Teleoperation in the presence of varying time delays and sandwich linearity in actuators

Automatica 49 (2013) 2813–2821

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Teleoperation in the presence of varying time delays and sandwich linearity in actuators✩ Farzad Hashemzadeh a,b,1 , Iraj Hassanzadeh a,b , Mahdi Tavakoli b a

Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran

b

Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada

article

info

Article history: Received 14 April 2012 Received in revised form 4 January 2013 Accepted 9 May 2013 Available online 22 June 2013 Keywords: Nonlinear teleoperation Varying time delay Sandwich linearity Actuator saturation Asymptotic stability

abstract In this paper, a novel control scheme is proposed to guarantee global asymptotic stability of bilateral teleoperation systems that are subjected to time-varying time delays in their communication channel and sandwich linearity in their actuators. This extends prior art concerning control of nonlinear bilateral teleoperation systems under time-varying time delays to the case where the local and the remote robots’ control signals pass through saturation or similar nonlinearities that belong to a class of systems we name sandwich linear systems. Our proposed controller is similar to the proportional plus damping (P + D) controller with the difference that it takes into account the actuator saturation at the outset of control design and alters the proportional term by passing it through a nonlinear function; thus, we call the proposed method as nonlinear proportional plus damping (nP + D). The asymptotic stability of the closed-loop system is established using a Lyapunov–Krasovskii functional under conditions on the controller parameters, the actuator saturation characteristics, and the maximum values of the timevarying time delays. To show the effectiveness of the proposed method, it is simulated on a variabledelay teleoperation system comprising a pair of planar 2-DOF robots subjected to actuator saturation. Furthermore, the controller is experimentally validated on a pair of 3-DOF PHANToM Premium 1.5A robots, which have limited actuation capacity, that form a teleoperation system with a varying-delay communication channel. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction In telerobotic applications with a distance between the local and the remote robots (e.g., telesurgery and space exploration), there will be a time delay in the communication channel of the system, which can destabilize the telerobotic (Sheridan, 1993). In practice, the communication delay can be time-varying and asymmetric in the forward and backward paths between the operator and the remote environment (Gao & Chen, 2007; Gao, Chen, & Lam, 2008). There are a number of control schemes for time-varying delay compensation in the literature, e.g., Hua and Liu (2010), Nuño, Basañez, and Ortega (2011) and Polushin, Liu, and Lung (2007). On the other hand, in almost all applications

✩ The material in this paper was partially presented at IEEE World Haptics Conference 2013 (IEEE WHC 2013), April 14–18 2013, Daejeon, Korea. This paper was recommended for publication in revised form by Associate Editor Yong-Yan Cao under the direction of Editor Toshiharu Sugie. E-mail addresses: [email protected] (F. Hashemzadeh), [email protected] (I. Hassanzadeh), [email protected] (M. Tavakoli). 1 Tel.: +98 411 3393751; fax: +98 411 3300819.

0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.05.012

of control systems including teleoperation systems, the actuator output (i.e., control signal) has a limited amplitude, i.e., is subject to saturation. Controllers that ignore actuator saturation may cause undesirable responses and even closed-loop instability (Kothare, Campo, Morari, & Nett, 1994). Although it may be possible to avoid actuator saturation by using sufficiently high-torque actuators in robots, the large size of the actuators will cause further problems in robot design and control. Therefore, it is highly desirable to develop control methods that take any actuator saturation into account at the design outset and, therefore, allow for efficient and stable control with small-size actuators that inevitably possess a limited output capacity. In order to address the stability of the position control loop for a single robotic manipulator subjected to bounded actuator output, several approaches have been proposed in the literature. An anti-windup approach is presented to guarantee global asymptotic stability of Euler–Lagrange systems in Morabito, Teel, and Zaccarian (2004). In Loria, Kelly, Ortega, and Santibáñez (1997), a controller is proposed involving a gravity compensation term plus a saturating function through which the position errors pass. A velocity and position feedback method with adaptive gravity compensation is reported in Zergeroglu, Dixon, Behal, and Dawson

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F. Hashemzadeh et al. / Automatica 49 (2013) 2813–2821

(2000) in which the velocity and position errors separately pass through two nonlinear saturating functions and the outputs are then added to an adaptive gravity compensation term. In Zavala-Rio and Santibáñez (2006), a brief review of PD plus gravity compensation controllers is provided. None of the above research has been done in the context of teleoperation systems. Going beyond anti-saturation control for a single robot, there has been some attention paid recently to actuator saturation in bilateral teleoperation systems (Ahn, Park, & Yoon, 2001). Combining wave variable with a nonlinear proportional controller, an architecture to handle actuator saturation is discussed in Lee and Ahn (2010) for the case where the delay in the communication channel is constant. In Lee and Ahn (2011), an anti-windup approach combined with wave variables is used for constant-delay teleoperation subjected to bounded control signals. In this paper, a control scheme is introduced to cope with actuator saturation in nonlinear bilateral teleoperation systems that are subjected to time-varying delays. Asymptotic stability of the position error in the teleoperation system is studied, resulting in conditions on the controller parameters, the actuator saturation characteristics and the maximum values of varying delays. The main advantage of the proposed nP + D controller is in unifying the study of actuator saturation and varying time delay in the same framework and in guaranteeing the stability of system in the presence of both time varying delays and actuator saturation. This paper is organized as follows. Section 2 states the preliminaries while the proposed control and main results are presented in Sections 3 and 4 respectively. In Section 5, simulation and experimental results are provided followed by the conclusions in Section 6. Notation. We denote the set of real numbers by R = (−∞, ∞), the set of positive real numbers by R>0 = (0, ∞), and the set of nonnegative real numbers by R≥0 = [0, ∞). Also, ∥X ∥∞ and ∥X ∥2 stand for the Euclidean ∞-norm and 2-norm of a vector X ∈ Rn×1 , and |X | denotes element-wise absolute value of the vector X . The L∞ and L2 norms of a time function f : R≥0 → Rn×1 are shown as ∥f ∥L∞ = supt ∈[0,∞) ∥f (t )∥∞ and

0.5 , respectively. The L∞ and L2 spaces   are defined as the sets f : R≥0 → Rn×1 , ∥f ∥L∞ < +∞ and   f : R≥0 → Rn×1 , ∥f ∥L2 < +∞ , respectively. For simplicity, we refer to ∥f ∥L∞ as ∥f ∥∞ and to ∥f ∥L2 as ∥f ∥2 . We also simplify the notation limt →∞ f (t ) = to f (t ) → 0. ∥f ∥L2 =

