Base Force/Torque Sensing for Position based Cartesian Impedance Control

Transcription

1 Base Force/Torque Sensing for Position ased Cartesian Impedance Control Christian Ott and Yoshihiko Nakamura Astract In this paper, a position ased impedance controller i.e. admittance controller is designed y utilizing measurements of a force/torque sensor, which is mounted at the root s ase. In contrast to conventional force/torque sensing at the end-effector, placing the sensor at the ase allows to implement a compliant ehavior of the root not only with respect to forces acting on the end-effector ut also with respect to forces acting on the root s structure. The resulting control prolem is first analyzed in detail for the simplified one-degree-of-freedom case in terms of staility and passivity. Then, an extension to the Cartesian admittance control of a root manipulator is discussed. Furthermore, it is shown how the steady state properties of the underlying position controller can e taken into account in the design of the outer admittance controller. Finally, a simulation study of the Cartesian admittance controller applied to a three-degrees-offreedom manipulator is presented. I. INTRODUCTION Impedance control is a prominent example for a compliant motion control algorithm used for autonomous manipulation and physical human-root interaction [1], []. Different implementations of the general impedance control concept have een proposed using either impedance or admittance causality of the controller. A controller with impedance causality sometimes called force ased impedance control usually requires a precise torque interface and thus can enefit greatly of integrated torque sensing and torque control [3], [4]. In many commercial roots this is not feasile and only a conventional position or velocity interface is provided. In that case, a compliant ehavior can still e implemented y integrating a force/torque sensor FTS at the end-effector and designing an outer loop admittance controller sometimes called position ased impedance control which provides the desired set-point for an inner loop position or velocity controller [5]. In this paper, we focus on the implementation of an admittance controller, which can e implemented on a position controlled root. However, y using a FTS mounted at the tip of the root, the compliant ehavior can only e achieved with respect to forces acting on the end-effector, while the root will e insensitive to forces acting along the root s structure. In contrast to this, the use of roots in partly unknown human environments requires a compliant ehavior This research is partly supported y Special Coordination Funds for Promoting Science and Technology, IRT Foundation to Support Man and Aging Society. Ch. Ott and Y. Nakamura are with Department of Mechano- Informatics, Graduate School of Information Science and Technology, University of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan {ott,nakamura}@ynl.t.u-tokyo.ac.jp of the root also for unplanned contacts at different points. While the application of force sensitive skins [6] or the integration of torque sensing [7], [8] are possile approaches to handle such situations, we investigate on an alternative approach in this paper. Our approach aims at integrating a force/torque sensor at the ase of the root instead of mounting it at the end-effector. This enales to perceive forces all along the root s structure independently of joint friction. However, since the forces measured at the ase are related to the root s motion, the manipulator dynamics must e taken into account in the design of the admittance controller. Apart from applications to fixed mounted manipulators, we expect that the same issue will also e relevant for implementing whole ody impedance controllers of humanoid roots. Feedack of the feet contact forces is often used in walking and alancing controllers of iped roots in order to control the interaction forces of the root with the ground [9]. However, in that case the force feedack is often designed in a pragmatic way and without rigorous theoretical justification or staility analysis. In the design of whole ody impedance controllers including a compliant ehavior of the lower ody with respect to forces acting on the main ody, we have to take account of the following key issues. Firstly, for position controlled roots it is necessary to incorporate the contact force measurements at the feet into the whole ody control, since these sensors provide an indirect measurement of all forces acting on the root. Secondly, for keeping the zero-moment-point within the support polygon of the feet, it is necessary to limit the contact forces and moments. Thirdly, for handling larger contact forces, a comination with stepping and walking technologies will e required. Within this paper, we treat the first of