IEEE TRANSACTIONS ON ROBOTICS, VOL. 33, NO. 1, … Human Robot... · In general, robots driven by ... Impedance/admittance controllers may be used in both robot-in-charge and human-in-charge,

IEEE TRANSACTIONS ON ROBOTICS, VOL. 33, NO. 1, FEBRUARY 2017 169

Adaptive Human–Robot Interaction Control forRobots Driven by Series Elastic Actuators

Xiang Li, Yongping Pan, Member, IEEE, Gong Chen, and Haoyong Yu, Member, IEEE

Abstract—Series elastic actuators (SEAs) are known to offer arange of advantages over stiff actuators for human–robot interac-tion, such as high force/torque fidelity, low impedance, and toler-ance to shocks. While a variety of SEAs have been developed andimplemented in initiatives that involve physical interactions withhumans, relatively few control schemes were proposed to deal withthe dynamic stability and uncertainties of robotic systems drivenby SEAs, and the open issue of safety that resolves the conflictsof motion between the human and the robot has not been sys-tematically addressed. In this paper, a novel continuous adaptivecontrol method is proposed for SEA-driven robots used in human–robot interaction. The proposed method provides a unified formu-lation for both the robot-in-charge mode, where the robot plays adominant role to follow a desired trajectory, and the human-in-charge mode, in which the human plays a dominant role to guidethe movement of robot. Instead of designing multiple controllersand switching between them, both typical modes are integratedinto a single controller, and the transition between two modes issmooth and stable. Therefore, the proposed controller is able todetect the human motion intention and guarantee the safe human–robot interaction. The dynamic stability of the closed-loop systemis theoretically proven by using the Lyapunov method, with theconsideration of uncertainties in both the robot dynamics and theactuator dynamics. Both simulation and experimental results arepresented to illustrate the performance of the proposed controller.

Index Terms—Adaptive control, safe human–robot interaction,series elastic actuator (SEA), smooth transition.

I. INTRODUCTION

IN RECENT years, rapid advances in computing and sensortechnologies have led to the development of a variety of ini-

tiatives that involve physical interactions with humans, such asrehabilitation robots [1]–[3] and exoskeleton or wearable robots

Manuscript received April 24, 2016; revised August 5, 2016; accepted Octo-ber 31, 2016. Date of publication December 13, 2016; date of current versionFebruary 3, 2017. This paper was recommended for publication by AssociateEditor R. Carloni and Editor A. Billard upon evaluation of the reviewers’ com-ments. This work was supported in part by the Seed Fund of the EngineeringDesign and Innovation Center, National University of Singapore, Singaporeunder Grant R-261-503-002-133, and in part by the Science and EngineeringResearch Council, Agency for Science, Technology and Research (A*STAR),Singapore under Grant 1421480015.

X. Li is with the Department of Mechanical and Automation Engineering,The Chinese University of Hong Kong, Hong Kong (e-mail: [email protected]).

Y. Pan, G. Chen, and H. Yu are with the Department of BiomedicalEngineering, National University of Singapore, Singapore 117583 (e-mail:[email protected]; [email protected]; [email protected]).

This paper has supplementary downloadable material available at http://ieeexplore.ieee.org.

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.2016.2626479

[4], [5]. Rehabilitation robots are developed for the treatment ofstroke and many other pathologies, and wearable robots are usedto augment the strength or complement the ability of humans.

In those applications involving physical human–robot inter-action, safety is always the most critical concern. Safe andhuman-friendly robotic applications have become key driversfor the research on a variety of series elastic actuators (SEAs).Unlike rigid actuators, SEAs are developed by introducing anelastic element between the load and the geared motor, such thatthe inertia and nonlinear frictions of the motor and the trans-mission are decoupled from the load, and external impacts andshocks are isolated from the gear transmission [6]. The firstSEA was proposed by Pratt and Williamson in [7], which con-sists of a motor, a gear transmission, a set of linear springs,and a sensor measuring the deflection of spring, where the forceoutput of SEA is calculated using Hooke’s law. Later, Robinsoncarried out further studies on the modeling and control designof the SEA in his Ph.D. research [8]. Inspired by the originalwork, much progress has been achieved in the development andapplications of SEAs [9]–[11].

In general, robots driven by SEAs can be modeled as elastic-joint robots [12]. A variety of position control schemes havebeen developed for robotic manipulators with elastic joints, in-cluding backstepping control [13], inversion control [14], andadaptive control [15]. Proxy-based approaches [16] were alsointroduced for elastic-joint robots in [17] and [18], which enablethe robot to track the desired trajectory with smooth and dampedresponses to unexpected impacts, and thus, achieve a robust andsafe performance. The aforementioned control methods [13]–[15], [17], [18] focus on the position control problem. For therobot driven by SEAs, which closely interacts with humans, theinteraction force between the human and the robot should beconsidered as an important safety measure, and it is necessaryto control not only the position but also the interaction force. Torealize the interaction control task, hybrid position/force con-trol [19] and impedance/admittance control [20]–[25] have beenproposed for different robots.

Existing control techniques for the human–robot interactiondo not provide much flexibility for collaboration tasks, requir-ing that the robot leads or follows the human by assessing theperformance of human in real time, e.g., the paradigm of “Assist-As-Needed (AAN)” [9], [26], [27] for robotic rehabilitation. Toachieve it, the robot control scheme should be able to transit be-tween a robot-in-charge mode, where the robot plays an activerole, and a human-in-charge mode, where the human is moredominant. While a hard switching of multiple controllers (e.g.,a proxy-based controller plus a torque controller [28]) may be

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170 IEEE TRANSACTIONS ON ROBOTICS, VOL. 33, NO. 1, FEBRUARY 2017

constructed to switch leader and follower roles during interac-tion, the hard switching may result in an overall discontinu-ous control input, which further causes the chattering move-ment of robot or even compromises the safety of human.Impedance/admittance controllers may be used in both robot-in-charge and human-in-charge, but the desired impedance hasto be adjusted according to the intention of human (i.e., inter-vening or not intervening), and, thus, the overall control inputis usually not continuous either. An adaption method was pro-posed for human–robot collaboration [29] to adjust the robot’srole to lead or follow the human, but it is limited to rigid-jointrobots. In addition, most aforementioned control schemes com-monly assume the robot dynamics to be exactly known [18],[22], [23] or do not take it into account [24], [26], [29]. Whilemuch progress has been achieved in adaptive control of con-ventional rigid-joint robots to deal with the uncertain dynamics[30]–[32], relatively few works have been reported for adaptivecontrol of compliantly actuated robots, and the dynamic stabilityhas not been systematically solved.

In this paper, a novel continuous adaptive control methodwhich integrates both modes of the human–robot interaction,i.e., robot-in-charge and human-in-charge, into a single con-troller, is proposed for SEA-driven robots. The proposed con-troller is able to automatically transit between both the modes,by monitoring the variation of interaction forces. The modeof robot-in-charge suits the scenarios where the human doesnot intervene or successfully follows the movement of robotsuch that the interaction force is not large, and the mode ofhuman-in-charge is activated when the human fails to followthe movement of robot or intends to lead the robot by exert-ing forces on the robot. The transition between the two modes issmooth and stable. Instead of designing multiple controllers andswitching between them, the proposed control scheme providesa unified formulation for both the robot-in-charge and human-in-charge modes. It is capable of detecting the human motionintention and resolving the conflicts of motion between the hu-man and the robot, which thus guarantees the safe human-robotinteraction. A unified stability analysis is performed for theclosed-loop system with consideration of uncertainties in boththe robot dynamics and the actuator dynamics. Both simulationand experimental results are presented to illustrate the perfor-mance of the proposed control method. The proposed adaptivecontroller can be widely used in various SEA-driven robot sys-tems, such as rehabilitation robots and robotic exoskeletons, forhuman assistance and augmentation.

