Analysis of Inequality Constraints Without Using Lagrange Multipliers With Applications to Classical Dynamical Systems

Abstract

The fundamental equation of mechanics (FEM) for constrained motion analysis provides a way to obtain control accelerations necessary to satisfy some set of holonomic or non-holonomic constraints. The methodology provides the control necessary to either perfectly satisfy or minimize the error in all the constraints and does not require computation of Lagrange multipliers. Furthermore, this framework is capable of addressing various types of constraints, and can treat systems that are under-, fully-, or over-constrained, conveniently. The FEM formulation has most commonly been applied to a variety of classes of equality constraints. Some attempts at extending this approach to inequality constraints have been presented in the literature, including applying slack variables to provide freedom in the constraint and diffeomorphisms to map equality constraints to bounded spaces. However, these approaches have different associated advantages and drawbacks. In order to bridge the benefits of both methodologies and mitigate their issues, this work proposes a treatment of holonomic inequality constraints within the framework of the FEM wherea class of functions built on the error and Gaussian distribution functions is leveraged to treat inequality constraints on several classical dynamical case studies. The proposed technique is applied to several classes of holonomic, solitary or one-sided inequalities, and bounding inequalities for a spring-mass-damper, an inverted pendulum, and an inverted pendulum on a cart, illustrating this approach’s broad applicability to mechanical systems.