bond graph based design of prosthesis for partially impaired hands

8/12/2019 Bond Graph Based Design of Prosthesis for Partially Impaired Hands

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Bond Graph Based Design of Prosthesis for PartiallyImpaired Hands

Anand Vaz

Department of Mechanical EngineeringSLIET, Longowal, District Sangrur

Punjab 148106, [email protected]

Shinichi Hirai

Department of RoboticsRitsumeikan University, Noji-higashi 1-1-1

Kusatsu, Shiga 525-8577, [email protected]

Abstract—In this work, concepts for the actuation of a prosthesis

for a partially impaired hand are developed systematically. The

partial impairment implies that the hand has lost one or more

fingers but retains the ability of its remaining natural fingers. It

is shown that the existing natural joints can be used for the

actuation of prosthetic finger joints and enable performance oftasks that would not have been possible otherwise. This is a

challenging task as motion has to be transmitted from the

remaining natural joints to the prosthetic joints. The joint axes

move with respect to each other during performance of tasks and

do not have any fixed relative orientation. String-tube actuation

mechanisms, developed earlier by the authors, are presented and

analyzed. Bond graphs are used for modeling and simulation of

the system dynamics of the proposed prosthetic mechanisms.

While the ideas for the proposed prosthesis are of immense help

for rehabilitation of the impaired, the approach used for

modeling will also be of interest to researchers in the areas of

robotics, system dynamics and control.

Keywords-bond graphs; hand prosthesis; modeling; simulation

I. I NTRODUCTION

The objective of this work is to present an alternativemethod to the modeling of a class of prosthesis for the humanhand. The prostheses have been proposed earlier by the authors[1], and are based on actuation of prosthetic fingers by theremaining natural fingers of a partially impaired hand.

Development of prosthesis for the human hand has been ofmuch interest in recent times. This activity is of significancedue to the role of prosthesis in rehabilitation of the affected

person [2]. The process of development involves a detailedstudy of the human hand and its amazing capabilities. The

fingers, their bones, joints, skin, palm, muscles, nerves,working under the guidance and control of the Central NervousSystem (CNS) make the hand extremely versatile [3], [4].Researchers in robotics have shown much interest and madesystematic contributions to the understanding of the humanhand. Systematic classifications of the hand postures, useful

both to the robotics and medical community are now available[3], [5]. This has also led to the development of robotic handswith the ability to grasp and manipulate objects in a dexterous manner [6], [5], [4], [7].

Modeling is an important aspect in the development of prosthetic mechanisms. Mathematical models help inunderstanding the behavior of the prosthesis during the processof design and performance evaluation. The mechanism andcontrols for the prosthesis can be designed effectively based onavailability of good models. It is of interest and importance tostudy, analyze and simulate the behavior, actuation principlesand working of such devices while they perform general tasks.The method of Bond graphs is an attractive and powerfultechnique for the modeling of such prosthetic systems. It offersa unified framework for the modeling of mechanisms, theactuation and control systems modeling due to its capability ofhandling multi-energy domains.

In the prosthesis proposed earlier by the authors, theactuation of joints on prosthetic fingers is carried out bycorresponding joints on natural remaining fingers [1], [8], [9].While performing tasks with the prosthesis, the axes of active natural joints have motion relative to corresponding passive

prosthetic joints which they actuate. The joint axes need not befixed or parallel with respect to each while working with the

prosthesis. This requires transmission mechanisms that areflexible yet positive displacement type, quite a challengingtask. The method of bond graphs has been applied to themodeling of string-tube based joint actuation mechanisms. It

provides a clear perspective into the system dynamics of such prosthetic devices and their actuation.

Organization of this paper is as follows. A brief backgroundfor the development of the prosthesis is laid out in section II.Principles for proposed prosthesis design are discussed insection III. The principles are realized using the string-tube

based mechanisms and modeled in section IV using bond

graphs. Derivation of system equations from the bond graphmodels is presented in the same section. Vector bond graphmodules for rigid body dynamics are developed and applied toa finger with two revolute joints as an example. A discussionon simulation results is also presented. Section V offers adiscussion on the salient features of the proposed approach. A

brief preview to the development of active prosthesis using the presented approach is initiated in section VI.



II. BACKGROUND

A scheme to identify joints on the hand is discussed below.Consider the skeletal link structure for the first three digits(fingers) of the hand, during the task of handwriting, as shownin Fig. 1 .

Fig. 1 Skeletal link structure of the first three fingers during hand writing.

Each digit starting from the thumb has been assigned analphabet.

TABLE I. IDENTIFICATION SCHEME FOR DIGITS

(Digit) Finger Alphabet

(1) Thumb T

(2) Index I

(3) Middle M

(4) Ring R

(5) Little L

The joints corresponding to each digit are further numberedstarting from the respective carpometacarpal joint at the wrist.

