.57% 4.41%  8.35% 17.57% 6.65%  17 18 19 20 21  1.68% 1.56%  1.38 0.97 0.93  2.58%  1.07 1.09 1.45 0.70  7.68% 7.78% 6.92% 17.58%  0.43  2.89%  25.53  3.19%  1.36  11.30%  22  0.22  4.34%  3.64%  0.97  13.79%  23  0.39  2.60%  50.98 63.77  4.56%  1.31  6.23%  5 6  0.22  7  0.18  8 9 10 11 12  0.17 0.22 0.13 0.21  13 14 15  1.77% 1.80%  10.55% 11.03% 10.93%  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 - a x i s end-point stiffness and Y - a x i s 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 - a x i s velocity (v ), Y - a x i s end-point x  stiffness (k ) and Y - a x i s equilibrium position (ysp) had on the tests run without the bump present. y  70  5.4 Contact Results  The tests chosen for these plots were four sets o f k and y p that were run at each o f the three y  E  values. Figure 5.18 is a summary plot o f the Y - a x i s equilibrium position, y p, M A E for four sets o f E  cases. This plot shows that M A E of y p increases as the X - a x i s velocity, v , increases. It also E  x  increases as the y p increases. The y p M A E decreases as the Y - a x i s end-point stiffness k E  E  y  increases for the data shown below.  0.5 +  0.45  o  Test #1,5,20 (k =800N/m, y =405mm) Test #3,15,25 (k =1400N/m, y " E P ( f Test #2,9,21 (k >800N/m, y =415mm) yd EPd Test #4,19,23 (k =1400N/m, y =415mm) yd  Epd  yd  p4  J  0.4  yd  0  5  m  m  c  )  x  Epd  0.35 0.3 J3 0.25  +  S-i  O  fcl O  W  0.2  0.15  +  X  +  o  0.1 0.05 0  0  20  40  60 v (mm/s)  80  100  Figure 5.18 - Summary of y p error E  Figure 5.19 is a summary plot o f the Y - a x i s end-point stiffness, k , M A E for four sets o f cases. y  This plot shows that M A E o f k does not appear to be correlated with X - a x i s velocity, v , or Y y  x  axis equilibrium position, y p, The observed k M A E increases as the commanded k increases E  as shown below.  y  y  5.4 Contact Results  100 r  71  o  9080-  +  Test # 1,5,20 (k =800N/m, y =405mm)  O ^ x  Test #3,15,25 (k =1400N/m, y =405mm) yd EPd Test #2,9,21 (k =800N/m, y =415mm)  ^  Test #4,19,23 (l =1400N/m, y =415mm)  yd  Epd  J  yd  c  Epd  yd  Epd  70-  B  6050-  a  o  40-  O  30x  20-  X  x +  +  +  1000  20  40  60  80  100  (mm/s) Figure 5 . 1 9 - Summary of k error y  Figure 5.20 is a summary plot of the force M A E for four sets o f cases. This plot shows that M A E o f force does not appear to be correlated with X - a x i s velocity, v , or Y - a x i s end-point x  stiffness, k . The Force M A E increases as the Y - a x i s equilibrium position, y p, increases for the y  data shown below.  E  72  5.5 Summary  2.51  i  i  i  + Test #1,5,20 (kyd =800N/m, EPd y =405mm) CDJ  J  O Test #3,15,25 (k =1400N/m, y =405mm) x Test #2,9,21 (kJ=800N/m, y =415mm) * Test #4,19,23 (I =1400N/m, y =415mm) yd  Epd  Epd  Epd  1.5  o  o  i-i  b  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 well. The manipulator delivers near identical behavior on each wipe o f the surface when the trajectory is repeated. Detailed results presented i n the Appendix and the data from the two above tables supports this assertion. Tests 6,7,8 were a group o f 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 o f 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. W h i l e 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 flow 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 2 0 % 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 y p due to the E  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 o f this work were to demonstrate the capabilities o f a P M C manipulator controlled with an equilibrium point hypothesis inspired controller. Particular interest was taken in the ability o f such a device to perform tasks that share characteristics with the activities o f daily living. 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 o f experiments it was shown that at low to moderate speeds (given the limitations o f the valves o f the air muscles) the controller tracks a demanding commanded trajectory, with some hysteresis induced i n the computation (rather than direct measurement) o f the actuator forces. The manipulator was designed to be large enough to carry out a wiping task. The muscles for this project were constructed i n the lab as suitable muscles were not available for purchase. Solenoid valves were chosen for metering air in and out o f the air muscles. Because the behavior o f these valves was not well suited to the size o f the muscles, orifice plates were designed and fit to each o f the valves to reduce the available maximum flow rate i n and out o f 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 o f these variables was achievable with simple PI control o f the mass flow in and out o f each muscle. The results o f the surface wiping tasks showed that it is possible to generate a wiping E P and stiffness trajectory that results i n the predicted normal force while wiping the surface. Additionally the mechanical compliance o f 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 i n the workspace. In addition, knowledge o f the precise location o f the contact object is not important as the mechanical compliance o f the manipulator compensates for small contact position errors. The particularly low cost o f implementation o f the technologies used i n this work is a promising factor i n the development o f affordable assistive robotic devices for i n 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 o f transition and contact tasks. This result holds where significant unmodeled disturbances are present, being easily handled b y a robot o f this type. The biggest trade-off is that it is not possible to generate superhuman stiffnesses with this manipulator were they desired. The usefulness o f this approach is supported b y the fact that the E P hypothesis fits a broad range of human motion tasks. The controller demonstrated in this work shows that it is possible to closely control the E P and joint stiffness values o f a manipulator. The success o f this controller was independently verified b y the external force measurement that showed the actual behavior o f the manipulator matched the expected behavior.  