multi- criteria optimal design of cable based ankle rehabilitation robot

7/23/2019 Multi- Criteria Optimal Design of Cable based Ankle Rehabilitation Robot

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Multi- Criteria Optimal Design of Cable based AnkleRehabilitation Robot

1. Introduction

An ankle rehabilitation robot has been conceptualized and designed to realize the rangeof motion, muscle strengthening and proprioception training exercises. The robotic device hasbeen designed to help the patient and the therapist in their cooperative efforts for the treatment ofimpaired ankle joint which may be a result of some injury or stroke. After analyzing the ankle jointanatomy and its motions, a parallel mechanism was selected for the robot. To mimic the humanankle joint and its muscles actuation, the proposed robot uses air muscles configured in a similarfashion to the actual muscle arrangement. The performance of a parallel robot greatly dependson its dimensions and configuration of actuators. To explore the advantages of these robots it isessential to obtain a set of kinematic parameters leading to optimal robot performance. Thedesign of robot has been evaluated on the basis of three main performance indices such as,workspace, condition number and 2-Euclidean norm of actuator forces, under some constraints.The performance criteria and the constraints have been discussed in detail to justify theirinclusion. The two Multi-objective Optimization Approaches (MOA), a weighted formula approachand a pareto optimal approach have been used and the results are compared and discussed.

2. Rehabilitation and robotics

Rehabilitation in a broader sense means a practice by which any form and grade of humanphysical disorder can be reinstated. The disorder could be the result of an injury or a stroke.Conventionally, rigorous and repetitive exercises are performed under the supervision of atherapist to restore range of motions and strength of limbs. These exercises improves motorfunctions by enhancing neuro-plasticity and neuro-recovery at the affected limbs. During arehabilitation treatment, cooperative efforts of therapist and the patient are required overprolonged sessions of treatment in a clinic. Moreover the patient is required to continue theprescribed exercises at home for a speedy recovery. It has been documented that usingconventional line of treatment the recuperation is slow and sometimes continues for more than ayear. The major drawbacks of such treatments is that the patient has to travel to attend theclinical sessions in his disability state and spend on his time and money. The workout at home ismonotonous and lacks motivation resulting in inadequate improvement. Similarly the therapisthas to perform strenuous and repetitive efforts with the patients and thus he can only attend alimited number of patients in a day. Rehabilitation treatment, is also not well structured andlargely depends on intuitive judgment of therapist on the progress of the patient.

Robotics can play an important role in the process of rehabilitation by assisting the therapistand the patient in reducing their efforts. Rehabilitation process can also be improved by acquiringprogressive data of patients improvement, which in turn can help the therapist to makesystematic decisions on choice of further exercises. Certain measure may be taken to make theworkouts more interesting and motivating as advised in []. There are certain other robotic aids,which are already in use such as MIT-MANUS and LOKOMAT etc.

Rehabilitation robots are different from the industrial robots in application and use and hencespecial care must be taken in their design. The industrial robots are more structured and meantfor a specific goal with minimum human interference. Moreover the operators for industrial robots

are also the trained technicians. On the other hand rehabilitation robots are generally multi-taskedand hence should be intelligent machines. Human augmented robots should be safe to use andmust be user friendly in operation since the user could be a layperson. Thus the design andcontrol of these robots is a challenging task requiring multi-discipline skills and in-depthknowledge of human joint anatomy and movements.


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3. Human ankle, its prob lems and physiotherapy

3.1 Ankle Complex

The ankle joint [1] is a combination of two joints (Fig.1) out of which the first joint is madeof three bones: the lower end of the tibia (shinbone), the fibula (the small bone of the lower leg)and the talus (the bone that fits into the socket formed by the tibia and the fibula). The talus sits

on top of the calcaneus (the heel bone). The talus moves mainly in one direction. It works like ahinge to allow foot to move up (dorsiflexion) and down (plantar flexion). The second joint issubtalar joint, in human anatomy it is also known as the talocalcaneal joint and this is a joint ofthe foot. It occurs at the meeting point of the talus and the calcaneus. This joint is responsible forthe inversion and eversion of the foot, but plays no role in dorsiflexion or plantarflexion of the foot.However it is very much a part of the ankle joint and thus can not be ignored.

There is one more joint called MTP (metatarsophalangeal) joint connecting fore and rearwith calcaneus, cuboid and Navicular bones as shown in Fig.1. The relative rotational motionaround this MTP joint realizes more natural exercises for toe raising and heel raising etc.

FibulaTalus

Tibia

Navicular

Calcaneus

Cuboid

Metatarsal

Phalange

Figure 1: Bones of the Ankle and sub talar joint

Fibula

There are ligaments on both sides of the ankle joint that hold the bones together.

There are many tendons that cross the ankle to move the ankle and move the toes.

Ligaments connect bones to bones while tendons connect muscles to bones this has beenshown in Fig. 2. The ankle joint is capable of rotations in all three planes (sagittal, frontal andtransverse planes) as shown in Fig. 1, where sagittal plane is defined byx and z and movementsin this plane occur around the y axis; transverse plane is defined byx and y and movements inthis plane occur around the z axis; and frontal plane is defined by y and z and movements in thisplane occur around thex axis. Various ankle movements [3]can be summarized as follows (Table1). The sub talar joint has a more constrained movement pattern owing to three articular facets

connecting the talus and calcaneus. Sub talar motion is described as pronation (lifting footupward) and supination (lifting foot downward). Thus the foot is the most complex bone structuresin the body since the functions of the ankle and sub talar joints are also interdependent with thoseof the talonavicular (MTP) and calcaneo-cuboid joints, as these joints share bones.


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Table 1. Ankle joint motions

Axes Name of the motion Range of Motion Torque Requirements (Nm)

3.2 Ankle injuries and physiotherapy

Ankle injuries [1] are one of the most common injuries in sports and daily active life andon an average 600 patients per annum are being treated in New Zealand only [2]. Youngsters aresubjected to ankle injuries from sports and whilst carrying excessive load whereas children andthe elderly gets it from walking on uneven surfaces and bone weakness. Non functionality of

ankle joint is also quite common in stroke surviving patient.Primary treatment for ankle injuries or non functionality includes rest, ice, compression

and elevation (RICE) of the affected foot along with stretching and exercise therapy, and partialweight bearing with aids to maintain mobility in the ankle. During ankle rehabilitation the entirefoot is dorsiflexed, plantarflexed, inverted and everted considering the combined ankle and subtalar joints.

Generally a complete rehabilitation program [4] including exercises for range of motions(ROM), muscle strengthening, and proprioceptive training which play an important role inspeeding up recovery for normal functioning of ankle. Achilles tendon should be put in stretchingexercise as soon as possible (within 48 to 72 hrs) after the injury and the ROM must berecovered. Once the ROM is achieved, strengthening of weakened muscles is essential for rapidrecovery and is a preventive measure against further reinjury. Manual therapy, such as toeraises, heel walks, and toe walks, toe curls, and marble pick-ups as well as resistance exercisesto ankle joints are attempted to regain strength and coordination of the muscles. As the patientachieves full weight bearing capability without pain, proprioceptive exercise is initiated for therecovery of balance and postural control using wobble boards . Finally, advanced exercises usinguneven surface wobble board should be performed to regain functions specific to normalactivities.

