Mohamed-Ali Belabbas

Erkan Bayram, Shenyu Liu, Mohamed-Ali Belabbas et al.

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 $\mathcal{Y}$. If the training set is expanded, most existing methods for finding a new control $u^*$ require starting from scratch, resulting in a quadratic increase in complexity with the number of points. To overcome this limitation, we introduce the concept of $\textit{ tuning without forgetting}$. We develop $\textit{an iterative algorithm}$ to tune the control $u^*$ when the training set expands, whereby points already in the paired set are still matched, and new training samples are learned. 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.

Erkan Bayram, Mohamed-Ali Belabbas, Tamer Başar

In this paper, we propose an iterative training algorithm for Neural ODEs that provides models resilient to control (parameter) disturbances. The method builds on our earlier work Tuning without Forgetting-and similarly introduces training points sequentially, and updates the parameters on new data within the space of parameters that do not decrease performance on the previously learned training points-with the key difference that, inspired by the concept of flat minima, we solve a minimax problem for a non-convex non-concave functional over an infinite-dimensional control space. We develop a projected gradient descent algorithm on the space of parameters that admits the structure of an infinite-dimensional Banach subspace. We show through simulations that this formulation enables the model to effectively learn new data points and gain robustness against control disturbance.

Erkan Bayram, Mohamed-Ali Belabbas, Tamer Başar

In our earlier work, we introduced the principle of Tuning without Forgetting (TwF) for sequential training of neural ODEs, where training samples are added iteratively and parameters are updated within the subspace of control functions that preserves the end-point mapping at previously learned samples on the manifold of output labels in the first-order approximation sense. In this letter, we prove that this parameter subspace forms a Banach submanifold of finite codimension under nonsingular controls, and we characterize its tangent space. This reveals that TwF corresponds to a continuation/deformation of the control function along the tangent space of this Banach submanifold, providing a theoretical foundation for its mapping-preserving (not forgetting) during the sequential training exactly, beyond first-order approximation.

Yinai Fan, Shenyu Liu, Mohamed-Ali Belabbas

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.

Ali AlBeladi, Girish Krishnan, Mohamed-Ali Belabbas et al.

Interest in soft continuum arms has increased as their inherent material elasticity enables safe and adaptive interactions with the environment. However to achieve full autonomy in these arms, accurate three-dimensional shape sensing is needed. Vision-based solutions have been found to be effective in estimating the shape of soft continuum arms. In this paper, a vision-based shape estimator that utilizes a geometric strain based representation for the soft continuum arm's shape, is proposed. This representation reduces the dimension of the curved shape to a finite set of strain basis functions, thereby allowing for efficient optimization for the shape that best fits the observed image. Experimental results demonstrate the effectiveness of the proposed approach in estimating the end effector with accuracy less than the soft arm's radius. Multiple basis functions are also analyzed and compared for the specific soft continuum arm in use.