Stability Analysis of Delayed 4-Channel Bilateral Teleoperation Systems
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1 Staility Analysis of Delayed 4Channel Bilateral Teleoperation Systems Noushin Miandashti Mahdi Tavakoli, IEEE Memer Department of Electrical and Computer Engineering, University of Alerta, Edmonton, AB T6G V4, Canada ABSTRACT This paper studies the staility of a delayed 4channel ilateral teleoperation system ased on the passivity framework. Assuming that the operator and the environment are passive systems, the staility of the teleoperation system is reduced to ensuring the passivity of a master control unit (MCU), a slave control unit (SCU), and the timedelayed communication channel. Each of these three locks is modeled as a transfer function matrix and passified using our proposed approach in a multiloop feedack (MLF) structure. We report conditions on the controllers of the 4channel architecture that are sufficient for passivity of MCU and SCU. Simulation results confirm the validity of these conditions for the staility of the teleoperation system. Index Terms: H.5. Information Interfaces and Presentation: User Interfaces Haptic I/O; I..9 Artificial Intelligence: Prolem Solving, Control Methods, and Search Control theory INTRODUCTION In the design of teleoperation systems, ensuring staility in the presence of communication time delay is of vital importance. A teleoperation system consists of a twoport network representing a teleoperator (comprising a master root, a slave root, their controllers, and a communication channel) coupled to two oneport networks representing a human operator and an environment. A teleoperator s twoport network model can e an impedance matrix relating velocities to forces, a hyrid matrix relating a mixed forcevelocity vector to another mixed forcevelocity vector, or an admittance matrix relating forces to velocities 4, 9. As discussed later, ased on this modeling, twoport network theory can e used to analyze the teleoperator s passivity and, therefore, the teleoperation system s staility. Passivityased staility analysis of ilateral teleoperation systems was first introduced in through scattering theory, and then presented in the wave variales framework in 8. A review of time delay compensation techniques for teleoperation systems can e found in. Assume that the human operator and the environment demonstrate passive ehaviors. In, 8, the emphasis is on passifying the communication channel assuming that the master and the slave when comined with their respective controllers are passive systems. Although the master and the slave roots are always passive 6, there is no guarantee that the master control unit (MCU) and the slave control unit (SCU) are also passive. To the est of the authors knowledge, the conditions under which the MCU and the SCU remain passive have not een reported efore. In this paper, we derive conditions for passivity of the MCU and the SCU while passifying the delayed communication channel. The wave variales for passifying a delayed communication channel were developed for a channel that can e modeled as a miandash@ualerta.ca mahdi.tavakoli@ualerta.ca IEEE World Haptics Conference 3 48 April, Daejeon, Korea /3/$3. 3 IEEE port network 8. In order to apply the wave variales method to a delayed 4channel teleoperation system, in which the communication channel has 4 inputs and 4 outputs, similar to the practice in 3, a weighted sum of force and velocity at each side of the teleoperation system is sent through the channel making the communication channel appear as a port network. However, this approach to delay compensation in the 4channel teleoperation system introduces nonphysical variales that complicate the teleoperator s passivity analysis. In this paper, without any need for physical interpretation of the signals that are involved in a delaycompensated 4channel teleoperation system, a transfer matrix ased approach to modeling and staility analysis is presented that is easier to follow compared to traditional twoport network ased passivity analyses. The rest of the paper is organized as follows. Section provides mathematical preliminaries required for this paper. In Section 3, the 4channel teleoperation system is modeled. In Section 4, the passification procedure for the delayed communication channel within the 4channel architecture is presented. Section 5 finds a condition for staility of the delaycompensated 4channel teleoperation system in terms of passivity of the MCU and the SCU. Simulation results are presented in Section 6 and Section 7 presents the conclusions. MATHEMATICAL PRELIMINARIES Definition. 