IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 2, APRIL 2015 233

Passivity and Stability of Human–Robot InteractionControl for Upper-Limb Rehabilitation Robots

Juanjuan Zhang and Chien Chern Cheah

Abstract—Each year, stroke and traumatic brain injury leavemillions of survivors with motion control loss, which results in greatdemand for recovery training. The great labor intensity in tradi-tional human-based therapies has recently boosted the researchon rehabilitation robotics. Existing controllers for rehabilitativerobotics cannot solve the closed-loop system stability with uncer-tain nonlinear dynamics and conflicting human–robot interactions.This paper presents a theoretical framework that establishes thepassivity of the closed-loop upper-limb rehabilitative robotic sys-tems and allows rigorous stability analysis of human–robot interac-tion. Position-dependent stiffness and position-dependent desiredtrajectory are employed to resolve the possible conflicts in motionsbetween patient and robot. The proposed method also realizes the“assist-as-needed” strategy. In addition, it handles human–robotinteractions in such a way that correct movements are encouragedand incorrect ones are suppressed to make the training processmore effective. While guaranteeing these properties, the proposedcontroller allows parameter adjustment to provide flexibility fortherapists to adjust and fine tune depending on the conditions ofthe patients and the progress of their recovery. Simulation and ex-periment results are presented to illustrate the performance of themethod.

Index Terms—Adaptive control, human–robot interaction, re-habilitation robotics.

I. INTRODUCTION

N EURAL damages in the form of stroke or traumatic braininjuries produce millions of new patients of paralysis each

year, which creates great need of therapeutic training for lostor impaired motion control restoration. The theory of synap-tic plasticity [1], which was later summarized as “neurons thatfire together, wire together” [2] has been proposed for morethan a century. According to this principle, simultaneous acti-vation of neurons leads to increase in synaptic strength betweenthem and, therefore, relearning of human movements through“reconnection” of broken neurological path [3] in case ofbroken-neurological-path-paralysis or “reorganization” of brain[4] in case of damaged-brain-motor-cortex-paralysis. Therefore,repeated and concurrent happening of human intentions to moveand limb movements is the key to recovery. Clinical one-on-onerehabilitative therapies have been introduced from the 1950s

Manuscript received July 5, 2013; revised September 23, 2014; acceptedJanuary 9, 2015. Date of publication February 16, 2015; date of current versionApril 2, 2015. This paper was recommended for publication by Associate EditorP. Soueres and Editor A. Kheddar upon evaluation of the reviewers’ comments.

J. Zhang is with the Department of Mechanical Engineering, Carnegie Mel-lon University, Pittsburgh, PA 15213 USA, and also with the School of Electri-cal and Electronic Engineering, Nanyang Technological University, Singapore639798 (e-mail: juanjuaz@andrew.cmu.edu).

C. C. Cheah is with the School of Electrical and Electronic Engi-neering, Nanyang Technological University, Singapore 639798 (e-mail:ecccheah@ntu.edu.sg).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2015.2392451

to realize this process. However, traditional human-based prac-tice is inefficient due to its great labor intensity [5] and badrepeatability. It totally relies on therapy and evaluation processconducted by one therapist attending one single patient at atime. Besides, the duration and frequency of these therapies arelimited by the strength of practitioners. Lab-intensive activitiesstand for one key application area for robotics. To overcomethe limitations of human-based practice, research on rehabil-itative robotics has started since early 1990s, among whichthe MIT-MANUS upper-limb therapeutic training robot was apioneer [6].

Robot systems can be classified into two general groups, iso-lated robotics that have no environmental contacts other thanbase support and nonisolated systems, which interact with theenvironment besides the base support, such as rehabilitationrobotics. Control theories for the former with systematic sta-bility analysis have already been established in the literature[7]–[10]. Control of rehabilitative systems, on the contrary, isdifferent from the traditional robotic control in the sense that itis not always a servo mechanism [11]. Besides stability, thereare two main objectives of a rehabilitative controller accord-ing to the training principle: encouraging human intentions tomove and ensuring correct human movements. At an early stage,patients are either fully paralyzed or generating untrustworthysignals, the training should be robot assisting, i.e., rehabilitativerobots work just like the assistive ones by treating all signalsand interactions from the human as disturbances and traditionalcontrol methods are applicable. However, when motion-relatedneurological connection is restored but weak, the training shouldbe robot-complementing, i.e., human intentions and movementsshould be encouraged and complemented rather than rejected[11]. In this case, traditional control methods are not suitable.

The primary control paradigm in robotic therapy developmentis the active assist exercise [12] that emulates physician-basedtherapies. In this type of controller, a human patient initiatesa movement, or a predetermined desired trajectory is knownto the robotic device. Then, the device will assist the user tomove the limbs and fulfill the task. Existing assistive controllersof rehabilitative systems are mostly directly or indirectly de-rived from motion and impedance manipulation [13]. Some aremodel-based controllers with force feedback [14], [15] or with-out force feedback [11]. Others are model-free methods withless predictable real-time interactions between patients and de-vices and, thus, less control of the training progress [16]. Moreearly devices employ motion-only-based control and handle in-teractions by disturbance rejection or uncertain modeling. Thismethod is proper at the early stage of therapy for some severelyparalyzed patients when functionality loss is almost total andpatients’ ability to move their limbs can be ignored. Amongthese controllers, simple PD control [5], [14] started with adominant role. Later devices, for both upper limb [17] and lowerlimb [18], use similar strategy, but instead of asserting torqueslinearly related to position errors, more sophisticated force fields

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234 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 2, APRIL 2015

are used to build a virtual moving wall in space along the desiredtrajectory. This demonstrates one form of the “assist-as-needed”paradigm, which is proved to be more effective in [19].

However, in these existing designs, stability of the systems isnot systematically solved with nonlinearity, uncertainty, chang-ing human dynamics or interaction forces [20], [21]. Besides,due to the fact that rehabilitative robotics work in close contactwith impaired human, safety is a critical concern during the op-eration. Various causes can incur injuries to human in interactionwith a robot [22] when the robot velocity, acceleration, or torqueexceeds certain values. Possible injuries include bone damage,joint damage, and soft tissue fracture, severity of which mayrange from minor to fatal [22]. The situation is especially seri-ous in case when robots work with motion-control-challengedhuman as in rehabilitation applications, when paralysis or spas-ticity causes high stiffness of human joint and inability to reactfast. To maintain human safety throughout the operation, sta-bility of the control systems must be ensured. Furthermore, theconflicting movements between human and robot should alsobe resolved in a stable manner to avoid large control forces andinstability.

This paper proposes an adaptive control framework for theupper-limb rehabilitative robotic systems that can easily real-ize training decisions made by therapists. The proposed methodhandles human–robot interaction in such a way that “correct”human movements are encouraged while incorrect movementsare compensated to ensure the efficiency of the training process.Position-dependent stiffness and desired trajectory are proposedto resolve the possible conflicts in motions between patientsand robots. The main contributions of this paper include: 1) Atheoretical framework which establishes system passivity andallows rigorous stability analysis of human–robot interactionin rehabilitative robotic system is presented. 2) The assist-as-needed policy is realized, while allowing customization of thecontroller based on different patients and their different stagesto cope with the patient cooperative training [23]. 3) The con-flicting movements between human and robot are reduced andcontained in a stable manner to align with rehabilitation phi-losophy. A preliminary version of this paper was presented inICRA2013 [24]. This paper is an extended version with more de-tails in the theoretical analysis, formal statements of theorems,and new experimental results.