II. BACKGROUND

A. SEA

A SEA is developed by placing an elastic element into theactuator [8]. The elasticity inside SEA converts the force con-trol problem into a position control one, and hence, improvesforce accuracy. Compared with stiff actuators, the elasticitygives SEAs several unique prosperities including low mechan-ical output impedance, tolerance to impact loads, and passivemechanical energy storage [10], which makes it suitable forhuman–robot interaction.

Fig. 1. Mechanical structure of the compliant actuator: (a) principle of theactuator design (CAD model); (b) prototype.

A compact SEA has been developed in our previous workfor portable rehabilitation robots [11]. Fig. 1 illustrates anupgraded version of our compliant actuator. It mainly consistsof a servomotor with one rotary encoder, a set of linear springs,a ball screw, and two potentiometers. The motion from motor isfirst transmitted to the ball screw via a coupler, which convertsthe rotatory motion of the shaft to linear motion of the ballscrew nut. Then, the motion of nut is transmitted to the outputcarriage through the linear springs, while the carriage drivesa robot joint using a pair of cables. The encoder is installedon the motor to measure the angular displacement of the motorand the ball screw, the linear potentiometer is installed tomeasure the displacement of the linear spring, and anotherrotary potentiometer is installed in the robot joint to measurethe joint angle. Compared with the previous version [11], thedeveloped SEA is able to generate high fidelity of force controland low output impedance, and also extend the force range andthe bandwidth.

B. Dynamic Model of Robot Driven by SEAs

Let q ∈ �n denote the joint configurations, and x ∈ �m de-note the position of robot end effector in task space, such asCartesian space or image space. The position of end effector isrelated to the joint-space configurations based on the forwardkinematics as [33]

x = Γ(q) (1)

where Γ(·) ∈ �n → �m denotes a nonlinear transformation.The task-space velocity of robot end effector is related to thejoint-space velocity q as [33]

x = J(q)q (2)

where J(q) ∈ �m×n is the Jacobian matrix from joint space totask space.

The dynamic model of SEA-driven robots, with the consid-eration of human–robot interaction, can be given by [12]

M(q)q +C(q, q)q +Dq q + g(q) +K(q − θ) = τ e (3)

Bθ +Dθ θ +K(θ − q) = τ (4)

where θ ∈ �n is the vector of motor rotor shaft positions,M(q) ∈ �n×n ,C(q, q)q ∈ �n , and g(q) ∈ �n denote the in-

LI et al.: ADAPTIVE HUMAN–ROBOT INTERACTION CONTROL FOR ROBOTS DRIVEN BY SERIES ELASTIC ACTUATORS 171

ertia matrix, the vector of centripetal and Coriolis torque, and thevector of gravitational torque of robot, respectively,K ∈ �n×n

is a diagonal stiffness matrix of SEA, τ e = JT (q)f e representsa vector of interaction torque, JT (q) ∈ �n×m is the transposeof J(q), f e ∈ �m is a vector of continuous interaction force,B ∈ �n×n is a diagonal and constant inertia matrix of actuator,Dq q ∈ �n and Dθ θ ∈ �n represent the viscous frictions forthe robot and the actuator, respectively, and τ ∈ �n is a vectorof the input torque exerted on the actuator.

Equation (3) describes the well-known rigid body dynamics,and (4) describes the dynamic model of SEA. The output torqueof SEA is defined as

τ o = K(θ − q) (5)

which is proportional to the offset between the rotor shaft posi-tion and the robot joint angle θ − q.

Some important properties of the dynamic model describedby (3) and (4) are listed as follows:

1) the matricesM(q) andB are positive definite;2) the matrix M(q) − 2C(q, q) is skew-symmetric;3) the friction matricesDq = diag(dq1 , . . . , dqn ) andDθ =

diag(dθ1 , . . . , dθn ) are diagonal and positive definite,where dqi and dθi (i = 1, . . . , n) are physical parameters;

4) the first four terms of robot dynamics (3) are linear in aset of physical parametersψq = [ψq1 , . . . , ψqnq ]

T ∈ �nq

as follows [31], [33]:

M(q)q +C(q, q)q +Dq q + g(q) = Y q (q, q, q, q)ψq

(6)where Y q (q, q, q, q) ∈ �n×nq is a known dynamic re-gressor matrix, and also [31]

λ

λ + p[M(q)q +C(q, q)q +Dq q + g(q)] = W q (q, q)ψq

(7)where λ is a positive constant, p is the derivative operator,

λλ+p is a first-order low-pass operator, and W q (q, q) ∈�n×nq is another known regressor matrix.

5) The actuator friction matrix in (4) is also linear in a set ofphysical parameters ψθ = [dθ1 , . . . , dθn ]T ∈ �n as

Dθ θ = Y θ (θ)ψθ (8)

whereY θ (θ) = diag(θ1 , . . . , θn ) ∈ �n×n is a known anddiagonal regressor matrix.

In this paper, the dynamic uncertainties in both the robotand the actuator are taken into consideration, in the sense thatthe dynamic models M(q), C(q, q), Dq , g(q), and Dθ areunknown. The stiffness K and the inertia matrix B of actuatorare assumed to be well defined. By referring to Properties 4 and5, two adaptation laws will be developed to estimate unknowndynamic parameters of the robot and the actuator, respectively.

Remark 1: Note that the construction of Y θ (·) is simpledue to its diagonal structure, and the construction of Y q (·)andW q (·) is relatively computationally involved for the higherDOFs robotic system. The complexity of robot regressors is anopen issue for all regressor-based methods [25], [30], which canbe alleviated by using the iterative algorithm [34]. ��

Fig. 2. Industrial manipulator driven by SEAs. (a) Robot-in-charge: The robotperforms the manipulation task itself. (b) Human-in-charge: The human detectsa moving obstacle and guides the robot to avoid the obstacle.

III. ADAPTIVE CONTROL FOR ROBOT DRIVEN BY SEAS

In this section, an adaptive human–robot interaction con-troller is proposed for SEA-driven robots, which smoothly inte-grates both modes of comanipulation, i.e., robot-in-charge andhuman-in-charge, into a single controller.

A. Two Modes of Human–Robot Interaction

The two modes of human–robot interaction are defined asfollows.

1) Robot-in-charge: The robot takes the lead to follow adesired trajectory, while the human does not interveneor manages to follow the moment of robot such that theinteraction force is not large.

2) Human-in-charge: The human exerts interaction forces onthe robot to take control actions, while the robot becomespassive to complement human’s intention and movement.

In our proposed system, the transition between the two modesis automatically achieved by monitoring the variation of inter-action forces. The concepts of robot-in-charge and human-in-charge are general and can be specified to suit different human–robot interaction scenarios.

