Learning and planning high-dimensional physical trajectories via structured Lagrangians

We consider the problem of finding sufficiently simple models of high-dimensional physical systems that are consistent with observed trajectories, and using these models to synthesize new trajectories. Our approach models physical trajectories as least-time trajectories realized by free particles moving along the geodesics of a curved manifold, reminiscent of the way light rays obey Fermat´s principle of least time. Finding these trajectories, unfortunately, requires finding a minimum-cost path in a high-dimensional space, which is generally a computationally intractable problem. In this work we show that this high-dimensional planning problem can often be solved nearly optimally in practice via deterministic search, as long as we can find a certain low-dimensional structure in the Lagrangian that describes our observed trajectories. This low-dimensional structure additionally makes it feasible to learn an estimate of a Lagrangian that is consistent with the observed trajectories, thus allowing us to present a complete approach for learning from and predicting high-dimensional physical motion sequences. We finally show experimental results applying our method to human motion and robotic walking gaits. In doing so, we furthermore demonstrate efficient path planning in a 990-dimensional space.