these prolems. Compared to previous works on this prolem, we aim at giving an adequate theoretical justification of the ase sensor feedack y deriving all the relevant dynamic equations and y presenting a staility analysis of the one-dof case. The use of ase mounted FTSs for identification and joint torque estimation has een well studied in the works of Duowsky et al. [1], [11], [1]. In [11], a method for estimating the dynamical parameters of a serial manipulator arm was presented. Due to the measurement of the ase force, no joint torque information was required in the identification procedure. In [1], the ase FTS was used for estimating the root s joint torques ased on known dynamical parameters. The estimated torque signal was used for implementing an inner torque control loop, which was augmented y an outer

2 PD position controller. In [13], a ase FTS was used in comination with a FTS mounted at the wrist for collision detection and identification in human-root interaction tasks. Kosuge et al. [14] integrated a ody force sensor on a moile root for cooperatively handling large ojects y multiple roots. In contrast to [1], we aim at incorporating the ase force measurement directly into the design of an admittance controller instead of implementing an inner loop torque controller. The desired impedance represents a dynamic relation etween external forces and the motion of the root. This impedance will e transformed into a dynamic relation etween the contact force at the ase and the root s motion. We will highlight some restrictions on the achievale closed loop dynamics which are due to the dislocation of the force sensing. A first version of the controller from this paper was already presented in [15]. In the present paper, the controller from [15] is refined y compensating for the steady state error of the underlying position controller. This refinement if achieved y modifying the outer admittance control loop ased on design ideas from [16], [17]. II. ROBOT MODEL INCLUDING THE BASE FORCE In this section, the general model of a root with n joints is discussed, in which an expression of the contact force at the ase is included. In contrast to interaction forces measured at the end-effector, the forces etween the root and its ase are internal forces. Therefore, we start with an extended model with a free-floating ase Fig. 1. By adding constraints on the ase motion, we can derive an explicit expression of the force and torque measured at the ase. F O x, ẋ x = fq Fig. 1. Model of a root manipulator mounted on a ase FTS. The fixed ase manipulator model is augmented y a free-floating ase link, for which the motion will e constraint. In this way, we can represent the reaction force at the ase as the constraint force. In the following, the position and orientation of the ase link is specified via local coordinates x R 6. The joint angles of the manipulator are denoted y q R n. Then, the model of the root with an free-floating ase link can e written as Mx, qẍ q ẋ + Cx, ẋ, q, q + ḡx q, q = F τ + τ ext, 1 wherein Mx, q R 6+n 6+n denotes the complete inertia matrix including the ase link [18]. The centrifugal and Coriolis terms are given via the matrix Cx, ẋ, q, q R 6+n 6+n. The gravity term is written as ḡx, q R 6+n. The joint torques τ R n are considered as the control inputs. The generalized force measured y the ase FTS is denoted y F R 6. The generalized external forces except for the generalized forces F exerted at the ase at the location of the FTS acting on the root are summarized y the vector τ ext. In case that the external torques are due to a generalized force R 6 acting at the end-effector, they can e written as τ ext = τ ext, = τ ext,m [ J T ] x, q J T, q x, q }{{} J T x,q with Jx, q R 6 6+n as the Jacoian matrix for the serial kinematic chain from the fixed world frame O to the end-effector. In the following, the external torques are split up into the two components τ ext, R 6 and τ ext,m R n acting on the ase link and the joints, respectively. In 1, the joint coordinates q are augmented y local coordinates of the ase link motion x in order to incorporate the contact force F into the equations of motion. Since the ase is attached to the ground via a stiff force/torque sensor, we have to augment 1 y an additional constraint, which prevents any motion of the ase link: x t = x, dx t dt = [ I ] }{{} Φ ẋ =. 3 q From this, one can see that the generalized force at the ase F is represented in 1 y the Lagrangian multipliers related to the constraint matrix Φ R 6 6+n from 3. In the following, this constraint will e incorporated into 1. In this way an expression of the generalized ase force can e derived. Therefore, we drop the dependence on the constant position and orientation x = x of the ase link and write Mx, q, Cx,, q, q, and gx, q in the form [ ] Mx M q M c q, q = M T c q Mq, [ ] Cx,, q, q = C q, q C 1 q, q, C q, q Cq, q ḡx, q = g q gq where Mq R n n is the joint level inertia matrix and M c q R 6 n represents the inertia coupling matrix etween the manipulator and the ase link. Notice that the classical root dynamics can e otained y pre-multiplying 1 y a matrix spanning the left nullspace of Φ T : Mq q + Cq, q q + gq = τ + τ ext,m. 4 In order to get an expression for the ase force as a function of the root s motion, we instead pre-multiply 1,

