Guaranteed cost control design for delayed teleoperation systems

Author's Accepted Manuscript

Guaranteed cost control design for delayed teleoperation systems Yuling Li, Rolf Johansson, Kun Liu, Yixin Yin

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PII: DOI: Reference:

S0016-0032(15)00332-4 http://dx.doi.org/10.1016/j.jfranklin.2015.08.011 FI2419

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

21 December 2014 6 June 2015 18 August 2015

Cite this article as: Yuling Li, Rolf Johansson, Kun Liu, Yixin Yin, Guaranteed cost control design for delayed teleoperation systems, Journal of the Franklin Institute, http: //dx.doi.org/10.1016/j.jfranklin.2015.08.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Guaranteed Cost Control Design for Delayed Teleoperation Systems✩ Yuling Lia,∗, Rolf Johanssonb, Kun Liuc , Yixin Yina a School

of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083, China b Department of Automatic Control, Lund University, P.O. Box 118, 22100 Lund, Sweden c ACCESS Linnaeus Centre and School of Electrical Engineering, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden

Abstract A procedure for guaranteed cost control design of delayed linear bilateral teleoperation systems with nonlinear external forces is proposed. The assumption that the external forces are nonlinear functions of velocities and/or positions of local devices, and one part of these forces satisfies a sector condition has been made. A virtual tool system is introduced to ‘observe’ the forces at the remote sides, the position and velocity information of the master, the slave and the virtual tool are feedbacked to the controllers, hence the proposed control scheme actually has a four-channel architecture. A delay-dependent stability criterion is formulated, and then a sub-optimal guaranteed cost controller is obtained by solving a convex optimization problem in the form of linear matrix inequalities (LMIs). The behavior of the resulting teleoperation system is illustrated in simulations. Keywords: Teleoperation, guaranteed cost control, LMI, delayed systems, stability

1. Introduction Teleoperation systems enable humans to extend their capacity to manipulate remote interfaces with better safety, at less cost, and even with better accuracy. Its rich applications vary from nuclear operations, space and underwater exploration, to medical surgery, see [1], [2] and references therein. Two main issues discussed about the control design of bilateral teleoperation systems are stability and transparency. The position/velocity tracking performance between the master and the slave and the accuracy of the haptic display of the environmental force to human operator are two criteria to indicate the “degree” of the transparency. However, due to the very nature of teleoperation, time delays associated with communication between the local and the remote sites are inevitable, and it is well-known that these communication time delays ∗ Corresponding

author Email addresses: [email protected] (Yuling Li), [email protected] (Rolf Johansson), [email protected] (Kun Liu), [email protected] (Yixin Yin)

Preprint submitted to Journal of The Franklin Institute

August 27, 2015

can severely degrade the stability and the transparency performance of the bilateral teleoperation systems. Handling the delays is especially problematic in the case of bilateral teleoperation, where the measurements are communicated in both directions to allow haptic feedback. The classical approaches to deal with delayed bilateral teleoperation systems are scattering theory [3] and wave variable formalism [4], both of which guarantee stable bilateral teleoperation with any passive environment and any passive human operator by rendering the communication channel passive. In addition to the above mentioned results, passivity-based controllers relying on damping injection, passive output synchronization and adaptive control were also developed [5, 6, 7, 8, 9, 10] recently. Several other approaches were proposed to deal with delays and are reviewed in [11]. Along with this passivity-based control, optimal control has also been introduced to improve the system performance. Control frameworks based on H∞ control [12, 13, 14], H2 /LQG control [15, 16, 17, 18, 19, 20, 21], or μ synthesisbased control [22, 23] can be found in the literature. However, when having a further look at these control approaches, it can be found that different assumptions are imposed on the control design and the performance analysis. For the passivity-based control methods, the assumption that both of the human operator and the environment be passive was imposed. While for the optimal control applications, most research work supposed that the environment force and the human force have known linear mass-spring dynamics. Obviously, the former utilizes little information about the external forces while the latter has to access much more information during the control design procedure. Nonetheless, both approaches seem to be restrictive in real applications since it is not so easy to get the exact dynamics of the external forces and the passivity assumption is not satisfied all the time. As a result, in this paper, we assume that external forces are nonlinear functions of velocities and/or positions of the local devices, and one part of these forces satisfies a sector condition. Note that only one part of these forces satisfies a sector condition with respect to the output, which means the other parts may not be passive. Thus passivity is actually not required in our case. Actually, some research on nonpassive input of teleoperation systems has been investigated recently, see [24, 25, 26] for more details. However, all the methods proposed by [24, 25, 26] assume that the input forces are with exact linear dynamics which have limitation in real applications. Furthermore, we use more specific passivity information for the passivity parts, i.e., the sector bounds to improve the control performance. By using this information, a guaranteed cost controller which not only guarantees that the closed-loop system be stable, but also limits its tracking error between the master and the slave to some upper bounds is proposed. The proposed control strategy actually preserves a fourchannel control architecture, since the states and the forces of the master and the slave are communicated during the operation. To simplify the stability analysis, nominal delays are introduced to the system. However, these nominal delays can be removed during real applications. Finally, the controller gains are given by the linear matrix inequality (LMI) technique. The rest of the paper is organized as follows. In Section 2, the teleoperation model and other preliminaries are generalized. The problem formulation is given in Section 3, while the controller design procedure, the stability analysis and the main results are given in Section 4, after which we give the simulation 2

