--- Submitted to 1993 IEEE Robotics and Automation Conference, May 2-7, 1993, Atlanta, GA, USA. Hybrid Control of Flexible Manipulators' with Multiple Contacts Jae Young Lew Wayne J. Book The George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Abstract - This paper proposes a hybrid position/force controller for flexible link manipulators that make contact with the environment at more than one point. The necessity for such a controller can be found in controlling a flexible bracing manipulator. The manipulator maintains contact at the bracing point to reduce its structural vibrations, and at the same time, the end effector may make contact with the workpiece to perform the task This approach requires hybrid control not only at the end effector but also at the bracing point. In this paper, the dynamic equations of the motion with mUltiple contact constraints are derived. The dynamic equations are transformed into two subs paces, the constrained and free subspace, using singular value decomposition of constraint equations. Each force and position controller are developed based on orthogonal- ity of these two subs paces. The asymptotic stability of the proposed hybrid controller is proven analytically. Finally, an experimental study is carried out to justify the feasibility and application of the proposed ideas. Introduction It is known that light weight manipulators can provide distinct advantages over conventional ones. The reduction of the component weight allows the actuators to move faster and carry heavier loads with longer links. However, in exchange for lighter weight, one must accept an increase in system flexibility. The inherent flexibility might result in undesirable structural vibration, which is a major concern for most applications. Active vibration feedback control by the joint actuator may reduce structural vibration. However, its performance is limited by the actuator bandwidth, and it could excite high modes in a complex multi-link system. Bracing a flexible manipulator may be one effective method to reduce the flexibility or damp out its structural vibration. The manipulator braces against a stationary frame while the end effector performs the fme motion control just as the human mm braces at the wrist for accurate writing. Bracing will secw'e the end point positioning by forming a kinematic closed chain. [Kwon,88]and [West,85] proved that bracing reduces the positioning uncertainty and increases the stiffness of the manipulator. This paper provides a hybrid position/force controller for a flexible manipulator that makes multiple contacts with the environment. The basic frame work of the proposed ideas is developed from the following previous literature. The author will briefly review them. [Book,84] introduced the idea of a bracing strategy for flexible link manipulators, and various bracing devices were compared. [West,85] showed advantages of bracing and designed a hybrid control for bracing manipulators. [Mason,81] presented a guideline for understanding constrained task at the end effector. The task broke down into subtasks that are defmed as the natural and artificial constraints. The position and force controllers are designed in each subtask by the use of a selection matrix. Hybrid control based on Mason's work was proposed and demonstrated by [Raibert,81]. Some researchers such as [yoshikawa, 87] improved the hybrid control by considering the dynamics of the manipulator. [Fisher,92] considered the kinematic stability condition of a hybrid control. However, most of the previous work on the hybrid control considers only the contact at the end effector. [McClamroch,86] took a different approach to control a constrained manipulator system. First, he modeled the constrained manipulator with differential equations and algebraic equations. He formulated this set of equations as a singular system. He proposed a systematic way to reduce system order and applied a nonlinear feedback controller to control the constrained systems. However, the application is limited to a rigid link manipulator with a single contact. [Mills,91] recognized the similarity between two approaches .like [Mason,81] and [McClamroch,86]. [Singh,85] used the singular value decomposition method to reduce the equations of motion for a class of constrained dynamic systems. However, the effects of constrained force dynamics are not fOlmulated in the problem. The outline of this paper is as follows. First, the dynamic equations of motion for a constrained flexible manipulator are derived using Lagrangian multipliers. Second, singular value decomposition is used to solve for the multipliers corresponding to the constrained forces. The equations of motion are represented as a standard fOlm without constraint forces which is more attractive to control designer. Third, the constraint free joint motion is obtained by taking the pseudo inverse of .-.. _._._._-_.- ---