3 y Φ and otain x F = τ ext, M c q q C 1 q, q q g q. 5 Moreover, sustituting q from 4 into 5, leads to F F M F = M c qm 1 q[τ + τ ext,m Cq, q q gq] + τ ext, C 1 q, q q g q. 6 The last three equations 4-6 asically represent three relations etween q, τ, and F, which will e relevant for the derivation and analysis of the admittance controller: A: 4 represents a relation q τ root dynamics. B: 5 represents a relation F q. C: 6 represents a relation F τ. From 5 and 6, it is ovious that the ase force, which is measured y the FTS, depends not only on the root s state q, q and the generalized external forces τ ext, ut also on the current joint torque τ, which is considered as the control input in our case. It should e mentioned that therefore the use of this force in the controller is from a theoretical point of view not unprolematic. This issue asically arises ecause we ignore the force sensor s elasticity in the model and treat it as an ideal force sensing element. However, for the controller design, one should avoid direct feedack from the force sensor measurement to the joint torque output of the controller as this feedack would not e well-defined. III. CONTROLLER DESIGN: THE ONE-DOF CASE In this paper, we focus on an admittance controller design. Therefore, we will use an underlying position controller for the root manipulator and design a compliant impedance ehavior in an outer loop ased on the measured forces at the ase. For this, in particular the relation etween the external forces and the measured contact force at the ase is of interest. The general relations for the n degrees-offreedom case are given in 5 and 6. Before discussing the design of a Cartesian admittance controller in section IV, we will analyze the simple one-degree-of-freedom case in this section in order to clarify the main design issues ased on a simple model. Consider the model shown in Fig., in which a single mass M is controlled via an actuator force F. Compared to the general model descried in Section II, we have the correspondence as shown in Ta. I. The actuator force F is determined y the output of an inner loop position controller for x R, which gets its set-point x d from an outer admittance controller. In the analysis of this section, we will assume that the underlying position controller has the form of a PD controller with velocity feed-forward term, i.e. F = Px x d Dẋ ẋ d, 7 with positive PD controller gains P > and D >. It can easily e verified that in this simple one-dof case the complete inertia matrix ecomes [ ] M + M M M =. M M M Fig.. Model of single mass, actuated y the force F and mounted on a ase force sensor. TABLE I CORRESPONDENCE BETWEEN THE GENERAL AND THE ONE-DEGREES-OF-FREEDOM CASE general case one-dof Coordinates q x Actuator force τ F External force τ ext, τ ext,m and thus the inertia coupling matrix M c q is given y M. By evaluating 4-6 for this one-dof case, we otain A Mẍ = F +, 8 B F = Mẍ, 9 C F = F. 1 Due to M c qˆ=m, the relation etween F and the actuator force F in 1 has a very simple form, i.e. the measured ase force is equal to the reaction force of the actuator. As a control goal, we assume a desired impedance relation in form of a second-order mass-spring-damper system M d ẍ + D d ẋ + K d x x =, 11 with M d >, D d >, and K d > as the desired inertia, damping, and stiffness, respectively. The point x R is the virtual equilirium position and is assumed constant. The desired ehavior 11 defines a dynamic relation etween ẋ and the external force. Since we want to realize this ehavior ased on the measurement of the contact force at the ase, we transform the desired impedance into a relation etween ẋ and the ase force F. This can e done y comining 11 and 9 to otain M d Mẍ + D d ẋ + K d x x = F. 1 From 1, one can see that the target inertia M d must always e larger than M, otherwise 1 would result in an unstale dynamics. Notice that 1 is independent of the underlying position controller 7 used for the implementation via admittance control. For an ideal position controller, the actual position x would ecome identical to its reference motion x d. One possile way for implementing 11 with a position controlled root is then to replace x in 1 y x d : M d Mẍ d + D d ẋ d + K d x d x = F. 13 This design strategy, shown in Fig. 3 so far did not involve the particular form of the underlying position controller.

