Skills transfer can exploit stiffness and manipulability ellipsoids, in the form of geometric descriptors representing the skills to be transferred to the robot. As these ellipsoids lie on symmetric positive definite (SPD) manifolds, Riemannian geometry can be used to learn and reproduce these descriptors in a probabilistic manner.

Another promising candidate for handling geometric structures in robotics is the framework of geometric algebra, whose roots can be found in Clifford algebra, which can be seen as a fusion of quaternion and Grassmann algebras. Geometric algebra provides a unification of various frameworks popular in robotics, such as screw theory, Lie algebra or dual quaternions, while offering great generalization and extension perspectives.

As a crude illustration of geometric algebra, think first of the way we represent the position of datapoints in a Cartesian space as a 3D vector. If we set some of these vector entries to 0, we can represent points in specific 2D planes of the 3D space (e.g, by representing a planar path in x-y by ignoring the z axis). In geometric algebra, we will store data in a vector of higher dimension (e.g., 32D), which will allow us to describe a richer variety of geometric objects, including constraints specified as spheres, circles, lines or segments. In analogy to the 3D vector, the non-zero values in this 32D vector will determine which geometric object it represents. Interestingly, both translation and rotation will be expressed in the same manner. Moreover, both states and actions will use the same formulation. These two properties provide a generic and homogeneous formulation for various robotics problems (kinematic and dynamic control, optimization, learning and planning).

Another informal parallel to illustrate the use of a 32-dimensional space is to consider the use of a unit quaternion to describe an orientation, whose components can be stored as a 4D vector. With this representation, for the very common task of moving both the position and orientation of a robot end effector, we need linear algebra for the translation part and quaternion algebra for the rotation part, which also most often involves a series a conversion from one to the other (e.g., to go through the different links of a kinematic chain). The use of dual quaternion is a step forward to avoid this situation, by embedding position information as another quaternion, resulting in a 8D vector that can be homogeneously treated with quaternion algebra. Geometric algebra follows a similar idea, but provides a more generic formulation, by using more dimensions to represent a richer set of objects. There are several variants described by a different number of dimensions, which offers to treat different categories of geometric problems, while keeping the same base formulation.

In contrast to quaternion algebra whose theory relies on a generalization of complex numbers (the imaginary i, j, k components of the quaternion), the theory behind geometric algebra relies on a simpler principle using basis vectors as a starting point to describe different directions, which are combined with the notion of inner and outer products. In standard linear algebra, the inner/dot product of two vectors gives a scalar, while the outer product of two vectors typically takes the form of a cross product for 3D vectors, whose results in another 3D vector whose length is proportional to the surface swiped by the two vectors (in the plane formed by these two vectors). In geometric algebra, the notion of outer product is generalized to any dimension. First, the result is not expressed as a vector, and an oriented surface is instead defined so that the ordering of the two vectors matters. Then, multiple consecutive outer products can be defined, representing an oriented volume for the outer product of 3 vectors. This composition of the original basis vectors allows the formation of a higher dimension space in which many different geometric objects can be defined in a uniform manner. Both geometric objects and transformations use the same algebra and can then also be represented in a homogeneous manner as elements of this higher dimension space. Practically, geometric algebra allows geometric operations to be computed in a very fast way, with compact codes, see above figure.

This representation is thus an interesting candidate to provide a single algebra for geometric reasoning, alleviating the need of utilizing multiple algebras to express geometric relations. Our work leverages the capabilities of geometric algebra in robot manipulation tasks. We show in [Löw and Calinon, 2023] that the modeling of cost functions for optimal control can be done uniformly across different geometric primitives, leading to a low symbolic complexity of the resulting expressions and a geometric intuitiveness. Our benchmarks also show that such an approach provides faster resolution of robot kinematics problems than state-of-the-art software libraries used in robotics. In [Löw, Abbet and Calinon, 2024], we present a software library to allow other researchers and practitioners to leverage the power of geometric algebra in their own research.

Learning and optimization problems in robotics are characterized by two types of variables: 1) task parameters representing the situation that the robot encounters, typically related to environment variables such as locations of objects, users or obstacles; and 2) decision variables related to actions that the robot takes, typically related to a controller acting within a given time window, or the use of basis functions to describe trajectories in control or state spaces. For each change of task parameters, decision variables need to be recomputed as fast as possible, so that the robot can fluently collaborate with users and can swiftly react to changes in its environment.