An example scenario of SEA-driven robot is illustrated inFig. 2. In the mode of robot-in-charge [see Fig. 2(a)], the envi-ronment is well known, and the robot is able to track the desiredtrajectory precisely, without the supervision from humans. Inthe mode of human-in-charge [see Fig. 2(b)], a moving obstaclesuddenly appears within the robot workspace, and the humanis able to react intelligently to the unforeseen change and guidethe robot for obstacle avoidance. The dual-mode comanipulationcan also be applied in sensory-feedback robotic manipulation[32] to improve robot’s performance, where the robot works inthe robot-in-charge mode when it performs the manipulationtask with sensory feedback, and the human takes control of therobot in the human-in-charge mode, if the robot moves outsidethe limited sensing zone.

Therefore, the proposed formulation provides a general solu-tion to address different control problems in the human–robotinteraction. The development of the proposed controller followsa backstepping approach [35]. The backstepping approach pro-vides a recursive method for stabilizing the system described by(3) and (4): First, based on the robot dynamics (3), a fictitious


Fig. 3. Illustration of weight factor w(fe ), where fe = [fe1 , fe2 ]T ∈ �2 :(a) w(fe ) transits smoothly between 0 and 1, and w(fe ) = 1 and w(fe ) = 0correspond to the modes of robot-in-charge and the human-in-charge, respec-tively; (b) side view.

desired input θd is proposed to achieve the robot-in-charge andhuman-in-charge modes. Next, based on the actuator dynamicequation (4), the control input τ is derived to realize the fic-titious signal, such that the actual shaft position of actuator θtracks the desired input θd .

B. Desired Input for Robot

First, a force region function is formulated to monitor thevariation of the interaction force f e as

h(f e) = ||f e ||2 −R2 (9)

where R is a positive constant that denotes the radius of theforce region. When f e is inside the force region, h(f e) ≤ 0,and vice versa.

Based on the force region, a weight factor is defined as

w(f e) = 1 − {min[0, [min(0, h(f e))]4 − [(κR)2 −R2 ]4 ]}4

[(κR)2 −R2 ]16

(10)where 0 < κ < 1 is a positive constant. An illustration ofthe weight factor w(f e) is shown in Fig. 3. From (10) andFig. 3, when f e is inside the force region such that h(f e) ≤ 0,w(f e) = 1. When f e exceeds the force region such thath(f e) > 0, it equals to 0. Since w(f e) ∈ C2 , the transitionbetweenw(f e) = 1 andw(f e) = 0 is smooth. Different valuesof weight factor will be utilized in the modes of robot-in-chargeand the human-in-charge, respectively.

By using the weight factor w(f e), a sliding vector sq for therobot dynamics is introduced as

sq = q − qr = q − w(f e)J+(q)xd + αqw(f e)J+(q)Δx(11)

where J+(q) is the pseudoinverse of J(q), Δx = x− xd , xdis the desired position, xd is the time derivative of xd , and qris a reference vector defined as

qr = w(f e)J+(q)xd − αqw(f e)J+(q)Δx (12)

where αq is a positive constant.

From (11), we have q = sq + qr and q = sq + qr . There-fore, the robot dynamics (3) can be rewritten as

M(q)sq + (C(q, q) +Dq )sq +M(q)qr

+ (C(q, q) +Dq )qr + g(q) +Kq

= Kθd +KΔθ + τ e (13)

where Δθ = θ − θd . The desired input θd is considered as afictitious input for the robot, and Δθ represents an input pertur-bation to the robot dynamics. The system in (13) can be viewedas being controlled by the input Kθd with the perturbation ofKΔθ.

From Property 4, M(q)qr + (C(q, q) +Dq )qr + g(q)in (13) can be expressed in terms of dynamic pa-rameters as M(q)qr + (C(q, q) +Dq )qr + g(q) =Y q (q, q, qr , qr )ψq . Hence, (13) leads to

M(q)sq + (C(q, q) +Dq )sq + Y q (q, q, qr , qr )ψq +Kq

= Kθd +KΔθ + τ e . (14)

We can now propose the desired input θd as

θd = q −K−1(w(f e)Kqsq − Y q (q, q, qr , qr )ψq ) (15)

where Kq is a diagonal and positive definite matrix, and ψq

represents the estimate of ψq , which is updated by

˙ψq = −Lq [Y T

q (q, q, qr , qr )sq +W Tq (q, q)ep ] (16)

where Lq ∈ �nq ×nq denotes a positive definite matrix, ep =W q (q, q)Δψq is the prediction error, and Δψq = ψq − ψq .The update law in (16) is designed as a composite updating law[36], [37], where the convergence of estimated parameters isdriven by both the sliding vector sq and the prediction error ep .That is, the parameter updating does not stop until both sq andep reduce to zero. Note thatW q (q, q)ψq is computed from (3)and (7) as:W q (q, q)ψq = λ

λ+p (τ e +K(θ − q)).The desired input θd integrates both modes of human–robot

interaction as follows:1) Robot-in-charge: When the interaction force f e is inside

the force region where h(f e) ≤ 0, w(f e) = 1, the slid-ing vector sq = q − J+(q)xd + αqJ

+(q)Δx, and thedesired input becomes

θd = q −K−1(Kqsq − Y q (q, q, qr , qr )ψq ) (17)

where the position control term αqJ+(q)Δx and the ve-

locity control term q − J+(q)xd in the sliding vectordrive the robot to track the desired position xd .

2) Human-in-charge: When the interaction force f e is out-side the force region where h(f e) > 0, w(f e) = 0,and hence qr reduces to 0 from (12) and also qr =0. Hence, Y q (q, q, qr , qr )ψq = M(q)qr + (C(q, q) +Dq )qr + g(q) → g(q), where M(q), C(q, q), Dq ,and g(q) represent the approximate models for M(q),C(q, q), Dq , and g(q), respectively. Then, the desiredinput is simplified as

θd = q +K−1 g(q). (18)


The desired input in (18) leads to the dynamic equationof the robot as: M(q)q + (C(q, q) +Dq )q + g(q) −g(q) = KΔθ + τ e . The use of the composite adaptationin (16) ensures the convergence of the prediction error[36], [37], thus g(q) → g(q). In addition, if the actualposition of rotor shaft θ tracks the desired input θd suchthat Δθ = θ − θd → 0, we have

M(q)q + (C(q, q) +Dq )q = τ e (19)

which is a damping system in the sense that the human isable to take control of the robot by providing the direc-tional information for the robot.

The impedance of robot varies from a high value to a lowvalue, when the desired input transits from the mode of robot-in-charge to the mode of human-in-charge, and then, the humancan easily adjust or guide the movement of robot while therobot provides assistive forces. In addition, the transition stagewhere 0 < w(f e) < 1 is useful for shared-control scenarios tocombine human control efforts and robot controllers, and theproportion of transition stage during interaction can be increasedby setting the parameter κ small. Since the weight factor w(f e)is continuous, the proposed desired input θd is also continuousthroughout human–robot interaction.

The radius of force region is adjustable and should be spec-ified to suit specific scenarios. It can be set large where therobot plays a more active role during human-robot interaction,and set small where the human is more dominant. For example,in robotic rehabilitation [5], when patients are fully paralyzedand the robot is expected to take the lead, a large force regionis desired such that most signals and interactions from humanare treated as disturbances. When motion-related functions arepartially restored, the force region can be scaled down such thatthe human motion intention is encouraged [27].

