Control of underactuated mechanical systems with servo-constraints

Abstract

This paper deals with a class of controlled mechanical systems in which the number of control inputs, equal to the number of desired system outputs, is smaller than the number of degrees of freedom. The related inverse dynamics control problem, i.e., the determination of control input strategy that force the underactuated system to complete the partly specified motion, is a challenging task. In the present formulation, the desired system outputs, expressed in terms of the system states, are treated as servo-constraints on the system, and the problem is viewed from the constrained motion perspective. Mixed orthogonal-tangent realization of the constraints by the available control reactions is stated, and a specialized methodology for solving the "singular" control problem is developed. The governing equations are manipulated to index three differential-algebraic equations, and a simple numerical code for solving the equations is proposed. The feedforward control law obtained as a solution to these equations can then be enhanced by a closed-loop control strategy with feedback of the actual servo-constraint violations to provide stable tracking of the reference motion in the presence of perturbations and modeling uncertainties. An overhead trolley crane executing a load-prescribed motion serves as an illustration. Some results of numerical simulations are reported.

Improved DAE formulation for inverse dynamics simulation of cranes

“…The time-specified system outputs lead to servo-constraints [12][13][14] (or control constraints [5,15]) on the system. Using the dependent coordinates y = [p T r T ] T , the m servoconstraint equations simplify to the following trivial form …”

“…The mathematical models of cranes, used in both the flatness-based and DAE formulations of the inverse simulation studies, are typically built using minimal sets of independent coordinates [5,[7][8][9][10][11][12], leading to the minimal-dimension constraint reaction-free dynamic equations. The equations are of relative high complexity, however, assisted with further complexity of the so-called servo-constraints [12][13][14] or control constraints [5,15] on the system, resulted from the specified motion requirements expressed in the independent variables.…”

“…The equations are of relative high complexity, however, assisted with further complexity of the so-called servo-constraints [12][13][14] or control constraints [5,15] on the system, resulted from the specified motion requirements expressed in the independent variables. The final governing equations for the inverse dynamics analysis are then of rather elaborate structure.…”

“…In the dependent variable formulation, the robot (crane without the load) dynamics and the load dynamics are initially described separately, and are then coupled through a simple passive constraint due to the inextensible hoisting cable. Since the load motion is explicitly specified, the followed formulation of the inverse dynamics problem simplifies compared to the independent variable formulations [5,[10][11][12] in many modeling and computational issues. The advantages are emphasized in the sequel with reference to a three-dimensional rotary crane model executing a specified motion of the load.…”

Simultaneous stabilization and trajectory tracking of underactuated mechanical systems with included actuators dynamics

“…second-order nonholonomic chained systems is given in [16], control of UNMS with servoconstraints is presented in [7][8][9]22], while control of holonomic and nonholonomic systems using sliding mode is presented by the direct Lyapunov approach [15]. Specific examples of simultaneous stabilization and/or trajectory tracking of UNMS are given in [3,14,27,33].…”