Robust control of cooperative underactuated manipulators

We propose in this work the first model-based robust control method for a team of underactuated manipulators jointly manipulating a load. The method is based on feedback linearization of the nonlinear dynamic coupling between the torques applied at the actuated joints and the Cartesian acceleration of the load, combined with a variable structure controller. Singularities in the control method are addressed, and a sufficient condition for a singularity-free controller implementation is obtained. Simulation and experimental results are presented to validate the theory presented. 1 Underactuated cooperative manipulators When a single mechanical manipulator is not able to manipulate a load because it is too bulky or too heavy, a team of manipulators acting cooperatively may be employed. Such cooperative systems have been extensively studied in the past, and several authors have provided elegant solutions to the problems of internal force control [13], load distribution [16], hybrid position / force control [14], teleoperation of multiple manipulators [15], joint flexibility [7], and distributed control [9]. One problem that has only recently been considered is that of controlling the position and orientation of a load carried by two manipulators, when either one of them or both are underactuated, i.e., are equipped with unactuated joints [8], [11], [12]. Unactuated joints are those joints of a manipulator which lack actuation, and they arise in practice either because of actuator failures or because of design considerations. In this paper we propose a model-based control method for the problem aforementioned. The method, based on the theory of variable structure control, is robust with respect to modeling uncertainties and external disturbances, a characteristic not present in the common linear PID controllers. Additionally, the control method is sufficiently general to allow for both set-point and trajectory tracking control. This way, complex control commands can be followed without the user having to switch between different controllers according to the control objective. These two characteristics, robustness and generality, distinguish this work from the authors’ previous ones, in which a linear controller [3], and two separate controllers for set-point or trajectory tracking control were proposed [8]. The contributions of the paper are the application, for the first time, of variable structure control to a team of underactuated manipulators, their modeling in terms of the active and passive joints’ contributions to the system’s dynamics, and a sufficient condition for singularity-free operation. The paper is divided as follows: in section 2 we review the main results from the kinematic modeling of cooperative underactuated manipulators, already presented by the authors in [8]. In section 3 we present their dynamic modeling, along with a sufficient condition for singularityfree controller implementation. In section 4 we present the robust control method. In section 5 we present simulation and experimental results obtained with a team of two 2-link underactuated manipulators to illustrate the validity of the control method. Finally, in section 6 we present our conclusions and directions for future work. IEEE/RSJ International Conference on Intelligent Robots and Systems, Victoria, B.C., Canada, Oct. 1998, pp. 443-448. 2 Kinematic modeling We present in this section the kinematic modeling of a team of underactuated manipulators. The presentation follows closely that of [8] and is concise for the sake of brevity. Without loss of generality, consider two identical n-link nonredundant manipulators grasping a common load (Figure 1). We denote by q1 the joint vector of the first manipulator, and by q2 the joint vector of the second one. The combined joint vector q is defined as