Moreover, when the force region is set arbitrarily large, theproposed control method is specified as a fully robot-in-chargecontroller, which is useful for applications where the desiredposition is well defined and the robot behavior is highly re-liable. When the force region reduces to a point, the pro-posed controller becomes a fully human-in-charge controller,and the robot is then treated as a human direct teaching sys-tem where the directional information is always provided bythe human. The force region with different sizes is illustratedin Fig. 4.

Substituting the desired input θd (15) into the dynamic equa-tion of robot (14) yields

M(q)sq + (C(q, q) +Dq + w(f e)Kq )sq

+Y q (q, q, qr , qr )Δψq = KΔθ + τ e . (20)

To analyze the stability of the dynamic equation of robot, aLyapunov-like candidate Vq is proposed as

Vq =12sTq M(q)sq +

12ΔψT

q L−1q Δψq . (21)

Fig. 4. Force region with different sizes: (a) when the size of h(fe ) ≤ 0increases, the robot is more dominant; (b) when the size decreases, the humanis more dominant.

Differentiating (21) with respect to time and substituting (20)into it, we have

Vq = sTq M q (q)sq +12sTq M q (q)sq − ΔψT

q L−1q

˙ψq

= sTq (12M q (q) −C(q, q))sq − sTq (Dq + w(f e)Kq )sq

+ sTq (KΔθ + τ e) − ΔψTq L

−1q

˙ψq

− sTq Y q (q, q, qr , qr )Δψq . (22)

Substituting the update law (16) into (22) and using Property 2yields

Vq = − sTq (Dq + w(f e)Kq )sq + sTq (KΔθ + τ e)

− ΔψTq W

Tq (q, q)W q (q, q)Δψq . (23)

If Δθ = 0 and τ e is bounded, then the sliding vector sq isbounded, and thus, the stability of robot dynamic equation canbe derived. Note that the aforementioned analysis is applicableto the robot subsystem. The stability analysis for the overallsystem and the convergence of Δθ → 0 will be given later.

C. Control Input for SEA

In this section, the control input for the actuator τ is developedsuch that the actual position of the actuator θ tracks the desiredinput θd and hence Δθ → 0.

First, another sliding vector is introduced for actuator as

sθ = θ − θr = θ − θd + αθ (θ − θd) = Δθ + αθΔθ. (24)

where αθ is a positive constant, and θr is a reference vectorwhich is defined as

θr = θd − αθΔθ. (25)

The control input of the actuator is now proposed as

τ = K(θ − q) −Kθsθ +Bθr + Y θ (θr )ψθ (26)


Fig. 5. Block diagram of the closed-loop system.

where Kθ is a positive definite matrix, and the uncertain dy-namic parameters ψθ are updated by

˙ψθ = −LθY T

θ (θr )sθ (27)

where Lθ ∈ �n×n is a positive definite matrix. The update lawin (27) is specified as a direct adaption law [31]. Hence, theconvergence of Δθ → 0 does not need the convergence of esti-mated parameters.

By using the sliding vector sθ , (4) is expressed as

Bsθ +Dθsθ +K(θ − q) +Bθr + Y θ (θr )ψθ = τ . (28)

By substituting (26) into (28), the closed-loop equation forthe human–robot interaction system is obtained as

Bsθ + (Dθ +Kθ )sθ + Y θ (θr )Δψθ = 0 (29)

where Δψθ = ψθ − ψθ . The block diagram of the closed-loopsystem is shown in Fig. 5.

A Lyapunov-like candidate is developed to analyze the sta-bility of the closed-loop system as

V = Vq +12sTθ Bsθ +

12ΔψT

θ L−1θ Δψθ + αθΔθTKθΔθ.

(30)Differentiating (30) with respect to time, and substituting (23),(27), and (29) into it yield

V = Vq + sTθ Bsθ − ΔψTθ L

−1θ

˙ψθ + 2αθΔθTKθΔθ

= − sTq (Dq + w(f e)Kq )sq + sTq (KΔθ + τ e)

− ΔψTq W

Tq (q, q)W q (q, q)Δψq

+ 2αθΔθTKθΔθ − sTθ (Dθ +Kθ )sθ . (31)

Note that sθ = Δθ + αθΔθ, (31) can be written as

V = − sTq (Dq + w(f e)Kq )sq + sTq KΔθ

+ sTq τ e − sTθ Dθsθ − ΔθTKθΔθ − α2

θΔθTKθΔθ

− ΔψTq W


= sTq τ e − sTθ Dθsθ − ΔθTKθΔθ

− ΔψTq W


− [sTq ΔθT ]P [sTq ΔθT ]T (32)

where P ∈ �4×4 is

P =

[Dq + w(f e)Kq − 1

2K

− 12K α2

θKθ

]. (33)

If the controller parameters αθ andKθ are chosen so that

α2θλmin [KθDq ] >

14λmax[K2 ] (34)

where λmin [·] and λmax[·] represent the minimum and the maxi-mum eigenvalues, respectively, thenP is positive definite. Next,(32) is written as

V +W = sTq τ e (35)

where

W = sTθ Dθsθ + ΔθTKθΔθ + [sTq ΔθT ]P [sTq ΔθT ]T

+ ΔψTq W

Tq (q, q)W q (q, q)Δψq (36)

and W is positive if condition (34) is satisfied. Integrating (35)over [0, t] yields∫ t

0sTq (ς)τ e(ς)dς = V (t) − V (0) +

∫ t

0W (ς)dς. (37)

Since W is positive, we have∫ t

0sTq (ς)τ e(ς)dς ≥ V (t) − V (0). (38)

The aforementioned inequality demonstrates the passivity ofthe dynamics between the input τ e and output sq [12]. Thepassivity shows that at any time the energy stored in the systemV (t) − V (0) is less than or equal to the energy introduced fromhumans. Therefore, the proposed control scheme guarantees thesafe human–robot comanipulation.

Now, we have the following theorem to state the boundednessof the sliding vector sq .

Theorem 1: The closed-loop system (20) and (29) gives riseto the boundedness of the sliding vector sq , if the interactionforce f e is bounded, and the control parameters αθ andKθ arechosen such that

2λmin [P ] − 1 > 0. (39)

Proof: If the interaction force f e is bounded, the torqueτ e = JT (q)f e is also bounded since J(q) is a trigonometric


function of q. Note that∫ t

0sTq (ς)τ e(ς)dς ≤ 1

2

∫ t

0||sq (ς)||2dς +

12

∫ t

0||τ e(ς)||2dς.

(40)From (37) and (40), we have

−V (0) +∫ t

0W (ς)dς ≤ 1

2

∫ t

0||sq (ς)||2dς +

12

∫ t

0||τ e(ς)||2dς.

(41)If condition (39) is satisfied, P is positive definite, thus W ispositive. From (36), W ≥ λmin [P ]||sq ||2 . Therefore, we have∫ t

0 λmin [P ]||sq (ς)||2dς≤ 1

2

∫ t

0 ||sq (ς)||2dς + 12

∫ t

0 ||τ e(ς)||2dς + V (0). (42)

which also implies that∫ t

0(2λmin [P ] − 1)||sq (ς)||2dς ≤

∫ t

0||τ e(ς)||2dς + 2V (0).

