A Generalized Sasaki Metric on the Second-Order Tangent Bundle
For researchers in geometric mechanics and robotics, this provides an intrinsic higher-order trajectory model on Lie groups, but the improvement is incremental over existing methods.
This paper defines a generalized Sasaki metric on the second-order tangent bundle and derives geodesic equations, showing that jet-constrained variational problems yield Riemannian quintics in tension. Applied to rigid body attitude dynamics, numerical simulations show modest reductions in actuator-relevant cost compared to Riemannian cubics with comparable tracking performance.
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.