results to show the effectiveness of the proposed method in Section 5. Finally, the discussion and the conclusion are included in Section 6 and Section 7, respectively. The notation used throughout the paper is standard. For a real symmetric matrix P , the notation of P  0 (P ≺ 0) is used to denote positive- (negative-) definiteness. I is used to denote the identity matrix with appropriate dimensions. ∗ represents a block matrix which is readily referred by symmetry. N denotes the set of all nonnegative integers while N+ represents the set of positive integers. 2. Preliminaries Consider a simple teleoperation system as follows: mm q¨m (t) + bm q˙m (t) + km qm (t) = fh (t) + um (t), ms q¨s (t) + bs q˙s (t) + ks qm (t) = fe (t) + us (t),

(1)

where qi , q˙i , q¨i ∈ Rn are the joint positions, velocities and accelerations of the master and slave devices with i = m or s representing master or slave robot manipulators respectively. Similarly, mi , bi , ki are the effective mass, damping and spring coefficients of the master and slave devices, respectively. External forces applied to the devices by the human operator and the environment are represented by fh , fe , respectively, while um , us stand for the control signals. The following assumptions have been made for simplicity: Assumption 1. 1. The forward and backward time delays in the communication channel are assumed to be symmetric and time varying functions with respect to time t, denoted by T (t), and satisfy 0 ≤ T (t) ≤ d,

(2)

where d > 0 is a constant. 2. The velocities are available for measurements. Defining xi = [xTi1 , xTi2 ]T := [qiT , q˙iT ]T (i = m, s), a minimal state-space realization of the system (1) is reformulated as x˙ i (t) = Ai xi (t) + Biw wi (t) + Biu ui (t), where



0 Ai = −m−1 i ki

(3)

     I 0 0 , Biw = , Biu = , m−1 m−1 −m−1 i bi i i

wm (t) = fh (t), ws (t) = fe (t). The external forces fh (t), fe (t) in the joint-space satisfy T (qm (t))Fh (t), fh (t) = Jm

fe (t) =

JsT (qs (t))Fe (t),

(4) (5)

where Fh (t) and Fe (t) are the forces applied to the end effector of the master and the slave, respectively, Jm (qm ) and Js (qs ) are the Jacobian matrices of the master and the slave, respectively. For the case that the Jacobian matrices and the dynamics of Fh (t), Fe (t) are unknown, we may suppose that the following assumption is satisfied. 3

Assumption 2. The human force fh and the environmental force fe are supposed to be memoryless functions described by equations of the form fh (t) fe (t)

= fh (t) − Fm (t, zm (t)), = fe (t) − Fs (t, zs (t)),

(6) (7)

where fh (t), fe (t) are some continuous uniformly bounded functions satisfying  ∞  ∞  2 fh (s)2 ds ≤ Λm , fe (s)22 ds ≤ Λs , (8) 0

0

and Fm (t, zm (t)), Fs (t, zs (t)) with zi (t) := q˙i (t) − vi (t) + Hi (qi (t) − qi (t)).

(9)

are the passive parts of the input forces, here, Hi = HiT ∈ Rn×n , and vi , qi are some continuous uniformly bounded functions representing reference velocities and reference positions. In addition, for each i = m, s, the function Fi (t, zi (t)) which is a nonlinear function of zi (t), is piecewise continuous in t, globally Lipschitz in zi (t) with Fi (t, 0) = 0 and satisfies the following sector condition for ∀t ≥ 0: (10) FiT (t)(Fi (t) − Li zi (t)) ≤ 0, where Li is constant real positive-definite matrix of appropriate dimensions. Remark 3. The forces fh (t), fe (t) do not satisfy passivity assumption due to the time-varying fh (t), fe (t). However, when fh (t) ≡ 0, fe (t) ≡ 0, Assumption 2 reduces to the prevailing passivity-type assumption [25]. Nonzero vi (t), qi (t) are used to model the external forces with nonzero and/or time-varying positions and velocities. 3. Problem Formulation Stability and transparency are two conflicting objectives for the control of teleoperation systems. For good transparency, the position error and the force error between the master and the slave should be minimized. Considering the scaling between the master and the slave, we define ep (t) = αp xs (t) − xm (t),

(11)

ef (t) = fh (t) + αf fe (t),

(12)

where αp > 0, αf > 0 are the scaling constants. To facilitate the transparency performance analysis in the framework of guaranteed cost control, the concept of virtual tool [19, 20] is introduced as follows, mv q¨v (t) + bv q˙v (t) + kv qv (t) = ef (t).