 ∞ 0

∥f (t )∥22 dt

Important properties of the nonlinear dynamic models (1) and (2), which will be used in this paper, are (Kelly, Santibáñez, & Loria, 2005; Spong, Hutchinson, & Vidyasagar, 2005) P-1 For a manipulator with revolute joints, the inertia matrix M (q) is symmetric positive-definite and has the following upper and lower bounds: 0 < λmin (M (q(t ))) I ≤ M (q(t )) ≤ λMax (M (q(t ))) I ≤ ∞ where I ∈ Rn×n is the identity matrix. P-2 For a manipulator, the relation between the Coriolis/centrifugal and the inertia matrices is as follows:

˙ (q(t )) = C (q(t ), q˙ (t )) + C T (q(t ), q(t )) . M P-3 For a manipulator with revolute joints, there exists a positive η bounding the Coriolis/centrifugal term as follows:

∥C (q(t ), x(t )) y(t )∥2 ≤ η ∥x(t )∥2 ∥y(t )∥2 . P-4 The time derivative of C (q(t ), q˙ (t )) is bounded if q˙ (t ) and q¨ (t ) are bounded. P-5 For a manipulator with revolute joints, the gravity vector G (q(t )) is bounded. (There exist positive constants γj such that every elements of the gravity vector, gj (q(t )) , j = 1, . . . , n,  satisfies gj (q(t )) ≤ γj .) 2.2. Actuator model with sandwich linearity In the following, sandwich linearity of the actuator as a vector function S (·) is introduced. It is assumed that n is the number of joints in the master and slave robots and the elements of S (X ), where X , [x1 · · · xn ]T , are sj (xj ) : R → R, j = 1, . . . , n, defined by

> M j, sj (xj ) = xj , < −Mj , 

if xj > Mj if − Mj ≤ xj ≤ Mj if xj < −Mj

Mj > 0.

(3)

With this definition of S (X ), it is possible to define different sandwich linearity characterizations for different joints of the manipulator. Note that the function sj (xj ) is only required to be linear for −Mj ≤ xj ≤ Mj and can be nonlinear (unbounded or bounded) for xj  > Mj . An example of such a function is saturation. It is imperative to have γj < Mj , j = 1, . . . , n, where γj is the upper bound of gj (q(t )). This condition implies that the actuators of each of the master and the slave manipulators have the capacity to overcome the corresponding robot’s gravity within their workspaces.

2. Preliminaries 3. Proposed control law 2.1. Teleoperation system dynamical model Consider the master (local) and slave (remote) robots with saturated inputs as follows: Mm (qm (t )) q¨ m (t ) + Cm (qm (t ), q˙ m (t )) q˙ m + Gm (qm (t ))

= τh (t ) − S (τm (t )) Ms (qs (t )) q¨ s (t ) + Cs (qs (t ), q˙ s (t )) q˙ s (t ) + Gs (qs (t )) = S (τs (t )) − τe (t ).

(1) (2)

Here, qi , q˙ i and q¨ i ∈ R for i ∈ {m, s} are the joint positions, velocities and accelerations of the master and slave robots, respectively. Also, Mi (qi (t )) ∈ Rn×n , Ci (qi (t ), q˙ i (t )) ∈ Rn×n , and Gi (qi (t )) ∈ Rn×1 are the inertia matrix, the Coriolis/centrifugal matrix, and the gravitational vector, respectively. Moreover, τh and τe ∈ Rn×1 are the torques applied by the human operator and the environment, respectively. Lastly, τm and τs ∈ Rn×1 are the control signals (torques) for the master and the slave robots skewed by the vector function S : Rn×1 → Rn×1 , which can be nonlinear. n×1

In this paper, a nonlinear Proportional plus Damping (nP + D) controller that incorporates gravity compensation is proposed for the master and the slave robots as

τm (t ) = −Gm (qm (t )) + P (qm (t ) − qs (t − T2 (t ))) + Km q˙ m (t )

(4)

τs (t ) = Gs (qs (t )) − P (qs (t ) − qm (t − T1 (t ))) − Ks q˙ s (t )

(5)

where P (X ) : R → R is a vector on linear function with elements pj (xj ) : R → R. The function pj (xj ) is required to be strictly increasing, bounded, continuous, passing through the origin, concave for positive x and convex for negative  x, with  continuous first derivative around the origin, such that pj (xj ) ≤ n×1

n×1

    xj  and pj −xj = −pj (xj ). Under the above assumptions, we will have the following properties for pj (xj ): P-I For any x, y ∈ R, pj (x) − pj (y) ≤ 2pj (|x − y|). P-II For any x, y ∈ R, if x < y then pj (x) < pj (y).





F. Hashemzadeh et al. / Automatica 49 (2013) 2813–2821

For any x, y ∈ R≥0 , pj (x + y) ≤ pj (x) + pj (y). For any x ∈ R, limε→0 pj (ε x) = limε→0 ε pj (x). For any x, y ∈ R, xpj (y) ≤ xpj (x) + ypj (y). For any x ∈ R, pj (x) ≤ min |x| , Nj where Nj , supx∈R pj (x). P-VII For any x(t ) ∈ R, time derivative of pj (x(t )) is bounded. P-VIII For any x ∈ R, xpj (x) ≥ 0. P-III P-IV P-V P-VI

Let us start by a few preliminary lemmas that will be needed in the proof of our first main result in Theorem I. For simplicity, we use Gm (·) , Gs (·) , Pm (·) and Ps (·) instead of Gm (qm (t )) , Gs (qs (t )), P (qm (t ) − qs (t − T2 (t ))) and P (qs (t ) − qm (t − T1 (t ))), respectively. Similarly, we denote by gmj (·), gsj (·), pmj (·) and psj (·) the jth element of Gm (·), Gs (·) , Pm (·) and Ps (·), respectively. Lemma I. Given gmj (·) ≤ γj < Mj , gsj (·) ≤ γj < Mj , pmj (·) ≤ Nj ≤ Mj − γi and psj (·) ≤ Nj ≤ Mj − γi , for positive-definite diagonal matrices Km and Ks the following inequalities hold: q˙ Tm (S (Gm (·) − Pm (·) − Km q˙ m ) − (Gm (·) − Pm (·))) ≤ 0 q˙ Ts (S (Gs (·) − Ps (·) − Ks q˙ s ) − (Gs (·) − Ps (·))) ≤ 0.

(6)

t

t −T (t )

Proof of Theorem I. Applying controller (4)–(5) to the system (1)–(2), we have following closed loop dynamics:

x (τ ) dτ





t

≤ t −T (t )

pi (x (τ )) dτ .

= −S (−Gm (qm (t )) + P (qm (t ) − qs (t − T2 (t ))) + Km q˙ m (t )) + τh (t ) Ms (qs (t ))¨qs (t ) + Cs (qs (t ), q˙ s (t ))˙qs (t ) + Gs (qs (t )) = +S (Gs (qs (t )) − P (qs (t ) − qm (t − T1 (t ))) − Ks q˙ s (t )) − τe (t ).