(43)Since 2λmin [P ] − 1 > 0, we can conclude that the boundednessof τ e ensures the boundedness of sq . �

In particular, in the robot-in-charge mode (||f e || < R,w(f e)= 1), we have sq = q − J+(q)xd + αqJ

+(q)Δx, and it canbe then derived from (43) that∫ t

0||Δx(ς)||2dς ≤

∫ t

0

η1

α2q

||f e(ς)||2dς +η2 + αq ||Δx(0)||2

α2q

(44)

where η1�= ||J(q)JT (q)||2

2λm in [P ]−1 and η2�= 2||J(q)||2 V (0)

2λm in [P ]−1 , which arepositive. Hence, the boundedness off e ensures the boundednessof tracking errors Δx.

In the human-in-charge mode (R ≤ ||f e || < Bf ,w(f e) = 0,where Bf is the upper bound of f e ), we have sq = q, and thus,inequality (43) leads to∫ t

0||q(ς)||2dς ≤

∫ t

0η3 ||f e(ς)||2dς + η4 (45)

where η3�= ||JT (q)||2

2λm in [P ]−1 and η4�= 2V (0)

2λm in [P ]−1 , are positive. Theinequality (45) implies that the boundedness of f e ensures theboundedness of q. That is, the joint-space velocity of robot islimited when the human guides the movement of robot.

Then, the following theorem states the stability of the overallSEA-driven robot system consisting of both the robot dynamicsand the actuator dynamics:

Theorem 2: The proposed control scheme described by (15),(16), (26), and (27) ensures the stability of the SEA-drivenrobotic system described by (3) and (4) and the convergenceΔθ → 0, if the interaction force f e is bounded, and the controlparameters are chosen such that condition (39) is satisfied.

Proof: The boundedness of f e ensures the boundedness ofτ e , and the boundedness of τ e ensures the boundedness of sqif condition (39) is satisfied, as shown in Theorem 1. Frominequality (38), we have

V (t) ≤∫ t

0sTq (ς)τ e(ς)dς + V (0). (46)

Since both sq and τ e are bounded, V is also bounded. From(30), the boundedness of V ensures the boundedness of sq ,Δψq , sθ , Δθ, and Δψθ . Since all state variables are bounded,the closed-loop system is stable.

In addition, the boundedness of sθ and Δθ ensures the bound-edness of Δθ. Hence, Δθ is uniformly continuous. Then, from(37), we have∫ +∞

0 {λmin [P ]ΔθT (t)Δθ(t)}dt ≤ ∫ +∞0 W (t)dt

= V (0) − V (+∞) +∫ +∞

0 sTq (t)τ e(t)dt (47)

where the right side is bounded. Hence, Δθ ∈ L2(0,+∞). Itthen follows [31], [33] that Δθ → 0. That is, the convergenceof θ → θd is guaranteed. �

The above stability analysis is not dependent on specificvalues of the weight factor w(f e), and thus, is applicable tothe robot-in-charge mode, the human-in-charge mode, andalso the transition stage. If the interaction force is furthernegligible, the proposed controller works in the robot-in-chargemode. Then, the following theorem states the convergence oftracking errors.

Theorem 3: The closed-loop system (20) and (29) gives riseto the convergence of tracking errors in the robot-in-chargemode, if the interaction force f e is negligible, and the controlparameters are chosen such that condition (34) is satisfied.

Proof: Since the interaction force is negligible, we haveτ e → 0. If condition (34) is satisfied, we have V = −W ≤ 0.Since V > 0, and V ≤ 0, V is bounded. The boundednessof V ensures the boundedness of sq , Δψq , sθ , Δψθ , andΔθ. The boundedness of sθ and Δθ ensures the bounded-ness of Δθ, and thus, Δθ is uniformly continuous. Since sq =q − J+(q)xd + αqJ

+(q)Δx in the robot-in-charge mode, wehave J(q)sq = Δx+ αqΔx. The boundedness of sq ensuresthe boundedness of Δx and Δx, and thus, the boundedness ofx. The boundedness of Δx and sq ensures the boundedness of qfrom (11). Then, both qr and qr are bounded, as seen from (12).From the robot dynamic (20), it is obtained that sq is bounded,since τ e → 0, and Δθ, sq , Δψq , qr , and qr are all bounded.Therefore, sq is also uniformly continuous.

From (37), it is easy to verify that∫ +∞0 {λmin [P ](ΔθT (t)Δθ(t) + sTq (t)sq (t))}dt

≤ ∫ +∞0 W (t)dt = V (0) − V (+∞) (48)

where V (0) − V (+∞) is bounded. Hence, we have sq ,Δθ ∈L2 . Then, it follows [31], [33] that sq ,Δθ → 0. The conver-gence of sq → 0 ensures that x→ xd and x→ xd as t→ ∞.That is, the convergence of tracking errors to zero is achieved.�

IV. ADAPTIVE OBSERVER FOR ROBOTS DRIVEN BY SEAS

Note that the control input (26) for the SEA is dependent onthe derivatives of the desired input θd and θd , and hence, the ac-celeration information q and its derivatives are required. In thissection, an observer is constructed to estimate the desired inputas θd , and thus, eliminate the requirement of the accelerationand its derivatives.


Observer techniques for LTI systems, e.g., classic Luenbergerobservers [38], [39], have been well explored. In this paper,we consider the problem of uncertain dynamics and proposean adaptive observer to estimate the desired input θd , whichfollows a similar development in [15] as{

˙θd = η + βeoη = B−1(τ −K(θ − q) − Dθoη +Keeo)

(49)

where θd denotes the estimated signal of θd , eo = θd − θdrepresents the observation error, η is an auxiliary variable,β is a positive constant, and Ke is a positive-definite ma-trix. The matrix Dθo denotes the approximate model forDθ , which is updated through another adaptation law for theobserver as

˙ψo = −LoY θ (η)z (50)

where z = θ − η, ψo = [ψo1 , . . . , ψon ] ∈ �n is the vector ofestimated parameters for the observer, andLo ∈ �n×n is a pos-itive definite matrix.

Note that θd is obtained by integrating ˙θd , while ¨

θd is ob-

tained by differentiating ˙θd with respect to time. By using the

estimated signals, a new sliding vector is then defined as

sθ = θ − ˙θr = θ − ˙

θd + αθ (θ − θd) = Δ˙θ + αθΔθ (51)

where the reference vector ˙θr is defined as

˙θr = ˙

θd − αθΔθ (52)

and Δθ = θ − θd .The control input for the SEA is now proposed as

τ = K(θ − q) −Kθ sθ +B¨θr + Y θ (

˙θr )ψθ (53)

where the uncertain parameters ψθ are updated by

˙ψθ = −LθY T

θ ( ˙θr )sθ . (54)

Using the estimated signals θd , ˙θd , and ¨

θd , τ in (53) does notrequire the acceleration information and its derivatives.

Equation (4) can be expressed in terms of sθ as

B ˙sθ +Dθ sθ +K(θ − q) +B¨θr + Y θ (

˙θr )ψθ = τ . (55)

Substituting (53) into (55) yields the closed-loop equation

B ˙sθ + (Dθ +Kθ )sθ + Y θ (˙θr )Δψθ = 0. (56)

Next, multiplying both sides of (49) with B and using (4),

we can get the closed-loop observer dynamics for ˙θd as

Bz +Dθz +Keeo = (Dθo −Dθ )η = −Y θ (η)Δψo

(57)where Δψo = ψo − ψo , and ψo is the dynamic model forDθ .