3.3 Ankle rehabilitation robotics

Construction of a suitable rehabilitation mechanism requires detailed study of human jointmovements and their control from the physiotherapy perspective. The direction, speed and rangeof- joint motion are altered by injury to muscles and ligaments. There is enormous potential tointegrate medical knowledge, human movement studies and mechanism synthesis in the domainof ankle rehabilitation devices.

Rehabilitation treatment specific to ankle joint is quite exigent for the reason that the anklejoint is the most complex part of the human skeleton [11]. Continuing physiotherapy is required tohelp recover ankle joint from the injuries or to enhance neuroplasticity for non functionality due tostroke. Physiotherapy requires rigorous repetitive movements of limbs about their respective

joints and robotic machines are useful in such applications once appropriately programmed. Forankle rehabilitation, typical passive devices such as elastic bands and the active units for strengthand proprioception exercises such as wobbles board and foam rollers are in use, but they allowonly simple and functional rehabilitation exercises. Robotic devices on the other hand areadvantageous owing to the fact that they are multifunctional to allow almost all types of exerciseson one machine and provides accurate motions and forces. These devices can be programmedby a physiotherapist and exercise modes could be selected depending on the type of injury and

Inversion 350

48X

Eversion 250 34

Dorsiflexion 200 40.7- 97.6Y

Plantarflexion 400 20.3- 36.3

Adduction 250 30Z

Abduction 200 30


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patients improvement. Thus, task oriented training by supervised robotic systems is helpful toprovide the patient with more diverse and useful exercises. It is realized that the parallel robotsare good choice to perform three rotational degrees of ankle movements, because they have thebenefits of high dynamic stiffness and required manipulability. So far only rigid link parallelmechanisms [6] have been conceptualized.

However all these devices have certain common limitations and drawbacks and to overcomethem this chapter proposes an air muscle actuated soft parallel mechanism as an improvementdue to its inherent advantages over RPM. Parallel mechanisms can be defined as Soft ParallelMechanisms (SPM) where cables or tendons are used as links and Rigid Parallel Mechanism(RPM) if all the links are rigid. SPM does not have any spherical joints and as a matter of fact ithas higher movability. Soft parallel devices are very light in weight and has higher payload toweight ratio. These could be designed to be wearable and portable. SPM has simpler dynamicmodel than their rigid-link counterparts because the inertias of their links (i.e., cables) can beignored. Soft parallel robots are being used recently in a variety of applications [1] ranging fromvery sophisticated medical and manufacturing applications to less precise construction andshipment activities. Wire flexibility of SPM poses some constraints on the workspace and an extravariable called tensionability is required to be considered during its kinematic design. Thismakes the design process more cumbersome and choice of a perfect design becomes verydifficult. It is therefore required that the design of the proposed device be optimized by carefullyconsidering all the constraints.

4. Cable based parallel robot for ankle rehabilitation

4.1 Existing devices based on parallel mechanisms

In order to implement ankle joint rehabilitation treatment, several devices have beendeveloped. Representatives of some commercial developments are Biodexs multi joint system3and balance system, which can facilitate ROM/strengthening, and balance exercises,respectively.

A 3-RSS/S parallel mechanism is proposed by Gengqian et al [4] and the design ofprototype has been validated using simulations. Girone et al. [5] suggested the Rutgers Anklethat can provide active six degrees of freedom (DoF) motions at an ankle with a Stewart platformstructure to cover various exercise modes. Using this, the patients foot is fixed firmly to the

platform of the Rutgers Ankle and is synchronized to virtual reality scenarios. Recently, theRutgers Ankle is extended to the Dual Stewart platform with 12 (DOF) motions for gaitrehabilitation. Even though presently most of the exercise modes are possible by the use of twoStewart platforms, balance/proprioception exercises can also be achieved with a single device.Since ankle movement in most exercises needs less than four (DOF) motions, Dai and Zhao [6]proposed a sprained ankle device using a three or four (DOF) parallel mechanism with a centralstrut, and analyzed the orientation and stiffness of the mechanism with considerations of thecentral strut. Yoon et al. [7] have designed a single platform-based reconfigurable robotmechanism with satisfactory MTP joint motions and less than six (DOF) motions which allowsmore natural foot and ankle motions to cover various exercise modes.

So far only platform type of robotic devices have been designed where foot is required tobe placed on top of a platform which is actuated from the bottom. There is a possibility ofundesirable shift in the ankle joint in all these designs. However in the proposed design care has

been taken to place actuators in a similar fashion as the actual muscle arrangement is, so thatthe ankle joint stays stationary.

4.2 Cable based parallel manipulators

A cable based robot utilizes the basic idea of the Stewart platform parallel link mechanism. Theinimitable feature of this approach is to use cables as the parallel links and to use winches as theactuators. A cable actuated robot can precisely contrive large loads with flexibility and has highstrength to weight ratio. These properties make a cable actuated robot applicable to a largevariety of applications ranging from building construction, cargo handling or inspection of an


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airplane (where the workspace of the robot must be very large) to applications as small asendoscopy, colonoscopy and surgery. Cable based Parallel robots (CBPR) are becoming apromising alternative to the rigid-link mechanisms in many applications, such as load lifting andpositioning, coordinate measurement, aircraft testing, haptic devices [8], and robot rehabilitation[9] etc. The reasons are quite obvious as,1) Using wires it is possible to place the moving platform on floor because the bottom of theplatform does not have any mechanism.2) Wire robot does not use any spherical joints and as a matter of fact it has higher movability andlower inertia in comparison with rigid link robots. On the other hand, vibration is sometimes aserious problem in cable drive systems. Coordinate control of multiple cables to increase theinternal force improves its mechanical rigidity and the vibration characteristics.3) Longer Motion stroke and greater range of motion (ROM) is possible since they have largerworkspaces because their spools can reel out a large amount of cables.4) Wire actuated parallel devices are very light in weight and are more convenient compared totheir counterpart.

CBPR have several other advantages [10] over rigid-link manipulators. First, all of their actuatorsare stationary, the transmission systems can be mounted on the fixed base and thus, they have ahigher payload-to-weight ratio, which makes them attractive for high-load and high speedapplications. Second, their special designs make them potentially inexpensive, modular, and very

easy to reconfigure. Finally, they have much simpler dynamics model than their rigid-linkcounterparts because the inertias of their links (i.e., cables) can be ignored.However, the draw back is the inherent addition of wire flexibility and vibrations. Also, it should benoted that unlike rigid links, cables are characterized by the unilateral property (can pull butcannot push moving platforms), and therefore the formulations and results obtained for thekinematics analysis, workspace analysis, trajectory planning, etc. of the rigid-link mechanismscannot be directly applied.