(13)

Denoting xv = [qvT , q˙vT ]T , the state representation of the dynamics (13) is described by : x˙ v (t)

=

Av xv (t) − Bv Fm (t) − Bv αf Fs (t) + Bv (fh (t) + αf fe (t)), (14)

4

where



0 Av = −m−1 v kv

   I 0 , Bv = . m−1 −m−1 v bv v

In this paper, we propose a control scheme which utilizes the information of the master, the slave, and the designed virtual tool system, i.e., um (t) us (t) where

= km x ¯(t − T (t)), = ks x ¯(t − T (t)),

(15) (16)

x¯(t) = [xTm (t), xTs (t), xTv (t)]T .

(17)

As we can see by (15) and (16), nominal delays are introduced between the local devices and its controllers. In addition, to implement (15) and (16), the virtual tool system should be located both at the master side and the slave side. The closed-loop system is depicted in Fig. 1, where Tp (t) represents the nominal delay, while T (t) is the real delay between the master and the slave through communication. Here, Tp (t) = T (t). fh (t)

qm (t), q
m (t)

Master

T p (t)

um (t)

qs (t − T (t)), q
s (t − T (t))

km

qv (t − T (t)) T p (t)

Virtual Tool

fe (t − T (t)) T (t) T (t)

Communication Channel

fe (t)

T (t) T (t) qs (t), q
s (t)

Slave

T p (t)

us (t)

qm (t − T (t)), q
m (t − T (t))

ks

fh (t − T (t)) T p (t)

qv (t − T (t)) Virtual Tool

Figure 1: Diagram of the closed-loop system Remark 4. The choice of the virtual system is very important since it will not be influenced by the controllers, hence it should be at least input-to-state stable [27] with fh (t), fe (t) as the input, xv (t) as the state. 5

Remark 5. It is noted that the proposed control scheme preserves four-channel architecture, even though the human force (or the environmental force) does not direct feedback to the slave (or the master). These forces from the remote side actually are ‘observed’ by the virtual tool system located at the local sides. Hence, as shown in Fig. 1, four communication paths have been established between the master and the slave. Note that the minimization of the tracking error ef , ep , and also the tracking error between the master/slave and the virtual tool system, is equal to the minimization of the norm of an augmented vector x(t), x(t) = [xTs (t), eTp (t), xTm (t) − xTv (t)]T = Π¯ where Π is a nonsingular matrix and ⎡ 0 Π = ⎣−I I

I αp I 0

(18)

⎤ 0 0 ⎦. −I

Hence, the control law (15) and (16) can be rewritten as u(t) =

Kx(t − T (t)),

(19)

T , ksT ]T Π−1 . where u = [uTm , uTs ]T , K = [km Up to now, from (3), (14), (18) and (19) we derive the system dynamics with the augmented vector x(t) as the state: x(t) ˙ = Ax(t) + BKx(t − T (t)) + DF (t) + D f  (t), (20) x(t) = x0 (t), ∀t ∈ [−d, 0],

where ⎡

As A = ⎣αp (As − Am ) αp (Am − Av ) ⎡

⎤ ⎡ 0 0 0 ⎦ , B = ⎣−Bmu Av Bmu ⎤

0 Am Av − Am

0 Bmw D = −D = ⎣ −Bmw + Bv

⎤ Bsu αp Bsu ⎦ , 0

−Bsw αp Bsw ⎦ . αf Bv

The main objective of this paper is to design a guaranteed cost control law um , us for the teleoperation system (1) with the human force and environmental force satisfying Assumption 2 such that the closed-loop system is stable with an associated cost C less than a guaranteed cost C  . The problem setup is shown T , qsT ]T denotes the position vector of the in Fig. 2. As shown in Fig. 2, q = [qm teleoperator, i.e., the position vectors of the master and the slave, T F (t) = [Fm (t), FsT (t)]T , f  (t) = [fh (t)T , fe (t)T ]T .

(21) (22)

T In this paper, the variable z = [zm , zsT ] is designed as

zm (t) zs (t)

= Hm qm (t) + q˙m (t) = Γm xm (t), = Hs qs (t) + q˙s (t) = Γs xs (t). 6

(23) (24)

f ∗ (t) −

⊗w ⊗

Teleoperator

Communication Channel

Controller


− T (t)) q(t − T (t), q(t

u F(t)

z(t)

Figure 2: Diagram of the system structure As a result, it remains a new problem how to obtain the controller gain K which guarantees stability of the controlled system (20) with external forces fh , fe under Assumption 2 with the following associated cost function (25) satisfying C(x, u) < C  :  ∞ C(x, u) = [xT (θ)Qx(θ) + uT (θ)Ru(θ)]dθ, (25) 0

where Q, R are given positive-definite matrices, and C  > 0 is the guaranteed cost bound. 4. Controller Design To analyze the stability of the controlled teleoperation system, we consider the following candidate Lyapunov-Krasovskii functional:  T