4 M for which the time derivative along the solutions of is given y V x, x d, ẋ, ẋ d = D d ẋ d Dẋ ẋ d + ẋ. 13 F x d M Position Control Fig. 3. Admittance control of the one-dof model using a FTS at the ase. If the admittance controller from 13 is replaced y 18, one can take account of the static error resulting from the position controller. However, for analyzing the staility properties we need to consider a particular controller structure. In the following, we assume the PD controller 7. Then, the closed loop system can e otained from 8 and 13. By using 7 and 1, we can eliminate F and F from 8 and 13 to otain Mẍ + Dẋ ẋ d + Px x d =, 14 M d Mẍ d + D d ẋ d + K d x d x = F Px x d + Dẋ ẋ d. 15 Let us first analyze the equilirium points of the system. Therefore, we assume that a constant external force is acting on the system. Then the unique equilirium point ˆx, ˆx d can e otained from as ˆx d = x + 1 K d 16 ˆx = x + K d + P K d P. 17 Using the new coordinates x = x ˆx and x d = x d ˆx d, the staility of the equilirium point in the sense of Lyapunov can e shown ased on the Lyapunov function V x, x d, x, x d = 1 T [ ] x M x x d M d M T [ ] P P P P + K d x x d 1 x x d x d + which is positive definite for M d > M. The time derivative of this function along the solutions of is given y V x, x d, x, x d = D d x d D x x d, from which staility of the equilirium point follows. Moreover, y invoking La Salle s invariance principle [19], also asymptotical staility can e shown. In the staility analysis, a constant external force was assumed. Regarding interaction with dynamic environments, one can additionally show passivity of the closed loop system with the external force as input and the velocity ẋ as output. This can e verified y considering the storage function V x, x d, ẋ, ẋ d = 1 Mẋ + 1 M d Mẋ d + 1 Px x d + 1 K dx d x,, From this, the passivity of the system with respect to the input-output pair ẋ, follows immediately 1. Notice that in the controller design so far, the outer loop admittance controller 13 was designed independently of the inner position controller. For a non-ideal position controller, the achieved impedance as a relation etween x and will e slightly distorted according to the properties of the inner loop position controller. In case of the PD controller 7, one can see from the steady state equation 17 that the achieved steady state ehavior corresponds to a stiffness value of K d P/K d + P. This stiffness tends to the desired value K d for large position controller gains P >> K d. However, it is possile to exactly compensate for the steady state error of the position controller if the gain P of the position controller is known. Then the stiffness term of the outer admittance control loop can e modified y adopting the techniques used in [16], [17]. If the stiffness term in 13 is replaced y K dp P K d x d x, with K d < P, the desired stiffness K d is achieved exactly. However, in this case the position controller gain P poses an upper limit for the achievale stiffness K d < P. The modified admittance controller is then given y M d Mẍ d + D d ẋ d + K dp P K d x d x = F. 18 Notice that this modification would not e necessary if instead of the underlying PD controller a controller with integral action is used. IV. CARTESIAN ADMITTANCE CONTROL OF A MULTI-BODY ROBOT In the previous section, the admittance controller design and its staility analysis have een presented in detail for a simple model. In this section, the same line of argumentation will e followed for designing a Cartesian admittance controller of a multi-ody root manipulator. Let the desired impedance e defined in Cartesian coordinates x = fq R 6, ẋ = Jq q, where fq represents the forward kinematic mapping and Jq R 6 6 the analytic Jacoian Jq := fq/ q. In the following derivations, we will consider the non-redundant case and assume that the Jacoian is non-singular and thus invertile in the relevant workspace. Extensions to the redundant case would additionally require to consider the effect of the nullspace dynamics see, e.g., [1], [] on the measurement of the ase FTS. As a desired impedance, we assume a mass-springdamper-like system of the form Λ d ẍ + D d ẋ + K d x x =, 19 1 A sufficient condition for a system with input u and output y to e passive [] is given y the existence of a continuous storage function S which is ounded from elow and for which the derivative with respect to time along the solutions of the system satisfies the inequality Ṡ yt u.