II. UPPER-LIMB REHABILITATIVE ROBOTICS DYNAMICS

The dynamics of an upper-limb rehabilitative robotic systemconsisting of a n-DOF robot manipulator connected to a humanarm at the end can be expressed as

M(q)q +[12M(q) + S(q, q)

]q + g(q) = τ + Jᵀ(q)fh (1)

where q ∈ �n is the joint displacement vector and q = dqdt ∈ �n

is the joint velocity vector; M(q) ∈ �n×n is the manipu-lator inertia matrix, which is symmetric and positive defi-nite; [ 1

2 M(q) + S(q, q)]q ∈ �n is the centripetal and Coriolistorques, in which S(q, q) ∈ �n×n is skew-symmetric, g(q) ∈�n is the vector of gravitational torques, τ ∈ �n stands forthe control input torques applied at the manipulator joints,fh ∈ �m is the force applied to robot end-effector by humanand J(q) ∈ �m×n denotes the Jacobian matrix relating jointspace and task space.

III. HUMAN–ROBOT INTERACTION INVOLVED CONTROL

In this section, we first identify three basic operation modesof the rehabilitative robotics based on the required capabil-ities. A controller consisting of a motion-related part (seeSection III-A) and an interaction-related part (see Section III-B)is then introduced to realize the operation modes, while dealingwith system uncertainties and changing human forces.

Based on the required capabilities of the rehabilitativerobotics, i.e., customizability, adjustability, and safety, the threebasic operation modes of the controller are identified as 1) ahuman-dominant mode in which the patient movements aretrusted, encouraged thus only slightly interfered by the robotsystem, 2) a robot-dominant mode in which patient motion con-trol is not reliable and desired movements are accomplishedmostly by the assistance of the robot system, and 3) a safety-stop mode in which system end-effector is stopped due to theexistence of conflicting human–robots movements and possibleinjuries to human if the robot stays active.

Human-dominant mode refers to the situation that human mo-tion control has been partially restored. In this case, it is impor-tant that the affected human limb gains enough exercise throughthe therapeutic process, i.e., free limb motion should be allowedto a great extent. Meanwhile, certain level of speed regulationis beneficial to avoid abnormal movements that are too slowand too fast. Both of these two actions above can strengthen theaccuracy and repeatability of human motion control in neuro-damaged patients’ therapeutic process. Robot-dominant mode,on the other hand, stands for a scenario that human movementsare either unreliable or inefficient. Thus, assistance from therobot system is always necessary to either finish the task or putthe human limb back to the correct track, both by moving thesystem closer to a desired trajectory. Trajectory tracking withregulation on both position and speed of the system is suitablefor this stage. Besides the two normal operation modes men-tioned above, sometimes, the system may drift to an extent suchthat the position errors are too big and resulting control torquesmay incur damages to human body. In this case, the robotic sys-tem should be stopped. It is also reasonable that the faster thesystem moves, the faster it should be stopped in this dangeroussituation. Thus, damping control should be employed here.

In practice, it is feasible and important for a normal reha-bilitative robot to possess the capability to work in all of thethree modes, determined by the system end-effector’s instantposition. As shown in Fig. 1, when the system end-effector isclose to the desired trajectory, it is in human-dominant region(H-DR), when it is in the robot-dominant region (R-DR), therobot will help to push the whole system back to human-dominant region; however, when the end-effector is drifted intothe outer safety-stop region (S-SR), the system is stopped.

To fit in the conditions of different patients and their variousrecovery status, the widths of human-dominant and R-DRs areexpected to be adjustable. At the early stage of the therapeu-tic process, human movements are minor or totally unreliable.Thus, human-dominant region should be reduced to a very smallarea so that robot-assistance is constantly present. The rehabili-tative robot actually works as an assistive robot in that case. Onthe other hand, when human motion control has been recovered,human region should be made larger to allow more freedom inhuman movements.

This scheme is also applicable to training of patients withspasticity. Similarly, depending on the severity of the spasticity,

ZHANG AND CHEAH: PASSIVITY AND STABILITY OF HUMAN–ROBOT INTERACTION CONTROL FOR UPPER-LIMB REHABILITATION ROBOT 235

Fig. 1. Operation regions that move along the desired trajectory xdes (t) de-termined by real-time position error ||Δx||. From inside to outside, there arethree main regions, namely human-dominant region (H-DR, ||Δx|| ∈ [0, ah ]),robot-dominant region (R-DR, ||Δx|| ∈ (ah , ar ]), and safety-stop region(S-SR, ||Δx|| ∈ (ar , +∞)). The increasing (IR-DR, ||Δx|| ∈ (ah , ai ]) anddecreasing (DR-DR, ||Δx|| ∈ (ai , ar ]) subregions of R-DR are identified byincreasing and decreasing stiffness (Section III-A.†2). The transitional subre-gion of S-SR (TS-SR, ||Δx|| ∈ (ar , at ]) exists to ensure smooth transition fromR-DR to stable S-SR (SS-SR, ||Δx|| ∈ (at , +∞). See Section III-A.†1).

the sizes of various regions can be fine-tuned during initialtrials. Patients with more severe spasticity should use largerR-DR so that reasonable amount of training can be receivedwithout constant stops in the safety-stop region. Actually, thedesired region can be suspended if the patient enters safety-stopregion to pause the training. The training can still be resumedby moving the arm back to the suspended region either throughself-effort or with help.

To deal with the specific issues related to the rehabilitationrobotics described by (1), the proposed controller is divided intotwo portions, a motion-related controller τm and an interaction-related controller τi , i.e.,

τ = τm + τi (2)

in which τm realizes the three main operation modes anddeals with modeling uncertainties, and τi deals with conflict-ing human–robot movements. Together, the two also ensure thestability of the closed-loop system.

A. Motion-Related Controller

The motion-related controller includes a stiffness control(proportional control) term ((3) term a), a sliding control term((3) term b) and an adaptive term ((3) term c). Stiffness controland sliding control terms together realize the three operationregions illustrated in Fig. 1, while the adaptive term deals withsystem uncertainties. The motion-related controller is expressedas

τm = −Mᵀ(x)Jᵀ(q)kp(δx)δx︸ ︷︷ ︸term a

−Kss︸ ︷︷ ︸term b

+ Yd(q, q, qr , qr )θ︸ ︷︷ ︸term c

(3)

where q ∈ �n and J(q) ∈ �n are the joint displacement vec-tor and Jacobian matrix respectively as defined in (1); x =[x1 , ..., xm ]ᵀ ∈ �m (m ≤ 3) is the position of the robot end-effector in task space, δx is a weighted position error, M(x) ∈�n×n is a modifier matrix, kp(δx) ∈ � is a nonnegative

position-dependent stiffness, Ks ∈ �n×n is a positive slidingcontrol gain, s ∈ �n is a sliding vector, qr ∈ �n is a virtualreference joint vector, Yd(q, q, qr , qr ) ∈ �n×h is a regressormatrix, and θ ∈ �h is an estimate of the unknown dynamicsparameter vector θ ∈ �h .

Different from classical proportional control, term a

uses a position-dependent gain kp(δx). Besides, instead ofthe direct position error Δx = x − xdes , in which xdes =[xdes,1 , ..., xdes,m ]ᵀ denotes the desired end-effector trajectoryin task space, a weighted position error δx is used, which isdefined as

δx = x − xd (4)

where xd is a weighted trajectory expressed as

xd = xi + w(Δx)(xdes − xi) = xi + w(Δx)xv (5)

in which xi is a time-invariant reference position, xv is thetime-varying part of xd , and w(Δx) is a Δx-based weight vec-tor. Note that when w(Δx) = 1, xd = xdes ; when w(Δx) = 0,xd = xi , and xd = 0.