6.1 General recommendations W i t h the successful demonstration o f a stiffness/EP controller on a planar manipulator now carried out, it is possible for this work to be expanded. There is a wealth o f information available from neuromotor control studies regarding the stiffness and E P trajectories that humans  6.2 Specific Recommendations for this Experimental W o r k  76  follow when carrying out tasks. Results from the observation o f humans can be directly implemented on this system that shares the capability o f simultaneously varying E P and stiffness. A variety o f 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 o f mass, stiffness and size most practical for a reaching manipulator to possess, (ii) methods o f user activation, (iii) a practical method for learning from human task examples. The method detailing the sizing o f air muscle parameters could also benefit from further investigation. The method used i n this work was only developed to allow for minimization o f the mounting length o f 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 o f the difference i n the mass o f air i n 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 b y 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 o f the wall are negligible. In this case they were neither. A i r muscle models including friction exist. One o f these models could be implemented to improve the force prediction capabilities o f the model. If an even greater increase i n 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 o f measuring end point force directly is redundancy and increased safety. W i t h the current configurations, additional tasks could be attempted. It would be interesting to determine the effect o f changing the mass o f 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 W o r k  Increasing the manipulator to a 3-degree of freedom device would allow for more realistic assistive task demonstrations.  77  Bibliography [I] Canada Census Bureau, " H o w Helathy are Canadians?," Statistics Canada Health Reports, Winter 1999, V o l . 1 1 , N o . 3 , pp. 47-61 [2] Sommers, A . R . , "Long-Term Care for the Elderly and Disabled - A N e w Health Priority," The New England Journal of Medicine, July 22, 1982, V o l . 307, N o . 4 p p . 221-226 [3] Jorm, A . F . , et al., "The disabled elderly living i n the community: care received from family and formal services," The Medical Journal of Australia, M a r c h 15, 1993, V o l . 158, pp.383-385 [4] N i r , Z., "The Biopsychosocial Adjustment o f a Disabled Elderly: A 1- Year Follow-up," Rehabilitation nursing, Fan/Feb, 2000, V o l . 25, N o . 1, pp. 13-23 . [5] Canada Census Bureau, "Health and Activity Limitation Survey, Back-up Tables Provinces and Territories", 1991 Census, Government o f Canada, British Columbia Edition [6] Kemper, P., "The Use o f Formal and Informal Home Care by the Disabled Elderly," Health Science Research, October, 1992, V o l . 72, N o . 4, pp. 421-451 [7] Ford, A . B., et al., "Impaired and Disabled Elderly i n the Community," American Journal of Public Health, September 1991, V o l . 81, N o . 9 pp. 1207-1209 [8] Boaz, R.F., "Improved Versus Deteriorated, Physical Functioning A m o n g Long-Term Disabled Elderly," Medical Care, 1994, Vol.32 N o . 6 pp.589-603 [9] Arthritis Society o f Canda Website: http://www.arthritis.ca [10] K . Ikuta and M . Nokata, "Safety Evaluation Method o f Design and Control for Human-Care Robots," 2003, Int. Journal of Robotics Research, vol. 22, pp. 28 1 -297 [II] Y . Yamada, Y . Hirawawa, S. Huang, Y . Umetani and K . Suita, "Human - Robot Contact in the Safeguarding Space," 1997, IEEE/ASME Trans. On Mechatronics, vol. 2, pp. 230-236 [12] Okada, M . , Nakamura, Y . , B a n , S., 2001, "Design o f a Programmable Passive Compliance Shoulder Mechanism", Proc. Of2001 IEEE International Conference on Robotics and Automation, pp.348-353 78  Bibliography  79  [13] B i c c h i , A . , R i z z i n o , S., Tonietti, G . , 2001, "Compliant Design for Intrinsic Safety: General Issues and Preliminary Design", Intelligent Robots and Systems, 2001. Proceedings. 2001 IEEE/RSJInternational  Conference on, V o l . 4, pp. 1864-1869.  [14] Sciavicco, L., Siciliano, B., 1996, Modeling and Control of Robot Manipulators, M c G r a w H i l l , New York. [15] N i c k e l , V . L . , J . Perry, and A . L . Garrett, "Development o f useful function in the severely paralyzed hand,", 1963, Journal of Bone and Joint Surgery, V o l . 45 A , N o . 5, pp. 933-952 [16] Tondu, B., Lopez, P., 2000, " M c K i b b e n Artificial Muscle Robot Actuators," IEEE Control Systems Magazine, pp. 15-38 [17] Medrano-Cerda, Gustabo A . Bowler, C o l i n J. Caldwell, Darwin G . , 1995 "Adaptive Position Control o f Antagonistic Pneumatic Muscle Actuators", I E E E International Conference on Intelligent Robots and Systems, v l , pp. 378-383 [18] Latash, M . L . , "Independent Control o f Joint Stiffness i n the Framework o f the EquilibriumPoint Hypothesis," 1992, Biological Cybernetics, Vol.67, pp. 377-384 [19] Feldman, A . G . , "Functional Tuning o f the Nervous System with Controls o f Movement or Maintenance o f a Steady Posture: 2. Controllable Parameters o f the M u s c l e , " 1966, Biophysics, vol. 11, pp. 565-578 [20] B i z z i , E. et al., "Does the Nervous System use Equilibrium-Point Control to Guide Single and Multiple Joint Movements?", 1992, Behavioral and Brain Sciences, vol.15, pp. 603-615 [21] Feldman, A . , Levin, M . , "The Origin and Use o f Positional Frame o f Reference i n Motor Control", 1995, Behavioral and Brain Sciences, vol.18, pp. 723-806 [22] B i c c h i , A . , R i z z i n o , S., Tonietti, G . , 2001, "Adaptive Simultaneous Position and Stiffness Control o f a Soft Robot A r m " , Intelligent Robots and Systems, 2001. Proceedings. 2001 IEEE/RSJ International Conference on, V o l . 4, pp. 1992-1997 [23] Colbrunn, R.W., Nelson, G . M . , Quinn, R.D., 2001, "Design and Control o f a Robotic L e g with Braided Pneumatic Actuators", Intelligent Robots and Systems, Proceedings, of IEEE/RSJ International Conference on, V o l . 2, pp. 992-998  Bibliography  80  [24] Noritsugu, T., Tanaka, T., "Applications o f Rubber Artificial Muscle Manipulator as a Rehabilitation Robot", IEEE/ASME Transactions on Mechatronics, V o l . 2, N o . 4., pp. 259-267 [25] Tondu, B., Boitier, V . and Lopez, P., 1994 "Naturally Compliant Robot-Arms Actuated B y M c K i b b e n Artificial M u s c l e s " , Proc. of the 1994 IEEE Int. Conf On Systems, Man and Cybernetics, San Antonio, T X , 3:2635-2640 [26] Colbrunn, R.W., 2000, "Design and Control o f a Robotic L e g with Braided Pneumatic Actuators," Graduate Thesis, Case Wester Reserve Universtiy [27] M i l l s , J . K . ; Lokhorst, D . M . , "Stability and Control o f Robotic Manipulators During Contact/Noncontact Task Transition," Robotics and Automation, IEEE Transactions on, V o l . 