4 The Mport network N M shown in Fig. with zero initial energy is passive if and only if t M f i (τ)v i (τ)dτ, t () i= for all admissile forces f i s and velocities v i s. f f k v Definition. 7 The system v k N M v k v M f k f M Figure : An Mport network. ẋ = f (x,u) () y = h(x,u) (3) is said to e passive if there exists a continuously differentiale positive semidefinte function V (x) (called storage function) such that u T y V = V x f (x,u), (x,u) Rn R (4) Moreover, it is said to e strictly passive if u T y > V. 7
2 Lemma. 7 The LTI minimal realization with G(s)=C(sI A) B Dis ẋ = Ax Bu (5) y = Cx Du (6) passive if G(s) is positive real; strictly passive if G(s) is strictly positive real. Definition 3. 7 An n n proper rational transfer function matrix G(s) is positive real if poles of all elements of G(s) areinres, for all real ω for which jω is not a pole of any element of G(s), the matrix G( jω)g T ( jω) is positive semide f inite, and any pure imaginary pole of jω of any element of G(s) is a simple pole and the residue matrix lim s jω (s jω)g(s) is positive semidefinite Hermitian. Applying the Definition 3 and Lemma to a transfer matrix G(s), which can represent a twoport network, leads to Raiseck s passivity criterion (see Appendix A for details). 3 SYSTEM MODELING The LTI dynamics of the operator and the environment are assumed to e = Fh Z hv m (7) = Fe Z e V s (8) where and F e are the operator force applied to the master root and the environment force applied to the slave root, Z h and Z e are the operator and the environment impedances, V m and V s are the operator and the environment velocities, and Fh and F e are the exogenous force inputs from the operator and the environment, respectively. The LTI models of the master and the slave roots are assumed to e Z m V m = F cm (9) Z s V s = F cs () where Z m = M m s and Z s = M s s are the impedances of singledof master and salve roots, respectively. Also, F cm and F cs are the control signals for the master and the slave roots, respectively. In the 4channel teleoperation architecture in Fig., these control signals are F cm = C m V m C 4 V md C 6 C d () F cs = C s V s C V sd C 5 C 3 d () where C m =(K dm K pm s ) and C s =(K ds K ps s ) represent local PD position controllers, C 6 and C 5 are local force controllers, C and C 3 are force feedack and feedforward controllers, and C to C 4 are position compensators working in conjunction with C s and C m, V md and V sd are desired velocities, and d and d are desired forces for the master and the slave, respectively. In this paper is considered as the environment force applied to the slave where in literature it is defined as the slave s force applied upon the environment. This change of notation is made to preserve symmetric structure of the 4channel teleoperation system. Z h Operator C 6 C m Z m Master V m C 4 C 3 C C Communication and Control V s Z s C s C 5 Slave Z e Environment Figure : 6 4channel ilateral teleoperation system structure 4 CHANNEL DELAY COMPENSATION IN 4CHANNEL TELE OPERATION Inspired y 3, for compensating for the communication delay in a 4channel teleoperation system, a weighted sum of force and velocity at each side of the teleoperation system must e considered as incoming signals such that the communication channel appears as a port network system. Once we do so, the 4channel teleoperation system in Fig. is remodeled as shown in Fig. 3. The transfer matrix relating the outputs of the delayed communication channel to its inputs are I V = e st e st V (3) Based on Definition 3, the channel transfer matrix is not positive real and thus the delayed communication channel is not passive. Realizing a passive communication channel can e carried out through two methods. The first method is ased