In the process of analyzing natural finger movementsduring prehension, one can observe that certain joints ondifferent fingers have corresponding movement relationships.For example, the movement relationship between the proximalinterphalangeal joint on the index finger (I3) and theinterphalangeal joint (T3) on the thumb is very pronouncedwhile writing. Another example is the correspondence betweenthe metacarpophalangeal joint at the index finger (I2) and thecarpometacarpal joint (T1) where the thumb joins the wrist,while grasping. Several correspondences between similar jointson different fingers can be observed on the same hand while

performing different tasks. We can take advantage of thesecorrespondences by utilizing the actuation capability of anatural joint on one finger to augment the movement of a jointon a prosthetic finger. Such actuation of prosthetic joints usingnatural joints on fingers does not increase the independentdegrees of freedom (DOF) of the severed hand as the motion of

prosthetic joints is dependent on corresponding natural joints.However, the use of the existing degrees of freedom, to provideactuation capability to prosthetic joints enables the

performance of tasks which would not have been possibleotherwise.

A. Essential characteristics of prosthesis.

For the impairment considered, a prosthetic device intendedto provide joint actuation capabilities for fingers should havethe following essential characteristics.

(1). It should provide for actuation of a prosthetic jointusing the abilities of an existing natural joint. The axes offinger joints move while performing a task. The axes are notfixed and also need not be parallel as the kinematic structurecontinuously changes. Further, torque has to be transmittedfrom the natural joints to the prosthetic joints, without loss ofmotion. In such challenging situations, the mechanism foractuation of the prosthetic joints has to be accommodating andflexible. Flexibility does not mean softness in this case, but acapability to change the configuration of the actuationmechanism continuously.

(2). It should be wearable, for example, like a glove on theimpaired hand. This would facilitate quick preparedness forany task. Once the task is done it can be removed for

relaxation.

(3). It should have an aesthetic appearance.

(4). The prosthesis should be affordable to the handicapped person. It should not be a financial burden to the person whohas already suffered a severe loss. This implies that themechanism should be as simple as possible, since simplicityand cost are almost directly related.

Realization of these essential requirements using the string-tube based joint actuation scheme is discussed in the nextsection.

T1

T2

T3

I1

I2

I3

I4

M3

M4

M2

M1

III. PRINCIPLES OF ACTUATION FOR PROPOSED PROSTHESIS

A. String-tube based joint actuation [1]

The human finger joints are composed of rigid skeletallinks (bones- phalanges and metacarpals) which are roughlyhinged so as to provide revolute motion. Although a joint is notstrictly constrained to a single degree of freedom, revolutemotion about one axis is usually prominent. Actuation of the

joints is performed by muscles which are connected to theseskeletal links. Based on an analogy with the human mechanismof joint actuation, the principle can be illustrated using a string-tube combination as shown in Fig. 2. The string-tubes play therole of muscles. Since strings can be used for actuation only intension, two of them are needed for the actuation of a joint asshown. The tube ends are fixed on one link (proximal) while

the corresponding string ends are connected to the next link(distal) of the same joint. Each string winds around a pulley atthe joint. The pulley may be considered to be a part of the distallink. The relative motion between a string and itscorresponding tube results in motion of the joint in onedirection.

The actuation of strings is required to be affected by anexisting natural joint and transmitted to the prosthetic joint. The

joint which is actuated by the natural joint is the active joint ,



while the corresponding prosthetic joint which gets actuated isthe passive one. When the sense of rotation of the passive jointis the same as that of the active joint , the configuration is saidto be a like configuration. When the sense is opposite, it is anunlike configuration. It may be noted that for the likeconfiguration, the direction of winding of string on the active and passive pulleys is in the same sense, whereas, for the unlikeconfiguration the direction of winding on the active joint pulleyis opposite to that of the passive joint pulley.

l P θ

Passive joint withlike configuration

String 2

Aθ

Active joint

String 1

Passive joint with

unlike configuration

String 2

Aθ

u P θ

Active joint

2 A

1 A

1 P

2 P

Like configuration

String 1

2 A

1 A

1 P

2 P

Unlike configuration Fig. 2 Like and unlike configurations

It may be observed that the passive joint can be convertedfrom a like to an unlike configuration, or vice-versa, by simply

rotating the passive joint by 18 about the vertical axis.0

For the task of handwriting, observe that joints I3 and T3 are in unlike configuration, whereas I3, M3 and R3 are in likeconfiguration with T1. Similarly in many other common tasksit is easy to observe a like configuration relationship among

joints I2, M2, R2 and L2, and also among joints I3, M3, R3 and L3.