5. Proposed Wearable Ankle Rehabilitation Device

A parallel robot for ankle joint rehabilitation treatments has been proposed in the present chapter.The robot is designed to provide three rotational degrees of freedom to the ankle joint for thenecessary ROM exercises. The device uses two parallel platforms, a fixed platform (FP) built-inwith a leg support structure and a moving platform (MP). Air muscles are used as actuators and

are mounted on leg support with their actuating end connected to MP through cables. Thesecables pass through sleeves provided in the FP. The leg support structure is fixed using somestraps which move over the knee and fixed at thigh. Lower end of the leg support remainsapproximately 100 mm above the ankle joint. MP remains below the foot and has a heel locatorand straps to fix foot. The moving platform of the robot is shaped to form a shoe of varying sizeand shape. This shoe has been attached to the leg support using a special mechanism whichprovides three rotational degrees of freedom to the shoe. Actuation of the moving platform or theshoe is realized using the air muscles and the cables. Apparently the air muscles can only pulland can not push hence to maintain the tension in all the cables it is desired to have redundantactuation. As a matter of fact all the cable based parallel robots have redundant actuation and ithas been mentioned in the literature that to achieve n-dof motion from the manipulator the robotshould have n+1 actuators. Hence to obtain 3-dof from the robot we have used a set of four airmuscles and four cables. Coordinated and antagonistic actuations of air muscles will ensure

desired changes in the wire lengths and pose of the moving platform subsequently for a range ofankle exercises. Coordinated and antagonistic actuations of air muscles will ensure desiredchanges in the wire lengths and pose of the moving platform subsequently for a range of ankleexercises.

6. Robot Kinematic Modeling

Robotic manipulators are required to perform specified motions in 3-D space. Toaccomplish this, the end effector is moved in the workspace along a predefined trajectory. Theposition and the orientation of the robot end effector at a specified workspace location can be


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obtained by controlling the displacements of its links. This calls for a mathematical model of therobot which can define the relationship between end effector motions and link displacements inthe domains of time and space. This mathematical model is called kinematic model and thederivatives of this model describes mechanics of the motion without taking forces into account.When the forces and/or torques are considered with the mechanics of motion the analysis istermed as dynamics. The kinematic and the dynamic study are essential tools for the design ofrobotic manipulators. Kinematic model helps us to completely describe the position andorientation of the end effector. The kinematic parameters of joints are of two types, fixedparameters and variables parameters and the kinematic model can be defined by providing theinformation about both types of kinematic parameters of the links.

Thus the kinematic model in general, describes the spatial motion of the end effector intime domain, about a fixed reference frame (or fixed platform). The kinematic study includes twotypes of models namely Forward Kinematic (FK) model and Inverse Kinematic (IK) model. Theforward kinematic model provides position and orientation of the end effector as a function ofvariable and constant kinematic parameters of all the links. Similarly the inverse kinematic modelis constructed to find the set of joint parameters that would bring the end effector in a desiredlocation in the workspace. The desired task of the end effector is specified in terms of its positionand orientation in the workspace. The joint variables to accomplish this task are found using IKanalysis. Joint variables, in turn are used to find the instantaneous coordinates of the endeffector using FK analysis.

It has been well established [] that for serial robots the inverse kinematic analysis isdifficult and provides more than one feasible solutions. However the forward kinematic analysis issimple and close form solution is possible. Conversely for parallel robots the IK solution exists inclose form but the FK solution is not possible. While doing a FK analysis for parallel robots oneends up with a set of highly coupled nonlinear equations for which unique solution is not possible.

Both types of kinematic models are normally required to study manipulator differentialmotion, its statics and to implement a desired control scheme for the end effector. Kinematicanalysis is also essential to estimate the feasible workspace of the robot and to perform asingularity analysis. A brief discussion on the kinematic modelling is presented in the followingsection.

6.1 Inverse Kinematic Analysis

As with other parallel mechanisms, the inverse kinematics of a cable-driven system isrelatively simple and provides a unique solution of cable lengths for given end effector pose. Theanalytic wire lengths have been determined in terms of the pose of the moving platform.

Figure 2. A sketch of cable and position vectors

of connectionpoints on FP and MP

YOF

bi

FOE,

O

ZO

Obi

XO

OPE

YE

XE

Eai

FP

MC

aiZE


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The diagram presented in Figure 2 shows the motion of the proposed SPM. The connection

points on MP and FP are denoted by ais and bis respectively. The connection points on fixedplatform are in the same plane (ZO= 0) and are placed at a radial distance rb from the coordinatesystem which is located at O. The position vectors of point bis on the FP are defined by

(1)

=0

sin

cos

ib

ib

iO r

r

b

where, rbis the radial distance from the base coordinate frame O. The variable i denotes the

angular position of point bion FP with respect to axis X0. The moving platform similarly has a setof connection points located at the corners of a rectangle and the coordinate frame attached toOE is about 60mm above the center of mass (MC) of the MP (with reference to the position ofankle joint which is approximately 60mm above moving plate level). The position vectors of thefour connection points on the moving platform ca be given as follows:

(2)

=

60

i

i

i

EY

X

a

Variable and are the coordinates of each connection point as shown in Figure 2. The

position vectors of the link lengths in terms of end poses can be expressed as a system of fourequations described below:

iX

iY

(3).4,...1=+= ibaRPL iO

i

E

E

O

E

O

i

O

where represents the position vector of point OEwith respect to O. is the rotational

transformation matrix of MP with respect to FP using a fixed axis rotation sequence of , and aboutX0, Y0, and Z0axes, respectively, and can be written as below.

E

OP EOR

(4)

++++

=

coscossincossin

cossinsinsincossinsinsincoscoscossincossincossinsinsinsincoscossincoscos

E

OR

6.2 Forward Kinematics

Forward kinematic (FK) mapping for parallel manipulators is difficult compared to serialmanipulators as it involves highly coupled nonlinear equations and their closed-form solutionderivation is a matter of interest to the researchers [55]. It is important to solve forward kinematicssince it is a key element in closed loop position and force control of parallel manipulators. It is anessential block in the trajectory control of a manipulator and hence accurate prediction of endpose from the available link lengths is essential. In the development of the proposed robot aSugeno based fuzzy logic model has been constructed to implement FK and it has been foundthat this model is significantly8 accurate and is also time efficient.

6.3 Geometric Modeling

The kinematic models establish the correlation between the joint displacements and theposition and orientation of end effector of a robot. This correlation can only be used for the staticcontrol of manipulator in the workspace. For a robotic manipulator the final desired position isimportant and at the same time the velocity by which it has traversed to reach to the final locationis also equally important. Thus it is essential to obtain a mapping between joint velocities and endeffector velocity. This mapping can be defined by a matrix, which is called the robot Jacobian


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matrix. The Jacobian matrix depends on robot configuration and linearly maps the cartesianvelocity in to joint velocities. It is interesting to note that the Jacobian matrix defined for theparallel robots corresponds to the inverse Jacobian of the serial robots. To determine theJacobian matrix of parallel robots two approaches, namely geometric approach and analyticalapproach can be used. In the present chapter we have used a geometric approach as discussedbelow:

To determine geometric Jacobian matrix using robot geometry, initially a relation betweenlink lengths and end effector pose is formulated. For the subject robot this relation is given byequation (3). Later on this relation is derived as described below to finally provide the Jacobianmatrix.

