Subspace-oriented Energy Distribution for the Time Domain Passivity Approach

Transcription

1 Subspace-oriented Energy Distribution for the Time Domain Passivity Approach Christian Ott, Jordi Artigas, Carsten Preusche Abstract The Time Domain Passivity Control Approach (TDPA) is a powerful tool to guarantee passive interaction between a robot and its environment. Rather than establishing fied control parameters to keep the system stable in any possible environment, the TDPA observes the energy flow due to the interaction and applies a dissipative term in the case the interaction becomes active. In a robot manipulator the rationale behind the Passivity Controller requires a criterion on how to distribute the energy to be dissipated among the multiple joints and must be adjusted according to the general control goal of the application. This paper presents a method for distributing the dissipation between decoupled subspaces of a redundant manipulator, prioritizing dissipation in the nullspace. The method allows to preserve passivity while avoiding disturbance of the general control goal defined in the taskspace. Thus the general control goal can, to some etent, be as well decoupled from passivity considerations and thus a less conservative controller can be achieved. The approach is sustained with a numerical simulation. I. INTRODUCTION Passivity is a common concept to guarantee stability in systems in which part of the plant is unknown or hard to model. This is especially the case in telerobotic scenarios dealing with time delay or in human-robot interaction, dealing with unknown or simplified models of the human. The TDPA represents a method to obtain passivity of a system without knowing the system dynamics and being less conservative compared to a passivity criterion applied in the Laplace domain. So, one of the advantages of the TDPA is that passivity is handled as an adaptable property rather than established in a design stage. It is adaptable in the sense that passivity is kept in the time domain, that is, dissipation only occurs whensoever it is needed. In turn, this often results in a less conservative design and higher system performance. The TDPA has been successfully applied in the fields of haptics [4], control of fleible manipulators, and bilateral control for teleoperation systems [1], [11], [12]. In all these concepts the step from the one degree of freedom (dof) case to a multi-dof general case inherits etra consideration on how to treat singularities or coupling between the degrees of freedom. Some approaches deal with the consistent representation of the 6-dof case in applications like haptics and bilateral control [6], [14]. Dealing with redundant serial kinematics, like the DLR light-weight robot, adds compleity due to the fact that the task space does not determine all dof in the joint space. This means that All authors are with the Institute of Robotics and Mechatronics at the German Aerospace Center, Wessling, Germany. the designer of the control architecture has to solve the problem that the dimension of the control space eceeds the dimension of the stability or passivity constraint(s). The operational space approach gives the opportunity to add additional constraints to solve the ambiguity of the mapping. Salaun et al. are using learned models to hierarchically solve parallel (sub-) tasks [13]. Using the passivity criterion to guarantee stability of the system leaves the designer with just a scalar value for the necessary energy dissipation, i.e. damping, and the problem of how to distribute the dissipative element among the energy ports of a system if there is more than one. A geometric solution for the 6 dof case has been presented in [1]. A constraint based on the geometry of the task solves the ambiguity in terms of distribution of the damping elements for the Cartesian directions. However this approach does not deal with the limitation of a maimal damping that can be added to a specific joint and the possibility to use the nullspace for energy dissipation. The paper is organized as follows. In Section II a brief review of the TDPC approach is given. Section III defines the dynamical model of a redundant robot manipulator as considered in the paper. The application of the TDPC approach to this model is outlined in Section IV. In Section V the method of how to distribute the TDPC damping between the task space and nullspace is developed, which presents the main contribution of this paper. The approach is verified in a simulation study in Section VI. Finally, Section VII concludes the paper. II. TIME DOMAIN PASSIVITY CONTROL FOR 1-DOF SYSTEMS The Time Domain Passivity approach (TPDA) [4] is based on an online correction of the passivity of a power network. The controlled variable is energy and the rule is that the system must remain passive. The passivity condition for a 1-port network is typically written as: t f(τ)v(τ)dτ Ψ(), (1) where f and v are power-conjugate variables such as force and velocity and Ψ() represents the initial stored energy. The (time domain) Passivity Controller (PC) is in charge of dissipating the amount of energy that overcomes the network passivity balance. The amount of virtual energy is monitored by a Passivity Observer (PO) over time, which estimates energy based on the signals that are available in the network port, typically the conjugated pair of force