(7)

(10)

(11)

To show the stability of the system (10)–(11), define xt = x (t + ψ), as the state of the system where x(t ) , [qm (t ) q˙ m (t ) qs (t ) q˙ s (t )], −Tmax ≤ ψ ≤ 0 and Tmax = max(T1max , T2max ), Chopra, Spong, and Lozano (2006) and Hale and Verduyn (1993). So Lyapunov–Krasovskii functional V (xt ) can be defined as V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ) + V4 (xt ) V1 (xt ) =

Lemma II. For any T (t ) ∈ R≥0 , x(t ) ∈ R≥0 , we have pi

In the above Mmin , min{Mj } and Nmax , max{Nj } for j = 1 · · · n. Also, dm , sup(x,y)∈Rn×1 ×Rn×1 ∥Gm (x) − P (y)∥∞ , ds , sup(x,y)∈Rn×1 ×Rn×1 ∥Gs (x) − P (y)∥∞ , and d , max {dm , ds }. Lastly, T1max , supt ∈R T1 (t ) and T2max , supt ∈R T2 (t ).

Mm (qm (t ))¨qm (t ) + Cm (qm (t ), q˙ m (t ))˙qm (t ) + Gm (qm (t ))

4. Main results



2815

1 2

(12)

q˙ Tm (t )Mm (qm (t )) q˙ m (t ) 1

+ q˙ Ts (t )Ms (qs (t )) q˙ s (t ) 2  t V2 ( x t ) = (−˙qm (τ ) τh (τ ) + q˙ s (τ ) τe (τ )) dτ

(13) (14)

0

Lemma III. For any vector functions qm (t ) ,



qm 1 (t ) · · · qm n (t )

T

T and qs (t ) , qs 1 (t ) · · · qs n (t ) and for any positive time-varying scalars T1 (t ) and T2 (t ), the following inequality holds: 

q˙ Tm (t ) (P (qm (t ) − qs (t )) − P (qm (t ) − qs (t − T2 (t ))))

≤ 2 |˙qm (t )|T



V3 ( x t ) =

t −T2 (t )

P (|˙qs (τ )|) dτ

˙ (t ) (P (qs (t ) − qm (t )) − P (qs (t ) − qm (t − T1 (t ))))  t T P (|˙qm (τ )|) dτ ≤ 2 |˙qs (t )|





Lemma IV. For any vectors A(t ) , [a1 (t ) · · · an (t )]T and B(t ) , [b1 (t ) · · · bn (t )]T and for any bounded time-varying scalar 0 ≤ T (t ) ≤ Tm , the following inequality holds:



A (t )

t −T (t )

P (B (τ )) dτ −



≤ Tm A (t )P (A(t )) . T

t

B (τ ) P (B (τ )) dτ T

t −T (t )

(9)

Theorem I. Assuming the human operator and the environment are passive, in the bilateral teleoperation system (1)–(2) with controllers (4)–(5), the velocities q˙ m and q˙ s and the position error qm − qs are bounded for any bounded time-varying time delays T1 (t ) and T2 (t ) provided that (1) (2) (3) (4)

T1max + T2max ≤ 2N max 2 (T1max + T2max ) < Ks 2 (T1max + T2max ) < Km γj + Nj ≤ Mj .

0



t

q˙ Ts (η) P (˙qs (η)) dηdγ .

(16)

t +γ

t

(−˙qm (τ ) τh (τ )) dτ + κm ≥ 0 and 0 t

(˙qs (τ ) τe (τ )) dτ + κs ≥ 0. 0

Considering property P-2, the time derivative of V1 (t ) is simplified to V˙ 1 (xt ) = q˙ Tm (t )(−S (P (qm (t ) − qs (t − T2 (t )))

Proofs of the above four lemmas are provided in the Appendix.

−d+Mmin

q˙ Tm (η) P (˙qm (η)) dηdγ

Note that based on the assumption of passivity of the operator and the environment, V2 (xt ) is a lower-bonded function. In other words, there exist positive constants κm and κs such that

where q˙ m (t ) and q˙ s (t ) are the time derivatives of qm (t ) and qs (t ), respectively.

t

(15)

t +γ

−T2max

(8)

 

t



+2

t −T 1 ( t )



0



−T1max

t

pj γj dγj

0

j =1

V4 ( x t ) = 2

qTs

T

qmj (t )−qsj (t )

n  

− Gm (qm (t )) + Km q˙ m (t ))) + q˙ Tm (t )(τh (t ) − Gm (qm (t ))) + q˙ Ts (t )(S (Gs (qs (t )) − P (qs (t ) − qm (t − T1 (t ))) − Ks q˙ s (t ))) + q˙ Ts (t )(−τe (t ) − Gs (qs (t ))).

(17)

The time derivatives of V2 (t ) and V3 (t ) are V˙ 2 (xt ) = q˙ m (t )τh (t ) − q˙ s (t )τe (t )

(18)

V˙ 3 (xt ) = (˙qm (t ) − q˙ s (t )) P (qm (t ) − qs (t )) .

(19)

By adding and subtracting each of q˙ m (t )P (qm (t ) − qs (t − T2 (t ))) and q˙ s (t )P (qs (t ) − qm (t − T1 (t ))) to and from V˙ 1 and noting that

2816

F. Hashemzadeh et al. / Automatica 49 (2013) 2813–2821

P (−X ) = −P (X ), it is possible to see that

Applying Lemma IV to (25), we get V˙ (xt ) ≤ −δm Km q˙ Tm (t )˙qm (t ) − δs Ks q˙ Ts (t )˙qs (t )

V˙ 1 (xt ) + V˙ 2 (xt ) + V˙ 3 (xt )

= q˙ Tm (t ){S (Gm (qm (t )) − P (qm (t ) − qs (t − T2 (t ))) − Km q˙ m (t )) − (Gm (qm (t )) − P (qm (t ) − qs (t − T2 (t ))))} + q˙ Tm (t ){P (qm (t ) − qs (t ))

+ 2(T1max + T2max )˙qTm (t )P (˙qm (t )) + 2(T1max + T2max )˙qTs (t )P (˙qs (t )). Defining ηm and ηs as

− P (qm (t ) − qs (t − T2 (t )))} + q˙ Ts (t ){S (Gs (qs (t )) − P (qs (t ) − qm (t − T1 (t ))) − Ks q˙ s (t )) − (Gs (qs (t )) − P (qs (t ) − qm (t − T1 (t ))))} + q˙ Ts (t ){P (qs (t ) − qm (t )) − P (qs (t ) − qm (t − T1 (t )))}.

