arXiv subjects

Jacob R. Goodman

Publications and source records attributed to Jacob R. Goodman.

12 recordsLinked to original sources

arXiv2026

Connection Towers and Sasaki Metrics on Higher-Order Tangent Bundles

Higher-order tangent bundles possess a rich tower of fibrations, suggesting the existence of geometric structures compatible with their iterated bundle structure. In this paper, we introduce the notion of a connection tower on a higher-order tangent bundle and study the geometric structures induced by such towers. In particular, we show that connection towers determine natural multiconnections, adapted splittings of the tangent bundle, and canonical vector bundle structures on higher-order tangent bundles. We then construct a specific connection tower induced by the Levi-Civita connection of a Riemannian manifold. This construction extends the classical Dombrowski connection map on the tangent bundle and leads naturally to a family of higher-order Sasaki metrics. We study the associated lifts of vector fields and derive explicit Lie bracket formulas for these lifts, together with structural identities for the induced multiconnection. Finally, we determine the Levi-Civita connection of the higher-order Sasaki metrics and derive explicit geodesic equations on the second- and third-order tangent bundles. We also obtain characterization results relating geodesics of the higher-order Sasaki metrics to geodesics on the base manifold.

math.DG↗

arXiv2026

A Generalized Sasaki Metric on the Second-Order Tangent Bundle

This paper constructs a connection map on the second-order tangent bundle induced by a linear connection on the base manifold and uses it to define a generalized Sasaki metric. The associated geodesic equations are derived, and jet-constrained variational problems are shown to yield Riemannian quintics in tension. The construction is then specialized to rigid body attitude dynamics with first-order actuator dynamics, producing an intrinsic higher-order trajectory model on the rotation group. Numerical simulations compare quintics in tension with Riemannian cubics as nominal trajectories and show modest reductions in actuator-relevant cost with comparable tracking performance.

math.MG↗

arXiv2026

Riemannian Metric Preconditioning for Trajectory Tracking

We introduce a rank-one Riemannian cometric update inducing a modification of the Riemannian metric that makes specific directions of motion cheaper to travel along. We establish basic completeness properties of this reward metric, and give an explicit characterization of its Levi--Civita connection. We propose a preconditioned trajectory-tracking strategy by adding the connection-difference term to a standard intrinsic PD control, and illustrate the construction on a connection control-affine system on the Special Euclidean group with a maze navigation experiment. When the nominal trajectory is an integral curve of the vector field used to define the reward metric, our methodology improves the overall tracking, which is demonstrated through simulation results.

math.MG↗

arXiv2024

Variational Lifting and Optimal Gauges on Riemannian Homogeneous Spaces

Building on standard Euler-Poincar\'e reduction on Lie groups, we develop a variational lifting framework for mechanical systems on Riemannian homogeneous spaces $H=G/K$. A mechanical action on $H$ is lifted to $G$ through a functional whose kinetic energy depends only on the horizontal component of the velocity, and we prove that its critical points project precisely onto those of the original action. The lifted functional possesses a natural gauge invariance under $H^1$ curves with values in the isotropy subgroup $K$. Consequently, every lifted critical curve decomposes into a smooth horizontal representative and an arbitrary vertical gauge, and the associated Euler-Poincar\'e equations with symmetry breaking are obtained in reduced form. We then introduce a second variational problem that selects a distinguished lift by minimizing the vertical kinetic energy. The optimal gauge is a length-minimizing geodesic on $K$, yielding an explicit decomposition of the total energy into projected and vertical contributions and a geometric interpretation in terms of holonomy for closed projected curves. The framework is illustrated first for pure quantum states on $\mathbb{CP}^{n}\cong SU(n+1)/S(U(1)\times U(n))$, where the gauge freedom corresponds to the isotropy of unitary lifts, and then for $S^2\cong SO(3)/SO(2)$ through an optimal camera-orientation problem for an axisymmetric satellite subject to an undesirable pointing region.

math.OC↗

arXiv2023

Reduction of Sufficient Conditions in Variational Obstacle Avoidance Problems

This paper studies sufficient conditions in 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. We provide necessary and sufficient conditions under which the resulting critical points, the so-called modified Riemannian cubics, are local minimizers. We then study the theory of reduction by symmetries of sufficient conditions for optimality in variational obstacle avoidance problems on Lie groups endowed with a left-invariant metric. This amounts to left-translating the Bi-Jacobi fields described to the Lie algebra, and studying the corresponding bi-conjugate points. New conditions are provided in terms of the invertibility of a certain matrix.

math.OC↗

arXiv2023

Reduction of Necessary Conditions for the Variational Collision Avoidance Problem

In this work, we study the reduction by a Lie group of symmetries of variational collision avoidance probelms of multiple agents evolving on a Riemannian manifold and derive necessary conditions for the reduced extremals. The problem consists of finding non-intersecting trajectories of a given number of agents, among a set of admissible curves, to reach a specified configuration, based on minimizing an energy functional that depends on the velocity, covariant acceleration and an artificial potential function used to prevent collision among the agents.

math.OC↗

arXiv2022

Reduction by Symmetry in Obstacle Avoidance Problems on Riemannian Manifolds

This paper studies the reduction by symmetry of a variational obstacle avoidance problem. We derive the reduced necessary conditions in the case of Lie groups endowed with a left-invariant metric, and for its corresponding Riemannian homogeneous spaces by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the Lie group. A number of special cases where the obstacle avoidance potential can be computed explicitly are studied in detail, and these ideas are applied to the obstacle avoidance task for a rigid body evolving on SO$(3)$ and for the unit sphere $S^2$.

math.OC↗

arXiv2022

Local Minimizers for Variational Obstacle Avoidance on Riemannian manifolds

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 and subsequently classified, with local uniqueness results obtained in both cases.

math.OC↗

arXiv2021

Geometric Control for Load Transportation with Quadrotor UAVs by Elastic Cables

Groups of unmanned aerial vehicles (UAVs) are increasingly utilized in transportation task as the combined strength allows to increase the maximum payload. However, the resulting mechanical coupling of the UAVs impose new challenges in terms of the tracking control. Thus, we design a geometric trajectory tracking controller for the cooperative task of four quadrotor UAVs carrying and transporting a rigid body, 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 position and attitude of the load exponentially converges to a given desired trajectory. We then show that this result leads to an uniformly converging tracking error for the original elastic model under some assumptions. Furthermore, under the presence of unstructured disturbances on the system, we show that the error is ultimately bounded with an arbitrarily small bound. Finally, a simulation illustrates the theoretical results.

math.OC↗

arXiv2021

Variational Obstacle Avoidance with Applications to Interpolation Problems in Hybrid Systems

We study variational obstacle avoidance problems on complete Riemannian manifolds and apply the results to the construction of piecewise smooth curves interpolating a set of knot points in systems with impulse effects. We derive the dynamical equations for extrema in the variational problem, and show the existence of minimizers by using lower-continuity arguments for weak convergence on an infinite-dimensional Hilbert manifold. We then provide conditions under which it is possible to ensure that the extrema will safely avoid a given obstacle within some desired tolerance.

math.OC↗

arXiv2021

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

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.

math.OC↗

arXiv2021

Variational Collision Avoidance on Riemannian Manifolds

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.

math.OC↗