Synchronized Control for Teleoperation with Different Configurations and Communication Delay

Transcription

1 Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec , 27 Synchronized Control f Teleoperation with Different Configurations and Communication Delay Hisanosuke Kawada, Kouei Yoshida and Tu Namerikawa Abstract This paper deals with a control f teleoperation with different configurations and communication delays. We propose a synchronized control with individual gains and power scaling in the task space. In this method, the end-effect motion and fce relationship between the master and slave robots can be specified freely in the task space and the control gains can be independently selected appropriately f the master and slave robots. The delay-independent asymptotic stability of the igin of the position and velocity errs is proven by using passivity of the systems and Lyapunov stability methods. The proposed control law achieves synchronization with power scaling. Several experimental results show the effectiveness of our proposed method. I. INTRODUCTION Teleoperation is the extension of a person s sensing and manipulation capability to a remote location. A typical teleoperation system consists of a master robot and a slave robot and they are coupled via communication lines. The first goal is that the slave robot should track the positions of the master robot and the second goal is that the environmental fce acting on the slave is accurately transmitted to the human operat. The most critical issues in the teleoperation are the time delay in communication line and the different configurations of the robots. The communication delay may destabilize the whole system. Then it is necessary to guarantee the stability f any communication delay [1]. Furtherme, the master and slave robots with different configurations act with different scales in several tasks involving teleoperation such as telesurgery and teleoperation of huge robotics f extra-vehicular activity. Then, the teleoperation with different configurations should be controlled on the endeffect motions in the task space and it is also necessary that the infmation of the motion and/ the fce should be scaled between the master and slave robots. This scaling is called as the power scaling [8]. Stabilization f a teleoperation with the communication delay was achieved by the scattering transfmation based on the idea of passivity [2] This is equivalent to wave variable fmulation [3]). Then, the additional structure with position control was proposed to solve the position drift [4], []. In [4] and [], however, the stability of the system is dependent on the communication delay. In [6], it is shown that multi passive systems connected communication netwk with time delay can be synchronized by the coupling control law based H. Kawada, K. Yoshida and T. Namerikawa are with Division of Electrical Engineering and Computer Science, Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa JAPAN. hisa@scl.ec.t.kanazawa-u.ac.jp, yoshida@scl.ec.t.kanazawa-u.ac.jp and tu@t.kanazawa-u.ac.jp on graph they [7]. This approach is called as synchronized control. Meover, the practical applicability of the result is demonstrated in the problem of teleoperation with any communication delay. The delay-independent asymptotic stability of the igin of the position and the velocity errs is guaranteed without the scattering transfmation. In the above approaches [2], [4], [], [6], however, the power scaling was not considered. In [8], [9], the power scaling f teleoperation with communication delay was achieved based on the scattering transfmation approach. However, the position drift is caused by the traditional scattering approach without explicit position control. In [1], the teleoperation is decomposed in shape and locked systems and passive feedfward action is used to implement a scaled codination between the master and slave robots. In this method, the communication delay was not treated. Furtherme, teleoperation with different configurations including different scale) and communication delay have not been dealt with. In this paper, we propose the synchronized control of teleoperation with