Predictive Controller Design for Bilateral Teleoperation Systems With Time Varying Delays Ya-Jun Pan, Carlos Canudas-de-Wit∗ and Olivier Sename
Abstract— In this paper, a new predictive approach is proposed for the impedance control of the bilateral teleoperation systems. In this approach, two predictors with the same structures are applied in the controller design both in master and slave sides. The controller can use the transmitting signals at current time, which are estimated by the predictors, instead of the measurable delayed signals only. Hence the influence of the delay on the whole system can be minimized and performance can be improved. System knowledge such as the human operator dynamics, the measurement of the instantaneous time varying delays and the environmental impedance model are incorporated as well. The system stability and the convergence of the proposed approach are analyzed.
I. I NTRODUCTION During the last several decades, many different teleoperation systems have been developed to allow human operators to execute tasks in remote or hazardous environments. In the case of bilateral teleoperation, the contact force of the slave manipulator with the environment is reflected back to the master side operator through the master manipulator. In the worst case, the induced communication time delay may destabilize the bilaterally controlled teleoperator [1]- [2], at best it will decrease the system closed loop performance. In literature, there are many control schemes proposed for dealing with the time delay in the teleoperation systems [3]. The passivity approach using the scattering theory can ensure the passivity of the whole system at the price of the system performance degradation. Furthermore, the digital implementation of the control law may break the passivity of closed loop structure [4]. In the pole placement approach for constant time delays, the closed loop system poles can be assigned arbitrary according to finite spectrum assignment approach if the time delay is a prior known [5]. In [6], a standard H ∞ control problem is formulated in which the time delay is treated as a perturbation to the system, which results in a conservative approach. Though the observer-based approach overcomes the limitation of modified smith predictors which suffer from the fact that the initial state of both the process and the delay element are ignored, it is assumed that the delays should be small and bounded [7]. The sliding mode control [8] provides another possibility for the time delayed system while many concerns should be on the practical implementation such as the high switching gain and the chattering phenomena. Most of the approaches as mentioned previously work well All authors are with Laboratoire d’Automatique de Grenoble, UMR CNRS 5528, ENSIEG-INPG, B.P. 46, 38 402, ST. Martin d’H`eres, FRANCE. Corresponding Author: Professor Carlos Canudas-de-Wit. Email:
[email protected]
as long as the time delay is constant and relatively well known. However, in most real network communications, the time delays may vary. Hence it is important to explicitly consider the existence of time varying delays during the control design. In this paper, a new predictive approach for the impedance control of the bilateral teleoperation systems is designed. We assume that some system knowledge are available, i.e, the human operator model, the measurement of the instantaneous time varying delay and the environmental impedance model. It is worth to utilize the known information in the controller design in order to avoid a too conservative design which leads to poor performance. The human operator can be modelled by an impedance operator [9]. One important issue is how to use the available information on the communication network time delays. Approaches that only require the upper bound of the time varying delay in the controller design often yield too conservative results. Hence, we use the measured time varying delays which can be instantaneously applied in the controller design. The study shown here is particularly addressed to the problem of wheeled ground vehicle teleoperation. As a result, the environmental impedance model in this paper is assumed to be nonlinear due to the friction caused in the rotational operation of the vehicle [10]. In the new approach presented here, two predictors with the same structures are applied in the master side and the slave side respectively. The controller can use the transmitting signals at current time, which are estimated by the predictors, instead of the measurable delayed signal only. Hence the influence of the delay on the whole system can be minimized. The system stability and the convergence of the proposed approach are analyzed. Finally, simulation results are presented to show the effectiveness of the proposed approach. A complete version of this paper, including experimental results, is available on request. II. P ROBLEM F ORMULATION This section describes the model of the teleoperation system addressing in particular the problem of vehicle teleoperation, and the control objectives. A. Model description In this paper, the following dynamics of the single DOF master and slave system is considered Mm v˙ m (t) = um (t) + Fh (t), (1) Ms v˙ s (t) = us (t) − Fe (t),