To prove the stability of the overall system, a Lyapunov-likecandidate is proposed as

V = Vq +12sTθ Bsθ +

12ΔψT

θ L−1θ Δψθ + αθΔθ

TKθΔθ

+12zTBz +

12ΔψT

o L−1o Δψo . (58)

Differentiating (58) with respect to time and substituting (23),(50), (54), (56), and (57) into it, one obtains

˙V = − sTq (Dq + w(f e)Kq )sq + sTq (KΔθ + τ e)

− ΔψTq W


− sTθ (Dθ +Kθ )sθ + 2αθΔθTKθΔ

˙θ

− zTDθz − zTKeeo . (59)

Note that Δθ = Δθ − eo , sθ = Δ˙θ + αθΔθ, and also z =

Δ˙θ + βeo . Therefore, (59) can be written as

˙V = sTq τ e − sTθ Dθ sθ − zTDθz

− ΔψTq W


− [sTq ΔθTΔ˙θT eTo ]P [sTq Δθ

TΔ˙θT eTo ]T (60)

where P ∈ �8×8 is given by

P =

⎡⎢⎢⎢⎢⎣Dq + w(f e)Kq − 1

2K 0 12K

− 12K α2

θKθ 0 0

0 0 Kθ12Ke

12K 0 1

2Ke βKe

⎤⎥⎥⎥⎥⎦ . (61)

If the parameters αθ ,Kθ , β, andKe are chosen so that

α2θλmin [KθDq ] >

14λmax[K2 ]

βλmin [Kθ ] >14λmax[Ke ]

λmin [Ke ]{βλmin [Kθ ] − 14λmax[Ke ]}{α2

θλmin [KθDq ]

− 14λmax[K2 ]} > 1

4α2θλmax[K2

θK2 ] (62)

then P is positive definite.The proposed adaptive tracking control scheme in (53) also

guarantees the passivity of the human–robot interaction system.Similarly, it can be shown that the boundedness of the interactionforce f e ensures the boundedness of the sliding vector sq , whenthe control parameters are chosen such that 2λmin [P ] − 1 > 0.In addition, the stability of the overall system and the conver-gence of Δθ → 0 are guaranteed. When the interaction force isfurther negligible such that τ e → 0, the controller works in themode of robot-in-charge, and the convergence of tracking errorsis also guaranteed, that is, x→ xd and x→ xd as t→ ∞.

Remark 2: In actual implementations, the control parame-ters are tuned according to the performance of the robot. Ingeneral, better velocity tracking can be achieved by tuning Kq


Fig. 6. Proposed control method was implemented in a 2-DOF robotic manipulator driven by SEAs, and the robot end effector was controlled to track thedesired trajectory as: xd1 = 0.2 + 0.2 sin (0.5t) m, and xd2 = 0.4 + 0.2 cos (0.5t) m. During the manipulation, interaction forces were exerted on the robotend effector to guide the movement of robot. (a) Path of robot end effector. (b) Tracking error Δx = x− xd . (c) Interaction forces fe and weight factor w(fe ).(d) Observation error eo = θd − θd .

and Kθ up, better position tracking and smaller steady-stateerror can be achieved with higher αq and αθ , and faster conver-gence of observation errors can be obtained by tuning β andKe

up. Finally, the adaptation gains Lq , Lθ , and Lo should startwith very small values and be tuned up when faster adaptationis desired. The control gains should be fine tuned together toachieve desired transient and steady-state performance. ��

Remark 3: The proposed controller does not require the ex-act knowledge of force limit. Instead, the force feedback is onlyrequired in the weight factor, which is specified as a measurefor the transition between both modes and embedded inside thecontrol scheme. Then, the usage of force feedback follows aqualitative manner, i.e., if the interaction force is not large, therobot-in-charge mode is in effect, and if the force is large, thehuman-in-charge mode is activated. In addition, the proposedmethod provides a general solution, in the sense that other sen-sory feedback, which is able to qualitatively describe the changeof human motion intention, can also be specified as the measureof transition instead of interaction force, such as EMG/EEGsensors for the measurement of electric activity from humanmuscles or brains, or inertial measurement units or 3-D sensorscapable of detecting human gestures. ��

Remark 4: Most control methods for human-robot interac-tion, e.g., proxy-based control [16], or DOB-based adaptive con-trol [40], commonly treat the interaction force as disturbance andreject it, which may be applicable to the robot-in-charge modeonly. For the human–robot collaboration, the robot should beable to follow the human’s guidance in the human-in-chargemode, by considering the interaction force as a position con-trol term. A hard switching of multiple controllers can be usedto switch leader and follower roles during interaction, but itmay result in an overall discontinuous control input. In adaptiveimpedance control [25], the adjustment of desired impedancefor both modes may encounter the similar issue. This paperpresents a continuous adaptive control method for human–robotinteraction, and the technical novelty is that a new sliding vec-tor (11) and a new reference vector (12) are proposed such thatboth the dynamic compensation and the desired input θd varywith the interaction force, to switch leader and follower rolesand thus suit the requirements of human–robot collaboration.The stability is also rigorously proved, with the considerationof nonlinearities and uncertainties in the dynamic model, thecoupling between the robot and the actuator, and the transitionbetween both modes. ��

V. APPLICATIONS

Numerical simulations and experimental results are presentedto demonstrate the effectiveness of the proposed control method.The simulation studies and experiments were carried out indifferent SEA-driven robotic systems.

A. Two-Link Robotic Manipulator

First, the proposed control method is implemented in the two-link robot as illustrated in Fig. 2. The two links have lengthsl1 = l2 = 0.42 m, distances between their joints and respectivecenter of masses lc1 = lc2 = 0.21 m, moments of inertia I1 =I2 = 0.5 kg · m2 , and masses m1 = m2 = 2 kg. The dynamicparameters of the SEAs were set as: the inertia matrix B =0.2I2 kg × m2 , where I2 ∈ �2×2 is an identity matrix, thefriction matrix Dθ = I2 kg × m2/s, and the stiffness matrix:K = 50I2 N · m/rad. The human interacts with the robot byexerting forces on the end effector of the robot, and the positionof the end effector is specified in Cartesian space.

The robot end effector started from the initial position at(0.19, 0.7) m, and was controlled to track the trajectory{

xd1 = 0.2 + 0.2 sin (0.5t) mxd2 = 0.4 + 0.2 cos (0.5t) m.

(63)

The force region in (9) was specified as:h(f e) = f 2e1 + f 2

e2 −22 ≤ 0, where fe1 and fe2 represent the forces in two coordi-nates, respectively, the radiusR = 2 N, and the parameter of theweight factor (10) was set as: κ = 0.7. The control parametersin the desired input θd (15) were set as: αq = 4, Kq = 20I2 ,and Lq = 5 × 10−5I5 where I5 ∈ �5×5 is an identity matrix,and the parameters of control input for actuator τ (53) were setas: αθ = 4,Kθ = 20I2 , and Lθ = 5 × 10−4I2 .

The path of the robot end effector is shown in Fig. 6(a), andthe tracking error is shown in Fig. 6(b), which reduces to zeroat around 2 s. During the manipulation, the human exerted aninteraction force on the robot end effector as

fei =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0, t ≤ t15 t−t1δt , t1 < t < t1 + δt

5, t1 + δt ≤ t ≤ t2 − δt5 − 5 t2 −tδt , t2 − δt < t < t2

0, t ≥ t2

(64)

where i = 1, 2, t1 = 8.5 s, t2 = 11.6 s, and δt = 0.004 s. Equa-tion (64) describes a very steep ramp input reaching the valueof 5 N in 0.004 s.