on the wave variales formulation 8. In Fig. 4, the outputs of the wave transformation lock at the master side are and at the slave side they are I = V u m ;v m = V u m (4) V = ( v s );u s = ( v s ) (5) Since u m (t)=u s (t T ) and v s (t)=v m (t T ), weget I = e st e st V V (6) V = e st V e st I (7) This will change the channel transfer function matrix from (3) to C(s)= e st e st (e st e st ) (e st e st ) (e st e st ) e st e st (e st e st ) (8) which is positive real according to Definition 3, and therefore is passive. The second method follows a purely transfer function ased approach. The ojective here is to change the original delayed channel (3) to ecome a passive transfer matrix. Thus, the prolem oils down to solving the following equations for unknown coefficients a,a,,,c and c, so that the resulting transfer matrix satisfies the positiverealness requirements. I = av e st ( V )ce st (9) V = a e st V c e st I () Note that if T =, we need to get I = and V = V. To satisfy this, it is necessary that a =,c =,a = c, = 7
3 Operator MCU SCU F e F I I C e I C 6 C 5 Z h C 3 C V V V V 4 Z e C Z m Z s V m C m s V s C Timedelayed communication channel Environment Figure 3: A 4channel ilateral teleoperation system in which the communication channel has een remodeled as a twoport network I Left Wave Transformer u m T V v m v s V u s Right Wave Transformer Figure 4: Delaycompensated communication channel. Operator v f MCU Teleoperator v v 3 v 4 Timedelayed f communication f 3 SCU f 4 channel Environment Figure 6: PN structure; passivity of the human operator, MCU, timedelayed communication channel, SCU, and environment is sufficient for passivity (and staility) of the teleoperation system. I V T V Figure 5: Another delay compensated communication channel. Sustituting these in (9)(), the new channel will e positive real if c = Thus, the new channel (9)() will have only one free parameter. Finally, the transfer matrix of this delaycompensated channel will turn out to e the same as (8), which meets the positiverealness conditions according to Niemeyer94 3 and is passive. 5 STABILIZATION OF A DELAYCOMPENSATED 4CHANNEL TELEOPERATION SYSTEM A mathematically involved and intractale approach to stailizing the delayed 4channel teleoperation system is to consider the teleoperator as a whole Vm Fh = G Vs total () where G total (s) is to e passified through the design of the controllers. However, this transfer matrix is far too complicated to analyze for passivity. In the context of ilateral teleoperation systems in the presence of constant timedelay, passivityased staility methods are attempt to passify the communication channel assuming that oth the MCU and the SCU are passive. However, the passivity of the MCU and the SCU needs to e guaranteed via proper control. Here, it is shown that there are certain conditions involving the values of the gains of the master and slave controllers in order for the passivity of the MCU and SCU to e guaranteed. Three different approaches to the passivity analysis are presented elow. The first two approaches, the port network (PN) and singleloop feedack (SLF) structures, are previously introduced in literature and we show that they cannot stailize the 4channel teleoperation system while using the third approach, multiloop feedack (MLF) structure that is proposed in this paper, the staility of the system will e guaranteed. 5. Approach : Port Network (PN) structure Theorem 5.. The teleoperator, i.e., the teleoperation system excluding the operator and the environment as shown in Fig. 6, is passive if the MCU, the time delayed communication channel and the SCU are passive. Proof. If the MCU, the time delayed communication channel and the SCU are passive. Based on Definition f v f ( v ) () f v f 3 v 3 (3) f 3 ( v 3 ) f 4 v 4 (4) The sum of () (4) gives f V f 4 V 4 (5) which implies the passivity of the teleoperator. The prolem faced in practice with the PN structure in the context of 4channel teleoperation is that since a weighted sum of oth force and velocity are exchanged etween the MCU, the channel, and the SCU, physical interpretation of these signals is not easy and writing the twoport network models ased on immitance parameters is rather difficult. It is preferale to pursue a solely transfer function ased approach that deals with system inputs and outputs regardless of their physical interpretation (or lack thereof). 73