V (t) = x (t)P x(t) + d where



S P  0, S = T1 S2

0



−d

t



t+s

 T  x(δ) x(δ) S dδds, x(δ) ˙ x(δ) ˙

 S2  0, S1  0, S3  0. S3

(26)

(27)

Taking the derivative of V (t) with respect to t along the trajectory of the system (20) yields V˙ (t)

=

 T  x(t) x(t) S x(t) ˙ x(t) ˙ T    t  x(s) x(s) −d S ds. x(s) ˙ x(s) ˙ t−d 2xT (t)P x(t) ˙ + d2

7



(28)

By employing Lemma 1 in [28], one could obtain  x(s) ds x(s) ˙ t−d ⎡ ⎤T ⎡ x(t) −S3 S3 ⎣ x(t − T (t)) ⎦ ⎣ S3 −S3

t x(s)ds −S2 S2 t−T (t) 

−d

≤

t



x(s) x(s) ˙



T

S

⎤ ⎤⎡ x(t) −S2T S2T ⎦ ⎣ x(t − T (t)) ⎦ . t x(s)ds −S1 t−T (t)

(29)

Following the free weighting matrix method [28], we add the following equation to (28) 2η T (t)N [Ax + BKx(t − T (t)) + DF (t) + D f  (t) − x(t)] ˙ = 0,

(30)

where N = [N1T , N2T , 0, N3T , 0]T , N1 , N2 , N3 are any invertible matrices with appropriate dimensions,  t η(t) = [x(t)T , xT (t − T (t)), ( x(s)ds)T , x˙ T (t), F T (t)]T , (31) t−T (t)

and employ the inequality obtained by completion of squares 2η T N D f  (t) ≤ η T (t)N D (N D )T η(t)/ρ + ρf  (t)T f  (t),

(32)

V˙ (t) ≤ η(t)T Φη(t) + ρf  (t)T f  (t),

(33)

one gets

where

⎡ Φ11 ⎢ ∗ ⎢ Φ=⎢ ⎢ ∗ ⎣ ∗ ∗

Φ12 Φ22 ∗ ∗ ∗

−S2T S2T −S1 ∗ ∗

Φ14 Φ24 0 Φ44 ∗

⎤ N1 D N2 D ⎥ ⎥ 0 ⎥ ⎥ N3 D ⎦ 0

(34)

with Φ11 = −S3 + d2 S1 + N1 A + AT N1T + (N1 D )(N1 D )T /ρ, Φ12 = S3 + N1 BK + AT N2T + (N1 D )(N2 D )T /ρ, Φ14 = d2 S2 − N1 + AT N3T + P + (N1 D )(N3 D )T /ρ, Φ22 = −S3 + N2 BK + K T B T N2T + (N2 D )(N2 D )T /ρ, Φ24 = −N2 + K T B T N3T + (N2 D )(N3 D )T /ρ, Φ44 = d2 S3 − N3 − N3T + (N3 D )(N3 D )T /ρ. Furthermore, by Assumption 2 and S-procedure we obtain

V˙ (t) ≤ η T Ξη + 2 FiT (Fi − Li zi ) − xT Qx − uT Ru i=m,s 

T

+ρf (t) f  (t),

(35) 8

where Ξ = {Ξij } = ΞT with Ξ11 = Φ11 + Q,

Ξ15 = Φ15 + ΥT ΓT LT ,

Ξ22 = Φ22 + K T RK,

Ξ55 = Φ55 − 2I,

Γ = diag{Γm , Γs },   α I −I 0 Υ= p , I 0 0

L = diag{Lm , Ls },

otherwise

Ξij = Φij , i, j ∈ N+ , 1 ≤ i, j ≤ 5.

By integrating both sides of the inequality (35) from 0 to t, one has  t V (t) − V (0) ≤ − [xT (θ)Qx(θ) + uT (θ)Ru(θ)]dθ + λmax {Ξ}η22 0  t +ρ (f  )T (s)f  (s)ds 0  t (f  )T (s)f  (s)ds. (36) < λmax {Ξ}η22 + ρ 0

˙ Now, in view of inequality (36) and suppose  we conclude that V (t) is  Ξ ≺ 0, negative outside the compact set Δ = {η η2 ≤ 2Λρ/ − λmax {Ξ}}, where

t Λ = 0 (f  )T (s)f  (s)ds ≤ Λm + Λs . Hence, it implies that η ∈ L2 , then V (t) ∈ L∞ , which implies that the state x(t) ∈ L∞ as well. In addition, it is easy to conclude that the cost C satisfies C < C  , where C  = V (0) + ρ(Λm + Λs ). Thus the following result is obtained. Theorem 6. For the teleoperation system (1) with external forces satisfying Assumption 2, a guaranteed cost controller (15) - (16) exists if there exist matrices X, Y , S¯1  0, S¯3  0, P¯  0,   S¯1 S¯2 ¯ S = ¯T ¯  0 S2 S3 and scalars ¯ > 0, ρ > 0 such ⎡ Ω11 Ω12 −S¯2T ⎢ ∗ Ω22 S¯2T ⎢ ⎢ ∗ ∗ −S¯1 ⎢ ⎢ ∗ ∗ ∗ Ω=⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

