A Class of Explicitly Solvable Vehicle Motion Problems

Abstract

Abstract-A small but interesting result of Brockett is extended to the Euclidean group SE(3) and is illustrated by several examples. The result concerns the explicit solution of an optimal control problem on Lie groups, where the control belongs to a Lie triple system in the Lie algebra. The extension allows for an objective function based on an indefinite quadratic form.Applying the result requires explicit knowledge of the Lie triple systems of the Lie algebra se(3). Hence, a complete classification of the Lie triple systems of this Lie algebra is derived.Examples are considered for optimal trajectories in 3 cases. The first case concerns cars moving in the plane. The second looks at motions that rigidly follow the Bishop frame to a space curve. The final example does not have a particular name as it does not seem to have been studied before.The appendix gives a brief introduction to Screw theory. This is essentially the study of the Lie algebra se(3).

Screw-Based Modeling of Soft Manipulators With Tendon and Fluidic Actuation

“…which is obtained by taking the derivative of (19) as seen from an observer sited on the (micro-)body frame. Solving equation 25, it can be shown that for the actuation field F a (X) to satisfy equation (25) for every possible value of the strain ξ n it has to be constant along X, or, in other words, the tendon/chamber paths have to be parallel to the midline of the arm and at a constant position with respect to it (Fig. 3-right).…”

“…for every possible ξ a n , which identifies the active DOFs subspace generated by a constant distribution actuation. To plan and follow configuration paths for a chain of sections composing a soft manipulator, it is important to know the geometric properties of the active DOFs subspace Since the completion algebra is the entire Lie algebra of the rigid motion se(3), the system m is called first-order controllable system or Brockett's system in the context of nonholonomic mechanics, of which explicit optimal control solution [24] has been recently extended to the Euclidean group SE(3) in [25]. Those results, made possible by the present screw-based model, could bring about significant improvements in the design and control of soft manipulators.…”

Motion Interpolation in Lie Subgroups and Symmetric Subspaces

“…The Lie triple systems of se(3) were classified in [4,7,12], details of symmetric subspaces of SE(3) can also be found in [14]. It was observed in [4], that most of the symmetric subspaces of SE(3) are linear spaces or the intersection of the Study quadric Q s with a linear subspace.…”

“…The theorem can be proved by straightforward inspection of all possible cases. All possibilities were found in [12,4] and [7]. To find points in the symmetric subspaces we need to be able to exponentiate elements of the Lie triple system.…”

Geometric methods and formulations in computational multibody system dynamics

“…Motivated by this observation, Stramigioli [150] uses this as the defining property of lower pairs. Recently, Selig [145] and Wu et al [158] seized on a concept by Brockett [28] called Lie triple systems. The latter refers to systems of vector fields that are closed under the triple Lie bracket (whereas a Lie algebra is closed under the Lie bracket).…”