Passive Control o Bilateral Teleoperated Manipulators Perry Y. Li Department o Mechanical Engineering University o Minnesota 111 Church St. SE Minneapolis MN 55455 pli@me.umn.edu Abstract The control o a bilateral teleoperated manipulator system is considered. The goal o the control are to i) coordinate the motions o the two manipulators according to a predened kinematic scaling, ii) render the dynamics o a locked system, and its response to orces rom the human operator and environment to approximate that o predened natural dynamics, iii) to provide or possible scaling o power. In addition, or saety reasons, the closed loop system need to remain passive. For linear dynamically similar systems, dynamics o the system can be block diagonalized into two decoupled mechanical systems the shape system that deals with the coordination error, and the locked system that describe the average motion o the two manipulators. The passive velocity eld control methodology is then applied to the shape system to regulate the coordination error at and at the same time preserves the passivity o the overall system. 1 Introduction Teleoperated manipulators are useul in applications, such as robotic surgeries and toxic waste cleanup, in which the work environment is either inaccessible or hostile to a human operator. They are also useul in extending the human operators' dexterity and their abilities to manipulate heavy and delicate objects. These are accomplished by providing suitable kinematic scalings between the motions o the operator and the manipulated tool, and by ampliying or attenuating the operator's orce and power. A typical setup or a teleoperated manipulator system consists o a work manipulator situated at the location where manipulation needs to take place and a master manipulator at a possibly remote site under the direct control o a human operator. These typically communicate electronically. A bilateral teleoperated manipulator system interacts with both the work environments and the human operator. To be successul, the motions o the work and master manipulators must mimic each other in a manner determined by the kinematic scaling. Thus, their motions must be coordinated. The interactions between the work robot and its environment, and between the master robot and the human operator should also be kinesthetically coupled so that human operator can eel and aect the work environment, and vice versa, in a natural manner. Thereore, the coordination control system or a bilateral teleoperator must be designed not to inhibit the eect o the work environment and operator's orces on the movement o the overall system. Since a teleoperated manipulator is a two-port system which simultaneously interacts with two environments (the human operator, and the work environment), an a-priori requirement must be that a broad class o environments with which the system interacts do not destabilize the system. It is particularly important when the teleoperator is required to provide orce or power amplication / attenuation. In applications where the teleoperated manipulator must deal with very delicate environment (e.g. in robotic surgery), saety isa rst priority. I the system is rendered passive with respect to a supply unction related to the mechanical power input, the interaction stability with any strictly passive work environment and human operator can be guaranteed using the amiliar passivity theorem [7], i.e. the interconnection between a passive and a strictly passive system is necessarily stable. In this paper, we propose a control methodology or linear, dynamically similar bilateral teleoperated manipulator systems that have the ollowing eatures i) the motions o the two robots are coordinated according to the prescribed linear kine-
matic scaling, ii) it can provide or bilateral power amplication / attenuation; iii) it ensures the passivity o the closed loop system with respect to a supply unction related to the mechanical power input rom the human operator and the environment; iv) the response o the closed loop system to operator and environment orces mimics that o an appropriately prescribed target mechanical system or which the work and the master manipulators are locked in place according to the prescribed kinematic scaling. In contrast to other teleoperator control strategies in the literature which rely heavily on the use o orce eedback, in the current paper, only kinematic eedback is used to guarantee stability