GRASPED OBJECT STIFFNESS DETECTION FOR ADAPTIVE CONTROL OF A PROSTHETIC HAND
A Thesis Presented to The Graduate Faculty of The University of Akron
In Partial Fulfillment of the Requirements for the Degree Master of Science
Ricardo Andrecioli
May, 2013
GRASPED OBJECT STIFFNESS DETECTION FOR ADAPTIVE CONTROL OF A PROSTHETIC HAND
Ricardo Andrecioli
Thesis Approved:
Accepted:
______________________________ Advisor Dr. Erik D. Engeberg
______________________________ Dean of the College Dr. George K. Haritos
______________________________ Co-Advisor or Faculty Reader Dr. Jiang Zhe
______________________________ Dean of the Graduate School Dr. George R. Newkome
______________________________ Depatment Chair or School Director Dr. Celal Batur
______________________________ Date
ii
ABSTRACT
Unfortunately, statistical analyses of amputee data shows an increase of the population with upper limb losses either by trauma or birth congenital defects. Several prosthesis options are commercially available, including electric powered prostheses. A review of surveys for upper limb prosthesis users have indicated improvement opportunities in the prosthesis design as well as improved functionality and controls. After review of literature, a PID sliding mode position controller and an adaptive PID sliding mode controller are presented for a prosthetic hand. The adaptive controller smoothly modulates the gains based on the detected stiffness of the grasped object. Three main control strategies will be compared: PID force control, sliding mode position and hybrid sliding mode force-position controllers. For each control option, an adaptive version will also be tested via benchtop experiments. In order to evaluate the performance of each controller under several grasping circumstances, a special manipulandum was designed to provide variable linear and nonlinear stiffness behavior, then each controller was then evaluated according to an experiment plan. The results from benchtop experiments indicate statistically significant improvements such as improved tracking response and reduced steady state error in the system response when using the adaptive controller for all three control cases considered. When comparing Force versus Position versus Hybrid Force-Position control, the latter when equipped of the adaptation method has presented the best results. Preliminary amputee experiments were also conducted iii
using the adaptive hybrid force-position controller in comparison to the constant gain controller as well as the amputees’ current prostheses for daily use. The results of these experiments show that the adaptive hybrid force-position sliding mode controller enabled the amputees to smoothly handle the manipulandum without breaking it.
iv
DEDICATION
In the beginning, I thought of this work as being dedicate to me, just for my selfish sense of accomplishment. I am not afraid to acknowledge that as I am also not afraid to say that I was wrong and I am glad that in the end I realized to whom this work was really intended to, He who has made me. At the very first when reading books and literature about amputees and upper limber powered prosthesis it was very difficult to me being a father myself to see pictures of little children who were born with a limb deficiency. At the same time I was attempting to see myself in an amputee’s condition, without a hand for instance, and came to me the conclusion we are never able to really feel all the difficulties they have to go through every day, in some cases have to give up their independency completely. And so I realized that this work was not for my only sense of accomplishment but actually something that someday could really help people. It does not matter if this work would help thousands of people or just one, it was worthy every minute of my time in the past six years I have been working to reach this moment for I believe that “whatever you do, whether in word or deed, do it all in the name of the Lord Jesus, giving thanks to God the
Father through him” (Colossians 3:17) and He will make the fruit of your work
abundant and prosperous.
v
No human being is able to ever reproduce such a skillful, complex, robust, smooth, powerful, adaptive and beautiful mechanical system such as the human hand as our Creator has been able to, however if we allow him to work through us we may achieve someday something that is close, but never the same. I would like to thank my dearly loved children Caio, Melissa and our baby which we don’t know if it is a boy or girl but sure is already loved as member of the family for I promised now Daddy will have more time with them. I also would like to send a special thank you to my beloved wife Regina whom was very supportive and is always next to my side, my long life partner, my best friend, my love. It is done baby!
vi
ACKNOWLEDGEMENTS
The author would like to thank J. Billock and the test subjects who volunteered their time for this research as well as Motion Control and the University of Akron for their generosity in providing the equipment and facilities necessary for the realization of this work.
vii
TABLE OF CONTENTS Page LIST OF TABLES ........................................................................................................ xi LIST OF FIGURES ......................................................................................................xii CHAPTER I. INTRODUCTION ....................................................................................................... 1 II. THE PROSTHETIC HAND ....................................................................................... 7 2.1 Sensors and Calibration ......................................................................................... 8 2.2 Variable Stiffness Manipulandum ........................................................................10 III. CONTROL PROBLEMS AND CHALLENGES .................................................... 13 3.1 The Tracking Problem and PID Control ............................................................... 13 3.2 Force Control Problem ........................................................................................ 14 3.3 PID Sliding Mode Controller ...............................................................................16 3.3.1 Stability Analysis ..................................................................................... 16 3.4 Position Control Problem .................................................................................... 18 3.5 Force-Position Control Problem .......................................................................... 20 IV. ADAPTIVE CONTROLLERS ............................................................................... 22 4.1 Stiffness Measurements ....................................................................................... 23 viii
4.2 Determination of Object Stiffness ........................................................................24 4.3 Definition of Stiffness-Gain Relationship ............................................................25 4.4 Force Control Problem ........................................................................................ 26 4.5 Adaptive Stability Analysis ................................................................................. 27 4.6 Position Control Problem .................................................................................... 30 4.7 Force-Position Control Problem ...........................................................................30 V. EXPERIMENTAL METHODS ............................................................................... 32 5.1 Benchtop Experiments ......................................................................................... 32 5.1.1 Force Control ...........................................................................................35 5.1.2 Position Control ....................................................................................... 36 5.1.3 Force-Position Control ............................................................................. 36 5.2 Calculation methods for system responses ........................................................... 37 5.3 Statistical Analysis Methods ................................................................................38 VI. BENCHTOP RESULTS ......................................................................................... 40 6.1 Force Control PID ............................................................................................... 40 6.2 Force Control APID ............................................................................................ 43 6.3 Position Control SMPID ......................................................................................52 6.4 Position Control ASMPID ................................................................................... 54 6.5 Force-Position Sliding Mode Control ..................................................................64 6.6 Force-Position Control ASMPID ......................................................................... 67 VII. CONCLUSION ..................................................................................................... 85 VIII. FUTURE DEVELOPMENT – AMPUTEE EXPERIMENTS ............................... 88 ix
8.1 Amputee Experiments Results ............................................................................. 91 BIBLIOGRAPHY ........................................................................................................ 94 APPENDICES............................................................................................................... 99 APPENDIX A: DATA ANALYSIS ALGORITHMS / MATLAB SOURCE CODES ....................................................................................................... 100 APPENDIX B: STIFFNESS ALGORITHM – C SOURCE CODE .............................. 110 APPENDIX C: MECHANICAL DESIGN OF THE MANIPULANDUM ................... 111 APPENDIX D: EXPERIMENTAL DATA ................................................................. 117 APPENDIX E: SIMULINK BLOCK DIAGRAMS ..................................................... 135
x
LIST OF TABLES Table
Page
1: Descriptive statistics for pre-experimental evaluation – Position PIDSM ................33 2: Descriptive statistics for pre-experimental evaluation – Position APIDSM ..............33 3: Power and sample size analysis (Alpha = 0.05, standard deviation = 0.347) .............33 4: ANOVA analysis for %OS for Force Controller. ..................................................... 48 5: ANOVA analysis for %Ess for Force Controller. ..................................................... 50 7: ANOVA for %OS for Position Controller. ............................................................... 60 8: ANOVA analysis for %Ess for Position Controller. ................................................. 61 9: ANOVA analysis for %Eabs for Position Controller. ..............................................62 10: ANOVA analysis for %OS for Force-Position Controller. .......................................81 11: ANOVA analysis for %Ess for Force-Position Controller. ...................................... 82 12: ANOVA analysis for %Eabs for Force-Position Controller. ..................................... 83
xi
LIST OF FIGURES Figure
Page
1:
Prosthetic hand and cosmetic glove from Motion Control, Inc. .............................7
2:
Hand position sensor calibration...........................................................................9
3:
Force versus position chart when grasping k2. Illustration of hysteresis that could affect stiffness calculation. ...................................................................................9
4:
Manipulandum used in the experiments..............................................................12
5:
Displacement of manipulandum when in use. ..................................................... 12
6:
Control diagram for the PID force controller for the Motion Control Hand. ........ 15
7:
Control diagram for the PIDSM position controller for the Motion Control Hand. .......................................................................................................................... 19
8:
Control diagram for the PIDSM hybrid force-position controller for the Motion Control Hand. ....................................................................................................20
9:
Force applied versus manipulandum displacement. Manipulandum setup with compression springs and spring rates from the manufacturer listed as K1= 0.52N/mm, K2 = 5.78N/mm and K3 = 11.38N/mm. ........................................... 22
10:
Manipulandum setup with K1 spring and magnet attached. ................................23
11:
Stiffness calculation algorithm flow chart...........................................................25
12:
Control diagram for the APID force controller. The control gains are adaptively dependent upon the detected stiffness of the system which changes when different objects are grasped. ............................................................................................ 27
13:
PID gains vs. object stiffness, arrows indicate increasing direction of stiffness. ..28
14:
Plane location for sliding manifold as stiffness changes...................................... 29
xii
15:
Control diagram for the APIDSM position controller. The control gains are adaptively dependent upon the detected stiffness of the system which changes when different objects are grasped. ....................................................................30
16:
Control diagram for the APIDSM hybrid force-position controller. The control gains are adaptively dependent upon the detected stiffness of the system which changes when different objects are grasped. ....................................................... 31
17:
Desired input for the grasping cycle. ..................................................................34
18:
Square Plot for DOE with 2 factors; Controller – 2 levels / Stiffness – 4 levels. .35
19:
Square Plot for DOE with 2 factors; Controller – 2 levels / Stiffness – 8 levels. .36
20:
Force Control, PID controller grasping spring k1 (left) and k2 (right)................. 41
21:
Force Control, PID controller grasping spring k3 (left) and k3M (right). ............ 42
22:
Force Control, PID controller grasping spring steel block...................................42
23:
Force Control, adaptive PID controller grasping spring k1. ................................43
24:
Force Control, adaptive PID controller grasping spring k2. ................................44
25:
Force Control, adaptive PID controller grasping spring k3. ................................45
26:
Force Control, adaptive PID controller grasping spring k3M. ............................. 46
27:
Force Control, adaptive PID controller grasping steel block. ..............................47
28:
Force Control, overshoot comparison by controller. ...........................................48
29:
Force Control, steady state error comparison by controller. ................................49
30:
Force Control, absolute (tracking) error comparison by controller. ..................... 50
31:
Force Control, PID gains vs. stiffness comparison by controller. ........................51
32:
Position Control, sliding mode PID controller grasping spring k1 (left) and k1M (right).................................................................................................................52
33:
Position Control, sliding mode PID controller grasping spring k2 (left) and k2M (right).................................................................................................................53 xiii
34:
Position Control, sliding mode PID controller grasping spring k3 (left) and k3M (right). ................................................................................................................ 53
35:
Position Control, sliding mode PID controller grasping steel block (left) and nothing (right). ...................................................................................................54
36:
Position Control, adaptive sliding mode PID controller grasping spring k1.........55
37:
Position Control, adaptive sliding mode PID controller grasping spring k1M. .... 55
38:
Position Control, adaptive sliding mode PID controller grasping spring k2.........56
39:
Position Control, adaptive sliding mode PID controller grasping spring k2M. .... 56
40:
Position Control, adaptive sliding mode PID controller grasping spring k3.........57
41:
Position Control, adaptive sliding mode PID controller grasping spring k3M. .... 57
42:
Position Control, adaptive sliding mode PID controller grasping nothing (empty). ............................................................................................................. 58
43:
Position Control, adaptive sliding mode PID controller grasping steel block. .....58