that the following LMI Ω14 Ω24 0 Ω44 ∗ ∗ ∗ ∗

Ω15 δ1 D D¯  D 0 0 δ3 D¯  δ3 D  −2¯ I 0 ∗ −ρI ∗ ∗ ∗ ∗

X 0 0 0 0 0 −Q−1 ∗

⎤ 0 YT ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥≺0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −R−1

holds, where Ω11 = −S¯3 + d2 S¯1 + δ1 AX T + δ1 XAT , Ω14 = d2 S¯2 + P¯ − δ1 X T + XAT δ3 , Ω22 = −S¯3 + BY + Y T B T , Ω44 = d2 S¯3 − δ3 X T − δ3 X,

Ω12 = S¯3 + δ1 BY + XAT , Ω15 = δ1 D¯  + XΥT ΓT LT , Ω24 = −X T + δ3 Y T B T , ¯ = −1 ,

9

(37)

 and the controller gains in (15) and (16) can be chosen as

 km = Y X −T Π. ks

The guaranteed cost bound is T  −1   −T    0  0 X 0 0 x(δ) X x(δ)  ¯ S C = dδds 0 X −1 0 X −T x(δ) x(δ) ˙ ˙ −d s (38) +x0 (0)X −1 P¯ X −T x0 (0) + ρ(Λm + Λs ), where Λm , Λs are given by (8). Proof. By the Schur complement, it can be seen that Ψ ≺ 0 implies Ξ ≺ 0, where ⎡ ⎤ Ψ11 Ψ12 −S2T Ψ14 Ψ15 N1 D I 0 ⎢ ∗ Ψ22 S2T Ψ24 N2 D N2 D 0 KT ⎥ ⎢ ⎥ ⎢ ∗ ∗ −S1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ Ψ44 N3 D N3 D 0 0 ⎥ ⎥ (39) Ψ=⎢ ⎢ ∗ ∗ ∗ ∗ −2I 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ −ρI 0 0 ⎥ ⎢ ⎥ −1 ⎣ ∗ 0 ⎦ ∗ ∗ ∗ ∗ ∗ −Q ∗ ∗ ∗ ∗ ∗ ∗ ∗ −R−1 with Ψ11 = −S3 + d2 S1 + N1 A + AT N1T ,

Ψ12 = S3 + N1 BK + AT N2T ,

Ψ14 = d2 S2 − N1 + AT N3T + P,

Ψ15 = N1 D + ΥT ΓT LT ,

Ψ22 = −S3 + N2 BK + K T B T N2T ,

Ψ24 = −N2 + K T B T N3T ,

Ψ44 = d2 S3 − N3 − N3T . Let N1 = δ1 N2 , N3 = δ3 N2 and pre-multiply and post-multiply the diagonal matrix diag{N2−1 , N2−1 , N2−1 , N2−1 , 1/, I, I, I}, one could get that the LMI (37) implies that Ξ ≺ 0, where P¯ = XP X T , S¯1 = XSX T , S¯2 = XS2 X T , S¯3 = X S¯3 X T ,

(40)

N2−1 .

(41)

T

¯ = 1/, Y = KX , X =

According to our earlier analysis, the closed-loop system is stable. Noting that T  −1   −T    0  0 X 0 0 x(δ) X x(δ) ¯ S V (0) = dδds 0 X −1 0 X −T x(δ) x(δ) ˙ ˙ −d s +x0 (0)X −1 P¯ X −T x0 (0), then (38) is obtained. It completes the proof. If Λm , Λs in (38) are given, we then are interested in finding the least upper bound of the cost C  in the form of (38) and also the optimal guaranteed cost controllers (15) and (16). However, in view of (38), it can be seen that it is not a convex optimization problem because of the nonlinear terms  −1   −T  X X 0 0 −1 ¯ −T ¯ S X P X and . 0 X −1 0 X −T Taking the above fact into account, we still can find a sub-optimal bound as follows. 10

Theorem 7. For the closed-loop system (1) with external forces satisfying Assumption 2, the control law (15) - (16) is a sub-optimal guaranteed cost control law if the following convex optimization problem min

¯ P¯ ,M,¯ X,Y,S, ,ζ,ρ

ζ + tr(M ) + ρ(Λm + Λs ),

(42)

subject to: 1. LMI (37), 2.

 S¯ P¯  0, S¯ = ¯T1 S2

3.



4.