(5)tqJq )(=&It is evident from the equation (3) that link lengths can be easily calculated for a given set

of end effector orientation. The magnitude of each vector isiOL

(6)iOT

i

O2

i

O LLL =

Using the time derivative of the kinematic constraint equations, the relation between the cablevelocity vector q and the twist of MP i.e. t can be related by the Jacobian matrix as follows:

(7))()()()(2 iE

E

OT

i

O

i

E

E

O

ii

o

i

E

E

O

i

T

i

E

E

O

ii aRPbaRPbaRPaRPLL &&&&& +++++=

Using linear algebra identity, if c and d are column vectors

(8)TTT cddccdd 22c T ==+which implies following

(9))()(22 i

E

E

OOT

i

O

i

E

E

OO

ii aRPbaRPLL &&& ++=

POSince the end effector is constrained to only rotational motion the time derivative of shall

become zero. Setting PO & = 0 and further simplifying we get,

(10))()( iE

E

OT

i

O

ii aRLLL && =

but since

(11)EO

E

O

E

O RR =&

(12)()( iE

EO

E

OT

i

O

ii aRLLL =& )also

(13)iE

E

O

i

OaRa=

(14))()( iO

E

OT

i

O

ii aLLL =&

(15))()( iO

E

OT

i

O

ii aLLL = &

(16))()( EOT

i

O

i

O

ii LaLL =&


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if t is the twist vector it is defined as

(17)

=

E

Ot

0

equation 12 can be arranged and compared with

(18)YtqX =& where the inverse Jacobian can be defined as

(19)Jtq=&or

(20)YXJ = 1

Here, is the angular velocity vector of the MP with respect to frame O. The inverse

Jacobian matrix of the cable robot can be written such that its ithrow is

T

E

O )(

4,....,2,1=

= iwhereL

LaJ

T

i

i

O

i

O

i . (21)

6.4 Workspace Analysis

The workspace analysis of parallel robots is different from that of serial robots.Translational and orientation workspace of serial robots can be assigned to different sets of jointsand can be studied separately. However for the parallel robots both kinds of workspaces areachieved through coupled motion of its links or joints and thus the total workspace can not bedecoupled. For parallel mechanisms with rigid links, workspace is the space where the inverse orforward kinematic solutions exist. However for a cable based robot, its workspace is the spacewhere sets of positive cable tensions exist because they are needed to constrain the movingplatform all the time regardless of any external wrench. In other words, if there is at least one setof positive cable tensions at a specific pose forming a force closure, then this pose belongs to theforce-closure workspace. Generally, force-closure workspace is a set of poses that the force-

closure condition is satisfied.Thus in mechanisms, it is not only necessary to solve the closure equations but it is also

essential to verify that equilibrium can be achieved with non-negative actuator (cable) forces. Thedesign and workspace of a 66 cable-suspended parallel robot has been discussed in [70-75],and workspace volume is characterized as the set of points where the centroid of the movingplatform can reach with tensions in all suspension cables at a constant orientation. The maincontribution of [71] is in establishing that for any geometry of platforms the largest workspacevolume occurs when the moving platform (MP) is the same size as the base platform (BP).

Workspace evaluation is generally performed [73] in the context of rigidity, tensionabilityand force/torque capacity, these terms are explained below:

6.4.1 Rigidity

A is rigid at a given pose under applied external forces and spine force only if all the

cables are in tension. Since the rigidity of a depends on the external load and hence for rigidityevaluation, dynamic forces should be also considered. Thus, rigidity analysis requires the motion,inertia and all other externally applied forces apart from kinematic constraints to be considered.For simplicity, another property called tensionability can be considered which only depends onthe kinematics and at the same time fulfills the criteria of rigidity of the manipulator.

6.4.2 Tensionability

Tensionability is defined as an essential property of and a manipulator is said to betensionable provided that there exists a finite spine force when the external loads are applied. On


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a separate note tensionability and sufficient spine force together fulfills the criteria for rigidity.However the reverse is not true i.e. if a manipulator is rigid but not tensionable and is inequilibrium then this equilibrium may be an unstable one, meaning that a small change in thepose of end effector may result in instability due to loss of tensionability. The stability of can alsobe explained in terms of stiffness of the device which will be discussed in later sections.

6.4.3 Pre-tension

A positive spine force exerts a finite tension in the cables which is called Pre-tension. This helpsin keeping all the cables tensionable in the workspace. However a smaller pre-tension may notyield tensionability and at the same time, excessive pre-tension may result in instability of themanipulator. There exists an optimum value for this pre-tension in the cables for which themanipulator shall be rigid and tensionable. The proposed device basically has four actuated linksand accommodates patients ankle joint (which acts as a central strut) in the parallel device. Theair muscles which are the actuating links are all in their half contracted positions initially tofacilitate the antagonistic actuation of moving platform. The cables connecting both platforms aregiven some Pre-tension, by adjusting the cable lengths. Force experienced by the central strutdue to this pre-tension in the cables is called spine force and is denoted by FS here. This helps inkeeping all the cables tensionable in the workspace. The static force and moment balance on theMP are:

(22)04

1

=+=i iS UF

(23)

Though there are no external moments applied to the MP, due to ankle stiffness, finite

moments are required to move ankle joint passively. To realize the ROM exercises airmuscles works antagonistically and applies the required moments at the end effector. To

find corresponding tensions in individual cable the dual relationship between kinematics

and statics can be used as follows:

0)(4

1

=+ =

ext

i

ii

o MUm

(24)UJM Text=

[ ]Tzyxext MMMM 000= (25)

Where U is the vector of cable forces present in each cable, is the Geometric

Jacobian matrix of the robot and is a 3 dimensional vector containing the required moments

given by (5). The rows in Eq. (5) represent the moments required to orient ankle joint about the

directions, respectively. Now, to obtain the equations for the force in each of the

cable, Eq. (4) is rearranged in the following manner.

J

extM

000 &, zyx

extMJU= (26)

where

1)( = TJJ (27)

Next, at each point within the possible workspace, the equation describing the force ineach cable is used to see if tension is obtainable. A Matlab program is written to search the entireworkspace and check for the condition of tensionability of cables and link length constraint.


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A novel approach to check for the condition of tensionability has been used. Theproposed robot is redundantly actuated, i.e. to achieve three degrees of freedom four actuatorsare used. Initially, using equation 7 the force vector for all the cable to achieve flexion trajectory iscomputed. This force vector has two pull and two push forces, since cables can not push thesecompressive forces are superimposed to the two pull forces.Similar treatment is given to othertrajectories such as inversion-eversion and adduction-abduction. Now using force superpositionprinciple, force vector can be obtained for a trajectory which has all three simultaneous rotations.The pretension in the cables takes care that all the cables remain under tension at all the times.

