Jacob R. Goodman

Bio: Jacob R. Goodman is an academic researcher from University of Michigan. The author has contributed to research in topics: Collision & Hilbert manifold. The author has an hindex of 1, co-authored 7 publications receiving 6 citations.

On the Existence and Uniqueness of Poincar\'e Maps for Systems with Impulse Effects

Abstract: The Poincare map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincare map for dynamical systems with impulse effects was introduced in the last decade and mainly employed to study the existence of limit cycles (periodic gaits) for the locomotion of bipedal robots. We investigate sufficient conditions for the existence and uniqueness of Poincare maps for dynamical systems with impulse effects evolving on a differentiable manifold. We apply the results to show the existence and uniqueness of Poincare maps for systems with multiple domains.

A Decentralized Strategy for Variational Collision Avoidance on Complete Riemannian Manifolds

Abstract: We introduce a variational approach for decentralized collision avoidance of multiple agents evolving on a Riemannian manifold, and we derive necessary conditions for extremal. The problem consists of finding non-intersecting trajectories of a given number of agents sharing only the information of relative positions with respect to their nearest neighbors, among a set of admissible curves, such that these trajectories are minimizers of an energy functional. The energy functional depends on covariant acceleration and an artificial potential used to prevent collision among the agents. We show the global existence of extrema for the energy functional. We apply the results to the case of agents on a compact and connected Lie group. Simulation results are shown to demonstrate the applicability of the results.

Variational Collision Avoidance on Riemannian Manifolds.

Abstract: This paper studies variational collision avoidance problems for multi-agents systems on complete Riemannian manifolds. That is, we minimize an energy functional, among a set of admissible curves, which depends on an artificial potential function used to avoid collision between the agents. We show the global existence of minimizers to the variational problem and we provide conditions under which it is possible to ensure that agents will avoid collision within some desired tolerance. We also study the problem where trajectories are constrained to have uniform bounds on the derivatives, and derive alternate safety conditions for collision avoidance in terms of these bounds - even in the case where the artificial potential is not sufficiently regular to ensure existence of global minimizers.

Geometric Control of two Quadrotors Carrying a Rigid Rod with Elastic Cables.

Abstract: This paper presents the design of a geometric trajectory tracking controller for the cooperative task of two quadrotor UAVs (unmanned aerial vehicles) carrying and transporting a rigid bar, which is attached to the quadrotors via inflexible elastic cables. The elasticity of the cables together with techniques of singular perturbation allows a reduction in the model to that of a similar model with inelastic cables. In this reduced model, we design a controller such that the rod exponentially tracks a given desired trajectory for its position and attitude, under some assumptions on initial error. We then show that exponential tracking in the reduced model corresponds to exponential tracking of the original elastic model. We also show that the previously defined control scheme provides uniform ultimate boundedness in the presence of unstructured bounded disturbances.

Local minimizers for variational obstacle avoidance on Riemannian manifolds

Abstract: This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories—called $ Q $-local minimizers and $ \Omega $-local minimizers—and subsequently classified, with local uniqueness results obtained in both cases.

Symplectic and Cosymplectic Reduction for simple hybrid forced mechanical systems with symmetries

Abstract: A BSTRACT . This paper discusses symplectic and cosymplectic reduction for autonomous and non-autonomous simple hybrid forced mechanical systems, re-spectively. We give general conditions on whether it is possible to perform symmetry reduction for simple hybrid Hamiltonian and Lagrangian systems subject to non-conservative external forces, as well as time-dependent external forces. We illustrate the applicability of the symmetry reduction procedure with examples and numerical simulations. and Hamiltonian systems, Reduction by symmetries, Time-dependent systems, Cyclic coordinates.

Variational Collision Avoidance on Riemannian Manifolds.

Abstract: This paper studies variational collision avoidance problems for multi-agents systems on complete Riemannian manifolds. That is, we minimize an energy functional, among a set of admissible curves, which depends on an artificial potential function used to avoid collision between the agents. We show the global existence of minimizers to the variational problem and we provide conditions under which it is possible to ensure that agents will avoid collision within some desired tolerance. We also study the problem where trajectories are constrained to have uniform bounds on the derivatives, and derive alternate safety conditions for collision avoidance in terms of these bounds - even in the case where the artificial potential is not sufficiently regular to ensure existence of global minimizers.