⎡ M ⎣ N

 S¯2  0, S¯1  0, S¯3  0, M  0, ¯ > 0, ρ > 0, S¯3

(43)

 ζ xT0 (0)  0, x0 (0) l1 X T + l1 X − l12 P¯

(44)

 l2

XT 0

T  N  X 0 + l2 0 XT



⎤

⎦  0, 0 − l22 S¯ X

(45)

¯ X, Y, M, ζ, ¯, ρ. Here, tr(•) denotes the trace of a matrix, has a solution P¯ , S, and  T  0  0 x0 (θ) x0 (θ) dθds = N N T , x˙ 0 (θ) x˙ 0 (θ) −d s l1 , l2 = 0 aregiven  scalars. In addition, the controller gains in (15) and (16) km are given as = Y X −T Π. ks Proof. It is noted that xT0 (0)P x0 (0) = xT0 (0)X −1 P¯ X −T x0 (0) < ζ and  0  −d

s

0



x(δ) x(δ) ˙

T



 x(δ) S dδds x(δ) ˙

 = = = <

0

−d



0

s

(46)

  T x(δ) x(δ) tr( dδdsS) x(δ) ˙ x(δ) ˙

tr(N N T S)  −1 T X tr(N 0

  −T X 0 S¯ X −1 0

tr(M ).

0

X −T

 N) (47)

Hence, it follows from the above analysis that C  ≤ ζ + tr(M ) + ρ(Λm + Λs ). On the other hand, by Schur complement, the inequations (46) and (47) are equivalent to 1.



 ζ xT0 (0) 0 x0 (0) X T P¯ −1 X

11

(48)

2.

⎡ ⎣

M N



XT 0

T N  0 ¯−1 X S XT 0



⎤

0 ⎦0 X

(49)

Note that the inequalities (48) and (49) are not LMIs. However, in view of the following inequalities X T P¯ −1 X > l1 X T + l1 X − l12 P¯ and



XT 0

  0 ¯−1 X S XT 0

which come from and

 T X ( 0

  T 0 X > l2 0 X

  0 X + l 2 XT 0

(50)  0 ¯ − l22 S, X

(51)

(X T − l1 P¯ )P¯ −1 (X T − l1 P¯ ) > 0

(52)

 T  0 −1 X ¯ ¯ S) S ( − l 2 XT 0

(53)

 0 ¯ > 0, − l2 S) XT

it can be seen that (48) and (49) guarantees that (46) and (47) hold. This completes the proof. Remark 8. Actually, for each pair (l1 , l2 ) in a certain domain, we can find the solution of problem (42). Hence, the minimal cost C can be considered as a function of (l1 , l2 ). Theorem 1 can be used to pick suitable (l1 , l2 ), i.e., suppose the problem in Theorem 1 has a feasible solution, Xf , S¯f , P¯f , hence  T    X X 0 0 (l1 , l2 ) ∈ {(l1 , l2 )|l1 XfT +l1 Xf −l12 P¯f > 0, l2 +l −l22 S¯f > 0}. 2 0 X 0 XT 5. Simulations and Results A master-slave teleoperation system (1) with the effective mass mm = ms = 10 kg, the damping bm = bs = 1 Ns/m, and the spring coefficients km = ks = 0 N/m was used to test the proposed method in numerical simulations. The parameters of the virtual tool system were picked as mv = 10, bv = 1 Ns/m, kv = 1 Ns/m. To show the capability of the controller (15) - (16) in stabilizing system (3) when condition (10) is satisfied, the maximum d of the delay T (t) is estimated firstly. By Theorem 6 we find the maximal allowable time delay d = 0.9999s. In the following simulations, the time-varying delay T (t) is set to be 0.2| sin(t)|s. Moreover, the initial states for the master, the slave and the virtual tool were chosen as qm (t) = et+0.2 , qs (t) = e2(t+0.2) , q˙m (t) = et+0.2 , q˙s (t) = 2e2(t+0.2) , qv (t) = 0, q˙v (t) = 0, ∀t ∈ [−0.2, 0]. Here, we consider the master and the slave with different initial positions and velocities to see the tracking performance.

12

k (t) 1

1.2 1

1

k (N)

0.8 0.6 0.4 0.2 0

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 3: External force k1 (t) To access the teleoperation system behavior we chose the human force and the environment force as follows, fh (t) fe (t)

= =

k1 (t) − khp tan−1 (qm (t)) − khv tan−1 (q˙m (t)), −1

k2 (t) − kep tan

−1

(qs (t)) − kev tan

(q˙s (t))