and passivity o the system. In a uture paper, the incorporation o orce eedback to urther improve perormance will be discussed. The paper is organized as ollows. In section 2, the control problem is ormulated. In section 3, we show that or linear dynamically similar teleoperated manipulators, the system dynamics can be decomposed into two decoupled mechanical systems a shape system and a locked system which determines the coordination error and the average motion o the system respectively, so that passive controllers can be designed independently or them. The control o the locked system is discussed in section 4 and the control o the shape system is discussed in section 5. For the shape system, using the Passive Velocity Field Control (PVFC) theory developed in [3, 4, 5, 6] is utilized. The main result is given in (6) and section 7 contains concluding remarks. 2 Problem Formulation Consider a linear teleoperated manipulator system consisting o two n,degree o reedom manipulators with dynamics given by M 1 q 1 = T 1 + F 1 M 2 q 2 = T 2 + F 2 (1) where T 1 ; T 2 2< n are the control orces, F 1 ; F 2 2 < n are the environment orces that the master and slave system encounter, and M 1 and M 2 2< nn are the inertia matrices or the manipulators. Let 2< nn be the desired bijective linear kinematic scaling so that wewould ideally like q 1 (t) =q 2 (t). We assume that the work and master manipulators are dynamically similar in the sense that there exists a scalar >, M 1 = T M 2 I the two manipulators are perectly coordinated, we would like the resulting locked system (expressed in the units and dimension o robot 2) to behave according to M L q L + C L (q L ; _q L ) _q L =,T F 1 + F 2 (2) where > is the desired power scaling, M L =,T M 1,1 ( + ) + M 2 = M 2 is the apparent inertia that appears to the environment o robot 2. The n n skew symmetric matrix C L (q L ;_q L )=,C L (q L ;_q L ) T =C L (q L ; _q L ); species the unorced natural dynamics o the system by ormally dening a connection [1] o the target system. For example, it can prescribe the preerred direction o travel. Notice that in (2), F 2 has the same eect as the orce,t F 1. Consequently, power amplication will be achieved when _q 2 = _q 1, since F T 2 _q 2 is equivalent tof T 1_q 1. The target system (2) can also be \pulled back" to be expressed in the units and dimensions o robot 1. In this case, M L q L + C L(q L ; _q L) _q L = F 1 + 1 T F 2 (3) where q L =,1 q L, M L = M 1 + 1 T M 2 and C L (q L ; q L)= 1 T C L (,1 q L ;,1 _q L) To ensure stable interaction with the human and the work environment, we require that the system must interact with them passively. For this purpose, dene the supply rate s( _q 1 ; _q 2 ; F 1 ; F 2 )=F T 1 _q 1+F T 2 _q 2 which is the sum o the power exerted by the orce F 2 and times o the power exerted by the orce F 1. We require that passivity o the interaction with respect to this supply rate be satised, i.e. s( _q 1 (); _q 2 (); F 1 (); F 2 ())d,c 2 (4) or some c 2<which would depend on the initial condition at t =. Notice that since >, i one o the interaction ports is open (i.e. either F 1 = o F 2 = ), the interaction o the system with the other port would be passive in the usual sense, i.e. with the mechanical power being the supply rate. Dene the scaled kinetic energy o the teleoperated system to be
( _q 1 ; _q 2 )= 2 _qt 1M 1 _q 1 + 1 2 _qt 2M 2 _q 2 (5) Notice that C L (q L ; _q L ) _q L in (2) is restricted to be skew-symmetric, thereore, the target system is passive with respect to the supply rate,t T F 1 + F 2 _q L. 3 Decomposition into shape and locked systems Let E = [q + 1, q 2 ]bethecoordination error and consider the constant coordinate transormation o the velocities VL = I _q1 ; (6) + I,I _q 2 {z } S and o the orces, TL = T S,T T1 ; E T 2 FL F E = S,T F1 F 2 Under these transormations, the dynamics o teleoperator block diagonalizes into M L _V L = T L + F L (7) M E E = T E + F E (8) with M L = + ( + ) M 2 M E = M 2 2 In addition, direct computation shows that the kinetic energy (5) can be written as ( _q 1 ; _q 2 )= 1 2 VT LM L V L + 1 2 _ E T M E (9) Let us consider (8) as a mechanical system with conguration coordinates E, inertia M E, control input T E and environment orce F E ; and (7) as another mechanical system with inertia M L, control T L and environment orce F L. We shall reer to (8) as the shape system since it determines the relative conguration o the two manipulators. Similarly, (7) will