44:
Position Control, overshoot comparison by controller. ....................................... 59
45:
Position Control, steady state error comparison by controller. ............................ 61
46:
Position Control, absolute (tracking) error comparison by controller. .................62
47:
Position Control, PID gains vs. stiffness comparison by controller. ....................63
48:
Force-Position Control, sliding mode PID controller grasping spring k1 (left) and k1M (right). .......................................................................................................65
49:
Force-Position Control, sliding mode PID controller grasping spring k2 (left) and k2M (right). .......................................................................................................65
50:
Force-Position Control, sliding mode PID controller grasping spring k3 (left) and k3M (right). .......................................................................................................66
51:
Force-Position Control, sliding mode PID controller grasping steel block (left) and nothing (right). ............................................................................................ 66
xiv
52:
Force-Position Control, adaptive sliding mode PID controller grasping spring k1. .......................................................................................................................... 67
53:
Force-Position Control, adaptive sliding mode PID controller grasping spring k1M. .................................................................................................................. 68
54:
Force-Position Control, adaptive sliding mode PID controller grasping spring k2. .......................................................................................................................... 68
55:
Force-Position Control, adaptive sliding mode PID controller grasping spring k2M. .................................................................................................................. 69
56:
Force-Position Control, adaptive sliding mode PID controller grasping spring k3. .......................................................................................................................... 69
57:
Force-Position Control, adaptive sliding mode PID controller grasping spring k3M. .................................................................................................................. 70
58:
Force-Position Control, adaptive sliding mode PID controller grasping steel block. .................................................................................................................70
59:
Force-Position Control, adaptive sliding mode PID controller grasping nothing. 71
60:
Desired position xD versus normal Force for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K1. ............................................. 72
61:
Desired position xD versus normal Force for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K1M. ..........................................72
62:
Desired position xD versus normal Force for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K2. ............................................. 73
63:
Desired position xD versus normal Force for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K2M. ..........................................73
64:
Desired position xD versus normal Force for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K3. ............................................. 74
65:
Desired position xD versus normal Force for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K3M. ..........................................74
xv
66:
Desired position xD versus normal Force for both ASMPID (top) and SMPID (bottom) empty hand grasping cycle. ..................................................................75
67:
Desired position xD versus normal Force for both ASMPID (top) and SMPID (bottom) for steel block. ..................................................................................... 75
68:
Motor input in Volts (V) for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K1. .................................................................76
69:
Motor input in Volts (V) for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K1M. .............................................................. 77
70:
Motor input in Volts (V) for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K2. .................................................................77
71:
Motor input in Volts (V) for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K2M. .............................................................. 78
72:
Motor input in Volts (V) for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K3. ................................................................. 78
73:
Motor input in Volts (V) for both ASMPID (top) and SMPID (bottom) for manipulandum setup with spring K3M. .............................................................. 79
74:
Motor input in Volts (V) for both ASMPID (top) and SMPID (bottom) empty hand grasping cycle. ........................................................................................... 79
75:
Motor input in Volts (V) for both ASMPID (top) and SMPID (bottom) for steel block. .................................................................................................................80
76:
Force-Position Control, overshoot comparison by controller. .............................81
77:
Force-Position Control, steady state error comparison by controller. ..................82
78:
Force-Position Control, absolute (tracking) error comparison by controller. .......83
79:
Force-Position Control, PID gains vs. stiffness comparison by controller. ..........84
80:
Picture of amputees grasping the manipulandum. ...............................................90
81:
Picture of amputees holding the manipulandum. ................................................ 91
82:
Amputee experiments results trial #1..................................................................92 xvi
83:
Amputee experiments results trial #2.................................................................. 93
84:
Force Controller ASMPID Simulink Code Diagram ......................................... 134
85:
Force Controller APID Simulink Code Diagram .............................................. 134
86:
Force Controller PID Simulink Code Diagram ................................................. 135
87:
Force Controller PID Simulink Code Diagram ................................................. 135
89:
Force-Position Controller ASMPID Simulink Code Diagram ........................... 136
88:
Force-Position Controller ASMPID Simulink Code Diagram ........................... 136
90:
Force-Position Controller SMPID Simulink Code Diagram .............................. 137
91:
Force-Position Controller SMPID Simulink Code Diagram .............................. 137
92:
Position Controller ASMPID Simulink Code Diagram ..................................... 138
93:
Position Controller ASMPID Simulink Code Diagram ..................................... 138
94: 95:
Position Controller SMPID Simulink Code Diagram ........................................ 139 Position Controller SMPID Simulink Code Diagram ........................................ 139
97:
Voltage to position linearization subsystem ...................................................... 140
96:
Motor input calculation (PID) subsystem ......................................................... 140
98:
Gain direct feed subsystem (SMPID only)........................................................ 141
99:
Gain calculation subsystem (ASMPID only) .................................................... 142
xvii
CHAPTER I INTRODUCTION
A study from 2004 has presented that each year, an estimated 158,000 persons are admitted to hospitals in the United States to undergo limb amputation. According to (Pezzin, et al., 2004) a comfortable, effective, and easy to use prosthesis makes a positive contribution to an amputee’s ability to accomplish social and physical roles independently. A more recent study from 2008 has estimated that approximately 185,000 persons undergo limb amputation each year in the United States. On studies presented in (Ziegler-Graham, et al., 2008), an estimated number of 1,757,000 persons were living with the loss of a limb in the year of 2010 in the USA. The number is expected to more than double by the year of 2050, a projected 3.6 millions persons will be living with a loss of a limb. From both studies, approximately 10% of the amputee population, i.e. 175,700 people, are related to an upper limb loss being the majority of it due to an accident or trauma. The advances on myoelectric signal processing and control has contributed significantly for the electrically alternative to the mechanical body powered systems in
1
commercial prosthesis. During muscular contractions, the amputees are able to generate a repeatable
gradually
varying
myoelectric
signal
that
can
be
captured
by
electrodes (EMG) then used to control the prosthesis. The majority of research has been focused on their limitations to comply with all the requirements for fine control. (Oskoei, et al., 2007) This thesis has as primary interest in the comparison between existing force and or position control in an electrically powered prosthetic hand (Muzumdar, 2004), as well as the implementation of other potential improvements such as adaptive and robust control. Previous publications have identified a great interest in electric prosthetic hands for future use in patients with an upper limb loss, as well as a great desire for improved functionality, specifically on adaptation to grasped object properties, as well as more lifelike function (P. Kyberd, et al., 2007) and (Biddiss, et al., 2007). A comparative survey of upper prosthesis acceptance and abandonment over the past 25 years (Biddiss, et al., 2007) indicated that improvements in electric powered prostheses should be in the areas of appearance, comfort, durability, functionality and control. These last areas are detailed as follows: Autograsp and responsive high speed Adaptive grasp and multi-articulated mechanisms Sensory feedback (tactile, slip, proprioceptive, etc) It is important to say that the human hand is a skillful system for object manipulation capable of controlling all the aspects of grasp such as force, position and stiffness. Development of biomechanical devices to mimic a human hand is a complex
2
task that requires deep understanding of human anatomy, muscle synergies and how the human central nervous system utilizes the hand’s biomechanics to transition between different environment conditions (Balasubramanian, et al., 2008). In addition to efficient control strategies, the design of the prosthesis is of importance, especially because it will most likely become a contact point with a person and therefore should be designed to guarantee safe hand motions as well as comfortable human touch through its physical interactions with human beings (Kajikawa, et al., 2010). For the control problem of an electrical prosthetic hand it is expected that the control strategy is capable of tracking position and/or force for an wide range of system stiffness. Despite the diversity of approaches, there are two fundamental methods, hybrid force/position control and impedance or stiffness control (Chan, et al., 1991). In this aspect it is important to mention that in human motor control tasks, sensory weighing is governed by the object stiffness (Mugge, et al., 2009), i.e. when handling stiff objects, deflections are negligible therefore position has little influence on the applied force, however when handling a soft object, deflections are large and allow position to contribute to estimated force. Maintaining this line of thinking, the use of hybrid force/position controllers are actually a straight implementation of this observed fact and therefore an advantage compared to impedance control when trying to mimic as close as possible the behavior of human hand. For robots to perform complex tasks there is a need for robust and stable force and or position control, however force control stability is particularly difficult to obtain
3
depending on the stiffness or compliance of the object being manipulated by the robot. The application of a Fuzzy model reference adaptive control approach replacing the need for environment stiffness detection presented in (Burn, et al., 2003) indicated significant improvements in the system response, however the creation of a model reference with the nonlinearities related to the human prehension problem could become extremely complex. The use of iterative learning impedance control for robotic manipulators has been also proposed in the tracking problem of position and force mainly due to the fact that manipulators are used to perform repetitive tasks, therefore given a targeted impedance or reference model (Cheah, et al., 1998) and (Yang, et al., 2011), however as mentioned before the derivation of model based systems are usually complex especially in nonlinear systems. In some particular robots such as continuum manipulators, the stability and vibration are of considerable concern when a flexible robot is in contact with its environment and therefore stiffness control has actually achieved better results than force and or position control (Mahvash, et al., 2011). In a problem with high environmental uncertainty, which is the case in control of a prosthetic electric hand, effective controls can only be achieved by application of an accurate environment stiffness detection and smooth gain adjustment technique; however it can become unstable when the stiffness varies significantly (Ow, 1997). In order to accurately detect and measure stiffness, tactile sensing is of essential importance. Studies have demonstrated that the manipulation of objects are compromised
4
when the “sense of touch is lost” (Westling, et al., 1984). An extensive investigation about human tactile sensing starting from its definition and brief discussion on the physiological aspects of sensing to technologies developed to improve touch sensing capability of robots is presented in (Dahiya, et al., 2010). The force, position and hybrid force-position control in object prehension using electric powered upper limb prostheses and traditional proportional and or derivative feedback control could be sometimes unstable or oscillatory for stiff objects. Many studies in the area of force and contact transition control have been developed. The application of a nonlinear PD controller in robot manipulators has presented improved force control with a rigid object (Xu, et al., 1995). In addition the force response behavior could vary depending on the input. In the realm of prosthetics, the SensorHand (Otto Bock, Minneapolis, USA) has a three-axis force sensor embedded in the thumb that could be used for force control and or grasped object slip prevention (Puchhammer, 2003). It is important to emphasize that the prosthesis control problem differs from structured robotics control such that different objects with different properties can be grasped and the controller does not know ahead of time what those objects are (Winges, et al., 2009). So, an intuitive improvement opportunity to make the prosthetic hand more versatile is to have the controller adaptively adjust the controller gains based on measurements of object stiffness; however, stiffness detection and measurement can be a complex task and so are the tactile sensors that are typically employed, see (Puchhammer, 2003), (Zhang, et al., 2006)and (Yussof, et al., 2008). In this research this problem will be addressed by a data fusion algorithm considering grasped object stiffness
5
(Tsetserukou, et al., 2008). The way this system could improve stability is by smoothly adjusting the gains in order to prevent unstable conditions. In this study a sliding mode controller will be considered and since it is expected that response tracking is of major interest in order to address the demand from users, a PID control will be added to the controller. An extensive analysis comparing a traditional force, position and hybrid force-position for the PID sliding mode (PIDSM) control strategies versus an adaptive version of the same controllers will be presented and the advantages and disadvantages of each will be discussed. The equipment used in this research is presented in Chapter II. Chapter III will present more aspects of the control problems and challenges related as well as discussion on other literature as well as stability analysis. The proposed adaptive alternative and related stability analysis will be presented in Chapter IV. The experimental methods and results are presented in Chapter V and VI. Finally the conclusions over the results and discussion for future research are presented in Chapter VII and VIII, respectively.