different configurations and communication delays based on [6]. The master and slave robots are modeled as the task-space dynamical systems. Using the task-space feedback passivation, the new outputs which contain endeffect position and velocity infmation are utilized to guarantee the passivity. Then the master and slave robots are coupled together by using the synchronized control with power scaling and non-negligible communication delays. In addition, the gains of the synchronized control can be independently selected at the appropriate value f the master and slave robots. Therefe, the end-effect motion and fce relationship between the master and slave robots can be specified freely in the task space. Using passivity of the systems and Lyapunov stability methods, the delay-independent asymptotic stability of the igin of the position and velocity errs is proven and the synchronization with power scaling is achieved. Finally several experimental results show the effectiveness of our proposed method. II. DYNAMICS OF TELEOPERATION Assuming absence of friction and other disturbances, the master and slave robot dynamics with n-degree-of-freedom in the joint space are described as M mq m) q m + C mq m, q m) q m + g mq m) = fi m + J T mf op M sq s) q s + C sq s, q s) q s + g sq s) = fi s J T s Fenv, 1) where the subscript m and s denote the master and slave indexes respectively, q m, q s R n 1 are the joint /7/$2. 27 IEEE. 246

2 46th IEEE CDC, New Orleans, USA, Dec , 27 angle vects, q m, q s, R n 1 are the joint velocity vects, q m, q s, R n 1 are the joint acceleration vects, τ m, τ s R n 1 are the input tque vects, F op R n 1 is the operational fce vects applied to the master robot by human operat, F env R n 1 is the environmental fce vects applied to the environment by the slave robot, M m, M s R n n are the symmetric and positive definite inertia matrices, C m q m, C s q s R n 1 are the centripetal and Ciolis tque vects and g m, g s R n 1 are the gravitational tque vects and J m, J s R n n are Jacobian matrices relating the end-effect velocity ẋ i R n 1 to the joint velocity q i as ẋ i t) = J i q i ) q i t), i = m,s. 2) F simplicity, we assume that Assumption 1: The J m q m ) and J s q s ) are nonsingular matrices at all times in operation. Above assumption is that we don t consider redundant robot and no kinematics singularities are encountered. Since the control at the task level, it is useful to rewrite the master and slave robot dynamics directly in the task space [12]. By further differentiation of 2) as ẍ i = J i q i) q i + J i q i) q i, i = m, s, 3) where ẍ m,ẍ s R n 1 are the end-effect acceleration vects and substitution 2) and 3) into 1), it is easy to see that the master and slave robots dynamics in the task space are described as fmmq m)ẍ m + C e mq m, q m)ẋ m + eg mq m) = Jm T fi m + F op fm sq s)ẍ s + C e sq s, q s)ẋ s + eg sq s) = Js T fi s F env, 4) where fm i = J T i M i J 1 i, Ci e = J T i C i M i J 1 i J i )J 1 i, eg i = J T i g i i = m, s. ) It is well known that the dynamics 4) have several fundamental properties as follows [12]. Property 1: The inertia matrices f M mq m), f Msq s) are symmetric and positive definite and there exist some positive constant m i1 and m i2 such that < m i1 I f M i q i ) m i2 I i = m, s. 6) Property 2: Under an appropriate definition of the matrices Cm q m, q m ) and Cs q s, q s ), the matrices N mq m, q m) = ḟm mq m) 2 e C mq m, q m) and N sq s, q s) = ḟm sq m) 2 e C sq m, q m) are skew symmetric such that z T N i q i, q i )z =, i = m, s, 7) where z R n 1 is any vect. Furtherme we assume f the stability analysis in later section as follows Assumption 2: The communication delays between the master and slave robots are any asymmetric constant value as T m and T s. Assumption 3: The human operat and the environment can be modeled as passive systems, respectively. Assumption 4: The operational and the environmental fce F op and F env are bounded by functions of the master and slave robots signals respectively. Assumption : All signals belong to L 2e, the extended L 2 space. III. CONTROL OBJECTS We would like to design τ m and τ s in 4) to achieve a task-space synchronization and