where vi and ui (i = m, s) are the actuators motor velocities and input force respectively. The subscripts “m” and “s” stands for master and slave respectively. Mi (i = m, s) is the mass. Fh (t) is the force applied at the master side by the human operator and Fe (t) is the force exerted on the slave by its environment. The human operator impedance is modelled as a sum of the human impedance and an external human force [9], Mhn s2 + Bhn s + Khn Fh (s) = xm (s) + Fhext (s), Mhd s2 + Bhd s + Khd {z } |
G (s )
s vm Human Operator
r vs Master
Communication
x m , v m , Fhext
Fhext
Fh −
(2)
um +
+
+ Fˆe (t )
H
Gm
Cm
B. Knowledge on the time varying delays In literature, some known information of the time varying delays is often used to facilitate the controller design. In [11], the internal delay dynamics representing the TCP transmission channel model of [12] is used. In this paper, the delays from master side to slave side and from slave side to master side could be sent in the same channel but in different directions. Hence it is assumed to be same and is denoted as T (t). There are many possibilities to achieve Tˆ(t) which will be used in the controller design, i.e. according to known protocol models, or direct measurement etc. In this paper we assume that the transmission time delay Tˆ(t) can be measured, for instance, dating of the transmitted variable. In details it is measured in the transmission channel by sending a local time signal t to the opposite side. In the slave side, it is measured by calculating the difference between the local slave current time (t) and the received signal delayed from the master side (t − T (t)), and vice versa. Hence Tˆ(t) = T (t). Note that this technique requires that the synchronization of the clocks of both systems. In this predictive approach, some un-accuracies in the time delay estimation can be tolerated. Assumption 1: The delay T (t) is bounded as 0 ≤ T (t) ≤ T ∗ < ∞, where T ∗ is a positive constant. 1 It is important to remark that the impedance approach used here will target a closed-loop impedance Ym (s) that has a “spring-like” force similar to the one generated by the auto-alignment torque, exact that is preserved independent to the advanced velocity, see equation (4)
Feext = 0
− Fe
us
T (t )
+
+
+
xˆm (t )
Predictor
Predictor vˆm (t )
H(s)
where σ1 , fn and δ are constants depending on the tire and road adhesion, vs (t) is the wheel’s rotational angular speed.
Environment s − Fe
Fig. 1. Block diagram of the teleoperator system for the passivity analysis.
+
where Mhn , Mhd , Bhn , Bhd , Khn , and Khd are constants, Fhext is the measurable external force applied by operator. In vehicle control, the wheeled ground contact force is actually a dynamic tire friction model. In [10], they proposed a model from the longitudinal and the lateral motion of a wheeled vehicle. In this paper, we simplify the model according to this existing work by neglecting the auto-alignment forces that are present manly at high speeds, and modelling only the rotational ones that dominate at lower velocities1 . This force can be approximated as a nonlinear function of the form vs (t) , (3) Fe (t) ≃ σ1 vs (t) + fn arctan δ
Slave
r Fh
Cs
Gs
E
vs
xm , vm T (t ) Master
v s , Fe
Slave
Fig. 2. The schematic diagram of the bilateral teleoperator system with the proposed control scheme including the bi-predictor.
C. Control objective The control objective of the system is to realize the following desired impedance control xm (s) = Ym (s) [(1 − α)Fh (s) + N (s)Fe (s)] , (4) xs (s) = Ys (s) [W (s)xm (s) − kf Fe (s)] . where Ym (s), N (s), Ys (s), and W (s) are design operators, and α and kf are design constants. They are chosen according to the user specification while preserving Fh 7→ passive closed-loop properties of the maps −Fe R∞ vm for the system (4), i.e. 0 F (t)v(t) > vs T 0 where F (t) = [Fh (t), Fe (t)] and v(t) = T [vm (t), vs (t)] . According to F ig.1, the following pre vm (s) Fh (s) sentation can be written = G(s) , vs (s) −Fe (s) (1 − α)Ym s −N Ym s where G(s) = , (1 − α)Ym Ys W s Ys kf s − Ys W N s and this passivity property is equivalent to the fol ∗ ≥ 0, lowing condition λmin G(jω) + [G(jω)] where λmin (·) denotes the minimum eigenvalue with re∗ spect to the corresponding argument and [G(jω)] = T [G(−jω)] denotes the conjugate of the transfer function G(jω). In this paper, the specifications are se2 lected as: Ym (s) = 1/ M s + B s + K , Ys (s) = m m m 1/ Ms s2 + Bs s + Ks , W (s) = Bw s + Kw , N (s) = β, α and kf are all constants. III. P REDICTOR - BASED C ONTROLLER D ESIGN As in F ig.2, the main idea of this control strategy to reach the objective (4), is to try to predict, at the master side, the undelayed contact force Fˆe (t) and at the slave side try to predict the undelayed motion of the master side x ˆm (t) and vˆm (t).