The position-dependence of kp and the addition of w(Δx)to desired trajectory together realize the three main operationmodes. The modifier matrix M(x) is added to eliminate theuse of x in the controller, which emerges due to the usageof w(Δx) in xd , for noise reduction in implementation. Themodifier matrix is defined as

M(x) = In − J+(q)A(x)J(q) (6)

where In is a n × n identity matrix, J+(q) is the pseudoinverseof J(q), A(x) ∈ �m×m is a transition matrix defined as

A(x) =

⎡⎢⎢⎢⎢⎢⎣

∂w

∂x1xdes,1 . . .

∂w

∂xmxdes,1

.... . .

...

∂w

∂x1xdes,m . . .

∂w

∂xmxdes,m

⎤⎥⎥⎥⎥⎥⎦

. (7)

The use of position-dependent stiffness kp , weight factorw(Δx), and modifier matrixM also applies to the sliding vectors in term b, which is proposed correspondingly as

s = q − qr

= q −M−1(x)[J+(q)xf − αJ+(q)kp(δx)δx] (8)

where α is a constant, qr is a virtual joint reference vector, xf is avirtual position reference vector defined as xf = w(Δx)xdes −A(x)xdes .

The regressor Yd(q, q, qr , qr ) in term c is defined by

M(q)qr +[12M(q) + S(q, q)

]qr + g(q) = Yd(q, q, qr , qr )θ

according to the linear-in-parameters property [25] of the systemdynamics. By using this property, the vector of unknown systemparameters θ can be estimated by the update law

˙θ = −LY �

d (q, q, qr , qr )s (9)

where L ∈ �h×h is a positive-definite and nonsingular squarematrix gain.

Note that this controller works with m ≤ 3. However, forbetter presentation of the controller characteristics, m = 2 is

236 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 2, APRIL 2015

used mostly in the graphic illustrations. The 3-D case will belater discussed and demonstrated by simulation.

At any instant time t when a neuro-damage-affected hu-man limb is in the process of finishing an action with thehelp of the robotic system, based on the direct position errorΔx between end-effector position x(t) and desired positionxdes(t), the workspace is divided into three portions: human-dominant region with radius ah , R-DR with outer radius ar , andsafety-stop region where human safety will be compromisedif robotic assistance is active. These three regions move alongthe desired trajectory together with instant xdes(t) as shown inFig. 1.

To meet these requirements of operation, certain condi-tions need to be satisfied. For proportional control term−Mᵀ(x)Jᵀ(q)kp(δx)δx in (3), kp(δx) is strictly asserted tozero in human-dominant region and nonzero in R-DR. Be-sides, with the increase of position error δx inside the robotregion, it is reasonable for kp(δx) to ramp up to rapidly restorethe system back to the previous region. However, the value of−Mᵀ(x)Jᵀ(q)kp(δx)δx will grow to a point where the controlinput torque becomes damaging to human body. Similar prob-lem happens to term −Kss. These concerns for human safetynurture the necessity of the third region safety-stop region. Inthis region, kp(δx) is again set to zero to cease stiffness control,and damping will be introduced to stop the motion.

These design requirements are realized by the employmentof the position-dependent trajectory weight vector (†1) and theposition-dependent stiffness (†2).

†1 The position-dependent weight vector w(Δx) is added torealize velocity tracking in human and R-DRs and dampingin safety-stop region by asserting xd(t) = 0, i.e., weightvector w(Δx) in (5) meets requirements

w(Δx) =

{1, H-DR ∪ R-DR

0, S-SR

in which the symbol ∪ denotes union operation.To ensure smooth transition instead of asserting w(Δx) = 0in whole safety-stop region, a transitional safety-stop sub-region (TS-SR) is added at the outer edge of R-DR, in whichw(Δx) smoothly changes from 1 to 0. The subregion inwhich w(Δx) = 0 is then named as stable safety-stop region(SS-SR). This smooth definition of w(Δx) is equation (10),as shown on the bottom of the page, where ar is a positiveconstant standing for outer radius of R-DR; at = ar + ht

stands for outer radius of transitional safety-stop region,with a small constant of design ht denoting the width of theregion. With this definition, an illustration of the shape ofw(Δx) for m = 2 is available in Fig. 2.While realizing velocity tracking and damping in differentregions, the addition of w(Δx) also potentially introduces xinto the controller if we define the sliding vector classically

Fig. 2. Illustration of the weight factor w versus Δx for m = 2, withar = 2, at = 4.5. w = 1 when ||Δx|| ∈ [0, ar ] in the human-dominant region(H-DR) and robot-dominant region (R-DR); w = 0 when ||Δx|| ∈ (at , +∞)in the stable safety-stop region (SS-SR). w smoothly transits from 1 to 0 when||Δx|| ∈ (ar , at ] in the transitional safety-stop region (TS-SR).

as

sq = q − J+(q)xd + αJ+(q)kp(δx)δx (11)

due to double differentiation of w(Δx) in xd(t). sq hereis labeled as an intermediate sliding vector. To avoid theuse of acceleration in the controller, the modifier matrix Mdefined in (6) is used as detailed below.Differentiating (5) with respect to time and using (7), wehave

xd = w(Δx)xdes + w(Δx)xdes

= A(x)x − A(x)xdes + w(Δx)xdes

= A(x)x + xf . (12)

From (6) and (12), the intermediate vector sq defined in (11)can be rewritten as

sq = [In − J+(q)A(x)J(q)]q − J+(q)xf

+ αJ+(q)kp(δx)δx

= Mq − J+(q)xf + αJ+(q)kp(δx)δx. (13)

Now, it is clear that the sliding vector that is defined as

s = M−1(x)sq (14)

can be rewritten into the format of (8), in which the referencejoint variable time derivative qr = M−1(x) [J+(q)xf −αJ+(q)kp(δx)δx] includes no x and therefore the term x iseliminated in qr and, hence, control input τ .With this weighted desired trajectory above, consideringthe fact that A(x) = 0m×m (m × m zero matrix) and, thus,M(x) = In for w =0 or 1 (i.e., in human-dominant, robot-dominant, and stable safety-stop regions), and the design

w(Δx) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1, ||Δx|| ∈ [0, ar ], i.e., H-DR ∪ R-DR

1 − {[||Δx||2/a2t − 1]4 − [a2

r /a2t − 1]4}4

[a2r /a2

t − 1]42 , ||Δx|| ∈ (ar , at ], i.e., TS-SR

0, ||Δx|| ∈ (at, +∞), i.e., SS-SR

(10)

ZHANG AND CHEAH: PASSIVITY AND STABILITY OF HUMAN–ROBOT INTERACTION CONTROL FOR UPPER-LIMB REHABILITATION ROBOT 237

requirement that kp(δx) = 0 in human-dominant andsafety-stop regions, the sliding control portion is

−Kss = −KsM−1(x)[q − J+(q)xd + αJ+(q)kp(δx)δx]

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−Ks [q − J+(q)xdes ],||Δx|| ∈ [0, ah ]i.e., H-DR

−Ks [q − J+(q)xdes

+αJ+(q)kp(δx)δx],

||Δx|| ∈ (ah , ar ]i.e., R-DR

−KsM−1(x)[q − J+(q)xd ],||Δx|| ∈ (ar , at ]

i.e., TS-SR

−Ksq,||Δx|| ∈ (at ,+∞)

i.e., SS-SR

(15)

which implements speed control in human-dominant re-gion and transitional safety-stop region, trajectory trackingcontrol in R-DR and damping control in stable safety-stopregion.†2 The position-dependent stiffness kp(δx) is used insteadof a constant one so that proportional control is only usedin R-DR, i.e.,

kp(δx) =

⎧⎪⎨⎪⎩

0, H-DR

positive value, R-DR

0, S-SR.

