Control Theoretic Approach to Fine-Tuning and Transfer Learning

Given a training set in the form of a paired $(\mathcal{X},\mathcal{Y})$, we say that the control system $\dot{x} = f(x,u)$ has learned the paired set via the control $u^*$ if the system steers each point of $\mathcal{X}$ to its corresponding target in $\mathcal{Y}$. Most existing methods for finding a control function $u^*$ require learning of a new control function if the training set is updated. To overcome this limitation, we introduce the concept of $\textit{tuning without forgetting}$. We develop $\textit{an iterative algorithm}$ to tune the control function $u^*$ when the training set expands, whereby points already in the paired set are still matched, and new training samples are learned. More specifically, at each update of our method, the control $u^*$ is projected onto the kernel of the end-point mapping generated by the controlled dynamics at the learned samples. It ensures keeping the end points for the previously learned samples constant while iteratively learning additional samples. Our work contributes to the scalability of control methods, offering a novel approach to adaptively handle training set expansions.

Geometric Heat Flow Method for Legged Locomotion Planning

We propose in this paper a motion planning method for legged robot locomotion based on Geometric Heat Flow framework. The motion planning task is challenging due to the hybrid nature of dynamics and contact constraints. We encode the hybrid dynamics and constraints into Riemannian inner product, and this inner product is defined so that short curves correspond to admissible motions for the system. We rely on the affine geometric heat flow to deform an arbitrary path connecting the desired initial and final states to this admissible motion. The method is able to automatically find the trajectory of robot's center of mass, feet contact positions and forces on uneven terrain.

A Homotopy Method for Motion Planning

We propose a novel method for motion planning and illustrate its implementation on several canonical examples. The core novel idea underlying the method is to define a metric for which a path of minimal length is an admissible path, that is path that respects the various constraints imposed by the environment and the physics of the system on its dynamics. To be more precise, our method takes as input a control system with holonomic and non-holonomic constraints, an initial and final point in configuration space, a description of obstacles to avoid, and an initial trajectory for the system, called a sketch. This initial trajectory does not need to meet the constraints, except for the obstacle avoidance constraints. The constraints are then encoded in an inner product, which is used to deform (via a homotopy) the initial sketch into an admissible trajectory from which controls realizing the transfer can be obtained. We illustrate the method on various examples, including vehicle motion with obstacles and a two-link manipulator problem.