0 50 100 150 200 250 300 350 400 450 5000

200

400

600

Sets of end effector orientation in the workspace

T1

0 50 100 150 200 250 300 350 400 450 500200

400

600

800

T2

0 50 100 150 200 250 300 350 400 450 5000

200

400

600

T3

0 50 100 150 200 250 300 350 400 450 5000

500

1000

T4

Forcesincables1,2,3and4(N)

Figure 3. Static cable forces at different end effector orientations

7. Design criterions of Cable based Ankle Rehabilitation Device (CARD)

In the light of unidirectional nature of the cable forces, the design of cable-basedmanipulator is more complicated than the rigid-link devices. There are certain criterias specific tothe cable based parallel robots which require more attention. These design criterion arediscussed below:

1. Maximum workspace criteria2. Near unity condition number criteria3. Singular value based criteria4. Minimum force norm based criteria5. Other criterias

An explanation of these measures is important to state their importance.

7.1 Workspace criterion

Workspace is vital parameter in the domains of kinematic analysis and should be givenproper attention. The feasible workspace volume depends on the geometrical configuration of therobot such as the size of the platforms and placement of connection points on them apart fromother constraints discussed in the previous section. Thus by changing the geometricalparameters it is possible to change the feasible workspace. It is desired that the workspace of arobot, under given constraints should be as large as possible for greater maneuverability.Further, unlike serial robots, workspace of parallel robots is unevenly shaped due to theircomplex kinematics. This further contributes in lowering the size of the feasible workspace. Apartfrom the size, the quality of the workspace is also important and it is desired that the workspace


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should be free from singularities. In the present study feasible workspace has been representedby an index as given below.

21

minmaxminmaxminmax ))()((

=

=

=

f

T

T

f

and

where

I

(28)

1 = workspace points satisfying tensionability criteria

2 = workspace points which are reachable with the restricted stroke length of the

actuator.In the proposed work robot design has been optimized to achieve maximum permissibleworkspace volume.

7.2 Condition Number criterion

As discussed in the previous sections, the Jacobian matrix of a robot maps joint rates

to the cartesian velocities of the manipulator. The condition number of this matrix is a measure ofits sensitivity to changes in the kinematic variables of the robot. A robot design with near unitycondition number is desirable since it minimizes the error in the end effector wrench due to inputerrors in joint torques. The condition number can also be used to evaluate the workspacesingularities. This number also reveals as how far a robot is from its present configuration to thenearest singular configuration. Stiffness of the end effector due to joint stiffnesses can also beobtained using condition number. Thus it is evident from the above discussion that the conditionnumber is a vital design parameter and the robot configuration should be optimally designed to

produce a minimum condition number close to unity. The condition number k is defined as the

ratio of the largest singular value

J

l to the smallest singular value s of the matrix for a

fixed orientation of the manipulator . The singular value can further be defined as the square root

of the eigenvalues of

J

TJJ JJTand . The range of condition number is described as below.


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( )

=

W

W

dw

dwk

GCN (31)

where k is the condition number at a given orientation and W is the feasible workspace. Since

it is difficult to calculate the exact solution to the integrals mentioned above, GCN is discretelydefined and expressed as

( )

W

k

GCN

n

i

== 1 (32)

where is the total number of n discrete feasible points constituting the workspace and the

numerator is the sum of condition numbers obtained at different points in the feasible workspacevolume grid. The GCN is bounded by the range

W

GCN1 (33)Here, when the GCN is a large number the entire workspace tends to be ill conditioned and whenthe GCN is near one the entire workspace is said to be well conditioned. GCN further depends on

the robot configuration which is defined by arrangements of connection points at both theplatforms and the link lengths. Hence there exists an optimum robot configuration for a good GCNand performance thereof,. To ensure that all the points in the workspace provides a conditionnumber within certain range, the maximum value of the condition number for a particular robotdesign over the entire workspace can be obtained and minimized. Once the maximum GCN isminimized, it can be ensured that1. The final GCN represents the average behavior of condition number over the feasibleworkspace.2. The condition number all over the feasible workspace is always less than the minimum GCNvalue.

7.3 Singular value based criterion

Singular values are important measure of kinematic behavior of the robot and provides

an assessment on its controllability. The manipulator loses or gains extra degrees of freedomwhen it enters in to a singular configuration. A kinematic singularity occurs when the determinantof the Jacobian matrix becomes zero in the workspace.

(34)0)det( =JReferring to equation (14), it is apparent that when J is singular and its null space is non-zero,

there will be certain non-zero cartesian vectors (t) resulting in zero joint vectors ( ). This further

means that, despite the joints being locked, the end effector can still have some infinitesimalmotion in a particular direction and gains one or more degrees of freedom. Hence it is essential tooptimize the design parameters of the robot, in order to minimize singular points in theworkspace. The condition number provides a fraction of maximum and minimum singular valuesand not the magnitude of these values in the workspace. Thus a criteria based on minimization of

maximum singular values shall yield a workspace free from singularities. However in the presentoptimization problem when GCN is minimized (which is the ratio of maximum singular value to theminimum singular value), the maximum singular value also get minimized.

q&


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7.4 Minimum actuator force based criteria

Since we intend to design a wearable robot for ankle joint rehabilitation treatment, it is

desired to keep the length of the robotic device similar to the length of the patients leg. Thelength of the robot is governed by the length of its actuators hence the actuators should be keptas small as possible. Further the size of these actuators depends on the cable forces calculatedin section 6.4. Longer air muscles are required for higher forces in the individual links. Thelengths of the actuators can be minimized by lowering the actuator force requirements. Onceagain it is apparent that the forces in the actuators are the function of robots geometricalparameters. By selecting connection points farther from the axis of rotation, the forces can begreatly reduces. To minimize the actuators force vector it is convenient to summarize the setvalues of force vector in a single number. Vector norms are generally used to represent thevector in a single value. Three types of vector norms are generally used such as, 1-norm, 2-normor -norm. 2-norm or euclidean length is preferred over the other two norms because it is moresensitive towards changes in larger force components. 1-norm is equally sensitive to all the forcecomponents and -norm is only sensitive to the changes in the largest force component. In the

present study, the higher forces in the actuators are required to be minimized to prevent thecables from breaking. Moreover, the higher actuator forces may also cause undesired cableelongation affecting the positional accuracy adversely. 2-Norm or the Euclidean distance of theactuator forces can be given as below:

UU = 02

(35)

7.5 Miscellaneous criterion

There are certain other design criteria which are specific to the application of the robot.Such criteria are size of the manipulator and the fixed platform, range of motions of the robot,material selection based on strength criteria, etc.

Despite their many advantages over serial robots, the parallel manipulators have someinnate problems owing to their close chain mechanism. A parallel robot basically has severalserial robots connected in parallel and its feasible workspace is the intersection of the individualworkspace contribution of its joints. Hence the feasible workspace of the parallel robots is small insize compared to serial robots. Moreover due to the complex kinematics of parallel robots, theshape of their workspace is quite irregular. Apart from the size and the shape, the workspace isalso affected by the kinematic singularities. The serial robots have only one kind of singularitywhereas the parallel robots due to their complex mechanism have three kinds of singularitiessuch as inverse kinematic singularity, direct kinematic singularity and the combined singularities.