4 5. Approach : SingleLoop Feedack (SLF) structure In this section, the staility analysis method proposed in 3, the SLF structure, is further investigated. As seen in Fig. 3, we have ( C 6 ) I ( C = V 5 ) m V = V s (6) Z tm Z tm Z ts Z ts C 3 C V m = V C C 4 V s = (7) where Z tm = Z m C m and Z ts = Z s C s. Manipulating (6)(7) will result in the singleloop feedack structure equivalent of the teleoperation system shown in Fig. 7, where C(s) is the transfer matrix (8) of the delaycompensated communication channel and C Z G(s)= tm C 4 Zts (8) y u y 3 u 4 y5 e Timedelayed communication channel e 4 Z h MCU SCU Z e e e 5 e 3 u y u 3 y 4 u 5 Theorem 5.. Assume that a teleoperation system is coupled to an environment and a human operator that are passive ut otherwise aritrary. If G(s) is strictly positive real, then for inputs with finite norms, the outputs in the system shown in Fig. 7 will have finite norms. (Note that the delay has een compensated for in the channel) Fh (C6) Fe (C5) I V G(s) C(s) V I CVm C4Vs C3Fh CFe Figure 7: SLF structure; equivalent singleloop feedack structure of the 4channel teleoperator (i.e., not including the human operator and the environment). A major drawack of SLF is that it guarantees inputoutput staility from the inputs involving and to the outputs involving V m and V s. However, we know that there are two more feedack loops involving Z h and Z e in a teleoperation system (not shown in Fig. 7) that can affect the staility of the closedloop system. In other words, only Fh and F e are true inputs to the system. Incorporating the dynamics of the human operator and the environment into Fig. 7 results in a complicated structure that cannot easily e analyzed for passivity using the tools listed in Section. 5.3 Approach 3: MultiLoop Feedack (MLF) structure MLF structure, which is the proposed approach of this paper will enale us to study the passivity of the 4channel teleoperation system easily compared to PN or SLF structures. The 4channel ilateral teleoperation system in Fig. 3 can e represented through ( an MLF structure as shown in Fig. 8, where e = V m, e = Fh I ), ( ( ( ) e 3 = V ), e 4 = V ), e 5 = V s, u =, u = F h, u 3 = ( ), ( ) ( ( ( ) u 4 = F e, u 5 =, y = V m Z h, y = Vm V ), y 3 = I V ), y 4 = I V s and y 5 = V s Z e. Theorem 5.3. The system in Fig. 8 is passive if Z e, SCU, timedelayed communication channel, MCU, and Z h locks are passive. Proof. Let V (x ),V (x ),V 3 (x 3 ),V 4 (x 4 ) and V 5 (x 5 ) e the storage functions of Z e, the SCU, the timedelayed communication channel, Figure 8: MLF structure; Equivalent multiloop feedack structure of the 4channel teleoperation system. the MCU and Z h. Assume the initial stored energy in each of these systems is zero. Based on Definition e T i y i V i (9) Fom the feedack loops in Fig. 8, it can e seen that Therefore, it is easy to show that e = u y (3) u e = y (3) u y 3 u3 y e 3 = (3) u 3 y 4 u4 y e 4 = 3 (33) u 4 y 5 e 5 = u 5 y 4 (34) e T y e T y e T 3 y 3 e T 4 y 4 e T 5 y 5 = u T y u T y u T 3 y 3 u T 4 y 4 u T 5 y 5 For the entire teleoperation system, let us define u = u u u 3 u 4 u 5 T, y = y y y 3 y 4 y 5 T Thus, u T y = u T y u T y u T 3 y 3 u T 4 y 4 u T 5 y 5 V V V 3 V 4 V 5 Taking V (x) =V (x )V (x )V 3 (x 3 )V 4 (x 4 )V 5 (x 5 ), we otain This concludes the proof. u T y V (35) 5.3. Passification of MCU and SCU With Z h and Z e assumed passive, ecause of Theorem 5.3, the teleoperation system stailization prolem will e reduced to passifying the communication channel, the MCU and the SCU. As explained in Section 4, the communication channel can e passified using either of the methods that were descried, i.e., the wave variales method in Fig. 4 or the transfer matrix ased method in Fig. 5. For ensuring passivity of the MCU and the SCU, controllers need to e designed such that they meet the definition of positive realness. 74