(26)

q˙ Tm (t )P (˙qm (t ))

ηm = (20)

ηs =

∥˙qm (t )∥22 q˙ Ts (t )P (˙qs (t ))

∥˙qs (t )∥22

,

∥˙qm (t )∥2 ̸= 0

,

(27)

∥˙qs (t )∥2 ̸= 0

(28)

Considering Lemma I, it is easy to see that there exist positive δm and δs such that

we have

V˙ 1 (xt ) + V˙ 2 (xt ) + V˙ 3 (xt ) ≤ −δm Km q˙ Tm (t )˙qm (t )

V˙ (xt ) ≤ −(δm Km − 2(T1max + T2max )ηm )˙qTm (t )˙qm (t )

+ q˙ Tm (t ){P (qm (t ) − qs (t )) − P (qm (t ) − qs (t − T2 (t )))} −δ

− (δs Ks − 2(T1max + T2max )ηs )˙qTs (t )˙qs (t ).

˙ (t )˙qs (t ) + ˙ (t ){P (qs (t ) − qm (t )) − P (qs (t )

T s Ks q s

qTs

− qm (t − T1 (t )))}

(21)

−1 q˙ Tm (t ){S (Gm (qm (t )) − P (qm (t ) Km ∥˙qm (t )∥22 − qs (t − T2 (t ))) − Km q˙ m (t )) − (Gm (qm (t )) − P (qm (t ) − qs (t − T2 (t ))))}, ∥˙qm (t )∥2 ̸= 0

−1 δs , q˙ Ts (t ){S (Gs (qs (t )) − P (qs (t ) Ks ∥˙qs (t )∥22 − qm (t − T1 (t ))) − Ks q˙ s (t )) − (Gs (qs (t )) − P (qs (t ) − qm (t − T1 (t ))))}, ∥˙qs (t )∥2 ̸= 0.

    

(22)

V˙ 1 (xt ) + V˙ 2 (xt ) + V˙ 3 (xt )

≤ −δm Km q˙ Tm (t )˙qm (t ) − δs Ks q˙ Ts (t )˙qs (t )  t + 2|˙qm (t )|T P (|˙qs (τ )|)dτ t

t −T1 (t )

P (|˙qm (τ )|)dτ .

(23)

t −T2 (t )

(24)

V˙ (xt ) ≤ −δm Km q˙ Tm (t )˙qm (t ) − δs Ks q˙ Ts (t )˙qs (t )



+

 Mj − dj  j = 1···n X1 , q˙ m (t ) : q˙ mj (t ) ≤



Considering (23) and (24) together, we have

+ 2|˙qs (t )|T



 (33)

t t −T2 (t )





(35)

− 2(T1max + T2max )ηm )˙qTm (t )˙qm (t ) ≤ 0.

P (|˙qm (τ )|)dτ

˙ (t )P (˙qm (t )) − 2

+ 2T2max q˙ Ts (t )P (˙qs (t )) − 2

Based on gmj (·)−pj (·)−Km q˙ mj (t ) ≤ gmj (·)−pj (·)+Km q˙ mj (t ) ≤ gmj (·) − pj (·) + Mj − dj ≤ Mj and the definition of δm in (22), we have δm = 1. Also, from (32), we know that ηm ≤ 1. Applying δm = 1 and ηm ≤ 1 to (30), the following inequality is found as a sufficient condition for (30):

Therefore (35) is a sufficient condition to have −(δm Km

t t −T1 (t )

(34)



Km > 2 (T1max + T2max ) .

P (|˙qs (τ )|)dτ

2T1max qTm



• Case 1: q˙ m (t ) ∈ X1 .

q˙ Ts (τ )P (˙qs (τ ))dτ .



(32)

where dj , sup(x,y)∈R×R gj (x) − pj (y). We distinguish the following two cases:

q˙ Tm (τ )P (˙qm (τ ))dτ + 2T2max q˙ Ts (t )P (˙qs (t ))

+ 2|˙qm (t )|T

     n            j=1 q˙ sj (t ) Nj   ηs ≤ min 1, .  ∥˙qs (t )∥22       

Km

t

−2

∥˙qm (t )∥22



t

t −T1 (t )





  Mj − dj X2 , q˙ m (t ) : q˙ mj (t ) > j = 1···n

V˙ 4 (xt ) = 2T1max q˙ Tm (t )P (˙qm (t ))



j =1

 

q˙ mj (t ) Nj   

Km

On the other hand, the time derivative of V4 (xt ) is

−2

ηm ≤ min 1,    

n   

To study the lower bounds of δm and δs for replacement in (30)–(31), let us consider two regions X1 and X2 as

t −T 2 ( t )



(30) (31)

Next, we will investigate conditions under which the inequalities (30) and (31) are satisfied. Given the definitions of ηm and ηs in (27) and (28) and using property P-VI of function P (·.), we have

Applying the result of Lemma III to the last two terms on the right hand side of (21), we get

+ 2|˙qs (t )|T

Now, let us find conditions on δm and δs such that V˙ (xt ) ≤ 0. It is possible to see from (29) that a sufficient condition for V˙ (xt ) ≤ 0 is

δm Km ≥ 2 (T1max + T2max ) ηm δs Ks ≥ 2 (T1max + T2max ) ηs .

where δm and δs are defined as

δm ,

(29)

t



t −T1 (t )



q˙ m (τ )P (˙qm (τ ))dτ

t

t −T2 (t )

q˙ Ts (τ )P (˙qs (t ))dτ . (25)

• Case 2: q˙ m (t ) ∈ X2 .   Then, gmj (·) − pj (·) − Km q˙ mj (t ) could be greater than Mj or less than Mj . So, 1. If gmj (·) − pj (·) − Km q˙ mj (t ) ≤ Mj , based on the definition of δm in (22), δm = 1.





F. Hashemzadeh et al. / Automatica 49 (2013) 2813–2821

2817

2. If |gmj (·) − pj (·) − Km q˙ mj (t )| > Mj , then si (gmj (·) − pj (·) − Km q˙ mj (t )) > Mj and using reverse triangle inequality, we will have the following inequality:

      gm (·) − pj (·) − si gm (·) − pj (·) − Km q˙ m (t )  j j j    > Mj − gmj (·) − pj (·) > Mj − dj . Using Lemma I, we have q˙ mj (t )((gmj (·) − pj (·)) − (sj (gmj (·) − pj (·) − Km q˙ mj (t ))))

= |˙qmj (t )||(gmj (·) − pj (·)) − (sj (gmj (·) − pj (·) − Km q˙ mj (t )))|.

(36)

Given that Mj − dj ≥ Mmin − d and based on the definition of

δm in (22), we get n   

q˙ mj (t ) (Mmin − d)

δm >



i =1

Km ∥˙qm (t )∥22

.

Knowing from (32) that ηm ≤

(37) Fig. 1. The stability region based on inequalities (35), (38) and (39).

 n   qmj (t )Nj j=1 ˙ ∥˙qm (t )∥22

and using (37), it is

possible to see that δm Km Nmax ≥ (Mmin − d) ηm . Using this, we can find following condition to satisfy the inequality (30). Mmin − d Nmax

≥ 2 (T1max + T2max ) .