9 , Iss.3, Jun 1993, pp.:335-345 [28] M i l l s , J . K . , David M . L . , 1993, "Control o f Robotic Manipulators during general task execution: A discontinuous Control Approach," The International Journal of Robotics Research, vol.12, No.2,pp.l46-163 [29] http://www.shadow.org.uk/index.shtml - Shadow Robot Company Webpage [30] Chou, C P . and Hannaford, B., 1996, "Measurement and M o d e l i n g o f Artificial Muscles", IEEE Transactions on Robotics and Automation, V o l . 12, pp. 90-102 [31] http://batman.mech.ubc.ca/~mal/Products.html - O R T S Website [32] Croft, E . A . , " M e c h 465 Course Notes," U B C [33] Klute, G . K . , 1999, Czerniecki, J . M . , Hannaford, B., " M c K i b b e n Artificial 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/ - Matrix V a l v e Website [35] Spong, M . W . , Vidyasagar, M . , 1989 Robot Dynamics and Control, Wiley, N e w Y o r k  Appendix A Air Muscle Equations A . l Air Muscle Equations for the Appendix From C h o u  dW = f  {P-P )dl  in  0  • ds, ={P-P )\ 0  dl, • ds, =P'dV  Equation A . 1  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 Where:  dW , = -fdL  Equation A . 2  0U  and,  dW  0Ut  Equation A . 3  = dW  in  The force in the muscle can be written as:  f = -P  dV  Equation A . 4  dL  where the length o f the muscle can be represented by Equation A . 5  L = b- cos(<9)  81  82  Appendix A  b • sin(0)  D=  Equation A . 6  Ml  The volume in the muscle is given by: 1 h V = —uD L - - ^ s i n 3  2  Am  2  Equation A . 7  (0)cos(0)  _ p,dV _ /Vde _ Fl>(2cos (<9)-sin (fl)) dL dL/ Ann / dtt 2  f  2  1  P'fr (3cos (fl)-l) 2  2  4;Z?J  Equation A . 9  2  Equation A . 10  cos (0) = ^ ' Z> 2  v  Equation A . 8  2  Ultimately yielding Z  2  W(3-=—1) 4;zw  2  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 Muscle Braiding Surgical Tubing  Dimensions  1/2" dia 3/16" O.D., 1/32" thick 4mm O.D., Plastic Tubing 0.75mm thick 1/4" O . D . - 3 / 1 6 " Large Brass Insert l.D. 1/4" O . D . - 1/8" Small Brass Insert l.D. Aircraft Cable 1/16" dia Aluminum Sleeves 1/16" dia O-Clamps  1/4" nominal dia  Muscle Braiding 2  l/4"dia  Quantity 18 1/2" length 8" length 25cm length  Supplier Product # Radar Inc. (Seattle) 625300113 Lancaster Medical Supplies N / A  152 584 Festo Columbia Valve 2 & Fitting B-405-3 Columbia V a l v e 1 & Fitting B-405-2 Steveston Marine N/A 2 loops & Hardware Steveston Marine N/A 4 & Hardware Acklands & F A R HC9-4 Grainger 3 2X3 Radar Inc. 3/4" Length (Seattle) 624900113  Make N/A N/A Festo Swagelo k Swagelo k N/A N/A Fairview Fittings  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 o f 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  Figure B.4 - Plastic and surgical tubing connected and plugged with brass inserts Slightly melt both ends o f 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. N o w 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. Pull 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 o f 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).  85  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. W h i l e holding the large mesh braiding pull the plastic tube until the junction of the two tubes comes out. N o w 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 like the others (see Figure B.7).  Figure B.6 - Exploded view Note: Top layout is exploded view o f 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 Kstiffmin %Kstiffmin>0 %The M a x i m u m obtainable torque is greater than TorqueMax %The working range o f the robot is greater than Deltheta %minimum ratio o f n/b>8 %max ratio o f n/b<21 % C is a vector that the solver tries to set <=0 %Lmount < Kstiffmax = 30; Kstiffmin = 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 (l/2)*(P(2)*(1750*P(2)*b 2+(Fmax+0)*pi*n 2)) (l/2); LPminCrossFmax = l/1050/P(l)*210 (l/2)*(P(l)*(1750*P(l)*b 2+(Fmax+0)*pi*n 2)) (l/2); FPmaxCrossb = P ( 2 ) * 7 0 0 0 * b 2 * ( 3 * b 2 / b 2 - l ) / ( 4 * p i * n 2 ) - 0 ; • FPminCrossb = P ( l ) * 7 0 0 0 * b 2 * ( 3 * b 2 / b 2 - l ) / ( 4 * p i * n 2 ) - 0 ; i f LPmaxCrossFmax >b Lmax=b; else Lmax=LPmaxCrossFmax; end i f LPminCrossFmax <b L m i n l = LPminCrossFmax; A  A  A  A  A  A  A  A  A  A  A  A  88  A  A  A  A  89  Appendix C  Fmaxl=Fmax; else L m i n l = b; Fmax 1 =FPminCrossb; end Lmaxs=[Lmin:.001 :Lmax]; Lmins=[Lmin:.001 : L m i n l ] ; Fmaxs=P(2)*7000*b 2.*(3.*Lmaxs. 2./b 2-l)./(4*pi*n 2)-0; Fmins=P(l)*7000*b 2.*(3.*Lmins. 2./b 2-l)./(4*pi*n 2)-0; TmaxDes=TorqueMax; L l =Lmount+r*DelTheta; L2=Lmount-r*DelTheta; DeltaF=TmaxDes/r A  A  A  A  A  A  A  A  F A = P(l)*7000*b 2*(3*(Ll) 2/b 2-l)/(4*pi*n 2)-0; F B = P(2)*7000*b 2*(3*(L2) 2/b 2-l)/(4*pi*n 2)-0; A  A  A  A  A  A  A  A  ifFB>Fmax FB=Fmax End F C = F A + DeltaF FD = FB -DeltaF K A = stiffnessl(n,b,Ll,P(l)*7000) K C = stiffnessl(n,b,L2,(FC+0)/(b 2*(3*(L2) 2/b 2-l)/(4*pi*n 2))) Kmin=r 2 * ( K A + K C ) A  A  A  A  A  K B = stiffnessl(n,b,L2,P(2)*7000) K D = stiffhessl(n,b,Ll,(FD+0)/(b 2*(3*(Ll) 2/b 2-l)/(4*pi*n 2))) Kmax=r 2 * ( K B + K D ) A  A  MaxTorqueAct = ( F B - F A ) * r figure plot(Lmins,Fmins,'r') hold plot(Lmaxs,Fmaxs); line([ b b],[Fmaxl Fmax]); line([LPmaxCrossFmax b],[Fmax Fmax]); line([Ll L 1 ] , [ F A F C ] ) ; line([L2 L2],[FB F D ] ) ; line([Lmount Lmount],[0 100])  C.2 solverbn.m  A  A  A  Appendix C  90  function[C,Ceq]=solverbn(X) n=X(l); b=X(2); r=X(3); Lmount=X(4); %this function solves the following constraints %The maximum obtainable stiffness is greater then Kstiffmax %The minimum obtainable stiffness is less then Kstiffmin %Kstiffmin>0 %The M a x i m u m 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; Kstiffmin = 1 5 ; 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 = LPminCrossFmax =  l/1050/P(2)*210 (l/2)*(P(2)*(1750*P(2)*b 2+(Fmax+0)*pi*n 2)) (l/2); l/1050/P(l)*210 (l/2)*(P(l)*(1750*P(l)*b 2+(Fmax+0)*pi*n 2)) (l/2); A  A  A  A  FPmaxCrossb = P ( 2 ) * 7 0 0 0 * b 2 * ( 3 * b 2 / b 2 - l ) / ( 4 * p i * n 2 ) - 0 ; FPminCrossb = P ( l ) * 7 0 0 0 * b 2 * ( 3 * b 2 / b 2 - l ) / ( 4 * p i * n 2 ) - 0 ; A  A  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;  A  A  A  A  A  A  A  A  A  A  Appendix C  91  Fmax 1 =FPminCrossb; end Lmaxs=[Lmin:.001 :Lmax]; Lmins=[Lmin:.001 : L m i n l ] ; Fmaxs=P(2)*7000*b 2.*(3.*Lmaxs. 2./b 2-l)./(4*pi*n 2)-0; Fmins=P(l)*7000*b 2.*(3.