5 with the symmetric and positive definite matrices Λ d R 6 6, D d R 6 6, and K d R 6 6 representing the desired inertia, damping, and stiffness, respectively. The virtual equilirium position is given y x R 6. The main advantage of using the ase sensor is that it allows to measure forces all along the root s structure, not only the the end-effector. Still, the aove desired impedance is defined with respect to Cartesian coordinates descriing the end-effector position and orientation. This means that for forces exerted in the vicinity of the end-effector, the perceived impedance will e close to 19. If the external forces are exerted far away from the end-effector, e.g. close to the ase link, then the perceived impedance ehavior will e different. However, under the assumption that a reliale contact point estimation is feasile, one could aim at adapting the compliance ehavior to the current point of contact. In order to compare the desired impedance with the equations of motion 4 and for comining it with 5-6, we rewrite the model 4 in Cartesian coordinates as Λqẍ + µq, qẋ + pq = J T qτ +, where Λq denotes the Cartesian inertia matrix Λq = JqM 1 qj T q 1 and the matrices µq, q and the Cartesian gravity term pq are given y µq, q = J T qcq, q MqJ 1 q JqJ 1 q and pq = J T qgq, respectively. The relation etween the ase force and the accelerations, i.e. 5, ecomes F = J T q Λ c qẍ µ 1 q, qẋ g q, 1 where the inertia coupling matrix Λ c q and µ 1 q, q are given y Λ c q = M c qj 1 q and µ 1 q, q = C 1 q, q M c qj 1 q JqJ 1 q, respectively. Finally, equation 6 takes the form F = M c qm 1 q[τ + J T mq µq, qẋ pq] + J T q µ 1 q, qẋ g q. Similar to the procedure in the one-dof case, we transform the desired impedance 19, which represents a relation etween the Cartesian velocity and the external forces, into an impedance relation etween the velocity and the generalized ase force. Therefore, we utilize 1 to otain Λ d J T D d J T qλ c q ẍ + qµ 1 q, q ẋ + K d x x = J q T F + g q. 3 Equation 3 presents the main component for the design of the controller. We are aiming again at an admittance controller with an inner position control loop. Instead of implementing the underlying position controller ased on the Cartesian dynamics, a joint level position controller While the assumptions made in this section would formally allow to represent the system dynamics in terms of x = f 1 q and ẋ only, we keep the dependence of the dynamic equations on the joint angles q since this formulation is closer to the actual implementation of the control law. can e used and comined with the Cartesian admittance y inverse kinematics as shown in Fig. 4. Let x d R n e the desired Cartesian position resulting from the admittance controller and q d = f 1 x d the corresponding set-point for the position controller, the admittance controller, which implements 19 ased on the measurement of the ase force, can e written as Λ d J T q d Λ c q d ẍ d + D d J T q d µ 1 q d, q d ẋ d + Position Control Inverse Kinematics 4 K d x d x = J T q d F + g q d. 4 q d x d τ F fq Λq R 6,6 Fig. 4. Cartesian admittance control of a manipulator mounted on a ase FTS. If the admittance controller from 4 is replaced y 8, one can take account of the static error resulting from the underlying position controller. Similar to the one-dof case, a correction of the steady state error due to a non-ideal position controller is possile y modification of the stiffness term in 4. Therefore, it is required that the steady state properties of the controller are known. If we consider for instance a PD controller with gravity compensation τ = Pq d q + D q d q + gq, 5 with positive definite gain matrices P R n n and D R n n, then the steady state error for a constant external force depends on the proportional gain matrix P. In steady state, the joint angle ˆq clearly fulfills ˆτ = P q d ˆq = J T ˆq. 6 The correction of the admittance control law can e done y following the methods proposed in [16], [17]. From 19 one can see that in the steady state of the desired impedance, the condition K d fˆq x = should hold. By comining this equation with 6, we get P q d ˆq = J T ˆqK d fˆq x. 7 The idea, adopted from [16], [17], is then to solve 7 for ˆq and use the resulting function ˆqq d, x in the implementation of the admittance. In this way, we otain Λ d J T q d Λ c q d ẍ d + ẋ d + D d J T q d µ 1 q d, q d K d fˆqq d, x x = J T q d F + g q d. 8

6 More details on how to solve an equation like 7 for ˆq can e found in [16], [17]. However, this solution requires that the controller gain P is larger than the Cartesian stiffness K d, i.e. 7 can e solved uniquely only if J T qk d Jq < P holds. Thus, the gain of the position controller represents an upper ound for the achievale stiffness, which is not surprising at all. V. SIMULATION RESULTS For the verification of the controller from Fig. 4, we present a simulation study of a planar three-degrees-offreedom root as shown in Fig. 5. For the inner loop position controller a PD controller with gravity compensation as in 5 is used. The proportional gain matrix P is chosen as a diagonal matrix. The two sets of proportional gains, which have een used in the simulations, are given in Ta. II. For the design of the damping gain matrix D the doule diagonalization design from [3] with a damping factor of.7 is