(54) (55)

with k1 (t), k2 (t) as the external input. The coefficients were chosen as khp = 1, kep = 1, khv = 10, kev = 10. Hence, the nonlinearity satisfies (10) with Lm = Ls = 10. In the simulation we chose the cost function (25) with Q = diag{1, 1, 100, 100, 100, 100}, R = diag{0.01, 0.01}. k1 (t) was chosen to be the rectangle signal depicted in Fig. 3. k2 (t) which represents that the environmental force at the slave side were considered as: k2 (t) = 10k1 (t). Based on the feasible solutions obtained by Theorem 6, we choose l1 = 1, l2 = 0.001, hence by Theorem 7 and solving the corresponding optimization problem by using the Yalmip toolbox [29], we obtain the suboptimal guaranteed cost control law (15-16) with the parameters as follows:   km = −24.9235 −38.8259 4.4129 −2.9555 18.9467 42.3063 , (56)   ks = 6.6772 0.1714 −22.8445 −39.3203 14.3150 39.4938 . (57) Applying the controllers to the described system, we obtained the simulation results showed in Fig. 4 - Fig. 6. It can be seen that the master and the slave response stably. From Fig. 4 and Fig. 5, we can see that the positions and the velocities of the master and the slave follow the virtual tool’s motion tightly. Specifically, when the slave interacts with the environment, the positions of the master and the slave move forward and backward with the increase and decrease of the human force respectively, and both of the master and the slave achieve their peak value at 20s and 60s (Fig. 4), the exact time when the non-passive force k1 (t) and k2 (t) turn to zero (Fig. 3). Good force tracking performance between the master and the slave is also achieved in this case, see Fig. 6. To have a further look at the control strategy, we also make simulations for the case when there are no nominal delays between the local devices, i.e., the master(or slave) and the local controller at the master (or slave) side. However, 13

Time delay T(t)=0.2|sin(t)| s 4 qm qs

Position (m)

3

qv 2 1 0 −1

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 4: Position tracking performance when there are nominal delays between local devices Time delay T(t)=0.2|sin(t)| s 3 vm vs

Velocity (m/s)

2

vv

1

0

−1

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 5: Velocity tracking performance when there are nominal delays between local devices in this case, the output of the virtual tools in the master side and the slave side are different, as shown in Fig. 7 and Fig. 8. The simulation results are shown in Fig. 9 - Fig. 11. It can be seen that the closed-loop system is stable when there are no nominal delays between the local devices. The position tracking performance, velocity tracking performance and force tracking performance are as good as the case when there are nominal delays between the local elements. This implies that the nominal delays can be removed in real applications. We now simulate the case when the human forces are non-passive, while the slave can move freely in the environment. The simulation results for the

14

Time delay T(t)=0.2|sin(t)| s 15 f

h

10

f

Force (N)

e

5 0 −5 −10 −15

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 6: Force tracking performance when there are nominal delays between local devices master and the slave using the designed control law (15-16) with the control gain (56-57) can be found in Fig. 12 - Fig. 14. It can be seen that the master and the slave response as similar as the case when both the human force and environmental force are non-passive, except that the positions and velocities of the master and slave are smaller than the ones in the case when the human and environmental forces are non-passive. It can be concluded that the input part k1 (t) and k2 (t) inject energy into the closed-loop system without degrading the stability. We further simulate two cases when there is no delay in the communication channel and when there are delays in the communication channel, but the actual delay bound is 0.6s, which is larger than the designed delay bound. For brevity, we only provide the position tracking results in both two cases. The simulation results for the master and slave using the designed controller (15-16) with controller gain (56 - 57) are shown in Fig. 15. Note that to compare the system performance with different delays, we also include the position tracking results for the case when the communication time delays are equal to the designed time delays in Fig. 15 . It is seen from from Fig. 15 that there is no big difference in the results when the delay switch from 0 to 0.2|sin(t)|. However, we also can find that there arises oscillation in the positions of the master, slave when when the actual delay is 0.6|sin(t)|. The positions of the master and the slave follow each other tightly and the closed-loop system is stable even though there is a lit bit oscillation when the actual delay bound is larger than the designed delay bound. Hence the designed controllers can still work even when the actual delays are larger than their designed delay bounds to some extent. When the actual delay bound becomes even bigger, the closed-loop system would turn to be unstable, and delay will destabilize the closed-loop system. Finally, we make simulation for the case that the slave is driven to a wall, hence the environmental force is chosen as fe (t) = k2 (t) − kep tan−1 (qs (t)) − kev tan−1 (q˙s (t)) where kep and kev is reset as 1000 an 0 respectively, and the

15

Time delay T(t)=0.2|sin(t)| s 6 Position of Virtual Tool in the master side Position of Virtual Tool in the slave side

Position (m)

5 4 3 2 1 0 0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 7: Position of virtual tools when there are no nominal delays between local devices

Time delay T(t)=0.2|sin(t)| s 1

Velocity (m/s)

Velocity of Virtual Tool in the master side Velocity of Virtual Tool in the slave side 0.5

0

−0.5 0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 8: Velocity of virtual tools when there are no nominal delays between local devices

16

Time delay T(t)=0.2|sin(t)| s 4 qm qs

Position (m)

3

qv 2 1 0 −1

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 9: Position tracking performance when there are no nominal delays between local devices

Time delay T(t)=0.2|sin(t)| s 3 vm vs

Velocity (m/s)

2

vv

1

0

−1

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 10: Velocity tracking performance when there are no nominal delays between local devices

