Analysis of Bilateral Teleoperation Systems under Communication Time-Delay

Transcription

1 Analysis of Bilateral Teleoperation Systems under Communication Time-Delay Anas FATTOUH and Olivier SENAME 1 Abstract In this article, bilateral teleoperation systems under communication time-delay are analyzed. Necessary and sufficient conditions guarantee the stability of the global system for any communication time-delay are firstly derived. Then, based on these conditions, weighting function on the slave system is obtained. Therefore, the problem of finding a position controller for the slave system which also guarantees the stability of the global system is reduced to a standard robust control problem: finding a controller which internally stabilizes a system and guarantees the H -norm of some transfer function less than one. Keywords: Bilateral teleoperation systems, Time delay systems, Robust stability. I. INTRODUCTION Force feedback can provide useful information to the operator of a teleoperation system. Assembly tasks can usually be completed much faster when the operator has a feel for the forces and torques caused by contact [1]. Other benefits include improved safety and less damage from overstressing the manipulator or material being handled [2]. However, when teleoperation is performed over a great distance, a time delay is incurred in the transmission of information from one site to another. This time delay can destabilize a bilaterally controlled teleoperator [3]. Many control schemes have been proposed in the literature to overcome the instability due to communication time delay in bilateral teleoperation systems. However, much of these works have been in the area of human performance and a few papers have dealt with control issues pertaining to this problem [4]. The main idea behind many studies in this area lies on the use of different methods to achieve desired master and slave compliance, then scaling factors on transmitted information are added to ensure the stability of the global system (see for example [5]). However, this approach reduces the bandwidth of the system which must lie in some interval for normal operation [6]. Passivity and scattering theory have also been largely used to ensure the stability of time-delay teleoperation systems (see for example [3], [7]). In this approach, all components of teleoperation system are assumed to be passive, otherwise some transformations have to be done in order to 1 Corresponding author. A. Fattouh is with Laboratoire d Automatique des Systèmes Coopératifs, Université de Metz, BP 8794, 5712 METZ Cedex 1, France Anas.Fattouh@lasc.univ-metz.fr O. Sename is with Laboratoire d Automatique de Grenoble, ENSIEG, BP 46, 3842 Saint Martin d Hères Cedex, France Olivier.Sename@inpg.fr make them passive. However, these transformations degrade the performance of the global system [8], [9]. In addition obtained control laws will not be directly applicable using digital controllers or digital communication channel as the discretization of a passive system is in generally not passive [1]. Leung et al. [4] have used H optimal control and µ-analysis and synthesis techniques to design robust controllers for bilateral teleoperation systems. However, communication time delay is treated as a disturbance on the system and not as a parameter of the system. In [15], the authors have proposed an H approach for impedance control of teleoperation systems. As in [4], where the control objectives are tracking and transparency, the delay is considered as an uncertainty, which leads to a conservative result. However, in this work a stability analysis with respect to time-delay is provided. Despite many recent studies on time-delay systems (see for instant [11]), a few results on these systems have been applied on bilateral teleoperation systems. In [12], [13], frequency sweeping test is used to derive conditions on PItype controller such that the global system is asymptotically stable. However, the study cannot be directly generalized for other types of controller. Authors have also proposed in [14] finite spectrum controller for bilateral teleoperation systems. However, the communication time-delay must be known and robustness is difficult to be analyzed. In this article, necessary and sufficient conditions ensuring the stability of a bilateral teleoperation system are derived in term of H -norm of some transfer function which must be less than one. In this case the stability of global system can be guaranteed for any communication time-delay by including this constraint (. < 1) in the design of the salve controller. This can be achieved using standard techniques in robust control (see for example [16], [17]). In addition, robustness on stability of the global system with respect to environment s variations can be also guaranteed if the H -norm of some transfer functions is less than one. Without loss of generality, the article considers only the one-dimensional case. In addition, all the transfer functions are assumed to be linear and time-invariant. A general representation of teleoperation systems is given in Section 2 followed by an analysis of this representation in Section 3. The robustness of bilateral teleoperation system is studied in Section 4. Section 5 presents simulation results that support the theoretical work and conclusion is drawn in Section 6.

