Nonlinear Bang-Bang Impact Control: A Seamless Control in All Contact Modes Maolin Jin 1 , Sang Hoon Kang 2 , Pyung H. Chang 3 and Eunjeong Lee 4 Dept. of Mechanical Engineering Korea Advanced Institute of Science and Technology 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea 1 mulan819@ kaist.ac.kr, 2 shkang@mecha.kaist.ac.kr, 3 phchang@kaist.ac.kr, 4 eunjeonglee@ieee.org Abstract - This paper presents stability analysis and experimental results of the nonlinear bang-bang impact controller for robotic manipulators. A stability condition has been derived based on the analysis in n L ∞ space and its physical interpretation has been given. The analysis shows that the stability of nonlinear bang-bang impact control depends on sampling time and the accuracy of inertia estimation. Stability is enhanced with the decrease of changes in the Coriolis, centrifugal, and disturbance forces. The stability condition is verified by simulations and experiments. Experiments show that overall performance is comparable to or better than existing impact control techniques which employ more complicated control strategies. Index Terms - Impact force control, Impedance control, Impedance/time-delay control, Robot joint friction, n L ∞ stability. I. INTRODUCTION It has been known to be very difficult for robots to interact with a variety of environments including a stiff one with a single simple control algorithm and gain [1]-[3]. In order to address this problem, a nonlinear bang-bang impact control ( NBBIC ) has been proposed by Lee [4]-[6]. Under NBBIC, a robot can successfully interact with an environment without changing control algorithm and control gains. The effectiveness of NBBIC was verified through a set of simulations and experiments for one degree of freedom system [4]-[6]. In this paper, a formal presentation of stability analysis of the nonlinear bang-bang impact control is presented for multi degree of freedom robotic manipulators. Sufficient stability conditions have been derived based on the analysis in n L ∞ space and their physical interpretation has been given. The stability conditions are verified via simulations and experiments. The overall impact control performance is experimentally compared with other representative impact control techniques. This paper is organized as follows: Section 2 describes hybrid Natural Admittance/Time-Delay Control ( NAC/ TDC ) with the proposed bang-bang impact control. Section 3 presents the stability analysis of the NBBIC and discusses its physical implications, while Section 4 validates the NBBIC stability theorem through simulation and experiment. Section 5 discusses conclusions and suggests future work. II. NONLINEAR BANG-BANG IMPACT CONTROL In this section, first, a brief description of Natural Admittance Control ( NAC ) [7] and Time Delay Control ( TDC ) [8]-[10] is presented along with their control laws. Second, the hybrid NAC/TDC is derived [4]-[6]. Lastly, a nonlinear bang-bang impact control strategy is explained for stability analysis [4]-[6]. A. Natural Admittance Control Under NAC the target dynamics is chosen to be smaller than the maximum target admittance which does not violate the passivity constraint [7]. The simplest form of natural admittance control is ( ) ( ) ( t t t t t t t v cmd des d des d τ( )=G θ ( )- θ( ))+K θ ( )-θ( ) +B θ ( )- θ()     (1) where { } () t t t t t dt = ∫ -1 cmd s s des d des d θ M τ +K (θ ( )- θ( )) + B (θ ( )- θ( ))    . (2) t ∈ θ() n R is a joint variable vector. The variable t represents time. t ∈ d θ () n R and t ∈ d θ ()  n R are the desired joint position and velocity vectors, respectively. ∈ s τ n R is an external torque vector measured, and () t ∈ τ n R is the control torque vector applied to the joints. × ∈ s M n n R , × ∈ des K n n R , and × ∈ des B n n R are the end-point mass, which can be estimated by system identification, and the desired stiffness and damping matrices, respectively. × ∈ v G n n R and t ∈ cmd θ ()  n R represent the diagonal constant velocity feedback gain matrix and the command joint velocity vector, respectively. In (1), the first term corrects deviations of the actual response from the modeled response, cmd θ  . The second term is the feedforward term which imposes desired dynamics implicitly. While a robot tries to achieve desired target dynamics, the desired dynamics generate virtual force composed of spring and damping forces on the end effector. Therefore, this virtual force should be accounted for in the force feedback loop through cmd θ  . Also, to mask undesirable dynamic effects such as friction, the sensed environment force s τ is fed back as a velocity command. However, like many other interaction controllers, NAC does not achieve the desired performance due to inherent nonlinear dynamics, modeling uncertainties and digital sampling. In order to enhance NAC by compensating the effect of uncertainties via time delay estimation, a hybrid NAC/TDC is developed in the next section. B. Hybrid Natural Admittance/Time-Delay Control The nonlinear dynamics of n degree of freedom robots are described by the following dynamic equation. ( ) ( ) ( ) () () () ( ), ( ) () () () t t t t t t t t = + + + + s τ M θ θ V θ θ G θ τ d   , (3) where ( ) () t × ∈ M θ nn R is an inertia matrix, () t ∈ θ n R is a joint Proceedings of the 2005 IEEE International Conference on Robotics and Automation Barcelona, Spain, April 2005 0-7803-8914-X/05/$20.00 ©2005 IEEE. 557