17

Time delay T(t)=0.2|sin(t)| s 15 f

h

10

f

Force (N)

e

5 0 −5 −10 −15

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 11: Force tracking performance when there are no nominal delays between local devices other parameters stay the same. In attempts to solve the corresponding feasibility problem induced by Theorem 6, however, the problem becomes unfeasible. This shows the conservatism of the applied method. 6. Discussion Design of teleoperation systems with passive or nonpassive input forces and delays that guarantee stable interaction and position/velocity coordination is a challenging task in recent years. In this paper, we investigate how a guaranteed cost controller can be obtained for teleoperation systems with nonpassive human operator and nonpassive environment. The symmetric time-varying delays in the communication channel is considered. In the context of nonpassive teleoperation design, some previously results can be found in the literature [30, 24, 31, 26, 32, 33]. Polushin [30] firstly removed the passivity assumption on input forces by designing a Proportional-Derivative (PD) -based control scheme for nonlinear teleoperation systems via the two-step construction of input-to-state stable subsystems, and then along this line, the force reflected teleoperation systems [34, 25] have been proved the boundedness of the position and velocity synchronization errors between master and slave sites. Non-passive human forces and passive environmental forces have also been considered under unsymmetrical time varying communication delays in [26], while the closed-loop stability condition is given by LMI technique. Various control schemes for teleoperation without introducing the passivity assumption were proposed in [32, 33]. The stability problem of teleoperation systems with non-passive input forces and constant input forces was also addressed in [24, 31]. In the above results with all the non-passive forces considered, however, most of the results just give the control structure which guaranteed the closedloop system is stable, while few ones considered how to pick the control gains with guaranteed system performance. Picking the control gains becomes even

18

Time delay T(t)=0.2|sin(t)| s 2 q

m

qs

Position (m)

1.5

qv 1 0.5 0 −0.5

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 12: Position tracking performance when the slave is in free motion

Time delay T(t)=0.2|sin(t)| s 3 vm vs

Velocity (m/s)

2

v

v

1

0

−1

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 13: Velocity tracking performance when the slave is in free motion

19

Time delay T(t)=0.2|sin(t)| s 5 f

h

f

e

Force (N)

0

−5

−10

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 14: Human forces and environmental forces when the slave is in free motion

Position tracking performance with different time delays 7 q (T(t)=0) m

Position(m)

6

qs(T(t)=0)

5

qm(T(t)=0.2|sin(t)|)

4

q (T(t)=0.2|sin(t)|) s

qm(T(t)=0.6|sin(t)|)

3

q (T(t)=0.6|sin(t)|) s

2 1 0

0

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 15: Position tracking performance with different time delays in communication channel

20

more complex for master-slave systems with high degrees of freedom, and it is very beneficial if the control gains can be chosen by some sort of automatic algorithm. In this paper, we obtain a holistic optimization-based procedure for the controller synthesis of teleoperation systems. However, some difficulties will arise in applying the proposed procedure for teleoperation systems. Firstly, the nominal delays are introduced between the master/slave and the virtual tool. Even though it is easy to add virtual delays in the system, making the virtual delays identical to the practical delays between the master and the slave is not so easy. Secondly, two identical virtual tools are needed in our method. Thirdly, both velocity and position coordinates should be available to measurement, hence, state estimation is necessary in cases without full state measurement. State estimation applied to the absolute stability problem can be found in the earlier work [35, 36]. Finally, there is no holistic way to choose free variables l1 and l2 , which makes the method more difficult in applications. It is noted that to study the problem of stabilization of time delay systems with input saturation, the closed-loop system can be also modeled as a time delay system with sector conditions [37]. However, the practical meaning of the problem formulation in our paper is totally different. The input saturation assumes that there is saturation limit on u(t) in (19) with unknown parameter K. In our case, the assumption that the external forces which are one part of the inputs of the system satisfy sector condition gives more flexibility on the controller design process. The virtual tool (13) in the control design could be any input-to-state stable system, and it of course could be different from the master and the slave. Furthermore, Theorem 6 and the Lyapunov-Krasovskii functional (26) and the equation (38) provide some margin for uncertainties and master-slave mismatch. 7. Conclusion In this paper, a guaranteed cost control design procedure for delayed bilateral teleoperation systems has been proposed. The case when the external forces are nonlinear and one part of them satisfies sector condition was considered. To reflect the force at the remote side, a virtual system is introduced, hence the proposed control scheme actually has a four-channel architecture. The problem setup is reduced to a guaranteed cost state feedback control design problem for a delay system after a few mathematical manipulations. Firstly, the delaydependent stability criterion is formulated, then the sub-optimal guaranteed cost controller is obtained by solving a convex optimization problem using LMI technique. To simplify the analysis, nominal delays are introduced between the local devices and their controllers. Simulation results show that the proposed control method guarantees that the closed-loop system is stable with a bounded cost. However, the proposed method is only applicable to linear teleoperators with symmetric time varying delays in the communication channel, and the upper bound of the delay is required to be known. The guaranteed cost is also related to the maximum delay, which is conservative in real applications. These problems will be under research and be reported in the near future.

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