(38)

Therefore if (35) and (38) are satisfied, then −(δm Km − 2(T1max + T2max )ηm )˙qTm (t )˙qm (t ) ≤ 0. Finally, conducting a similar analysis to find a condition for (31) to hold will result in (38) and following inequality: Ks > 2 (T1max + T2max ) .

(39)

Using the above analysis, it is possible to see that if Mmax satisfies (38) and Km and Ks fulfill inequalities (35) and (39), thenV˙ (xt ) ≤ 0 meaning that all terms in V (xt ) are bounded. Therefore, q˙ m , q˙ s and qm − qs ∈ L∞ and the poof is complete.  Note that the parameter d defined   in Theorem I is equal to Nmax + γmax where γmax , max γj . Using the inequalities (35), (38) and (39), a schematic representation of the stability condition in terms of T1max and T2max is shown in Fig. 1. The stability condition (38) in terms of Mmin and Nmax is shown in Fig. 2. Theorem II. In the bilateral teleoperation system (1)–(2) with the controller (4)–(5), the absolute values of the velocities |˙qm (t )| and |˙qs (t )| and the position error |qm (t ) − qs (t )| tend to zero asymptotically in free motion (i.e., τh (t ), τe (t ) → 0) if all conditions in Theorem I are satisfied and both T˙1 (t ) and T˙2 (t ) are bounded. Proof of Theorem II. Integrating both sides of (29), it is possible to see that q˙ m (t ) and q˙ s (t ) ∈ L2 . Based on the result of Theorem I, V (t ) is a lower bounded decreasing function. Therefore, q˙ m (t ), q˙ s (t ) and qm (t ) − qs (t ) ∈ L∞ . Using the fact that t qm (t ) − qs (t − T2 (t )) = qm (t ) − qs (t ) + t −T (t ) q˙ s (t )dt and

t

Fig. 2. The stability region provided by (38) in Mmin versus Nmax plane given T1max and T2max .

... and q¨ s (t ) → 0. Let us investigate the boundedness of q i (t ) for i ∈ {m, s}. The closed-loop dynamics found from combining the open-loop system (1) and (2) with the controllers (4) and (5) is q¨ m (t ) = (Mm (qm (t )))−1 {−Cm (qm (t ), q˙ m (t ))˙qm (t ) − Gm (qm (t )) − S (−Gm (qm (t )) + P (qm (t ) − qs (t − T2 (t ))) + Km q˙ m (t ))} q¨ s (t ) = (Ms (qs (t )))−1 {−Cs (qs (t ), q˙ s (t ))˙qs (t ) − Gs (qs (t )) − S (−Gs (qs (t )) + P (qs (t ) − qm (t − T1 (t ))) + Ks q˙ s (t ))}. Differentiating both sides with respect to time and given d dt

(Mi (qi (t )))−1 = −(Mi (qi (t )))−1 {Ci (qi (t ), q˙ i (t )) + CiT (qi (t ), q˙ i (t ))}Mi (qi (t )) i ∈ {m, s}

2

q˙ (t )dt ∈ L∞ , we have qm (t ) − qs (t − T2 (t )) ∈ L∞ . Since t −T2 (t ) s the gravity terms gm and gs are bounded, using property P-1 and P-2 of system dynamics and given the boundedness of S (τm (t )) and S (τs (t )), it can be seen that q¨ m (t ) and q¨ s (t ) ∈ L∞ . Because q˙ m (t ) ∈ L2 and q¨ m (t ) ∈ L∞ , using Barbalat’s lemma we have that q˙ m (t ) → 0. Similarly, it can be reasoned that q˙ s (t ) → 0. ... q Now, ... if q¨ m and q¨ s are continuous in time, or equivalently m (t ) and q s (t ) ∈ L∞ , then q˙ m (t ) and q˙ s (t ) → 0 ensures that q¨ m (t )

(40)

(41)

and based on properties P-1 and P-3 and given the boundedness of q˙ s and q˙ m , it is easy to see that dtd (Mm (qm (t )))−1 and d dt

(Ms (qs (t )))−1 are bounded. Given that P˙ (·) is bounded and us-

ing properties P-1, P-3 and P-4 of system dynamics and the boundedness of qs (t ) − qm (t − T1 (t ))..., qm (t ) ... − qs (t − T2 (t )), q˙ m , q˙ s , q¨ m , q¨ s , T˙1 and T˙2 , it can be seen...that q m and ... q s are bounded. Given that q˙ m (t ) and q˙ s (t ) → 0 and q m (t ) and q s (t ) ∈ L∞ , using Barbalat’s lemma we have that q¨ m (t ) and q¨ s (t ) → 0.

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F. Hashemzadeh et al. / Automatica 49 (2013) 2813–2821

Considering the dynamic equation of the master and slave robots in (10) and (11), having shown that q¨ i (t ) → 0 and q˙ i (t ) → 0, i ∈ {m, s}, it is easy to see that Gm (qm (t )) → S (Gm (qm (t )) − P (qm (t ) − qs (t − T2 (t ))))

(42)

Gs (qs (t )) → S (Gs (qs (t )) − P (qs (t ) − qm (t − T1 (t )))) .

(43)

Given S (Gi (qi (t ))) = Gi (qi (t )), we find that P (qm (t ) − qs (t − T2 (t ))) → 0

(44)

P (qs (t ) − qm (t − T1 (t ))) → 0

(45)

and using assumptions of P (·),

|qm (t ) − qs (t − T2 (t ))| → 0 |qs (t ) − qm (t − T1 (t ))| → 0.

(46) (47)

It is possible to see the asymptotic zero convergence of velocities and the position tracking error and proof is completed.  Remark I. Based on the assumption M −(N +γ ) T1max + T2max ≤ min 2Nmax max in Theorem I, it is possible to max see a trade-off between the robustness to the maximum values of time delays and the tracking performance in controller design. For instance, if Nmax is lowered, then the position difference between the master and the slave robots contributes less to the control signal, resulting in an increase in the settling time for the position tracking response. At the same time, as Nmax is lowered, the maximum admissible values of time delays increase; i.e., the robustness of the system stability to larger time delays improves. The above trends in performance and stability are understandable once one pays attention to the control laws (4)–(5). When Nmax has a small value, the nonlinear proportional terms in (4)–(5) are suppressed, leaving more ‘‘room’’ for the derivative signals q˙ (t ) to contribute to the control signal, i.e., the velocity gains Km and Ks are allowed to be larger. It is clear from the stability conditions (35) and (39) that larger Km and Ks allow for larger values for the maximum time delays. At the same time, with small Nmax , settling time of position tracking increases, degrading the performance of the teleoperation system. Therefore, it can be concluded that, for a fixed Mmin , there is a trade-off between stability and performance of the system and this trade-off can be tuned by changing Nmax . Remark II. The discussion in this paper has focused on jointspace dynamics of the master and the slave robots and joint-level position tracking. It is noteworthy that the dynamics of a robot in the task space are similar to the joint-space dynamics (1)–(2) and the inertia, Coriolis/centrifuge, and gravity matrices/vectors in task-space dynamics also have the same properties (Lee & Ahn, 2011). Therefore, the proposed controller and stability results in this paper can be used in the task space in the same way that they are used in the joint space. This is advantageous because, in different applications, one may be facing with different saturation effects. For instance, for safety reasons, a low-level controller may be enforcing preset limits on the (Cartesian) forces applied by the end-effector of a robot; this would be a case of actuator saturation in the task space. 5. Simulation and experiment results In this section, simulation and experimental results for the proposed teleoperation controller are provided. 5.1. Simulation on a teleoperated pair of 2-DOF planar robots To verify the theoretical results of this paper, the master and the slave manipulators are considered to be a pair of 2-DOF