With the addition of w(Δx) to facilitate the system passivityand stability, weighted error δx is used for proportionalcontrol portion instead of director error Δx.In order to obtain a continuous and differentiable stiffnessthat meets the requirements above, an auxiliary function isfirst defined as

fp(δx) = 1 − exp[−a(||δx||2 − a2h)] (16)

and the position-dependent stiffness is described accord-ingly as

kp(δx) = k1 [max(0, fp(δx))]2 exp[−a(||δx||2 − a2h)](17)

in which a and k1 are positive constants of design, andah is the radius of human-dominant region as mentionedearlier. An illustration of the shape of this stiffness functionis available in Fig. 3. Note that there are two subregionsfor R-DR, an increasing one (IR-DR) and a decreasing one(DR-DR). The outer radius of the former ai is adjustable bya and ah according to

ai = {||δx|| :∂kp(δx)

∂δx= 0, ||δx|| > ah}.

The outer radius of the decreasing region, as well as thewhole R-DR ar is also controllable by k1 , a, and ah .Supposing the controller processor has a minimum dis-tinguishable signal size or quantization step size p, i.e.,a scalar signal y ≡ 0 if |y| < p, then ar is determined byar = {||δx|| : kp(δx) = p, ||δx|| > ai}.Note that human-dominant region provides the patients withopportunity to exercise and R-DR guarantees human actioncorrectness and reinforces human motion control. Thus, theexistence of both regions is a key feature to the rehabilitative

Fig. 3. Illustration of the position-dependent stiffness kp versus δx form = 2, with ah = 1, a = 0.3, k1 = 2. (a) A 3-D view. (b) A 2-D view:intersection of kp and surface δx2 = 0. When ||δx|| ∈ [0, ah ] ∪ (ar , +∞) inhuman-dominant region (H-DR) and safety-stop region (S-SR), kp = 0; When||δx|| ∈ (ah , ar ] in R-DR, kp > 0. Increasing robot-dominant region (IR-DR)

denotes the subregion of R-DR in which∂ kp

∂ ||δ x || > 0, i.e., kp increases as ||δx||increases. Decreasing robot-dominant region (DR-DR) denotes the subregion

with∂ kp

∂ ||δ x || < 0.

process and the relative sizes of them should reflect the re-covery extent of human motion control. When human limbmovement is less reliable, ah should be set smaller to ensureless human free motion and more robotic assistance; sim-ilarly, when human gains better motion control, ah shouldbe set larger to allow more human exercise and little robothelp as needed to further boost patients’ recovery. When ah

is set to zero, the system is robot-dominant, which is usefulat the beginning, i.e., robot-assisting stage, of the therapy.Besides, the general width of R-DR can be easily controlledby a. As a increases, the width of R-DR drops rapidly.The fixed reference position xi of the weighted trajectoryxd is chosen to be sufficiently far away from the desiredpath xdes so that ||Δx|| > ar always guarantees ||δx|| > ar

in operation, i.e., in this region kp(δx) = 0.With the kp(δx) and xi defined above, the stiffness controlportion is

−Mᵀ(x)Jᵀ(q)kp(δx)δx =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

0,||Δx|| ∈ [0, ah ]i.e., H-DR

−Jᵀ(q)kp(δx)δx,||Δx|| ∈ (ah , ar ]i.e., R-DR

0,||Δx|| ∈ (ar ,+∞)i.e., S-SR

(18)

in which proportional control with nonzero stiffness is de-ployed in R-DR only.

With stiffness and desired trajectory defined above, τm

is proposed as (3). Note that J+(q) and M(x) = In −J+(q)A(x)J(q) are involved in the controller which poten-tially causes issues at the singularities of them. For J+(q), itis handled by specifying a desired trajectory xdes(t) such thatall singularities are in stable safety-stop region, in which casekp(δx) and xd already reduce to zero and M = In , the motionrelated controller reduces to

τm = −Ksq + Yd(q, q, 0, 0)θ

thus the singularity issue of the controller due to J(q) iscontained. Since matrix A is 0m×m in human-dominant,

238 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 2, APRIL 2015

robot-dominant, and stable safety-stop regions, M = In is al-ways invertible in these main operating regions. For the casethat A = 0m×m in the transitional safety-stop region, sinceM = In − J+(q)A(x)J(q) is a general matrix that varies ac-cording to robot kinematics and tasks, it is difficult to derivea general analytic inverse matrix. The inverse matrix has to beanalyzed according to the particular robot structure and tasksto contain singularity issue (if any) by trajectory planning orsaturating the inverse matrix. Since transitional safety-stop re-gion is a very narrow zone between robot-dominant and stablesafety-stop regions that only for the purpose of smoothing thetransition, saturating M−1 will not affect the performance inthe three main regions.

B. Interaction-Related Controller

The interaction-related controller has two objectives, to en-sure the closed-loop system stability together with the motion-related controller, and to reduce the conflicts between humanand robot movements. τi is proposed as a function of the hu-man interaction force fh , the position error Δx, and the slidingvector s as

τi = Jᵀ(q)Cx(fh ,Δx, s) (19)

in which Cx is the task-space interaction controller.Research work on human science [26] has shown that hu-

man central nervous system may use a composite variable [25]consisting of tracking errors and their temporal derivatives inmotion control. This composite variable can serve as either anerror prediction or a criterion to be minimized. In a limb re-habilitative process assisted by a robot system, when humanlimb is well functioning and the robot is unactuated, the inter-action force received by the robot should be fully responsiblefor achieving both position and speed objectives set for it, i.e.,minimizing the composite variable. For the controller proposedin this paper, this composite variable is represented by the taskspace sliding vector sx , a combination of the transformed posi-tion and velocity errors as

sx = J(q)s = J(q)M−1(x)sq

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

δx, ||Δx|| ∈ [0, ah ], i.e., H-DR

δx + αkp(δx)δx, ||Δx|| ∈ (ah , ar ], i.e., R-DR

J(q)M−1(x)sq , ||Δx|| ∈ (ar , at ], i.e., TS-SR

x, ||Δx|| ∈ (at ,+∞), i.e., SS-SR.

(20)

In this case, the direction of resulting force Cx(fh ,Δx, s) + fh

should be aligned with the opposite of this variable.In the safety-stop region, since the whole system is supposed

to be damped to static status, fh is directly canceled by thecontroller. On the other hand, in human-dominant region, hu-man movements are considered reliable to certain extent, andthe controller is then introduced with three modes based onγ ∈ [0, π], which is the magnitude of directional difference be-tween vector−sx and fh as demonstrated in Fig. 4, to encourageactive human movements. When fh orientates close enough to−sx with γ ≤ β, in which β is a threshold adjustable in [0, π

2 ],human motion is considered reliable, and the interaction forceis retained. This region is called the retaining region (RR). Oth-erwise, if fh direction is out of this area, but still maintainsan angle difference less than σ (adjustable in [0, π

2 ]) with oneretaining region edge (denoted by unit vector either −s′x1 or

Fig. 4. Interaction force handling regions determined by the magnitude ofdirectional difference γ between interaction force fh and the opposite of taskspace sliding vector sx . When γ ≤ β , β ∈ [0, π

2 ], system is in retaining region(RR); when γ ∈ (β, β + σ], σ ∈ [0, π

2 ], system is in projecting region (PR);otherwise, system is in canceling region (CR).

