S.N Van Den Bink - Modelling and Control of a Robotic Arm Actuated by Nonlinear Artificial Muscels

8/21/2019 S.N Van Den Bink - Modelling and Control of a Robotic Arm Actuated by Nonlinear Artificial Muscels

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Modelling and control of a

robotic arm actuated by

nonlinear artificial musclesS.N. van den Brink

DCT 2007.002

Master of Science Thesis

Committee: Prof. Dr. H. Nijmeijer

Dr. Ir. N. van de Wouw

Dr. Ir. A.G. de Jager

Coaches: Ir. R.M.G. Rijs

Ir. H.C.A. Dirkx

Technische Universiteit EindhovenDepartment of Mechanical EngineeringDynamics and Control Group

Philips Applied TechnologiesMechatronicsHome robotics project

Eindhoven, 25th January 2007


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Abstract

Modern industrial robots are designed according to the principal design rules to con-struct light and stiff. These design rules are not suitable for robots to be used in adomestic environment. Industrial robots are stiff and strong and, for these reasons,dangerous to humans. To allow for robots to be safe in a domestic environment, theyneed to be compliant. So, in case a domestic robot comes in contact with a human,this does not lead to injuries.Several options are available to make robots compliant. One option is to usecompliant actuators, for example a McKibben muscle. This is a pneumatic artificialmuscle with great resemblance to a biological muscle. A McKibben muscle consistsof a 15 cm piece of bicycle tube, surrounded by a nylon braid of20cm. The endingsof the tube and the braid are connected using a fitting. Inflating the muscle yields adecrease in muscle length and an increase in diameter and stiffness.

To perform research on the applicability of McKibben muscles as robotic actuators,a robotic arm is constructed atPhilips Applied Technologies in Eindhoven. It has twodegrees of freedom and is actuated by four McKibben muscles. The research goalsare, firstly, to construct a predictive model of the robotic arm including McKibbenmuscles and, secondly, to construct controllers that guarantee accurate motion controlof the robotic arm, using model knowledge on the robotic arm and the muscles.A predictive model of the McKibben muscle is proposed, which takes into account thechange in muscle shape as well as in stiffness. This model is used in the total roboticarm model, also including the pneumatics and the rigid body dynamics of the arm.Dedicated experiments are performed to identify the parameters of the muscle modeland the robotic arm model. Other experiments are used to validate whether themodels give a correct prediction of the experimental robotic arm behavior. It appearsthat the robotic arm behavior is predicted correctly within a limited pressure range.The main model error is in the description of the muscle stiffness.

The knowledge gained by the experiments and the modelling is used to define a con-trol strategy. By driving the muscles in pairs, the robotic arm is stabilized completelyusing linear feedback controllers. Each muscle pair requires two controllers: one tocontrol the robotic arm position and one to control the pressure in both muscles.The stiffness variation as a function of the muscle length of the McKibben muscles iscompletely compensated for by applying a notch-filter in the position controller. Thecontrollers are implemented on both a one- and two-degree-of-freedom variant of theexperimental system. Closed-loop experiments show the effectiveness of the controlstrategy, where the feasible bandwidth of this controlled system is4Hz.To improve the predictive capacities of the robotic arm model, future research on theforce balance inside a muscle and the subsequent stiffness is required and advised.

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Samenvatting

Moderne industrile robots zijn ontworpen volgens het principe om licht en stijf con-

strueren. Deze ontwerp principes zijn echter niet geschikt voor huiselijke robots.Industrile robots zijn stijf en sterk en zijn daarom gevaarlijk; ze staan om die rededan ook altijd afgeschermd van mensen. Om de toepassing van robots in een woon-omgeving mogelijk te maken, zullen deze compliant (minder stijf) moeten zijn zodatze meegeven als er tegenaan wordt gestoten. Dit om verwondingen te voorkomen.Er zijn verschillende manieren om een robot compliant te maken. En mogelijkheidis om robots aan te drijven met actuatoren die niet stijf zijn, bijvoorbeeld eenMcKibben spier. Dit is een kunstmatige pneumatische spier die qua gedrag erg veellijkt op een biologische spier. Een McKibben spier bestaat uit15cm fiets binnenbandmet daarom heen een nylon netje van20 cm. De uiteinden van de band en het netjezijn aan elkaar bevestigd met een afsluitdop. Door de spier op te blazen wordt dezekorter, dikker en stijver.

Om de toepasbaarheid van McKibben spieren als robot actuatoren te onderzoeken, isbijPhilips Applied Technologiesin Eindhoven een robot arm gebouwd. Deze heeft tweegraden van vrijheid en wordt aangedreven door vier McKibben spieren. De onder-zoeksdoelen zijn, ten eerste, om een voorspellend model van de robot arm inclusiefde McKibben spieren te ontwerpen en, ten tweede, om met kennis van de robot armen van de spieren, regelaars te ontwerpen die gecontroleerde bewegingen van de ro-bot arm garanderen.Een model voor de McKibben spieren is bedacht, dat zowel de vorm verandering alsde verandering in stijfheid beschrijft. Dit model is vervolgens gebruikt in het totalerobot model, inclusief pneumatiek en de dynamica van de robot arm.Met behulp van experimenten is het gedrag van de spieren en van de robot arm inkaart gebracht. Deze experimenten zijn gebruikt om de parameters in het model afte schatten en om te verifiren of de modellen een realistische beschrijving geven

van de werkelijkheid. Uit validatie experimenten blijkt dat het gedrag van de robotarm in een beperkt druk goed voorspeld wordt. De grootste model fout zit in hetvoorspellen van de stijfheid van de McKibben spieren.

De kennis, verworven door de metingen en het model, is gebruikt om een regelstra-tegie te formuleren. Het blijkt dat, als de spieren in paren worden aangestuurd, hetmogelijk is om met behulp van eenvoudig lineaire regelaars de robot arm volledig testabiliseren. Voor elk paar spieren zijn twee regelaars nodig; n voor de positie vande robot arm en n voor de druk in beide spieren. Essentieel in het ontwerp van depositie regelaar is een notch filter, die de variatie in stijfheid van de spieren volle-dig gecompenseerd. Dit is getest op de robot arm voor zowel een configuratie met n

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als voor een configuratie met twee graden van vrijheid. Experimenten aan het gere-

gelde systeem tonen de effectiviteit van de voorgestelde regelstrategie aan, waarbijde maximale bandbreedte4Hz is.Om het model van de robot arm beter te laten voorspellen, is verder onderzoek naarde krachtbalans in de spieren en de daaruit volgende stijfheid nodig en aanbevolen.

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Preface

This report describes the results of my graduation project at the Dynamics andControl group of the Technische Universiteit Eindhoven. The project is carried outunder authority of the Mechatronics department of Philips Applied Technologiesin Eindhoven. During my assignment I received a lot of help and support by manypeople. Herewith I would like to show my gratitude.

In the first place, I would like to thank my coaches Rob Rijs and Bart Dirkx fromPhilips Applied Technologies and Nathan van de Wouw and Henk Nijmeijer fromTechnische Universiteit Eindhoven for their support, input and critical commentsduring my research. I appreciate that very much.In the second place, but invaluable, I would like to thank the people and studentsat the Mechatronics department of Philips Applied Technologies for numerous dis-cussions, pleasure and for showing me the true capabilities of engineering.Finally, I like to show my gratitude towards my parents Marcelle and Wim, for theirgreat support during almost ten years of studying, and to my girlfriend Esther, forher patience and comfort.

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Contents

Abstract i

Samenvatting iii

Preface v

Nomenclature xi

1 Introduction 11.1 Problem statement 41.2 Report outline 6

2 Description of the robotic arm and the muscles 72.1 Setup of the robotic arm 72.2 Sensors and control platform 82.3 Pneumatics and the binary valves 92.4 McKibben muscles 10

2.4.1 Construction 102.4.2 Working principle 10

2.5 Contribution of the thesis 12

3 Physical modelling 153.1 Robotic arm model 153.2 Input signals and the binary valves 183.3 Pneumatics 21

3.3.1 Pressure and flow relations 213.3.2 Splitter 243.3.3 Formulation of the pneumatic model 26

3.4 McKibben muscles 273.4.1 Braid kinematics 273.4.2 Muscle volume 283.4.3 Pressure in the muscle 293.4.4 Nominal muscle length 303.4.5 Force exerted by the muscle 313.4.6 Muscle stiffness and damping 323.4.7 Formulation of the muscle model 33

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3.5 Equations of motion of the robotic arm 34