6
CHAPTER II THE PROSTHETIC HAND
The Motion Control Hand® (Motion Control, Inc. Salt Lake City, USA) has a single degree of freedom with thumb being coupled to forefingers via a four-bar linkage system. Strain gages located on the thumb are used to determine the normal and shear forces when grasping objects.
Figure 1: Prosthetic hand and cosmetic glove from Motion Control, Inc.
7
The distance between the thumb and forefingers is defined as the position of the hand. A picture of the hand is shown on Figure 1. According to (E. Engeberg, et al., 2008), the state equations for the hand position are given by x1
x2
x2
B J
x2
K
n
x J 1
J
E
D.
(1)
J
The states x1 and x2 are the position and velocity respectively. J, B and K are the effective inertia, damping and stiffness of motor and linkage bars. D is the sum of disturbances produced by nonlinearities such as Coulomb friction and backlash. E is the voltage input to the motor and n is a torque constant that includes the armature resistance and relation between motor angle and hand position.
2.1
Sensors and Calibration
A Hall Effect sensor was added to the hand in order to measure the position of the hand (distance between fingers). The low cost and small size of the sensor and implementation with minimum hardware modification were the predominant factors in the selection of the sensor. When calibrating the sensor, the distance between thumb and finger or hand position versus the output voltage from the Hall Effect sensor presented a nonlinear behavior. In order to minimize the problem complexity, the sensor voltage output was linearized to improve signal processing via use of curve fitting.
8
The curve fitting analysis on the output voltage presented itself exponentially relative to the distance between the sensor and permanent magnet. Steel blocks with known sizes were grasped with the prosthetic hand and the output voltage from the sensor was then recorded and later plotted as presented in Figure 2.
80 70 Position (mm)
60 50 40 30 20 10 0
-6
-4
-2
0 Voltage (V)
2
y = 30.101e0.196x R² = 0.9877 4
Figure 2: Hand position sensor calibration.
Figure 3: Force versus position chart when grasping k2. Illustration of hysteresis that could affect stiffness calculation.
9
6
As it can be seen in Figure 2, the exponential behavior of the sensor is demonstrated by the curve fitting results and the R squared value of 0.9877 indicated an excellent fit. The adjusted equation
= 30.101
.
, where y represents the actual hand
position (distance between finger and thumb) and x is the output voltage from the sensor, was used in the implementation of the controller. One disadvantage of the sensor was its high hysteresis which can be seen on Figure 3, this could compromise the measurement accuracy of positions and therefore all relative calculations that would have use of position measurements.
2.2
Variable Stiffness Manipulandum
To assist in the experiments, a manipulandum was designed and built in order to simulate adjustable, variable, and nonlinear stiffness conditions. The system consisted of 2 levers connected via a single degree of freedom rotational joint. The 2 levers had each a pocket or recess to accommodate a mechanical compression spring. The springs can be easily interchanged within the manipulandum, thus yielding adjustable linear stiffness. In the design of the manipulandum, it was required to have a light weight system yet sturdy and durable. For this reason aluminum was the primary material used in the construction of the manipulandum. To create a smooth low friction rotation joint, a nylon
10
bushing was considering with an interference fit between the bushing and housing in one arm of the manipulandum. The two arms were then connected with a steel shaft (ground pin) with clamps on both ends. The geometry of the manipulandum was so to optimize the maximum displacement possible given the maximum open position of the prosthetic hand, with normal force ranging from nearly zero to 100N. The resulting manipulandum design is a robust solution that has been used in eight publications thus far; (Engeberg, et al., 2013), (Engeberg, 2013), (Engeberg), (Engeberg), (Engeberg, 2010), (Engeberg), (Engeberg, et al., 2011), (Andrecioli, et al., 2010) and (Andrecioli, et al., 2012). Three different compression springs were used in this research, these springs will be referenced throughout this work as K1, K2 and K3. The spring rates according to the manufacturer are listed as K1 = 0.52N/mm, K2 = 5.78N/mm and K3 = 11.38N/mm. In addition to the variable stiffness achieved by mechanical springs, a permanent magnet can be attached to the frame facing the back of one of the levers. The magnet position can be adjusted by increasing or decreasing the distance or radius with respect to lever’s pivot point thus yielding a range of nonlinear force that was used in combination to the springs to produce additional nonlinear stiffness components to the manipulandum. This was very helpful in order to simulate a brittle object, for example, an egg. The spring-magnet stiffness condition will be defined for notation purpose as K1M, K2M and K3M being the combination of K1, K2 and K3 with the magnet respectively. In order to assist calibrating the manipulandum stiffness, an A1321 (Allegro Micro) Hall Effect sensor was added to measure the actual manipulandum displacement
11
and LSP-10 load cells (Transducer Techniques, Temecula, USA) were used to measure the grasping force. The stiffness of the manipulandum based on the spring setup is shown on Figure 4 and Figure 5. Detailed CAD drawings for the manipulandum are given in Appendix C.
Levers
Load Cell Spring
Magnet
Hall Sensor
Figure 4: Manipulandum used in the experiments.
Figure 5: Displacement of manipulandum when in use.
12
CHAPTER III CONTROL PROBLEMS AND CHALLENGES
3.1
The Tracking Problem and PID Control
In most systems, depending on the gain chosen, force or position control in object prehension using electric powered upper limb prostheses and traditional proportional control can be sometimes oscillatory and or result in large steady state error depending on stiffness or compliance of the grasped objects. As mentioned previously, the position control of a prosthetic hand could become significantly difficult particularly when handling objects with very high stiffness values which yields small or no deformation of the object. In addition, the use of electromyogram (EMG) control could have noise-to-signal ratios considerably high, further increasing the complexity of the problem. In a similar manner, force control of a prosthetic hand could also become significantly difficult. Previous studies (Andrecioli, et al., 2010) have shown this to happen when handling objects with low stiffness values. Low stiffness objects result in high object deformation for low force levels. This in combination with a high noise-to-signal ratio from the EMG control could make the force control problem extremely challenging as well.
13
As discussed in (Winges, et al., 2009) it is important to emphasize that the prosthetic hand control problem differs from well-defined robotics control in a way that awide range of objects with different properties can be grasped and the controller does not know ahead of time what those objects are. As successfully presented on previous publications, see (Chang, 2002), (Chang, et al., 1998), (Burn, et al., 2003), (Drissen, 2006), (Kajikawa, et al., 2010) and (Ouyang, et al., 2004) an improvement opportunity to make prosthetic hands more versatile is to have the controller adaptively adjust the feedback gain. The manner this system could improve stability in this thesis is by adjusting the feedback gain based on the grasped object stiffness. The overall control problem of a powered prosthetic hand has similar problems related to proportional control and the influence of feedback gains used (Eker, 2006) because robotic system dynamics are very sensitive to the stiffness of the object being grasped and the tuning of the gains are usually made in order to favor to either a compliant or a stiff object.
3.2
Force Control Problem
As mentioned previously, force control of a prosthestic hand is an extremely challenging problem. The nonlinear behavior of the system makes it very difficult, especially in a tracking problem when PID control logic is used in combination with EMG. An example of a PIDSM force controller is shown on Figure 6.
14
d dt
Figure 6: Control diagram for the PID force controller for the Motion Control Hand. However the varying nature of the objects being handled by the prosthetic hand adds a certain degree of difficulty regarding tuning the system gains to work with a wide range of object stiffness. In force control, it was initially observed that higher gains would make the system to respond satisfactorily when handling compliant objects, i.e. low percentage of overshoot and steady state error, however this condition when grasping stiff objects would be very oscillatory, with high percentages of overshoot. On the other hand, lower gains would improve the system response for stiff objects at a cost of increasing significantly the steady state error when handling compliant objects. Therefore, the compromise between percentage of overshoot (%OS) and steady state error (%Ess) will be a consistent situation for this particular constant gain controller, especially because of the wide range of objects and stiffness that a human is expected to handle on a daily basis. In ddition to the challenges mentioned above, the controllability of the hand when empty is poor, especially when using EMG. Since no force is involved when the hand is empty, this become the case for position control which is discussed in the the next session.
15
3.3
PID Sliding Mode Controller
For a tracking control problem it is expected that integral action will improve steady state error. Derivative action is also desired for optimal system response on a tracking problem, however care must be taken because EMG is noisy in nature. Due to the nonlinear nature of the problem, a sliding mode (SM) controller will be considered in combination with a PID sliding manifold in order to ensure good signal tracking.
3.3.1 Stability Analysis
According to (Khalil, 2001), let the system from (1) be written as = = ( , )+ =
+
(2)
Since a PID sliding mode (PIDSM) controller is being implemented and it is desired to track the input signal then augmented by the integrator
( ) or just =
for simplicity of notation. The system is
.
(3)
By applying the change of variables =
(4)
=
(5)
16
we obtain the augmented system
note
( ,
=
= .
= = )+
(6) +
To minimize the tracking error, , we must drive the sliding manifold to zero: = Thus,
and
+
+
= 0.
(7)
must be chosen such that the polynomial +
+
(8)
is Hurwitz. To be consistent with the control diagrams, the polynomial in (8) needs to be rewritten as + where
=
and
=
+
(9)
.
Because any random object can be grasped by amputees using prosthetic hands, it is assumed that the exact parameters of the system model are unknown and the best estimate for the system is given by )+
= ( ,
=
+
(10)
Assuming also the difference between actual and estimated model is bounded ( ,
)
( ,
17
)
( ,
)
(11)
The best continuous estimate for the input (
=
,
to maintain
)
= 0 is given by
(12)
Therefore, to robustly deal with the disturbances and model inaccuracies, a nonlinear term must be added to the input so that = where
is known and
>
( )
(13)
. In order to prevent chattering the signum function is
replaced by a saturation function, thus (15) can be written as =
( ).
(14)
Since it is required the controller to bring the states of the system onto the sliding manifold,
must enforce the constraint | |
(15)
Thus from (8) – (14) we obtain =
3.4
( ,
)+
+
+
+
| |
(16)
Position Control Problem
In the position control problem for electric powered prosthetic hands, tuning the gains and optimizing the response is also a great challenge. The system still can present an oscillatory behavior depending on the gains chosen and the object stiffness. The control diagram for PID position control is given on Figure 7.
18
Similar to the force control problem, the nonlinear behavior of the system presents several complications in applying a standard proportional or PID control logic. Therefore, as discussed previously, the implementation of a PID sliding mode controller is also suggested for the position control problem.