The design of cable based parallel robots is again more challenging since they have anadded constraint on the workspace in the form of tensionability. The feasible workspace of theserobots is the euclidean space where the robot manipulator can reach with positive tension in all itscables. Thus apart from the singularities the condition of tensionability is also required to be

checked to evaluate a feasible workspace for the manipulator.The proposed robot is to be used for ankle joint rehabilitation treatments and the subjects

are supposed to fix their foot and the ankle into the moving platform. It is evident that the devicehas to be robust to take care of the physical differences in the shape and size of different patientsfoot and ankle. Differences in the foot of different subjects also amounts to the small variations inthe robot kinematic and dynamic parameters. Furthermore, when in actual use, the effectivelocation of the ankle joint in the robotic device also changes to some extent. This is due to thefact that the ankle joint is made up of two joints (as explained in section 3.1) and different anklemotions are realized by one of these joints.


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The kinematic and geometric parameter variations in the joint space results in thecorresponding cartesian parameters variations. Owing to this fact, the Jacobian matrix of therobot, which relates the joint and cartesian rates is required to be well conditioned. It is apparentthat if the Jacobian matrix is not well conditioned then the small changes in the joint variables willresult in very large changes in the cartesian variables. Which will further lead to difficulty inmanipulator control and large trajectory following errors. Nevertheless, by choosing optimalgeometric parameters of the robot, the condition number of the Jacobian matrix can be improvedand the design can be made robust to parametric variations.

Another problem with cable based parallel robots is that the cables can not be subjectedto very high forces. Higher forces in the cables may cause breakage or undesired elongation ofthe cable which will adversely affect the positional accuracy. Moreover to achieve higher forces,the required length of the air muscles also increases proportionally (as discussed in section 7.4)which is not desirable. Optimizing the geometrical parameters of the robot, the force requirementscan be minimized.

It can be concluded from the above discussion that the design of the robot is required tobe optimized to maximize its workspace, and minimize the condition number and the actuatorforce requirements. The dimensional analysis and optimization of the robot design may includethe key design parameters such as the shapes and sizes of the parallel platforms, locations of theconnection points of the cables on both these platforms and the lengths of cables. Such a designoptimization is essential to reduce the above mentioned shortcoming of the robot while

maintaining its inherent merits.

8. Design Optimization

Though the parallel robots have been extensively used recently in a wide range ofapplications, their potential has not been completely exploited. The reasons are obvious and havebeen discussed in the previous section. To address the issues of the workspace, singularity andthe robust design of parallel robots, normally trial and error approach is used. This approach isbased on rigorous experiments or simulation runs and intuitive judgments on the resultsthereafter. The main drawback of this approach is that with an increase in number of tunableparameters the required number of simulation runs increases exponentially. Moreover tuning ofall the design parameters simultaneously is difficult and time consuming. Some of theresearchers have tried to optimize one or more of the design objectives using some numerical

methods. Several performance indices such as manipulability, isotropy, dexterity index,conditioning index, global conditioning index and global isotropy index have been defined bydifferent researchers (Khatami & Sassani, 2002) to compare the performances of competingmanipulator designs.

In a recent study (Hassan & Khajepour, 2008) the actuator forces in a cable basedparallel manipulator have been optimized using a minimum norm solution. Author have usedDykstras projection method to optimally distribute the actuator forces among the cables and theredundant limbs. Though the force distribution among links have successfully been optimized toprovide minimum norm solution, the geometrical parameters have not been tuned to minimize theactuator forces. Genetic algorithms (GA) have been used by (Sergiu et al.,2006 ) to optimize theworkspace of a 2-DoF parallel robot using a mono-objective function. A novel kinematic designmethod has been implemented by (Liu, 2005) and various performance indices such as globalconditioning workspace, global conditioning index and global stiffness index have been used to

obtain the design parameters. A multi-criteria optimization based on conventional weightedaverage approach has been performed by (Lemay & Notash, 2003). The authors have proposeda combination of GA and simulated annealing algorithms to optimize workspace, dexterity andmass & size of the manipulator simultaneously. Workspace and stiffness of a modular parallelrobot have been studied and optimized by (Merlet, 2003). The author has proposed a branch andprune type algorithm to improve the robot performance. A new performance index called spaceutilizationhas been proposed by (Stock & Miller, 2003) to evaluate the optimal kinematic designof a linear Delta robot. Exhaustive search minimization method has been used to optimizemobility, workspace and manipulability. Global conditioning index has been optimized as a resultof altering the length of links by (Khatami & Sassani, 2002). The authors have used a nested


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implementation of two GAs to obtain a mini-max genetic solution. GA have been used withconstraints defined as penalty functions by (Su et al., 2001) to minimize the minimum conditionnumbers in the entire workspace. To validate and verify the algorithm, results obtained from GAhave further been compared with the Quasi-Newton method. Architectural optimization of a 3-dofparallel robot has been performed by (Tsai & Joshi, 2000) to maximize the global conditioningindex.

8.1 Mathematical Formulation

The optimization problem is formulated as below.

(1) Minimize Global Condition Number (GCN)

,()(),....,( 821 jiPiGCNqqqf += ) (36)

( )

=

W

W

dW

dWk

iGCN )( (37)

Here, qsare geometrical variables, i is the number of robot designs and , k is the

condition number at a given orientation and W is the feasible workspace. Further is

a penalty term defined as

Wj),( jiP

maxmin

max

)),((

)),((),(

=

jiJ

jiJjiP

(38)

minwhere the smallest singular value and max is defined as largest singular value of the

robots jacobian matrix .J

(2) Maximize the workspace volume

)])()([(),,,( minmaxminmaxminmax =if (39)

where ,, are roll, pitch and yaw angles for the moving platform

(3) Minimize Tension norm in the workspace

extMJU = (40)

where

1)(

= TJJ (41)

and

[ ]Tzyxext MMMM 000= (42)


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and U is the vector of cable forces present in each cable, is the Geometric Jacobian

matrix of the robot and is a 3 dimensional vector containing the required moments.

J

extM

8.2 Kinematic Constraints

8.2.1 Workspace Constrain ts

2,...,0

2,0,

2,0

85

3241

qqand

qqqq

(43)

2811 ,...., rr (44)

stroke length check for mthlink

(45)

maxmin

mmm lll

8.2.2 Link Interference Constrain ts

Link interference restriction between any two adjacent links

)(),( nmnm RRLLDist + (46)

8.2.3 Tensionability Constraints

Tensionability constraint

0 (47)mU

Checking maximum tension in a link

1max][ mU (48)

121, and are constants

81 ,...., rr are the radial distances of the connection points on both platforms from their

respective origins.

nm LandL are adjacent links connecting two parallel platforms.

nm RandR are the radius of cables of adjacent links.

is the vector of cable tension in linkmU

Here, qisare geometrical variables, i is the number of binary string in GA and . Both

moving and fixedplatforms provide eight geometrical variables (qis) as shown in the Figure 2.Sizes and separation of the two platforms have been kept constant thereby fixing r1, r2, and r3.Though the device is left-right symmetric but we have evaluated all the kinematic parameters

Wj


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independently to see whether the points are required to be placed symmetrically. It seems thatdue to the specific trajectory, the manipulator has to follow, q5and q8 are not symmetric.This is a multivariable constrained optimization problem where location of points at platforms orgeometrical parameters are varied to obtain a desired GCN. To solve such problemconventionally direct search or gradient decent methods are used but these techniques becomeless efficient when the search space is large and is finely discretized. Further these methodssometimes get trapped into a local minima and fail to provide a global optimum solution. It hasbeen observed that when the objective function does not change over certain points insuccession, these traditional methods become less effective. We propose to use a modifiedGenetic Algorithms in the present optimization problem. GA works with population of points andprocess them simultaneously and hence is more likely to give a global solution. As is evident fromFigure 4-6, that values of condition number are quite similar in the neighborhood of any givenpoint and the variation is very smooth. Thus discretizing the workspace does not lead to anyinformation loss and this fact further supports our choice of GA.

