Control of Uncertain Nonlinear Multibody Mechanical Systems

Abstract

Descriptiotis of real-life complex multibody mechanical systems are usually uncertain. Two sources of uncertainty are considered in this paper: uncertainties in the knowledge of the physical system and uncertainties in the "given" forces applied to the system. Both types of uncertainty are assumed to be time vaiying and unknown, yet bounded. In the face of' such uncertainties, what is available in hand is therefore just the so-called "nominal system," which is our best assessment and description ofthe actual real-life situation. A closed-form equation of motion for a general dynamical system that contains a control force is developed. When applied to a real-life uncertain multibody system, it causes the system to track a desired reference trajectory that is prespecified for the nominal system to follow. Thus, the real-life system's motion is required to coincide within prespecified error bounds and mimic the motion desired of the nominal system. Uncertainty is handled by a controller based on a generalization ofthe concept of a sliding surface, which permits the use of a large class of control laws that can be adapted to specific real-life practical limitations on the control force. A set of closed-form equations of motion is obtained for nonlinear, nonautonomous, uncertain, multibody systems that can track a desired reference trajectory that the nominal system is required to follow within prespecified error bounds and thereby satisfy the constraints placed on the nominal system. An example of a simple mechanical system demonstrates the efficacy and ease of implementation ofthe control methodology.

Dynamic modeling of SCARA robot based on Udwadia–Kalaba theory

“…The explicit dynamical equation of the constrained system called as the Udwadia-Kalaba theory is denoted in the following general form 5,9 M€…”

“…6,7 Moreover, it is not an issue for Udwadia-Kalaba equation based on the Gauss' principle of least constraint, and the optimal value of the generalized variables solution can still be obtained by the Moore-Penrose generalized inverse when the number of the constraint equations is not equal to the number of generalized variables. Udwadia and colleagues [8][9][10][11] also effectively dealt with the dynamics and control of nonlinear uncertain systems to benefit the basic research and further application of this method. It has been studied in some application fields, such as industrial robot, 12 parallel robot, 13 flexible multibody systems, 14 railway vehicles collision, 15 machine fish, 1 control of tethered satellites, 16 rigid multibody systems, 17 and mobile robots.…”

Concise method to the dynamic modeling of climbing robot

“…This section shows how to obtain the explicit equations of motion for a constrained dynamical system using a simple straightforward three-step procedure: 18,24 1. In terms of the generalized coordinates, equations of motion of the unconstrained dynamical system written using Newtonian or Lagrangian mechanics are considered.…”

An approach to the dynamic modeling and sliding mode control of the constrained robot

“…Professors Udwadia and Kalaba [6][7][8][9] proposed the equation of the multi-body system motion under the constraint condition, which is one of the important achievements in Lagrange mechanics field. This equation is applicable to a variety of constraints, such as holonomic and non-holonomic constraints.…”