Controlled Lagrangians and Potential Shaping for Stabilization of Discrete Mechanical Systems

Stabilization of systems with one degree of underactuation with energy shaping, a geometric approach

A geometric formulation for stabilization of systems with one degree of underactuation which fully solves the energy shaping problem for these system is given. The results show that any linearly controllable simple mechanical system with one degree of underactuation is stabilizable by energy shaping, possibly via a closed-loop metric which is not necessarily positive-definite. An example of a system with one degree of underactuation is provided for which the stabilization by energy shaping method is not achievable using a positive-definite closed-loop metric.

Controlled Lagrangians and Stabilization of Euler--Poincaré Mechanical Systems with Broken Symmetry I: Kinetic Shaping

We extend the method of controlled Lagrangians with kinetic shaping to those mechanical systems on semidirect product Lie groups with broken symmetry, more specifically to the Euler--Poincaré equations with advected parameters. We find a matching condition for the controlled Lagrangian for such systems whose configuration manifold is a general semidirect product Lie group $\mathsf{G} \ltimes V$. Our motivating examples are a bottom-heavy underwater vehicle and a top spinning on a movable base. Their configuration space is the special Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \ma…

Controlled Lagrangians and Stabilization of Discrete Mechanical Systems I

Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic matching procedure. Specifically, new terms in the cont…

Controlled Lagrangians and Stabilization of Euler--Poincaré Mechanical Systems with Broken Symmetry II: Potential Shaping

We apply the method of controlled Lagrangians by potential shaping to Euler--Poincaré mechanical systems with broken symmetry. We assume that the configuration space is a general semidirect product Lie group $\mathsf{G} \ltimes V$ with a particular interest in those systems whose configuration space is the special Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \mathbb{R}^{3}$. The key idea behind the work is the use of representations of $\mathsf{G} \ltimes V$ and their associated advected parameters. Specifically, we derive matching conditions for the modified potential exploiting t…

受控拉格朗日量与具有对称破缺非完整约束的欧拉-庞加莱方程的镇定

Controlled Lagrangians and Stabilization of Euler--Poincaré Equations with Symmetry Breaking Nonholonomic Constraints

本文将受控拉格朗日量方法推广到带平流参数的非完整欧拉-庞加莱方程，通过引入依赖平流参数的准速度消去拉格朗日乘子并导出匹配条件，最终为摆式冰鞋模型设计了镇定滑动平衡的控制律。

The inverse problem of the calculus of variations and the stabilization of controlled Lagrangian systems

We apply methods of the so-called `inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the inverted pendulum on a cart and the inertia wheel pendulum. By making use of a condition that follows from Douglas' classification, we derive feedback controls for which the control system is variational. We then use the energy of a suitable controlled Lagrangian to provide a stability criterion for the equilibrium.