5 Relating inputs to corresponding outputs, the transfer matrices of the MCU and the SCU satisfy where Vm Fh = G V MCU I, I V = G V SCU s C 6 Z G MCU = m C m Z m C m C 3 C (C 6 ) C Z m C m Z m C m C4 G SCU = Z s C s C C 4(C 5 ) Z s C s C 5 Z s C s Z s C s (36) (37) (38) Assuming that C, C 3, C 5, C 6 (ut not C and C 4 ) are scalar gains and C to C 6 are to e designed, applying Theorem 5.3 leads to ( C 6 )K dm ω >, ( C 5 )K ds ω > (39) RC ( jω)((k pm M m ω ) j K dm ω) > (4) RC 4 ( jω)((k ps M s ω ) j K ds ω) > (4) (K pm M m ω ) C 3 K M ( C 6 )K dm (RC )(K pm M m ω ) (IC )K dm ω (K pm M m ω )C 3 K M ( C 6 ) ((RC ) (IC ) )K M Kdm ω ( C 6 )(IC )(K pm M m ω )K dm ω ( C 6 )(RC )Kdm ω (4) (K ps M s ω ) C K S ( C 5 )K ds (RC 4 )(K ps M s ω ) (IC 4 )K ds ω (K ps M s ω )C K S ( C 5 ) ((RC 4 ) (IC 4 ) )K S Kds ω ( C 5 )(IC 4 )(K ps M s ω )K ds ω ( C 5 )(RC 4 )Kds ω (43) (A) (B) M m.3 K pm K dm C C 6 C 4.3s /s M s.6 K ps K ds C 3 4 C 5 3 C.6s /s M m.3 K pm K dm C C 6 C 4.3s /s M s.7 K ps 3 K ds 3 C 3 C 5 3 C.35s 3 3/s Tale : The masses and controllers gains used in the simulation. (A) passive and (B) nonpassive. ation system: K mi,k pi,k di >, for i=,4 (45) C <, C 3 <, C 6 >, C 5 > (46) M = K p = C 3 M m K pm C 6 (47) M = K p = C M s K ps C 5 (48) K dm ( ( C 3 ) C 3 ) ( C 6 )K d (49) K ds ( ( C ) C ) ( C 5 )K d4 (5) In the simulation study that follows, the aove controllers for the 4channel teleoperation system are used. However, as mentioned efore, conditions (39)(43) are general and can e tested for any given set of controllers. Remark Fig. 9 depicts a loop feedack variant of the 4 channel teleoperation system. It can e shown that it is passive if the timedelayed communication channel and G MS are passive (note that since the operator and the environment are assumed passive, ased on Definition 3 the matrix will e passive as Zs Z e well). The comination of the MCU and SCU is given y C 6 Z tm Ztm C 5 G MS = Z ts Zts C 3 C (C 6 ) C Z tm Z tm (5) C C 4(C 5 ) Z ts C 4 Z ts where K M =(K pm M m ω ) Kdm ω and K S =(K ps M s ω ) Kds ω. Inequalities (39)(43) are general sufficient conditions for ensuring passivity of the delayed 4channel teleoperation system in the sense that any given set of controllers (C,,C 6 and proportionalderivative C m and C s ) for any given system (M m and M s ) can e checked for passivity, and are reported for the first time in this paper. The transparency conditions for the 4channel teleoperation system (without delay) shown in Fig. are Z h Z e G MS V m V s C = C s Z s,c 4 = (C m Z m ),C = C 6,C 3 = ( C 5 ) (44) I V Timedelayed communication channel V It is easy to see that (44) and (39)(43) are incompatile. This means that using the controllers (44), which guarantee transparency under zero delay, may not result in a stale teleoperation system when there exists time delay. Though the controllers (44) could not ensure passivity, let us assume that C and C 4 have similar structures to C s Z s and C m Z m, respectively. This results in PIDlike controllers C = K m s K p s K d and C 4 = K m4 s K p4 s K d4 with free parameters K mi,k pi,k di, i =,4. Applying Definition 3, we get the following conditions to ensure the staility of the 4channel delayed teleoper Figure 9: loop structure of 4channel teleoperation system Again, the communication channel can e passified using either of the methods in Section 4. It is easy to show that the passivity of G MS is equivalent to the passivity of MCU and SCU in the structure shown in Fig SIMULATION RESULTS In this section, the passivity conditions (45)(5) found in the previous section will e verified via simulations. For checking 75
6 the passivity of the 4channel teleoperation system, a passivity oserver has een incorporated into the simulation to calculate the dissipated energy in the teleoperator. This dissipated energy is given y t t E dissipated = (τ)v m (τ)dτ (τ)v s (τ)dτ (5) The teleoperator is passive if this integral is nonnegative at all times. The 4channel teleoperation system in Fig. 3 has een simulated in MATLAB/Simulink. The time delay in the communication channel is set to.5 sec. A pair of DOF master and slave roots modeled y point masses is considered. Both the master and the slave are connected to LTI terminations with transfer functions s. This termination is passive as for s = jω we have Re( s )= ω > when ω >. A sinewave (with unit amplitude and Hz frequency) Fh is used. E dissipated non passive passive. 