It has been observed [3] that GA is a global optimizer and converges to the optimumsolution very quickly but its local search capabilities are not equally good. Though it convergesvery fast to the possible solution but then to fine tune the solution it takes time and sometimesmight not converge to the desired accuracy. To overcome this problem we propose to use one ofthe gradient descent algorithms (for local optimization) in the selection operator of GA. In thepresent work after a fixed accuracy is achieved from GA cycles, the selection process shifts to the

gradient based selection to fine tune results.

8.2. Multi-criterion Optimization Techniques

8.2.1 The Conventional Weighted-Formula Approach: Transforming a Multi-objective Problem into aSingle-objective Problem

In the literature, by far the most used approach to cope with a multi-objective problemconsists of transforming it into a single-objective problem. This is typically done by assigning anumerical weight to each objective (evaluation criterion) and then combining the values of theweighted criteria into a single value by either adding or multiplying all the weighted criteria. Thatis, the quality Q of a given candidate model is typically given by one of the two kinds of formula:

Q = w1x c1+ w2x c2+ . . . + wnx cn (49)orQ = c1

W1x c2

W2x . . . x cn

Wn(50)

where wi, i=1,n, denotes the weight assigned to criteria ci, and n is the number ofevaluation criteria. Let us mention some examples of this approach in the context of the twomultiobjective scenarios discussed in the Introduction. The first scenario involves rule inductionalgorithms for classification, where it is common to evaluate the quality of a candidate rule bymeasuring two or more criteria. An example is:Q = w1xAccuracy + w2xcost (51)

An instance of the general formula structure (1) combining cost and accuracy into asingle measure of rule quality. This formula and its variations are used in several rule inductionalgorithms [Bruha & Tkadlec 2003], [Furnkranz & Flach 2003]. Another example is:

Q = costx w + accuracyx (1-w) (52)

An instance of the general formula structure (2) used in [Kaufmann & Michalski 1999] toproduce one of the evaluation criteria used in a lexicographic approach for rule evaluation. Thesecond scenario involves attribute selection algorithms for classification, where it is also commonto evaluate the quality of a candidate attribute subset by measuring two or more criteria. Anexample is:Q = x Acc + x (1 (S + F)/2) (53)

This formula was used by [Cherkauer & Shavlik 1996] to measure the quality of acandidate attribute subset in an attribute selection method following the wrapper approach, where


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Acc (accuracy) was measured by the validation-set accuracy of a decision tree built with theselected attributes, S was the decision tree size and F was the number of attributes (features) inthe candidate attribute subset. Other examples of attribute selection methods following thewrapper approach and combining two or more attribute subset quality criteria in a weightedformula can be found in [Liu & Motoda 1998].

8.2.2 The Lexicographi c Approach

The basic idea of this approach is to assign different priorities to different objectives, andthen focus on optimizing the objectives in their order of priority. Hence, when two or morecandidate models are compared with each other to choose the best one, the first thing to do is tocompare their performance measure for the highest-priority objective. If one candidate model issignificantly better than the other with respect to that objective, the former is chosen. Otherwisethe performance measure of the two candidate models is compared with respect to the secondobjective. Again, if one candidate model is significantly better than the other with respect to thatobjective, the former is chosen, otherwise the performance measure of the two candidate modelsis compared with respect to the third criterion. The process is repeated until one finds a clearwinner or until one has used all the criteria. In the latter case, if there was no clear winner, onecan simply select the model optimizing the highest-priority objective. A well-known algorithmusing this approach is the AQ18 rule induction algorithm [Kaufmann & Michalski 1999] and its[SIGKDD Explorations. Volume 6,Issue 2 - Page 78] variants. AQ18 uses a lexicographic

evaluation functional (LEF) defined by a sequence of pairs , whereeach ci, i=1,,n, represents the value of a performance criterion for a given candidate model,and each ti, i=1,,n, represents the tolerance associated with ci. This tolerance is specified by athreshold indicating the maximum value that the performance criterion ci for a given candidatemodel is allowed to deviate from the value of ci for the best current candidate model. Moreprecisely, let M1 and M2 be two candidate models being compared, and assume, without loss ofgenerality, that M1 has a better value of ci than M2. If the difference between M1s ci value andM2s ci value is greater than ti, then M1 is immediately considered better than M2, without theneed to check lowerpriority objectives. Otherwise the difference between M1 and M2 is notconsidered significant, and in order to select the best candidate model one has to use theremaining lower-priority objectives. In [Kaufmann & Michalski 1999] this approach is used with aLEF where a predictive accuracy-related measure (a combination of completeness andconsistency gain) is used as the highest-priority criterion (c1), and rule description simplicity as

the next criterion (c2); but of course other criteria could be used.

8.2.3 The Pareto Approach

The basic idea of the Pareto approach is that, instead of transforming a multi-objectiveproblem into a single-objective problem and then solving it by using a single-objective searchmethod, one should use a multi-objective algorithm to solve the original multi-objective problem.Intuitively, this approach makes sense. One should adapt the algorithm to the problem beingsolved, rather than the other way around. In any case, this intuition needs to be presented inmore formal terms, which is done in the following. Let us start with a definition of Paretodominance. A solution s1 is said to dominate (in the Pareto sense) a solution s2 if and only if s1 isstrictly better than s2 with respect to at least one of the criteria (objectives) being optimized ands1 is not worse than s2 with respect to all the criteria being optimized. Mathematically, assuming

without loss of generality that all criteria ci, i=1,k, are to be maximized, a solution s1dominates a solution s2

iff there exists ci such that s1(ci) > s2(ci) and for all ci, i=1,k, s1(ci) >= s2(ci),

where s1(ci) denotes the quality of solution s1 with respect to criteria ci and k is thenumber of criteria being optimized. A solution si is said to be non-dominated if and only if there isno solution sj that dominates si. Note that the Pareto approach never mixes different criteria into asingle formula all criteria are treated separately.


