Analytical Dynamics

Abstract

Constrained motion is of paramount importance in the design and analysis of mechanical systems and central to the study of analytical dynamics. The problem of constrained motion was first posed over two hundred years ago, and it has been worked on vigorously ever since. This book offers a fresh approach to the subject. Eminently readable, it is written as an introduction to analytical dynamics, with emphasis on fundamental concepts in mechanics. The connection between generalized inverses of matrices and constrained motion is a central theme. The book begins with a description of the motion of a particle subjected to holonomic and nonholonomic constraints and presents explicit equations of motion. Examples are provided throughout the book, and carefully formulated problems at the end of each chapter reinforce the material covered. This computationally appealing approach will be useful to students in engineering and the applied sciences.

A unifying framework for robot control with redundant DOFs

“…This control-law is especially interesting as it has a clear physical interpretation (Udwadia and Kalaba 1996;Bruyninckx and Khatib 2000;Udwadia 2003): the metric used is consistent with principle of virtual work of d'Alembert. Similarly as before we can compensate for coriolis, centrifugal and gravitational forces in joint-space, i.e., setting u 1 = C + G + u 0 .…”

“…The solution to this point-wise optimal control problem (Spo 1984;Spong et al 1986) can be derived from a generalization of Gauss' principle, as originally suggested in (Udwadia 2003). It is also a generalization of the propositions in (Udwadia and Kalaba 1996;Bruyninckx and Khatib 2000). We formalize this idea in the following proposition.…”

“…Often, we require a solution which has a kinematic interpretation; such a solution is usually given by a metric like N = (M · M) −1 = M −2 . In other cases, the control force u may be required to comply with the principle of virtual displacements by d'Alembert for which the metric N = M −1 is more appropriate (Udwadia and Kalaba 1996;Bruyninckx and Khatib 2000). In practical cases, one would want to distribute the forces such that joints with stronger motors get a higher workload which can also be achieved by a metric such as N = diag(τ −2 1 ,τ −2 2 , .…”

“…The major insight is that tracking control can be reformulated in terms of constraints, which in turn allows the application of a generalization of Gauss' principle of least constraint 1 (Udwadia and Kalaba 1996;Bruyninckx and Khatib 2000) to derive a control law. This insight leads to a specialized point-wise optimal control framework for controlled mechanical systems.…”

“…This insight leads to a specialized point-wise optimal control framework for controlled mechanical systems. While it is not applicable to non-mechanical control problems with arbitrary cost functions, it yields an important class of opti-1 Gauss' principle least constraint (Udwadia and Kalaba 1996) is a general axiom on the mechanics of constrained motions. It states that if a mechanical system is constrained by another mechanical structure the resulting accelerationẍ of the system will be such that it minimizes (ẍ − M −1 F) T M −1 (ẍ − M −1 F) while fulfilling the constraint.…”