View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Institute of Transport Research:Publications Base Force/Torque Sensing for Position based Cartesian Impedance Control Christian Ott and Yoshihiko Nakamura Abstract— In this paper, a position based impedance con- of the robot also for (unplanned) contacts at different points. troller (i.e. admittance controller) is designed by utilizing While the application of force sensitive skins [6] or the measurements of a force/torque sensor, which is mounted at the integration of torque sensing [7], [8] are possible approaches robot’s base. In contrast to conventional force/torque sensing at the end-effector, placing the sensor at the base allows to to handle such situations, we investigate on an alternative implement a compliant behavior of the robot not only with approach in this paper. Our approach aims at integrating respect to forces acting on the end-effector but also with a force/torque sensor at the base of the robot instead of respect to forces acting on the robot’s structure. The resulting mounting it at the end-effector. This enables to perceive control problem is first analyzed in detail for the simplified forces all along the robot’s structure independently of joint one-degree-of-freedom case in terms of stability and passivity. Then, an extension to the Cartesian admittance control of friction. However, since the forces measured at the base a robot manipulator is discussed. Furthermore, it is shown are related to the robot’s motion, the manipulator dynamics how the steady state properties of the underlying position must be taken into account in the design of the admittance controller can be taken into account in the design of the controller. outer admittance controller. Finally, a simulation study of the Apart from applications to fixed mounted manipulators, Cartesian admittance controller applied to a three-degrees-of- freedom manipulator is presented. we expect that the same issue will also be relevant for im- plementing whole body impedance controllers of humanoid I. INTRODUCTION robots. Feedback of the feet contact forces is often used in walking and balancing controllers of biped robots in order to Impedance control is a prominent example for a compliant control the interaction forces of the robot with the ground [9]. motion control algorithm used for autonomous manipulation However, in that case the force feedback is often designed in and physical human-robot interaction [1], [2]. Different im- a pragmatic way and without rigorous theoretical justification plementations of the general impedance control concept have or stability analysis. been proposed using either impedance or admittance causal- In the design of whole body impedance controllers includ- ity of the controller. A controller with impedance causality ing a compliant behavior of the lower body with respect to (sometimes called ”force based impedance control”) usually forces acting on the main body, we have to take account requires a precise torque interface and thus can benefit of the following key issues. Firstly, for position controlled greatly of integrated torque sensing and torque control [3], robots it is necessary to incorporate the contact force mea- [4]. In many commercial robots this is not feasible and only a surements at the feet into the whole body control, since these conventional position or velocity interface is provided. In that sensors provide an indirect measurement of all forces acting case, a compliant behavior can still be implemented by in- on the robot. Secondly, for keeping the zero-moment-point tegrating a force/torque sensor (FTS) at the end-effector and within the support polygon of the feet, it is necessary to limit designing an outer loop admittance controller (sometimes the contact forces and moments. Thirdly, for handling larger called ”position based impedance control”) which provides contact forces, a combination with stepping and walking the desired set-point for an inner loop position or velocity technologies will be required. Within this paper, we treat controller [5]. the first of these problems. Compared to previous works In this paper, we focus on the implementation of an ad- on this problem, we aim at giving an adequate theoretical mittance controller, which can be implemented on a position justification of the base sensor feedback by deriving all the controlled robot. However, by using a FTS mounted at the relevant dynamic equations and by presenting a stability tip of the robot, the compliant behavior can only be achieved analysis of the one-DOF case. with respect to forces acting on the end-effector, while the The use of base mounted FTSs for identification and joint robot will be ”insensitive” to forces