Design of a simple and modular 2-DOF ankle physiotherapy

Applied Bionics and Biomechanics 8 (2011) 101–114DOI 10.3233/ABB-2011-0010IOS Press

101

Design of a simple and modular 2-DOFankle physiotherapy device relying on ahybrid serial-parallel robotic architecture

Christos E. Syrseloudisa,∗, Ioannis Z. Emirisa, Theodore Lilasb and Artemis Maglaraa,c

aDepartment of Informatics and Telecommunications, University of Athens, Athens, GreecebDepartment of Shipping Trade and Transport, University of the Aegean, Chios, GreececReflexion LTD, Piraeus, Greece

Abstract. The aim of this work is to propose a new 2-DOF robotic platform with hybrid parallel-serial structure and to undertakeits parametric design so that it can follow the whole range of ankle related foot movements. This robot can serve as a human anklerehabilitation device. The existing ankle rehabilitation devices present typically one or more of the following shortcomings:redundancy, large size, or high cost, hence the need for a device that could offer simplicity, modularity, and low cost of constructionand maintenance. In addition, our targeted device must be safe during operation, disallow undesirable movements of the foot,while adaptable to any human foot. Our detailed study of foot kinematics has led us to a new hybrid architecture, which strikesa balance among all aforementioned goals. It consists of a passive serial kinematics chain with two adjustable screws so thatthe axes of the chain match the two main ankle-axes of typical feet. An active parallel chain, which consists of two prismaticactuators, provides the movement of the platform. Thus, the platform can follow the foot movements, thanks to the passivechain, and also possesses the advantages of parallel robots, including rigidity, high stiffness and force capabilities. The lack ofredundancy yields a simpler device with lower size and cost. The paper describes the kinematics modelling of the platform andanalyses the force and velocity transmission. The parametric design of the platform is carried out; our simulations confirm theplatform’s suitability for ankle rehabilitation.

Keywords: Ankle rehabilitation, serial robot, parallel robot, design, kinematics, simulation

1. Introduction

Typically, people with kinetic problems are requiredto work with a physiotherapist for a number of ses-sions. Ankle injuries are one of the most commonsuch phenomena. The aim of this work is to pro-pose a new 2-DOF robotic platform with hybridparallel-serial structure, to serve as a human anklerehabilitation device. The existing ankle rehabilitationdevices present typically one or more of the followingshortcomings: redundancy, large size, heavy weight, orhigh cost, hence the need for a device that could offer

∗Corresponding author. E-mail: [email protected].

simplicity, modularity, and low cost of constructionand maintenance.

Simple rehabilitation devices, found in almost everyphysiotherapist clinic, are meant for functional reha-bilitation. Such devices are elastic bands, foam rollersand wobble boards [21]. Elastic bands are the simplestdevices, each made of multi-shaped strips of elastic.Foam rollers act as unstable surfaces and are usedto improve balance and proprioception (one’s innerperception of own body status). Wobble boards arecircular discs with a hemispherical pivot in the centerof one of the sides [15] (Fig. 1). Unfortunately, thesedevices have several shortcomings.

Moving and exercising in the clinic is problematicfor patients with serious injuries or patients in remote

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102 C.E. Syrseloudis et al. / Design of a simple and modular 2-DOF ankle physiotherapy device

Fig. 1. A wobble-board rehabilitation exercise [15].

rural areas. Home exercising usually involves simplemechanical devices loaned to patients by clinics. Thesedevices lack quantitative diagnostic and networkingcapabilities that would allow therapists to remotelymonitor the patient’s progress. Also they are rarelyinteractive, making exercising repetitive and boring,since the patient must perform repeatable and identi-cal motions for a long time. This is also a repetitive,boring and time consuming task for the physiothera-pist. In addition none of the existing simple devicescover all requirements of the rehabilitation tasks.

As a result, the interest for an automatic mechanicaldevice that would provide physiotherapy exercises issignificant and is actually increasing. Robotics offerspowerful and secure methods for designing devicesfor physiotherapy exercises adapted to ankle injuries.Possessing tele-operation capabilities and remote datarecording will be useful to patients that have severeinjuries and cannot easily move, or live in remote ruralareas.

1.1. Previous work

There have been a number of robotic devices pro-posed for ankle physiotherapy. Important work has

been carried out at Rutgers University by Girone et al.[10] with the development of a haptic interface forhuman ankle rehabilitation (Fig. 2). This haptic inter-face has been based on a 6-DOF Stewart platform thatapplies variable forces and virtual reality exercises onthe patient’s foot, including remote control operation.It has been tested successfully in orthopaedic rehabili-tation, post-stroke rehabilitation, and rehabilitation ofmusculo-skeletal injuries.

However the Stewart platform has certain disadvan-tages. It is redundant for this application, since theankle has fewer degrees of freedom [3, 8]. Also, theactuators used are noisy, the controller is oversized andthe cost of the device is consequently high. In addition,in the rehabilitation program, there is no reference as towhat extent the special characteristics of each patient’sfoot can be considered.

The work of Dai et al. [5] is based on the study ofankle injuries, and ankle functional anatomy, which isrepresented in an orientation image space. A particulararea of this orientation space is selected to generate adesirable orientation range of motion for ankle rehabil-itation. Three parallel tripod-type ankle rehabilitationmechanisms were proposed and their mobility and con-straints were analyzed.

