This paper provides an overview of recent developments on design of hybrid controllers for continuous-time control systems that can be described by linear or nonlinear differential state equations. Hybrid controllers provide a generalization of classical feedback controllers for linear and nonlinear systems. The benefit of hybrid controllers, that they can be used to achieve closed-loop performance objectives that cannot be achieved using classical linear or nonlinear controllers, is emphasized. This paper introduces hybrid controllers in the form of a switching control architecture and provides a summary of recently developed control approaches that utilize this control architecture. We provide a conceptual framework for these results, identify limitations of the results, and discuss the current status of hybrid control design approaches.

We study feedback control laws that can stabilize an equilibrium position of an underactuated planar PPR robot manipulator. The manipulator operates in a horizontal plane, and it consists of two actuated prismatic links, an actuated revolute link, and an unactuated end effector that is elastically mounted on the third link. The fact that the end effector is unactuated and is elastically mounted on the manipulator, and thus can be controlled only through coupling with the manipulator dynamics, provides a major obstacle to control design for this problem. The control objective is to design a feedback controller that can achieve arbitrary positioning and arbitrary attitude of the end effector while suppressing any motion of the end effector relative to the third link. A nonlinear control model is constructed for this underactuated manipulator and a time-invariant discontinuous feedback law is constructed that achieves the objectives with exponential convergence rates. The effectiveness of the proposed feedback law is illustrated through simulations.

We synthesize an adaptive motion control law for a free-flying robot with no external forces or moments. The basic idea is to make use of the articulated part of the space robot to control the position and orientation of the end-effector in an inertial frame. The inertia parameters for the robot end-effector and load are assumed to be a priori unknown. The articulated-body model is linear in the unknown parameters, so that an adaptive control law is developed. A novel feature of our approach is that the parameter estimates are obtained using momentum integrals only. In addition, we use unit quaternions to represent orientation errors. The stability properties of the adaptive control law is shown using Lyapunov stability theory. Computer simulations of an example 12 degrees of freedom space robot system are presented.

A design procedure for simultaneous position and force control is developed, using Lyapunov's direct method, for manipulators in contact with a rigid environment that can be described by holonomic constraints. Many manipulators that interact with their environment require taking into account the effects of these constraints in the control design. The forces of constraint play a critical role in constrained motion and are, along with displacements and velocities, to be regulated at specified values. Lyapunov's direct method is used to develop a class of position and force feedback controllers. The conditions for gain selection demonstrate the importance of the constraints. Force feedback has been shown not to be mandatory for closed-loop stabilization, but it is useful in improving certain closed-loop robustness properties.>

A feedback control problem for regulation of contact force and position in constrained robot systems is considered. The nonlinear differential-algebraic equations which describe the dynamics of a constrained robot system are linearized about a constrained equilibrium. A method of obtaining an equivalent state realization for the resulting linear differential-algebraic equations is developed. This method is numerically efficient since it is based on a singular value decomposition. A linear quadratic optimal control problem associated with the linearized differential-algebraic equations is solved. The resulting linear feedback control law guarantees good regulation of both contact force and position of the constrained robot. Since the method is based on the linearization of the nonlinear differential-algebraic equations, it is valid only in a neighborhood about the point of linearization. Two robot examples are considered in order to illustrate the proposed method.>

A production system is modeled as a stochastic discrete-event process. The production system produces discrete parts from an infinite supply of raw materials. The production time for a part is exponentially distributed and can be affected by a control variable-the production rate, which can be adjusted only at production initiation epochs for the discrete parts. A short-term cost criterion is introduced to accommodate two primary economic factors: cost due to the control effort and cost based on missing production deadlines. This short-term cost function depends only on quantities in the production of a single part, but with a correction term to achieve better long-term performance. Two cases are considered in detail: one case where the penalty on missing the part production deadline is asymmetric and another case where the penalty is symmetric. In each case, the optimal operating policy is determined by minimizing the short-term cost function. Properties of the resultant optimal control policy are investigated; the closed-loop behavior of the production system under the optimal control policy is discussed for each case.>

A mathematical formulation for the time-optimal contour following problem, defined by a unilaterally constrained manipulator, is presented. The formulation includes a careful development of the manipulator dynamics and of conditions for avoidance of impact between the end effector and the constraint surface. By a priori specification of the paths for the unconstrained motion segments, a parameterization approach is used to simplify the optimal control formulation, so that solution procedures can be identified. The methodology is applied to a simple contour-following problem for a planar Cartesian manipulator.>

A robotic contour following problem, defined by a unilaterally constrained manipulator, is presented. Our approach explicitly takes into account inequality constraints and resulting contact forces as part of the system dynamics. Possible impact at an entry time is also discussed. A phase plane technique is applied to the timeoptimal motion planning problem subject to conditions of impact avoidance. The contact force is incorporated into the optimal planning problem.

A new stochastic model of tool wear is proposed that is compatible with on-line control and feedback of tool wear measurcments. Properties of the model are given for one class of machining conditions. Conditions are given for agreement with the Taylor tool life formula in the mean.

There are many robot system configurations where external contact forces on a robot play an important role in the system dynamics. Such contact forces arise due to constraints on the motion of the robot. Mathematical models of such robot system naturally give rise to a mathematical system of differential equations and algebraic equations, that can be viewed as a singular system of differential equations. Such singular models are developed for several robot system configurations. A general reduction approach is presented that can, in principle, form the basis for the development of planning and feedback control schemes for such robot system configurations.