3.5.1 Robot kinematics 343.5.2 Robot dynamics 36

3.6 Discussion 37

4 Identification and model validation 39

4.1 Muscle identification 39

4.2 Validation of the equilibria 45

4.3 Robotic arm identification 48

4.4 Validation of the linearized model 52

4.5 Time domain model validation 55

4.6 Discussion 56

5 Feedback control of the robotic arm 59

5.1 Single joint control strategy 59

5.2 Single joint control 62

5.2.1 Angular controller design 63

5.2.2 Pressure controller design 63

5.2.3 Performance 67

5.3 Dual joint control 67

5.3.1 Dual joint control strategy 69

5.3.2 MIMO system identification 70

5.3.3 Loop shaping of the controller 71

5.4 Feedback control versus reinforcement learning 75

5.5 Discussion 78

6 Conclusions and recommendations 79

A Specifications of the robotic arm parts 83

A.1 Additional photos of the robotic arm 83

A.2 Binary valves 86

A.3 Festo SDE1 pressure sensor 88

A.4 Novotechnik SP2500 potentiometer 89

A.5 Electrical scheme of the switching board 90

B McKibben muscles properties 91

B.1 Advantages and disadvantages 91

B.2 Literature on McKibben muscles 92

B.2.1 McKibben muscle models 92

B.2.2 Control strategies for systems with PAMs 94

B.3 Example of a McKibben muscle used as actuator 95

B.4 The tube of a McKibben muscle 97

C Pulse width modulation 99

C.1 Formulation of the PWM algorithm 101

C.2 Saturation function 103

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D Considerations on the pneumatic model 105

D.1 Pneumatic test setup 105D.2 Flow modelling 107D.3 Valve modelling 108D.4 Simulation of the pneumatic test setup 109D.5 Two DoF pneumatic model 113

E Considerations on the muscle model 117E.1 The model of Chou Hannaford 117E.2 The model of Petrovic 120E.3 The model for a Shadow McKibben muscle 120E.4 Analytical solution for the stiffness parameters 122

F Derivation of the multibody model 123

F.1 Kinematics of the pneumatic robot arm 126F.2 Mass and inertial properties 128F.3 Spring 129F.4 Muscle forces 131F.5 Two DoF equation of motion 132F.6 One DoF equation of motion 135F.7 Inertial parameters of both bodies 137F.8 Generalized muscle forces 2 to 6 139

G Derivation of the static muscle parameters 141

H Tuning of the controllers 147H.1 Controller part definitions 147

H.2 Frequency response functions of the pressure loop 148H.3 Nichols diagram of the open-loop pressure control 149H.4 Tuning of the 1DoF angle controller for joint 1 150H.5 Dual joint control scheme 153H.6 MIMO system identification 154H.7 Interaction between the joint angles 156H.8 Stability of the dual joint controller 157

I Implementation of the robotic arm model 159

Bibliography 169

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Nomenclature

Variables

A cross-sectional area m2A direction cosine matrix F force NH angular momentum kg m2 s1

J inertia kg m2

L half muscle fibre length mQ generalized force N mRs specific gas constant kg J s

1

T temperature KT time interval sV volume m3

W work J = N m

b damping N s m1

c stiffness correction parameter d diameter me unit vector f frequency Hzf force direction vector g gravitational constant m s2

h muscle length mk stiffness N m1

k PWM sample l length mm mass kgp pressure Pa = N m2

q mass flow kg s1

q generalized coordinate s open valve time sr radius mt time su plant input / control signal Vv binary valve signal Vx model statey model outputz model input / flow control signal V

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power spectrum

input or decoupling matrix output matrix

pneumatic gain flow resistance factor m2

filter parameter braid angle in ChouHannaford model rad muscle damping parameter dynamic viscosity Pa s angle of the robotic arm rad muscle stiffness parameter Pa muscle stiffness parameter

density kg m3

mean density kg m3 damping ratio valve threshold s biased muscle stiffness parameter natural eigen frequency rad s1

angular velocity vector rad s1

C controllerH frequency response function (FRF)HOL open loop FRF KI interaction coefficient M model linearization of overz rad V1

Mp model linearization ofpAB overzp bar V1

P experimentally identified FRF ofoverz rad V1Pp experimentally identified FRF ofpAB overzp bar V1R robotic armS sensitivity

Subscripts

A muscle AB muscle BC muscle CD muscle DA B cumulative muscles A and BC D cumulative muscles C and DS spring

c capacitye errorf muscle fitting and connectorin flow ink samplel low-passm musclen notch

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out flow out

s splittersv sensors and valvesth thresholdw working point0 nominal

Notation

C set of complex numbersR set of real numbers

x equilibrium ofxx perturbation ofx

x estimation ofxx vectorxx columnxX matrixX

Abbreviations

AI artificial intelligenceBW bandwidthCL closed-loopCM center of massDoF degree-of-freedomFRF frequency response function

MIMO multiple input multiple outputODE ordinary differential equationOL open-loopPAM pneumatic artificial musclePID proportional differentiating integrating controllerPWM pulse width modulationRL reinforcement learningSISO single input single output

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1Introduction

This thesis considers the modelling and control of a robotic actuator to be used ina domestic environment. The commonly known robotic actuators are industrialactuators, which are designed for application in industrial robots. In general, in-dustrial robots are unsafe for humans and not practically applicable in a domesticenvironment.

The difference between industrial and domestic robotics is a direct consequence oftheir specifications. A differentiation in robotics based upon the robots applicationis given in[Kara, 2006]and shown in Figure1.1. Industrial robots are designed tocombine speed and accuracy while domestic robots must be able to be mobile,interactive1 and safe. As a result, industrial robots are stiff, heavy (except for the

moving parts) and powerful; domestic robots are small and light but less accurateand less powerful.

One of the issues domestic robot research focusses on, is the actuation. Comparedto industrial actuators, biologic actuators (muscles) have the ideal properties for do-mestic robots: strong, flexible, light and efficient. One type of actuators, approach-ing the biological muscle properties, are pneumatic artificial muscles or PAMs.The best known PAM is the McKibben muscle2, a light and strong actuator withnonlinear characteristics. A disadvantage is that McKibben muscles are difficult tocontrol.

Modern industrial motion control

The design of industrial robots is characterized by lightweight and stiff structures.As a result the eigenmodes are located at high frequencies, which directly affectsthe feasible bandwidth of a robot. The mechanics are optimized for linearity toallow for the application of well-known classical linear control techniques.

In Figure1.2,two examples of modern industrial robots are given. TheABB IRBis less advanced and general applicable, while the ASML Twinscan is extremelyadvanced and dedicated for one specific task.

1 Interactive in thecontext of a domestic robot hastwo meanings. Firstly, domestic robotsare character-ized by a higher level of autonomy in a loosely structured operating environment; secondly, domesticrobots must be able to interact on some level (on/off button up to talking) with people.

2 The first known application of a McKibben muscle is in a orthotic device for a polio patients, devel-oped by J.L. McKibben, see[Nickel, 1963]for details.


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Robotics

Industrial Robotsautomatically controlled

reprogrammable multipurposemanipulator in three or more

directionsrobots for manufacturing

Service Robotssemi or fully autonomous robotto perform services useful to the

well-being of humansall but manufacturing robots

Professional RobotsServicing humans, their

environment and equipmentmaintenance, repair,

inspection, surgery

Domestic Robotshome care, entertainment,disabled assistance, hobby

vacuum cleaners, toys,wheelchair robots

Figure 1.1:A differentiation of robotics, as defined by[Kara, 2006]. Examples of the appli-cation area of each group are indicated in italic font.

With both robots, high accuracies and short settling times can be achieved; strikingare the accelerations and masses involved. One would probably expect this to bepossible due to very complex controllers; in reality straightforward linear feedback

controllers are used. The high performance level is achieved by putting effort intothree areas: robot design, modelling the robot behavior and, in case of the ASMLTwinscan, knowledge of the robot environment.

Modelling the robot dynamics and the internal disturbances enables for a controllerdesign that is optimized for speed and accuracy and, more important, to apply feed-forward control. Feedforward tremendously reduces settling times and trajectorytracking capabilities.A dry and clean environment is a minimal requirement for each robot. Often arobot is placed on (active or passive) suspensions to prevent floor vibrations fromentering the robot. If accurate knowledge of the environment is required, the robotis placed in a cleanroom. In this way, the system behavior is kept constant; forexample dilatation due to changes in temperature, pressure or humidity are pre-

vented.

Robots in a domestic environment

In the last decades, a vast amount of research regarding domestic robotics has beenperformed. This research is induced by the expectation that gradually a shift fromrobotics from an industrial environment to a domestic environment is happening[UNECE, 2004]. The main research focus is on three areas: autonomous behavior,(human) functionality and compliant3 actuators.

3 Compliance has two interpretations. The technical definition means the inverse of stiffness: weak-ness. In common language, the meaning of compliance refers to accommodating behavior.

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Introduction

(a)ABB IRB (b)ASML Twinscan

Figure 1.2:Example of two modern industrial robots. The ABB IRB [ABB, 2006]is a sixaxes industrial robot, which can be used for a variety of tasks (painting, assembly, mate-rial handling, palletizing, etc). Its weight is approximately1000 kg and its maximum

payload is 60 kg; the maximum allowable acceleration is 1.5g. The IRB does not havespecial requirements concerning the environment it is used in.The ASML Twinscan[ASML, 2006]is a lithographic system for semiconductor produc-tion. The mass of the reticle stage is20kg, it is accelerated up to 10g and positioned withan accuracy of3 nm with 30 ms settling time. The Twinscan can only be used within acleanroom.