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B

A D

C

E F

Simplicity

accuracy

Figure: Pareto dominance illustration

For an understanding of the concept of Pareto dominance, let us say that the twoobjectives to be maximized are the accuracy and the simplicity of a model. In the figure, model Ais dominated by model B and by model D; model C is dominated by model D; and model E isdominated by model F. Models B, D and F are non-dominated solutions. They form the so-calledPareto front. Once Pareto dominance has been defined, the next step is to understand the crucialdifferences between a multi-objective algorithm based on Pareto dominance and a single-objective algorithm. There are two related crucial differences. First, there is a difference in thekind of output expected from each of these two kinds of algorithm. A multi-objective algorithmshould return to the user a set of non-dominated solutions, rather than just a single solution as ina single-objective algorithm. simplicity accuracy Second, and related to the first difference, thesearch performed by a multi-objective algorithm should explore a considerably wider area of thesearch space and keep track of all non-dominated solutions found so far, in order to find as many

solutions in the Pareto front as possible. Of course, this makes a Pareto-based multi-objectiveoptimization algorithm more complex than its single-objective counterpart. Some Pareto-basedmulti-objective optimization data mining algorithms are discussed in [Kim et al. 2000],[Bhattacharyya 2000], [Pappa et al. 2002].

9. Genetic Algorithm Methodology

The fitness function and parameter selection have been discussed here and for a detailedreading on GA [7,10] is recommended.

9.1 Generation

Initial population of 100 binary coded strings of 96 bits was generated using Knuths randomnumber generator [7]. Eight variables q1,q8 as shown below, one for each connection pointlocation have been defined. The binary string of 96 bits has 12 bits assigned for each of the eightvariables thus the solution accuracy of the order of 10

-05can be achieved.

32132132132132132132132187654321

1001..01010..10101..01010..11011..00110..11010..01101..1qqqqqqqq


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numbers and value of GCN and penalty term for each string (each string corresponds to a robotdesign) is evaluated from the feasible workspace to further find the fitness function.

9.2 Fitness Evaluation

In order to evaluate each string in the population, its fitness must be calculated. Generally GA isused for maximization problems and hence for present case of minimization of GCN a specialfitness function (16) has been used. The binary strings are decoded to real numbers and value ofGCN and penalty term for each string (each string corresponds to a robot design) is evaluatedfrom the feasible workspace to further determine the value of the fitness function.

9.3 Reproduction

The population initially selected may not have all the good strings in terms of their fitness values.Thus good strings have been analyzed and their multiple copies based on individual fitness havebeen selected in the mating pool. The Roulette-Wheel Selection method [3] has been used forreproduction. After an assumed accuracy is achieved and a near global optimal point has beenidentified the selection process shifts to the gradient based selection and multiple copies of binary

strings in proportion to their gradient values are stored in the mating pool. Thus strings withlarger gradients have more copies in the mating pool. This enables the algorithm to fine tune theglobal optimal point and perform local search more effectively.

9.4 Crossover and Mutation

A four point crossover with 0.95 probability has been used and mutation has been performed with0.01 probability. Generally crossover probability is kept close to one so that all the parent stringsmay get a chance to crossover. Mutation helps in finding a global optimum solution but to avoid arandom search its probability is kept low.

2

1

q5

q6q7

q8

r7

3

2

1q4

q3

A0

3

r3

r4

4q1

q2r2

r1

r5

r6

r8

Figure 2. Rectangular end platform and circular fixed

platform with their respective geometrical

arameters

4


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0 500 1000 1500 2000 2500 30002.2

2.4

2.6

2.8

3

Works ace oints

Co

nditionnumber

Figure 3: Condition number behavior over theworkspace

10. Results and discussions

After four initial GA iterations, GCN value is converged to 3.385 which is a near

optimal value and later on using modified GA, it is further reduced to 2.5572 in 18 iterations.Further reduction in GCN was not observed, possibly because five out of eight variablesapproached their limiting values.

0 5 10 15 20 25 300

2000

4000

6000

8000

Global Condition Number

2ndNormofCable

Forces



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05

1015

2025

0.7

0.8

0.9

1

0

2000

4000

6000

8000

globalCond

itionNumbe

r

FeasibleWorkspace

CableForc

eNorm


The geometrical parameters obtained were [36, 36, 4, 6, 75, 74, 75, 52.4] and as can beseen that except for q3, q4 and q8 all other variables are in their extremities. The feasibleworkspace was once again checked randomly with optimum robot configuration for conditionnumber. It was found (Figure 3) that the range of condition number variation in the entire feasibleworkspace was from 2.3322 to 2.9241, which is within permissible limits. This also shows that therobot configuration obtained is stable and has better manipulability in its feasible workspace forgiven range of orientations. The condition number was checked at each point in the workspace,which is represented by a set of three angles, however due to difficulty in plotting results withrespect to three orientations, results have been provided for three fixed yaw orientations ()shown in Figures 4-6. These results again confirms that the condition number is well within

desired limits and varies smoothly in the vicinity for the entire feasible workspace.It appears fromthe results that the optimum variables obtained are symmetric and hence they could be groupedtogether to reduce complexity.

Figure 5 . Condition number vs orientation at ( )0

20= Figure 6. Condition number vs orientation at 020=


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Figure 4. Condition number vs orientation at ( )0

0=

0 1000 2000 3000 4000 5000 6000 7000

0

5

10

15

20

25

30

Cable Force Norm

GlobalConditionN

umber



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0 1000 2000 3000 4000 5000 6000 70000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

PermissibleWorkspace

Cable Force Norm

Figure 4. Permissible workspace vs static cable forces

Table 1. Final kinematic design parameters

q1 44.7415 degq2 65.7147 degq3 59.9703 degq4 70.9323 degq5 41.6373 degq6 29.9708 degq7 32.7772 degq8 29.9708 degr1 90 mmr2 114.8 mm

r3 86 mmr4 94.5 mmr5 85 mmr6 104.5 mmr7 77 mmr8 86.4 mm


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11. CONCLUSIONS

A new soft parallel mechanism (SPM) for ankle joint rehabilitation has been proposed in thechapter, after carefully studying the complexities of human ankle joint and its motions. Theproposed device is an improvement in terms of simplicity, rigidity and payload performance. Theproposed device is very light in weight (total weight is less than 1 Kg excluding the weight ofsupport mechanism) and is quite inexpensive. The workspace study is carried out for future

applications of task space trajectory control. Main objective of this work was to optimize thekinematic design so that the performance of this robotic prototype can be enhanced. Anoptimization problem was formulated wherein the optimum GCN was obtained (constrained by apenalty term) by altering the configuration of connection points at the two parallel platforms.Genetic algorithm is slightly modified to perform local search efficiently and for this a gradientbased selection operator is used in place of Roulette-Wheel Selection after a near optimal valuefor GCN is obtained. GA finds the near optimum global solution very quickly and thereaftergradient descent selection operator further narrows down the result. Simulation run at 3000random points in the feasible workspace shows (Figure 3) that the resulting robot configuration isquite robust and variation as well as range of condition number is within acceptable limits. Since itis not possible to obtain GCN value equal to unity, the final values obtained for GCN and P (which are 2.5572 and 2.9241) are reasonably acceptable.

multi- criteria optimal design of cable based ankle rehabilitation robot

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