The stiffness analysis and mechanism synthesisstemming from ankle physiotherapy motion led to arobotic device for sprained ankle rehabilitation. Theseare three or four actuator platforms and therefore they

Fig. 2. The Rutgers Ankle haptic interface [10].

C.E. Syrseloudis et al. / Design of a simple and modular 2-DOF ankle physiotherapy device 103

are redundant. Also, the rotation of the moving plat-form is performed about a vertical pivot strut, which isnot a desirable characteristic for foot movements.

Yoon and Ryu [27] proposed an ankle rehabilitationdevice based on a reconfigurable parallel robot. It isa 4-DOF parallel robot with two moving platforms. Itenables all desired ankle and foot motions, includingtoe and heel raising as well as the traditional anklerotations. This is because the mechanism can gener-ate the relative rotation between the front and the rearplatforms as well as pitch and roll motions. To per-form each mode of range of motion, strengthening,and proprioception exercises, a unified position-basedimpedance control is developed taking into account thedesired position and velocity. However, this platformis quite complex and heavy and as a result is ratherdifficult in construction and transfer.

Recently, a 2-DOF redundantly actuated parallelmechanism for ankle rehabilitation was proposed bySaglia et al. [22]. The mechanism has the advantage ofmechanical and kinematics simplicity when comparedto existing multi-degree of freedom parallel mecha-nism prototypes, while at the same time it is fullycapable of carrying out the exercises required in anklerehabilitation protocols. The proposed device allowsplantar/dorsiflexion and inversion/eversion using actu-ation redundancy to eliminate singularity, and toimprove the workspace dexterity. However, this deviceis over-actuated, which means that there are redundantactuators. This also increases significantly the cost.

Syrseloudis et al. [23] analyzed the motions of thefoot according to the 2-axes ankle model [8] as wellas the shape of the workspace which it covers. Exper-iments on the foot motions of several healthy humansubjects were carried out by the use of a Mephisto 3Dscanner and an MTi motion sensor of the XSens MotionTechnologies. The workspace that the feet covers, aswell as the forces [14], velocities and accelerationsthat it achieves when it moves over the entire allowedregions have been studied in detail.

A parallel Tripod (3-RPS) [13] with an extra rota-tion axis on the moving platform as a possible anklerehabilitation device was studied by Syrseloudis andEmiris [24] (Fig. 3). The Tripod has two rotational(pitch, roll) and one translational (z) degrees of free-dom. As the yaw angle changes significantly duringthe foot movements on the platform, an extra rotationaxis was added on the moving platform to provide thenecessary extra yaw angle, since the Tripod’s origi-nal yaw is negligible. Although this device can follow

Fig. 3. Tripod based ankle rehabilitation device.

the foot movements very satisfactorily, it is not simpleenough for our purposes, hence our effort to design arobot with fewer DOFs.

There are more general devices offering rehabil-itation. One example is by the TELEDOC project[2], where they developed a complete computerisedhome rehabilitation system for the upper arm. Lastly,Toth and Ermolaev [26] have used standard, full scaleindustrial serial robots for the physiotherapy of spastichemiparetic stroke patients. Culmer et al. [4] proposeda 3-DOF serial robot and a dual serial robot for therehabilitation of the upper limb. This robotic structureis irregular and its stiffness capabilities are insufficientfor the foot.

1.2. Our contribution

Existing robotic mechanisms for ankle rehabilitationare typically redundant, oversized, too heavy, or quiteexpensive. This paper addresses the need for a designemphasizing the following features:

• adaptability to all human feet,• simplicity, including a limited number of actua-

tors leading to a smaller size and weight, low costof construction and maintenance,

• mechanical adaptability and modularity for easydisassembling and transferring,

• safety, especially with respect to prohibited move-ments that can cause injuries to the human user.

For the rehabilitation task, the patient needs to sit ona bench while his shank is vertically fixed. Therefore,


the foot must move while exercising only the ankle.The rehabilitation robot should be able to perform deli-cate motions so that it will not hurt the patient. Howeveroccasional load from the patient, like in the case ofstepping onto the device, can exert much more loadthan the limiting forces. Therefore an over-design isrequired for the robot actuators which should be com-bined with additional safety force sensors in order toachieve safe operation. This issue would make a com-mon serial robot unsuitable for the required operationand would make a parallel robot oversized.

Based on the biomechanics literature, in particularthe description of the kinematics structure of the ankle,we adopted the standard 2-axes ankle model [8], there-fore leading us to a 2-DOF design. To fully respect thekinematics structure of the ankle, and obtain a simpledevice with two DOFs, we should add modularity viasome mechanically adjustable capabilities.

These requirements led us to a new architectureof a hybrid serial-parallel robotic platform that canapply ankle rehabilitation exercises while respectingthe aforementioned features. The robotic device iscomposed of a passive serial kinematics chain with twoadjustable lengths and two prismatic actuators, whichwork in parallel, providing the necessary movementsto the platform. The structure of the robot has beenbased on our study of foot kinematics as describedin Syrseloudis et al. [23]. Our objective was thatthe moving platform follows precisely the allowablemovements of the foot without redundant characteris-tics. This is achieved with the assistance of two screwsand in this way the device becomes mechanicallyadaptive to the two main ankle axes. This mechani-cal enhancement led to the elimination of additionalactuators.