static fce reflection f the teleoperation with different configurations and communication delay. Let us define the position tracking errs of the end-effect with power scaling as e mt) = α 1 x st T s) x mt) e st) = αx mt T m) x st) where α R is a positive scalar and it expresses a motion scaling effect. x m,x s R n 1 are the end-effect position vects. Then the control objects in this paper are: 1) the synchronization of teleoperation is achieved as e i t),ė i t) as t i = m,s 9) and 2) static fce reflection is achieved with ẍ i t) = ẋ i t) =, i = m,s as 8) βf op = F env 1) where β R is a positive scalar and it expresses a fce scaling effect. IV. CONTROL DESIGN To achieve the synchronized teleoperation system, we design the master and slave robot controllers. A. Feedback Passivation the master and slave robot inputs are given as, fi m =J T m { f M mq m)λẋ m e C mq m, q m)λx m+eg mq m)+f m} fi s =J T s { f M sq s)λẋ s e C sq s, q s)λx s+eg sq s)+f s} 11) where F m and F s are the additional inputs required f synchronized control in the next section and Λ = diagλ 1,,λ n ) R n n is a positive definite diagonal control gain matrix. Substituting 11) into 4), the master and slave robot dynamics are represented as fmmq m)ṙ m + e C mq m, q m)r m = F op + F m fm sq s)ṙ s + e C sq s, q s)r s = F env + F s 12) where the vect r m and r s are the new outputs of the master and slave robots and these are defined by linear combinations of the end-effect position and velocity vects as r mt) = ẋ mt) + Λx mt) 13) r st) = ẋ st) + Λx st). Fig. 1 shows a block diagram of the master and slave robots with feedback passivation. Then we have the following lemma. Lemma 1: Consider the systems described by 12) and define the inputs of the master and slave robot dynamics as F m = F m + F op and F s = F s F env and the outputs as r m and r s respectively. Then, under Assumption 1, the 247

3 46th IEEE CDC, New Orleans, USA, Dec , 27 1) Feedback passivation 11) Direct Kinematics FP FP Fig. 1. the master and slave dynamics with nonlinear compensation Fig. 2. The synchronized control architecture systems with the above input and outputs are passive such that t r T i z) F i z)dz β, i = m,s. 14) Proof: This result follows easily by using following positive definite energy functions f the systems as V i r i t)) = 1 2 rt i t) M i q i )r i t). i = m,s. 1) Using feedback passivation as 11), the master and slave robot dynamics are passive with respect to the new outputs 13) which contain both the end-effect position and velocity infmation. Thus the teleoperation can be controlled in the passivity framewk f the end-effect position and velocity signals by the new outputs. B. Synchronized Control Law with Power Scaling We propose the following synchronized control law considering different configurations and power scaling, { F m t) = K m α 1 r s t T s ) r m t)) 16) F s t) = K s αr m t T m ) r s t)) where K m and K s are defined as follows { K m = K K s = k s K 17) and K R n n is a positive definite diagonal control gain matrix and R and k s R are positive control gains f the master and slave robots respectively and α R is a positive motion scaling fact. Compared with the conventional method in [6], where the gains of the master and slave side should be equal as K m = K s = K due to the stability analysis, the gains of proposed control law 16) can be independently selected at the appropriate value f the master and slave robots. In addition, the delayed feedback and feedfward signals are multiplied by a motion scaling fact. Fig. 2 shows a block diagram of the proposed teleoperation system. The Master+FP and Slave+FP show the master s and the slave s reduced dynamics in 12). V. STABILITY ANALYSIS In this section we analyze the proposed synchronized control law with a power scaling and individual control gains previously and show that the control objectives are successfully fulfilled. Theem 1: Consider the system described by 12), 16) and 17). Then, under Assumption 1-, the igin of the position tracking errs given by 8) e m,e s and its derivatives ė m, ė s are delay-independent asymptotically stable. Therefe the teleoperation is synchronized and power scaled. Proof: Define