Fig. 3. First joint (up) and second joint (down) positions of the 2-DOF planar robot teleoperation with actuator saturation and variable time delay controlled by the proposed nP + D controller.

planar robots with revolute joints. In this simulation, it is assumed that the control signals for the both the master and the slave robots are subjected to actuator saturation at levels +1 and −1. Also the single-way forward and backward time delays in the communication channel are chosento be random variables with uniform distributions over 0 0.3 second. The random nature of time delays makes it possible to show the effectiveness of the proposed method for fast-varying time delays. Note that the singleway time delays used in this experiment, which vary between 0 and 300 ms, are reasonable in Internet-based teleoperation applications because the round-trip delay has been shown to have an average of 350 ms if the distance between the local and the remote robots is up 10,000 km (Oboe, 2003). The input human forces, FhX and FhY , applied to the master robot are zero at the beginning and starts to increase after 0.5 and 1.5 s, reaching 10 N at 1 and 2 s and staying at that level until 2 and 3 s, respectively. After that, they decrease until 2.5 and 3.5 s, respectively, when they reach zero. The application of this human force results in the joint positions profiles for the master and the slave robots shown in Fig. 3. To show the efficiency of the proposed method, the propose nP + D controller has been applied to the 2-DOF planar robots assuming saturation on joint actuators as well as the previouslydiscussed time varying delays in the communication channel. The nonlinear function P (·) in the controllers (4) and (5) is chosen as 0.2 tan−1 (·). Note that all conditions listed for Theorem I are satisfied with the function tan−1 (·); see the required properties of P (·) in Section 3. Also, Km and Ks are set to 3. This means that, in this simulation, the controllers (4) and (5) become

τm (t ) = −Gm (qm (t )) + 0.2 tan−1 (qm (t ) − qs (t − T2 (t ))) + 3q˙ m (t ) τs (t ) = Gs (qs (t )) − 0.2 tan−1 (qs (t ) − qm (t − T1 (t ))) − 3q˙ s (t ) .

(48)

Note that in this simulation, Mmin = 1, Nmax = π /10, γmax = 0.05 and T1max = T2max = 0.3, for which it is easy to see that the sufficient condition (38) is satisfied. As mentioned in Remark I, the maximum value of the nonlinear function P (·) in the control laws (4)–(5), Nmax , can affect the performance of the closed-loop system. To show this, in Fig. 4, the position tracking errors between the master and the slave robots are shown for three different values of Nmax . Evidently, smaller values of Nmax make the closed-loop response slower and the settling time larger.

F. Hashemzadeh et al. / Automatica 49 (2013) 2813–2821

2819

Fig. 5. The experimental teleoperation setup consisting of two PHANToM Premium 1.5A robots and the schematic of the PHANToM robot with its corresponding joint angles.

1

Fig. 4. First joint (up) and second joint (down) position errors between master and slave robots for different values of Nmax .

q1 (Rad.)

0.5

Mater Slave

0 -0.5 -1

0

5.2. Experiment on a teleoperated pair of 3-DOF PHANToM robots

τm (t ) = −Gm (qm (t )) + 0.1p(5(qm (t ) − qs (t − T2 (t )))) + q˙ m (t ) τs (t ) = Gs (qs (t )) − 0.1p (5 (qs (t ) − qm (t − T1 (t )))) − q˙ s (t ).

20

q2 (Rad.)

25 Master Slave

0.5 0 -0.5

0

5

10

15 Time (Sec.)

20

0.8

25 Master Slave

0.6 0.4 0.2 0 0

5

10

(49)

(50)

where p(·) is the same as that used in (49), i.e., p(x) , sgn(x) min {|x| , 1}. In Fig. 7, the step responses of the slave robot’s joint

2 http://www.ece.ualberta.ca/~mtavakol/Automatica_submission/.

15 Time (Sec.)

1

-0.2

In the above, we choose the nonlinear function in the controllers to be p(x) , sgn(x) min {|x| , 1} with sgn(·) being the signum function. As a result, when the position errors qm (t )− qs (t − T2 (t )) (the same holds for the position error qs (t ) − qm (t − T1 (t ))) is between −0.2 and 0.2 radian, it appears linearly through a gain of 0.5 in the control signal. Otherwise, its contribution to the control signal is maxed out at −0.1 or 0.1. Note that the function 0.1p(5x) meets all the required properties listed in Section 3. Also note that the approximate levels of the actuator saturation for the first, second and third joint of the PHANToM robot in generalized coordinate are M1 = 0.29, M2 = 0.29 and M3 = 0.23 N m, respectively. Also γmax = 0.07, Mmin = 0.28, T1max = T2max = 0.1, Nmax = 0.1 and it is easy to see that for these values the sufficient condition (38) for stability is satisfied. Experimental results of the proposed nP + D controller in terms of joint position tracking between the master and the slave robots are shown in Fig. 6. Let us study the effect of Nmax , which is the maximum value of the nonlinear function P (·) in (4)–(5), on the performance of the teleoperation system in experiments. To show the slave’s joint position trajectory in response to a step set-point (corresponding to a fixed position for the master), the control signal (50) is applied to the slave robot for different values of Nmax :

τs (t ) = Gs (qs (t )) − Nmax p (5 (qs (t ) − qm (t − T1 (t ))))

10

1.5

q3 (Rad.)