−s′x2), it is in the projecting region (PR) and is projected towhichever edge that is closer. Other than these two regions, fh

is considered wrongly orientated, i.e., human movements arenot reliable. Interaction is directly canceled and this area is thecanceling region (CR). Finally, R-DR serves as a transition area.

With these discussions, the interaction-related controller isdeveloped such that

Cx(fh ,Δx, s) + fh

=

⎧⎪⎨⎪⎩

μs(s)cx(fh), ||Δx|| ∈ [0, ah ], i.e., H-DR

μs(s)μx(Δx)cx(fh), ||Δx|| ∈ (ah , ar ], i.e., R-DR

0, ||Δx|| ∈ (ar ,+∞), i.e., S-SR

(21)

where

cx(fh)

=

⎧⎪⎪⎨⎪⎪⎩

fh , γ ∈ [0, β], i.e., RR

−s′x ||fh || cos2[(γ − β) · π

2σ

], γ ∈ (β, β + σ], i.e., PR

0, γ ∈ (β + σ, π], i.e., CR

(22)

with −s′x equals to −s′x1 or −s′x2 , whichever has a closer direc-tion with fh . The term

μx(Δx) = sin2[(ar − ||Δx||) · π

2(ar − ah)], ||Δx|| ∈ (ah , ar ] only

(23)is a position-based coefficient to ensure smooth transition be-tween human-dominant and safety-stop regions; the saturationfunction

μs(s) =

⎧⎪⎨⎪⎩

1, ||sx || ≥ ws

sin2(||sx ||π2ws

), ||sx || < ws

is added as a second coefficient of cx(fh) to assert force-dominant controller when s has a norm bigger than a smallpositive constant ws , and sliding-vector-dominant controllerotherwise so that the controller is smoothed at s = 0. Without

ZHANG AND CHEAH: PASSIVITY AND STABILITY OF HUMAN–ROBOT INTERACTION CONTROL FOR UPPER-LIMB REHABILITATION ROBOT 239

μs(s), the resulting force magnitude ||Cx + fh || is discontin-uous at sx = 0, while it is smooth and continuous with μs(s).Instead of direct projection to −s′x direction, a second orderprojection cos2(•) is used in the controller to ensure its differ-entiability.

With definitions above, the control result Cx(fh ,Δx, s) + fh

is uniformly continuous on s,Δx and also fh , and the term

sᵀJᵀ(q)[Cx(fh ,Δx, s) + fh ] = sᵀx [Cx(fh ,Δx, s) + fh ]

=

⎧⎪⎨⎪⎩

μs(s)sᵀxcx(fh), ||Δx|| ∈ [0, ah ], i.e., H-DR

μs(s)μx(Δx)sᵀxcx(fh), ||Δx|| ∈ (ah , ar ], i.e., R-DR

0, ||Δx|| ∈ (ar ,+∞), i.e., S-SR

(24)

where

sᵀxcx(fh)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

−||sx ||||fs || cos(γ),γ ∈ [0, β]

i.e., RR

−||sx ||||fs || cos(β) cos2[

(γ−β )·π2σ

],

γ ∈ (β, β + σ]

i.e., PR

0,γ ∈ (β + σ, π]

i.e., CR.

(25)

Since β ≤ π2 , (24) is never greater than zero.

When m = 3, all mathematical descriptions of the controllerstill hold, while the shapes of various regions change from 2-Dto 3-D. For motion-related controller, human-dominant regionchanges from a disk to a ball and R-DR changes from a ring toa hollow ball. For interaction-related part, regions change fromcircular sectors into cones.

IV. PASSIVITY AND STABILITY OF THE SYSTEM

In this section, passivity of the closed-loop system with theproposed controller and convergence of various state variables indifferent operation modes are analyzed using Lyapunov method.It is shown that system passivity can be ensured and the positionof the end-effector either converges to the human-dominant re-gion, or stops in safety-stop region depending on the interactionsbetween the human and robot.

A. System Passivity

By using (8) and substituting (2), (3), and (19) into (1), theclosed-loop system dynamic equation becomes

M(q)s +[12M(q) + S(q, q)

]s + Mᵀ(x)Jᵀ(q)kp(δx)δx

+ Kss + Yd(q, q, qr , qr )Δθ

= Jᵀ(q)[Cx(fh ,Δx, s) + fh ] (26)

where Δθ = θ − θ. With no interaction-related controller as-serted, i.e., Cx(fh ,Δx, s) = 0, the time integral of the inner

product between an output y = s and (26) yields∫ t

0sᵀ(ς)Jᵀ(ς)fh(ς)dς

=∫ t

0{sᵀ(ς)M(q(ς))s(ς) + sᵀ(ς)

[12M(ς) + S(ς)

]s(ς)

+ [q(ς) − J+(ς)xd(ς))]ᵀ[M−1(ς)]ᵀMᵀ(ς)Jᵀ(ς)kp(ς)δx(ς)

+ [M−1(ς)αJ+(ς)kp(ς)δx(ς)]ᵀMᵀ(ς)Jᵀ(ς)kp(ς)δx(ς)

+ sᵀ(ς)Kss(ς) + sᵀ(ς)Yd(ς)Δθ(ς)}dς. (27)

By using the skew-symmetric property of S(q, q), the first twoterms on the right-hand side of (27) can be expressed as∫ t

0{sᵀ(ς)M(q(ς))s(ς) + sᵀ(ς)

[12M(ς) + S(ς)

]s(ς)}dς

=12sᵀ(t)M(q(t))s(t) − 1

2sᵀ(0)M(q(0))s(0). (28)

The third term of (27) can be simplified as∫ t

0{[q(ς) − J+(ς)xd(ς))]ᵀ[M−1(ς)]ᵀ

Mᵀ(ς)Jᵀ(ς)kp(ς)δx(ς)}dς

=∫ t

0{[J(ς)(q(ς) − J+(ς)xd(ς))]ᵀkp(ς)δx(ς)}dς

=∫ t

0{[x(ς) − xd(ς)]ᵀkp(ς)δx(ς)}dς

=∫ t

0{δxᵀ(ς)kp(ς)δx(ς)}dς

=∫ δx

0{d(δxᵀ)kp(δx)δx}|t0

= Pt(δx(t)) − Pt(δx(0)) (29)

where the pseudopotential function Pt(δx) is expressed as

Pt(δx) =16a

k1 [max(0, fp(δx))]3 .

The last step of (29) comes from the fact that

∂Pt(δx)∂δx

=16a

· k1 · 3

·[max(0, fp(δx))]2 exp[−a(||δx||2 − a2h)]a · 2 · δx

= kp(δx)δx

and, thus, ∫ δx

0{d(δxᵀ)kp(δx)δx} = Pt(δx).

Next, the fourth term can be written as∫ t

0{[M−1(ς)αJ+(ς)kp(ς)δx(ς)]ᵀ

Mᵀ(ς)Jᵀ(ς)kp(ς)δx(ς)}dς

=∫ t

0α[δx(ς)]ᵀkp(ς)2δx(ς)dς. (30)

240 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 2, APRIL 2015

Fig. 5. Illustration of the pseudoenergy versus the weighted position errorδx for m = 2: ah = 1, a = 0.3, k1 = 2. For ||δx|| ∈ [0, ah ], i.e., human-dominant region, Pt (δx) = 0. For ||δx|| ∈ (ar , +∞), i.e., safety-stop region,Pt (δx) is saturated. For ||δx|| ∈ (ah , ar ] in the R-DR, Pt (δx) smoothlychanges from 0 to its maximum.