*Lmins 2 . / b 2 - l ) . / ( 4 * p i * n 2 ) - 0 ; A  A  A  A  A  A  A  A  TmaxDes=TorqueMax; L1 =Lmount+r*DelTheta; L2=Lmount-r*DelTheta; DeltaF=TmaxDes/r F A = P(l)*7000*b 2*(3*(Ll) 2/b 2-l)/(4*pi*n 2)-0; F B = P(2)*7000*b 2*(3*(L2) 2/b 2-l)/(4*pi*n 2)-0; A  A  A  A  A  A  A  A  i f F B > Fmax FB=Fmax end F C = F A + DeltaF F D = F B - DeltaF K A = stiffnessl(n,b,Ll,P(l)*7000) K C = stiffnessl(n,b,L2,(FC+0)/(b 2*(3*(L2) 2/b 2-l)/(4*pi*n 2))) Kmin=r 2 * ( K A + K C ) A  A  A  A  A  K B = stiffnessl(n,b,L2,P(2)*7000) KD = stiffnessl(n,b,Ll,(FD+0)/(b 2*(3*(Ll) 2/b 2-l)/(4*pi*n 2))) Kmax=r 2 * ( K B + K D ) A  A  A  A  A  MaxTorqueAct = ( F B - F A ) * r C(l) C(2) C(3) C(4) C(5) C(7) C(6) C(8)  = Kstiffmax - K m a x ; %ensure the maximum stiffness is possible = K m i n - Kstiffmin; %ensure the minimum stiffness is possible = -Kstiffmin %ensure the minimum stiffness is positive = TorqueMax - MaxTorqueAct; %ensure the max torque is achievable = 0; = n/b-21; %check the b/n ratio = 8 - n/b; %check it on the other side = DelTheta*r + Lmount -b ; %check the theta range on the right side  Ceq=[]  C.3 Stiffnessl.m  92  Appendix C  function stiffnessl=stiffhessl(n,b,L,P)  A=l/(4*pi*n 2); B=b 2/(4*pi*n 2); Phi=B-3*A*L 2; Vol=B*L-A*(L 3); stiffnessl=(P+101000)A^ol*Phi 2+P*6*L*A; A  A  A  A  A  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 - B O M - 1  3  o  1  «  S  3< f5  13  5  .2 S ffl °P>A E 2 •S o  -a  1  % c*t ° ca  3 ,"  2 ou  00  !  > T  S 2  o  o  g  <  * D o g  t- B  h  i  l  00  ~  D -  ,1 .a >»  -^t oo on  o Si  s;  •I «  55 55  2  'S  3  c  ^  c  D U  1  ' O °  o 03  ! oo  M-  > <^ d- — §  >  f - IX,  W  Table D.2  - BOM-2  °3 •a  : o  «  °  ; —  CJ ^>  OflO  5  S  "  C/3 OH  O O  x>  CN  CJ o  -k-' "2 73  cu — f CJ a, CJ  8  8  CZ) M O  —  u  • •  *  iuate uate She gra<iuate uate She  31  o  ca i—  60 w»  a CJ  CJ  •a  o  c  UBC  73 3  UBC  c  -a Ocd  CJ  a .1  •a  I  l<  l3 73  6  1 op  5  CJ  o  2  Appendix D  96  Table D.3 - B O M - 3  c  ll  |3  3  M  13  T  CJ  JO Z  3 BOON  g-  o "  s  S3  eg  i  GO ON *c3 « . cd ca U CU  cd cd  • P  1  6b  i-a c  3  Cu  cd  2  CO  Q  £fc2  uc u  Cfl  O-  o o  Z  OS  3  la  ca M o o ta. t/3  00  c  Z  I  H  1)  55 55  .  Appendix E Detailed Machining Drawings  97  Appendix E  98  Figure E . l - Drawing 1  Appendix E  99  Figure E.2 - End Effector  Appendix E  100  Figure E.3 - L i n k 2 and sprocket  Appendix E  101  Figure E.4 - Pulley 1 and big gear  102  Appendix E  Figure E.5 - Pulley 2 and big gear  Appendix E  103  Figure E . 6 - Link drive assembly  104  CVJ  SHOP  DRAW 1NG  US D I G I T A L  Appendix E  CC >—  ro  CC 0 1—  > —  o  •  i —  OO  — — Q_ =tt=  CC CJ  O - -  CJ  — —  >  LLJ  1  0  L U  < J  L U  '  ::>• O  C J  L U  1  '—  CC L U  C J  < C CO L_i_  CC CP oo L U L_i_ cCj O O CJ CJ z : 2: LU 00  ITEM  CC  — CC — CD CQ  +  : •  ro  CC  -  CNJ  0  1—  CVJ  1  ce  -•••!  1—  0  C J  Cl_  ,  ^ -  1 1  <=t 1  O  CQ  co  >  ce <=c  L U  00  no : •  1 1  <J  o  OO  1  CC  —  (  LJ-J  1  -=E 1—  1—1—1  ce  «=C Q_  1  >_ CQ  -:  L U  CC  <c  <c  1  cj  00  LO m 1— ^ cj  =H=  I—  -c  C J  LiJ  m  >  O  C J  1 Z-  •< 1 Q1_CC CC  cm  — cCJ>_  0  >— cz> cz> - H - H - H> -H < = > — 0  CO  CZ  CC LiJ  Figure E.7 - Encoder and gear  X  X X  X  X  X  X  X  LiJ  Z  UJ  O  Li  J  LiJ  1 TT.0 0 o u s X X  '  —  UJ  1 rn 1 -z. 1— -< -- O  Appendix E  105  X LJL. LZJ  Q_  CC CC L_u  ro O CVI  =z> <  CVJ  -:::  co  CC  CC  oo  L_i_l C O  C_J  Q_  o  1—  —  X  cvj  CVJ  O — CQ  oo  X  u_  CQ  CC 1 i ! Z D  oo X CVJ  ro  o -.  iii  1— X  >_  CC  —  oo -<  oo  CQ  CQ  l — CC Q_  ^3"  > o  ICC LaJ  Z D  ^1 La_  X  CC LaJ CQ X LaJ  oo  Z D  O  CVJ  >  z~  =3  is:  -=c  X  Q_  —  CQ  -  —^  X  1  O O  —  ::>  :—  ro  -<  cz> < z> CVJ  Z D  cv  LO  1— Q_  CVJ  O  i  =H=  O O  CC LO CC  ro  O  Cl_  O o zc zc  LO  -z  LaJ 1  CC Q  <c  <c  co  LO LO LO |  J  =H=  1  CJD  <c  Li_l  o  CJ>.  -  U_l C C I S Cl_  CC  r-i  _  —o >  i —  oo  — o> c > — <o o <_•> o -H-H-H-H X X X  0 Figure E.8 - B o x  x  X X  X  x  ><:  0  CO LU  OO CTi uJ L U _  LJ ^  CC OO L U L U laJ C/3 I ZH  O O X  x  O  -<  I 1 ^  3: LU  n_  1  <c —- O  106  Appendix E  co  CC O  CO CVJ  oo ce O I— Q_  <=c  o OQ  CQ  CJ  o  CC  CJ  CC Q  C_j  CJ  oo  o  CC  cc  Q_  CO CO CO  LO LO I  =t=t=  CJ CJ UJ  CO  O cc CL_  CJ  oo  O  O  O  c=>  -H-H-H-H  Figure E.9 - Adaptor 1  x  X  X  X  x  x  x  x  0  X  x  107  Appendix E  Figure E. 10 - Adaptor 2  Appendix E  108  Figure E . l 1 - Adaptor 3  Appendix E  109  Figure E.12 - Link 2  110  Appendix E  Figure E . l 3 - Timing belt sprocket 2  111  Appendix E  Figure E . 1 4 - S h a f t 2  112  Appendix E  Figure E . l 5 - Pulley 1  113  Appendix E  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>  <r>  z^: <c  CD CD CD  CC I—  zz> >  CC  O  I—  <=c  —  COCD CVJ  1  CL —) -:  —  -S  1— u_  LU I  <c zc oo  ej  OO  —i -S t  O  <_  CO  _Q O  —o  CC UJ  >-  —  z<:  :CC — <c  CD CD  o_  o  CQ  -:  UJ  LZJ  LZJ  «=C CC  1—  -<c  cn LO LO  =8=  1  1  <L> <L>  |— :>  OO  UJ  zc > c_> O  >  UJ CC  "—  —  o>  cz> <z:o  o  X  x  X  X X  -  o  oo  —  OO C2t OO L U L U  o < 0-  CJ ^ ^  -H-H-H-H x  0  O  OO LU L U L iJ CO  o X  x  Figure E . 1 9 - S h a f t 1  LO  1 , 1  CC  oo  ::::> —  i nz — <_> s  I 1 ^  <c —-  LU 1—  O  Appendix E  117  Figure E.20 - L i n k 1  Appendix E  118  Figure E.21 - Fitting  Appendix E  119  Figure E.22 - Small gear  Figure E.23 - Top  Appendix E  121  CC  o  o  o  CC Li_l CQ  CQ CJ  oo  CJ OO  CC  CC  O O  CC LHJ  o  LO  u~>  ^3"  CJ  CJ  o  =1* CJ  o CC Q_  CC  M O  _l_lt I— —<=3<0O <0o . CO Q CO UJ uu <_> z :  -H-H-H  x x XXX XXX  Figure E.24 - Side  0  O  122  Appendix E  CC  cc O  CVJ  o  CC L-lJ CO  CQ  CJ  oo  00  CJ  CQ  CC  O  CC  CO  CC  Cl_  CC Q  C_j  LO CVJ  LO LO LO I  -=r  CO  CJ  CO  CJ  O  UJ  CC  CC  oo  o  O OO O -H-H-H-H X  XQ  X  LO CVJ  Figure E.25 - Base  X  X  X  X  X  X  X  X  or 00  UJ  I  Appendix F Assembly Instructions Step 1: End Effector (see drawing # 2) •  First attach the back adapter to one side o f the force transducer. Use three M 3 screws and tight it gently. The adapter can be mounted on any o f the two sides and i n 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 o f 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 o f 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 o f the big gears on pulley 1 using two dowel pins 3/32. M a k e 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 i n the pulley and fixe it with four 4-40 A l l a n 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 l s o make sure all the previous steps are done correctly. Believe me, you don't want to this twice. •  First introduce link 2 i n the shorter slot o f link 1.  •  Insert shaft 2 inside link 2 and the sprocket. The shaft is larger than link 1 and both his tips should be out o f the holes. Link 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 i n 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 w i l l not disengage themselves when operating the robot.  •  Insert shaft 1 b y the upper side. The shoulder o f 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.  •  Finally bolt pulley 1 on link 1, using four A l l a n 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 w i 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) i n the base.  Appendix F  •  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: T o t a 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.  125  Appendix G Sensors and Calibrations G.l Experimental Equipment Specifications Pneumatic Actuators Each pneumatic actuator is approximately 33 c m i n length. Inside, each consists o f rubber surgical tubing (3-mm diameter) and is covered b y a tough plastic weave. W h e n the surgical tubing is being inflated, this provides a radial force and the weave contracts, resulting i n 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 i n 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 o f 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 o f 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 o f two types o f pressure transducers. The transducers by Sensotec have a range o f 150 psig and came with calibration papers. AutoTran transducers have a minimum range o f 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 o f the pulley. In the final set-up they were attached to the shafts o f toothed gears that meshed with gears on the respective pulleys. Coupled with the gears, they provide % degree o f 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 o f one pulley translates to the rotation o f one link in the same plane o f motion. The inner link is directly attached to its pulley and the second link is attached at the end o f the first link 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 o f 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 i n thickness. Voltmeter The Fluke 801 OA digital multimeter was used to read out the voltage while doing calibrations.  Digital Scale Weighing o f all o f the components used in the calibration o f the force transducer was done by a Toledo S M - F digital scale. The accuracy was one tenth o f a gram.  Appendix G  128  G.I.J Solenoid Valves Matrix Solenoid Valve M o d e l 821 3/3 N C , Identification code G N K 8 2 1 2 0 3 C 3 K K 3 Port, 3 W a y H i g h Frequency V a l v e www.matrix.to.it/pd009.htm  Description - The Pneumatic Solenoid Valves 820 Series The research about materials and new technological solutions allowed the realization o f a shutter solenoid valve with an extremely simple operation principle and with avant-garde dynamic characteristics. The mass o f the moving elements has been reduced to the minimum and every inner friction has been eliminated: i n this way, we obtained response times o f milliseconds and an operation life over 500 million cycles. Due to the possibility o f controls o f 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 o f 200 H z . O n the contrary, solenoid valves with speed-up control have a response time lower than 1 ms, both i n opening and i n closing, with a maximum operation frequency o f 500 H z . Besides high-speed characteristics, solenoid valves 820 Series offer flow rate values up to 180 dm3/min ( A N R ) , with feeding pressure from 0 to 8 bar. Controlling 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 " T Y P E = P I C T ; A L T = P n e u m a t i c scheme"}Control Direct - P F M PWM 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 i n opening 24 < 6ms X X / K K < 3 ms Response time i n closing 24 < 2 ms X X / K K < 1 ms M a x i m u m frequency 100 H z 200 H z Weight 130 g Product life expectancy > 500 M i s cycles F l o w rate (at 6 bar) 90 N l / m i n - Control tension X X / K K N o . Outlets 1 Outlet N o . Electrical controls 2 Controls Port connection Integrated cables IP 62 L = 500mm / 100mm Control tension Speed-up in tension ( 2 4 V D C ) 0.8 W Operating pressure 0 - 8 bar  Materials B o d y in P P S , Flanges i n A l , Seals in N B  Appendix G  129  G.1.2 Sensotec Pressure Transducers Sensotec M o d e l L M 150, serial numbers 70258, 702583 150psig range www.sensotec.com/pdf/lm.pdf Sensotec offers the M o d e l L M pressure transducer as a low cost alternative with good performance for high volume applications. Each unit is constructed o f welded stainless steel for durability i n dry rugged environments. Both gas and liquid pressure overloads o f up to 50% over capacity are safely accepted.  Performance Output Resolution  Pressure Range Accuracy (min.) 1.8mV/V(nom) infinite  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  150psig ±0.5% F.S.  -65°F to 250°F  Electrical  Input 10VDC Bridge Resistance 350 o h m s * * Electrical Termination (std.) Cable 3 ft.  Mechanical  Media  Overload-Safe Pressure Port Wetted Parts Material Type Case Material * Consult Sensotec on units below 150psi * * 5000 ohm below 150psi  Gas, Liquid 5 0 % over capacity V4-I8NPT female Stainless steel Gage Stainless steel  Appendix G  130  G.1.3 AutoTran Pressure Transducers Autotran M o d e l 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 o f 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 i n a wide variety o f applications. There are no welds, no O-rings, and no silicone oil 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 3wire, 1-5V version.  Specifications { P R I V A T E } Pressure Range: Accuracy: Stability: Thermal Effects: Compensated Range: Operating Temperature: M e d i a Compatibility: Input Supply: Supply Current: Load Resistance (Voltage Output): Load Resistance (Current Output): Output Signal: Zero Offset: Electrical Connection: Housing: Connections: Dimensions:  Oto 100 PSI (0 to 7 Bar) < l%ofFS +/- 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 ) A n y 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 2 0 m A two wire +/- 2 % 24" 3-wire cable (1-5V), 2 4 " 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 i a (54.8 m m L x 21.4 mmDia)  Appendix G  131  G.1.4 Force Transducer P T (Precision Transducers) M o d e l S T 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 2mV/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 5 V to 15V ac/dc M a x i m u m 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 - -40 to +100°C operating temperature -Tracks from 0 to 100,000 cycles/sec -Ball bearing option tracks to 10,000 R P M  -Small size - L o w cost - Single +5V supply -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 + 5 V D C supply. The SI and S2 encoders are available with ball bearings for motion control applications or torque-loaded to feel like a potentiometer for front-panel manual interface.  