Two string-tube pairs, almost inextensible, are used foractuation of the passive prosthetic joint by the active jointwhich is attached to a natural finger joint. String 1 starts from

point 1 A and goes around the base pulley centered at the jointaxis, through the tube and on to the passive joint pulley, where

it is attached at point1

P . String 2 starts from the point2

A and

goes around another base pulley centered at the active jointaxis, through tube 2, and on to the passive joint , where it is

attached at point . Since a string can actuate a finger link

while it is in tension only, the strings used here are required to be taut always. The strings can be passed through flexibletubing which offers high impedance to axial compression, asdiscussed later. This can ensure constant string length betweenthe pulleys, while maintaining appropriate string tension, evenif the center distance between the active and passive joint axeschanges. With such an arrangement, the axes of the active and

passive joints need not be parallel or fixed.

2 P

IV. BOND GRAPH MODELING OF PROSTHETIC JOINT

ACTUATION

Models for the system dynamics of the string-tube based prosthetic mechanisms have been developed along with the principles of working. Fluid based actuation also follows asimilar line of development and is therefore not discussedseparately. The models are represented using the pictorial

description of Bond graphs. Bond graph models are based onthe interaction of power between the elements of the system.Cause-effect relationships are also depicted and help inderiving system equations in an algorithmic manner from the

bond graph itself. The system equations are in the first order

form state space form of ( ) (functiond

effect causedt

= ) , and

are suitable for numerical integration as well as for

development of control systems and analysis based on moderncontrol theory. Details about the method and the art ofconstructing Bond graphs can be found in [10] and [11].

A detailed discussion on the bond graph constructionfollows for the unlike configuration. The dynamics of likeconfiguration can also be treated in the same manner.

Consider the active joint connected to the natural finger inFig. 2. When the active joint is moved clockwise, as in closing

the grip, A

θ decreases – clockwise, and a moment A

τ is

applied about the active joint. :Se A

τ is the element showing

the effort variable, torque A

τ applied at junction 1 Aθ

, as a

result of movement of the active joint. Bonds connected to

junction 1 Aθ share the common flow variable

Aθ . The moment

of inertia of the active finger link, about the joint axis, is shown

by . The string 2 on the active finger link experiences a

pull due to tension along the string. This force of magnitude

:I A J

2 F , is tangential to the base pulley at the joint, and is

represented by the2

0 F junction. The transformer element

relates this force to the moment it generates about the

active joint axis. Due to its power conserving nature

also relates the active joint angular rate

2:TF A

r

2:TF

Ar

Aθ to the speed of

winding of the string2 A

s on the base pulley. Under the

assumption that no force is lost in transmission, the stringapplies this force at the fixed point

2 P on the passive thumb

link. This results in the development of a counter clockwisemoment about the passive joint axis and consequently in the

movement of the passive prosthetic joint. Junction 1 P θ

represents the common flow variable P

θ . The moment of

inertia of the passive finger link, about its joint axis, is modeled

by :I P J .

Two paths have been shown in the Bond graph between the

1 Aθ

and 1 P θ

junctions. These are on account of the two strings

1 and 2. Initial tensions in the string-tube system are also

included. Now we consider the opening movement. The active joint

is moved counter clockwise. A

θ increases, and a counter

clockwise moment A

τ is applied about the active joint axis.

The string 1 on the active finger link is pulled and a force due

to string tension is experienced at1

A , along the string. This

force is represented by the junction1

0 F

. The radius of the base

pulley at the active joint for string 1 can be different from1 Ar



that for string 2. It has been shown intentionally to be so, using

the transformer element , which depicts the relationship

between the force and the moment generated by it.

also relates the joint angle rate

1:TF A

r

1:TF A

r

Aθ to the speed of winding of

the string1

s . Under the assumption that no force is lost in

transmission, the string 1 applies this force at point on the

passive thumb link. This results in the development of aclockwise moment about the passive prosthetic joint axis. Notethe difference of the signs between the transformer moduli inthe two cases.

1 P

The relationship between the joint angular rates A

θ and P

θ

given by both the string paths should be the same if the stringsare considered to be inextensible. For string 1,

1

1 1 1 1

1

, , A

A A A P P P

A P

r s r s r

r

θ θ θ

θ = = − ∴ =

P − (1)

For string 2,

2

2 2 2 2

2

, , A

A A A P P P

A P

r s r s r

r

θ θ θ

θ

= − = ∴ = −

P (2)

Hence

1 2

1 2

A P

A A P

r r

r r

θ

θ = − = −

P (3)

which is a constant ratio. Amplification of torque, ormovement, from the active to the passive joint can be achieved

by designing this ratio according to the requirement.

If the string-tube combination is assumed to be

inextensible, a derivative causality results at the element :I P J .