The proposed design decomposes the serial andparallel robots phenomena in a way that the passivemechanical structure accepts the high loads and theactuators are matched only for the patient exercise.Previous designs trying to decompose the above issuesuse a spherical joint [5]. This approach decouplesaccidental forces from treatment torques hence a newproblem arises. The machine axes of rotation are off-set from the ankle axes of rotation therefore causingundesired movements to the patient. In the proposeddesign we manage to decouple accidental forces fromtreatment torque; we also match patients ankle rotationaxes. Therefore treatment suitable devices can be a sixdegree Stewart platform and the proposed hybrid serialparallel design.

Load comparison on actuators shows the benefits ofour design namely: intrinsically safe operation, smalleractuators and two instead of six actuators. This leadsto reduced cost, weight and also to simplicity. Com-pared to a possible software adjustment for the Stewartplatform, the main difference of our hybrid design isthe manual adjustments by the therapist, before patienttreatment. Thus our design is safer because it avoidswrong software input parameters.

Our 2-DOF hybrid robotic device possesses the min-imal number of actuators and small controller size,consequently decreasing the construction cost. Thepassive serial chain allows the parallel actuators towork almost exactly like the two DOFs of the humanfoot; it is possible to adapt to any patient’s foot mechan-ically before the session starts. Hence there is no needof very sophisticated task planning software and con-trol during operation. This device is more secure, forexample than a Stewart platform, because the Stew-art platform may perform non-allowable movements.Lastly, the proposed device can be easily transportedand can operate at a remote-rural area.

The main objective of this paper is the parametricdesign of the new robotic platform which is basedon the analysis and resulting specifications fromSyrseloudis et al. [23], while additional measurementsof specific points that define the two ankle axes of sev-eral feet have been carried out. These are used for thedefinition of the bounds of the two adjustable lengthsof the robot. The robot has been, finally, simulatedin MATLAB, where its kinematics behavior is graph-ically represented, thus confirming its usefulness inrehabilitation.

This paper has the following structure: first the kine-matics of the foot is briefly introduced and Section 2.In Section 3, the architecture and the kinematics ofthe new hybrid serial-parallel robotic platform is ana-lyzed and in Section 4, the parametric design of theplatform is carried out. In Section 5 specific motionsof the platform are graphically represented via simula-tions. We conclude with the suitability of the proposedarchitecture as an ankle rehabilitation device.

2. Foot kinematics

The structure and kinematics of the human foot aredescribed in this section. The human ankle has a com-plex multi-joint structure [3, 8]. The central bone is thetalus. Its surrounding bones are the calcaneus, the nav-


icular and the cuboid; they are responsible. The upperpart of the talus articulates with the shank segmentthrough the tibia and fibula bones. This is the upperankle joint (denoted UAJ). It supports the rotationaldorsiflexion/plantarflexion motion. The movementsbetween the fore bones are strictly coupled for therotation of the ankle joint in 3-dimensional space.

Motion of the foot with respect to the talus isregarded as a rotation about the (fixed) subtalar joint(denoted STJ); this supports the rotation called inver-sion/eversion. The main rotations of the foot aredepicted in Fig. 4 and the two axes of the ankle inFig. 5.

The UAJ is assumed to pass through the specificpoints P3 (lateral malleolus) and P4 (medial malleolus)while the STJ passes through the points P6 (calcaneuspoint) and P7 (navicular point), see Fig. 5 [8]. Thelower limb is assumed to be composed of three rigidlinks capable to rotate between each other: the shank,the talus and the foot configuring a serial manipulatordescribed in Dul and Johnson [8] and in Syrseloudiset al. [23]. The size of the foot bones, their relativepositions as well as the orientation of the rotationaxes, jointly determine the foot kinematics. Many fac-tors influence the joint rotation, e.g. the shape of thearticular surfaces, and the position of rotation axes.Constraint and resistance on the foot motions are dueto ligaments, capsules and tendons.

A serial manipulator equivalent to the foot isobtained as follows. The shank-link connects the cen-ter K1 of the P1P2 segment and the center K2 of theP3P4 segment. The talus-link connects the center K2of the P3P4 segment and the center K3 of the P6P7 seg-

ment. Finally, the foot-link connects the center K3 ofthe P6P7 segment with point P8, which is on the fifthmetatarsal. The knee-axis which is defined by P1–P2,is assumed to be fixed. The rotational Upper AnkleAxis and the Subtalar Axis are defined by the P3–P4and P6–P7 axes, respectively. The main rotations ofthe foot about the ankle are defined according to theright-hand rule in Fig. 6. In particular, plantarflexionis the negative and dorsiflexion is the positive rotationaround the UAJ (Z2) axis. Similarly, inversion is thenegative and eversion is the positive rotation aroundthe STJ (Z3) axis.