Then, based on the system parameter estimate update law (9),the final term of (27) can be expressed as

∫ t

0{sᵀ(ς)Yd(ς)Δθ(ς)}dς

=∫ t

0{Δθ(ς)L−1Δθ(ς)}dς

=12Δθ(t)ᵀL−1Δθ(t) − 1

2Δθ(0)ᵀL−1Δθ(0). (31)

Combining (28)–(31), we have∫ t

0sᵀ(ς)Jᵀ(ς)fh(ς)dς

= V (t) − V (0)︸ ︷︷ ︸Stored Energy

+∫ t

0W (ς)dς

︸ ︷︷ ︸Dissipated Energy

(32)

where

V =12sᵀM(q)s + Pt(δx) +

12ΔθL−1Δθ (33)

and

W = sᵀKss + αδxᵀk2p (δx)δx. (34)

The pseudopotential Pt(δx), as illustrated in Fig. 5, has no lo-cal minimum and only global minimum inside human-dominantregion, and has

Pt(δx) ≥ 0.

Then, since instant stored energy V (t) in the system is composedof quadratic forms 1

2 ΔθL−1Δθ, sᵀM(q)s and the nonnegativePt(δx), it is thus clear that

V (t) ≥ 0. (35)

Next, the energy dissipation rate W (s, q) is nonnegative from(34). Therefore, we can prove the theorem as follows.

Theorem 1: The input Jᵀ(q)fh and output y = s of theclosed loop system described by (26) and (9) satisfy the passivityproperty.

Proof: Equation (32) can be written as∫ t

0sᵀ(ς)Jᵀ(ς)fh(ς)dς

= V (t) − V (0)︸ ︷︷ ︸Stored Energy

+∫ t

0W (ς)dς

︸ ︷︷ ︸Dissipated Energy

≥ −V (0) = −k20

in which k20 is a nonnegative value denoting the initial system

energy. Therefore, the input Jᵀ(q)fh and output y = s havepassive relationship. �

In practical rehabilitative robotic systems, due to the exis-tence of human–robot interaction, the term sᵀJᵀ(q)fh will notbe zero, which may result in increase of errors and storageenergy. This problem is solved by the interaction-related con-troller τi that leads to sᵀJᵀ(q)[Cx(fh ,Δx, s) + fh ] ≤ 0 as in(24). Therefore, it is established that

V = −W ′ = −W + sᵀJᵀ(q)[Cx(fh ,Δx, s) + fh ] ≤ 0 (36)

in which W ′ is the net energy dissipation rate.

B. Convergence

Different from the traditional control problem in which thereare no external forces involved, the human–robot interactionmay push the system end-effector into various regions. There-fore, the convergence of the state variables needs to be inves-tigated depending on the robot end-effector position that thesystem starts in or enters. With the passivity conclusion, we canprove the following theory.

Theorem 2: The closed-loop system described by (26) and(9) guarantees the convergence of the position of end effectorto the human-dominant region if it is in human or R-DRs. Ifposition of the end-effector enters the stable safety-stop region,then it stops in this region.

Proof: For all cases, the function V represents a Lyapunov-like function, with V ≥ 0 and V = −W ′ ≤ 0 from (35) and(36). Hence, it is clear that V (t) ≤ V (0), i.e., it is upperbounded. Since V is positive-definite in s and Δθ, and V isnegative semidefinite, we have that s and Δθ are bounded.

1) Human-Dominant and Robot-Dominant RegionsWhen robot end-effector enters or starts in these regions,it is known that

w(Δx) = 1, A(x) = 0m×m ,M−1(x) = In

V =12sᵀM(q)s + Pt(δx) + ΔθᵀL−1Δθ

V = −sᵀKss − αδxᵀk2p (δx)δx

+ sᵀJᵀ(q)[Cx(fh ,Δx, s) + fh ]

s = [q − J+(q)xdes + αJ+(q)kp(δx)δx]

and the closed-loop dynamics is

M(q)s +[12M(q) + S(q, q)

]s + Kss + Jᵀ(q)kp(δx)δx

+ Yd(q, q, qr , qr )Δθ − Jᵀ(q)[Cx(fh ,Δx, s) + fh ] = 0.

(37)

ZHANG AND CHEAH: PASSIVITY AND STABILITY OF HUMAN–ROBOT INTERACTION CONTROL FOR UPPER-LIMB REHABILITATION ROBOT 241

First, the boundedness of s and Δθ is alreadyestablished. Next, kp(δx)δx is bounded by defini-tion. qr = J+(q)[xf − αkp(δx)δx] is bounded if xf =w(Δx)xdes − A(x)xdes is bounded, which is true if xdesis bounded. Then, according to (8), q is also bounded.Since q is bounded and J(q) is composed of trigonomet-ric functions of q, x = J(q)q is bounded. By usingM(x),qr has been released from the dependence on x. Therefore,the boundedness of x, w(Δx), w(Δx), xdes , M(x), andkp(δx)δx suggests the boundedness of qr . Thus, regressorYd(q, q, qr , qr ) is bounded. Next

V = −2sᵀKss − 2αk2p (δx)δxᵀδx

− 2αδxᵀkp(δx)δx∂kp(δx)

∂δxδx + sᵀJᵀ(q)

[Cx(fh ,Δx, s) + fh ]

+ sᵀJᵀ(q)[Cx(fh ,Δx, s) + fh ] + sᵀJᵀ(q)

[Cx(fh ,Δx, s) + fh ]

is bounded due to the boundedness of s, δx, δx, kp(δx),∂kp (δx)

∂δx , Cx(fh ,Δx, s) which is ensured by the defini-tion of Cx(fh ,Δx, s), and fh due to the human motioncapability limitation. According to Barbalat Lemma [25],V → 0 as t → ∞, which implies s → 0 and kp(δx)δx →0 as t → ∞. Thus, it is clear that q − J+(q)xdes → 0 ast → ∞. We then have, x → xdes , and x → H-DR.Therefore, velocities of the robot inside human-dominantor R-DRs converge to the desired ones xdes(t) and itsend-effector goes to human-dominant region with radiusah unless some strong external interference force it intothe safety-stop region, in which case the analysis will bedetailed in the next section.

2) Safety-Stop RegionIf the robot end-effector enters this region accidentally orstarts within it, it is known that Cx(fh ,Δx, s) + fh = 0,kp(δx)δx = 0, and Pt(δx) = Pt,sat with Pt,sat represent-ing the Pt(δx) value in its saturation region, which yields

s ={M−1(x)[q − J+(q)xd ], ||Δx|| ∈ (ar , at ], i.e., TS-SR

q, ||Δx|| ∈ (at,+∞), i.e., SS-SR

and

V ={−[q − J+(q)xd ]ᵀKs [q − J+(q)xd ], ||Δx||∈(ar , at ]

−qᵀKsq, ||Δx||∈(at ,+∞).