Mechanical Notes B a l l Bearing: j { P R I V A T E } Acceleratio jn j Vibration  10,000 rad/sec  Sleeve Bushing j 10,000 rad/sec  2  j 20 g. 5 to 2 K H z  [  2  |20 g. 5 to 2 K h z  !  j Shaft Speed  10,000 R P M max. continuous j 100 R P M max. continuous  j Shaft Rotation  N/A  {Acceleration  ! 5 0 K rad/sec  j Continuous & reversible  |N/A  2  | 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 Bearing Life (Weight j Shaft Runout  ' ;  j  (40/P) = Life i n millions o f JN/A revs, where P = Radial load in pounds.  j  0.7 oz.  J0.7 oz.  |  jo.0015 T.I.R. max.  j  \ 0.0015 T.I.R. max.  Materials & Mounting: { P R I V A T E } Shaft i Bushing Connector Hole Diameter :  Brass or stainless \ Brass G o l d plated J 0.375 in. +0.005 - 0  Panel Thickness  0.125 in. max.  Panel Nut M a x . 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 o f 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 m V / V ) * ( 1 5 0 psig) = 85.725 psig Calibration Equation: V = offset + P*calibration*8*G/150 psig => G is factor to be determined 4.01 V = - 2 . 5 2 +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 m V / V ) * ( 1 5 0 psig) = 92.643 psig Calibration Equation: V = offset + P*calibration*8*G/150 psig => G is factor to be determined 4.01 V = - 2 . 5 2 +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) 0 -2.29 0 10 -1.52 10.66905545 20 -0.84 20.09107844 30 -0.14 29.79021976 40 0.56 39.48936107 50 1.41 51.26688982 60 2.08 60.55035365 70 2.94 72.46644155 80 3.52 80.50287293  Voltage(#2) Voltage(#3) -1.13 -1.1 -1.58 -1.54 -2.01 -1.97 -2.37 -2.35 -2.87 -2.84 -3.4 -3.38 -3.82 -3.8 -4.36 -4.35 -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# #1 #4 #6 #11 #12  Part Mass (g) Part weight(N) 2709.7 26.582157 3059.5 30.013695 1653.2 16.217892 1198.8 11.760228 525.7 5.157117  part# none 12 11 6 12+6 1 4 4+12 4+11  Weight(N) 0 5.157117 11.760228 16.217892 21.375009 26.582157 30.013695 35.170812 41.773923  03/19/02 voltmeter(V) w/ horizontal shift (V) 2.47 2.51 2.31 2.35 2.1 2.14 1.96 2 1.79 1.83 1.63 1.67 1.52 1.56 1.35 1.39 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 M1 M2 M3 M4  length (m) end pt. 1 (offset) end pt. 2 0.302 0.388 0.363 0.33 0.366 0.39 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 | length(m) muscle end pt. 1 (offset) slope 0.536254 0.0002638 M1 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 C a l c u l a t i o n : Slope of M l = 0.0002638*(0.388-0.302)/(0.536254-0.302) = 9.685E-05 These results are tabulated below: muscle M1 M2 M3 M4  length(m) end pt. 1 (offset) slope 0.302 9.685E-05 0.363 -1.012E-04 0.366 2.692E-04 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 o f the below trajectories was followed this the manipulator in free-space. N o wall was present for any o f 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 o f 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 c m s _ 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 c m s _ l 1 0 0 K 4 1 . 5 _ c o m p 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 c m s _ l 4 0 0 K 4 1 . 5 _ c o m p 20_7.5cms_800K_40.5_comp 21_7.5cms_800K_41.5_comp 138  Appendix H  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  139  Appendix I Summary of Transition Tests y versus Path  y__ versus Path  410  410  405  6 E 400  y no wall  395  J  a  y with wall • y,EPd 390  5  10 Path (mm)  15  20  5  k versus Path y  1200  10 Path (mm)  Force versus Path  Actual no wall Actual wall Desired  1150 1100 1050 1000 950 900 850 800  0  5  10 Path (mm)  15  20  5  Test #1 - 30 degrees, v=0.2cm/s  Figure 1.1 - Transition test #1  140  10 Path (mm)  15  20  Appendix I  141  y versus Path  y__ versus Path  410  ^EP  410  405  6 6 400  y no wall — - y with wall  395  a  • "  390  5  y  EPd  10 Path (mm)  15  20  5  k versus Path y  1200  10 Path (mm)  15  Force versus Path  Actual no wall Actual wall Desired  1150 1100 1050 c Z  1000 950 900 850 800  5  10 Path (mm)  5  20  0  Test #2 - 60 degrees, v=0.2cm/s  Figure 1.2 - Transition test #2  5  10 Path (mm)  15  20  Appendix I  142  y versus Path  y__ versus Path  410  410  405  405  - M ^ - ""  E E 400  £ 400  y-'t  395  - — y no wall — - y with wall  /[•?  J  y -j  y  390  5  a  EPd  15  20  390  5  k versus Path y  1200 1150  10 Path (mm)  395  10 Path (mm)  15  20  15  20  Force versus Path  Actual no wall — - Actual wall Desired  1100 1050 B z  1000 950 900 850 800  5  10 Path (mm)  15  20  0  Test #3 - 90 degrees, v=0.2cm/s  Figure 1.3 - Transition test #3  5  10 Path (mm)  143  Appendix I  y versus Path J  y^,, versus Path  a  410  . • \y -~ y~ ^  —  y no wall a _ - y with wall J  y  EPd-  10 Path (mm)  15  20  5  k versus Path y  1200  10 Path (mm)  Force versus Path  Actual no wall Actual wall Desired  1150 1100 1050 c  z;  1000 o  950  PL,  900 850 800  5  10 Path (mm)  15  20  -10  0  Test #4 - 30 degrees, v=0.5cm/s  Figure 1.4 - Transition test #4  5  10 Path (mm)  15  144  Appendix I  y__ versus Path  y versus Path  ^EP  410  410 y no wall J  a  — - y with wall •'EPd y__,  405  405  E 400  E 400  395  395  390  5  10 Path (mm)  15  390  20  k versus Path y  1200  10 Path (mm)  15  20  Force versus Path 15  Actual no wall — - Actual wall Desired  1150  5  Actual Predicted 10  1100 1050  o &  950  0  900 850 800  0  5  10 Path (mm)  15  20  -10  Test #5 - 60 degrees, v=0.5cm/s  Figure 1.5 - Transition test #5  5  10 Path (mm)  15  20  Appendix I  145  y versus Path  y ^ versus Path  ^EP  410  410  405  /~  jr /  S / /• r rz-r  395  y no wall a — - y with wall a y J  /•'r  J  390  _A  E P d  10 Path (mm)  10 Path (mm)  k versus Path y  1200 1150  15  15  Force versus Path  - — Actual no wall — - Actual wall Desired  1100 1050 c Z  1000 950 900 850 800  5  10 Path (mm)  15  20  0  Test #6 - 90 degrees, v=0.5cm/s  Figure 1.6 - Transition test #6  5  10 Path (mm)  15  146  Appendix I  y versus Path a  V™ versus Path  J  ^EP  410  y no wall a y with wall a J  —  J  y  5  10 