The foot kinematics can be approximated by termsof serial manipulator kinematics. We use homogeneousmatrix transformations according to the Denavit-Hartenberg notation, denoted DH [6]. The assignedrelative frames Oi between the moving links are shownin Fig. 6. Ti

i+1 is the transformation matrix from Oi+1into Oi defined as follows:

T i+1i =

⎡⎢⎣

c(θi) −c(α)s(θi) s(αi)s(θi) aic(θi)s(θi) c(αi)c(θi) s(αi)c(θi) ais(θi)

0 s(ϑ) c(ϑ) di

0 0 0 1

⎤⎥⎦(1)

where c(*) = cos(*), s(*) = sin(*) and i = 1, . . . , 4.The transformation matrix from the last into the first

coordinate system is given by the relation:

T 41 = T 2

1 T 32 T 4

3 . (2)

The last coordinate system is that of the foot, systemO4X4Y4Z4 on Fig. 6. For a point P = [x y z 1]T on thissystem, the above transformation into the first (shank)

Fig. 4. Main rotations of the foot about the two axes of the ankle.


Fig. 5. Main rotation axes of the ankle.

Fig. 6. Serial kinematics chain of the foot and D-H frames assign-ment.

coordinate system can be expressed as P01 = [x1 y1 z11]T where

P01 = T 41 P. (3)

These equations give a parametric formula of theD-H parameters in the movement of P with respect

to the fixed coordinate system of the shank. The D-Hparameters are defined as follows, by referring toFig. 6:

• αi is the twist angle between the Zi, Zi–1 axes,• ai is the length of the common normal to the Zi,

Zi–1 axes,• di is the offset between the common normals ai

and ai–1,• θi is the rotation angle between Xi, Xi−1.

The independent variables of the model are theangles θ2 (dorsiflexion/plantarflexion), and θ3 (inver-sion/eversion), whereas θ1 is constant. The zeroconfiguration of the equivalent serial manipulator istaken to be when the foot is in the erect standingpose. The values of the variable angles about the zeroconfiguration are taken to lie in the following ranges:−40◦ ≤ θ ≤ 25◦ and −20◦ ≤ θ3 ≤ 20◦ [19, 23]. Thisanalysis concerns the right leg while movements of theleft leg are assumed to be the mirror-image of the rightleg [8]. The parameters αi, ai, di depend on the footanatomy and size.

In Dul and Johnson [8] the transformation matricesexpressed in Euler angles were estimated for a malesubject. Standard instruments were used to measurethe distances between the bony landmarks. After thecalculation of several internal distances using the trian-gulation technique, the redundant distance method wasused for the calculation of the transformation matri-ces between the foot and the talus, and between thetalus and the shank frames. From these data, a kine-matics model of the foot was based on homogeneousmatrix transformations in Euler angles. We used theMaple computer algebra software (version 9.5) to per-form (partly symbolic) calculations on the distances,and obtained the desirable D-H parameters [23].

Taking into account the previously mentionedmotions, it might seem conceivable that a serial robotwould be able to meet the requirements [4, 26]. How-ever, industrial serial robots are huge and a 3-DOFserial robot with structure similar to that of the foot,and having a platform-shape end-effector, has someserious drawbacks. The size of the serial chain mustbe quite small and actuators should be mounted on thejoints. This makes the robotic structure irregular withinsufficient stiffness capabilities.

To overcome this problem, we propose a 2-DOFparallel robot with a passive serial kinematics chain,which constraints the movements (Fig. 7) and a parallelchain, which provides the movements. The objective


Fig. 7. The 2-DOF hybrid parallel-serial robotic platform for ankle rehabilitation.

is to incorporate the advantages of parallel robots asthey have rigidity, high manipulability and heavy loadshandling.

3. A 2-DOF ankle rehabilitation platform

This section discusses the proposed hybrid serial-parallel robotic architecture and its kinematicscharacteristics.

The robot consists of a base platform and a movingplatform like most parallel robots. The latter is wherethe patient’s foot shall be placed. A vertical strut con-nects the base of the robot with a passive serial chain.The serial chain has structure similar to that of the footand provides the necessary constraints on the move-ments. It has one revolute and one cylindrical jointwhich support the rotations about the two main rota-tion axes of the ankle, see Fig. 7. R1 is a revolute-jointwhich is collinear with the Upper Ankle Joint (UAJ)and C1 is a cylindrical-joint which is collinear withthe Subtalar Joint (STJ) of the foot. The serial chainis connected with the moving platform and has twoadjustable screws so that the corresponding lengths D1,D2 can be adjusted according to the axes position ofeach individual patient’s foot. The parallel chain con-

sists of two prismatic actuators which are connectedwith the moving platform through S-Joints (points A1,A2) and with the base platform through U-Joints (pointsB1, B2).

3.1. Kinematics modeling of the platform

We start by modeling the kinematics of our device.The mobility of the platform is modeled by applyingthe Grubler formula for spatial structures [17]. Thetotal number of its degrees of freedom N is given asfollows:

N =6(n − j − 1)+∑

fi =6(8 − 9 − 1)+14=2.

(4)

where n represents the total number of rigid bodies ofthe mechanism, including the base, j is the total numberof joints, and fi the number of degrees of freedom ofjoint i. Of course the summation is understood over alljoints.

Initially we assign the base coordinate frameObxbybzb on the fixed base and the moving frameOpxpypzp on the moving platform as shown in Fig. 7.The two frames are parallel when the moving plat-form is in zero position. As we have a serial kinematics


chain it is useful to implement the Denavit-Hartenbergmethod [6] for the assignment of the relative refer-ence frames on the passive serial chain and therefore toobtain the overall kinematics formula of the platform,see Equation (1).