Figure 7: Control diagram for the PIDSM position controller for the Motion Control Hand. Observation from previous tests indicated the influence of the gain relationship to grasped object stiffness in the system response to be the opposite of the relationship observed in the force control problem, i.e. higher gains would actually cause system oscillations when handling compliant objects instead of stiff ones as in force control. On the other hand, lower gains would make the system less prone to oscillation, but as stiffness increases the steady state error is expected to increase as well. Therefore, similar to the force control problem, there will have to be a compromise between %OS and %Ess when setting the gains for a wide range of object stiffness. By using very small gain values, the controllability of the hands seems to improve when the hand is empty, however grasping solid stiff objects larger in size than the desired input can cause the hand to “crush” the object, saturating the error and the hand motor power amplifier. In some instances the saturation of the power amplifier would cause the hand to lock in a certain position. This eventually implies a waste of energy
19
reducing the batteries and increasing fatigue of the hand components. An alternative proposal for solving this issue is presented next.
3.5
Force-Position Control Problem As is widely known, both position and force control strategies have advantages
and disadvantages. In order to obtain the benefits of both control strategies while minimizing the disadvantages, a hybrid force-position control strategy has been proposed. A hybrid controller has been previously proposed and successfully implemented for the control of a prosthetic hand in (E. Engeberg, et al., 2008) except the approach used consider force-velocity controller instead of force-position control. The control diagram for the hybrid control system is given on Figure 8.
d dt
Figure 8: Control diagram for the PIDSM hybrid force-position controller for the Motion Control Hand.
The desired force input is subtracted from the force feedback and scaled by a gain Gf. This then is the desired position input, xD, for the position PID sliding mode control that will generate the input to the hand.
20
The logic behind this controller is that when the hand is empty, the force is virtually zero and the system behaves just as a positon controller obtaining better controllability of the hand. When grasping solid stiff objects, as the force increases the position input decreases preventing a “crushing” condition. Because the PID sliding mode logic is in the position loop, it is expected that the system would behave similarly to the position controller with regards of the gains and object stiffness relationship. Even though this control logic seems to solve some of the issues presented so far, i.e. position control or when grasping high stiffness solid objects, the gain tuning problem for a wide range of objects and the implication that would have on %OS and %Ess is still an open problem to be discussed on the next section. The actual Simulink implementation of the controller is given on Appendix E.
21
CHAPTER IV ADAPTIVE CONTROLLERS
As previously mentioned, the control problem of a prosthetic hand of a complex nature due to the nonlinearities involved in the mechanical system as well as the uncertain environment presents a challenging problem. Adaptive control of robotic mechanisms in uncertain environments has been approached in (Burn, et al., 2003). Due to the direct influence of the total system stiffness (combination of hand and object) in the system response it is expected that a system adaptation based on the grasped object stiffness would solve some of the problems presented so far. Therefore, object stiffness detection is the core problem in order to develop an improved control strategy
Figure 9: Force applied versus manipulandum displacement. Manipulandum setup with compression springs and spring rates from the manufacturer listed as K1= 0.52N/mm, K2 = 5.78N/mm and K3 = 11.38N/mm.
22
to the control problem discussed so far and the performance of the algorithm proposed in section 4.2 is critical to this work.
4.1
Stiffness Measurements
In order to verify the different stiffness created by the combination of springs and magnet, a short experiment was performed and the normal force and manipulandum travel or displacement data was recorded. Linear force-displacement curves for different linear springs are shown in Figure 9 and nonlinear stiffness KM1 is shown in Figure 10 while using K1 in parallel with the magnet.
Figure 10: Manipulandum setup with K1 spring and magnet attached.
23
4.2
Determination of Object Stiffness
This research uses an improved algorithm for stiffness calculation when compared to the algorithm presented in (Andrecioli, et al., 2012). The improvements consisted mainly of an approximation of the derivative of force with respect to position or displacement, the algorithm used in (Andrecioli, et al., 2010) was mainly based on the force change (
) divided by position change (
), where the starting condition is
defined when contact with grasped object is made. Such approach will result in averages results for the stiffness and potentially smoothing or reducing nonlinearities present in some of the objects grasped. By approximating this derivative, an instantaneous measure of stiffness can be estimated for a given an increment in force and or displacement. Thus, it is expected a better tracking of nonlinear stiffness objects. The algorithm calculation is described below:
defining
=
as actual normal force and
,
(17)
as actual hand position, the sensor sampling is
done at a certain frequency, in this particular case 1kHz, and data is stored in matrices or data files. The subscript index
represents the most recent normal force or position
reading from the sensors. The subscript index
1 represents the previously recorded
reading for either normal force or position. The parameters are calculated as presented on the flowchart on Figure 11.
24
It is important to note that there are no biological sensors in the human body capable of sensing only torque or force, these are perceptual variables derived from receptors in the skin, muscles and tendons. The estimates of stiffness from two sources, even though redundant, when combined could reduce the uncertainty in perception and improve stiffness detection (Wu, et al., 2011).
x1t | FN t
x1 t x1 t
x1t
FN |
F
N t
F
N t
x1 x1t
1
F 1
N
F
N t
x1t x1 t
1
FN t FN t
1
FN t 1
FN
t 1
Figure 11: Stiffness calculation algorithm flow chart
4.3
Definition of Stiffness-Gain Relationship
The gain-stiffness relationship given by (3) will be used for this study. For cohesion of notation, the capital letter G with subscript n will be used to refer to the gains of the system, i being for integral, d for derivative and p for proportional gains.
25
Gn max , if K Gn ( K )
an K
K max
bn , if K min
Gn min , if K where the values for (
and
) and a compliant ( .
K
K max ,
(18)
K min
are determined by defining the desired gain for a stiff ) object and then solving the set of equations for
and
is initially small then increased as needed according to the stiffness increases or
decreases depending on the controller chosen. It is important to mention at this point that the measured stiffness is the actual system stiffness which is the combination of the grasped object and the prosthetic hand and therefore it can change with variable gains as well.
4.4
Force Control Problem
The control
diagram
for
the adaptive
force
controller is presented
onFigure 12. The normal force and hand position are used to calculate the object stiffness. The stiffness obtained is then used to adjust the proportional, integral and derivative gains.
26
d dt
Figure 12: Control diagram for the APID force controller. The control gains are adaptively dependent upon the detected stiffness of the system which changes when different objects are grasped.
4.5
Adaptive Stability Analysis
The stability analysis for the adaptive integral control can be proven similarly to the regular sliding mode PID controller. Consider the sliding manifold for the PID controller given in (8). The gains of the system
,
and
must be chosen in a
manner that the polynomial in (9) is Hurwitz in order to guarantee system stability. For the adaptive controllers, these gains are linear functions of the grasped object stiffness. Mathematically, these functions are essentially changing the slope of the sliding manifold as described in equation (8) to favorably affect the dynamics of the prosthetic hand.
27
Figure 13: PID gains vs. object stiffness, arrows indicate increasing direction of stiffness.
However as long as these gains are such as the polynomial in (9) is Hurwitz and this can be achieved by saturating the gains at given minimum and maximum values, and the magnitude of the input to the plant is saturated at the constant value , the same stability analysis presented previously applies. This fact is graphically presented on Figure 13.
28
20 15 10
e
5 0 -5 -10 -15 -20 20 10
20 10
0 0 -10
de/dt
-10 -20
-20
edt
K = Low K = High
Figure 14: Plane location for sliding manifold as stiffness changes. The stiffness will change the direction or the 3-dimensional slope of the manifold in the 3D space (if it was a 2D problem it would be changing the slope of the manifold). Assuming at initial conditions for a given stiffness the error is anywhere in the sphere and the sliding manifold is the one given by zero stiffness (minimum gains), if the stiffness changes while the system is in reach phase, it will extend or prolong the reach phase as the sliding manifold changes with the new stiffness, but as the system enters the sliding manifold, it will stay there and the error will then be driven to zero. If the stiffness changes while the system has already reached the sliding manifold, the worst case scenario the manifold will change and the system will start again with the reaching phasing until it enters the the new manifold and start driving the error to zero. Such explanation can be graphically visualized on Figure 14.
29
4.6
Position Control Problem
For the same reason presented on 0, the position control is also considered for an adaptive version. The control diagram for the adaptive force controller is presented on Figure 15. In the same way, the normal force and hand position are used to calculate the object stiffness. The stiffness obtained is then used to adjust the proportional, integral and derivative gains.
Figure 15: Control diagram for the APIDSM position controller. The control gains are adaptively dependent upon the detected stiffness of the system which changes when different objects are grasped.
4.7
Force-Position Control Problem
Even though the control logic presented on 0 solves the issue of the extremes, i.e. empty hand and high stiffness solid objects, the gain tuning problem for a wide range of objects and the implication that would have on %OS and %Ess is still an open problem to be discussed. As previoulsly discussed, the use of gain adaptation based on object
30
stiffness presented several advantages in control of electric powered prosthetic hands. The control logic diagram for the proposed hybrid force-position adaptive controller is given on Figure 16.
d dt
Figure 16: Control diagram for the APIDSM hybrid force-position controller. The control gains are adaptively dependent upon the detected stiffness of the system which changes when different objects are grasped.
As it can be seen on Figure 16, the system consists basically of an outer proportional force-feedback loop providing the input to the inner PID adaptive position control, where the slope of the sliding manifold is adaptatively adjusted based on the object stiffness measured according to (2) and the algorithm proposed in Figure 11. Even though the stiffness values could be anything from zero to nearly infinite, in order to keep the gains bounded, a saturation function was applied for each proportional, integral and derivative gain to prevent a “zero” gain situation that would make the hand unresponsive to any comand or input as well as to prevent extremely high gains that could cause crushing and premature wear of the hand mechanisms, which is one of the major factors in user disatisfaction.
31
CHAPTER V EXPERIMENTAL METHODS
5.1
Benchtop Experiments
In order to evaluate and compare the system response and performance for all the proposed control strategies, several benchtop experiments were planned. These experiments consisted of several replications of grasping cycles using the manipulandum presented in 0. For calculation of amount of replications, the standard deviation for each controller was determined. The highest standard deviation was used to ascertain the sample size. This approach would indicate how many replications to consider in each test. For example, the number of experiments needed to be performed if it is expected to detect a 0.5% difference in the error when comparing 2 or more springs. The results of the short experiment for the position controller are given in Table 1 and Table 2. From Table 1 it can be seen that 0.347 is the highest standard deviation for absolute error (Eabs) with the PIDSM controller. Since more than 2 levels will be considered in the experiments, a power and sample size analysis for a one-way ANOVA for the highest standard deviation observed is shown on Table 3. This gives the largest sample size for a given power of 95%.
32
Table 1: Descriptive statistics for pre-experimental evaluation – Position PIDSM Variable
Spring
Mean
StDev
K1
0.4032
0.1176
K2
0.0341
0.0379
K3
0.014900
0.001664
K1
2.691
0.272
K2
5.329
0.231
K3
5.634
0.347
Mean %OS
Mean Eabs
Table 2: Descriptive statistics for pre-experimental evaluation – Position APIDSM Variable
Spring
Mean
StDev
Mean %OS
K1
0.292
0.277
K2
0.1609
0.1230
K3
0.01857
0.00576
K1
2.215
0.284
K2
2.7753
0.0814
K3
2.6653
0.0663
Mean Eabs
Table 3: Power and sample size analysis (Alpha = 0.05, standard deviation = 0.347) SS Means
Size
Power
Actual Power
Difference
0.125
13
0.90
0.906052
0.5
0.125
16
0.95
0.958506
0.5
0.500
5
0.90
0.962302
1.0
0.500
5
0.95
0.962302
1.0
1.125
3
0.90
0.961583
1.5
1.125
3
0.95
0.961583
1.5
33
Considering a difference of 0.5% or greater in the responses, the analysis results indicated that 13 to 16 samples would give a power of 90% to almost 96%, respectively, in regards of Eabs, this range of samples should be enough for the other responses as well, since all had lower standard deviation values. Hence, 13 to 16 trials were considered in the experiments.