(a)QRIO (b)Asimo (c)Partner robot

Figure 1.3: Three modern humanoids. The QRIO[Sabe, 2005; Fujita, 2005] is a hu-manoid entertainment robot; it remembers people, learns, shows emotions and is capableof having simple conversations[Tanaka, 2004]. QRIO has 38DoF, is 0.58 m high andweighs 6.5 kg. The Asimo[Honda, 2003]has been introduced in 2000, the height is 1.20m, it weighs 52 kg, has 26DoF and is capable of walking, running and climbing stairs. ThePartner Robot from Toyota, is 1.20 m high and weighs 35 kg, [Toyota, 2004]. It has soft lipswhich enable for playing trumpet. Also a rolling variant of the partner robot exists.

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First, autonomous in the given context means that a robot is capable of perform-

ing one or several tasks independently; an example is autonomous vacuum clean-ers. Autonomy can also be considered in a wider perspective, by taking into ac-count the ability to interact with humans and learning capabilities; this is a vividresearch area[Bekey, 2005]. An example of autonomous humanoid is the QRIO asgiven in Figure1.3a.Secondly, the functionality research focusses on possible domestic robot layouts.Well known are humanoids, robots with the overall appearance and behavior basedon that of the human body. The reason to adopt human features for a robot layoutis simple: our world is designed for humans. To have a robot functioning well inour world, it is desirable to resemble us.In Figure1.3, three humanoid examples are given. These humanoids are builtusing stiff materials and electric actuators, i.e. according to industrial constructionprinciples. This makes them less suitable for the domestic environment that they

are built for. For safety reasons, it is better not to have industry-like robots in ahuman environment. Imagine a person crossing the path of an industrial robot.Since this robot is unable to detect the person and act accordingly, the robot willmove on, conflicting serious injury on the person.To avoid injuries, different design rules apply for robots in a domestic environ-ment: lots of sensors for detecting unsafe situations, less rigid structures with lowmoving masses and compliant actuators. This brings us to the third domestic robotresearch area: compliant actuators.

In[Bax, 2003], several alternative compliant actuators are considered. One of theactuators rated with the best suitability for application in a domestic environmentis a PAM. An extensive overview of several PAM types, is given in [Daerden, 2002].

Two well-known muscle types are the pleated pneumatic muscle [Daerden, 1999]and the McKibben muscle. In[Tondu, 2000]a small historic overview of the Mc-Kibben muscles is given. McKibben muscles are the actuators of the robotic arminvestigated in this report. In AppendixB.1,the advantages and disadvantages ofMcKibben muscles are discussed.

1.1 Problem statement

The problem statement of this thesis project focusses on the modelling and controlof a robotic system, actuated by four pneumatic artificial muscles. To investigatethis, a two-degree-of-freedom (2DoF) robotic arm is built at Philips Applied Tech-nologies in Eindhoven; an impression is given in Figure 1.4. The robotic arm isdeliberately not designed according to the principal design rules to construct lightand stiff[Koster, 2000]; the dynamics of the joints are coupled and the movingmasses are relatively large compared to the stiffness. This is done to resemble apossible layout of a domestic robot. The robotic arm consists of two revolute jointswhich are connected in series. Each joint is driven by a pair of McKibben musclesin an antagonist setup.McKibben muscles are known for their nonlinear behavior. Physical modelling ofthis behavior is considered to be difficult. Because of such complex dynamics, thecontrol of robots actuated by McKibben muscles, has focussed on various complexcontrol strategies. Classical feedback control has never been applied, simply be-cause a linear approach seemed inadequate to handle the nonlinear characteristics

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Introduction

of McKibben muscles. This thesis will focus on the classical loop-shaping approach

to design a feedback controller. The problem statement of this thesis is formulatedas follows:

A predictive model for the two DoF robotic arm should be built, based on physi-cal principles and dedicated experimental identification, where special attentionshould be given to the modelling of the artificial muscles. This model should bevalidated by experiments and used to construct linear feedback controllers, us-ing a classical feedback approach, allowing for both stabilization and trackingtasks. Finally, these controllers should be implemented and evaluated on theexperimental setup.

The modelling part of the thesis is started by performing dedicated tests in orderto understand the behavior of the system and the McKibben muscles. Based upon

these experiments and the resulting increased insight, a model based on physicalrelations must be formulated. An experimental model must be gained by mea-suring frequency response functions. Based upon the models and measurementresults, a control strategy is chosen and controllers are designed.The control part of the thesis starts with a robotic arm with lower complexity. Byfixing the rotation in revolute joint 1 , the robotic arm is a DoF system, actuatedby muscles A and B; see Figure1.4a. If theDoF robotic arm is successfully con-trolled, it is expanded with the second joint as shown in Figure1.4b.

In [Maas, 2005], the results of a complex control strategy, called reinforcementlearning (RL), are presented. This is an artificial intelligence technique, which at-tempts to stabilize and control theDoF robotic arm by predicting the robotic armbehavior using an experimental model, generated by reinforcement learning. Partof the current thesis is to compare the (black box) RL control approach with the

muscle A

muscle B

body 2

joint 2

(a)

C

D

body 1A

B

joint 1

spring

(b)

Figure 1.4:Impression of the DoF and 2DoF robotic arm.

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(model based) feedback control approach. To be able to make a fair comparison,

improving the robotic arm is not in the focus of the assignment.

1.2 Report outline

In Chapter2, the robotic arm and the pneumatic and electronic components aredescribed. As McKibben muscles are not commonly known actuators, its maincharacteristics and working principles are explained in detail in the second chapter.The physical model for both theDoF as for the 2DoF robotic arm is presented inChapter3. All aspects determining the behavior of the robotic arm are modeledindividually. To understand the physical principles of every model part, dedicatedexperiments are performed. These experiments are also used to verify the modelparts and to identify the model parameters.

The model has multiple purposes. Firstly, it allows for estimating the possiblebandwidth and accuracy of the controlled robotic arm on a model level and it canbe used as part of the control loop (for example in a feedforward or in an input output linearization control strategy). Secondly, the influence of every parame-ter on the behavior can be estimated. In other words, the model can be used forpredicting the robots behavior, which can support domestic robot design using ar-tificial muscles as actuators.

The model is validated in both the time- and frequency-domain in Chapter4. Thetransient behavior and equilibrium points of the experimental robotic arm and themodel are compared in the time domain, the linear input output behavior iscompared in the frequency domain. The input output behavior is determined bymeasuring the frequency response functions of the robotic arm and by simulationof the model.In Chapter5, the control approach and results of both the DoF and 2DoF con-trolled system are presented. The frequency response functions as measured inChapter4, are used to determine the lay-out of the controllers. By taking the sec-ond DoF into account, the system changes from SISO to MIMO. The complexityincreases due to the interaction of the multiple inputs on the outputs. The scalingfrom aDoF to a 2DoF system, the MIMO system identification, the control ap-proach and the control results are discussed. Finally, the model-based controller iscompared with the black-box AI controller.The conclusions and recommendations of this thesis are presented in Chapter6.Propositions are made for future work considering the motion control of robots ina domestic environment.

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2Description of the robotic

arm and the muscles

2.1 Setup of the robotic arm

The robotic arm consists of two revolute joints. Both joints are driven by two Mc-Kibben muscles which work in an antagonist setup. Each muscle is driven by twosolenoid binary valves, so there are eight valves needed to drive the two joints; inFigure2.1a, an image of the robotic arm in an outer position is given, in Figure2.1ba schematic impression of the pneumatic robot arm in its initial position is given.The location and designation of the four muscles, an additional spring, eight valves,four pressure sensors and the two joints are indicated.

In biological systems, the muscles act relatively close to the rotation point. In caseof the robotic arm, the McKibben muscles act also relatively close to the rotationpoints. This makes it difficult to generate a large momentum. On the other hand,small elongations of the muscle are sufficient to generate large rotations of thejoints.

Body 1 rotates in revolute joint 1, driven by muscles C and D . Its rotation is givenby the angle between body 1 and the mounting table surface. Body 1 consists oftwo bars: an aluminium and a steel part (blue respectively grey in Figure 2.1b). Onbody 1 , muscles A and B , four valves and two pressure sensors are mounted.

To support body 1, a spring is mounted between the fixed world (green part in

Figure 2.1b) and body 1. This spring functions as a gravity compensator for themass imbalance of body 1. Without the spring, muscle D is limited in its workingarea as a relatively large force is required to keep body 1 in position.

Body 2 is mounted at the end of body 1 , where revolute joint 2 is located. Therotation of body 2is given by the angle of body 2with respect to body 1. Body2consists of an aluminium bar with a small additional mass on top. Due to thevertical setup of body 2and its limited mass, static balance can easily be enforcedby muscles A and B ; no gravity compensation is required here.

In the following section, the functionality of the control platform of the robotic armis explained. The setup of the pneumatic system is given in Section2.3 and theproperties and behavior of McKibben muscles are explained in Section2.4. This


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A

B

D

joint 1

joint 2

power source

air supply

switching board

valves

valves

pressuresensors

spring

table

(a)

C

D

A

B

body 1

body 2

joint 1

joint 2spring

(b)

Figure 2.1: The pneumatic robot arm as available at Philips Applied Technologies. Thefour muscles are indicated by A, B, C and D. In Figure (a), muscles A and Dare com-pletely inflated while muscles muscles B andC are completely stretched. In Figure (b) theequilibrium position is given, in case the pressure in the muscles is equal in each pair, i.e.pA =pB andpC =pD .

chapter is ended by a review on the consulted literature. More information, pho-tographs and specifications of the robotic arm and its components, can be foundin AppendixA.