The Ooxoyozo frame is the base frame of the serialchain and is placed arbitrarily on the strut with the axiszo collinear with the UAJ axis, and xo collinear withzb, as shown in Fig. 7. The origin O2 of the O2x2y2z2frame is the Op of the platform frame and the axis z2 isparallel with z1 (C1). The total transformation matrixT is given by the multiplication:

T = TbT10 T 2

0 Tr. (5)

where Tb is a constant homogeneous rotation matrixdefining the relative rotation of the Ooxoyozo frame intothe base Obxbybzb frame. The points on the platformcoordinate frame need to be multiplied with an extrarotation matrix Tr in order to be transformed into thelast D-H frame of the serial chain. The D-H parameterscan be computed on the host computer in which theorientation and position of the two axes of rotation aregiven as inputs. Consequently the inverse kinematicproblem is described from the following two equations:

L21 = ‖B1A1‖2 = ‖T · A1 − B1‖2, (6)

L22 = ‖B2A2‖2 = ‖T · A2 − B2‖2. (7)

where L1, L2 are the length of the actuated links.It is well known that the Jacobian matrix is a key

part for the study and design of robots. It appears inthe equations which describe the relation between thevelocities, accelerations and forces of the end-effectorand the actuated joints. Furthermore, it is used for theallocation of singular configurations. In our case, thevelocities of the actuators and the moving platform areconnected through the equation:[

L1L2

]= J−1 [ νx νy νz ωx ωy ωz ]T . (8)

where J−1 is a 2 × 6 pose-dependent inverse Jacobianmatrix, νx, νy, νz are the linear velocities and ωx, ωy, ωz

the angular velocities of the end-effector. By followingthe normalized Plucker vector-based procedure for theinverse Jacobian calculation of a general parallel robotas it is described in detail in Merlet [17], we get:

J−1 =[

n1 n1 × A1Op

n2 n2 × A2Op

]. (9)

where n1, n2 are the unit vectors of B1A1, B2A2.

3.2. Velocity and force transmission

When the linear actuators are activated their veloc-ities and applying forces are transferred onto themoving platform. The relations between the forces-velocities of the actuators and the moving platformare expressed by the Jacobian matrix and are pose-dependent. The wrench force vector F and the actuatorforce vector τ are related in the following way:

F = J−T τ. (10)

Also the relation between the linear velocities L

of the actuators and the vector of linear and angularvelocities X of the moving platform is:

L = J−1X. (11)

Given the desired velocities as well as the force andmoments that should be handled by the platform, theextreme values of the velocities and forces of the linearactuators are computed in the design phase. Here wefollow the kinetostatic capability analysis of a parallelmanipulators described in Kim and Choi [12] where thecalculation of the magnitude bounds of the force andvelocities of the end-effector is reduced to an eigen-value problem. Matrix M is essential here and is definedas follows:

M = J−T J−1 =[

A B

BT C

]. (12)

The linear forces and torques have different unitsit is reasonable for the bounds computation of force-torques at the end-effector to be decoupled into twoconstraint maximization subproblems, one for theforces f and one for the torques m. By use of matrix Mthe two maximization subproblems can be rearrangedinto the following eigenvalue subproblems:

A · f = af2 · f, (13)

C · m = am2 · m. (14)

where A, C are the 3 × 3 submatrices of M and f, m the3 × 1 vectors of linear forces and torques on the end-effector. By Equation (13), a three-dimensional forcetransmission ellipsoid is defined which has principalaxes given by f and radii given by af . In the same way,Equation (14) defines a torque transmission ellipsoidwith principal axes defined by m, and radii defined by


am. The bounds of force and torques magnitudes arenow given by the inequalities:

‖f‖af max

≤ ‖τ‖ ≤ ‖f‖af min

, (15)

‖m‖ammax

≤ ‖τ‖ ≤ ‖m‖ammin

. (16)

where ‖ · ‖ is the Euclidean norm, afmax and afmindenote the square roots of the maximum and minimumeigenvalues of A, and ammax and ammin are those ofthe maximum and minimum eigenvalues of C. In theabove, τ is the 2 × 1 forces vector applied by the linearactuators.

Following a similar procedure as in the force trans-mission analysis, velocity transmission analysis isdecoupled in two subproblems, one for the linear ν

and one for the angular ω velocities magnitudes.

A · ν = aν2 · ν, (17)

C · ω = aω2 · ω. (18)

where A, C are the 3 × 3 submatrices of M and ν, ω

the 3 × 1 vectors of linear and angular velocities of theend-effector. The bounds of the velocities magnitudesare therefore given from the inequalities

aνmin‖ν‖ ≤ ‖L‖ ≤ aνmax‖ν‖, (19)

aω min‖ω‖ ≤ ‖L‖ ≤ aω max‖ω‖. (20)

where aν max and aν min denote the square roots ofthe maximum and minimum eigenvalues of A, andaω max and aω min are those of the maximum and mini-mum eigenvalues of C. L is the 2 × 1 vector of linearvelocities applied by the actuators. Matrix M is posedependent and concerns one specific configuration. Bydiscretizing the whole workspace of the robot and com-puting the global extreme eigenvalues of M, the globalmagnitude bounds of velocities and forces are com-puted.