2 and velocity whose product is power. The PC is designed as an impedance or admittance virtual damping element, whose magnitude is modulated according to the amount of observed active energy. The causality of the PC is chosen according to the causality of the system where it is connected to. In the impedance configuration, force is modified to dissipate energy and velocity is conserved. In the admittance configuration, velocity is the modified command. R v 2 f 2 _ v λ f 1 _ f 2 f 1 PC v 1 PO Fig. 1: Cascade network connection of an environment, G, a PC in impedance and a 1-dof robot, R. The figure combines network and diagram flow (inside the PC) representations. The analysis is here restricted to systems that have a sampling rate substantially faster than the system dynamics. With this assumption and using the port variables of network G as defined in Fig. 1, the PO for a one-port network can be written as: Ψ obs (k) = ΔT f 1 (j)v 1 (j). (2) k= where Ψ obs (k) is the observed energy and ΔT is the sample time of the system. The network is passive as long as Ψ obs (k), k. In the impedance configuration, the PC is defined as follows: f 2 = f 1 λv (3) v = v 1 = v 2 Ψ obs(k)ψ pc (k 1) λ = ΔTv 2 else G if X X : = Ψ obs (k)ψ pc (k 1) < & v >. Being λ the damping coefficient, Ψ pc the energy dissipated by the PC, computed as Ψ pc (k) = ΔT k λ(j)v2 (j), f 2 the force input to the robot and f 1 the output from the network G. III. PHYSICAL MODELING The dynamical model of a robot manipulator with n joints and q R n as the vector of generalized coordinates is given by M(q) q C(q, q) q g(q) = τ, (4) where M(q) R n n is the inertia matri, C(q, q) q R n is the vector of centrifugal and Coriolis forces, and g(q) R n is the gravitational torque vector. The vector of generalized joint torques τ = τ c τ e R n is given by the sum of the actuator torques τ c (control input) and the eternal torques τ e. Depending on the application, the eternal torque τ e can represent interaction with a real physical environment or the force from a virtual reality environment. In this paper, we consider kinematically redundant robots, in which the number of joint angles is larger than the number m < n of Cartesian task coordinates = (q) R m. Accordingly, the mapping from joint velocities q to Cartesian velocities is given by the Jacobian matri J(q) R m n = J(q) q, J(q) := (q) q. () Within this paper, we assume that the Jacobian matri J(q) has full row-rank, i.e. kinematically singular configurations are ecluded. In addition to the Cartesian coordinates, we are interested in a representation of the redundant nullspace motion. Therefore, we introduce additional r = n m nullspace velocity coordinates v n R r. As shown in [8], [9], this can be done based on a matrin(q) R r n, which has full row-rank r and is designed such that the etended Jacobian matri J(q) according to ( [ ] = vn) J(q) q J(q) = q (6) N(q) is non-singular. Due to the assumption that J(q) has fullrow rank, such a matri N(q) can be constructed in a straight-forward way. Therefore, we first compute a full row-rank nullspace base matri Z(q), fulfilling the property J(q)Z(q) =, with one of the algorithms described in [2], []. Then, N(q) can be chosen as N(q) = (Z(q)M(q)Z(q) T ) 1 Z(q)M(q), (7) where the inertia matri M(q) is used as a metric [9]. As discussed in [7], the nullspace velocities v n defined in this way are in general non-integrable. Therefore, while v n describe the nullspace motion at a velocity level, there eist no appropriate nullspace coordinates n(q) such that N(q) = n(q)/ q. As a consequence, we will use a system formulation based the joint angles q and the velocity coordinates and v n. The robot dynamics can then be written in these new coordinates as ( ( Λ(q) μ(q, q) = vn) vn) J(q) T (τ g(q)), (8) wherein Λ(q) = J(q) T M(q) J(q) 1 = [ ] Λ (q) Λ n (q) μ(q, q) = Λ(q)( J(q)M(q) 1 C(q, q) J(q) ) J(q) 1. Due to the particular choice in (7), the generalized inertia matri Λ(q) has a block-diagonal structure 1. Therefore, we can write the total kinetic energy of the robot as the sum of two terms, which can be related to the kinetic 1 The block-diagonal elements in Λ(q) are given by Λ (q) := (J(q)M(q) 1 J(q) T ) 1 and Λ n(q) := Z(q)M(q)Z(q) T.