2.2 Sensors and control platform

For the control and data acquisition of the robotic arm, a dSPACE system isused, [dSPACE, 2000]. The controllers are defined in MATLAB SIMULINK,[Mathworks, 2005] and uploaded to the dSPACEsystem.A switching board is built to enable the communication between the dSPACE sys-tem and the robotic arm. Moreover, the power supply to the sensors and the valvesis embedded on the switching board.

The eight valves applied in the robotic arm are controlled by eight binary signalsfrom the dSPACE output. On the switching board, the dSPACE output signals areamplified to24V using field emitting transistors or FETs.Four pressure sensors are used to measure the pressure in the four muscles. Theangle of both joints is measured using two linear high-quality potentiometers. Onlyfour analog inputs are available on the dSPACE system, which is less then the sixavailable analog sensor signals. Two jumpers on the switching board enable toselect which sensor signals are linked to the d SPACEinputs.The specifications of the pressure sensors and the potentiometers are respectivelygiven in AppendixA.3 and A.4. The electronic scheme of the switching board isgiven in AppendixA.5.

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Description of the robotic arm and the muscles

2.3 Pneumatics and the binary valves

The pneumatic system, required for driving the McKibben muscles, consists of anumber of hoses, a pressure reduction valve, a capacity of10 liter and the valves.To control the pressure in a muscle, two valves are required: one valve controls theflow towards the muscle, the other valve controls the flow from the muscle into theopen. As the valves are placed in series, the flow-in valve is overruled if the flow-outvalve is enabled. In conclusion, the flow towards a muscle has three possible states:flow in, flow out and no flow.In Figure2.2,a schematic overview of the complete pneumatic system is shown.As can be seen, eight 3/2-valves are used; i.e. each valve has three connectors andtwo possible states. In AppendixA.2, the specifications of the valves are given;most important is the response time: 2 ms.The eight valves are driven by four flow-in signals vin and four flow-out signals

vout. The eight binary control signals are given in matrix notation by:

v=

vin,A vin,B vin,C vin,Dvout,A vout,B vout,C vout,D

T. (2.1)

The capacity prevents large pressure fluctuations in the air supply due to air lossesin the robot pneumatics. The pressure reduction valve ensures the pressure in thecapacity to be maximally1bar. The original supply pressure is approximately6bar.The air hose leaving the capacity is split into four hoses, one hose towards eachmuscle.

air in

air out

stop

reduction valve

v = 1v = 0

3/2 valve:

capacity

muscle A

sensor A

muscle B

sensor B

muscle C

sensor C

muscle D

sensor D

vin,A vout,A

vin,B vout,B

vin,C vout,C

vin,D vout,D

Figure 2.2: Schematic representation of the complete pneumatic system. The eight valvesare depicted for low control signals, i.e. v = 0. The two valves controlling one muscle are

placed in series. As a result, the flow-in valve of muscle iis overruled ifvout,i= 1.

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Figure 2.3:A deflated McKibben muscle from the Shadow Robot Company as used in thepneumatic robot arm. Visible are the braid en the two fittings with the air hose connectors.

2.4 McKibben muscles

A deflated McKibben muscle as used in the robotic arm is shown in Figure2.3;itis produced by the Shadow Robot Company[Shadow, 2006]. Besides this muscletype, several other types are available. The first commercially available McKibbenmuscle was the Rubbertuator from Bridgestone, shown in Figure2.6a. It is wellknown for its application in a wall climbing robot[Pack, 1996]. A large number ofpublications refer to Rubbertuators, unfortunately they are out of production so thepresented results can not be verified.Festo produces the pneumatic muscleMAS[Festo, 2006]. It has a different working

area than the Shadow muscle, consequently it behaves differently. The pressurerequired is between2 and 6 bar and the stroke is smaller; an image is shown inFigure2.6b. Finally, Merlin Robotics [Merlin, 2006] produces a McKibben musclewhich is almost similar to the Shadow muscle.

2.4.1 Construction

A McKibben muscle consists of four parts, a rubber tube surrounded by a braid(see Figure2.4) and two fittings. The fittings serve as gas closures, enable for themounting and contain the air hose connectors. The braid consists of nylon threadswoven into a framework of pantographs. The nylon threads have a fixed lengthand are considered to be stiff compared to rubber tube, this means that the nylon

fibres do not change in length when the muscle is pressurized or a when externalforce is applied to the muscle. As the threads are woven symmetrically, the threadorientation does not change, i.e. the fittings do not rotate with respect to each otherwhen the muscle is being pressurized or extended.

2.4.2 Working principle

In Figure2.5,the form of a McKibben muscle is shown as a function of the pres-sure. At0.0bar1, the McKibben muscle is deflated. The rubber tube and the nylonbraid do not touch; the muscle length is determined by the rubber tube. Inflating

1 Pressure relative to the environmental pressure

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Figure 2.4: Structure of a McKibben muscle: a isobutylene (rubber) tube is surroundedby a braid. The weaving of the threads is axially symmetric and the separate threads arenot connected to each other. The tube and the braid are fixed together at the ends by two

fittings (not shown). Image from[Shadow, 2006].

the muscle, results in an increase in muscle length, while the tube diameter re-mains constant; this is typical behavior for cylindrical balloons, see[Mller, 2004].The maximum muscle length is reached when the tube touches the braid com-pletely at the lowest possible pressure, this is at approximately 0.3bar. If braid andtube touch, the muscle becomes stiff and is able to deliver a force. Further inflationof the muscle results in a decrease in length and an increase in muscle diameter.This is a typical behavior for all PAM types. The shortening of the muscle occursdue to the structure of the braid. Pressurizing the muscle forces the tube to in-crease in volume. To increase the volume, either the diameter, the length or bothmust increase. The braids weaving structure forces the diameter to increase. The

0.0 0.3 0.7 1.0

200

185

170

rubbertube

nylonbraid

fitting

trendline

p[bar]

h

0

[mm]

Figure 2.5: The shape and length h0 of an unloaded muscle as function of the relativepressurep. Inflating the muscle causes an increase, followed by a decrease in muscle length.

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(a) (b) (c) (d)

Figure 2.6: In Figures (a) and (b), respectively, a Rubbertuator and a M AS [Festo, 2006]are shown. Figures (c) and (d) show a Shadow muscle at respectively0.3and1.0 bar.Clearly visible is the increase in diameter and decrease in length at1.0bar. Compared tothe other muscle types, the Shadow muscle is less cylindrically shaped.

braid also determines that an increasing diameter results in a shortening of the

pneumatic muscle. The combination of an increasing diameter and a decreasinglength, shows that the volume of a muscle hardly changes during inflation. Thisalso means that the additional air due to inflation, mainly results in an increase inpressure. The change in shape for a Shadow muscle is shown in Figures2.6c and2.6d at respectively0.3and1.0bar.The maximum allowable pressure in a Shadow McKibben muscle is 3.5 bar. Incase of the robotic arm, it is chosen to resemble biologic muscle behavior; as thepressure determines the stiffness of the muscles, it is decided to use a workingpressure of1.0bar.The working area of a muscle starts as the tube touches the braid; from this pointon the muscle becomes stiff. Normally, the working area starts at 0.3 bar (see Figure2.5). Applying a pretension to the muscle (by stretching) lowers the pressure at

which the working area starts; an additional advantage is that the energy dissipationdue to friction in the braid is lowered. In the robotic arm, the muscles are mountedwith pretension and, consequently, the working area starts at approximately 0.2 bar.A larger working area of the muscles is useful as this allows for the control of thejoint angle of the robotic arm over its entire range.

2.5 Contribution of the thesis

An overview of the consulted literature concerning McKibben muscles, is given inAppendixB.2. This overview is divided in two parts: the modelling of McKibbenmuscle behavior and the control of systems actuated by McKibben muscles.

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An example of a McKibben muscle used as robot actuator is given in AppendixB.3.

In this example, the three types of muscle behavior as discussed in the literaturereview are illustrated:

The active behavior comprehends the change in muscle length, as a functionof the muscle pressure. This type of behavior is particular important in thepressure range below 1.5 bar. Existing models do not consider the activebehavior explicitly; as a result, their predictive capacities below 1.5 bar ispoor.

The passive behavior describes the stiffness and damping of the muscle.In most cases, the stiffness is derived using the ChouHannaford model,[Chou, 1994]. No explicit relations are found on the damping of McKibbenmuscles and in most some cases, viscous damping is assumed.

The volumetric behavior describes the change in muscle volume. In exist-ing models, the volumetric behavior is based on cylindrical shaped muscles.When considering the Shadow muscles (which are applied in the roboticarm), a barrel shaped muscle is a more likely description.

Existing McKibben muscle models do not explicitly make a distinction betweenthese three types of behavior. In this work, an alternative model describing the be-havior of McKibben muscles will be proposed. Explicit relations for the three typesof behavior will be derived. The purpose of this model is to gain insight in theworking principles of a McKibben muscle. The new model must also describe thebehavior below1.5 bar and take into account a different muscle shape. The pro-posed model will be identified using dedicated experiments and will be subjectedto a thorough validation procedure.