The desired input for the grasping cycle is given on Figure 18. As it can be seen, the input was composed of 4 main sections: a step section in both directions (open/close), ramp, sinusoidal with 2 different frequencies and a last component of EMG signal.
Figure 17: Desired input for the grasping cycle.
The manipulandum was set up on the bench top resting on a static position, a rubber glove was used to cover the fingers and improve the grasp preventing the manipulandum to slip during the grasping cycle. In some conditions, a steel block was used as the object grasped.
34
Measurement readings for the position and force were recorded during the grasping cycle then later evaluated to measure the performance of the controller. While performing the benchtop experiments, all signals were sampled by an NI PCI 6229 data acquisition card at 1kHz for processing in Simulink.
5.1.1 Force Control
The experiment consisted of full factorial design (Montgomery, 2001) with only two factors. The first factor being the controller or control logic (ASMPID/SMPID) and second being object stiffness, which consisted of setting the manipulandum with different springs (K1, K2, K3, K3M) as well as using a solid steel block. The square plot is given on Figure 18.
k1
k2
k3
k3M
Block
Controller
ASMPID
SMPID
Stiffness Figure 18: Square Plot for DOE with 2 factors; Controller – 2 levels / Stiffness – 4 levels.
35
5.1.2 Position Control
The experiment consisted of full factorial design with only two factors. The first factor being the controller or control logic (ASMPID/SMPID) and second being object stiffness, which consisted of setting the manipulandum with different springs (K1, K2, K3, K1M, K2M, K3M) as well as using a solid steel block and empty. The square plot is given on Figure 19.
k1
k1M
k2
k2M
k3
k3M
None
Block
Controller
ASMPID
SMPID
Stiffness
Figure 19: Square Plot for DOE with 2 factors; Controller – 2 levels / Stiffness – 8 levels.
5.1.3 Force-Position Control
The experiments consisted of several grasping cycles for a given trajectory while the manipulandum and electric hand were resting in a static position. The conditions are the same ones used for the position controller experiment and presented on the design of experiment (DOE) square plot in Figure 19. The derivation of the calculations for the
36
system responses are presented next. The actual MatLab codes for the calculation are presented on Appendix A.
5.2
Calculation methods for system responses
Based on the experiments planned above, it is important to define the calculation methodology to measure the response and performance of each controller, specially due to the variety of aspects being evaluated on a control system such as percentage of overshoot (%OS), rise time, percentage of steady state error (%Ess), percentage of tracking or absolute error (%Eabs), just to mention a few. The methods and calculations are given below. Let index
be the index representing the test condition, let be one of the several
step input included in the grasping cycle for the
test and the time index. Let also
be the time index at the beginning of the step and
the time index at the end of the step.
( )>
input when
( ) and negative step
( )
1.5)/*Check if force if greater than 1.5, ie contact stablished*/ { if (iniflag = false)/*auxiliary variable to flag if contact was previously stablish*/ { iniflag = "true";/* if not then set it to true then set initial conditions*/ y2[0]=u2[0]; /* Write – x @ contact or x0*/ y3[0]=u0[0]; /* write fn @ contact or F0*/ y1[0] = u4[0]/u5[0];/*calculate stiffness u4[0]=F-F0 and u5[0]=x-x0*/ } if (u5[0] > 1.5) /*Check if Abs value of (x-x0) > 1, minimum displacement increment to considered for updating variables and recalculate stiffness*/ { y2[0]=u2[0]; /*–Update x0*/ y3[0]=u0[0]; /* update F0*/ y1[0] = u4[0]/u5[0];/*update stiffness calculation*/ } else { y1[0] = u8[0];/*Ouput previous filtered stiffness calculated*/ } } else { iniflag = "false"; /*reset contact flag*/ y2[0]=u2[0]; /* No contact to be recorded*/ y1[0]=0; /* Zero stiffness*/ y3[0]=u0[0];/*f0 same as f*/ y5[0]=0; /* Write dx*/ y6[0]=0; /* Write df*/ }
110
APPENDIX C
MECHANICAL DESIGN OF THE MANIPULANDUM
111
112
113
114
115
116
APPENDIX D
EXPERIMENTAL DATA Force Controller
Spring k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k2 k2
Controller ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID
Stiffness 0.1401 0.0610 0.1084 0.0649 0.1064 0.1287 0.1527 0.1073 0.1492 0.1991 0.1351 0.1351 0.0821 0.1231 0.0340 0.1557 0.1563 0.0434 0.1030 0.1048 0.0507 0.0741 0.1082 0.1364 0.0310 0.0154 0.1601 0.0566 0.5197 0.5236
Gp 1.1104 1.1409 1.1226 1.1394 1.1234 1.1148 1.1055 1.1230 1.1069 1.0876 1.1123 1.1123 1.1327 1.1170 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.9638 0.9623
Gd 0.0635 0.0654 0.0643 0.0653 0.0643 0.0638 0.0632 0.0643 0.0632 0.0620 0.0636 0.0636 0.0649 0.0639 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.0540 0.0539
117
Gi 0.6989 0.6996 0.6996 0.6989 0.6996 0.6991 0.6986 0.6997 0.6989 0.6943 0.6981 0.6985 0.6995 0.6995 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.6429 0.6436
%OS 27.5 11.3 21.0 29.1 11.5 12.1 20.4 12.4 14.5 15.9 23.4 15.4 12.8 8.7 12.3 1.4 1.4 0.6 1.3 0.2 0.2 0.3 3.3 1.7 0.7 0.2 1.6 1.4 5.2 4.5
%Ess 22.5 10.2 12.1 24.5 12.1 22.9 6.7 11.0 11.7 8.7 18.8 11.0 8.6 8.9 56.1 35.4 35.5 48.9 51.8 55.9 57.7 53.4 49.7 59.1 54.9 53.6 52.0 47.7 3.8 4.6
%Eabs 21.5 33.5 19.7 39.7 32.4 20.2 13.9 35.0 16.0 15.7 19.5 15.7 27.6 14.6 28.8 27.3 27.1 35.4 34.4 29.8 31.4 36.7 32.3 29.6 28.2 29.7 27.6 30.1 4.3 4.9
Spring k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3
Controller ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID
Stiffness 0.4082 0.3704 0.3656 0.6065 0.5389 0.4430 0.5351 0.5294 0.6474 0.3862 0.3929 0.4107 0.3345 0.6574 0.4792 0.5717 0.6771 0.6553 0.6751 1.0335 0.2950 0.6629 0.7849 0.8349 0.7872 0.7814 0.9104 1.2095 1.0369 0.6977 1.1339 0.6895 1.4460 0.9114 1.1024 1.2867 1.1812 1.1206 1.3328 1.2399 1.4557 0.1023 0.5324 0.5089 0.6339 1.5211 0.5069
Gp 1.0069 1.0215 1.0234 0.9303 0.9564 0.9935 0.9579 0.9601 0.9145 1.0154 1.0128 1.0060 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.8130 0.6975 0.7642 0.8951 0.7267 0.8982 0.6061 0.8126 0.7388 0.6676 0.7084 0.7318 0.6498 0.6857 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000
Gd 0.0568 0.0577 0.0578 0.0518 0.0535 0.0559 0.0536 0.0537 0.0508 0.0573 0.0572 0.0567 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.0442 0.0368 0.0411 0.0495 0.0386 0.0497 0.0308 0.0442 0.0394 0.0348 0.0375 0.0390 0.0337 0.0360 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800
118
Gi 0.6663 0.6702 0.6657 0.6235 0.6397 0.6604 0.6354 0.6410 0.6136 0.6674 0.6643 0.6600 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.5545 0.4839 0.5248 0.6048 0.5018 0.6067 0.4279 0.5544 0.5091 0.4657 0.4906 0.5050 0.4548 0.4768 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050
%OS 5.2 2.4 3.4 2.7 3.5 3.4 3.7 4.2 3.2 5.6 5.9 3.7 3.1 0.4 0.7 0.1 0.8 1.0 0.5 0.3 0.7 1.2 0.6 0.3 2.1 1.2 6.6 7.5 8.5 4.1 5.7 0.1 4.4 4.1 4.9 4.2 6.3 6.4 3.6 4.9 0.5 0.9 0.6 0.1 0.3 0.4 0.3
%Ess 4.7 3.9 4.2 3.6 5.4 3.9 2.9 4.9 3.7 4.2 5.2 2.9 15.6 13.3 12.6 14.7 14.3 13.6 13.5 13.6 14.8 14.4 14.3 13.7 16.8 14.9 6.1 4.7 5.3 3.0 4.8 26.7 3.4 3.1 4.4 3.1 3.1 2.4 3.2 3.7 8.3 8.3 9.1 8.1 10.6 9.8 7.8
%Eabs 4.6 4.3 4.1 4.8 5.2 4.6 5.0 4.5 5.3 4.7 4.2 5.3 15.6 15.6 18.4 14.0 12.6 13.1 13.4 14.2 14.4 12.3 15.6 15.5 14.4 12.1 6.4 5.6 4.2 3.1 3.7 19.9 3.5 3.2 3.6 4.1 3.9 3.4 4.1 4.1 27.5 13.3 13.2 13.5 11.9 11.9 13.9
Spring k3 k3 k3 k3 k3 k3 k3 k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M Block Block Block Block Block Block Block Block Block Block Block Block
Controller SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID
Stiffness 1.1905 0.6950 1.4561 1.5138 1.4748 1.3279 1.2869 1.2382 0.9995 1.1236 1.1045 1.4847 0.6386 0.6471 1.3564 0.8864 0.6898 0.6591 0.7648 0.8548 1.0284 0.6343 0.6271 1.1932 1.2675 0.7096 1.2097 0.6409 1.2047 0.7228 0.7147 0.6523 0.6005 1.3080 0.6400 3.1134 0.7122 0.5508 1.0170 0.6435 1.0160 0.8454 0.7992 0.8551 0.7868 0.7630 0.6940
Gp 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.6864 0.7786 0.7306 0.7380 0.5912 0.9180 0.9146 0.6407 0.8223 0.8982 0.9099 0.8692 0.8344 0.7675 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.1092 0.8894 0.9518 0.7717 0.9160 0.7721 0.8380 0.8559 0.8343 0.8606 0.8698 0.8964
Gd 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.0360 0.0420 0.0389 0.0394 0.0299 0.0510 0.0508 0.0331 0.0448 0.0497 0.0505 0.0479 0.0456 0.0413 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.0016 0.0492 0.0532 0.0416 0.0509 0.0416 0.0458 0.0470 0.0456 0.0473 0.0479 0.0496
119
Gi 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.4771 0.5303 0.5042 0.5087 0.4189 0.6183 0.6167 0.4486 0.5603 0.6049 0.6124 0.5877 0.5675 0.5267 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.1054 0.6013 0.6394 0.5294 0.6176 0.5296 0.5699 0.5808 0.5676 0.5836 0.5893 0.6056
%OS 0.8 0.4 0.2 0.1 0.2 3.4 3.4 4.0 4.0 4.1 3.5 5.3 4.6 0.1 9.8 4.6 5.5 6.4 4.7 5.2 7.0 0.2 0.5 0.1 0.2 0.4 2.5 0.4 0.3 1.9 0.2 0.2 0.3 0.4 0.7 1.0 8.7 7.5 7.1 7.9 5.1 7.1 7.6 6.3 6.7 6.5 7.2
%Ess 10.2 9.2 9.1 8.9 8.7 10.4 9.7 3.6 2.2 3.1 2.8 4.1 2.5 26.0 5.0 2.1 2.8 3.4 3.3 3.7 2.6 7.7 7.8 8.5 10.0 9.4 10.4 8.5 7.7 10.1 10.1 9.8 8.0 8.7 10.1 12.3 1.1 2.4 1.4 1.4 0.7 0.4 0.6 0.7 0.6 0.5 0.7
%Eabs 12.3 12.6 12.4 12.8 12.9 18.8 16.8 4.8 3.3 3.4 3.0 3.9 3.5 19.0 4.2 3.4 2.8 3.2 3.0 5.2 3.6 16.6 16.3 14.8 11.4 11.6 15.5 14.3 15.2 14.6 12.2 11.8 14.8 12.9 12.4 11.2 2.8 2.3 3.1 2.5 2.4 2.3 2.1 2.3 2.4 2.3 2.3
Spring Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block
Controller ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID
Stiffness 0.4919 0.4795 10.5845 1.1557 0.8416 0.8994 2.4879 0.6193 1.0514 0.9496 0.6124 1.2598 0.4480 1.1756 1.3417 0.6023
Gp 0.9745 0.9792 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000
Gd 0.0547 0.0550 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800 0.1800
120
Gi 0.6534 0.6538 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050 0.0050
%OS 11.4 11.5 28.9 24.0 26.4 13.9 12.2 7.8 5.5 14.2 0.0 23.2 18.1 14.0 31.3 5.9
%Ess 3.3 7.2 20.4 21.2 23.0 6.9 7.0 7.8 8.4 7.8 25.9 20.1 10.7 9.2 16.8 5.7
%Eabs 7.3 9.4 14.7 13.7 13.1 6.8 7.3 7.1 7.8 7.7 20.3 10.1 13.0 17.9 7.9 7.3
Position Controller Spring k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2
Controller ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID
Stiffness 0.0373 0.0462 0.0505 0.0537 0.0613 0.0561 0.0540 0.0538 0.0573 0.0519 0.0567 0.0642 0.0562 0.0651 0.1454 0.1612 0.1282 0.3807 0.2891 0.4049 0.1615 0.2181 0.2233 0.1418 0.2340 0.2565 0.0895 0.1848 0.3529 0.3894 0.4637 0.4538 0.4302 0.4288 0.4547 0.2882 0.3561 0.4890 0.4051 0.3799 0.3711 0.4076 0.2370 0.3915 0.4285
Gp 0.2418 0.2446 0.2460 0.2470 0.2494 0.2478 0.2471 0.2471 0.2482 0.2464 0.2480 0.2504 0.2478 0.2507 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.3420 0.3524 0.3675 0.3631 0.3598 0.3552 0.3634 0.3217 0.3401 0.3747 0.3544 0.3454 0.3454 0.3548 0.2300 0.2300 0.2300
Gd 0.0316 0.0320 0.0322 0.0323 0.0326 0.0324 0.0323 0.0323 0.0324 0.0322 0.0324 0.0327 0.0324 0.0328 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0451 0.0465 0.0487 0.0481 0.0477 0.0471 0.0481 0.0423 0.0449 0.0497 0.0469 0.0457 0.0456 0.0469 0.0300 0.0300 0.0300
121
Gi 0.1553 0.1565 0.1571 0.1576 0.1587 0.1579 0.1576 0.1576 0.1581 0.1573 0.1580 0.1591 0.1580 0.1592 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.2001 0.2048 0.2117 0.2098 0.2083 0.2062 0.2099 0.1911 0.1994 0.2150 0.2058 0.2018 0.2017 0.2060 0.1500 0.1500 0.1500
%OS 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.0 0.1 0.5 0.4 0.1 0.6 0.5 0.5 0.1 0.1 0.0 0.2 0.0 0.0 0.0 0.0 0.1 0.3 0.0 0.0 0.1 0.0 0.0 0.2 0.1 0.0 0.2 0.1 0.1 0.1 0.5 0.7 0.9 0.6 0.3 0.0 0.0 0.0
%Ess 2.5 2.2 2.3 2.2 2.1 2.3 2.3 2.1 2.2 2.3 2.0 2.1 2.5 1.9 2.7 1.8 2.1 2.1 2.3 2.2 2.1 2.2 2.2 2.3 2.4 2.1 2.2 2.3 2.6 2.3 4.5 2.2 2.3 2.6 1.9 2.5 2.6 4.4 2.6 2.7 2.5 2.6 2.9 3.0 2.6
%Eabs 4.3 4.4 4.3 4.3 4.3 4.3 4.4 4.3 4.2 4.4 4.4 4.1 4.6 4.4 4.6 4.6 4.6 4.6 4.7 4.7 4.6 4.6 4.6 4.8 4.7 4.6 4.6 4.5 4.2 4.1 4.9 4.3 4.2 4.2 4.2 4.2 4.2 4.8 4.4 4.4 4.3 4.3 5.5 5.4 5.3
Spring k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k1M k1M k1M k1M k1M k1M k1M k1M
Controller SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID
Stiffness 0.3128 0.3087 0.3641 0.3193 0.3846 0.3421 0.3989 0.3428 0.3216 0.3578 0.5152 0.4292 0.4274 0.4398 0.4562 0.5102 0.5325 0.5080 0.4988 0.5789 0.5058 0.5343 0.5642 0.4989 0.4849 0.5991 0.4131 0.5345 0.5604 0.7188 0.6390 0.6673 0.6129 0.7570 0.6438 0.6307 0.6299 0.7704 0.6606 0.3171 0.3407 0.3248 0.3195 0.3321 0.3307 0.4305 0.3813
Gp 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.3500 0.3532 0.3584 0.3588 0.3684 0.3786 0.3721 0.3703 0.3925 0.3736 0.3770 0.3834 0.3751 0.3707 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.3168 0.3226 0.3162 0.3140 0.3225 0.3152 0.3372 0.3345
Gd 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0464 0.0468 0.0475 0.0476 0.0489 0.0503 0.0495 0.0492 0.0524 0.0496 0.0503 0.0511 0.0498 0.0492 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0418 0.0426 0.0418 0.0415 0.0426 0.0417 0.0447 0.0443
122
Gi 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.2039 0.2053 0.2077 0.2079 0.2122 0.2168 0.2139 0.2131 0.2232 0.2145 0.2162 0.2191 0.2152 0.2132 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1890 0.1916 0.1887 0.1878 0.1915 0.1883 0.1983 0.1970
%OS 1.3 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.3 0.1 0.2 0.1 0.0 0.2 0.6 0.5 0.3 0.1 0.2 0.2 0.4 0.8 0.4 0.7 0.4 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.0 0.1 0.1 0.0 0.1 0.1 0.1 0.1 0.1 0.1
%Ess 2.1 2.9 2.5 2.4 2.6 2.6 2.8 2.4 2.6 2.6 2.8 2.1 2.3 2.6 2.5 2.3 2.2 2.0 1.8 1.9 2.2 2.4 2.0 2.7 2.1 3.0 3.1 3.1 3.0 3.0 3.1 3.0 3.1 3.0 3.1 3.0 3.0 3.0 3.2 2.4 2.3 2.5 2.7 2.4 2.5 2.1 2.3
%Eabs 5.3 5.5 5.4 5.3 5.4 5.4 5.5 5.3 5.4 5.6 5.4 5.8 5.1 5.1 4.8 4.7 4.8 4.9 4.8 4.8 4.8 4.8 4.8 4.6 4.6 5.8 5.7 5.9 5.9 5.8 5.9 5.9 5.9 5.9 5.9 5.9 5.9 5.9 5.9 4.7 4.8 4.8 4.8 4.9 4.8 5.3 4.6
Spring k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M
Controller ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID
Stiffness 0.4052 0.4788 0.4553 0.3768 0.4290 0.3872 0.5842 0.5867 1.0484 0.7789 1.1842 1.0091 0.8770 1.5446 1.1325 0.7901 0.8237 1.1141 0.8188 0.7730 0.4251 0.3582 0.3841 0.4548 0.4640 0.4622 0.3986 0.4400 0.3899 0.3647 0.3385 0.3884 0.3826 0.3569 0.6918 0.6480 0.7054 0.5138 0.6610 0.7329 0.8107 0.6105 0.9180 0.7059 0.6798 0.6484 0.7145
Gp 0.3402 0.3577 0.3455 0.3253 0.3434 0.3346 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.3515 0.3342 0.3393 0.3464 0.3573 0.3423 0.3387 0.3448 0.3398 0.3255 0.3213 0.3354 0.3210 0.3218 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300
Gd 0.0450 0.0474 0.0458 0.0430 0.0454 0.0443 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0465 0.0442 0.0449 0.0459 0.0474 0.0454 0.0449 0.0458 0.0450 0.0431 0.0425 0.0445 0.0425 0.0426 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300
123
Gi 0.1995 0.2073 0.2019 0.1929 0.2009 0.1971 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.2046 0.1968 0.1991 0.2024 0.2072 0.2005 0.1989 0.2017 0.1993 0.1929 0.1910 0.1974 0.1909 0.1913 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500
%OS 0.1 0.2 0.0 0.4 0.1 0.1 0.1 0.1 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.0 0.1 0.1 0.2 0.0 0.2 0.1 0.2 0.6 0.2 0.3 0.4 0.4 0.4 0.3 1.3 0.8 0.6 0.1 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.1