A. Proposed controller The controller of the teleoperation system is designed as um (t) = −αFh (t) − Bm vm (t) −Khm xm (t) + β Fˆe (t) i ˆe (t − Tˆ(t)) , +β F (t − T (t)) − F e us (t) = (1 − kf )Fe (t) − Bs vs (t) − Ks xs (t) (5) +Bw vˆhm (t) + Kw x ˆm (t) i +Bw vm (t − T (t)) − vˆm (t − Tˆ(t)) h i +Kw xm (t − T (t)) − x ˆm (t − Tˆ(t)) ,
where Tˆ(t) is the estimate of the time delay and can be v ˆs (t) ˆ measured, and Fe (t) = σ1 vˆs (t)+fn arctan . Fˆe (t− δ Tˆ(t)) and x ˆm (t− Tˆ(t)) are the estimated signals artificially delayed by the measured time delay and Fe (t − T (t)), vm (t−T (t)), xm (t−T (t)) are the measured signals Fe , vm , xm that have been sent from one side to another through the communication channel. Because the signal Fe (t) is not available in the master side and xm (t) is not available in the slave side, we have to use the delayed signal instead for the error correction purpose. Hence the main motivation of our paper is to estimate the signals at current time t. B. State Space Representation In order to design the predictors, we need to write state space model for the master and slave closed-loop dynamics. 1) Master equations: Substituting the human operator model (2), the environmental contact force model (3) and the master side controller in (5) to the master actuator dynamics, = σ1 s, and defining σ(s) (t)) Fnm (t) = arctan vˆsδ(t) + arctan vs (t−T − δ v ˆs (t−Tˆ (t)) , usm (t) = x ˆs (t)+xs (t−T (t))−ˆ xs (t− arctan δ ˆ ˆ ˆ T (t)), where vˆs (t− T (t)) and x ˆs (t− T (t)) are the estimated signals artificially delayed by the measurable time delay Tˆ(t), we have (Mm s2 + Bm s + Km )xm (s) (1 − α)(Mhn s2 + Bhn s + Khn ) = xm (s) + L {βfn · Mhd s2 + Bhd s + Khd Fnm (t) + (1 − α)Fhext (t) + βσ(s) ◦ usm (t) (6)
where L(·) is the Laplace transform operator. Define ym (t) = zm4 (t) = xm (t), then we have an observable canonical state space form for (6) z˙ m (t) ym (t)
= Am zm (t) + bm usm +bf Fhext (t) + bnm Fnm (t), = c m zm .
(7)
2) Slave equations: Substituting the environmental contact force model (3) and the slave controller in (5) to the
slave actuator dynamics, and defining uss (t) = x ˆm (t) + vs (t) ˆ xm (t − T (t)) − x ˆm (t − T (t)) and Fns (t) = arctan , δ (Ms s2 + Bs s + Ks )xs (s) = (Bw s + Kw )uss (s) −kf σ(s)xs (s) − kf fn Fns (s), (8) Define ys = zs2 (t) = xs (t). From (8), we have z˙ s (t) ys (t)
= As zs (t) + bs uss (t) + bns Fns (t), = cTs zs .