Most references on control strategies for robots actuated by PAMs vary from fuzzy-logic, gain scheduling, backstepping control, sliding mode control, learning controlto feedforward control. These control strategies are rather exotic, compared tolinear feedback control. In order to apply PAMs on a larger scale, a simple buteffective control strategy is required. In[Pack, 1994;Schrder, 2003], it is proventhat a linear control strategy is possible, although their performance is low.No references are found on a control strategy for robots with PAMs based on loop-shaping. This is probably due to the nonlinear behavior of the muscles: a lin-ear control approach may not seem to represent a good option as stability can notbe guaranteed in a non-local sense. Also the required experimental modelling inthe frequency domain might be problematic due to nonlinear behavior. How, weconjecture that a loop-shaping approach toward designing high-performance con-trollers for robots with PAMs is very well possible.

The contribution of this thesis is threefold. Firstly, a model is proposed which givesa better understanding of the observed McKibben muscle behavior. Secondly, it isproven that good closed-loop performance is possible by using linear controllers;and thirdly, a decoupling is proposed which maximizes the application range of theMcKibben muscles.

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3Physical modelling

In this chapter, the modelling of the robotic arm is discussed. The robotic armmodel is divided into four parts: the input conversion, the pneumatic system withthe valves, the McKibben muscles and the robot dynamics. The purpose of themodel is twofold: firstly, understanding the behavior of the McKibben muscles andof the robotic arm; secondly, to support the control design. In Section3.1,the com-plete robotic model is presented; in Sections3.2to3.5, the four model parts areelaborated on. This chapter is ended by a discussion.

3.1 Robotic arm model

To gain insight in the robotic arm behavior, it is sufficient to consider the DoFrobotic arm (with joint 1 fixed). For control purposes, a model describing the 2DoFrobotic arm is required. In this chapter, a model describing the behavior of theDoF robotic arm is presented, but the scaling to a 2DoF model is kept in mind.

The structure of the robotic arm model is visualized in Figure 3.1; the four modelparts are indicated by the circles. The arrows refer to the physical input and outputquantities of the respective model parts. The physical quantities are given by fourinput signals:

u= uA uB uC uD

T, (3.1)

inputconversion

pneumaticsand valves

robotdynamics

muscle Amodel

B C

D

u v

p

q F

h

Figure 3.1: Structure of the robotic arm model. The model consists of four parts. Asthere are four muscles present in the 2DoF robotic arm, the part describing the musclebehavior is embedded four times; in case of theDoF robotic arm model, the muscle modelis implemented two times.


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four binary valve-in signals and four binary valve-out signals:

v= vin,A vin,B vin,C vin,D

vout,A vout,B vout,C vout,D

T, (3.2)

four flows in and four flows out of the muscles:

q=

qs,A qs,B qs,C qs,Dqout,A qout,B qout,C qout,D

T, (3.3)

four pressures in the four muscles:

p=

pA pB pC pDT

, (3.4)

four muscles forces:

F = FA FB FC FD T , (3.5)four muscle lengths:

h=

hA hB hC hDT

, (3.6)

and two angles of the robotic arm:

=

1 2T

. (3.7)

The inputs to the 2DoF robotic arm are the four control signals inu. The outputsare given by the two angles in and the four pressuresp in the muscles. The stateof the robotic arm model is completely described by nine states:

x = [x1 x2 x3 x4 x5 x6 x7 x8 x9]T

= [pc pA pB 2 2 pC pD 1 1]T , (3.8)

with 1and 2the angular velocities; the pressure in the capacityx1= pcis definedin the pneumatic model part.

If revolute joint 1 of the robotic arm is rigidly fixed, only revolute joint 2remains;this yields theDoF robotic arm. In theDoF robotic arm, only muscles A and Band joint 2have a function. When considering Figure3.1,this means all quantitiesindexed by C and Dand angle 1 become redundant; i.e. the states of the DoFrobotic arm are given by:

x = [x1 x2 x3 x4 x5]T

= [pc pA pB 2 2]T .(3.9)

The dynamic model of theDoF robotic arm model is given by the following first-order differential equation:

x =

RsT V1c

qin qs(u)

RsT

qs,A (u) qout,A (u)

x2 hA Vh + Vd2 d2h A

V1

ARsT

qs,B (u) qout,B (u)

x3 hB Vh + Vd2 d2h B

V1

B

x5

QA+QB +m2g l12sin(x4)

Jzz2+m2l

212

1

. (3.10)

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Physical modelling

In the first relation of(3.10), Vc, T andRs define the volume of the capacity, the

absolute temperature and the specific gas constant of air respectively. In (3.10), weindicate the dependency of the variables on the input u. Let us now also explicatetheir dependencies on the states.The two flows are defined by qin =qin(x1)and qs =qs(x1, x2, x3, u)respectively.The flows in the second relation are given by qs,A = qs,A (x1, x2, x3, uA , uB ) andqout,A =qout,A (x2, uA ) and the flows in the third relation are given byqs,B =qs,B (x1, x2, x3, uA , uB ) and qout,B = qout,B (x3, uB ). All flow relations aredefined by the pneumatic model as derived in Section 3.3.In the second and third relation,d2= d2(x4)represent diameter of the subsequentmuscle and VA = VA (x4)and VB = VB (x4)indicate the muscle volumes. Theserelations are defined in the muscle model as derived in Section3.4.The lengths ofmuscles A and Bare given by respectivelyhA =hA (x4)andhB =hB (x4)and theseare defined by the kinematics of the robotic arm as derived in Section 3.5.

In the fifth relation,QA = QA (x2, x4)and QB = QB (x3, x4)represent the gener-alized muscle forces depending on the pressure and the length of the subsequentmuscle, these are derived in Section3.5. The mass of body 2 is given bym2 andJzz2indicates the inertia of body 2with respect to its center of mass; l12is the dis-tance of the center of mass of body 2with respect to joint 2.

The first part of the robotic arm model (Figure3.1), is the input conversion whichconverts the input signals u into binary signalsv . The binary valves as applied inthe robotic arm have two possible states, i.e. open or closed. As a result, the valvesbehave highly nonlinear. By applying a pulse width modulation (PWM) algorithmto the valves (which is possible due to their fast switching time), the valves approxi-mate linear continuous behavior. For this reason, PWM is added to the robotic arm(and to the model); this is explained in Section3.2.

In Figure3.2,a new muscle model is proposed that describes the behavior of onemusclei {A B C D }. In this figure, also the interactions with the pneumaticsand the robot dynamics are indicated. The relation defining the pressurepiis con-sidered to be part of the muscle model. From the muscle pressure pi, the stiffnesski, dampingdiand nominal lengthh0,iare derived.The muscle pressuresp are also an input to the pneumatic model; together withthe valve signals v. The pneumatic model is the second part of the robotic armmodel; it considers the flowsqtowards and out of the muscles and the pressure inthe capacitypc. The pressure model is discussed in Section3.3.The third part of the robotic arm model, considers the muscle model as presentedin Figure3.2. The internal variables of muscle i {A B C D }, are given by thevolumeVi, the nominal lengthh0,iand stiffnesskiand dampingdiof the muscle.In the volumetric behavior, the muscles volume is defined; the passive behaviordefines the muscles stiffness and damping. The active behavior gives the nominallength of the muscle. The nominal length is the length the muscle takes for a givenpressure, if no force is applied to the muscle. Remark that the nominal length ingeneral does not equal the actual muscle length hi.The state of a muscle is defined by its pressure pi and actual length hi. The mus-cles pressure dynamics is given in the second and third relation of (3.10), using theflowqi and the muscles volume Vi; in Figure3.2,this is indicated by the pressureresponse. The actual muscle length hi is not defined by the muscle model, but isenforced on the muscle by the dynamics of the robotic arm. The muscle model isderived in Section3.4.

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mus

clemod

el

pneumaticsand valves

pressureresponse

activebehavior

passivebehavior

robotdynamics

volumetricbehavior

forceresponse

robotkinematics

ki di

qi

pi

pi

pi

h0,i Fi

hi

hi

hi

Vi

Figure 3.2:The structure of the muscle model and its interaction with the pneumatics andthe robot dynamics, see also Figure3.1. For clarity, the conversion from the joint angles tothe muscle lengths using the robot kinematics is indicated; remark that the robot kinematicsare also incorporated in the robot dynamics.

The fourth part of the robotic arm model, considers the dynamics of the roboticarm; these are defined by the equations of motion (EOM). The inputs to the EOMare the muscle forces F, which are defined in the force response of the musclemodel, as indicated in Figure3.2. The outputs of the EOM are the revolute jointangles . Using the kinematic relations, the actual muscle lengthsh are derivedfrom these angles. The EOM of the robotic arm are given in the fourth and fifthrelation of (3.10); these are derived in Section3.5.

The robotic arm is presented in this section. The individual parts as indicated inFigures3.1and3.2are derived in Sections3.2to3.5. This chapter is completed by

a brief discussion on the model setup and on the adopted numerical method forsimulation of the model, see Section3.6.