In this section, experimental results for the proposed control method are reported. In the experimental setup shown in Fig. 5, two 3-DOF PHANToM Premium 1.5A robots are connected via a communication channel with varying time delays with uniform distributions between 81 and 100 ms. Please note that, in addition to the experimental results reported below, a video showing experiments on this teleoperation test-bed accompanies this paper.2 In the proposed nP + D method, the following controllers are used:

5

15 Time (Sec.)

20

25

Fig. 6. Joint position tracking between the master and the slave using the proposed nP + D control method.

positions are shown for three different values of Nmax . Evidently, a smaller Nmax , which corresponds to a reduced contribution of the tracking error qs (t ) − qm (t − T1 (t )) to the slave robot’s control signal, leads to a slower step response (i.e., larger settling time). This is a result that is consistent with Remark I. 6. Conclusion and future work In this paper, we developed a novel method to cope with actuator saturation in bilateral teleoperation systems that are subjected to time-varying time delays in their communication channels. The proposed controller, which we call the nP + D method, is similar to the conventional P + D controller except for the fact that we have replaced the proportional term by a nonlinear function through which the position errors pass. This makes the proposed nP + D method capable of handling actuator saturation and guaranteeing position tracking even in the presence of time-varying time delays in the communication channel. We analyzed the stability of the system using a Lyapunov–Krasovskii functional and showed asymptotic position tracking between the master and the slave robots. The derived stability conditions involve relationships between the nP + D controller parameters, the actuator saturation characteristics, and the maximum values of the time-varying delays. We simulated the proposed controller on a teleoperated pair of 2-DOF planar robots subjected to actuator saturation and time-varying delays. We also experimentally tested the proposed controller on a teleoperated pair of 3-DOF PHANToM Premium robots, which are naturally subject to actuator saturation.

2820

F. Hashemzadeh et al. / Automatica 49 (2013) 2813–2821

q1s(t) (Rad.)

1

Proof of Lemma II. Based on the definition of integral, we know that

0.5

 0

t t −T (t )



-0.5 0

1

2

3

4

5

n→∞

q2s(t) (Rad.)

= lim

n→∞

1

n

x t − T (t ) + k

(A.4)

  n−1  T (t ) n

k=0

x t − T (t ) + k

pi

T (t )



n

.

(A.5)

Using the properties P-III and P-IV of pi (·) and knowing that x (τ ) and T (t ) are positive,

0.5 0

t



0

1

2

3

4 5 Time (Sec.)

6

7

8

pi

t −T ( t )

1.5 q3s(t) (Rad.)

n

k=0



pi (x (τ )) dτ

1.5

≤ lim

1

n→∞

0.5

= lim

0 -0.5

T (t )

t t −T ( t )

Time (Sec.)

x (τ ) dτ = lim

 n −1  T (t )

n→∞

0

1

2

3

4 5 Time (Sec.)

6

7

8

x (τ ) dτ

n −1 

k=0

In the future, it is possible to account for the simultaneous presence of time-varying time delay, actuator sandwich linearity, and a third source of problem in teleoperation control such as model uncertainties, unmodeled dynamics, external disturbances, etc.

Proof of Lemma I. The  assumption  in the lemma  can be summed up as gmj (·) − pmj (·) < Mj and gsj (·) − psj (·) < Mj . Regarding increasing monotone property of function sj (·), if q˙ mj ≥ 0 then

 sj gmj (·) − pmj (·) − kj q˙ mj ≤ sj gmj (·) − pmj (·) and if q˙ mj < 0     then sj gmj (·) − pmj (·) − kj q˙ mj > sj gmj (·) − pmj (·) . Based on the fact that gmj (·) − pmj (·) belongs to the linear part of   sj , sj gmj (·) − pmj (·) = gmj (·) − pmj (·); therefore, for all q˙ mj q˙ mj sj gmj (·) − pmj (·) − kj q˙ mj

 





 − gmj (·) − pmj (·) ≤ 0 

(A.1)

˙ (S (Gm (·) − Pm (·) − Km q˙ m ) − (Gm (·) − Pm (·))) n  = q˙ mj (sj (gmj (·) − pmj (·) − kj q˙ mj )

qTm

n



x t − T (t ) + k

 

x t − T (t ) + k

pi



T (t )





n T (t )



n

t



pi (x (τ )) dτ

and proof completed.

(A.6) 

Proof of Lemma III. Considering property P-I of pj (·),

     pj qm (t ) − qs (t ) − pj qm (t ) − qs (t − T2 (t ))  j j j j   ≤ 2pj qsj (t ) − qsj (t − T2 (t ))   t   = 2pj  q˙ sj (τ ) dτ 

(A.7)

t −T2 (t )

and similarly

(A.8)

Using property P-II of pj (·) and knowing that for any q˙ ∈ R and 

 t

for any positive T (t ),  t −T (t ) q˙ (τ ) dτ  ≤ pj

   

t

 

t −T (t )

q˙ (τ ) dτ 



t

 ≤ pj

t −T (t )

t

t −T (t )

|˙q (τ )| dτ , then

 |˙q (τ )| dτ .

pj

t t −T (t )

|˙q (τ )| dτ





t

≤ t −T (t )

pj (|˙q (τ )|) dτ .

Considering (A.7), (A.9) and (A.10),

and therefore q˙ Tm (t ) (P (qm (t ) − qs (t )) − P (qm (t ) − qs (t − T2 (t ))))

(A.3)

(A.9)

Considering Lemma II,



=

n  (˙qmj (t )(pj (qmj (t ) − qsj (t )) j =1

and proof completed.

x t − T (t ) + k

t −T (t )

(A.2)

Using the similar study, q˙ Ts (S (Gs (·) − Ps (·) − Ks q˙ s ) − (Gs (·) − Ps (·))) ≤ 0

n

k=0

     pj qm (t ) − qs (t ) − pj qm (t ) − qs (t − T2 (t ))  j j j j  t ≤2 pj (|˙q (τ )|) dτ

j =1

− (gmj (·) − pmj (·))) ≤ 0.

n

T (t )

t −T1 (t )

Here, Proofs of Lemmas I, II, III and IV are provided.



n

lim

n→∞

 n−1  T (t )

     pj qs (t ) − qm (t ) − pj qs (t ) − qm (t − T1 (t ))  j j j j   t   q˙ mj (τ ) dτ  . ≤ 2pj 

Appendix



= pi

T (t )

n−1  T (t )

t −T (t )

Simulation and experimental results of the proposed nP + D control method have demonstrated its efficiency.

pi

k=0

=

Fig. 7. Step response of slave robot’s joint positions for different values of Nmax .







− pj (qmj (t ) − qsj (t − T2 (t )))))

(A.10)

F. Hashemzadeh et al. / Automatica 49 (2013) 2813–2821

≤

n  (|˙qmj (t )||pj (qmj (t ) − qsj (t )) j =1

− pj (qmj (t ) − qsj (t − T2 (t )))|)   t n   pj (|˙qsj (τ )|)dτ =2 |˙qmj (t )| t −T (t )

j =1

= 2 |˙qm (t )|T



t

t −T2 (t )

P (|˙qs (τ )|) dτ .