2 II. SYSTEM REPRESENTATION A bilateral teleoperation system can be represented by the block diagram of Fig. 1 and consists of five subsystems: the human operator, the master manipulator, the communication channel, the slave manipulator and the environment. The variables of the system are: F h is the force applied by human operator, F e is the contact force with the environment and, are the position of master and slave manipulators respectively. Fig. 2. Considered structure of bilateral teleoperation system. Fig. 1. Block diagram of bilateral teleoperation system. The operator commands a position forward to the environment through the master, the communication channel and the slave. Likewise, the force sensed at the environment is transmitted back to the human operator through these blocks. Since the teleoperator is controlled bilaterally, the arrows in Fig. 1 can be reversed. In this case the operator commands force forward to the environment and environment s position is sent back to the master. With this understanding, the teleoperation system is acting as a hybrid system, where the human operator determines the switching between the position and force control modes [18]. Generally, three controllers are designed for this system: two local controllers for master and slave manipulators in order to achieve desired master and slave compliances and second slave controller such that, in steady state, the slave position is equal to the master position and the global system is asymptotically stable. In the next section, block diagram of Fig. 1 will be studied in detailed. The communication channel will be considered as a pure time-delay. The local controllers are assumed to be already designed and integrated in the master and slave transfer functions. Then, the stability of the global system will be analyzed and constraint on the design of second salve controller will be obtained such that the global system is asymptotically stable for any communication time-delay. III. STABILITY ANALYSIS In view of previous section, let P m and P s be the transfer functions of master and slave manipulators with local controllers, Z e be the environment s impedance, h 1 and h 2 be time delays of communication forward and backward channels respectively and C be the second slave controller. With these notations, bock diagram of Fig. 1 can be redrawn as shown in Fig. 2. Definition 1: Consider the bilateral teleoperation system of Fig. 2. This system is said to be asymptotically stable if: 1) The transfer function from to is asymptotically stable with unitary gain. 2) The transfer function from F h to is asymptotically stable for any communication time-delay. A. Analysis of Condition 1 in Definition 1 The block diagram of the system from to is shown in Fig. 3 where W 1 (s) is a weighting function reflecting the desired performance. Fig. 3. Master Slave positions system. Based on Fig. 3, the transfer function from to is given by where T s := = C P s 1 + C P (1) s P s = P s 1 + P s Z e (2) From robust control theory, condition 1 in Definition 1 can be rewritten as: find C such that T s is internally stable and W 1 T s < 1 (3) where the constraint of unitary gain for T s is included in the weighting function W 1 (s). B. Analysis of Condition 2 in Definition 1 From block diagrams 2 and 3, the transfer function from F h to can be described in the block diagram of Fig. 4 where h = h 1 + h 2. Based on Fig. 4, the transfer function from F h to is given by T m := P m = F h 1 + P m T s Z e e sh (4)

3 Fig. 4. Operator force Master position system. Our objective now is to find a constraint on the transfer function T s such that T m given in (4) is asymptotically stable for any time delay h. To this end, let s recall the following result [19]. Lemma 1: Let P(s) and Q(s) be two polynomials in complex variable s satisfying: 1) deg s [P (s)] < deg s [Q(s)], 2) Q(s) is stable, then, the polynomial Q(s) + P (s) e sτ is stable for all τ if and only if Q(jω) > P (jω), ω R (5) In view of above lemma, we have the following proposition. Proposition 1: Consider the system of Fig. 4. Assume that P m and T s are strictly proper and stable, then the transfer function T m given in (4) is asymptotically stable for any time delay h if and only if W 2 T s < 1, where W 2 = P m Z e (6) Proof: Let P m (s) = Nm(s) D m(s) T s (s) = Ns(s) D s(s) Then transfer function T m given in (4) can be rewritten as follows N m D s T m = D m D s + N m N s Z e e sh (8) As P m and T s are assumed to be strictly proper and stable then the two conditions of Lemma 1 are satisfied. Therefore, using Lemma 1, system (4) is asymptotically stable for all time delay h if and only if (7) D m D s (jω) > N m N s Z e (jω), ω R (9) which is equivalent to condition (6). Remark 1: It should be noted that assumptions on P m and T s in Proposition 1 are always satisfied as we initially assumed that P m is the transfer function of the master with a local controller, and T s represents the transfer function from to which has already been stabilized by C. In view of (3), (6) and Remark 1, the following proposition can be easy obtained. Proposition 2: Consider the bilateral teleoperation system in Fig. 2 and transfer functions (1) and (6). This system is asymptotically stable according to Definition 1 if and only if C such that T s is internally stable and WT s < 1 (1) where W is a weighting transfer function satisfying W(jω) = max W 1 (jω), W 2 (jω) }, ω R (11) In the next section, the robustness of the proposed control scheme will be analyzed. IV. ROBUSTNESS ANALYSIS In this section, the following problem is considered: given a bilateral teleoperation system as in Fig. 2 and assuming that the environment impedance Z e belongs to some admissible set Ξ, find conditions on T s such that the system is asymptotically stable according to Definition 1 for all Z e Ξ, where the term admissible will be defined later. Notice that communication time-delay is not included in this problem as our stability criterion (5) is delay independent. According to Definition 1, robustness in stability of system in Fig. 2 is equivalent to the robustness in stability of T s : Xm and T m : F h. Consider the system of Fig. 3, define the following family of transfer functions P s = Ps : Z e Ξ } (12) The set Ξ is said to be admissible, if Ps for nominal impedance Z e and P s have the same unstable poles [2]. Proposition 3: Consider the system of Fig. 3 with the family of transfer functions (12) and Ξ is admissible. Assume that the system is internally stable for nominal impedance Z e, then the system is internally stable for all Z e Ξ if W 3 T s < 1 (13) where W 3 is a weighting transfer function satisfying P s (jω) W 3 (jω) max Z e Ξ P s n (jω) 1, ω R (14) and P n s = P s for nominal impedance Z e. proof: Using (14), the family of transfer functions (12) can be written as follows: P s = (1 + W 3 ) P n s (15) where P s n = P s for nominal impedance Z e and is a variable stable transfer function satisfying < 1. From (15) and robust control theory [2], the robust stability condition for T s : Xm is given by (13). In the next proposition, the robustness in stability of T m : F h is considered.