(9)
Remark 1: Note that by construction the Am , As matrices have stable eigenvalues because the target model and the human impedance are stable operators. This will be useful later during the stability analysis of the predictors. C. Predictor design The predictor is designed on the basis of the stable statespace representation given previously. For that, measurable delayed signal from the transmission channel, will be used. Note that at both sides, some variables will be delayed artificially according to the time delay model, such as x ˆs (t − Tˆ(t)) in master side and x ˆm (t − Tˆ(t)) in slave side. Here we assume that the measurement of the time delays are accurate, i.e. Tˆ(t) = T (t). One possibility to design the predictor is that we can only design the estimate of the slave dynamics in the master side and vice versa. The difficulty existing in this method is that only the delayed version of the transmit signals usm (t), uss (t) are available. To cope with this problem, we would rather duplicate the predictor equations at both sides here. At total, we need to build four predictors: two at the master side and two at the slave side. They have however the same structure, and need to have the same initial conditions. Hence one of the working hypothesis is that both systems have synchronized clocks. They have the following structure ˆ˙ m (t) = Am z ˆm (t) + bm usm (t) z +bf Fhext (t − T (t)) + bnm Fnm (t), ˙ ˆs (t) = As z ˆs (t) + bs uss (t) + bns Fˆns (t), z
where usm (t), Fnm (t) and uss (t) are all available in the master side and the slave side according to their definitions, Fˆns (t) = arctan vˆsδ(t) , and the signal vˆs (t) needed in Fnm (t) and Fˆns (t) is calculated from vˆs (t) = zˆ˙ s2 . Similarly, the signal vˆm (t) in the construction of us (t) for (5) can also be calculated from vˆm (t) = zˆ˙ m4 . In the predictor design, it is possible to apply feedback delayed signal to improve the prediction accuracy. This study is to ˜i (t) = z ˆi (t) − zi (t), the error be finished soon. Defining z dynamics are: ˜˙ m (t) = Am z ˜m (t) + bf Fhext (t − T (t)) − Fhext (t) , z i h ˜˙ s (t) = As z ˜s (t) + bns Fˆns (t) − Fns (t) . z
Defining dm (t) = Fhext (t − T (t)) − Fhext (t), ds (t) = Fˆns (t) − Fns (t), the following properties hold: |dm (t)| =
|Fhext (t−T (t))−Fhext (t)| = |T (t)F˙hext (ξ)| ≤ T ∗ ρf , where ξ ∈ [t − T (t), t], T ∗ is the upper bound of the delay T (t) and F˙ hext (t) ≤ ρf where ρf is a constant. Hence if ρf is a small constant if Fhext (t) is slowly time varying. This means that dm (t) ∈ L∞ [0, ∞). ds (t) is actually an atan(·) function, which is bounded. Hence ds (t) ∈ L∞ [0, ∞), i.e. |ds (t)| ≤ kc = const. ds (t) also satisfies the Lipschitz condition when vˆs (t) or vs (t) is near the origin, vˆs (t) vs (t) |ds (t)| = arctan − arctan δ δ kl0 |ˆ vs (t) − vs (t)| ≤ kl kˆ zs (t) − zs (t)k = kl k˜ zs k, r 2 Bs +kf σ1 are constants. where kl0 and kl = kl0 1 + Ms
then, the estimation error norm k˜ zs (t)k, tends, in finite time, to a ball Br defined as Br = {˜ zs (t) : k˜ zs (t)k ≤ γkc = r} . 2) |ds (t)| ≤ kl k˜ zs k. Assume that As is stable, and there exists P ∈ R2×2 , P = P T > 0, ε, ζ > 0 satisfying T As P + P As + (ε−1 kl + ζ)I P bns < 0. (13) bTns P −ε−1 then k˜ zs (t)k → 0. Proof: The error dynamics in the slave side is
≤
IV. S TABILITY A NALYSIS A. The convergence of the predictor Note first that the matrices Am and As are both stable from construction, as discussed in remark 1. Theorem 1: Assume that Am is stable, and there exists P ∈ R4×4 , P = P T > 0, γ > 0 satisfying T Am P + P Am + I P bf < 0, (10) bTf P −γ 2
˜˙ s (t) z
