Learning from Demonstration: Variable Impedance Control. A tensor-based formulation of GMM/GMR on the Riemannian manifold

In the last years, realistic applications have brought robots to complex environments such as museums, hospitals, and homes. These are unstructured, dynamic, uncertain settings inhabited by humans. This growth in robot applications and technologies has changed the classic point of view of robots. Nowadays, a lot of attention has been paid to the problem of providing the robot with abilities close to the human level. Robots must be able to adapt to unstructured environments and react online to sudden variations in the surroundings. Robots interact with the domain around them, performing complex manipulation tasks (e.g., assembly operations: peg-in-hole, slide-in-the-groove, bolt screwing, pick and place) and human collaboration tasks. In this context my work proposes an approach to improve the learning skills of a robot. A powerful tool to modulate the behaviour of robots in response to changes in the environment is impedance control. Robots must use contact forces and variable impedance control to perform complex handling tasks to adapt to the surrounding situation. In order to achieve this adaptability, it is possible to use the learning methods by which the robot will be able to perform manipulation tasks even in unknown scenarios. My work proposes a Learning from Demonstrations (LfD) approach using contact force detection and time-varying Cartesian stiffness to perform manipulation tasks by learning the stiffness skills based on force. My approach, built on Riemannian manifolds, takes advantage of the geometry of non-Euclidean spaces, which are ubiquitous in robotics to represent the orientations of the rigid body, the matrices of inertia, Manipulability ellipsoids, or controller gain matrices. My work first estimates the sequence of stiffness matrices through human demonstrations of the task that is then used together with the forces detected on the end-effector to build a probabilistic model of the task. The latter is the Gaussian mixture model (GMM) that is used during the reproduction phase to obtain the stiffness sequence through the Gaussian Regressor (GMR). The stiffness matrix $\tilde{K}^\mathcal{P} \in \mathcal{S}^{m}_+ $ is symmetric and positive definite (SPD), so to learn force-based variable impedance skills my approach uses a tensor-based formulation of GMM/GMR on the Riemannian manifold $\mathcal{S}^{m}_+$ to directly learn and reproduce $\tilde{K}^\mathcal{P}$ which exploits the geometry of the SPD space. I validated my work in simulation using a 2D and 3D system in a couple of real-world scenarios, analyzing the limits of the approach. The proposed approach gets good results