Within the MEMMO and LEARN-REAL projects, we investigate the roles of offline and online learning for optimal control. The problem is formalized as an optimization problem with a cost function to minimize, parameterized by task parameters and decision variables. The formulation transforms the minimization of a cost function into the maximization of a corresponding probability density function. Hence, the problem of finding the minima of a cost function is transformed into the problem of sampling the maxima of the density function. The proposed approach does not require gradients to be computed and can find multiple optima, which can be used for global optimization problems.

The density function is modeled in the offline phase using a tensor train (TT) that exploits the structure between the task parameters and the decision variables. Tensor train factorization is a modern tool for the representation of complex nonlinear functions in variable separation form, with robust techniques such as TT-Cross to find the representation. It allows conditional sampling over the task parameters with priority for higher-density regions. This property is used for fast online decision-making, with local Gauss-Newton optimization to refine the results, see the above Figure for an overview.

The proposed tensor train for global optimization (TTGO) approach has several advantages for robot skills acquisition. First, the problem formulation makes it a ubiquitous tool. It allows the distribution of computation into offline and online phases. It can cope with a mix of continuous and discrete variables for optimization thus providing an alternative to mixed-integer programming. By construction, the decision variables also stay within a bounded domain, which can be advantageous in some problems (e.g., joint angle limits, control bounds). Another important advantage of TTGO is that the underlying TT-cross method used to approximate the distribution in TT format can easily be extended to various forms of human-guided learning, by letting the user sporadically specify task parameters or decision variables within the iterative process. The first case can be used to provide a scaffolding mechanism for robot skill acquisition. The second case can be used for the robot to ask for help in specific situations.

In the reference below, we demonstrated the capability of the approach for trajectory optimization within a varied set of control and planning problems with robot manipulators.

In robotics, ergodic control extends the tracking principle by specifying a probability distribution over an area to cover instead of a trajectory to track. The original problem is formulated as a spectral multiscale coverage problem, typically requiring the spatial distribution to be decomposed as Fourier series. This approach does not scale well to control problems requiring exploration in search space of more than 2 dimensions. To address this issue, we propose the use of tensor trains, a recent low-rank tensor decomposition technique from the field of multilinear algebra. The proposed solution is efficient, both computationally and storage-wise, hence making it suitable for its online implementation in robotic systems. The approach is applied to a peg-in-hole insertion task requiring full 6D end-effector poses (see second reference below).

To facilitate the acquisition of manipulation skills, task-parameterized models can be exploited to take into account that motions typically relates to objects, tools or landmarks in the robot's workspace. The approach consists of encoding a movement in multiple coordinate systems (e.g., from the perspectives of different objects), in the form of trajectory distributions. In a new situation (e.g., for new object locations), the reproduction problem corresponds to a fusion problem, where the variations in the different coordinate systems are exploited to generate a movement reference tracked with variable gains, providing the robot with a variable impedance behavior that automatically adapts to the precision required in the different phases of the task. For example, in a pick-and-place task, the robot will be stiff if the object needs to be reached/dropped in a precise way, and will remain compliant in the other parts of the task.

Our ongoing work explores the extension of the task-parameterized principle to a richer set of behaviors, including coordinate systems that take into account symmetries (e.g., cylindrical and spherical coordinate systems) and nullspace projection structures.

Linear quadratic tracking (LQT) is a simple form of optimal control that trades off tracking and control costs expressed as quadratic terms over a time horizon, with the evolution of the state described in a linear form. This constrained problem can be solved by expressing the state and the control commands as trajectories, corresponding to a least squares solution. A probabilistic interpretation of the LQT solution can be built by using the residuals of this estimate.

This approach allows the creation of bridges between learning and control. For example, in learning from demonstration, the observed (co)variations in a task can be formulated as an LQT objective function, which then provides a trajectory distribution in control space that can be converted to a trajectory distribution in state space. All the operations are analytic and only exploit basic linear algebra.

The proposed approach can also be extended to model predictive control (MPC), iterative LQR (iLQR) and differential dynamic programming (DDP), whose solution needs this time to be interpreted locally at each iteration step of the algorithm.