3.2 Input signals and the binary valves

The binary valves used in the robotic arm are fast-switching valves. In AppendixA.2,it is shown that the response time of the valves is less then 3 ms. This is veryquick compared to other time constants in the robotic arm. For this reason thedynamic behavior of the valves is not taken into account, i.e. the change in valveflow resistance due to a change in valve state is considered to be instantaneous.

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Physical modelling

The eight binary valves are used to control the four muscle flows. Each valve has

two states, either open (v = 1) or closed (v = 0). The pair of valves controlling theflow to one muscle, have only three different states: flow in, flow out and no flow.The state of the valves is determined by the valve control signal v(t), see(2.1). Thestate of the four valve pairs is determined by the flow control signal:

u(t) =

uA (t) uB (t) uC (t) uD (t)T

. (3.11)

The flow control signals are continuous and can be positive and negative (u R4).A positive value means flow into the muscle while a negative value means flow outof the muscle.The discrepancy between the continuous signal u(t)and the binary signalv(t)isfixed by applying a pulse width modulation (PWM) algorithm. PWM yields thebinary valves to approximate linear continuous behavior (like servo valves) while in

reality the binary valves behave highly nonlinear. The PWM algorithm makes useof the fast switching time of the binary valves. Besides, the PWM algorithm is usedto convert a single flow signal uiinto two valve signalsvin,i and vout,i:

uiPWM vin,i vout,i for i A B C D . (3.12)

The PWM algorithm yields an output with equal power compared to the input.This is realized by defining a PWM frequency fPWM . Every PWM sample, a pulse isgenerated of which the duration depends on the sampled value ofu(t).First, the sampling timetk and the threshold vth(a saturated

1 sawtooth function)are defined:

k(t) =

t fPWM

(3.13a)

tk(t) = k(t)

fPWM(3.13b)

uth(t) = t fPWM k(t) (3.13c)vth(t) = sat{,1}(uth(t)) . (3.13d)

The saturation, as introduced in(3.13d), functions as a dead-zone which takes thelimited valve switching time into account. It is of no use to open the valve for atime smaller then its switching time, as this will not result in a flow. Besides, thissaturation prevents nervous behavior in an equilibrium point of the controlled

robotic arm.

Next, the sampled time tk is used to apply a zero-order-hold to u(t). By applyingtwo different saturation functions,u(tk(t))is divided into a flow in and a flow outsignal:

uk,in(t) = sat{0,1}(u(tk(t))) (3.14a)

uk,out(t) = sat{1,0}(u(tk(t))) . (3.14b)

1 The saturation function y = sat{a,b}(x)is defined as x being saturated by the boundary values aandb. Ifx [a, b]the saturation function yields y = x, ifx > b, then y = band ifx < a, theny= a. The implementation ofsat{a,b}(x)and some examples are given in AppendixC.2.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

0.5

1

t[s]vin,i

(t)

vth(t)ui(t)

ui(tk)

Figure 3.3:Example of the PWM algorithm withfPWM = 20Hz and= 0.05s.

The valve signals v are derived by comparing both sampled signals uk,in(t) anduk,out(t)to the thresholdvth(t):

vin(t) = uk,in(t) > vth(t) (3.15a)

vout(t) = uk,out(t) vth(t),this yieldsvin,i(t) = 1. The values used in this example are fictitious. The com-

plete derivation and implementation of the PWM algorithm is given in Appendix C.

Two errors are introduced by the PWM. Firstly, the input signal u(t)is sampled bya zero-order-hold at timestk. This error is linear with fPWM . Secondly, the sampledinput signal is saturated by a value . Both errors are corrected for in the controlloop.

Also the flow signal u(t) is saturated between1 and 1, however this is not anerror introduced by the PWM algorithm. Values ofu(t)smaller than 1or largerthan1will result in an opened valve for the complete duration of the PWM sample.As such, this saturation is part of the pneumatic system and the saturation occursif the valve signals are oversteering.

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Physical modelling

3.3 Pneumatics

The general setup of the pneumatic system is explained in Section 2.3; in this sec-tion a model describing the pneumatic behavior is discussed. To explain certainchoices made in the modelling process, the pneumatics of theDoF robotic arm istreated first. Next, this model is extended to the 2DoF robotic arm.No temperature dependency is taken into account in the model, the temperature isconsidered to be constant and at20 C= 293K. The pressure system is modelledas a series of first-order systems. No mass effects of the gas flow (like shockwaves)are taken into account as these process are too quick to play an important role inthe behavior of the robotic arm.The complete pneumatic model is based on relative pressures; as a reference theatmospheric pressure patm is taken. This is a rather common approach as mostpressure sensors measure relative pressures only. This choice does not affect the

model as only pressure differences are important.The gas medium used in the robotic arm is pressurized air; which is considered tobe an ideal gas. The equation of state describing an ideal gas is known as Boyle-GayLussacs law:

p V =RsT m , (3.16)

withpthe pressure,Vthe volume,Rsthe specific gas constant of (in this case) air,Tthe absolute temperature andmthe mass of air in the volume.Air is a compressible gas and, therefore, the density depends largely on the abso-lute pressurepabs. This is shown by rewriting (3.16)into:

=

m

V =

pabsRsT . (3.17)

3.3.1 Pressure and flow relations

In Figure3.4, a graphical interpretation of the DoF pneumatic model is given,the model lay-out differs from the actual layout, which is given in Figure2.2. Fourpressures and six flows can be distinguished. The out valves are placed on the otherside of the muscle, this is done to simplify the implementation of the valves in themodel.Another difference between the actual pneumatics and the model is that the air-in valves are not shown. In AppendixD, it is shown that modelling the valves as

small volumes gives numerical problems in the resulting model. The same holdsfor the splitter; it can not be modeled as a small volume with multiple flows to it.This is solved by combining the valve signals vin,i and the splitter into one relationdefining the splitter pressureps. The latter is elaborated on in Section3.3.2.

Pressures

In Figure3.5, a volume with an in-going flow qin and out-going flow qout is de-picted. The pressure is related to the flow by the time derivative of (3.16):

pdV

dt +V

dp

dt = RsT

dm

dt . (3.18)

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pin qin qs

splitter

qs,A

qs,B

pc, Vc

pA , VA

pB , VB

vout,A

vout,B

qout,A

qout,B

ps

vin,A vin,B

Figure 3.4: Schematic interpretation of the pneumatic model for the 1DoF model. Theflow directions are indicated by the arrows. The black solid lines represent hoses and thedotted lines represent binary signals. The following parts can be distinguished: two musclesystems given in yellow and red, the external inputs given in blue, the splitter is orange andthe capacity is grey.

p , V

qin qout

Figure 3.5:Pressure model.

q

p1 p2

Figure 3.6:Flow model.

lh

2rhq v

Figure 3.7:Laminar Poiseuille flow.

2rtq

Figure 3.8:Throttle valve flow.

air supply capacity hose 1 valve hose 2 muscle

qinpin q1 q2

pc pm

p1 p2

Figure 3.9:Schematic representation of the pneumatic test setup.

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Physical modelling

The variation of mass in the volume depends on the air flowing in and out:

dmdt

= qin qout. (3.19)

Substituting(3.19) in (3.18) and rearranging terms gives a first-order differentialequation for the pressure:

p =

RsT(qin qout) pdV

dt

V1. (3.20)

From this relation the pressures in the muscles and the capacity will be derived.

The capacity has a constant volume ofVc = 10 l; this means the dVdt

term in relation(3.20) is equal to zero. In Figure3.4,the flows towards the capacity are indicated.The pressure in the capacity is now given by:

pc = RsT V1c

qin qs

. (3.21)

The pressure in the splitterpsis defined in a different manner as will be explainedin Section3.3.2and the pressures in the muscles are part of the muscle model.

Flows

The flow through a hose depends on the pressure difference between the two sidesof the hose, as shown in Figure3.6.Two possible relations are considered in mod-elling the flow; namely, a laminar Poiseuille flow (Figure 3.7) and the flow througha throttle valve (Figure3.8).In order to determine the type of flow present in the robotic arm, a pneumatic test

setup has been built. The pneumatic test setup is schematically depicted in Figure3.9. It is used to determine the types of flow through the hoses. Modelling thepneumatic test setup and comparing the simulations with the measurements hasshown that each flow through a hose can be modelled by the flow through a throttlevalve. This is discussed in more detain in AppendixD.

The flow through a throttle valve according to[Polytech, 1997] is given by:

q = sign (p1 p2)

|p1 p2| . (3.22)The throttle parameter depends only on the throttle radius rt as indicated inFigure3.8:

= 0.59 2r2t . (3.23)

The densityof the air depends on the absolute pressure in the hose as defined by(3.17). As the pressure in the hose is not exactly known, it is estimated by the meanof the two pressures at each side of the hose:

= p1+p2+ 2patm

2RsT . (3.24)

When considering the robotic arm, the flows being defined by throttle valves canbe understood. For flowqinthe throttle parameter in is defined by the reductionvalve. Forqsand qoutthe throttle parameterssand outare defined by the binaryvalves. The flowsqs,A andqs,Bare defined in Section3.3.2.