be reerred to as the locked system since it determines V L which is the average velocity o the two manipulators. Notice that V L = _q 1 = _q 2 when the two manipulators are coordinated. The signicance o (9) is that the energy o the teleoperator system in (5) is exactly the sum o the energies o the shape and o the locked system. Proposition 3.1 I the locked system in (7) and the shape system in (8) are individually controlled using T L and T E respectively, such that each system is individually passive i.e. 9 c L ;c s s.t. 8 F L ; F E, and 8t, F T LV L d,c 2 L ; F T E _ Ed,c 2 s ; (1) then, the teleoperator system (1) is passive and satises (4). Thus, the control o the 2n,DOF system (1) reduces to the independent control o the locked and shape systems, (7)-(8), while maintaining their individual passivity properties. The objective or the shape system is to regulate (E; _ E) at (; ) while the objective or the lock system is to mimic the target system dynamics in (2). 4 Lock system control Notice that F L =,T F 1 + F 2 which is exactly the second term on the RHS o (2), the target system dynamics. Moreover, when (E; E) _ = (; ), V L = _q 1 = _q 2, Thereore, i we dene the locked conguration o the two manipulators (in the robot 2 units and dimensions) to be q L = + q 1 + q 2 then a possible control or the locked system is T L =,C L (q L ; V L )V L (11) which duplicates the desired target dynamics (2). In addition, i E(t)!, then the dynamics o robot 2 also converges (2), and the dynamics o robot 1 converges to (3). Since C L (q 2 ; _q 2 ) is skew-symmetric, by considering 1 2 VT L (t)m LV L (t), the locked system (7) under the control (11) can easily be shown to be passive in the sense o (1). 5 Shape system control PVFC Having dened T L in (11), the remaining task is to dene the shape system control T E in (8) so that E(t)! and the passivity requirement (1) is satised. To this end, we apply the passivevelocity eld control (PVFC) methodology developed in [3, 4, 5, 6]. Whereas the common approaches to regulation and tracking o mechanical system oten involves introducing damping terms, an interesting aspect o the PVFC approach is that a storage unction directly
related to the physical energy o the system which is conserved in the absence o environment input orces, and the rate o change o which is exactly the environment energy input. Using this storage unction, the passivity property o the closed loop system can be easily obtained. Roughly speaking, the PVFC approach to the regulation o (E; _ E) at the origin consists in 1. Dening a velocity eld V d (E) so that i = V d (E), then E(t)!. 2. Dening an augmented mechanical system which consists o the shape system, and an additional energy storage element, such as a ywheel. The augmented system will now have n+ 1 coordinates. 3. Extending the velocity eld V d (E) 2 < n to a velocity eld on the the conguration space o the augmented system < n+1, V da (E) 2< n+1. 4. Dening the inputs to the augmented system so that the energy in the shape system is exchanged with the energy in the ywheel, and vice versa; and at the same time causes (t)! (t)v d (E(t)), where (t) is associated with the amount o total energy stored in the augmented system. Once, (t)! (t)v d (E(t)) is accomplished, it can be shown that E(t)!. Step 1 Dene a desired velocity eld on < n. Following [4, 6, 2] we dene the desired velocity eld to be a bounded negative gradient vector eld o a suitable navigation unction, W < n! < +. For our purpose, W (E) can be dened to be g(r) kek. Figure 1 W (E) = r Z kek g(r)dr; where g < +! < + is a once dierentiable unction shown in Figure 1. In this example, W (E) behaves quadratically or small kek and linearly or large A desired velocity eld on < n, V d (E) 2< n is now dened to be the gradient vector eld o W (E) or some metric V d (E) =,K @W T @E where K 2< nn is positive denite and symmetric. Step 2 Dene an augmented shape system ME E_ TE = + M T E {z } {z } M a2< nn T E;a2< n+1 FE (12) Here, 2<is interpreted as the velocity o a ywheel which has an inertia M >, T E;a 2< n+1 is the coupling orce between the shape system and the ywheel which is to be determined. The kinetic energy o the augmented shape system can be dened to be a ( ;)= 1 2 _ E T M E + 1 2 M 2 (13) Step 3 Dene an augmented velocity eld or the augmented system (12) V da (E) = Vd (E) V d (E) where V d (E) satises or some constant E >, and 8E 2< n, V d (E) T M E V d (E)+M V d (E) 2 =2E (14) Thus, i ( (t);(t)) T is a scaled multiple o the V da, then, the energy o the augmented system remains constant. Energy is merely transerred between the shape system and the ywheel. Since V da () isrequired to be bounded, V d (E) can be ound by solving (14) once a suciently large E is chosen. Step 4 Dene the coupling orce T E;a In this step, we rst dene the inverse dynamic unction, w a (E; ;)=M a d dt V da(e(t)); and two (n+ 1)(n+ 1)skew-symmetric matrices G(E; ;)= 1 2E wa VdaM T a, M a V da w T a R(E; ;)= M a V da _q T a M T a, M a _q a V T dam a The coupling control T E;a is dened to be T E;a = G(E; E;) _ +R(E; E;) _ where is a positive gain constant. (15)