acting along the robot’s torque estimation has been well studied in the works of structure. In contrast to this, the use of robots in partly Dubowsky et al. [10], [11], [12]. In [11], a method for esti- unknown human environments requires a compliant behavior mating the dynamical parameters of a serial manipulator arm was presented. Due to the measurement of the base force, This research is partly supported by Special Coordination Funds for Promoting Science and Technology, ”IRT Foundation to Support Man and no joint torque information was required in the identification Aging Society”. procedure. In [12], the base FTS was used for estimating the Ch. Ott and Y. Nakamura are with Department of Mechano- robot’s joint torques based on known dynamical parameters. Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan The estimated torque signal was used for implementing an {ott,nakamura}@ynl.t.u-tokyo.ac.jp inner torque control loop, which was augmented by an outer (6+n)×(6+n) PD position controller. wherein M¯ (xb, q) ∈ R denotes the complete In [13], a base FTS was used in combination with a inertia matrix including the base link [18]. The centrifugal FTS mounted at the wrist for collision detection and iden- and Coriolis terms are given via the matrix C¯ (xb, x˙ b, q, q˙ ) ∈ (6+n)×(6+n) tification in human-robot interaction tasks. Kosuge et al. R . The gravity term is written as g¯(xb, q) ∈ [14] integrated a body force sensor on a mobile robot for R(6+n). The joint torques τ ∈ Rn are considered as the cooperatively handling large objects by multiple robots. control inputs. The generalized force measured by the base 6 In contrast to [12], we aim at incorporating the base FTS is denoted by F b ∈ R . The generalized external forces force measurement directly into the design of an admittance (except for the generalized forces F b exerted at the base at controller instead of implementing an inner loop torque the location of the FTS) acting on the robot are summarized controller. The desired impedance represents a dynamic by the vector τ ext. In case that the external torques are due 6 relation between external forces and the motion of the robot. to a generalized force F ext ∈ R acting at the end-effector, This impedance will be transformed into a dynamic relation they can be written as between the contact force at the base and the robot’s motion. T We will highlight some restrictions on the achievable closed τ ext,b J b (xb, q) τ ext = = T F ext , (2) loop dynamics which are due to the dislocation of the force τ ext,m J q (xb, q) sensing. A first version of the controller from this paper was T J (xb,q) already presented in [15]. In the present paper, the controller | {z } (6×(6+n)) from [15] is refined by compensating for the steady state with J(xb, q) ∈ R as the Jacobian matrix for the error of the underlying position controller. This refinement serial kinematic chain from the fixed world frame O to the if achieved by modifying the outer admittance control loop end-effector. In the following, the external torques are split 6 n based on design ideas from [16], [17]. up into the two components τ ext,b ∈ R and τ ext,m ∈ R acting on the base link and the joints, respectively. II. ROBOT MODEL INCLUDING THE BASE FORCE In (1), the joint coordinates q are augmented by local In this section, the general model of a robot with n joints is coordinates of the base link motion xb in order to incorporate discussed, in which an expression of the contact force at the the contact force F b into the equations of motion. Since the base is included. In contrast to interaction forces measured at base is attached to the ground via a stiff force/torque sensor, the end-effector, the forces between the robot and its base are we have to augment (1) by an additional constraint, which internal forces. Therefore, we start with an extended model prevents any motion of the base link: with a free-floating base (Fig. 1). By adding constraints on the base motion, we can derive an explicit expression of the ∗ dxb(t) x˙ b xb(t)= x , = 0 ⇒ I 0 = 0 . (3) force and torque measured at the base. b dt q˙ Φ From this, one can see that the| {z generalized} force at the base F b is represented in (1) by the Lagrangian multipliers Φ ∈ R6×(6+n) F ext related to the constraint matrix from (3). In the following, this constraint will be incorporated into (1). In this way an expression of the generalized base force can be derived. Therefore, we drop the dependence on the constant x = f(q) ∗ position and orientation xb = xb of the base link and write F ¯ ∗ ¯ ∗ 0 ∗ b M(xb , q), C(xb , , q, q˙ ), and g(xb , q) in the form ∗ M b(q) M c(q) xb, x˙ b M¯ (x , q) = , b M T (q) M(q) O c ¯ ∗ 0 Cb(q, q˙ ) C1(q, q˙ ) C(xb , , q, q˙ ) = , Fig.