Path (mm)  EPd  15  20  5  k versus Path y  1200  10 Path (mm)  Force versus Path  Actual no wall Actual wall Desired  1150 1100 c c  z  1050 1000  o  0*  950 900 850 800  5  10 Path (mm)  15  20  0  Test #7 - 30 degrees, v=lcm/s  Figure 1.7 - Transition test #7  5  10 Path (mm)  15  147  Appendix I  y versus Path J  y  a  versus Path  /  yy  y no wall J  a  — - y with wall y  5  10 Path (mm)  EPd  15  20  5  k versus Path y  1200  10 Path (mm)  15  20  15  20  Force versus Path  — Actual no wall - Actual wall Desired  1150 1100 1050 z  1000 950 900 850 800  5  10 Path (mm)  15  20  Test #8 - 60 degrees, v=lcm/s  Figure 1.8 - Transition test #8  5  10 Path (mm)  148  Appendix I  y versus Path  y__ versus Path  410  410  405  6 S 400  r  y no wall  395  a  _ - y with wall a J  -0/  y  390  10 Path (mm)  15  20  5  k versus Path y  1200 1150  EPd  10 Path (mm)  15  20  Force versus Path 15  Actual no wall — - Actual wall Desired  -  10  1100  — r \ 1050 6 Z  — "X  1000  \  o  \  950 -\  900  \  —  850 800  5  10 Path (mm)  15  20  -10  0  Test #9 - 90 degrees, v=lcm/s  Figure 1.9 - Transition test #9  5  10 Path (mm)  Actual Predicted 15  20  Appendix J Summary of Contact Tests  149  Appendix J  150  y-position versus x without bump  y-position versus x with bump  versus x without bump 1600 1500  ? z  —  versus x-position with bump 1600  Actual Desired  1500  1400  1400  1300  1300  1200  S  z  1100  1100 1000  900  900  800  800 -50  0 X(m)  50  700 -100  100  Force versus x-position without bump  -50  0 x (mm)  50  Force versus x-position with bump  -50  0 x (mm)  Test #1 - v =15mm/s, Y =405mm, k =800N/m x  Actual Desired  1200  1000  700 -100  —  ep  Figure J . l - Contact test #1  y  100  Appendix J  151  y-position versus x without bump  y-position versus x with bump  100  k versus x-position with bump  k versus x without bump 1600  Actual Desired  1500  E  Z w  1600  1400  1400  1300  1300  1200  1200  1100  1100  1000  1000  900  900  800  800  700 -100  Actual Desired  1500  700 — -100 1  -50  0 X(m)  50  100  Force versus x-position without bump 30  -50  0 x (mm)  50  Force versus x-position with bump  Actual Predicted  25 20 Z  g  15  o  10 5 0 — -100 1  -50  0 x (mm)  50  100  0 x (mm)  Test #2 - v =15mm/s, Y =415mm, k =800N/m x  ep  Figure J.2 - Contact test #2  y  50  100  Appendix J  152  y-position versus x without bump  y-position versus x with bump  versus x without bump  k^ versus x-position with bump  1600  1600  1500  1500  1400  1400  1300  1300  1200  B  z  1100  >  1000  1200 1100 1000  900  900  800 700 — -100  Actual Desired  Actual Desired  800  1  700 — -100 1  -50  0 X(m)  100  50  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  Test #3 - v =15mm/s, Y =405mm, k =1400N/m x  50  ep  Figure J.3 - Contact test #3  y  100  Appendix J  153  y-position versus x without bump  y-position versus x with bump  100 k versus x without bump  ^ versus x-position with bump 1600 1500 1400 1300  e  z  1200 1100 1000 900  Actual Desired -100  -50  0 X(m)  50  800 700 -100  100  Force versus x-position without bump  0 x (mm)  — -50  0 x (mm)  v =15mm/s, Y =415mm, k =1400N/m x  50  Force versus x-position with bump  50  Test #4  Actual Desired  ep  Figure J.4 - Contact test #4  y  100  Appendix J  154  y-position versus x without bump  y-position versus x with bump  k versus x-position with bump  ky versus x without bump 1600 1500  —  1600  Actual Desired  1500  Z  1300  1300  1200  1200  1100  1100  1000  1000  900  900  800  800  700 -100  Actual Desired  1400  1400  B  —  700 — -100 1  -50  0 X(m)  50  100  Force versus x-position without bump  -50  0 x (mm)  50  100  Force versus x-position with bump 30 —  25  Actual Predicted  20 Z | 15 o 10 5 0'— -100  -50  0 x (mm)  Test #5 - v =30mm/s, Y =405mm, k =800N/m x  ep  Figure J.5 - Contact test #5  y  50  100  Appendix J  155  y-position versus x without bump  y-position versus x with bump 430  EPd  y  425  — ^EPa y.  420 415  6" B >,  410  I—  405 400 390 — -100 1  395 390 — -100 1  -50  0 x (mm)  50  100  ky versus x without bump  -50  0 x (mm)  50  100  ^ versus x-position with bump 1600  Actual Desired  Actual Desired  1500 1400 1300 1200 1100 1000 900 800 700 — -100 1  -100  -50  0 X(m)  50  100  -50  0 x (mm)  Force versus x-position with bump  Force versus x-position without bump  -100  0 x (mm)  Test #6 - v =30mm/s, Y =410mm, k =800N/m x  50  ep  Figure J.6 - Contact test #6  y  50  100  Appendix J  156  y-position versus x without bump  y-position versus x with bump  ky versus x without bump 1600  Actual Desired  1500  S  z  ^ versus x-position with bump 1600 1500  1400  1400  1300  1300  1200  1200  1100  >, 1100  1000  1000  900  900  800  800  700'— -100  -50  0 X(m)  50  700 -100  100  Force versus x-position without bump  —  -50  0 x (mm)  50  Force versus x-position with bump  Test #7 - v =30mm/s, Y =410mm, k =800N/m x  Actual Desired  ep  Figure J.7 - Contact test #7  y  100  Appendix J  157  y-position versus x without bump  y-position versus x with bump  100  100  ky versus x without bump 1600  Actual Desired  1500  e  z  ^ versus x-position with bump 1600  1400  1400  1300  1300  1200  1200  >. 1100  1100  1000  1000  900  900  800  800  700 -100  -50  0 X(m)  50  Actual Desired  1500  700 — —100  100  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  -50  0 x (mm)  Test #8 - v =30mm/s, Y =410mm, k =800N/m x  50  ep  Figure J.8 - Contact test #8  y  100  Appendix J  158  y-position versus x without bump  y-position versus x with bump  100  ky versus x without bump 1600 1500  —  ^ versus x-position with bump 1600  Actual Desired  1500  1400  Actual Desired  1400  1300 Z  —  1300  1200  e  z  1100  1200 1100  1000  1000  900  900  800  800  700 -100  700 -100  -50  0 X(m)  50  100  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  Test #9 - v =30mm/s, Y =415mm, k =800N/m x  50  ep  Figure J.9 - Contact test #9  y  100  Appendix J  159  y-position versus x without bump  y-position versus x with bump  k versus x-position with bump  k versus x without bump  —  1600  Actual Desired  Actual Desired  1500 1400 1300 1200 1100 1000 900 800  700'— —100  -50  0 X(m)  50  100  700 — —100  0 x (mm)  -50  Force versus x-position with bump  Force versus x-position without bump  -100  0 x (mm)  Test #10 - v =30mm/s, Y =405mm, k =1 lOON/m x  50  ep  Figure J. 10 - Contact test #10  y  50  100  Appendix J  160  y-position versus x without bump  y-position versus x with bump  ky versus x without