It implies that this element is unable to contribute a statevariable to the system, and is dependent. In other words, themotion of the passive finger is dictated by that of the active finger. This is as expected. Considering the string-tube to beinextensible is too ideal a property. In reality, the string lengthespecially between the active and passive joint pulleys issubject to extension and the tube is subject to axialcompression. Both the stiffness and internal damping effects of

the string-tube can be modeled using elements C : s K and

:R s R as shown in .

s K depends on the Young’s modulus of

the string-tube along its length. Inertia of the string-tubecombination is neglected in this model. The stiffness andinternal damping properties for the combination of string-tube

pairs are nonlinear and have to be determined experimentally.However, the nature of causality remains unaffected, as theforce of tension produced by the string-tube is determined by

its properties. In this case, it is assumed that the surface speedof the string is the same as that of the pulley at contact.

In the bond graph of , the surface speeds of the active and passive pulleys, through which strings 1 and 2 pass, are shown.Efforts due to string tensions have been represented by

common effort junctions1

0 F

for string 1, and 02 F for string 2.

is the effort variable tension, decided by properties of string

1, and applied on junction

12e

10

F . Similarly, is the effort

variable tension, decided by properties of string 2, and applied

on junction 0

17e

2 F .

jointP:R R

1 P θ

:I P

J

1:C

TF

1 P s

2 P r

1 P r −

e

TF

1 F 2 P

s2

F

jointA:R R

1 Aθ

A J :I

1:C

TF

1 A s

2 Ar −

1 Ar

e

TF

1 F 2 A

s2

F

eS

A: τ

20

F

10

F

2 F

1 F

2 s∆

1 s∆

21

s∆

11

s∆

2C :

s K

2R :

s R

1C :

s K

1R :

s R

1 2

3

4 5

6 7

8 9

10

11

12

13

1415

16

17

18

19

20 21

1 p

2 p

4q

6q

8q

9q

Fig. 3 Bond graph for the string-tube based joint actuation.

Bearing friction at the joints may be treated as a nonlinear phenomenon, even though the magnitude may be very smallfor rolling bearing elements. An example of such nonlinear

behavior may be given by Coulomb friction or viscous friction.However, the characteristics of friction are not a point ofemphasis here.

A. Deriving system equations

The system of Fig. 3 has 4 state variables. is the angular

momentum of the active finger about the active joint axis.

is the angular momentum of the passive finger about the

passive joint axis.

1 p

2 p

4q

1 s= ∆ is the extension of string 1, and

6

q2

s= ∆ is the extension of string 2. Two additional states are

obtained from the activated C elements. These are8 A

q θ = and

9 P q θ = . These do not contribute to the dynamics of the system.

andi

ei

f represent the effort and flow respectively in bond i.

The set of effects contributed by the elements to the systemare given below

11 2

; A

A P

p p f f 2

P J J

θ θ = = = = (4)

4 4 6function( ), and function( )e q e 6q= = (5)

in the general case where elastic behavior of the string-tubemay be non-linear. For linear elastic behaviour

14 4 6; and s

e K q e K q2 6 s= = (6)

Similarly,

( ) ( )5 5 7function , and functione f e= =7

f

2 7 s

(7)

and for the particular case of linear viscous behavior ofinternal damping,

15 5 7, and s

e R f e R f = = (8)

The rate of string extensions are given by



15 1 1 1 1 1 1 1 A P A A P P A P

A P

p f s s s r r r r 2 p

J J θ θ = ∆ = − = + = + (9)

17 2 2 2 2 2 2 2 A P A A P P A P

A P

p f s s s r r r r 2 p

J J θ θ = ∆ = − = − − = − − (10)

Tensions developed in strings due to the above extensionsare

1 1

11 4 5 4 1 1 s s A P

A P

p p F e e K q R r r 2

J J ⎛ ⎞= + = + +⎜⎝ ⎠

⎟ (11)

2 2

1

2 6 7 6 2 2 s s A P

A P

p p F e e K q R r r 2

J J

⎛ ⎞= + = − +⎜

⎝ ⎠⎟ (12)

Since the strings are able to transmit tensions only, thefollowing conditions apply:

(13)1 12 12

2 17 17

0, i.e., 0, if 0, and likewise

0, i.e., 0, if 0

F e e

F e e

= = <

= = <

Bearing friction at joints is given by

( ) ( )20 1 21 2

jointA 1 jointP 2

function , and function ,

for the general case, ,

for linear viscous friction characteriscs

e f e

R f R f

= =

= =

f

2 A

(14)

The above effects are produced by functions of cause applied to the system as given below.

1 1 20 1 1 2 A Ae p e r F r F τ = = − − + (15)

(16)2 2 21 1 1 2 P P e p e r F r F = = − − +

2

14 4 1 1 1 A P

A P

p p f q s r r 2

J J = = ∆ = + (17)

16 6 2 2 2 A P

A P

p f q s r r 2 p

J J = = ∆ = − − (18)

Equations (15), (16), (17) and (18) represent the dynamicsof the system in the first order state space form

( ) (functiond

effect causedt

= ) .