(38)

Similar to the analysis of human and R-DRs, with theassumption of bounded human input, it can be proved thatV → 0 as t → ∞ based on the definition of the controller.Therefore, in stable safety-stop region, q → 0 as t → ∞,i.e., x → 0 as t → ∞, due to the quadratic property of Von q. It means that damping is realized in stable safety-stop region. On the other hand, in transitional subregion,as t → ∞, [q − J+(q)xd ] → 0, i.e., x → xd . This means

velocity tracking toward xd in the transitional region isrealized.Therefore, when the system is inside the stable stop region,the velocities converge to zero rapidly and the system stopsinside this region unless it hits the transitional region first.If by any means, system enters the narrow transitionalregion, the system end-effector velocity converges to theweighted desired velocity, which leads the system intoeither robot-dominant or stable safety-stop regions. Sincew(Δx) reduces to zero, the weighted desired velocity alsoreduces to zero when system approaches the stable stopregion. �

V. SIMULATION AND EXPERIMENTS

Simulation and experiments were performed to demonstratethe functionality and performance of the proposed controller.Simulation was based on a 3-DOF RRR-robot to demonstratethe controller capability in 3-D space, i.e., m = 3, and on curvedtrajectory (see Section V-A). Experiments were conducted ona system consisting of 2-DOF planar robot connected in serieswith a supported human arm to demonstrate the system capa-bility in 2-D space (see Section V-B), i.e., m = 2. Results anddiscussions are also given.

A. Simulation

A 3-link RRR manipulator with end-effector subjected toexternal force to emulate human–robot interaction was used forsimulation. The three links have lengths l1 = l2 = l3 = 0.5 m,distances between their joints and respective center of masseslc1 = lc2 = lc3 = 0.25 m, moments of inertia I1 = I2 = I3 =I3 kg · m2 , and masses m1 = m2 = m3 = 2 kg. θ1 , θ2 and θ3denote the angular positions of the three joints.

A circular desired trajectory, centered at [x1cx2cx3c ]ᵀ =[0.50.50.5]ᵀ m, with orientation vectors [x1ax2ax3a ]ᵀ =[ 1√

31√3

1√3]ᵀ, [x1bx2bx3b ]ᵀ = [ 1√

6− 2√

61√6]ᵀ, and radius r =

0.3 m, as described below was used for simulation:

xdes,i(t) = xic + r cos[2πξ(t)]xia

+r sin[2πξ(t)]xib , (i = 1, 2, 3)

in which the speed distribution function ξ(t) was specified as

ξ(t) =

{ 2t2

T 2 , t ∈ [0, T2 ]

1 − 2(T −t)2

T 2 , t ∈ (T2 , T ]

to ensure zero start and end velocities. T = 15 s is the total sim-ulation duration. The time-invariant trajectory reference pointwas chosen as xi = [0 0 0]ᵀ.

The control parameters used were ah = 0.05 m, ht = 0.01,k1 = 5e4, a = 200, α = 0.5, Ks = 50 · I3 , L = 04×4 , β = 0rad, σ = 0.5π rad, and ws = 0.01. A constant force fh =[10 10 10]ᵀ N was applied to demonstrate the controller ca-pability under human interference. The outer radius of R-DRwas calculated as ar ≈ 0.322 m at p = 0.1 with the pa-rameters provided. The initial joint position q0 was set as[2.3 2.7 1.2]ᵀ rad, which meant an initial end-effector posi-tion x0 = [0.5430 0.3698 0.6078]ᵀ m by forward kinematics.These initial position and desired trajectory were chosen suchthat the initial position error, Δx0 = 0.3366 m, was larger thanar , i.e., the system started in safety-stop region, as the path laterapproached x0 and the system entered robot-dominant mode.

242 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 2, APRIL 2015

Fig. 6. Three-dimensional RRR-robot simulation results. (a) The desired and simulated trajectories in space. (b) The desired and simulated position trajectoriesof all three axis in time domain. (c) The instant distance of robot end-effector and desired central trajectory Δx in time. From (a) and (b), it is seen that the systemend-effector started at a location far away from the initial desired position xdes (0) by design, which resulted in controller running safety-stop (S-S) mode at thebeginning. As the desired trajectory xdes approached the end-effector later (intentionally by trajectory planning), controller entered robot-dominant (R-D) modeat tr . At th , the system first entered human-dominant region and started H-D Mode. R-D mode occasionally was still activated in this period to ensure that thesystem stayed in human-dominant region.

Results of simulation, which include the desired and actual tra-jectories, velocities and position errors in time, are shown inFig. 6.

At the beginning, being in safety-stop region, the device end-effector did not move (see Fig. 6.b), i.e., safety-stop controlmode functioned correctly. Later, the device entered R-DR.Robot-dominant mode was then activated and the end-effectorstarted to follow the desired human-dominant region. The re-gion error with respect to the human-dominant region definedas er (t) = max(||Δx(t)|| − ah , 0) is available in Fig. 6(c).

From the simulation, the controller is able to work with 3-Dhighly curved desired trajectory in different operation modeswith force interference. Note that due to the smooth transitionbetween different operation regions of the controller, the behav-ior of the closed-loop system is not hard-switched at boundariesof various regions. Therefore, the end-effector did not start tomove toward the human-dominant region immediately after itenters the R-DR (see Fig. 6(c). Instead, it started to move whenΔx is smaller (t = 3.31 s). After the system entered human-dominant region for the first time, er was not always exactlyzero, but sometimes slightly bigger due to force interference, inwhich cases the system entered robot-dominant mode and wasagain pulled back to human region. From t = 6.025 s, whenthe system stabilized in human-dominant mode, the root meansquare of region error er was about 8.86e–4 m.

B. Experiments

The proposed controller was implemented on an upper-limbrehabilitative robot that consisted of a 2-DOF planar robot ma-nipulator connected to an unactuated planar arm support in series(see Fig. 7). A six-axis force sensor was installed between therobot end-effector and support to measure interaction forces.

The purpose of the experiments was to test the functionalityof the controller and it was not a clinical or medical trial totest the biological response of patients. Thus, only one healthysubject (21 years, 1.72 m, 68 kg, male) was involved. Experi-mental methods employed were approved by the university andcorresponding informed consent was provided by the subject.

In all experiments with data reported, the following pa-rameters were used: k1 = 0.5, a = 0.3, Ks = 0.5 · I2 , α = 1,

Fig. 7. Experimental testbed: An upper-limb rehabilitative robotics composedof a 2-DOF planar robot, force sensor, and a 2-DOF passive arm support.

L = 1e–3 · I3 , initial system parameter estimate θ(0) =[0, 0, 0]ᵀ, p = 1e–3, β = π

6 rad, σ = π2 rad, ws = 0.01, and

ht = 0.01 m. A straight desired trajectory with zero initial andfinal velocities and accelerations, from [0.4 0.3] m to [−0.1 0.3]m, between time t = 0 s and t = 50 s, was set as

[xdes,1(t)

xdes,2(t)

]=

[0.4 − 4e-5Δt3 + 1.2e-6Δt4 − 9.6e-9Δt5

0.3

]

with the fixed trajectory reference xi = [0 0]ᵀ and Δt = t − t0 .Three sets of tests were conducted. Test I investigated the

assistive performance of the controller by its trajectory track-ing ability starting within the R-DR when human region radiuswas set as ah = 0 m. Outer radii of increasing and decreasingR-DRs were computed as ai = 0.0191 m and ar = 0.0494 m.Test II investigated the rehabilitative performance by human-dominant region following and human action encouragement inhuman region. The human-dominant region radius was set asah = 0.05 m, which meant ai = 0.0525 m and ar ≈ 0.0703 maccordingly. The system started within human-dominant regionand was intentionally pushed by the subject toward −x2 direc-tion (sideways) in this test. Trajectories, positional errors, andvelocities of Test I and II are shown in Figs. 8 and 9, respectively.The root-mean-squared region and velocity errors are given inTable I. In both cases, the actual trajectories tracked the desiredtrajectory (I) or human-dominant region (II) closely. Region and

ZHANG AND CHEAH: PASSIVITY AND STABILITY OF HUMAN–ROBOT INTERACTION CONTROL FOR UPPER-LIMB REHABILITATION ROBOT 243

Fig. 8. Experimental results for assistive performance test, i.e., ah = 0. (a) Desired and actual trajectories in space. (b) Position tracking errors of both x and ydirections in time domain. (c) Desired and actual velocities of both directions in time domain, in which v1 denotes velocity on x1 axis and v2 is velocity on x2axis.