3 energy due to the Cartesian motion Ψ (q, ) and due to the nullspace motion Ψ n (q, v n ), i.e. 1 2 qt M(q) q = 1 2 T Λ (q) Ψ (q, ) 1 2 vt n Λ n(q)v n. (9) This decoupling of the kinetic energy into the sum of two parts will be utilized later in the design of different damping elements. Notice however, while this decoupling works for the kinetic energy, the Cartesian dynamics and the nullspace dynamics are still coupled via the dependency of q and the nonlinear terms in (8) [7]. For further reference, we define the following energy based norms on the different velocity coordinates: q 2 M := q T M(q) q = 2, 2 := T Λ (q) = 2Ψ (q, ), v n 2 n := v T nλ n (q)v n = 2Ψ n (q,v n ). IV. TIME DOMAIN PASSIVITY FOR MECHANICAL MUTI-BODY SYSTEMS The general epression for a M-port PO can be written as: Ψ obs (k) = ΔT (f 1 (j)v 1 (j)...f M (j)v M (j)). (1) j= The energy flow due to an environment interacting with a robot can be regarded as a multi-port network, where the number of ports is in general the dimensions of the eternal torques τ et : Ψ obs (k) = ΔT q T (j)τ e (j), (11) j= The question arises on how to dissipate energy in case Ψ obs (k), among the multiple joints of a robot manipulator. One could think of handling passivity for each degree of freedom independently. This would however result in a conservative approach since, as can be deduced from (11), an active dimension in the environment does not necessary imply that the whole environment is active. The PC must be built according to a criteria on how to distribute that energy. Ψ F q 1 τ e,1 τ pc,1 q n τ e,n τ pc,n Ψ obs Ψ pc q S d v pc F pc P S T τ pc τ e q e,1 τ e,1 q e,n τ e,n Ψ obs Fig. 2: Cascade network connection of a multi-dimensional environment, G, a multi-port PC in impedance causality, P, and a multiple dof robot plus controller, F. Fig. 2 shows the network representation of the general structure of a PO/PC connected to a multi-body system. G The n port network F contains the robot and a generic controller whose specification depends on the application, e.g. a PD controller plus a trajectory generator or a Cartesian compliance. The network G is a generic environment which represents the source of energy to be dissipated, e.g. an environmental disturbance, a virtual reality (VR) or a teleoperator. The multi-port PC in impedance causality is represented by P and regulates the interaction between G and F. S is a transformation from the joint space to the space where the PC is implemented and, as will be seen, depends on the specific PC approach. This representation is general and allows multiple solutions, such as the approaches presented in this paper. The presentation in this paper is limited to an impedance configuration of the PC. In that case the following relations are established: q = q e, (velocity input) τ = τ c τ e τ pc, (torque output) (12) where τ pc is the dissipation torque vector whose epression will be specified in the following subsections and represents the correction upon τ e such that the interaction with G becomes passive. Thus the control input to the robot becomes τ c = τ c τ pc, where τ c is the contribution of the generic controller contained in F. A. Joint Level Approach The first solution is sought entirely in the joint space. An active environment in the joint space can be for instance a disturbance on some joint. Thus the PC is computed also in the joint space ands contains just an identity transformation. The dissipation can be distributed at the joint level by using the inertia matri of the manipulator. Thus and analogous to (3), the PC for a multi-dimensional environment can be written as: τ pc (k) = d q (k)m(q(k)) q(k) (13) 1 Ψ obs (k)ψ pc (k 1) d q (k) = ΔT q(k) 2 if X q M else X q : = Ψ obs (k)ψ pc (k 1) < & q(k) M >, where the Ψ pc (k1) is the update of the dissipated energy by the PC, i.e. Ψ pc (k) = ΔT d q (j) q(j) 2 M j= The additional torque τ pc from the PC is added to the control torque τ c of a nominal control action, i.e. τ c = τ cτ pc. The controller in (13) guarantees passivity for any active environment G. However, it does not enforce any rule on how active energy must be dissipated beyond projecting it on the joint space mass matri. Therefore, the distribution of the damping among the multiple degrees of freedom is handled implicitly, which can lead to undesired disturbances of a given control goal (e.g. a task space related control goal).