(A.11)

Similarly

˙ (t ) (P (qs (t ) − qm (t )) − P (qs (t ) − qm (t − T1 (t ))))  t ≤ 2 |˙qs (t )|T P (|˙qm (τ )|) dτ

qTs

(A.12)

t −T 1 ( t )

and proof completed.



Proof of Lemma IV. Given property P-V of function pj (·), n  

aj (t )pj bj (τ ) − bj (τ ) pj bj (τ )









j =1 n  

.

(A.13)

AT (t )P (B (τ )) − BT (τ ) P (B (τ )) ≤ AT (t )P (A(t )) .

(A.14)

≤

aj (t )pj aj (t )





j =1

Therefore

Integrating both sides based on dτ from t − T (t ) to t, t



t −T (t )

AT (t )P (B (τ )) dτ −



t

t −T (t )

BT (τ ) P (B (τ )) dτ

t

 ≤

t −T (t )

AT (t )P (A(t )) dτ .

(A.15)

That can be simplified to A (t ) T



t

P (B (τ )) dτ −



2821

Hua, C. C., & Liu, X. P. (2010). Delay-dependent stability criteria of teleoperation systems with asymmetric time-varying delays. IEEE Transactions on Robotics, 26, 925–932. Kelly, R., Santibáñez, V., & Loria, A. (2005). Control of robot manipulators in joint space. Springer. Kothare, M. V., Campo, P. J., Morari, M., & Nett, C. N. (1994). A unified framework for the study of antiwindup designs. Automatica, 30(12), 1869–1883. Lee, S. J., & Ahn, H. S. (2010). Synchronization of bilateral teleoperation systems with input saturation. In International conference on control, automation and systems (pp. 1375–1361). Lee, S. J., & Ahn, H. S. (2011). A study on bilateral teleoperation with input saturation and systems. In International conference on control automation and systems (pp. 161–166). Loria, A., Kelly, R., Ortega, R., & Santibáñez, V. (1997). On global output feedback regulation of Euler–Lagrange systems with bounded inputs. IEEE Transactions on Automatic Control, 42, 1138–1143. Morabito, F., Teel, A. R., & Zaccarian, L. (2004). Nonlinear antiwindup applied to Euler–Lagrange systems. IEEE Transaction on Robotics and Automation, 20(3), 526–573. Nuño, E., Basañez, L., & Ortega, R. (2011). Passivity-based control for bilateral teleoperation: a tutorial. Automatica, 47, 485–495. Oboe, R. (2003). Force reflecting teleoperation over the Internet: the JBIT project. Proceedings of the IEEE, 91(3), 449–462. Polushin, I. G., Liu, P. X., & Lung, C. H. (2007). A force-reflection algorithm for improved transparency in bilateral teleoperation with communication delay. IEEE/ASME Transactions on Mechatronics, 12(3), 361–374. Sheridan, T. B. (1993). Space teleoperation through time delay: review and prognosis. IEEE Transactions on Robotics, vol. 9(no. 5), 592–606. Spong, M. W., Hutchinson, S., & Vidyasagar, M. (2005). Robot modeling and control. Wiley. Zavala-Rio, A., & Santibáñez, V. (2006). Simple extensions of the PD-with-gravitycompensation control law for robot manipulators with bounded inputs. IEEE Transactions on Control Systems Technology, 14(5), 958–965. Zergeroglu, E., Dixon, W., Behal, A., & Dawson, D. (2000). Adaptive set-point control of robotic manipulators with amplitude-limited control inputs. Robotica, 18, 171–181.

Farzad Hashemzadeh was born in Maku, Iran, in 1981. He obtained his B.Sc. in Biomedical Engineering from Amirkabir University of Technology in 2003 and M.Sc. in Electrical Engineering from University of Tehran in 2006. He received his Ph.D. degree in Control Engineering from University of Tabriz in 2012. In 2011 he held one year sabbatical research, at the TBS lab in the University of Alberta, Canada. In 2012 he joined the Electrical and Computer Engineering Department of the University of Tabriz. His research interest includes robust and nonlinear control, Haptics and time delayed control of telerobotic systems.

t

BT (τ ) P (B (τ )) dτ

(A.17)

Iraj Hassanzadeh received his Ph.D. in Electrical Engineering, Control, Robotics, from University of Tabriz, Iran, in conjunction with the University of Western Ontario, London, Canada, and M.Sc. degrees from the University of Tabriz, in 2002 and 1994, respectively. He received his B.Sc. degree from the University of Tehran, Iran, in 1991. He has been working as a Post-doctoral fellow in Mechatronics and Robotics fields at Ryerson University, Toronto, Canada, for almost 2 years during 2004–2005. Since 2002, he has been with the faculty of Electrical and computer Engineering, University of Tabriz, Iran. His research interests include robotics, visual servo, tele-robotics, control theory, applications and power system.

Ahn, H.S., Park, B.S., & Yoon, J.S. (2001). A teleoperation position control for 2-DOF manipulators with control input saturation. In IEEE international symposium on industrial electronics (pp. 1520–1525). Chopra, N., Spong, M. W., & Lozano, R. (2006). Synchronization of bilateral teleoperation with time delays. Automatica, 44(8), 39–52. Gao, H., & Chen, T. (2007). H-infinity estimation for uncertain systems with limited communication capacity. IEEE Transactions on Automatic Control, 52(11), 2070–2084. Gao, H., Chen, T., & Lam, J. (2008). A new delay system approach to network-based control. Automatica, 44(1), 39–52. Hale, J. K., & Verduyn, S. M. (1993). Applied mathematical sciences, Introduction to functional differential equations. New York: Springer.

Mahdi Tavakoli received his B.Sc. and M.Sc. degrees in Electrical Engineering from Ferdowsi University and K.N. Toosi University, Iran, in 1996 and 1999, respectively. He then received his Ph.D. degree in Electrical and Computer Engineering from the University of Western Ontario, London, ON, Canada, in 2005. In 2006, he was a post-doctoral research associate at Canadian Surgical Technologies and Advanced Robotics (CSTAR), London, ON, Canada. In 2007–2008, and prior to joining the Department of Electrical and Computer Engineering at the University of Alberta as an assistant professor, Dr. Tavakoli was an NSERC Post-Doctoral Fellow with the BioRobotics Laboratory of the School of Engineering and Applied Sciences at Harvard University, Cambridge, MA, USA. Dr. Tavakoli’s research interests broadly involve the areas of robotics and systems control. Specifically, his research focuses on haptics and teleoperation control, medical robotics, and image-guided surgery.

t −T (t )

t −T (t )

≤ T (t )AT (t )P (A(t )) .

(A.16)

Given property P-VIII of function pj (·) , A (t )P (A(t )) > 0 and T

so A (t ) T



t t −T (t )

P (B (τ )) dτ −



≤ Tm AT (t )P (A(t )) and proof is completed.

t

t −T (t )

BT (τ ) P (B (τ )) dτ



References