applied resulting in a configuration dependent damping matrix..5 m.5 m 1. kg 1. kg ϕ 1. kg y x Fig. 5. Simulation model: A planar three-dof manipulator mounted on a ase FTS. The link length of all three segments is set to.5m. The inertia of the links is represented y a single mass located in the center of the link segments. The external force acts in the horizontal x-direction. As Cartesian coordinates, the end-effector position x, y and orientation φ are used. TABLE II GAINS OF THE JOINT POSITION CONTROLLER Joint 1 3 Prop. Gain L [Nm/rad] Prop. Gain H [Nm/rad] TABLE III IMPEDANCE PARAMETERS Direction x y ϕ Inertia 5 Ns /m 5 Ns /m 5 Nms /rad Stiffness 1 N/m 1 N/m 1 Nm/rad Damping 31.3 Ns/m 31.3 Ns/m 9.9 Nms/rad In this simulation study, we compare the two admittance controllers ased on 4 and on 8 and we will oserve the influence of the underlying position controller on the closed loop ehavior. In all simulations, the desired impedance is chosen according to 19 with diagonal matrices for the desired inertia, damping, and stiffness. The values of the diagonal elements are given in Ta. III. The external excitation is chosen as a stepwise external force acting on the end-effector in x-direction see Fig. 5. In the first simulation, we use the admittance controller from 4 and the parameter set L from Ta. II. The initial configuration for the simulation can e seen in Fig. 5. The resulting step response for a force step of 1N in x-direction is shown in Fig. 6. The desired step response in x-direction according to the parameters in Ta. III is shown y the lack dotted line, while the simulation result is shown y the lack solid line. While the transient ehavior is similar to the desired ehavior, one can oserve a steady-state error, which results from the non-ideal position controller with a finite proportional gain P. This can also e seen y oserving the motion in y- and φ-direction, which should remain zero according to the desired ehavior. In the second simulation, we now replace the admittance control law 4 y 8. In order to solve 7 for ˆqq d, q, a first order approximation is used. The results are shown in Fig. 7. One can see that the modified stiffness term in the admittance controller eliminates the steady state error due to the position controller. Clearly, for the implementation of 8 it must e assumed that the value of the proportional gain of the underlying position controller is known. In the transient phase, the quality of the position controller still influences the accuracy. The deviation of the Cartesian coordinates in y- and φ-direction from the equilirium during the transient phase can e explained y the effects of a nonideal underlying joint position controller. This is verified y a third simulation in which the admittance controller 8 is comined with a joint position controller with higher gains, which are given y the set H in Ta. II. The corresponding simulation result is shown in Fig. 8. One can see that for the higher proportional gains in the position controller the desired impedance is realized much etter during the transient phase. x x [m, rad] 1 x time [s] Fig. 6. Simulation result with the admittance controller 4 and an underlying position controller with the lower proportional gains set L in Ta. II. The desired step response in x-direction is given y the lack dotted line. The simulation result for the Cartesian motion in x-, y-, and φ-direction are shown y the lack solid line, the lue dashed line and the red dashed-dotted line, respectively.

7 x x [m, rad] 1 x time [s] Fig. 7. Simulation result with the admittance controller 8 and an underlying position controller with the lower proportional gains set L in Ta. II. The desired step response in x-direction is given y the lack dotted line. The simulation result for the Cartesian motion in x-, y-, and φ-direction are shown y the lack solid line, the lue dashed line and the red dashed-dotted line, respectively. x x [m, rad] 1 x time [s] Fig. 8. Simulation result with the admittance controller 8 and an underlying position controller with the higher proportional gains set H in Ta. II. The desired step response in x-direction is given y the lack dotted line. The simulation result for the Cartesian motion in x-, y-, and φ-direction are shown y the lack solid line, the lue dashed line and the red dashed-dotted line, respectively. VI. SUMMARY AND OUTLOOK In this paper, we analyzed the admittance control prolem of a root manipulator, in which the force/torque sensor is mounted at the ase of the root. This has the advantage that external forces acting all along the root s structure are perceived y the sensor. The desired impedance still is given in terms of a dynamic relation etween external forces at the tip and the end-effector motion, ut the interaction of the root with its environment is not restricted to the endeffector. The contriution of this paper is a generalization of the controller from [15] y taking the steady state properties of the underlying position controller into account in the design of the outer admittance control loop. The design idea was exemplified y a detailed analysis of the one- DOF case. The Cartesian impedance control prolem for a general multi-degrees-of-freedom root was discussed and verified y a simulation study. 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