4 B. Cartesian Space Approach Since the control goal is often established in the Cartesian space, it is reasonable to specify the dissipation criteria also in the Cartesian space. In [1] the energy to be dissipated was projected on the force vector resulting from the interaction with a VR. While this method allows intuitive interaction with the VR, the dissipation capacity is limited by the norm of the velocity along the force vector. Indeed, the dissipation capacity of a manipulator in a sampling lag depends on the joint configuration, the mass distribution, the velocity and the maimum torques that the robot can apply. A reasonable way to distribute the energy along the joints is by using the mass distribution of the manipulator. Thus and analogous to (3), the PC for a multi-dimensional environment can be written as: F pc (k) = d (k)λ (q(k)) (k) (14) 1 Ψ obs (k)ψ pc (k 1) d (k) = ΔT (k) 2 if X else X : = Ψ obs (k)ψ pc (k 1) < & (k) >, where Ψ pc (k 1) := ΔT k j= d (j) (j) 2 is the update of the dissipated energy by the PC. The PC is computed in the Cartesian space and the transformation S from Fig. 2 is given by the Jacobian matri. Notice that for a nonredundant manipulator this solution is equivalent to the joint level case (apart from singular configurations). However, for a redundant robot the damping is only implemented in Cartesian space, while the dissipation capacity according to the nullspace motion is not eploited. This demerit will be removed in the detail in the following section. V. SUBSPACE ENERGY DISTRIBUTION As outlined in the previous section, the PO/PC concept aims at controlling the energy balance between a robot and its environment by adjusting additional damping elements. While the energy balance condition is a scalar criteria, we have additional freedom in the distribution of the damping among the individual power ports. In the following we will show how this freedom can be used in order to avoid loss of performance in selected motion subspaces as good as possible. The selection of relevant motion subspaces of interest clearly depends on the considered application. In here we will restrict the presentation to the significant case of a redundant robot manipulator for which we distinguish between damping of the Cartesian motion and damping of the nullspace motion. For the design of the damping elements, we will consider the following two criteria in the given order: 1) Primarily use dissipation in the null-space, as this is less disturbing in a task space related control goal. 2) Allow for dissipation in the Cartesian space if necessary. A. Combined taskspace and nullspace PC By the particular choice of the nullspace velocity v n via N(q) as given by (7), the kinetic energy is given by the sum of a Cartesian energy Ψ (q, ) and a nullspace energy Ψ n (q,v n ) (c.f. (9)). Related to the velocity representation by the Cartesian velocity and a nullspace velocity v n, we introduce a corresponding representation ofτ c by a Cartesian force F R m and a nullspace force F n R r. According to (6), these generalized forces are related to the actuator torques via τ c = J(q) T F N(q) T F n. As a consequence of this change of control inputs, we can write the overall energy balance as d dt = qt τ = q T τ c q T τ e = T F v T nf n q T τ e P c The passivity controller concepts aims at achieving a given target dissipation P d. The generalized forces are chosen as damping elements, in which the inertia distribution is used as a natural metric, i.e. by choosing F = d Λ (q) (1) F n = d n Λ n (q)v n, (16) with non negative damping factors d c and d n, we get the power dissipated by the PC as P c = d T Λ (q) d n v T nλ n (q)v n 2 v n 2 n In discrete time implementation, the energy dissipated by the PC until time step (k 1) can thus be computed by k1 ( Ψ pc (k 1) = ΔT d 2 d n v n 2 n) j= Net, the damping factors d (k) and d n (k) have to be chosen such that P c (k) = P d (k) with P d (k) := min((ψ obs (k)ψ pc (k 1))/ΔT,), (17) while considering prioritization of d n (k) over d (k). The available energies related to the Cartesian motion and the nullspace motion suggest upper bounds d,ma = 1/(2ΔT) and d n,ma = 1/(2ΔT) for the damping gains. A straightforward strategy for the damping factor design thus would be 2 ( ) Pd d n = min v n 2,d n,ma (18) n ( Pd d n v n 2 ) n d = min 2,d,ma, (19) in which the damping gain d n is used primarily for dissipation, and d is utilized only if d n is in saturation. 2 For better readability, the discrete time step argument (k) is dropped in the following equations.. (j)