In addition the observed variables are,

18 9; and A

A P

pq q 2

P

p

J J θ = = = = θ (19)

B. Bond graph modules for rigid body mechanics

The fingers of the hand may be considered to be made up ofalmost rigid links (bones called phalanges). The joints between

links are generally revolute, though not in a strict kinematicsense. The joints are spherical in the kinematic sense but have a prominent revolute motion about an axis. The rigid constraintsat joints are relaxed due to the presence of soft tissue and fluid.We can consider a joint to be revolute about its axis. It isconvenient to use the notation of vector bond graphs as itmakes the representation quite compact.

We shall differentiate between a scalar and vector bond bytheir relative thickness as shown in Fig. 4. In this work, avector bond graph is an ordered collection of three scalar

bonds. Hence the dimension three is not indicated explicitly onthe vector bond. This is to avoid congestion in figures. Thus, if

the flow vectorC

f r = then { } { }1 2 3

T T

Cx Cy Cz f f f v v v= ,

where { }1 2 3

T

f f f f = and { }T

Cy Cz v v .

Similarly, if the effort vector

C Cxr v=

C F e = then

{ } { }1 2 3

T T

Cx Cy Cz e e e F F F = , where { }1 2 3

T

e e e e=

and { }T

C Cx Cy Cz F F F F = .

Scalar bond Vector bond

e

Power e f = ⋅

e

Power T T e f f e= =

Fig. 4 Convention for scalar and vector bonds

The fundamental equations of motion for rigid bodies can be represented using bond graphs as shown in Fig. 5. Both

translation and rotation for the rigid body are combined in one bond graph.

00

1 Bω

00

1C r

MTF

MTF 00

n F

01

0 F

Fig. 5 Bond graph representing translation and rotation of the rigid body.

It shows that the translational momentum of the entire rigid body can be considered to be concentrated at the center of itsmass, and it changes according to the resultant of the forcesapplied on it.

{ }0

0 C

d 0

P r

dt = ∑ F (20)

It also clearly represents the cause-effect relationship between torque acting on the rigid body and its angularmomentum about its center of mass (CM) C . The total torqueacting on the rigid body about C causes a change in the angularmomentum of the rigid body about C . The effect is the rotation

of the body with angular velocity 0

0 C ω , and is decided by the

inertial properties of the rigid body.

The equation for rotation of the rigid body, due to forcesacting on it, as represented in the bond graph can be written as

{ }0 0 0 0 0

0C C C P P C

d I r F

dt ω τ ⎡ ⎤ = × =⎣ ⎦ ∑ (21)

It may be noted that the Bond graph is integrally causalled.

0

C :I I ⎡ ⎤⎣ ⎦ [ ]I:

0 T

C nr ⎡ ⎤×⎣ ⎦

0

1

T

C r ⎡ ⎤×⎣ ⎦

0

e:Sτ

0

1eS : F

0

eS :

n F

1



Since the terms of the inertia tensor are expressed in

the frame 0 they change due to rotation of the body withrespect to frame 0. We know that the components of the inertiatensor for the rigid body are constant if expressed in a framefixed on the body itself.

0

C I ⎡ ⎤⎣ ⎦

The orientation of frame C with respect to frame 0, given

by the rotation matrix , is obtained by the integration of the

matrix differential equation

0

C R

0 0 0

0C C R ω ⎡ ⎤= ×⎣ ⎦

C R (22)

This means that if the orientation of the rigid body at time

is given as0t ( )0

0C R t , the orientation at time t can be

obtained as from (22). The angular velocity vector( )0

C R t 0

0 C ω

is necessary. There are well known dependencies among the

elements of due to its orthonormality. The columns of

are unit vectors of frame C along its coordinate axes expressed

in the frame 0. Due to the angular velocity

0

C R0

C R

0

0 C ω of the body,

these unit vectors undergo a change of orientation, at time t ,with respect to frame 0. The columns are obtained fromintegration of

0 0 0

0ˆ ˆ , 1, 2, 3

i iC C C u u iω ⎡ ⎤= × =⎣ ⎦ (23)

where, . In terms of Bond graphs,

this relationship may be expressed using

1 2 3

0 0 0 0ˆ ˆ ˆC C C R u u u⎡= ⎣ C

⎤⎦

[ ]1:C elements and

effort activated bonds connected to junction 00

1C ω

as shown.

The state variable vector associated with each element [ ]1:C

is . Thus the orientation matrix can be constructed

again. This approach has redundancy in it, caused by the

orthonormal nature of . The third column vector of can

be obtained from the previous two columns by vector crossmultiplication, instead of numerical integration.

0 ˆiC u 0

C R

0

C R0

C R

The relative orientation between two links, say A and B,

can be obtained from (24) once their orientations and ,

with respect to frame 0, are known.