4 Proposition 4: Consider the system of Fig. 2 with the family of transfer functions (12) and Ξ is admissible. Assume that the system is internally stable for nominal impedance Z e, then the system is internally stable for all Z e Ξ if W 4 T s < 1 (16) where W 4 is a weighting transfer function satisfying } W 4 (jω) = max W 3 (jω), max W (jω), Z e Ξ ω R (17) and W and W 3 are given by (11) and (14) respectively. proof: If (16)-(17) are satisfied, then we have W 3 T s < 1 (18) WT s < 1, Z e Ξ (19) As (18) is satisfied, then T s is asymptotically stable for all Z e Ξ (Proposition 3). In addition, as (19) is satisfied, then (1) is satisfied for all Z e Ξ, therefore, T m is asymptotically stable for all Z e Ξ. Remark 2: The design of C such that (1) is satisfied for nominal stability and (16) is satisfied for robust stability can be carried out using µ-analysis and synthesis toolbox of MATLAB [21]. V. SIMULATION RESULTS Consider the following dynamics of the master and the slave manipulators Mm v m = F h + τ m M s v s = F e + τ s (2) where v m and v s are the velocities for the master and the slave respectively, τ m and τ s are the respective motor torques, M m and M s are the respective inertias, F h is the operator torque and F e is the environment torque. In order to stabilize the above system, Anderson and Spong [3] have proposed the following PI control law τ m = B m v m B s1 (v m v s ) K s (vm v s )dt τ s = B s2 v s α f F e + B s1 (v m v s ) +K s (vm v s )dt where M m = kg, M s = 1kg B m = 3N/m, B s2 = N/m, Z e = 1, α = and K s and B s1 are the parameter of the PI controller which must be chosen such that the closed-loop system is stable. In the presence of communication time delay h, Niculescu et al. [12] have shown that for K s = 5 and B s1 = 2.8, the closed-loop system is stable for all h >. However, when the admittance of the environment changes to Z e = 2, the system becomes unstable. In this case, choosing K s = 12 and B s1 = 2.8, the closed-loop system is stable for all h < 27 sec. Authors in [14] have used finite spectrum techniques to design a controller for system (2). The following control law has been found which stabilize the system for any known time delay h. τ m = 7v m + u(t) τ s = ( v m v s ) + 5 ( v m v s )dt ( u(t) =.292δ+116 δ(δ+1) Fh (t) F e (t) ).46 δ+1 u(t h) + h ( 14.3e 17.5θ e θ) u(t + θ)dθ where v m (t) = v m (t h 1 ), Fe (t) = F e (t h 2 ) and δ is the differential operator. The same example is considered here. The proposed method will be used to find a controller C for nominal stability then for robust stability. Based on the above discussion, the master and slave transfer functions with local controllers are given by 1 Pm =.4s 2 +3s+5 (21) 1 P s = s 2 +s+2.8 The impedance of the environment is modelled as follows A. Design for Nominal Stability Z e = 2s + 3 (22) In this subsection, a solution to (1) is calculated. From (2), (6), (11), (21) and (22), we have s P s = 2 +s+2.8 s s s s W = W 2 = 2s+3 s 2 +3s+5 (23) Now we are looking for a controller C which internally stabilize P s and such that WC P s < 1. Using MAT- LAB, the following controller has been found C = s s s s s s s s The bode diagram of WC P s is shown in Fig. 5. It is clear from this figure that WC P s < 1. Fig. 6 and 7 show master and slave positions for variable communication time delay and nominal environment (Z e = 2s + 3) and non-nominal environment (Z e = 5s + 2) respectively. Note that in the two cases the slave position pursuits the master position. It should be noted that when slave is not in contact with the environment, the system becomes unstable (see Fig. 8). In the following subsection, a robust controller is designed to overcome this problem. B. Design for Robust Stability In this subsection, a solution to (16)-(17) is calculated. Firstly, assume that Z e = as + b (24)