5 5 Time(s) Figure : Passivity oserver E dissipated used for the teleoeprator s passivity analysis According to (45)(5), it is expected that the passivity of the 4 channel teleoperator should depend on the controller gains. Fig. shows that when the controllers gains are chosen to meet conditions (45)(5), e.g., as listed in Tale (A), the teleoperator is passive. However, for a set of masses and controllers gains that do not satisfy (45)(5), e.g., as listed in Tale (B), the teleoperator will ecome nonpassive. 7 CONCLUSION AND FUTURE WORK In this paper, the staility of the 4channel teleoperation system in the presence of time delay is studied. When delay exists in the communication channel, addressing the passivity (and staility) of the teleoperation system as a whole is too complicated and computationally intractale. Modifying the structure of the 4channel teleoperation system as shown in Fig. 8 (or Fig. 9) will enale us to study the requirements of passivity on a modular and tractale asis. On the other hand, it was discussed that passifying the communication channel alone will not guarantee the staility of the entire teleoperation system. Assuming that oth the operator and the environment are passive, controllers for the master and the slave still need to satisfy a set of requirements found in this paper. As for the future work, a similar passivity analysis will e applied to the 4channel multilateral teleoperation systems in the presence of time delay. ACKNOWLEDGEMENTS This research was supported y the Natural Sciences and Engineering Research Council (NSERC) and Quanser, Inc. A APPENDIX Raiseck s passivity criterion 5. The necessary and sufficient condition for passivity of a port network with immitance parameters p are. The pparameters have no RHP poles. Any poles of the pparameters on the imaginary axis are simple, and the residues of the pparameters at these poles satisfy the following conditions (k ij denotes the residue of p ij and kij is the complex conjugate of k ji ): k k k k k k, with k = k 3. The real and imaginary part of the p parameters satisfy Rp Rp 4Rp Rp (Rp Rp ) (Ip Ip ) Proof. p ( jω) p G( jω)= ( jω) p ( jω) p ( jω) is positive real if G( jω)g T ( jω)= p ( jω)p ( jω) p ( jω)p ( jω) = p ( jω)p ( jω) p ( jω)p ( jω) ((RP RP ) j(ip IP )) RP RP ((RP RP ) j(ip IP )) is positive semidefinite. Since positive semidefiniteness is equivalent to having nonnegative leading principle minors, the conditions in 3 will e met. Also the following matrix must e positive semidefinite hermitian. (k ij denotes the residue of p ij and k ij is the complex conjugate of k ij ): k k lim s jω (s jω)g(s)= k k Applying the positive semidefinite hermitian conditions will leave us with the same conditions in. This concludes the proof. REFERENCES R. Anderson and M. Spong. Bilateral control of teleoperators with time delay. Automatic Control, IEEE Transactions on, 34(5):494 5, May 989. P. Arcara and C. Melchiorri. Control schemes for teleoperation with time delay: A comparative study. Rootics and Autonomous Systems, (38):49 64,. 3 A. Aziminejad, M. Tavakoli, R. Patel, and M. Moallem. Transparent timedelayed ilateral teleoperation using wave variales. Control Systems Technology, IEEE Transactions on, 6(3): , 8. 4 B. Hannaford and J.H. Ryu. Timedomain passivity control of haptic interfaces. Rootics and Automation, IEEE Transactions on, 8():,. 5 S. Haykin. Active network theory. AddisonWesley, P. F. Hokayem and M. W. Spong. Bilateral teleoperation: An historical survey. Automatica, 4():35 57, 6. 7 H. Khalil. Nonlinear Systems. Prentice Hall,. 8 G. Niemeyer and J. J. E. Slotine. Telemanipulation with time delays. International Journal of Rootics Research, 3(9):873 89, 4. 9 G. Niemeyer, G. Verghese, and T. Sheridan. Design issues in port network models of ilateral remote manipulation. In Rootics and Automation, Proceedings., IEEE International Conference on, volume 3, pages 36 3, 989. J.H. Ryu, B. Hannaford, C. Preusche, and G. Hirzinger. Time domain passivity control with reference energy ehavior. volume 3, pages , 3. 76
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