0

A R0

B R

(24)0 0 A T

B A B R R R=

In order to illustrate the method, these developments areapplied to an example of a finger with two revolute joints,shown in Fig. 6. However the formulation discussed above is avery general approach, not specific to a 2 DOF finger only.

1

0

O F

3

0

O F

0 X

0Y

1Y

1 X

2Y

2 X

1C

2C

link 1

link 2

0

1τ

0

2τ

1O

2O

3O

Fig. 6 A finger with two revolute joints.

The bond graph for the system is shown in Fig. 7. Iftranslational constraints are rigidly maintained, derivativecausality appears at the power bonds connected to the

translational inertia elements. It occurs due to the imposition ofkinematic constraints which result in the dependence of the

momenta of masses1 and

2 on the angular momenta of

links 1 and 2 about their respective centers of mass. In naturalfinger systems, the presence of soft tissue and fluid relax the

joint constraints by introducing their own properties of stiffnessand dissipation. This amounts to the introduction of bond graphelements representing stiffness and dissipation at respective

joints as shown in the integrally causalled bond graph of Fig. 7.

0

0 2

1ω

0

0 2

1C r

MTF

MTF 03

0O F

02

0O F −

[ ]2I: M

2 3

0 T

C Or ⎡ ⎤×⎣ ⎦

2 2

0 T

C Or ⎡ ⎤×⎣ ⎦ 0

2τ −

3

0

eS :

O F

00 1

1ω

00 1

1C r

MTF

MTF 02

0O F

01

0O F

[ ]1I: M

1 2

0 T

C Or ′⎡ ⎤×⎣ ⎦

1 1

0 T

C Or ⎡ ⎤×⎣ ⎦

00 0

1Or

0

0

00

f :SOr =

0

1τ

0

2τ −

02

0τ −

00 2

1Or ′

0 1

[ ]12C: K

[ ]12R: R

00 11 Or

0 1

[ ]01C: K

[ ]01R: R

00 2

1Or

2

0

2:I

C I ⎡ ⎤⎣ ⎦

1

0

1:I

C I ⎡ ⎤⎣ ⎦

1

2τ −

MTF

0

1

T R

TF0

1 2ω

[ ]0 0 11

2 z τ −

2θ

TF

[ ]0 0 10

1 z τ

1θ

1

1 2ω

Fig. 7 Bond graph shows stiffness and dissipation at joints. Application

of joint torques is also shown.



The transmission of motion from the active to the passive joint can be observed in Fig. 7 for the actuation in the threedimensional system. The string-tube connecting an active and

passive joint decides the string tensions and hence the torquesexperienced on the joints based on the extension of the stringswith respect to the tubes. This extension is decided by theangular motions of the joints connected by the string-tubes.

C.

Simulation and discussionSimulations were carried out based on the bond graph

model derived earlier. The active joint is coupled with theindex finger at joint I3, and the passive prosthetic joint islocated at the thumb joint T3. The system is in the unlikeconfiguration similar to the one shown in Fig. 2. The initial

posture is such that A

θ = 60° and P

θ = 120°. It implies that the

finger and thumb are in the closed position. The active finger is

supplied a torque ( )0.0005sin A

t τ π = N-m so that the fingers

may open and close. The parameters chosen are:

, ,5 21 10 . A

Kg m J −= × 5 21.2 10 . P

Kg m J −= ×1

51 10 / s

K N m= × ,

2

51 10 / s

K N m= × , , ,

, , ,

1100 . /

s R N s= m m

m

2100 . /

s R N s=

1 0.01 Ar m= 2 0.01 Ar m= 1 0.01 P r m= 2 0.01 P r = ,, . The simulation is

for a duration of 2 s. Results of the simulation are shown anddiscussed below. Fig. 8 clearly shows that the joints operate inthe unlike configuration.

jointA0.0001 . . R N m s=

jointP0.0001 . . R N m s=

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3

-2

-1

0

1

2

3 Angular velocities of finger joints

time (s)

A n g u l a r v e l o c i t i e s ( r a d / s )

dθ A

/dt

dθ P

/dt

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.5

1

1.5

2

2.5

3 Angular displacement of fingers

time (s)

θ A ,

θ P

( r a d )

θ A

θ P

Fig. 8 Simulations plots showing angular velocities and displacements of joints. Opening and closing of the joints in unlike configuration is considered.

Tension increases initially in string 1 till opening iscompleted and then reduces. The closing movement results inincrease of tension in string 2. Initial tension is absent in thestring tube system while considering the result of Fig. 9.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025

0.03Tension in string 1

time (s)

T e n s i o n ( N )

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025

0.03 Tension in string 2

time (s)

T e n s i o n ( N )

Fig. 9 Tension in strings. No pretensioning has been done here.