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It should be remarked that the flows do not depend on the length of the hoses as

is the case in a Poiseuille flow. In the current system setup, the flow behavior isdominated by the diameter of the valves only. In case the hoses would become verylarge, the Poiseuille flow behavior would become dominant.

Using Figure3.4and(3.22), the flows can be defined by:

qin = insign (pin pc)

|pin pc| (3.25)

qs = ssign (pin pc)

|pc ps| . (3.26)

The flows out of the muscles depend on one pressure only: the muscle pressure.Additionally, the flow out of the muscle depends on the status of the out valves.

This is taken into account by:

qout,A = vout,Aout

pA (3.27)

qout,B = vout,Bout

pB . (3.28)

3.3.2 Splitter

In theDoF robotic arm, the hose leaving the capacity is split into two hoses, oneto each muscle, see Figure3.4.As a result, the flow to each muscle depends largelyon the flows towards the other muscle. The most logical way to model this behaviorwould be to assume one flow to enter and four flows to leave a small volume. The

splitting of the flow would be given by a pressure relation as given by (3.20).In AppendixD, it is shown that modelling such small volumes leads to stiff differ-ential equations resulting in numerical problems. Instead of using small volumes(modelled by first-order dynamics), it is decided to combine the binary valves andthe splitter into a zeroth-order (algebraic) relation. This means that the pressure inthe splitter changes instantaneously upon a change in the valve state.

When both in-valves are closed, the pressure in the spitter is equal to the pressurein the capacity. When one of the two valves is opened, the pressure in the splitteris equal the pressure in the opened muscle. If both valves are opened, the pressurein the splitter is equal to the lowest of the muscle pressures (as qsis determined bythe largest pressure difference). This behavior is shown in Table3.1;it can be fitted

into a mathematical relation by usingvin,A andvin,B , which are defined in(3.15):

ps=

1 vin,A vin,A pc pB

pA min(pA , pB )

1 vin,B

vin,B

. (3.29)

To determine the flowqs,itowards each muscle, a proportionality factor which ful-fills conservation of mass is required:

qs =i

qs,i. (3.30)

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Physical modelling

ps

vin,A

0 1

vin,B

0 pc pA

1 pB min(pA , pB )

Table 3.1:The pressurepsin the splitterdepends on the air-in signals.

qs,A

qs

vin,A

0 1

vin,B

0 0 1

1 0 (pcpA)(pcpA)+(pcpB)

Table 3.2: The flow from the splitter tomuscle Adepends on the air-in signals.

The proportionality is explained forqs,A using Table3.2. In case ofvin,A = 0, no

flow towards muscleA

is possible. In casevin,A = 1whilevin,B = 0, the flowqsisdetermined by the pressure in muscle Aaccording to (3.29)and the flow towardsmuscle Ais given byqs,A =qs.

In case ofvin,A = vin,B = 1, the muscle with the lowest pressure determines qsaccording to (3.26)and (3.29). The flowqsis divided intoqs,A andqs,Bproportionalto the pressurespAandpB . The complete relation defining the flow towards muscleAis then given by:

qs,A = vin,A (pc pA )

(pc pA ) +vin,B (pc pB ) qs. (3.31)

To prevent numerical problems, in the denominator only(pc pB )is multiplied byvin,A . Consequently, in casevin,A =vin,B = 0the denominator will not equal zero.It is assumed thatpAwill never be equal topcwhich is a reasonable assumption aspcincreases when both valves are closed.

The same reasoning as above applies for muscle B; i.e. the flow qs,B towards muscleBis given by:

qs,B = vin,B (pc pB )

vin,A (pc pA ) + (pc pB ) qs. (3.32)

par value par value par value

rt,in 0.40mm in 4.194 107 m2 Rs 287J kg1 K1rt,s 0.52mm s 7.088 107 m2 T 293Krt,out 0.72mm out 1.359 106 m2 Vc 0.01m3

Table 3.3:The parameters used in the pneumatic model.

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3.3.3 Formulation of the pneumatic model

The formulation of the completeDoF pneumatic model is given by combining(3.21) (3.32) into:

pc = RsT V1c

qin qs

(3.33a)

ps =

1 vin,A vin,A pc pB

pA min(pA , pB )

1 vin,B

vin,B

(3.33b)

qin = insign (pin pc)

|pin pc| (3.33c)

qs = ssign (pc ps)

|pc ps| (3.33d)

qs,A = vin,A (pc pA )qs(pc pA ) +vin,B (pc pB ) (3.33e)

qs,B = vin,B (pc pB )qs

vin,A (pc pA ) + (pc pB ) (3.33f)

qout,A = vout,Aout

pA (3.33g)

qout,B = vout,Bout

pB (3.33h)

= p1+p2+ 2patm

2RsT . (3.33i)

It might appear that the pressures pA and pBare forgotten in the above relations.However, as these are part of the muscle model, they are stated in the muscle modelin Section3.4.7.The parameters used in the pneumatic model are defined in Table3.3.In AppendixD.5,a pneumatic model is proposed for the 2DoF robotic arm. Thismodel is similar to the model as given in (3.33), where two adjustments are re-quired:

the splitter is defined in two steps to take the four muscles into account and this requires the flows towards the muscles qs,i to be defined in a slightly

different fashion.

Remarks

In case vin,A = vin,B = 1, it is possible that air flows from the muscle with thehighest pressure to the muscle with the lowest pressure. This behavior is takeninto account in(3.31) and (3.32), however problems may occur if the pressure inone or both muscles becomes higher than the pressure in the capacity. An increasein muscle pressure is possible while all valves are closed, due to a changed in lengthof the muscle as will be explained in Section3.4.2.The relation in (3.29) yields the wrong result for ps in one very specific case: ifvin,A = vin,B = 1and one or both muscle pressures are higher than the pressurein the capacity. This is caused by themin(pA , pB )term in(3.29), it assumes pc

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(a) (b) (c) (d)

kr kb

kgh h

d1

L

d2

cc= nd2

Figure 3.10:The derivation of the muscle kinematics. The parameters of the Shadow Mc-Kibben muscles are given byL = 127.4mm, n = 1.25andd1 = 25mm.

The threads of the braid makes up a framework of parallelograms. Sincethe number of parallelograms in the braid is constant, the braids behavioris uniform with the behavior of one single parallelogram, this is shown inFigure3.10c.

The muscles kinematics can be seen as a bar mechanism consisting of four bars,forming a parallelogram. The length of each bar equals half the nylon thread lengthL. The kinematic relations of this bar mechanism are given by:

c = 4L2 h2 (3.35)d2 =

c

n =

1

n

4L2 h2 (3.36)

h =

4L2 (nd2)2 . (3.37)

A muscle is completely stretched if its shape is cylindrical, i.e. ifd2 = d1 as alsoshown in Figure2.5. As a result, the maximum muscle lengthhmax is defined bythe diameter d1. This yields hmax 235 mm, which is in accordance with themuscles maximum length.It should be noted that the kinematic relations do not directly depend on the pres-sure in the muscle, i.e. they can be used for both the stretched and the unloaded

case.

3.4.2 Muscle volume

In Section2.5,it is stated that the volume of a Shadow McKibben muscle is bestdescribed by a barrel-shaped volume. Two barrel formulations are considered, anelliptical and a parabolic shape [Weisstein, 2006]; the elliptical shape is consideredto describe the muscle shape best.In Figure 3.11,the three parameters describing an elliptical barrel are given. Param-eterd1is the fitting diameter, the variable muscle length hequals the barrel lengthand the barrel diameter is equal to d2. Parameter d2 is a function of the muscle

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Physical modelling

h

d1d2

Figure 3.11:Three barrel parameters.

lengthh(t), as given by(3.36). The volume of an elliptic barrel is given by:

V =V (h(t) , d2(h(t))) (3.38)

V = h

60

3 d21+ 4 d1d2(h) + 8 d

22(h)

. (3.39)

The actual muscle length h(t) is enforced on the muscle by the robot dynamics. Re-lations(3.39) and(3.36)define the volumetric behavior block in the muscle modelof Figure3.2.

3.4.3 Pressure in the muscle

The pressure in a muscle depends, among other things, on its volume. In Section

3.3, it is shown that the pressure in the muscle satisfies the following differentialequation:

pi =

RsT

qs,i qout,i

pi dVidt

V1i for i {A B C D } . (3.40)

Considering the dependency of the barrels volume in(3.38), the pressure in a mus-cle as defined by (3.40)is rewritten into:

pi =

RsT(qs,i qout,i) pi

V

h +

V

d2

d2h

i

dhidt

V1i , (3.41)

fori {A B C D }, with V

h =

60

3 d21+ 4 d1d2+ 8 d

22

(3.42)

V

d2=

h

15

d1+ 4 d2

(3.43)

d2h

= h n

4L2 h2 . (3.44)

The inputs to (3.41)are the flowsqs,iandqout,idefined in Section3.3and the mus-cle volume as defined by the volumetric behavior. Relation (3.41) defines thepres-sure responseblock in the muscle model in Figure3.2.