4. Parametric design of the robot

This section presents the parametric design of ourmechanism, since we have chosen the robotic archi-tecture.

Design concerns the calculation of the geometricparameters of the robot satisfying our requirements[16, 17]. These requirements are, for example, theworkspace covered by the end-effector, the desired

velocities and accelerations, and the specification ofthe force capabilities for load handling.

Based on the above foot analysis, the following val-ues were selected for initial dimensioning of the device:

• Moving platform: 0.40 × 0.20 m so that it canaccept all or at least the majority of human footsizes.

• Base platform: 0.60 × 0.40 m. The first axis zo isplaced 50 cm above the fixed base.

The bounds of rotation axes of the serial chainwere defined according to the range of the feet rota-tions described in Nigg et al. [19] and Syrseloudiset al. [23] and so they are: −40◦ ≤ z0 ≤ 20◦, and−20◦ ≤ z1 ≤ 20◦. The STJ axis, and so the z1 has themean foot orientation as it is given in Isman and Inman[11] and forms an angle of 23◦ with the xpzp plane andan angle of 41◦ with the xpyp plane.

The design of the platform was based on the designframework described in Syrseloudis et al. [23]. Thiswork has been extended and in order to complete thedesign of the robot, additional measurements on thefoot of several human subjects have been conducted.Coordinates of specific points of the foot, shown inFig. 5, have been measured, utilizing a Microscribecoordinate measuring device (Fig. 8) [18]. For theexperimental measurements the right feet of 19 adultmales and females have been used in the erect standingpose.

The UAJ is defined by points P3 (lateral malleolus)and P4 (medial malleolus), while the STJ is defined bypoints P6 (calcaneus point) and P7 (navicular point)(Fig. 5). For calculating the bounds on distance D1 thepoints P3, P4 and P6 were projected on the horizontalplane. The vertical distance of the projected point P6from the projected line P3P4 defines distance D1. Thecomputed values of D1 were found to be in the range:

0.035 m ≤ D1 ≤ 0.056 m. (21)

with mean value 0.0483 m and standard deviation0.0068 m. For calculating the range of distance D2, themean value of the height of points P3 and P4 from thehorizontal plane was computed. The resulting valuesare in the following range:

0.054 m ≤ D2 ≤ 0.093 m. (22)

with mean value 0.0729 m and standard deviation0.0102 m. Figure 9 depicts graphically the values ofthe computed distances D1, D2 per human subject.


Fig. 8. The microscribe 3-dimensional coordinate measurementdevice.

Points A1, A2 on the moving platform have beenassigned coordinates (0.15, – 0.06, 0) and (0.15, 0.06,0) of the moving frame, respectively. If the pointsare far from the rotation axis then the actuators applysmaller forces but larger velocities. Reversely, if thepoints are near the rotation axis then the actuatorsapply forces with larger values and smaller velocities.Therefore, the points are selected to be in the middle

of the platform in order to balance the amounts of thevelocities and forces exerted by the actuators. The coor-dinates of the base platform points B1, B2 have beenassigned to (0.10, –15, 0) and (0.10, 0.15, 0) respec-tively, on the base reference frame nearer to the originso to avoid singularities. The coordinate units are in m.When the platform moves through the entire range ofrotations and parameters D1, D2 take all values in theabove intervals, then the length of the legs are foundin the range: (0.31 m, 0.56 m).

Having computed the kinematics parameters of therobot as well as the desired end-effector velocities andwrench forces, the actuator velocities and forces canbe computed. The values of these parameters are trans-ferred to the platform and depend on the geometriccharacteristics of the platform through the inverse Jaco-bian. The platform must handle torque values up to200 Nm as described in Syrseloudis et al. [23]. Theresult is coming from Maganaris et al. [14] in whichthey studied the tension torque that the soleus and tib-ialis anterior muscles can exert. These are two of themain dorsiflexor-plantaflexor muscles and the maxi-mum measured torque was about 121 Nm. In Fukunagaet al. [9] the maximum measured torques of the wholeplantarflexor and dorsiflexor muscle groups were about143 Nm. Therefore a desired upper bound of 200 Nmincludes a wide range of foot torque capabilities. Inorder for the platform to achieve these torque bounds,

Fig. 9. Foot distances D1 (black bars), and D2 (white bars), measured for 19 human subjects.


the actuator forces must be greater than 675 N accord-ing to Equation (16).

Similarly, for velocities calculation used the upperbounds in angular velocities of the platform studiedin Syrseloudis et al. [23]. An MTi motion sensor wasused for the measurement of the foot angular veloc-ities among several human subjects. The MTi sensorwas fastened on the sole of the foot under the ankleand angular velocity in the following three move-ments were measured: dorsiflexion/plantarflexion,inversion/eversion and full rotations of the foot. Themaximum recorded angular velocity was 9.3 rad/secand therefore the upper bound of 10 rad/sec is adopted.According to Equation (20), the linear actuators shouldachieve velocities at least 2.1m/sec in order to reach thebound of 10 rad/sec.

Now the design of the robot is completed and fromthe dimensioning of the robot its individual parts canbe mechanically designed and prepared. Also from theactuators force-velocity bounds the motor characteris-tics providing the robot motion can be selected.