Pretensioning with a force of 10 N has been considered in

Fig. 10.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 29.98

9.99

10

10.01

Tension in string 1

time (s)

T e n s i o n ( N )

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 29.98

9.99

10

10.01

Tension in string 2

time (s)

T e n s i o n ( N )

Fig. 10 Tension in strings when pretensioning of 10 N is considered.

The transaction of power is depicted in Fig. 11. The powerinput to the system by the natural index finger through theactive joint is distributed to accelerate the prosthetic joints andmove them, and also overcome opposing friction.Pretensioning of strings does not alter the power transactions

for the case of joint friction characteristics considered.



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

0

2

4

6

8

10

12x 10

-4Power transactions

time (s)

P o w e r ( W )

input power to accelerate active finger to accelerate passive finger Power lost in bearing RPower lost in bearing L

Fig. 11 Power transactions. The distribution of input power for actuationof the finger joints and the power losses in bearings.

V. SALIENT FEATURES

Redundancy is a very important and characteristic feature provided by nature to the human body. The proposed prostheticmechanisms utilize this feature effectively. Using the ability ofremaining natural fingers to provide movement capability to

prosthetic joints is an effective way of channelizing thisredundancy to some extent. The ideas for the mechanisms forthe prosthetic devices are inspired by existing naturalmechanism of the human hand.

The function of feedback control during joint actuation is performed by the remaining natural fingers. Sensing andcontrol abilities of these natural members, a part of theneuromuscular system, integrated with the CNS perform therole of controller. Additional hardware for control is therefore

not required in the passive versions of such prosthetic devices.

Due to the flexible string-tube combination, the active and passive joint axes can have movement relative to each other.The active joint axes need not be parallel or fixed with respectto corresponding passive joint axes. The string-tubecombination maintains constant string length between jointseven while they move. This is especially a very useful featuresince it permits movement of the joints and transmission at thesame time.

The active and passive joints are physically implemented inthe form of modular buttons that are buttoned on to a glovewith necessary skeletal framework. Such a buttoning conceptmakes the mechanism easy to assemble and disassemble very

quickly.

Actuation of prosthetic joints using natural joints on fingersdoes not increase the independent DOF of the severed hand.This is due to the fact that motion of passive joints is dependenton active natural joints. However, the use of the existingdegrees of freedom, to provide actuation capability to

prosthetic joints enables the performance of tasks which wouldnot have been possible otherwise.

Fig. 12 Tension in strings when pretensioning of 10 N is considered.

Each passive joint derives its movement from one or moreactive joints on the same hand in a modular way. Thus

provision of extra passive joints is equivalent to adding degreesof freedom to the prosthetic mechanism. It is well known inrobotics literature that such DOF add to the dexterity of the

prosthetic device.

Every disability within the scope of the proposed prostheticmechanisms requires a separate tailor made prosthetic device.This is based on the nature of loss that requires rehabilitation.Training and practice with the prosthesis is therefore inevitable.

It is appropriate to mention that the prosthetic devicessubstituted in place of the lost natural fingers have limitedcapability, for example, sensing texture of objects, etc. cannot

be performed by them.

Evaluation of such a device comprises of (a) functionalcapability to perform key essential tasks (b) Study of powerand its constitutive variables (velocities and forces) utilized toactuate active and passive joints during actual implementationof tasks. The Bond graph approach helps in developing modelsfor system dynamics such that these power variables can beeffectively analyzed.

VI. ACTIVE PROSTHESIS: A PREVIEW

It is interesting to note that the string-tube based realizationof the prosthetic mechanism can also be viewed from a controlsystem perspective. The passive system is actuated by efforts

proportionate to the extension and its rate. The string-tubeextension is related to the error between joint angles undercertain assumptions. Hence the control effort on the passive side is proportionate to the error in joint angles and itsderivative (PD control).

The mechanisms discussed in this paper so far were passiveversions. No energy external to the natural hand is used in thesedevices. Our representation of the system dynamics using bondgraphs also facilitates the development of active versions ofsuch prosthetic devices. Active prostheses are useful foraugmentation of weak finger muscles. Energy external to thehuman hand is applied in active devices. Power to provideactuation of such muscles can be drawn from an externalelectric source, for example. The active joint connected to aweak finger requires very limited effort and power for its



actuation. The input motion to the active joint is provided bythe remaining natural finger. Consider the system of Fig. 13which is obtained by simplification of Fig. 3.

From (26), (27) and (30),

join t P P P D P P J K K R

θ θ θ ε ε = + − θ (31)

It is clear that the passive subsystem follows the joint angle

trajectory A

θ of the active joint, while reflecting a fraction of

the effort P

α τ felt by the passive subsystem. The element

shown in the bond graph is non conservative, unlike

the power conserving transformer TF , and is used to scale power between the active and passive subsystems.