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3.4.4 Nominal muscle length

The active behavior of the muscle model defines a relation for the nominal lengthof the muscle as a function of the pressure p. In Figure3.2,this is indicated bytheactive behaviorblock. To construct such a model, a fourth thought experimentis performed. The nominal length is considered to represent an equilibrium inthe face of three forces working in the muscle. These forces are indicated by threesprings in Figure3.10d:

The air in the muscle acts like a gas spring. The gas spring stiffness kgdepends on the pressure, an increase in pressure yields an increase inkg(p).As the air pushes from the inside on the rubber tube and braid, it tends toelongate the muscle.

The nylon threads keep the tube from expanding; as a result it opposes elon-gation of the muscle. Inflating the muscle sharpens the curvature of themuscle (at the endings yielding the orientation of the nylon threads to changewhat is opposed by the braid; i.e. the braid stiffness kb(p)increases with themuscle pressure.

The material behavior of the rubber tube is viscoelastic; for the sake of sim-plicity elastic behavior with stiffness kr is assumed, which is considered tobe independent of the pressure. The rubber tube opposes elongation.

To model the nominal muscle behavior, it is assumed that the braid and the rubbertube always touch. This assumption enables for the following boundary condition.In case the muscle pressurep= 0bar, the nominal diameterd20= 0mm, yieldinga nominal muscle lengthh0equal to the braid length2L; i.e.h0(p= 0) = 2L.A relation determining the nominal length equilibrium is proposed, which fulfils

the boundary condition and the three phenomena as described above:

h0 = 2L kg(p)kr+kb(p)

. (3.45)

Extensive research to expressions for kg(p), kb(p) and kr has unfortunately notresulted in a balanced analytical solution for (3.45). It is decided to adopt(3.45)and assumingkg(p)and kb(p)to be linear with the muscle pressure and kr to beconstant. The resulting relation is normalized to two parameters: s1 [m] and s2[Pa]. The nominal length for musclei {A B C D } is now defined by:

h0,i = 2L s1pis2+pi

. (3.46)

The nominal behavior is experimentally identified by measuring the length anddiameter of five different unloaded muscles. Each muscle is mounted on one sideand unconstrained on the other side; the length and the diameter are measured forpressures between 0.0and 1.0bar. In Figure3.12, the results of this experimentare given in three plots h0(p),d20(p)andd20(h0).Based on heuristics, parameterss1and s2of(3.46)are fitted using the experimentshown in the left part of Figure3.12; this yields s1 = 0.117 m and s2 = 0.232bar. The model is also plotted in Figure3.12. The model-based nominal diameterd20is derived by substituting (3.46)into (3.36). Despite the variations between theseveral muscles and the model assumptions, the nominal muscle length gives anaccurate fit on the experiments. Using the braid kinematics also gives an accuratefit for the nominal diameter.

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Physical modelling

0 0.2 0.4 0.6 0.8 1 1.2140

150

160

170

180

190

200

210

0 0.2 0.4 0.6 0.8 1 1.238

40

42

44

46

48

50

52

150 160 170 180 190 200 21038

40

42

44

46

48

50

52

muscle 1muscle 2muscle 3muscle 4muscle 5model

p[bar]p[bar] h0[mm]

h0

[mm]

d20

[mm]

d20

[mm]

Figure 3.12:Experimental result for the nominal length h0 and nominal diameterd20 asa function of the pressure p, using five muscles. These measurements are compared to the

modelled nominal length as given in (3.46), withs1 = 0.117m ands2 = 0.232bar. Themodel based nominal diameterd20 = d2(h0)is derived using(3.36)withn = 1.25andL= 0.1274m. The measurements in the left figure show that the rubber tube touches thebraid at approximate0.3bar. Also some hysteresis in h0 can be seen; each measurementhas two lines, one for increasing pressure (upper line) and one for decreasing pressure.

In Figure2.5, it can be seen that the assumption, that the braid and the rubber tubealways touch, is only true forp > 0.3bar. As a result, the muscle model is onlydefined for muscle pressures which exceed0.3bar. It is noted that no clear physicalinterpretation can be given to s1and s2. Relation(3.46)defines theactive behaviorblock in the muscle model of Figure3.2.

3.4.5 Force exerted by the muscle

From the robotic arm point of view, a muscle behaves like a spring damper sys-tem. In the dynamics of the robotic arm, this spring damper behavior is consid-ered to generate a forceFi, which implemented for each musclei {A B C D }.The force response of muscleicombines the active and the passive muscle behav-ior and is given by:

Fi = k(pi, hi)

hi(t) h0(pi)

+d(pi)hi(t), (3.47)

withk(pi, hi)the stiffness andd(pi)the damping.

The stiffness part of the muscle force is given byk(pi, hi)and the elongation termhi(t)h0(pi). The elongation is defined by the difference between the actual lengthand the nominal length of the muscle. Herewith, the active muscle behavior (re-lated to the pressure as described in previous section) enters the muscle force rela-tion.

The damping part of the muscle force is given by d(pi)and the time derivative ofthe muscle length byhi(t). The nominal length or its time derivative plays no directrole in the damping force as only the actual motion determines the damping. Thisis easily understood; imagine a sudden change in pressure. This causes a changein nominal length and stiffness, resulting in a muscle force. The force resultsin a motion of the load and a change in muscle length, this motion is subject todissipation. Relation (3.47) defines theforce responseblock in the muscle model.

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3.4.6 Muscle stiffness and damping

The stiffnessk is a function of the pressure pi and the actual muscle lengthhi. InAppendix E.3,a model describing the stiffness of a Shadow McKibben muscle isderived as:

k(pi, hi) =

+pi

0+1hi+2h2i

. (3.48)

The muscle stiffness behavior is defined by the pressure and the square of theactual muscle length. To compensate for the fact that the stiffness is only definedforp >0.3bar, a biason the pressure is introduced.In AppendicesE.1 andE.2, the two stiffness models as defined in[Chou, 1996]and [Petrovic, 2002a] are given. These two models are used in the derivation of(3.48). Two reason are given to redefine the muscle stiffness model:

The new model takes the muscle volume into account by the barrel descrip-tion and the braid kinematics; existing models assume a cylindrical volumewith fixed diameter.

The new model is adjusted for pressures below1.5bar by introducing a bias, the existing models are only predictive for pressures exceeding 1.5bar.

No further research on modelling the dampingdusing other models has been per-formed in this work. It is assumed that the damping in the muscle is determinedby the damping in the rubber tube and by the friction in the nylon braid. The latterincreases as the pressure increases. For this reason, a static damping is assumedwith an additional pressure-dependent damping. This is given by:

d(pi) = 0+1pi. (3.49)

The minor interest in the muscle damping is motivated by the dynamic muscle testas discussed in Section4.1and by the work [Klute, 2002]. The order of magnitudeof the stiffness and damping is:

O(k) = 104O(d) = 101 . (3.50)

The force generated by the muscle has a stiffness and a damping part:F =Fk+Fd.The contribution ratio of the stiffness and damping depends on the frequency= of the angle = sin(t)of the muscle elongation and is given by:

Fd()

Fk=

d cos(t)

k sin(t) d

k . (3.51)

Remind that in this simple analysis, a linear damping and stiffness are assumed. Inorder for the damping to play a significant role, Fd/Fk >0.1. Sincek 1000 dthedamping is only significant if > 20Hz. As the bandwidth of the robotic arm isnot expected to exceed10Hz, the modelling of the damping is not given extensiveattention. Resuming, the contribution of the muscle damping to the behavior ofthe robotic arm is minimal.Still, some damping must be modeled to ensure energy dissipation in the roboticarm model. The easiest way would be assuming a constant viscous damping ofthe muscles; however, a pressure dependant viscous damping as given in (3.49) isused. Relations(3.48) and(3.49)define the passive behavior block in the musclemodel of Figure3.2.

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Physical modelling

3.4.7 Formulation of the muscle model

Combining relations in this section, gives the muscle model for i {A B C D }:d2,i =

1

n

4L2 h2i (3.52a)

Vi = hi

60

3 d21+ 4 d1d2,i+ 8 d2,i

2

(3.52b)

pi =

RsT

qs,i qout,i

pi hi Vh

+ V

d2

d2h

i

V1i

(3.52c)V

h

i

=

60

3 d21+ 4 d1d2,i+ 8 d2,i

2

(3.52d)

Vd2

i

= hi

15

d1+ 4 d2,i

(3.52e)

d2h

i

= hi n

4L2 h2i(3.52f)

h0(pi) = 2L s1pis2+pi

(3.52g)

k(pi, hi) =

+pi

0+1hi+2h2i

(3.52h)

d(pi) = 0+1pi (3.52i)

Fi = k(pi, hi) hi h0(pi)+d(pi)hi. (3.52j)The inputs to this model are the two flowsqs,iand qout,i(resulting from the pneu-matic model as derived in Section3.3)and the actual muscle lengthhi(determinedby the robot dynamics which will be derived in Section 3.5). The outputs of the mus-cle model are the pressure pi and the forceFi. In Chapter4, the passive muscleparameters are identified using an experimental setup.

Remarks

In Figure3.12,two phenomena that are not taken into account in the muscle modelcan be seen. Firstly, the five muscles used in the experiment show a large variabilityin their nominal behavior, approxi

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S.N Van Den Bink - Modelling and Control of a Robotic Arm Actuated by Nonlinear Artificial Muscels