Fig. 7. Performance of the proposed controller with different settings of control parameters: (a) very small parameters; (b) small parameters; (c) well-tunedparameters; (d) too large parameters.

Fig. 8. Transition of the proposed controller between robot-in-charge (w(fe ) = 1) and human-in-charge (w(fe ) = 0).

Fig. 9. Online adjustment of desired impedance: a high impedance is specified for robot-in-charge (w(fe ) = 1), while a low impedance is specified forhuman-in-charge (w(fe ) = 0).

When the interaction force exceeded the force region suchthath(f e) > 0, the weight factorw(f e) decreased from 1 to 0 asshown in Fig. 6(c). Then, the proposed controller transited fromthe mode of robot-in-charge to the mode of human-in-charge,and the robot became passive. The human pushed the endeffector away from the desired trajectory xd , to guide the robotto avoid suddenly appearing obstacles as illustrated in Fig. 2.When the human force f e disappeared such that h(f e) ≤ 0, thecontroller transited to the mode of robot-in-charge again, whichdrove the robot end effector to track the desired trajectory.

To eliminate the requirement of acceleration information andits derivatives, the observer (49) was constructed to estimate thedesired input θd . The observer parameters were set as: β = 100,Ke = 80I2 , and Lo = 20I2 . The initial estimate of Dθo was

set as: Dθo = 0. The estimated signals θd , ˙θd , and ¨

θd wereused instead of θd , θd , and θd in the control input τ , and theobservation error is shown in Fig. 6(d), which is less than 0.2 radthroughout the course of manipulation.

A series of simulations were further carried out to test theperformance of the proposed controller with different settingsof control parameters, which include the followings:

1) very small parameters (αq = 1, αθ = 1, Kq = 15I2 ,Kθ = I2 , β = 1, Ke = 8I2 , and Lq , Lθ , Lo remainthe same);

Fig. 10. Single-link robot driven by SEA, where the human interacts with therobot by exerting forces on the end point of the robot link.

2) small parameters (αq = 1, αθ = 4, Kq = 15I2 , Kθ =20I2 , β = 100, Ke = 80I2 , and Lq , Lθ , Lo remain thesame);

3) well-tuned parameters (all remain the same);4) too large parameters (αq = 10, αθ = 4, Kq = 20I2 ,Kθ = 20I2 , β = 100, Ke = 80I2 , and Lq , Lθ , Lo re-main the same).

Random noises were added in simulations and treated asmeasurement noises, and the results are shown in Fig. 7. Whenthe control parameters were set very small [setting 1)] such that


Fig. 11. Experiment 1: The torque region was specified sufficiently large such that the weight factor remained 1, and the proposed controller was treated as afully robot-in-charge controller. (a) Angular position of robot. (b) Tracking error Δq = q − qd . (c) Output torque of actuator τo = K (θ − q). (d) Observationerror eo = θd − θd . (e) Control output τ .

conditions in (62) were not satisfied, the closed-loop systemwas unstable, as shown in Fig. 7(a). When conditions in (62)were satisfied but the parameters were still small [setting 2)],the tracking errors in steady state were very large, as shownin Fig. 7(b). Tracking errors were reduced by using well-tunedparameters [setting 3)] as shown in Fig. 7(c). However, if theparameters were set too large [setting 4)], noises were amplified,resulting in an unstable system as in Fig. 7(d).

To illustrate the transition performance of the proposed con-troller, a comparison was also performed with an online ad-justment of desired impedance in adaptive impedance control[25]. In the comparison, the interaction force was specified asa sinusoid as: f e = [2.1 sin (0.4t), 0]T . The desired impedancewas switched from a large value in the robot-in-charge mode(w(f e) = 1), to a low value in the human-in-charge mode(w(f e) = 0). The comparison results are shown in Figs. 8 and9, where the control output for the proposed controller is alwaysbounded, which ensures a stable and smooth transition betweenboth modes, but the control output of the online adjustment isdiscontinuous, resulting in large position errors around t = 7 s[see Fig. 9(b)].

B. One-DOF Robot Arm

Then, the proposed control method was implemented in aone-DOF robot arm as shown in Fig. 10. The single-link robotis driven by our SEA introduced in Section II-A, with a cali-brated stiffnessK = 0.0024 N · m/rad, and the human interactswith the robot by exerting forces on the end point of the robotlink. Due to hardware limitations, it is unable to measure theinteraction force directly. Based on the general formulation, theoutput torque of actuator is measured to reflect the change ofhuman motion intention as follows.

1) In the robot-in-charge mode, the human does not inter-vene such that τ e is very small, and τ o = K(θ − q) ≈M(qd)qd +C(qd , qd)qd +Dq qd + q(qd) (qd denotesthe desired joint angle) in steady state from (3), whichis also small because of the light robot link and the lowdesired velocity.

2) In the robot-in-charge mode, the human guides the robot’smovement, resulting in a large relative position θ − q andthus a large output torque of actuator.

First, a torque region function was formulated similarly as

h(τo) = τ 2o −R2 (65)

where R denotes the radius of torque region. When the torqueτo is inside the torque region such that h(τo) ≤ 0, the controller

works in the mode of robot-in-charge. When the torque is verylarge such that h(τo) > 0, the controller works in the modeof human-in-charge. The desired angle for the robot qd wasspecified as a sine wave trajectory as

qd = 0.28 sin (πt) rad. (66)

In the first experiment, the radius of the torque region R wasset sufficiently large as: R = 14.3 Nm, and thus, the weightfactor w(τo) remained 1. The proposed controller was specifiedas a fully robot-in-charge controller, where the robot alwaysplayed a dominant role. The control parameters in the desiredinput θd (15) were set as: αq = 10, Kq = 100, and Lq = 10−5 .To eliminate the requirement of acceleration information andits derivatives, an observer was developed in (49) to estimatethe desired input as θd , and the parameters of observer were setas: Ke = 100, and β = 60, and Lo = 2. Then, the parametersin the control input τ in (53) were set as: αθ = 18, Kθ = 120,and Lθ = 10−5 . The angular position of the robot is shown inFig. 11(a), and the tracking error is shown in Fig. 11(b) whichis less than 0.1 rad. The output torque is shown in Fig. 11(c).The observation error is shown in Fig. 11(d), which is around0.02 rad. The results show that the robot successfully trackedthe desired trajectory in the mode of robot-in-charge.

In the second experiment, the radius R was set arbitrarilysmall such thatR→ 0, the weight factorw(τo) remained 0, andthe proposed controller was specified as a fully human-in-chargecontroller. The robot became passive, and the human guided themovement of robot by providing the directional information forthe robot. The control parameters remained the same. The an-gular position of robot is shown in Fig. 12(a), which impliesthat the robot was controlled by the human, regardless of the de-sired time-varying trajectory. The error between the robot jointangle and the desired angle is shown in Fig. 12(b). The outputtorque of SEA is shown in Fig. 12(c). Since the human keptexerting force on the robot end effector, the relative angulardisplacement between the robot link and the rotor shaft of ac-tuator θ − q was very large, and the output torque of SEA wasalso very large. The observation error shown in Fig. 12(d) isabout 0.02 rad.