TF NC

jointP:R R

1 P θ

:I P

1:Ce

jointA:R R

1 Aθ

A J :I

1:C

e

TF NC eS

A: τ 0

P τ

P τ θ

ε

1θ ε

C : P

K R : D

K θ ε

P τ

Aθ A

θ

P τ

P θ

ref τ P

θ ref P τ α τ =

Aθ

A p P p

Active subsystem passive subsystemIntermediate subsystem

Controlling of a passive joint based on inputs from morethan one active natural joints, using this principle, is alsofeasible. In this case the individual active joints provide motioninputs to the intermediate subsystem. The passive joint wouldmove according to the resultant effort applied on it by the

intermediate subsystem, as the 1 P θ

junction representing the

motion of the passive joint is an effort summing junction. Thediscussion initiated in this section offers scope for furtherexploration into analysis and design of active prostheticsystems.

Fig. 13 Active control of a prosthetic system

The input torque Aτ is supplied by the neuromuscularcapability of the natural finger based on the task requirement

decided by the CNS. The active side has a small inertia A

J .

The passive inertia P

J may be larger in comparison to the

inertia on the active side. The effort P

τ applied on the passive

subsystem is scaled down by a factor α , and reflected back onthe active subsystem, by the intermediate subsystem. Thereflected effort felt on the active side is

R EFERENCES [1] A. Vaz and S. Hirai, “Actuation of a Thumb Prosthesis using Remaining

Natural Fingers,” Proceedings of the IEEE/RSJ Int. Conf. on Intelligent Robots and Systems (IROS 2003), Las Vegas, October 27- 31, 2003. pp.1998-2003.

[2] J. Pillet, and A. Didierjean-Pillet, “Aesthetic Hand Prosthesis: Gadget orTherapy? Presentation of a New Classification,” Journal of HandSurgery (British and European Volume), 26B, no. 6, pp. 523-528, 2001.

(25), 0 ref P

τ α τ α = <

[3] T. Iberall, C. L. MacKenzie, “Opposition Space and HumanPrehension,” in Dextrous Robot Hands, S. T. Venkataraman and T.Iberall, Ed. New York: Springer-Verlag, 1990, pp. 32-54.

1≤ [4] Iberall,. T, “Human Prehension and Dexterous Robot Hands.” Int. J. Robot. Res., vol. 16, no. 3, pp. 285-299, June 1997.

The flow variables A

θ from the active side, and P

θ from

the passive side, are visible to the intermediate subsystem, and

are used to determine the effort P τ required to be applied onthe passive subsystem. In this example, the intermediate

subsystem acts like an elastic coupling having stiffness P

K and

damping D

K . It applies efforts of different scale,ref P

τ α τ = on

the active subsystem, and P

τ on the passive subsystem.

Deriving system equations from the bond graph,

[5] M. R. Cutkosky, R. D. Howe, “Human Grasp Choice and Robotic GraspAnalysis,” in Dextrous Robot Hands, S. T. Venkataraman and T. Iberall,

Ed. New York: Springer-Verlag, 1990, pp. 5-31.[6] M. T. Mason, and J. K. Salisbury, Jr., Robot Hands and the Mechanics

of Manipulation, The MIT Press, Cambridge, Massachusetts, 1985.

[7] A. Bicchi, “Hands for Dexterous Manipulation and Robust Grasping: ADifficult Road Toward Simplicity,” IEEE Trans. Robot. Automat., vol.16, pp. 652-662, Dec. 2000.

; ; ; P P

A P

A P K P R

A P

p pe K e K

J J Dθ θ

θ θ ε = = = = ε (26)

[8] A. Vaz and S. Hirai, “Modeling Contact Interaction of a Hand Prosthesiswith Soft Tissue at the Interface,” Proceedings of the IEEE Int. Conf. onSystems, Man and Cybernetics (SMC 2003), Washington D. C., October5- 8, 2003. pp. 4508-4513.

P P P K R P De e K K

θ θ τ ε = + = + ε (27)

A P

A P

A P

p p

J J θ

ε θ θ = − = − (28)

[9] A. Vaz and S. Hirai, “Bond Graph Modelling of a Hand ProsthesisDuring Contact Interaction,” Proceedings of the IASTED Int. Conf. on

Applied Simulation and Modelling (ASM 2003), Marbella, Spain,September 3- 5, 2003. pp. 313-318.

joint A A P A p R

Aτ α τ θ = − − (29)

[10] D. C. Karnopp, D. L. Margolis, and R. C. Rosenberg, System Dynamics: Modeling and Simulation of Mechatronic Systems, third

edition, Wiley-Interscience, 2000.

joi nt P P P p R

P τ θ = − (30)

[11] A. Mukherjee, R. Karmakar, Modeling and Simulation of EngineeringSystems Through Bondgraphs, Narosa Publishing House, New Delhi,2000.

bond graph based design of prosthesis for partially impaired hands

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