Fig. 9. Experimental results for rehabilitative performance test with ah = 0.05 m. (a) Desired and actual trajectories in space. (b) Region errors er =max(||Δx|| − ah , 0) versus time. (c) Desired and actual velocities in time domain.

TABLE IRMS VALUES FOR REGION ERROR, er = max(||Δx|| − ah , 0),

AND VELOCITY TRACKING ERROR, WHICH IS DEFINED AS ev = ||x − xdes ||,IN 2-D EXPERIMENTS

Test I Test II

RMS er 0.01305 m 0.00147 mRMS ev 0.0102 m/s 0.0035 m/s

velocity tracking errors were both smaller in Test II comparedwith I.

Test III demonstrated the safety-stop capability of the con-troller using ah = 0.05 m. The end-effector started in human-dominant region and was intentionally pushed into the safety-stop region by a large human force. The system stopped in thesafety-stop region after forced into this area (see Fig 10).

High oscillations and errors in the data were due to slow re-sponse and high frictions of the testbed, which had a controldelay of 0.05 s. Part of the velocity errors also came from thedifferentiation of position data. Tests I and II demonstrated thesystem capabilities of trajectory-tracking and region-recoveryin assistive and rehabilitative modes and showed that the sys-tem possessed the ability to interact with the human in a sta-ble manner. Test III confirmed the safety-stop capability of the

(a)(b)

Fig. 10. Experimental results for safety-stop performance test with ah = 0.05m and ar = 0.0703 m. (a) The force spike applied at the handle by the subjectin time domain. (b) The desired and actual spatial trajectories.

controller. In Test I, velocity errors were relatively larger sincethe system operated in R-DR, where velocity tracking was lessdominant. In contrast, due to the existence of human-dominantregion and large amount of time in it, where velocity controlwas dominant, Test II had smaller velocity errors. System per-formance is expected to improve using a faster hardware withless sensory and actuation delays.

244 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 2, APRIL 2015

In actual rehabilitation, one therapy does not fit the condi-tions of different patients at different stages. The system shouldthus provide flexibility for therapists to adjust and fine tune.The proposed controller, while possessing all the properties de-scribed, allows parameter adjustment. When implementing thiscontroller on a specific device, the parameter tuning processshould start with the therapist’s assessment of the patient’s con-ditions. If the patient is at early stage of the therapy and hasminimal motion control, the radius of human-dominant regionah should be set very small to ensure maximum robot assis-tance. Then, along the training process, with the improvementof patient’s motion control ability, ah should be progressivelyset larger to encourage patient’s active movements. With ah

fixed, the size of R-DR, denoted by ar , should be determined toavoid injuries from large input torque or excessive human jointmovements. Similarly, in interaction-related controller, β and σshould be made smaller when the patient has less motion controland then adjusted to a larger value when the patient improves.Based on the region sizes determined by the therapist above,location of the end-effector and direction of the human interac-tion then define the “correct” and “incorrect” movements. Forexample, an end-effector position in human-dominant region isconsidered correct and deviation from it is considered incorrect;an interaction force directed to −sx is correct, while a forceopposite to −sx is incorrect. These clinically chosen parame-ters are then fed into the controller system to be realized by ourcontrol framework.

Other parameters k1 , Ks , L, α, ht , ws , are control perfor-mance related that should be tuned by control engineers beforethe start-up of the device and then fixed or limited to a smallfine-tuning zone for the clinical training. Choice of k1 shouldbe balanced between maximum human bearable torque and theregion tracking performance. Then, Ks and α are fine tunedtogether to achieve desired region-tracking, velocity-tracking,and steady-state error performance of the controller. Short re-gion restoration rise time can be achieved by tuning k1 up andbetter velocity tracking can be done by tuning Ks up. Smallersteady-state error can be achieved by higher α. Finally, the pa-rameter adaptation gain L should start with very small valuesand be tuned up when faster adaptation is desired.

The capabilities of adjustability, customizability, safety, andstability of the controller provide a dependable control platformfor clinical upper-limb rehabilitative training using robotic sys-tems. This controller is motion-dominant with the compensationof interaction-handling. Motion-dominant robot-assisted reha-bilitation training is an intuitive imitation of traditional ther-apies, which is still used in a lot of upper-limb rehabilitativedevices [6], [5], [27]. It has been shown to be an effective sub-stitute for traditional therapist-based training [28] with compa-rable training results and reduced human labor intensity. Otheralternative methods start to show promises in rehabilitation, forexample, the manipulation of joint energy input [29], whichis especially popular for lower limb applications [30] in stancephase of walking due to the high-power-density nature of humanlocomotion.

VI. CONCLUSION

In this paper, a controller infrastructure for upper-limb re-habilitative robotics used in reaching and tracking training,which realizes three control modes, namely human-dominantmode, robot-dominant mode, and safety-stop mode, has been

developed. With the employment of position-dependent stiff-ness and position-dependent desired trajectory, the proposedcontroller possesses the capability of automatic smooth tran-sition between different operation modes to realize “assist-as-needed” strategy. By this controller, the passivity of upper-limbrehabilitative robotic system has been established, the stabilityof the closed-loop system has also been solved with systemnonlinearities, uncertainties, and varying human–robot interac-tions; besides, the conflicts between the movements of robot andhuman user have been reduced by the proposed controller in astable manner. With this controller, the human–robot interactiverehabilitative system moves mostly inside human-dominant re-gion to ensure human exercise of proper amount and will bepulled back if entering R-DR. Built-in safety-stop functionalityis realized and triggered in case of over-sized position errorsto avoid damage to human limbs. While guaranteeing all theproperties described, the proposed controller allows parameteradjustment, which provides the therapists with the freedom todesign patient-specific training processes. Computer simulationand pilot tests on hardware have illustrated the performances ofthe aforementioned capabilities of the proposed controller.

Although this paper focuses on rehabilitation robotics, theproposed controller with the safety-stop scheme can be appliedto robotic devices that involve physical human–robot interactionto ensure safety.

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Juanjuan Zhang was born in Shandong, China. Shereceived the B.E. degree with a major in electricaland electronic engineering and a minor in comput-ing from Nanyang Technological University, Singa-pore, in 2007 with first class honors. She is currentlyworking toward the dual-Ph.D. degree from the De-partment of Mechanical Engineering, Carnegie Mel-lon University, Pittsburgh, PA, USA, and the Schoolof Electrical and Electronic Engineering, NanyangTechnological University, Singapore.

Her research interests include adaptive control, re-habilitation robots, and bipedal robots.

Chien Chern Cheah was born in Singapore. He re-ceived the B.E. degree in electrical engineering fromNational University of Singapore, Singapore, in 1990,and the M.E. and Ph.D. degrees in electrical engineer-ing, from Nanyang Technological University, Singa-pore, in 1993 and 1996, respectively.

He is an Associate Professor with the Schoolof Electrical and Electronic Engineering, NanyangTechnological University. He is an Associate Editorfor Automatica.