5 B. Constraint handling The damping gain design in (18) and (19) allows to realize the desired dissipation as long as the kinetic energy in the robot motion is high enough (so that not both of the gains reach their maimum values). In practice, the maimum damping factors related to the available kinetic energy of the robot, however, are not the only limitation for the design of the damping elements. The design in (18) and (19) will result in large gains if the kinetic energy of the robot is small. Even if this implies small velocities, it is a priori not guaranteed that the actuator limitations on the maimum joint torques are satisfied. In order to integrate actuator limitations, we reformulate the damping design as a constrained optimization problem. Our main objective is the implementation of a given target dissipation P d by minimizing the error function e(d,d n ) := P c (d,d n ) P d. Additionally, we aim at keeping the Cartesian damping as small as possible (second objective) and minimizing the overall damping gains (third objective). To this end, we combine these three objectives into one single quadratic aggregate objective function J(d,d n ) = α 1 e(d,d n ) 2 α 2 d 2 α 3 d 2 n, (2) whereα 1, α 2, andα 3 are weights for the different objectives, which should fulfill α 1 > α 2 > α 3 >. By this choice of the weights, dissipation is generated mainly in the nullspace and secondarily in the Cartesian direction if needed. Actuator torque limits τ c > τ min and τ c < τ ma can be added by formulating the additional linear inequality constraints τ min [ ] ( ) J(q) T Λ (q) N(q) T d Λ n (q)v n d n τ ma In addition, it is useful to define constraints on the damping gains d d,ma and d n d n,ma. The minimization of the objective function J(d,d n ) under consideration of the previously described constraints defines a quadratic programming (QP) problem, which can efficiently be solved by state of the art solvers, like qpoases [3]. Notice that the unconstrained optimization would result in damping only within the nullspace, while the additional inequality constraints effectuate a distribution of the dissipation onto the different subspaces. C. Haptic interaction As shown in [1], in haptic interaction tasks it can be useful to perform the dissipation in one specific preferred Cartesian direction ξ. The above optimization problem can be easily etended to this case. Instead of (1), we utilize F = Λ (q)(d ξ P d (I P)) = d ξ Λ (q)p d Λ (q)(i P), d ξ (q, ) d (q, ) where P := ξξ T /(ξ T ξ) is a projection matri into the direction of ξ and d f and d are damping gains. The controller dissipation is then given by P c = d ξ T d ξ (q, )d T d (q, )d n v n 2 n. The objective function is formulated in the same way as before J(d,d ξ,d n ) = α 1 (P c (d,d ξ,d n )P d ) 2 α 2 d 2 α ξd 2 ξ α 3d 2 n, with α 1 > α 2 > α ξ > α 3 > such that dissipation is primarily generated in the nullspace, secondarily in the direction of ξ and thirdly in the remaining Cartesian directions. The actuator torque limitations and damping factor constraints can be formulated again as linear inequality constraints a m A d a M, with d ξ d n A = d (q, ) T J(q) 1 d ξ (q, ) T J(q) 1 v T nλ n (q)n(q) 1 a m = τ min, a M = VI. CASE STUDY τ ma d,ma d ξ,ma d n,ma T. For evaluation of the proposed algorithms, a simulation of a planar four degrees-of-freedom robot is presented. Fig. 3 shows a sketch of the model and the model parameters. As Cartesian coordinates, the position coordinates of the endeffector have been chosen. Therefore, one has m = 2, and consequently a degree of redundancy of r = nm = 2..m.kg.kg.kg y.kg Fig. 3: Simulation model for the case study. Two case studies are here presented to show the operation of the controller and the gain in performance. In the first case study a joint-level PD controller, with proportional gains K p = (2,,3,2)Nm/rad and derivative gains K d = 2.7 (K p./diag(m(q))), is used to trace a trajectory intentionally chosen such that it results in significant kinetic energy in both subspaces (task space and nullspace). Fig. shows the kinetic energies of each subspace, as defined,