a function f the system with respect to state vect xt) = [ ] rm,r T s T,e T m,e T T s as V msx) =αkm 1 rt m t) M f mq m)r mt) + α 1 ks 1 rt s t) M f sq s)r st) + αe T mt)λke mt) + α 1 e T s t)λke st) + 2αkm 1 { Fop T ζ)rmζ)}dζ + 2α 1 ks 1 {Fenv T ζ)rsζ)}dζ + αrmζ)kr T mζ)dζ t T m + α 1 rs T ζ)kr sζ)dζ 18) t T s First we prove that the function V ms is positive definite. In 18), Mm and M s are positive definite by Property 1), α is positive, K and Λ are positive definite and the operat and the environment are passive by Assumption 3) as 2α 1 k 1 s 2αk 1 m n o Fenvζ)r T sζ) dζ, 19) n F T opζ)r mζ) o dζ. 2) Thus the function V ms is positive definite. The derivative of this function along trajecties of the system with the Property 2 are given by V ms = 2αkm 1 rmf T m + 2ks 1 α 1 rs T F s + 2αe T mλkė m + 2α 1 e T s ΛKė s + αr T mkr m + α 1 r T s Kr s αr T mt T m )Kr m t T m ) α 1 r T s t T s )Kr s t T s ) 21) Substituting 16) into 21), we can get V ms =2αe T m t)λkėmt) + 2α 1 e T s t)λkėst) α{r mt) α 1 r st T s)} T K{r mt) α 1 r st T s)} α 1 {r st) αr mt T m)} T K{r st) αr mt T m)}. 22) Substituting 13) and using 8), it can be rewritten as V ms = αė T mt)kė m t) αe T mt)λkλe m t) α 1 ė T s t)kė s t) α 1 e T s t)λkλe s t) 23) Thus the derivative of the Lyapunov function V ms is negative semi-definite. To show the unifmly continuity of V ms, we 248

4 46th IEEE CDC, New Orleans, USA, Dec , 27 consider the derivative of V ms as follows, V ms = 2αë T mkė m 2αe T mλkλė m 2α 1 ë T s Kė s 2α 1 ė T s ΛKΛe s 24) The V ms is unifmly continuous, if the ë m,ë s,ė m,ė s, e m and e s are bounded. Since V ms is lower-bounded by zero and V ms is negative semi-definite, we can conclude that the signals r m,r s,e m and e s are bounded. Note that Laplace transfm of 13) yields strictly proper, exponentially stable, transfer functions between r m,r s and x m,x s are given as, 1 1 ) X i s) = diag,, R i s) i = m,s 2) s + λ 1 s + λ n where s is the Laplace variable, the R i s) and X i s) are the Laplace transfm of the r i and x i respectively. Since r m,r s L and 2), the outputs of the system 2) will have the property ẋ m,ẋ s,x m,x s L [12] and ė m,ė s L. From Assumption 4, the operational and the environmental fce are bounded by the function of the r m,r s as F op, F env L. From 16), It is easy to see that F m, F s L. Then we can get that τ m,τ s L. From 4), the master and slave robot accelerations are bounded as ẍ m,ẍ s L which gives us that the signal ë m and ë s are bounded. Therefe V ms is bounded and V ms is unifmly continuous. Applying Barbalat s Lemma [13] we can see that V ms x) as t. The igin of the position tracking errs given by 8) e m, e s and its derivatives ė m, ė s are the delay-independent asymptotically stable. Therefe the teleoperation is synchronized and power scaled. In the steady state, we can show that the contact fce is transmitted to the master side. Proposition 1: Consider the system described by 12), 16) and 17). Then, under the Assumption 1- and the steady state as follows ẍ i t) = ẋ i t) =,x i t) = constant, i = m,s, 26) we obtain that the scaled contact fce is accurately transmitted to the master robot side as follows βf op = k s KΛαx m x s ) = F env 27) where β= α ks ) R is a positive fce scaling fact. Proof: In the steady state 26), the master and slave dynamics 12) are reduced to F op = F m = KΛα 1 x s x m) 28) F env = F s = k skλαx m x s) The above equations give α k s F op = k s KΛαx m x s ) = F env. 29) Therefe the scaled contact fce is accurately transmitted to the master robot side. Remark 1: From Theem 1 and Proposition 1, we can conclude the following properties α > 1 : The motion of the slave is scaled up α < 1 : The motion of the slave is scaled down α = 1 : The slave is operated in a same scale motion β affects similarly the fce of the slave) The asymptotic stability is guaranteed when the scaling facts and controller gains are finite. Hence the proposed method can independently specify the scale of the motion and the fce relationship between the master and slave robots by using α and β. The main advantages of the proposed control strategy are to scale with different fact of the signals of the motion and the