%Ess 2.3 1.7 1.7 2.3 1.6 2.4 3.3 3.2 3.2 3.1 3.0 3.2 3.0 3.1 3.2 3.0 3.1 3.2 3.0 3.1 2.0 2.6 2.5 1.4 1.9 2.4 2.2 2.2 2.1 2.3 2.7 1.8 2.3 2.2 3.1 3.3 3.4 3.1 3.5 2.8 3.5 3.5 3.3 3.4 3.6 3.3 3.3
%Eabs 4.6 4.3 4.6 4.5 4.7 4.4 5.3 5.3 5.3 5.4 5.2 5.3 5.2 5.4 5.2 5.4 5.3 5.4 5.4 5.2 5.2 5.4 5.5 5.5 5.4 5.7 6.0 5.7 5.8 6.2 6.1 6.0 6.2 6.2 6.0 6.2 6.2 6.1 6.2 6.2 6.2 6.2 6.3 6.2 6.2 6.1 6.0
Spring k2M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open
Controller SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID
Stiffness 0.7207 0.4292 0.5331 0.5371 0.5630 0.5181 0.5570 0.5642 0.3646 0.4524 0.4900 0.3966 0.4922 0.5711 0.5487 0.7120 0.7064 0.7129 0.7098 0.6111 0.9381 0.8975 0.7057 0.7186 0.7181 0.6762 0.6194 0.5739 0.6446 0.0057 0.0006 0.0003 0.0005 0.0004 0.0001 0.0003 0.0003 0.0002 0.0002 0.0000 0.0002 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000
Gp 0.2300 0.3500 0.3639 0.3628 0.3637 0.3704 0.3633 0.3697 0.3320 0.3456 0.3636 0.3367 0.3616 0.3768 0.3660 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2317 0.2301 0.2300 0.2301 0.2300 0.2299 0.2300 0.2300 0.2300 0.2300 0.2299 0.2300 0.2299 0.2299 0.2300 0.2300 0.2300 0.2300
Gd 0.0300 0.0464 0.0485 0.0483 0.0484 0.0492 0.0484 0.0493 0.0439 0.0459 0.0483 0.0446 0.0481 0.0502 0.0487 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0302 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300
124
Gi 0.1500 0.2039 0.2103 0.2098 0.2102 0.2131 0.2101 0.2130 0.1958 0.2020 0.2101 0.1980 0.2092 0.2161 0.2112 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1507 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1499 0.1500 0.1499 0.1499 0.1500 0.1500 0.1500 0.1500
%OS 0.1 0.1 0.1 0.2 0.1 0.2 0.4 0.5 0.7 0.4 0.6 0.5 0.3 1.2 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.9 0.2 0.3 0.2 0.2 0.1 0.3 0.1 0.2 0.2 0.1 0.1 0.6 0.4 0.3 0.2 0.0 0.4
%Ess 3.1 2.1 2.5 2.4 2.5 1.8 2.9 2.7 2.0 2.2 1.4 2.3 1.9 2.4 1.9 3.3 3.4 3.4 3.4 3.4 3.4 3.4 3.3 3.1 3.1 3.4 3.3 3.3 3.2 2.1 1.9 2.1 2.5 2.3 2.4 2.7 2.1 2.5 2.4 2.8 2.4 2.5 2.6 2.5 2.1 2.0 2.2
%Eabs 6.2 5.8 5.8 5.7 5.9 5.8 5.9 5.8 5.8 6.0 6.1 6.1 6.1 6.1 6.0 6.2 6.2 6.2 6.1 6.1 6.2 6.2 6.2 6.2 6.1 6.0 6.0 6.0 6.0 4.7 4.5 4.6 4.5 4.5 4.5 4.7 4.5 4.6 4.5 4.6 4.5 4.6 4.6 4.5 4.5 4.5 4.6
Spring Open Open Open Open Open Open Open Open Open Open Block Block
Controller SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID SMPID
Stiffness 0.0000 0.0002 0.0004 0.0004 0.0009 0.0023 0.0034 0.0091 0.0108 0.0110 2.2801 2.0690
Gp 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.4448 0.2300
Gd 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0604 0.0300
125
Gi 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.2474 0.1500
%OS 0.3 0.0 0.3 0.0 0.2 0.0 0.3 0.2 0.5 0.3 1.2 0.5
%Ess 2.2 2.4 2.3 2.3 2.5 2.3 2.3 2.2 2.5 2.4 5.2 5.2
%Eabs 4.6 4.5 4.6 4.5 4.6 4.6 4.7 4.6 4.7 4.6 5.1 6.0
Force - Position Controller Spring k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1 k1M k1M k1M k1M k1M k1M k1M
Controller ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID
Stiffness 0.3015 0.0848 0.1207 0.1388 0.1259 0.1230 0.1190 0.1228 0.1115 0.0969 0.0877 0.0867 0.0922 0.0849 0.0891 0.1073 0.1042 0.1098 0.1329 0.0282 0.1068 0.0569 0.0641 0.0771 0.0632 0.0613 0.0645 0.0133 0.0537 0.0730 0.0122 0.0616 0.0722 0.0137 0.0751 0.0726 0.0812 0.0850 0.4265 0.4497 0.4370 0.4493 0.3923 0.4384 0.4511
Gp 0.3260 0.2569 0.2684 0.2742 0.2701 0.2692 0.2679 0.2691 0.2655 0.2608 0.2579 0.2576 0.2593 0.2570 0.2583 0.2641 0.2631 0.2649 0.2723 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.3404 0.3338 0.3370 0.3319 0.3263 0.3229 0.3329
Gd 0.0429 0.0336 0.0352 0.0359 0.0354 0.0353 0.0351 0.0352 0.0348 0.0341 0.0337 0.0337 0.0339 0.0336 0.0338 0.0346 0.0345 0.0347 0.0357 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0451 0.0443 0.0447 0.0441 0.0432 0.0429 0.0442
126
Gi 0.1930 0.1620 0.1672 0.1698 0.1679 0.1675 0.1669 0.1675 0.1659 0.1638 0.1625 0.1623 0.1631 0.1621 0.1627 0.1653 0.1648 0.1656 0.1689 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1997 0.1968 0.1981 0.1959 0.1933 0.1919 0.1964
%OS 1.0 1.9 1.3 2.2 0.9 1.4 0.5 2.6 0.9 0.5 0.4 0.8 0.4 0.2 0.3 0.2 0.7 0.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.2
%Ess 2.0 3.3 1.9 3.4 2.1 2.1 2.4 2.8 2.8 2.4 2.0 2.5 2.6 2.1 2.2 2.6 2.8 2.8 2.7 2.9 2.9 2.2 2.9 2.7 2.8 2.5 2.6 2.6 2.8 2.6 3.0 2.7 2.4 2.5 2.7 2.4 2.5 2.5 3.0 3.4 3.5 3.5 3.2 2.9 3.4
%Eabs 11.6 4.5 4.2 4.4 3.9 4.0 4.0 4.1 3.8 3.8 4.1 3.9 4.0 4.1 3.9 3.9 4.0 3.9 3.9 4.7 4.8 4.6 4.8 4.7 4.7 4.7 4.7 12.0 4.8 4.7 12.0 4.7 4.8 11.2 4.8 4.8 4.7 4.7 5.0 5.4 5.5 5.3 5.4 5.4 5.4
Spring k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k1M k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2
Controller ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID
Stiffness 0.4155 0.4642 0.3568 0.4330 0.3997 0.4899 0.4347 0.4553 0.4303 0.4603 0.3788 0.4556 0.4480 0.4868 0.0585 0.4601 0.4964 0.4779 0.6098 0.1784 0.5499 0.3453 0.5337 0.5265 0.4532 0.5372 0.4996 0.6822 0.1381 0.5599 0.6631 0.3026 0.3234 0.3249 0.3443 0.2835 0.2859 0.3475 0.3438 0.3403 0.2475 0.3494 0.3481 0.3315 0.0707 0.2968 0.3061
Gp 0.3288 0.3681 0.3201 0.3307 0.3293 0.3450 0.3300 0.3373 0.3408 0.3328 0.3198 0.3427 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.3262 0.3326 0.3318 0.3390 0.3198 0.3194 0.3390 0.3385 0.3378 0.3081 0.3360 0.3389 0.3327 0.2525 0.3230 0.3268
Gd 0.0436 0.0486 0.0424 0.0438 0.0436 0.0458 0.0438 0.0448 0.0452 0.0442 0.0424 0.0455 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0429 0.0438 0.0437 0.0447 0.0421 0.0421 0.0447 0.0446 0.0445 0.0405 0.0443 0.0448 0.0439 0.0330 0.0425 0.0430
127
Gi 0.1945 0.2119 0.1905 0.1953 0.1947 0.2018 0.1950 0.1983 0.1998 0.1963 0.1904 0.2008 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1931 0.1959 0.1956 0.1988 0.1902 0.1900 0.1988 0.1986 0.1983 0.1850 0.1975 0.1989 0.1961 0.1600 0.1917 0.1933
%OS 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.4 0.4 0.2 0.4 0.4 0.7 0.2 0.3 0.3 0.3 0.7
%Ess 3.6 3.4 3.3 3.2 3.2 3.2 2.9 3.1 2.8 3.4 3.6 2.7 4.0 3.6 3.5 3.8 3.9 3.7 3.7 3.7 3.5 3.6 4.1 3.5 4.3 4.2 4.1 4.1 3.6 4.2 3.7 2.7 2.7 2.8 2.6 2.2 2.4 2.6 2.7 2.1 2.7 3.1 2.5 2.7 2.9 3.0 3.2
%Eabs 5.5 15.4 5.4 5.3 5.4 5.4 5.2 5.0 5.0 5.2 5.1 5.3 6.0 6.2 11.4 6.1 5.9 6.0 6.0 11.7 6.1 13.2 6.0 5.8 12.3 6.1 5.7 6.1 11.2 6.3 6.0 4.1 4.1 4.0 4.0 4.1 4.0 4.0 4.0 4.1 4.2 4.2 4.1 4.3 10.9 4.5 4.5
Spring k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2 k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M
Controller ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID
Stiffness 0.2543 0.2540 0.2523 0.4362 0.4367 0.4305 0.1141 0.4489 0.1202 0.1141 0.4623 0.5360 0.5543 0.6064 0.5230 0.5626 0.5708 0.6065 0.5886 0.2913 0.5890 0.5507 0.4454 0.4438 0.4548 0.4591 0.4592 0.4775 0.4556 0.4875 0.1151 0.4366 0.4218 0.4695 0.4699 0.4161 0.4359 0.4336 0.4922 0.4971 0.4152 0.1642 0.5953 0.6858 0.1086 0.1073 0.0933
Gp 0.3109 0.3109 0.3100 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.3483 0.3513 0.3408 0.3393 0.3490 0.3496 0.3474 0.3511 0.2630 0.3364 0.3394 0.3559 0.3581 0.3354 0.3393 0.3361 0.3541 0.3564 0.3426 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300
Gd 0.0409 0.0409 0.0408 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0462 0.0466 0.0451 0.0450 0.0463 0.0464 0.0461 0.0467 0.0345 0.0446 0.0449 0.0472 0.0475 0.0444 0.0450 0.0446 0.0470 0.0473 0.0453 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300
128
Gi 0.1862 0.1862 0.1858 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.2032 0.2045 0.1998 0.1992 0.2035 0.2038 0.2028 0.2045 0.1648 0.1979 0.1992 0.2066 0.2076 0.1974 0.1992 0.1978 0.2058 0.2069 0.2006 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500
%OS 0.4 1.2 1.2 0.0 0.0 0.1 0.1 0.0 0.0 0.1 0.1 0.0 0.1 0.0 0.1 0.1 0.1 0.1 0.0 0.1 0.1 0.0 0.1 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.0 0.1 0.1 0.2 0.4 0.2 0.4 0.4 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1