6 in (9). In case the robot interacts with an active environment, the controller presented in the previous section will first try to dissipate the energy available in the null space and then in the Cartesian space. Eternal Energy (J) Time (s) Fig. 4: Active response due negative damping. Kin. Energy(J) Ψ Ψ n Ψ Time (s) Fig. : Kinetic energies of the first case study. An active virtual environment built in the form of an eternal negative damping τ e = J(q) T d env with a damping gain of d env = Ns/m is applied to the endeffector of the robot. Fig. 4 shows the active response of this virtual environment. The system becomes marginally stable with evident oscillatory behavior. The goal of the POPC is to render the overall system passive by choosing appropriate controller damping. The reference trajectory is shown in the fourth plot of Fig. 6. After applying the combined task-space and null-space PC the system becomes stable thanks to the action of the modulated dampers, d and d n, both shown in first and second plots of Fig. 6. The constrains are set such that d n and d are limited to 1/s and the maimum torques applied by the PC to 6.N m, also visible in Fig. 6. It can be seen how dissipation first takes place in the nullspace d n and after that in the Cartesian space, when the first reaches its dissipation capacity (see, e.g., Fig. 6, around t = s). The dissipation capacity of each subspace is limited naturally by the kinetic energy of that space and the maimum damping value. The shift from one subspace to the other may also happen when the maimal PC torques are reached. This feature can be as well observed in Fig. 6 (e.g. around t = 3s). Indeed, the freedom in distributing the dissipation between different subspaces allows imposing constraints without compromising (or at least compromising less) passivity. The net case study aims at showing performance of the system with a control goal specified in the Cartesian space. Performance is evaluated through the deviation of the position command in the task-space due to an active disturbance. Indeed, dissipating in the nullspace should be transparent to the general control goal. To this end, a controller was set to render a Cartesian compliance at the end-effector of the robot with stiffness and damping given by K p = (2,2)N/m and K d = (1,1)Ns/m. The command was a constant position. A disturbance in the form of a pulse of length T p = 3sec and amplitude A p = 2Nm in the first joint was then applied which caused a deviation of the end-effector position. Figure 7 shows the kinetic energies of each sub-space, the active behavior of the environment d(1/s) dn(1/s) τd(nm) q (rad) Kin. Energy (J) Time (s) cmd actual Ψ (q, ) Fig. 6: Case Study 1: Joint space trajectory with Cartesian negative damping. Controller on. due to the impulse, and the position error at the end effector Δ = des actual. kin. energy (J) energy (J) pos.error (m) Ψ (q, ) E env. Epc time (s) Fig. 7: Case Study 2: Cartesian compliance with disturbance at joint 1. Controller off. Fig. 8 shows the effect of the controller. The damping gain d n becomes saturated during the impulse, and, as can be seen, this dissipation is most of the time enough, to let the Cartesian space undamped. The benefit of the method is clearly seen in the position error, which becomes by the order of two magnitudes smaller. VII. CONCLUSION The proposed controller allows the amount of damping to be modulated according to an active source as proposed by the TDPC framework. A source of activity can be a virtual environment due to the discretization and a phase lag in the causality, a communication delay such as in the field of