bump 1600  ^ versus x-position with bump 1600  Actual Desired  1500 1400  1400  1300  1300  1200  1200  '» 1100  1100  1000  1000  900  900  800  800  700 — -100  Actual Desired  1500  1  700 — -100 1  -50  0 X(m)  50  100  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  Test #11 - v =30mm/s, Y =410mm, k =1 lOON/m x  50  ep  Figure J . l 1 - Contact test #11  y  100  Appendix J  161  y-position versus x without bump  y-position versus x with bump  100  ^ versus x without bump 1600  ^ versus x-position with bump 1600  —— Actual — Desired  1500  1500  1400  1400  1300  1300  1200  1200  1100  1100  1000  1000  900  900  800  —  Actual Desired  800  700 — -100 1  700 — -100 1  -50  0 X(m)  50  100  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  Test #12 - v =30mm/s, Y =410mm, k =1 lOON/m x  50  ep  Figure J.12 - Contact test #12  y  100  Appendix J  162  y-position versus x without bump  y-position versus x with bump  ^ versus x without bump  ^ versus x-position with bump 1600  Actual Desired  1500  —  Actual Desired  1400 1300 1200 1100 1000 900 800 700 — -100 1  700 — -100 1  -50  0 X(m)  50  100  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  -100  Test #13 - v =30mm/s, Y =410mm, k =1 lOON/m x  50  ep  Figure J.13 - Contact test #13  y  100  Appendix J  163  y-position versus x without bump  y-position versus x with bump  100.  ky versus x without bump  —  ky versus x-position with bump 1600  Actual Desired  Actual Desired  1500 1400 1300  ? z  1200 1100 1000 900 800  -100  -50  0 X(m)  50  100  700 -100  Force versus x-position without bump  • — -100  -50  0 x (mm)  -50  0 x (mm)  Force versus x-position with bump  Actual Predicted 50  100  Test #14 - v =30mm/s, Y =415mm, k =1 lOON/m x  50  ep  Figure J.14 - Contact test #14  y  100  Appendix J  164  y-position versus x with bump  y-position versus x without bump  k versus x without bump  ^ versus x-position with bump  y  1600 1500 1400 1300 1200 1100 1000 900  700 — -100  800  Actual Desired  1  Actual Desired  700 — -100 1  -50  0 X(m)  50  100  Force versus x-position without bump  -50  -50  0 x (mm)  Force versus x-position with bump  0 x (mm)  Test #15 - v =30mm/s, Y =405mm, k =1400N/m x  50  ep  Figure J. 15 - Contact test #15  y  100  Appendix J  165  y-position versus x without bump  y-position versus x with bump  100  ky versus x without bump  ^ versus x-position with bump  1600  1600  1500  1500  1400  1400  1300  1300  1200  1200  1100  1100  1000  1000  900  900  800 700'— —100  800  Actual Desired -50  0 X(m)  50  100  —  700 -100  Force versus x-position without bump  0 x (mm)  -50  50  Force versus x-position with bump  -50  0 x (mm)  Test #16 - v =30mm/s, Y =410mm, k =1400N/m x  Actual Desired  ep  Figure J.16 - Contact test #16  y  100  Appendix J  166  y-position versus x without bump  y-position versus x with bump  k versus x without bump  E  Z  ^ versus x-position with bump  1600  1600  1500  1500  1400  1400  1300  1300  S  1200  z  1100  1200 1100  1000  1000  900  900  800 700 -100  800  Actual Desired  —  700 — -100  Actual Desired  1  -50  0 X(m)  50  100  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  -50  Test #17 - v =30mm/s, Y x  50  0 x (mm)  =410mm, k =1400N/m ep  Figure J.17 - Contact test #17  y  100  Appendix J  167  y-position versus x without bump  y-position versus x with bump  ^ versus x without bump  ^ versus x-position with bump  1600 1500  1600  ^^_^_ _ ^_____^ =  1500  =  1400  1400  1300  1300  1200  B  Z >  1100  1200 1100  1000  1000  900  900  800 700 -100  — -50  0 X(m)  Actual Desired  50  800 100  —  700 -100  Force versus x-position without bump  -50  0 x (mm)  50  Force versus x-position with bump  -50  0 x (mm)  Test #18 - v =30mm/s, Y =410mm, k =1400N/m x  Actual Desired  ep  Figure J. 18 - Contact test #18  y  100  Appendix J  168  y-position versus x without bump  y-position versus x with bump  ^ versus x without bump  ky versus x-position with bump  1600  1600  1500  1500  1400  1400  1300  1300  1200  1200  £ Z  1100  >>  1100  1000  1000  900  900  800 700 — -100  —  Actual Desired  800  1  -50  0 X(m)  50  100  Actual Desired  700 -100  Force versus x-position without bump  0 x (mm)  -50  Force versus x-position with bump  30 25 20 Z  1o 15 It  10 5 0— -100  Actual Predicted  1  -50  0 x (mm)  50  100  Test #19 - v =30mm/s, Y =415mm, k =1400N/m x  50  ep  Figure J.19 - Contact test #19  y  100  Appendix J  169  y-position versus x without bump  y-position versus x with bump  ky versus x without bump  ky versus x-position with bump 1600  Actual Desired  Actual Desired  1500 1400 1300  s  z  1200 1100 1000 900 800  -100  -50  0 X(m)  50  100  700 -100  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  Test #20 - v =75mm/s, Y =405mm, k =800N/m x  50  ep  Figure J.20 - Contact test #20  y  100  Appendix J  170  y-position versus x without bump  y-position versus x with bump  ky versus x without bump 1600  ^ versus x-position with bump 1600  Actual Desired  1500  1500  1400  Z  1300  1200  f  1200  1100  %  1100  1000  1000  900  900  800  800  700 -100  Actual Desired  1400  1300 B  —  700 — -100 1  -50  0 X(m)  50  100  Force versus x-position without bump  -50  0 x (mm)  Force versus x-position with bump  -50  0 x (mm)  Test #21 - v =75mm/s, Y . =415mm, k =800N/m x  50  ep  Figure J.21 - Contact test #21  y  100  Appendix J  171  y-position versus x without bump  y-position versus x with bump  ky versus x without bump  a z  k versus x-position with bump  1600  1600  1500  1500  1400  1400  1300  1300  1200  £ Z  1100  w  ^  M  1200 1100  1000  1000  900  900  800 700 -100  Actual Desired -50  0 X(m)  50  800 100  700 -100  Force versus x-position without bump  -100  Actual Desired -50  0 x (mm)  Force versus x-position with bump  100  Test #22 - v =75mm/s, Y =405mm, k =1400N/m x  50  ep  Figure J.22 - Contact test #22  y  100  Appendix J  172  y-position versus x without bump  y-position versus x with bump  ky versus x without bump  ky versus x-position with bump  1600  1600  1500  1500  1400  1400  1300  1300  1200  1200  1100  1100  1000  1000  900  900  800 700 — —100  800  Actual Desired  Actual Desired  700 — -100 1  -50  0 X(m)  50  100  Force versus x-position without bump  0 x (mm)  -50  0 x (mm)  Force versus x-position with bump  50  Test #23 - v =75mm/s, Y =415mm, k =1400N/m x  50  ep  Figure J.23 - Contact test #23  y  100  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080794/manifest

Comment

Related Items