fce exchanged between the master and slave side and to guarantee the convergence of the position errs. Note that the controller gains and k s should be selected in the appropriate range. The smaller and k s may deteriate the tracking perfmance and the larger ones may exceeds the physical constraints of the control inputs. Therefe, the fce scaling fact should be selected considering β = α ks under the restriction on the magnitudes of and k s. VI. EVALUATION BY CONTROL EXPERIMENTS In this section, we verify the efficacy of the proposed teleoperation methodology. The experiments were carried out on a couple of different configuration robots as shown in Figs. 3, 4 and. The master robot is planar seriallink arm with 2DOF and the slave robot is planar parallellink arm with 2DOF. The inertia matrices, Ciolis matrices and Jacobian matrices of the master and slave robots are identified as follows Master:» θ M m = m1 θ m2 cosq m1 q m2), θ m2 cosq m1 q m2) θ m3» θ C m = m2 q m2 sinq m1 q m2), g θ m2 q m1 sinq m1 q m2) m =, lm1 sinq J m = m1) l m2 sinq m2) l m1 cosq m1) l m2 cosq m2) θ m1 =.149[kgm 2 ], θ m2 =.764[kgm 2 ], θ m3 =.686[kgm 2 ] Slave: θs1 + 2θ M s = s3 cosq s2) θ s2 + θ s3 cosq s2), θ s2 + θ s3 cosq s2) θ s2 θs3 sinq C s = s2) q s2 θ s3 sinq s2) q s1 + q s2), g θ s3 sinq s2) q s1 s = ls1 sinq J s = s1) l s2 sinq s1 + q s2) l s2 sinq s1 + q s2), l s1 cosq s1) + l s2 cosq s1 + q s2) l s2 cosq s1 + q s2) θ s1 =.366[kgm 2 ], θ s2 =.291[kgm 2 ], θ s3 =.227[kgm 2 ]. Here the slave robot is larger than the master robot. Then we should scale up the motion from the master robot to Slave arm Fig. 3. Remote environment Master arm Experiment setup 249

5 46th IEEE CDC, New Orleans, USA, Dec , 27 Fig. 4. Master arm Fig.. Slave arm the slave robot. We also measure the operational and the environmental fce i.e. F env,f op in 4)) using the fce senss. We use a real-time calculating machine dspace Inc.) and 1 [ms] sampling rate is obtained. The control programs are written in MATLAB and SIMULINK, and implemented using the Real-Time Wkshop and dspace Software which includes Control Desk, Real-Time Interface and so on. We use the environment of an aluminum wall covered by a rubber as shown in Fig. 3. The controller parameters K, Λ, and k s are selected as follows Λ =, K =, k m = 1, k s = 6. 3) The scaling fact is selected considering the ratio of the length of the link l s1 /l m1 = ) as α = 1.. Hence we expected that the motion of the slave robot is 1. times and the fce of the slave robot is also β = α ks = 9 times as much as those of the master robot. All experiments have been done with an artificial constant communication delays of T = T m = T s =. [s]. Two kinds of experimental conditions are given as follows. Case 1: The slave moves without any contact. Case 2: The slave moves in contact with the environment. In Case 1, we show the experimental comparison between the equivalent gains as = k s = 1 and the individual gain as 3). Figs. 6-9 show the results in Case 1. Fig. 6 shows time responses of the end-effect s X-positions in case of the equivalent gains as = k s = 1 and Fig. 8 shows time responses of the end-effect s X-positions in case of the individual gain as 3). From Figs. 6 a) and 8 a), the responses of the slave robot are almost α times as much as those of the master robot. Figs. 6 b) and 8 b) are modification of the above experimental data f comparison where the master robot responses are multiplied by α and shifted to.[s] to cancel the communication delay. In Fig. 6 b), there are the steady state errs due to physical coulomb friction and/ some disturbance of the robots. If gain K is larger, the errs are expected to become smaller. However, it is not possible to choose the larger gains, because the control gains are equivalent as = k s = 1 and the input tque command f the master robot are behind to its physical maximum tque the physical maximum tque of the master are about.3 [Nm]) as shown in Fig. 7. On the other hand, in Fig. 8 b), the positions of the slave robot accurately track those of the master robot and the synchronization with power scaling between the master and slave robots is achieved. Because the controller gains