6 Closed loop dynamics Consider the closed loop dynamics o the augmented shape system in (12) under the coupling control (15) ME E_ =[G+R] + M FE (16) where we have omitted arguments in the matrices G(E; ;) and R(E; ;) to avoid clutter. To state the convergence properties o (16), let us dene the scaled velocity eld tracking error (t) e (t) (t) =,(t)v (t) da (E(t)) (17) where (t) = a( E(t); _ 1 (t)) 2 2E with a ( (t);(t)) being the energy o the augmented shape system as dened in (13), E is the energy associated the augmented desired velocity eld V da (E) dened in (14). Notice that (t)v da in (17) is the velocity i (;) T is aligned with V da (E(t)) and the energy is the current energy level a ( (t);(t)). Theorem 6.1 Consider the closed loop augmented shape system (16) to be an input-output system with input F e and output. Then, 1. The system is passive with respect to the supply rate F e (t) T (t), i.e. or any F e () and or all t, R t F e() T ()d,c 2. 2. The kinetic energy a ( (t);(t)) dened in (13) satises d dt a( (t);(t)) = F e (t) T (t). 3. I F e (t) =, or all t, then, except or a measure zero set o initial conditions (E(); E();()), _ i) e (t) (t)! exponentially, and ii) E(t)! exponentially. Proo 1) and 2) is a direct consequence that G and R in (15) are skew symmetric. The proo o (3) can be ound in [3, 6]. To prove the convergence o e (t) (t)! and E!, the Lyapunov unctions W = 1 2 et (t) Ma e (t) and W + W(E) are utilized where W (E) is the navigation unction, is a suitable positive scalar. The control law combining (11) and (15) is 1 1 T 1 _q @ T 2 A = S T CL 1 n(n+1) a M _ (n+1)n G + R S a @ _q 2 A (18) S where S a =, and S is the coordinate transorm in (6). Notice that the matrix in (18) is 1 skew symmetric so that the passivity o the control system is clearly evident. 7 Conclusions In this paper, we consider the control o linear dynamically similar bilateral teleoperator systems. The overall control system is passive with respect to a supply rate that is modied rom the mechanical power to take into account power amplication and attenuation. The control is achieved by use o a decomposition into a shape system and a lock system so that the overall passivity isachieved by guaranteeing the passivity o the decomposed components. The Passive Velocity Field Controller was then applied to stabilize the shape system. In the present paper, only kinematic eedback is used. In a uture paper, we will investigate the use o energy storage aorded by PVFC (i.e. the ywheel) to passively eliminate the eect o the environment orces on the coordination error. Reerences [1] S. Kobayashi and K. Nomizu. Foundations o dierential geometry, volume 1. Wiley, 1963. [2] Daniel E. Koditschek. Total energy or mechanical control systems. In Jerry Marsden, P. S. Krishnaprasad, and J. Simo, editors, Control Theory and Multibody Systems, volume 97, pages 131{ 157. American Mathematical Society, 1989. AMS series on Contemporary Mathematics. [3] Perry Y. Li and Roberto Horowitz. Passive velocity eld control o mechanical manipulators. In Proceedings o the 1995 IEEE International Con. on Robotics and Automation, volume 3, pages 2764{277, April 1995. Nagoya, Japan. [4] Perry Y. Li and Roberto Horowitz. Application o passive velocity eld control to contour ollowing problems. In Proceedings o the 1996 IEEE Conerence on Decision and Control, Dec 1996. Kobe, Japan. [5] Perry Y. Li and Roberto Horowitz. Passive velocity eld control, part 1 Geometry and robustness. In IEEE Transaction on Automatic Control, 1997. To appear. [6] Perry Y. Li and Roberto Horowitz. Passive velocity eld control, part 2 Application to contour ollowing problems. In IEEE Transaction on Automatic Control, 1997. To appear. [7] M. Vidyasagar. Analysis o nonlinear dynamic systems. Prentice Hall, second edition, 1993.