In the third experiment, the torque region h(τo) was specifiedwith the radius ofR = 1.17 Nm, and the parameter of the weightfactor was set as κ = 0.67. Therefore, when the output torque τoexceeded the torque region due to the interaction with human,h(τo) > 0,w(τo) = 0, the controller transited from the mode ofrobot-in-charge to the mode of human-in-charge, and vice versa.The control parameters also remained the same. The angular


Fig. 12. Experiment 2: The torque region was specified arbitrarily small such that the weight factor remained 0, and the proposed controller was treated as afully human-in-charge controller. (a) Angular position of robot. (b) Tracking error Δq = q − qd . (c) Output torque of actuator τo = K (θ − q). (d) Observationerror eo = θd − θd . (e) Control output τ .

Fig. 13. Experiment 3: The torque region was specified as h(τo ) = τ 2o − 1.172 ≤ 0, and the proposed controller transited from the mode of robot-in-charge

to the mode of human-in-charge when the weight factor reduced to 0, and vice versa. (a) Angular position of robot. (b) Tracking error Δq = q − qd . (c) Outputtorque of actuator τo = K (θ − q). (d) Observation error eo = θd − θd . (e) Control output τ .

Fig. 14. Hard-switching of a proxy-based controller for robot-in-charge (w(τo ) = 1) and a damping controller for human-in-charge (w(τo ) = 0).

position of robot and the tracking error are shown in Figs. 13(a)and 13(b), respectively. At the beginning, the controller wasworking in the mode of robot-in-charge, and the robot was ableto track the desired trajectory where the tracking error was lessthan 0.1 rad. During the manipulation (t = 2 ∼ 3 s, 6 ∼ 7 s,etc.), the human exerted force on the robot, which led to a largeoutput torque τo , the weight factorw(τo) reduced to zero, and therobot became passive. Therefore, the human was able to pushthe robot away from the desired trajectory. When the humancontrol effort removed, w(τo) smoothly increased to 1, and thecontroller transited to the mode of robot-in-charge again, whichdrove the robot to track the desired trajectory. The output torqueof SEA is shown in Fig. 13(c). The observation error is shown inFig. 13(d), which is less than 0.08 rad. The experimental resultsare also shown in the video submission.

In the experiment, a comparison was also performed withthe hard-switching of a proxy-based controller and a dampingcontroller [41]. The proxy-based controller works in the robot-in-charge mode (w(τo) = 1), and the damping controller is ac-tivated in the human-in-charge mode (w(τo) = 0). The resultsare shown in Fig. 14. As seen from Fig. 14(a), the proxy-basedcontroller was able to deal with the interaction torque when itwas not large (at t = 0.7 s and t = 1.8 s). However, when theinteraction torque became larger such that the control schemewas switched to the damping controller, the overall controloutput was discontinuous during the transition stage, which then

resulted in a series of chattering at t = 4 s, t = 6 s, and t = 12 s[see Fig. 14(b)].

VI. CONCLUSION

In this paper, a novel adaptive control method has beenproposed for robotic systems driven by SEAs. In the case ofminor human control efforts (robot-in-charge), the proposedcontroller allows the robot to carry out the manipulation byitself. On the other hand, when the human cannot follow themovement of robot or the human intends to take control of therobot (human-in-charge), the robot becomes passive such thatthe human is able to exert forces to guide the movement ofrobot. The smooth and automatic transition among both modesaddresses the potential conflicts between the human and therobot, and hence guarantees the safe human–robot interaction. Ithas been rigorously proved that the closed-loop system is stable,with consideration of uncertainties in both the robot dynamicsand the actuator dynamics. Both simulation and experimentalresults are presented to show the effectiveness of the proposedcontrol method. The proposed formulation provides a generalsolution to improve the performance of robot and also ensurethe safety of human. It can be applied in various SEA-drivenrobot systems, including industrial robots and healthcare robots,to fulfil the requirements of different human–robot interactiontasks.


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Xiang Li received the Bachelor’s and Master’s de-grees from Beijing Institute of Technology, Beijing,China, in 2006 and 2008, respectively, and the Ph.D.degree from Nanyang Technological University, Sin-gapore, in 2013.

He is a Research Assistant Professor with the De-partment of Mechanical and Automation Engineer-ing, The Chinese University of Hong Kong, HongKong. His research interests include robot control,visual servoing, human–robot interaction, and cellmanipulation.

Dr. Li has served as a Reviewer for several international journals includ-ing Automatica, IEEE TRANSACTIONS ON ROBOTICS, IEEE TRANSACTIONS

ON MECHATRONICS, and Asian Journal of Control. He received the HighlyCommended Paper Award in the Third IFToMM International Symposium onRobotics and Mechatronics.


Yongping Pan (M’14) received the B.Eng. degree inautomation and the M.Eng. degree in control theoryand control engineering from Guangdong Universityof Technology, Guangzhou, China, in 2004 and 2007,respectively, and the Ph.D. degree in control theoryand control engineering from South China Universityof Technology, Guangzhou, in 2011.

He was a Control Engineer with Santak ElectronicCo., Ltd., US Eaton Group, Shenzhen, China, and theUS Light Engineering Co., Ltd., Guangzhou, from2007 to 2008. From 2011 to 2013, he was a Research

Fellow with the School of Electrical and Electronic Engineering, NanyangTechnological University, Singapore. He is currently a Senior Research Fellowwith the Department of Biomedical Engineering, National University of Sin-gapore, Singapore. He has authored or coauthored more than 70 peer-reviewedacademic papers, including more than 45 in reputable referred journals. His re-search interests include automatic control, computational intelligence, robotics,and automation, where he is currently focusing on biomimetic adaptive andlearning control with applications to rehabilitation robots.

Dr. Pan has been an Associate Editor or Editorial Board Member of seveninternational refereed journals. He also serves as a Reviewer for more than 40international refereed journals.

Gong Chen received the B.E. degree in mechanicalengineering and automation, with a minor in com-puter science, from Shanghai Jiao Tong University,Shanghai, China, in 2011 and the Ph.D. degree inbiomedical engineering with National University ofSingapore, Singapore, in 2016 under the supervisionof Dr. Haoyong Yu.

He was a Research Fellow with the Singapore In-stitute for Neurotechnology, National University ofSingapore. His research interests include rehabilita-tion robots, compliant actuator, and control.

Haoyong Yu (M’10) received the B.S. and M.S. de-grees in mechanical engineering from Shanghai JiaoTong University, Shanghai, China, in 1988 and 1991,respectively, and the Ph.D. degree in mechanical en-gineering from Massachusetts Institute of Technol-ogy, Cambridge, MA, USA, in 2002.

He was a Principal Member of Technical Staff withDSO National Laboratories, Singapore, until 2002.He is currently an Assistant Professor with the De-partment of Biomedical Engineering and a PrincipalInvestigator of the Singapore Institute of Neurotech-

nology (SINAPSE), National University of Singapore. His research interestsinclude medical robotics, rehabilitation engineering and assistive technologies,system dynamics, and control.

Dr. Yu received the Outstanding Poster Award at the IEEE Life SciencesGrand Challenges Conference 2013. He served on a number of IEEE Confer-ence Organizing Committees.

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IEEE TRANSACTIONS ON ROBOTICS, VOL. 33, NO. 1, … Human Robot... · In general, robots driven by ... Impedance/admittance controllers may be used in both robot-in-charge and human-in-charge,