˜Ts P z ˜s with P ∈ Define the Lyapunov candidate V (t) = z 2×2 R being a positive definite symmetric matrix. Case (1): |ds (t)| ≤ kc . The proof is similarly as in the proof in Theorem 1. Hence it is omitted here. Case (2): |ds (t)| ≤ kl k˜ zs k. The derivative of V will be V˙
= ≤
then, the estimation error norm k˜ zm (t)k, tends, in finite time, to a ball Br defined as Br = {˜ zm (t) : k˜ zm (t)k ≤ γT ∗ ρf = r} . Proof: The error dynamics in the master side is ˜˙ m (t) = Am z ˜m (t) + bf dm (t), z
(11)
where dm (t) ∈ L∞ [0, ∞) and |dm (t)| ≤ T ∗ ρf . Define the ˜Tm P z ˜m with P ∈ R4×4 being Lyapunov candidate V (t) = z a positive definite symmetric matrix. The derivative of V is as the following V˙
=
≤
˜Tm (ATm P + P Am + γ −2 P bf bTf P + I)˜ z zm 2 1 ˜m − z ˜Tm z ˜m + γ 2 d2m − γdm − bTf P z γ ˜Tm (ATm P + P Am + γ −2 P bf bTf P + I)˜ z zm ˜m + γ 2 d2m . −˜ zTm z
Using the Schur Complement, if (10) holds for P = ˜m + γ 2 d2m . The P T > 0 and γ > 0, then V˙ ≤ −˜ zTm z inequality above means that the error is bounded, i.e. the estimation error norm k˜ zm (t)k, tends, in finite time, to a ball Br defined as Br = {˜ zm (t) : k˜ zm (t)k ≤ γT ∗ ρf = r}. Note that, since Am is a stable matrix, there always exist P = P T > 0 and γ > 0 such that (10) holds. Theorem 2: Two cases are considered in this theorem: 1) ds (t) ∈ L∞ [0, ∞), i.e. |ds (t)| ≤ kc . Assume that As is stable, and there exists P ∈ R2×2 , P = P T > 0, γ > 0, satisfying T As P + P As + I P bns < 0, (12) bTns P −γ 2
˜s (t) + bns ds (t). = As z
≤
˜Ts (ATs P + P As )˜ ˜Ts P bns ds (t) z zs + z ˜s +bTns ds (t)P z T T ˜s (As P + P As + εP bns bTns P + ζI)˜ z zs ˜s + ε−1 d2s2 −ζ˜ zTs z ˜Ts (ATs P + P As + εP bns bTns P z ˜s , +ε−1 kl I + ζI)˜ zs − ζ˜ zTs z
where ε and ζ are positive constants. Using the Schur Complement, if (13) holds for P = P T > 0 and γ > 0, ˜s , which means that the error z ˜s (t) will then V˙ ≤ −ζ˜ zTs z be zero finally, i.e. k˜ zs (t)k → 0. For both cases, since As is a stable matrix, the existence of P = P T > 0 and γ > 0 which satisfy (12) and (13) is ensured. B. Closed-loop stability vsr
r According to the desired target (4) with xrm , vm , xrs and as state variables, the target control law should be
r r (t) − Km xrm (t) + βFer (t) um (t) = −αFhr (t) − Bm vm r r u (t) = (1 − kf )Fe (t) − Bs vsr (t) − Ks xrs (t) s r +Bw vm (t) + Kw xrm (t).
The state space representation of the target system becomes z˙ rm (t)
r = Am zrm (t) + bm zs2 (t) + bf Fhext (t) r +bnm Fns (t) (14)
z˙ rs (t)
r r = As zrs (t) + bs zm4 (t) + bns Fns (t), (15)
v r (t) r where Fns (t) = arctan( s ). The tracking error between δ the closed-loop system (7)-(9) and the target (14)-(15) is e˙ m (t) e˙ s (t)
= Am em (t) + bm es2 (t) + bm ηm1 (t) +bnm ηf (t) + bnm ηm2 (t) (16) = As es (t) + bs em4 (t) + bs ηs (t) + bns ηf (t), (17)
where ei (t) = zi (t) − zri (t) (i = m, s), ηm1 (t) = z˜s2 (t) − vr z˜s2 (t − T (t)), ηf (t) = arctan( vδs ) − arctan( δs ), ηm2 (t) =
(t)) arctan( vˆsδ(t) ) − arctan( vsδ(t) ) + arctan( vs (t−T ) − δ v ˆs (t−T (t)) arctan( ), ηs (t) = z˜m4 (t) − z˜m4 (t − T (t)). δ Stability of this error equation are studied next according to the two different cases concerning the assumption of the uncertain terms ηm2 (t) and ηf (t). Case 1: In this case, we consider that ηm2 (t), ηf (t) ∈ L∞ [0, ∞), i.e. |ηm2 (t)| ≤ 2kc and |ηf (t)| ≤ kc . Then (16) and (17) can be rewritten as
˙ e(t)
A=
D=
Am 02×3 bs
bnm bns
bm 02×1
= Ae(t) + Dη(t), (18) 04×1 bm em (t) , e(t) = , As es (t) ηf (t) ηm1 (t) bnm 04×1 , η(t) = ηm2 (t) . 02×1 bs ηs (t)
Note that the system specifications ensure a stable A. Theorem 3: Assume that A is stable, and there exists P ∈ R6×6 , P = P T > 0, γ > 0 satisfying T A P + PA + I PD < 0, (19) DT P −γ 2 I then, the tracking error norm of the closed loop system (18), tends, in finite time, to a ball Br defined as q 2 + η2 = r . Br = e(t) : ke(t)k ≤ γ 5kc2 + ηm1 s Proof: The proof is similar as in Theorem 1. Case 2: In this case, we consider that