5. Design evaluation of the robot

This section evaluates the design of the proposeddevice. In order to study the robot better and for visu-alization purposes, a simulation has been developedin a Matlab GUI. The workspace shows that the plat-form movements match with the workspace of the foot

because rotates about the two main rotation axes ofthe ankle and therefore the resulting movements are inagreement with that of the foot. The platform in twodifferent poses is depicted in Fig. 10.

Next, the force transmission from the actuators tothe moving platform was graphically representedaccording to Equation 10. We assumed that the linearactuators apply the maximum linear forces of 675 Nproducing a pure dorsiflexion rotation. The torquevector of the moving platform with respect to the dor-siflexion angle is depicted in Fig. 11, while the unitsof the torques are in Nm.

In the second step, we assumed that the linearactuators apply the maximum linear forces of 675 Nperforming a pure eversion rotation. The torque vec-tor of the moving platform with respect to the eversionangle is depicted in Fig. 12. The units of the vectorare the same as before. It is obvious that the resultingtorques are smaller than 200 Nm satisfying the designrequirements.

For the study of the velocity transmission of therobot, the platform was rotated with the maximumangular velocity of 10 rad/sec performing a pure dor-siflexion. The linear velocities of the actuators withrespect to the dorsiflexion angle are graphically calcu-lated (Equation 11) in Fig. 13a, b respectively. Actuator1 refers to B1A1 and actuator 2 refers to B2A2.

Similarly the platform was assumed to rotate withthe same maximum angular velocity, but in a pure ever-sion rotation. The corresponding linear velocities of the

70

60

50

40

30

20

10

050

4030

2010

-10-20

-30-40

-50 -50 -40 -30 -20-10 0 10 20 30

3020

100-10

-20-30

-40-50 -50 -40 -30 -20 -10 0 10 20 30 40 5040

40

50

50

70

60

50

40

30

20

10

0

0

Fig. 10. The robotic platform in 2 different poses.


Dorsiflexion angle [deg]

Tor

que

[Nm

]

Fig. 11. Tx, Ty: lower curve, Tz: upper curve.

actuators with respect to the eversion angle are nowdepicted in Fig. 14a, b. Linear velocities are expressedin m/sec and angular velocities in rad/sec. It is clear thatthe resulting velocities are smaller than the required2.1 m/sec.

The simulation show that the robot can follow themovements of the foot about the two main axes of theankle. Finally, the graphically representation of force-torques and velocities values of the robot are found to

Eversion angle [deg]

Tor

que

[Nm

]

Fig. 12. Tx,Tz,Ty: top (on the left), middle and bottom curve, resp.

lie within the bounds specified by the design require-ments, evaluating the design of section 4.

6. Conclusion

Let us conclude with a summary of results and futurework.

(a)

(b)

Actuator 1 Velocity


Vel

ocit

y[m

/s]

Actuator 2 Velocity


Vel

ocit

y[m

/s]

Fig. 13. Actuators velocity in a pure dorsiflexion rotation.


(a)

(b)

Actuator 1 Velocity


Vel

ocit

y[m

/s]

Actuator 2 Velocity


Vel

ocit

y[m

/s]

Fig. 14. Actuators velocity in a pure eversion rotation.

In this paper a new 2-DOF hybrid serial-parallelrobot was presented. The proposed robotic device canfollow the movements of the foot with respect to theankle and, consequently, can be used as a rehabilita-tion device for ankle injuries treatment. The structureof the robotic platform has been decided after thedetailed study of foot kinematics. The main objectivewas to overcome the drawbacks of the existing deviceswhich are typically redundant, oversized, heavy, orexpensive.

The resulting robotic structure is a platform withthe minimal number of actuators so as to minimizethe size and cost. The hybrid serial-parallel designallows the parallel actuators to work almost exactlylike the 2 DOF’s of the human foot. The addition oftwo extra adjustable screws enable the platform to beadjusted according to the characteristics of each indi-vidual patient’s foot. The adjustment is performed priorto the rehabilitation session.

The above result in the following advantages: thetask planning software and the kinematics control dur-ing operation are simplified. The device is considerablysafer in comparison to a redundant platform because,in the latter case, the second platform may perform

a non-allowed movement. Finally, design of the plat-form has been carried out according to the foot motionsrequirements.

In the future, the mechanical design, the actuatorselections, the development of the human robot inter-face will be performed. Extensive stiffness analysisaccording to finite element analysis (e.g. ANSYS soft-ware package (ANSYS)) or virtual joint method (VJM)[20] will be implemented for the study of the robot’sbehavior in loading conditions. This analysis is nec-essary for the kind and thickness of materials whichwill be used, mainly for the strut and the passive serialchain which sustains the main torque amounts. Also,the serial kinematics chain can become more easilyadjustable to the kinematics characteristics of typicalfeet. The dynamic modeling of the platform is alsoa subject for research. Compliant control algorithmssuch as force control, impedance control etc., which areappropriate for the rehabilitation task, will be studiedin detail based on special injury treatment. The con-struction of the platform and its use in rehabilitationexercises on actual patients is our final task.

