Kinematic reduction and the Hamilton-Jacobi equation

Abstract

A close relationship between the classical Hamilton-\ud Jacobi theory and the kinematic reduction of control systems by\ud decoupling vector fields is shown in this paper. The geometric interpretation\ud of this relationship relies on new mathematical techniques\ud for mechanics defined on a skew-symmetric algebroid. This\ud geometric structure allows us to describe in a simplified way the\ud mechanics of nonholonomic systems with both control and external\ud forces.Peer ReviewedPreprin

Variational discretization for optimal control problems of nonholonomic mechanical systems

“…This Lie bracket verifies the usual properties of a Lie bracket except the Jacobi identity (see [1] for example). …”

Optimal Trajectory Tracking of Nonholonomic Mechanical Systems: a geometric approach

“…As when we work in tangent bundles, it is possible to determine the Christoffel symbols associated with the connection ∇ G D by ∇ G D eB e C = Γ A BC e A . Note that the coefficients Γ C AB of the connection ∇ G D are (see [1] for details)…”

A variational-geometric approach for the optimal control of nonholonomic systems

“…where γ : I ⊂ R → D is a ρD-admissible curve [2]. In terms of control inputs, Equation (6) can be rewritten as…”

“…Locally D (2) is described by the vanishing of the constraintsq i − (ρD) i A y A = 0 on T D, where local coordinates on T D are (q i , y A ,q i ,ẏ A ) and coordintes on D (2) are determined by (q i , y A ,ẏ A ) where the inclusion from D (2) to T D, denoted by i D (2) : D (2) → T D is given by…”