We present a novel approach to enhance the performance of sampling-based Model Predictive Control (MPC) in constrained optimization by leveraging products of experts. Our methodology divides the main problem into two components: one focused on optimality and the other on feasibility. By combining the solutions from each component, represented as distributions, we apply products of experts to implement a project-then-sample strategy. In this strategy, the optimality distribution is projected into the feasible area, allowing for more efficient sampling. This approach contrasts with the traditional sample-then-project method, leading to more diverse exploration and reducing the accumulation of samples on the boundaries. We demonstrate an effective implementation of this principle using a tensor train-based distribution model, which is characterized by its non-parametric nature, ease of combination with other distributions at the task level, and straightforward sampling technique. We adapt existing tensor train models to suit this purpose and validate the efficacy of our approach through experiments in various tasks, including obstacle avoidance, non-prehensile manipulation, and tasks involving staying on manifolds. Our experimental results demonstrate that the proposed method consistently outperforms known baselines, providing strong empirical support for its effectiveness.

Contact-rich manipulation plays a vital role in daily human activities, yet uncertain physical parameters pose significant challenges for both model-based and model-free planning and control. A promising approach to address this challenge is to develop policies robust to a wide range of parameters. Domain adaptation and domain randomization are commonly used to achieve such policies but often compromise generalization to new instances or perform conservatively due to neglecting instance-specific information. \textit{Explicit motor adaptation} addresses these issues by estimating system parameters online and then retrieving the parameter-conditioned policy from a parameter-augmented base policy. However, it typically relies on precise system identification or additional high-quality policy retraining, presenting substantial challenges for contact-rich tasks with diverse physical parameters. In this work, we propose \textit{implicit motor adaptation}, which leverages tensor factorization as an implicit representation of the base policy. Given a roughly estimated parameter distribution, the parameter-conditioned policy can be efficiently derived by exploiting the separable structure of tensor cores from the base policy. This framework eliminates the need for precise system estimation and policy retraining while preserving optimal behavior and strong generalization. We provide a theoretical analysis validating this method, supported by numerical evaluations on three contact-rich manipulation primitives. Both simulation and real-world experiments demonstrate its ability to generate robust policies for diverse instances.

Robot manipulation usually involves multiple different manipulation primitives, such as Push and Pivot, leading to hybrid and long-horizon characteristics. This poses significant challenges to most planning and control approaches. Instead of treating long-horizon manipulation as a whole, it can be decomposed into several simple manipulation primitives and then sequenced using PDDL planners [1, 2, 3] or Large Language Models [4, 5]. Although such manipulation primitives usually have lowto-medium-dimensional state and action spaces, the breaking and establishment of contact make it tough for most motion planning techniques. Gradient-based techniques suffer from vanishing gradients when contact breaks, while sampling-based techniques struggle with the combinatorial complexity of multiple contact modes, i.e., sticking and sliding. This leads to time-consuming online replanning in the real world for contact-rich manipulation, limiting the real-time reactiveness of robots in coping with uncertainties and disturbances. Learning manipulation primitives that can quickly react to the surroundings, therefore, makes a lot of sense. Since the learned manipulation primitives will be sequenced by symbolic planners, which have no information about the geometric/motion level, the learned manipulation primitive should be robust to diverse instances with varied physical parameters, such as shape, mass, and friction coefficient. For example, once the push primitive is scheduled by the high-level symbolic planner, it should be able to Figure 2: Illustration of DA, DR and DC.

Recent advances in robot skill learning have unlocked the potential to construct task-agnostic skill libraries, facilitating the seamless sequencing of multiple simple manipulation primitives (aka. skills) to tackle significantly more complex tasks. Nevertheless, determining the optimal sequence for independently learned skills remains an open problem, particularly when the objective is given solely in terms of the final geometric configuration rather than a symbolic goal. To address this challenge, we propose Logic-Skill Programming (LSP), an optimization-based approach that sequences independently learned skills to solve long-horizon tasks. We formulate a first-order extension of a mathematical program to optimize the overall cumulative reward of all skills within a plan, abstracted by the sum of value functions. To solve such programs, we leverage the use of tensor train factorization to construct the value function space, and rely on alternations between symbolic search and skill value optimization to find the appropriate skill skeleton and optimal subgoal sequence. Experimental results indicate that the obtained value functions provide a superior approximation of cumulative rewards compared to state-of-the-art reinforcement learning methods. Furthermore, we validate LSP in three manipulation domains, encompassing both prehensile and non-prehensile primitives. The results demonstrate its capability to identify the optimal solution over the full logic and geometric path. The real-robot experiments showcase the effectiveness of our approach to cope with contact uncertainty and external disturbances in the real world.

The convergence of many numerical optimization techniques is highly dependent on the initial guess given to the solver. To address this issue, we propose a novel approach that utilizes tensor methods to initialize existing optimization solvers near global optima. Our method does not require access to a database of good solutions. We first transform the cost function, which depends on both task parameters and optimization variables, into a probability density function. Unlike existing approaches, the joint probability distribution of the task parameters and optimization variables is approximated using the Tensor Train model, which enables efficient conditioning and sampling. We treat the task parameters as random variables, and for a given task, we generate samples for decision variables from the conditional distribution to initialize the optimization solver. Our method can produce multiple solutions (when they exist) faster than existing methods. We first evaluate the approach on benchmark functions for numerical optimization that are hard to solve using gradient-based optimization solvers with a naive initialization. The results show that the proposed method can generate samples close to global optima and from multiple modes. We then demonstrate the generality and relevance of our framework to robotics by applying it to inverse kinematics with obstacles and motion planning problems with a 7-DoF manipulator.

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, implemented with a 7-axis Franka Emika Panda robot. In this experiment, ergodic exploration allows the task to be achieved without requiring the use of force/torque sensors.