5 Magnitude (db) 1 ( ) and 1 ( ) and Fig. 5. Frequency (rad/sec) Bode diagram of WC P s Fig. 8. Master - Slave positions: without contact. with a [,4] and b [,6]. By evaluating (17), we have found W 4 = 1s + 1 s (25) Now we are looking for a controller C which internally stabilize P s and such that W4C P s < 1. Using MAT- LAB, the following controller has been found C =.4s4 +.96s s s s 4 + 6s s s Fig. 9 and 1 show master and slave positions for variable communication time delay and nominal environment (Z e = 2s + 3) and for Z e = respectively. Note that in the two cases the slave position pursuits the master position. Fig. 6. Master - Slave positions: nominal environment impedance ( ) and Fig. 7. Master - Slave positions: non-nominal environment impedance. ( ) and Fig. 9. Master - Slave positions: nominal environment impedance.

6 ( ) and Fig. 1. Master - Slave positions: without contact. VI. CONCLUSIONS In this article, the nominal and robust stability of bilateral teleoperation system with communication time delay has been characterized by a weighting transfer function on the complementary sensitivity transfer function of slave system. Thus, standard techniques in robust control have been used to design the position controller for slave system. In future works, deep study on the weighting transfer function will be done. [12] S.-I. Niculescu, D. Taoutaou, and R. Lozano, On the closed-loop stability of a teleoperation control scheme subject to communication time-delays, in Proc. 41st IEEE Confer. on Decision & Control, (Las Vegas, Nevada, USA), pp , 22. [13] D. Taoutaou, S.-I. Niculescu, and K. Gu, Robust stability of teleoperation schemes subject to constant and time-varying communication delays, in Proc. 42nd IEEE Confer. on Decision & Control, (Hyatt Regency Maui, Hawaii, USA), pp , 23. [14] A. Fattouh and O. Sename, Finite spectrum assignment controller for teleoperation systems with time delay, in Proc. 42nd IEEE Confer. on Decision & Control, (Hyatt Regency Maui, Hawaii, USA), 23. [15] A. Fattouh and O. Sename, H -based impedance control of teleoperation systems with time delay, in 5th IFAC Workshop on Time Delay Systems, (INRIA, Rocquencourt, France), 23. [16] B. A.Francis, A course in H control theory. Springer-Verlag, [17] K. Zhou, J. C. Doyle, and K. Glover, Robust and optimal control. Prentice-Hall Inc., [18] O. Khatib, A unified approach for motion and force control of robot manipulators, IEEE Journal Robotics Automations, no. 1, pp , [19] L. E. El sgol ts and S. B. Norkin, Introduction to the theory and applications of differential equations with deviating arguments, vol. 15 of Mathematics in Science and Engineering. Academic Press, New York, [2] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback control theory. Macmillan Publishing Company, 199. [21] G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, µ-analysis and synthesis toolbox for use with MATLAB. The MathWorks, Inc., 21. REFERENCES [1] D. Stokic, M. Vukobratovic, and D. Hristic, Implementation of force feedback in manipulation robots, Int. Journal of Robotic Research, vol. 5, pp , [2] N. R. Parker, S. E. Salcudean, and P. D. Lawrence, Application of force feedback to heavy duty hydraulic machines, in IEEE Int. Conf. on Robotics & Automation, (Atlanta, Georgia), [3] R. J. Anderson and M. W. Spong, Bilateral control of teleoperators with time delay, IEEE Trans. on Automatic Control, vol. 34, no. 5, pp , [4] G. M. H. Leung, B. A. Francis, and J. Apkarian, Bilateral controller for teleoperators with time delay via µ-synthesis, IEEE Trans. on Robotics and Automation, vol. 11, pp , [5] J. H. Park and H. C. Cho, Sliding-mode controller for bilateral teleoperation with varying time delay, in Proc. of the 1999 IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics, (Atlanta, USA), pp , [6] M. Uebel, M. Ali, and I. Minis, The effect of bandwidth on telerobot system performance, IEEE Transactions on Systems, Man and Cybernetics, vol. 24, no. 2, p , [7] G. Niemeyer and J.-J. E. Slotine, Towrds force-reflecting teleoperation over the internet, in IEEE Int. Conf. on Robotics & Automation, pp , [8] S. Munir, Internet based teleoperation, tech. rep., The George W.Woodruff School of Mechnanical Engineering, Georgia Institute of Technology, 2. [9] N. A. Tanner and G. Niemeyer, Pratical limitations of wave variable controllers in teleoperation. To be published, 24. [1] G. M. H. Leung and B. A. Francis, Bilateral control of teleoperators with time delay through a digital communication channel, in Proc. 31st Confer. on Decision and Control, (Tucson, Arizona, USA), pp , [11] S.-I. Niculescu, Delay effects on stability. A robust control approach, vol Springer-Verlag: Heidelbeg, 21.