7 kin. energy (J) d(nrad/s) dn(nrad/s) energy (J) pos.error (m) Ψ (q, ) E env. Epc time (s) Fig. 8: Case Study 2: Cartesian compliance with disturbance at joint 1. Controller on. teleoperation, or an eternal disturbance due to a collision. All those sources can be modeled as active networks which inject energy into the system through the interaction medium and can have an impact on the performance or even make the system unstable. Typically the Cartesian or joint level controller must be robust enough to cope with any disturbance. This often results in control designs characterized by high damping values and as a consequence mobility can be impaired. The scheme presented in this article starts from an ideal scenario, that is, with no disturbances and passive environments. Indeed, the general controller is assumed to be designed within this ideal scenario, omitting any passivity considerations related to the interaction with active environments. The dual-space PO/PC dissipates energy from the environment when the environment becomes active. The method eploits the redundancy of the manipulator by prioritizing damping in the null-space, with the benefit that the dissipation becomes transparent in the task-space. Moreover, the method presented herein allows constraints in maimum torques and damping values without compromising passivity. Our future work will aim on the implementation of the algorithm in a 6DOF haptic or tele-operation eperiment. REFERENCES [1] Jordi Artigas, Jee-Hwan Ryu, and Carsten Preusche. Time domain passivity control for position-position teleoperation architectures. Presence: Teleoperators and Virtual Environments, 19(): , 21. [2] Y-C. Chen and I. D. Walker. A consistent null-space based approach to inverse kinematics of redundant robots. In IEEE International Conference on Robotics and Automation, pages , [3] H. J. Ferreau, H. G. Bock, and M. Diehl. An online active set strategy to overcome the limitations of eplicit MPC. International Journal of Robust and Nonlinear Control, 18(8):816 83, 28. [4] B. Hannaford and J-H. Ryu. Time-domain passivity control of haptic interfaces. IEEE Transactions on Robotics and Automation, 18(1):1 1, 22. [] M.Z. Huang and H. Varma. Optimal rate allocation in kinematically redundant manipulators - the dual projection method. In IEEE International Conference on Robotics and Automation, pages 72 77, [6] O. Khatib. A unified approach for motion and force control of robot manipulators: The operational space formulation. IEEE Journal of Robotics and Automation, 3(1):43 3, [7] Ch. Ott, A. Kugi, and Y. Nakamura. Resolving the problem of non-integrability of nullspace velocities for compliance control of redundant manipulators by using semi-definite lyapunov functions. In IEEE International Conference on Robotics and Automation, pages , 28. [8] J. Park. Analysis and Control of Kinematically Redundant Manipulators: An Approach based on Kinematically Decoupled Joint Space Decomposition. PhD thesis, Pohang University of Science and Technology (POSTECH), [9] J. Park, W.K. Chung, and Y. Youm. On dynamical decoupling of kinematically redundant manipulators. In IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 149 1, [1] Carsten Preusche, Gerd Hirzinger, Jee H. Ryu, and Blake Hannaford. Time Domain Passivity Control for 6 Degrees of Freedom Haptic Displays. In IEEE/RSJ International Conference on Intelligent Robots and Systems, pages , October 23. [11] J-H. Ryu, D-S. Kwon, and B. Hannaford. Stable teleoperation with time-domain passivity control. IEEE Transactions on Robotics, 2(4): , 24. [12] Jee-Hwan Ryu, Jordi Artigas, and Carsten Preusche. A passive bilateral control scheme for a teleoperator with time-varying communication delay. Mechatronics, 2(7): , 21. Special Issue on Design and Control Methodologies in Telerobotics. [13] C. Salaun, V. Padois, and O. Sigaud. Control of redundant robots using learned models: An operational space control approach. In IEEE/RSJ International Conference on Intelligent Robots and Systems, pages , 29. [14] S. Stramigioli, A. van der Schaft, A. Maschke, and B. Melchiorri. Geometric scattering in robotic telemanipulation. IEEE Transactions on Robotics and Automation, 18(4):88 96, 22.