In parallel, an identification method of foot kinemat-ics parameters is being developed, which estimates the


foot axes and kinematics parameters [25], by recordingfoot movements with a coordinate measuring device,and with no use of any internal angle measurement,extending the ideas from [7]. Identification of the ankleaxes and foot kinematics parameters is useful in therehabilitation exercise. In this way, a general device(e.g. Stewart platform) can suitably perform the tra-jectory planning or a mechanically adjustable devicecan be tuned properly such as a future version of therobot proposed here with more adjustable parameters.

Finally, the simplicity of the described design makesit easy for our device to be enhanced with tele-operation capabilities and remote data recording inorder to be useful to patients that cannot move, or livein remote areas.

Acknowledgments

The authors thank Dr. C.N. Maganaris (ManchesterMetropolitan University) for his support in the studyof ankle kinematics. The second author has carried outpart of this work while on leave at the SALSA team ofINRIA. This work is supported by the General Secre-tariat of Research and Technology of Greece througha PENED 2003 program, contract Nr. 70/03/8473,co-funded by the European Social Fund (75%) andNational resources (25%). The latter includes a con-tribution by private company Reflexion Ltd (10%).

References

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(IARP) 22nd JOINT COORDINATING FORUM Madrid,Report on Robotics Research in Spain, Available at: http://www.iai.csic.es/iarp/sapr/12 SPAIN IARP JCF 2003.pdf

[3] A.J. van den Bogert, G.D. Smith and B.M. Nigg, In vivo deter-mination of the anatomical axes of the ankle joint complex:An optimization approach, J Biomechanics 27(12) (1994),1477–1488.

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[7] D. Daney and I. Emiris, Robust parallel robot calibrationwith partial information, Proc IEEE Intern Conf Robotics &Automation, Seoul, Korea, 2001, pp. 3262-3267.

[8] J. Dul and G.E. Johnson, A kinematics model of the humanankle, J Biomedical Engineering 7 (1985), 137–143.

[9] T. Fukunaga, R.R. Roy, F.G. Shellock, J.A. Hodgson andV.R. Edgerton, Specific tension of human plantarflexors anddorsiflexors, J Applied Physiology 80 (1996), 158–165.

[10] M. Girone, G. Burdea, M. Bouzit, V. Popescu and J.E.Deutsch, A Stewart platform-based system for ankle telere-habilitation, Autonomous Robots 10 (2001), 203–212.

[11] R.E. Isman and V.T. Inman, Anthropometric studies of thehuman foot and ankle, Bulletin Prosthetic Research 10–11(1969), 97–129.

[12] H.S. Kim and Y.J. Choi, The kinetostatic capability analysis ofrobotic manipulators, Proc IEEE/RSJ Intern Conf IntelligentRobots and Systems, Kyongju, Korea, 1999.

[13] K. Lee and D.K. Shah, Kinematic analysis of a three degreesof freedom in parallel actuated manipulator, J Robotics andAutomation 4(3) (1988), 354–360.

[14] C.N. Maganaris, V. Baltzopoulos, D. Ball and A.J. Sargeant,In vivo specific tension of human skeletal muscle, J ApplPhysiol 90(3) (2001), 865–872.

[15] C.G. Mattacola and M.K. Dwyer, Rehabilitation of the ankleafter acute sprain or chronic instability, J Athletic Training37(4) (2002), 413–429.

[16] J.-P. Merlet, Designing a parallel robot for a specificworkspace, INRIA Sophia-Antipolis, France, TechnicalReport 2527, 1995.

[17] J.-P. Merlet, Parallel Robots, 2nd edn., Springer, The Nether-lands, 2006.

[18] Microscribe 3D Digitizer, Available at: http://www.ghost3d.com/gt systems.htm

[19] B.M. Nigg, V. Fisher, T.L. Allinger, J.R. Ronsky and J.R.Engsberg, Range of motion of the foot as a function of age,Foot & Ankle 16(6) (1992), 336–343.

[20] A. Pashkevich, D. Chablat and P. Wenger, Stiffness analy-sis of overconstrained parallel manipulators, Mechanism andMachine Theory 44 (2009), 966–982.

[21] PerformBetter, Available at: http://www.performbetter.com[22] J. Saglia, N.G. Tsagarakis, J.S. Dai and D.G. Caldwell,

A high performance 2-dof over-actuated parallel mechanismfor ankle rehabilitation, Proc IEEE Intern Conf Robotics &Automation, Kobe, Japan, 2009.

[23] C. Syrseloudis, I. Emiris, C. Maganaris and T. Lilas,Design framework for a simple robotic ankle evaluation andrehabilitation device, Proc 30th IEEE Intern Conf Engi-neering in Medicine & Biology, Vancouver, Canada, 2008,pp. 1410–1413.

[24] C. Syrseloudis and I. Emiris, A parallel robot for anklerehabilitation: evaluation and its design specifications, 8thIEEE Intern Conf Bioinformatics & Bioengineering, Athens,Greece, 2008, pp. 1–6.

[25] C.E. Syrseloudis, D. Daney and I.Z. Emiris, Identification offoot kinematics parameters, Submitted, 2011.

[26] A. Toth and I. Ermolaev, Robots with patients, Engineer IT(Nov/Dec 2006), 60–62.

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Design of a simple and modular 2-DOF ankle physiotherapy