f the master and slave robots can be independently selected appropriately as 3). Figs show the results in Case 2. As shown in Figs. 1 and 11, when the slave robot is pushing the environment -3 [s]), the contact fce is faithfully reflected to the operat. The operat can perceive the environment through the fce reflection. The environmental fce responses are roughly β times as much as operational fce responses. Fig. 12 is modification of the above experimental data f comparison where the position of master robot responses are multiplied by α, the fce of master robot responses are multiplied by β and shifted to the right. From Fig. 12, the environmental fces on contact are accurately transmitted to the operat. VII. CONCLUSIONS In this paper, we proposed the task-space synchronized control f teleoperation with different configurations and communication delays. In this method, the motion and fce relationship between the master and slave robots could be specified freely in the task space. The delay-independent asymptotic stability of the igin of the position and velocity errs was proven by using passivity of the systems and Lyapunov stability methods. The synchronization with power scaling and non-negligible communication delays was achieved. Finally several experimental results showed the effectiveness of our proposed method. REFERENCES [1] P. F. Hokayem and M. W. Spong, Bilateral teleoperation: An histical survey Automatica, Vol. 42, No. 12, pp , 26. [2] R. J. Anderson and M. W. Spong, Bilateral Control of Teleoperats with Time Delay, IEEE Trans. on Automatic Control, Vol.34, No., pp , [3] G. Niemeyer and J. -J. E. Slotine, Telemanipulation with Time Delays, The Int. J. of Robotics Research, Vol. 23, No.9, pp , 24. [4] N. Chopra, M. W. Spong, R. Ortega and N. E. Barabanov, On Tracking Perfmance in Bilateral Teleoperation, IEEE Trans. on Robotics, Vol. 22, No. 4, pp , 26. [] T. Namerikawa and H. Kawada, Symmetric Impedance Matched Teleoperation with Position Tracking, In proc. of the 4th IEEE Conference on Decision and Control, pp , 26. [6] N. Chopra and M. W. Spong, On Synchronization of Netwked Passive Systems with Time Delays and Application to Bilateral Teleoperation, In proc. of the SICE Annual Conference 2, pp , 2. [7] R. Olfati-Saber and R. M. Murray, Consensus Problems in Netwks of Agents With Switching Topology and Time-Delays, IEEE Trans. on Automatic Control, Vol. 49, No. 9, pp , 24. [8] K. Kosuge, T. Itoh and T. Fukuda, Scaled telemanipulation with communication time delay, In proc of the IEEE International Conference on Robotics and Automation, pp , [9] C. Secchi, S. Straimgioli, C. Fantuzzi, Power scaling in pt- Hamiltonian based telemanipulation, In proc. of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp , 2. [1] D. Lee and P. Y. Li, Passive Bilateral Control and Tool Dynamics Rendering f Nonlinear Mechanical Teleoperats, IEEE Trans on Robotics, Vol. 21, No., pp.936-9, 2. [11] H. Kawada and T. Namerikawa, Passivity-based Synchronized Control of Teleoperation with Power Scaling, In proc. of the SICE-ICASE International Joint Conference 26, pp , 26. [12] C. C. de Wit, B. Siciliano and G. Bastin Eds), They of Robot Control, Springer, [13] H. K. Khalil, Nonlinear systems, Prentice-Hall,

6 46th IEEE CDC, New Orleans, USA, Dec , x mx a) Time response α x mx t - T ) b) Scaled and shifted data Y-Axis [mm] start end start Remote environment -18 Master Slave end X-Axis [mm] Fig. 6. Position data in Case 1 equivalent gain as = k s = 1 ) Fig. 1. Trajecties in Case 2 proposed method) Tque [Nm] a) Master robot Tque [Nm] b) Slave robot Fig. 7. Input tque command in Case 1equivalent gain as = k s = 1) Tque [Nm] x mx a) Time response α x mx t - T ) b) Scaled and shifted data Fig. 8. Position data in Case 1 individual gain as 3) ) a) Master robot Tque [Nm] a) Slave robot Fig. 9. Input tque command in Case 1 individual gain as 3) ) X-Position [mm] X-Fce [N] X-Position [mm] X-Fce [N] x mx F opx F envx Y-position [mm] Y-Fce [N] x my x sy Fig. 11. Time responses in Case 2 α x mx t-t) β F opx t-t) F envx F opy F envy Y-position [mm] α x my t-t) x sy Y-Fce [N] β F opy t-t) F envy Fig. 12. Shifted and scaled response results of Fig