%Ess 2.2 2.1 2.9 3.9 3.8 3.9 3.8 3.8 3.8 3.7 3.7 3.9 3.8 3.9 3.7 3.7 3.7 3.7 3.3 3.6 3.5 3.4 3.6 3.2 3.7 3.4 3.0 3.6 3.3 3.4 3.6 3.2 3.1 3.2 3.2 3.4 3.1 3.6 3.4 3.1 3.0 4.1 3.9 3.9 3.9 3.9 4.0
%Eabs 4.2 4.2 4.0 5.3 5.3 5.4 12.2 5.3 11.6 11.8 5.3 5.3 5.3 5.4 5.2 5.3 5.3 5.3 5.3 8.4 5.3 5.4 5.7 5.7 5.7 5.8 5.8 5.4 5.6 5.7 11.9 5.7 5.7 5.7 6.2 5.4 5.6 5.4 5.6 5.6 5.6 12.1 6.2 6.2 11.9 11.8 11.9
Spring k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k2M k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3 k3
Controller SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID
Stiffness 0.5447 0.6616 0.7424 0.6434 0.7052 0.6544 0.7160 0.7579 0.8177 0.6755 0.7441 0.1886 0.8561 0.7012 0.7074 0.6679 0.7064 0.6700 0.1533 0.6567 0.6750 0.6422 0.6022 0.6112 0.2089 0.6901 0.6475 0.6297 0.2093 0.5533 0.6802 0.6097 0.5720 0.1519 0.6340 0.1542 0.6386 0.7191 0.8120 0.8088 0.8610 0.7144 0.7487 0.1677 0.2210 0.8141 0.9097
Gp 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.4211 0.4297 0.4254 0.4220 0.4236 0.2764 0.4125 0.4191 0.4096 0.4030 0.4058 0.2870 0.4268 0.4148 0.4115 0.2900 0.3875 0.4226 0.4071 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300
Gd 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0565 0.0576 0.0568 0.0565 0.0565 0.0364 0.0552 0.0562 0.0547 0.0538 0.0541 0.0380 0.0572 0.0554 0.0549 0.0383 0.0516 0.0567 0.0542 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300
129
Gi 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.2362 0.2400 0.2379 0.2365 0.2371 0.1709 0.2323 0.2353 0.2308 0.2279 0.2291 0.1758 0.2387 0.2332 0.2317 0.1770 0.2209 0.2369 0.2296 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500
%OS 0.1 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.0 0.1 0.0 0.1 0.0 0.0 0.1 0.1 0.0 0.2 0.1 0.0 0.0 0.2 0.3 0.3 0.1 0.2 0.3 0.0 0.1 0.0 0.0 0.1 0.0 0.1 0.1 0.0 0.0 0.0 0.1 0.1 0.1 0.1
%Ess 3.9 3.8 4.1 3.8 3.8 3.7 4.0 4.2 3.8 4.1 4.0 4.3 3.7 2.9 2.7 2.8 3.1 2.8 2.9 2.9 2.7 2.7 2.9 2.8 2.7 2.7 2.6 2.8 2.6 2.5 2.7 2.7 4.1 4.1 4.2 3.9 4.1 4.0 4.1 4.2 4.2 4.1 3.8 4.0 4.0 4.1 4.2
%Eabs 6.1 6.1 6.1 6.2 6.1 6.3 6.3 6.2 6.1 6.1 6.0 11.5 6.1 4.0 4.0 4.1 3.9 3.9 11.3 4.0 4.0 4.2 4.2 4.3 9.8 4.0 4.1 3.9 10.2 4.2 4.3 4.3 5.6 11.7 5.6 11.7 5.5 5.6 5.5 5.6 5.6 5.4 5.5 11.2 11.7 5.7 5.7
Spring k3 k3 k3 k3 k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M k3M Open Open Open Open Open
Controller SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID
Stiffness 0.2285 0.8216 0.7061 0.7914 0.4330 0.4862 0.5799 0.5657 0.5182 0.6452 0.6285 0.5851 0.5421 0.5710 0.5951 0.1778 0.5963 0.5555 0.5848 0.6440 0.5329 0.1243 0.5939 0.2984 0.6883 0.6709 0.9567 0.8809 0.8485 0.8646 0.7500 0.1833 0.6317 0.6141 0.7291 0.1202 0.6442 0.7212 0.7683 0.1786 0.9889 0.8276 0.0282 0.0074 0.0054 0.0021 0.0007
Gp 0.2300 0.2300 0.2300 0.2300 0.3597 0.3751 0.3844 0.3732 0.3681 0.4004 0.3857 0.3860 0.3701 0.3805 0.3901 0.2837 0.3791 0.3755 0.3771 0.3865 0.3743 0.2675 0.3909 0.3500 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2389 0.2323 0.2316 0.2306 0.2301
Gd 0.0300 0.0300 0.0300 0.0300 0.0476 0.0497 0.0511 0.0496 0.0488 0.0534 0.0514 0.0514 0.0492 0.0506 0.0519 0.0373 0.0505 0.0500 0.0501 0.0515 0.0497 0.0351 0.0521 0.0500 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0312 0.0303 0.0302 0.0301 0.0300
130
Gi 0.1500 0.1500 0.1500 0.1500 0.2082 0.2152 0.2194 0.2144 0.2121 0.2267 0.2201 0.2202 0.2131 0.2177 0.2220 0.1741 0.2171 0.2155 0.2161 0.2204 0.2149 0.1668 0.2224 0.3500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1540 0.1510 0.1507 0.1502 0.1500
%OS 0.0 0.1 0.0 0.1 0.1 0.0 0.0 0.1 0.2 0.0 0.0 0.0 0.1 0.0 0.0 0.1 0.1 0.1 0.0 0.2 0.1 0.1 0.0 9.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 2.4 0.2 0.6 0.6 0.7
%Ess 4.1 4.2 4.1 4.0 3.4 3.2 3.2 3.5 3.7 3.2 3.9 3.6 3.8 3.6 3.7 3.5 3.9 3.5 3.2 3.5 3.3 3.5 3.0 15.0 4.1 4.1 3.9 4.2 3.9 4.1 4.0 4.0 4.4 4.2 4.1 4.2 4.2 4.3 4.1 4.3 4.3 4.2 3.2 1.9 2.2 2.2 2.4
%Eabs 11.8 5.7 5.5 5.6 4.9 5.0 5.2 5.2 5.2 5.2 5.2 5.1 5.2 5.1 5.2 10.5 5.2 5.1 5.1 5.1 5.0 10.7 5.2 16.1 6.0 6.2 6.1 6.1 6.1 6.1 5.9 11.8 6.1 6.1 6.1 11.8 6.1 6.1 6.1 11.8 6.0 6.1 4.4 4.3 4.4 4.4 4.4
Spring Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Open Block Block Block Block Block Block Block Block Block Block Block Block Block Block
Controller ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID ASMPID
Stiffness 0.0005 0.0008 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0001 0.0001 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2.8441 2.9239 2.3265 2.6333 1.9970 2.3524 2.4710 2.3955 0.8533 2.5321 2.0234 2.3085 2.2568 2.2308
Gp 0.2300 0.2302 0.2299 0.2299 0.2299 0.2299 0.2299 0.2299 0.2299 0.2299 0.2300 0.2299 0.2299 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.4331 0.4883 0.4753 0.4900 0.4851 0.4699 0.5239 0.4854 0.3018 0.5041 0.4562 0.4770 0.4980 0.4846
Gd 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0585 0.0664 0.0646 0.0668 0.0661 0.0638 0.0716 0.0660 0.0402 0.0686 0.0619 0.0648 0.0677 0.0660
131
Gi 0.1500 0.1500 0.1499 0.1499 0.1499 0.1499 0.1499 0.1499 0.1499 0.1499 0.1500 0.1499 0.1499 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.2419 0.2670 0.2612 0.2679 0.2657 0.2587 0.2832 0.2657 0.1826 0.2742 0.2525 0.2619 0.2714 0.2655
%OS 0.8 0.2 0.4 0.4 0.6 0.3 0.2 0.4 0.4 0.2 0.1 0.1 0.2 0.1 0.0 0.2 0.2 0.2 0.3 0.1 0.1 0.0 0.1 0.0 0.1 0.2 0.2 0.3 0.2 0.3 0.1 0.3 0.1 0.3 0.4 0.7 0.6 0.7 0.4 0.3 0.8 0.6 0.5 0.4 0.7 0.6 0.4
%Ess 2.5 2.1 2.4 2.2 2.1 2.2 2.3 2.2 2.6 2.6 2.0 2.5 2.2 2.5 3.1 2.6 2.4 2.8 2.5 2.6 2.3 3.0 2.8 2.8 2.7 2.8 3.0 2.8 2.4 2.8 2.4 2.6 3.0 7.5 7.7 7.4 7.2 7.8 7.4 7.4 7.7 7.5 7.7 7.6 7.7 7.4 7.4
%Eabs 4.4 4.4 4.4 4.4 4.5 4.4 4.4 4.5 4.4 4.4 4.3 4.4 4.4 4.5 4.6 4.6 4.6 4.6 11.6 4.6 11.3 11.3 4.6 4.6 11.6 11.6 4.6 11.7 4.6 4.5 4.5 11.7 11.6 5.6 5.7 5.6 5.6 5.7 5.8 5.8 5.7 11.7 5.6 5.6 5.6 5.5 5.5
Spring Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block Block
Controller ASMPID ASMPID ASMPID ASMPID ASMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID SMPID
Stiffness 1.6154 2.9039 2.1108 2.3599 2.1571 2.3150 2.5122 3.0067 2.4337 2.6996 2.3892 3.7183 3.0720 1.0044 1.7571 0.9917 2.0367 2.1622 2.6597 2.3849 3.0881 2.3509 3.2761 2.4015
Gp 0.4472 0.5183 0.4925 0.4879 0.4780 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300 0.2300
Gd 0.0606 0.0709 0.0671 0.0665 0.0649 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300 0.0300
132
Gi 0.2483 0.2808 0.2690 0.2669 0.2623 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500 0.1500
%OS 0.4 0.5 1.0 0.6 0.4 0.1 0.1 0.2 0.4 0.1 0.1 0.1 0.2 0.3 0.1 0.1 0.5 0.4 0.4 0.3 0.3 0.4 0.1 0.4
%Ess 7.2 7.2 7.2 7.4 7.2 8.7 8.6 8.1 8.5 8.4 8.3 8.2 8.2 8.1 8.0 7.2 8.4 8.3 8.4 8.4 8.8 8.3 8.1 8.6
%Eabs 5.6 5.4 5.7 5.7 5.5 6.5 6.5 6.4 6.4 7.5 6.5 7.5 7.6 11.0 6.3 5.8 6.6 6.4 6.5 6.6 6.5 6.6 6.5 6.6
APPENDIX E
SIMULINK BLOCK DIAGRAMS
133
Figure 85: Force Controller APID Simulink Code Diagram Figure 84: Force Controller ASMPID Simulink Code Diagram
134
Figure 87: Force Controller PID Simulink Code Diagram Figure 86: Force Controller PID Simulink Code Diagram
135
Figure 88: Force-Position Controller ASMPID Simulink Code Diagram Figure 89: Force-Position Controller ASMPID Simulink Code Diagram
136
Figure 91: Force-Position Controller SMPID Simulink Code Diagram Figure 90: Force-Position Controller SMPID Simulink Code Diagram
137
Figure 93: Position Controller ASMPID Simulink Code Diagram Figure 92: Position Controller ASMPID Simulink Code Diagram
138
Figure 94: Position Controller SMPID Simulink Code Diagram 139
Figure 95: Position Controller SMPID Simulink Code Diagram
Figure 96: Motor input calculation (PID) subsystem
Figure 97: Voltage to position linearization subsystem
140
Figure 98: Gain direct feed subsystem (SMPID only)
141
Figure 99: Gain calculation subsystem (ASMPID only)
142