˜s (t − T (t))k, kl k˜ zs (t) − z kl0 |vs (t) − vsr (t)| ≤ kl1 ke(t)k + kl2 |ηs (t)|,
h i
Bs +kf σ1 w where kl1 = kl0 0 0 0 B 1 −
= Ms Ms r 2 2 Bs +kf σ1 w w + and kl2 = kl0 B kl0 1 + B Ms Ms Ms . Then (16) and (17) can be rewritten as
|ηm2 (t)| |ηf (t)|
≤ ≤
˙ e(t) = Ae(t) + bηf (t) + Dη d (t),
(20) T
where A and e are same as in (18), b = [bnm , bns ] , ηm1 (t) bm bnm 04×1 D= , η d (t) = ηm2 (t) . 02×1 02×1 bs ηs (t)
Theorem 4: Assume that A is stable, and there exists P ∈ R6×6 , P = P T > 0, ε, γ, > 0 satisfying T A P + P A + ε−1 kl1 I + I P D Pb DT P −γ 2 I 0 < 0, (21) T b P 0 −ε−1 then, the tracking error norm of the closed loop system (20), tends, in finite time, to a ball Br defined as
Proof: The proof is similar as in the proof of the case (2) in Theorem 2. V. S IMULATION S TUDIES In the simulation, the parameters of the system (1) are selected as Mm = 0.03 kg · m2 , Bm = 0.5N ms, Km = 2 N m, α = 0.5, Ms = 0.04 kg · m2 , Bs = 0.5 N ms, Ks = 4 N m, kf = 2, Bw = 0.2 N ms, Kw = 0.5, Mhd = 1, Bhd = 3, Khd = 1, Mhn = 0.5, Bhn = 0.5, Khn = 1, β = 1, σ1 = 0.2, δ =0.1 and fn = 0.1. Hence, Ym (s) = 1/ 0.03s2 + 0.5s + 2 , Ys (s) = 1/ 0.04s2 + 0.5s + 4 and W (s) = 0.2s + 0.5. The external operator force is Fhext (t) = 5 sin(t). The initial conditions of the system in state space form are xm (0) = 0.1 rad, vm (0) = 0 rad/sec, xs (0) = 0.1 rad and vs (0) = 0 rad/sec. The initial ˆm (0) = [0, 0, 0, 0.1]T and conditions of the predictor are z ˆs (0) = [0, 0.1]T . The delays are T = 0.4 + 0.2sin(t) sec. z The passivity of the system desired target is satisfied according to λmin [G(jω) + G∗ (jω)] ≥ 0. A. Compared with the scheme without predictors Here the proposed scheme is compared with the conventional case when there is no prediction applied. In F ig.3, the profiles of the positions xm and xs are shown. Note that the tracking errors of the proposed scheme are smaller than that of the scheme without prediction, in which the system is still stable because the passivity of the system with the delays in this simulation is still not broken. Note that the tracking errors of the master sides are smaller than those of the slave sides. This is because the minimum disturbance attenuation level in slave side (γmin = 0.049) is smaller than that of the master side (γmin = 29), hence the estimation errors of the predictors, which is reflected on the tracking errors of the whole system. B. Robustness with respect to time delay uncertainties We further study the robustness of the proposed scheme on the time delay uncertainties. Two cases are considered here. In the first case, the measured delays are different as the real ones, i.e. it is artificially set to be Tˆ(t) = 0.5 + 0.1sin(t) sec, while the real time delay is T (t) = 0.4 + 0.2sin(t) sec. As in Fig.4 (a) and (b), the proposed scheme contains the robust property on the time delay estimation. In the second case, the real time delay from slave to the master side is set to be 0.5 + 0.3sin(t) sec which is different with the time delay from master to slave side. As in Fig.4 (c) and (d), the system is still stable and hence the proposed scheme seems to be robust with respect to time delay uncertainties. The theoretical proof of robustness is under study. C. Robustness on the known model uncertainties
In this case, we consider that the human operator model Br = {e(t) : and the environmental impedance model are not accurate q and there are model uncertainties. Additive uncertainties are 2 + η 2 ) + (γ 2 + ε−1 k )η 2 = r . γ 2 (ηm1 ke(t)k ≤ l2 s m2 introduced to the master and slave system respectively, i.e.
(a)
∆Fhext (t) = 0.3sin(t) and ∆Fe (t) = 0.2sin(t). As shown in Fig.5, though the tracking error is larger, the system is still stable when there are disturbance, which means that robust stability is obtained with the proposed approaches. The theoretical proof needs further investigation.
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Fig. 5. The evolution of the positions (∆Fhext (t) = 0.3sin(t) and ∆Fe (t) = 0.2sin(t)): (a) xm (t); (b) exm (t); (c) xs (t); (d) exs (t).
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ACKNOWLEDGEMENTS 0
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Fig. 3. The evolution of the master and slave positions: (a) xm (t); (b) exm (t) = xm (t) − xrm (t); (c) xs (t); (d) exs (t) = xs (t) − xrs (t).
This study was realized within the framework of the NECS (http://www-lag.ensieg.inpg.fr /canudas/necs)-CNRS project. The authors would like to thank the CNRS for funding the project. R EFERENCES
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Fig. 4. The evolution of the tracking errors: (a) exm (t) and (b) exs (t) when Tˆ(t) 6= T (t); (c) exm (t) and (d) exs (t) when the real delay from master to slave side is 0.5 + 0.3sin(t) sec.
VI. C ONCLUSIONS A new predictor-based approach has been proposed for bilateral force reflecting teleoperation system. Compared with the scheme without predictors, the proposed approach can achieve better system performance while ensuring the system passivity and stability in the existence of time varying delays. In the case that the time varying delays, the known human operator and the environmental impedance model are not accurate, the system can also maintain stability which further shows the robustness of the scheme.
[1] W. H. Kwon and A. E. Pearson, “Feedback stabilization of linear systems with delayed control,” IEEE Transactions on Automatic Control, vol. 25, no. 2, pp. 266–269, 1980. [2] A. Feliachi and A. Thowsen, “Memoryless stabilization of linear delay-differential systems,” IEEE Transactions on Automatic Control, vol. 27, no. 2, pp. 586–587, 1981. [3] C. Melchiorri, “Robotic telemanipulation systems: an overview on control aspects,” in Proceedings of the 7th IFAC Symposium on Robot Control, Wroclaw, Poland, 2003, pp. 707–716. [4] R. J. Anderson and M. W. Spong, “Bilateral control of teleoperators with time delay,” IEEE Transactions on Automatic Control, vol. 34, no. 5, pp. 494–501, 1989. [5] A. Fattouh and O. Sename, “Finite spectrum assignment for teleoperation systems with time delay,” in Proceedings of the IEEE Conference on Control and Decision, Hawaii, USA, 2003. [6] G. M. H. Leung, B. A. Francis, and J. Apkarian, “Bilateral controller for teleoperators with time delay via µ-synthesis,” IEEE Transactions on Robotics and Automation, vol. 11, pp. 105–116, 1995. [7] T. J. Tarn and K. Brady, “A framework for the control of time delayed telerobotic systems,” in Proceedings of the IFAC Robot Control, SYROCO, Nantes, France, 1997, pp. 599–604. [8] H. C. Cho., , J. H. Park, K. Kim, and J. O. Park, “Sliding mode based impedance controller for bilateral teleoperation under varying time delay,” in Proceedings of the 2001 IEEE International Conference on Robotics and Automation, Seoul, Korea, May 2001, pp. 1025–1029. [9] C. Canudas-de-Wit and P. Billot, “Human-friendly control design for drive-by-wire steering vehicles,” in Proceedings of the IFACAdvances in Automotive Control Workshop, Karlsruhe, Germany, March 2001, pp. 28–30. [10] E. Velenis, P. Tsiotras, C. Canudas-de-Wit, and M. Sorine, “Dynamic tire friction models for combined longitudinal and lateral vehicle motion,” Submitted to Vehicle System Dynamics, 2003. [11] E. Witrant, C. Canudas-de-Wit, D. Georges, and M. Alamir, “Remote output stabilization under two channels time-varying delays,” in the 4th IFAC Workshop on Time Delay Systems, Rocquencourt, France, 2003. [12] V. Misra, W. B. Gong, and D. Towsley, “Fluid-based analysis of a network of AQM routers supporting TCP flows with an applicqtion to RED,” in Proceedings of ACM SIGCOMM’00, Stockholm, Sweden, September, 2000.
