How many ways are there to climb a staircase in a given number of steps? Infinitely many, if we focus on the continuous aspect of the problem. A finite, possibly large number if we consider the discrete aspect, i.e. on which surface which effectors are going to step and in what order. We introduce NAS, an algorithm that considers both aspects simultaneously and computes all the possible solutions to such a contact planning problem, under standard assumptions. To our knowledge NAS is the first algorithm to produce a globally optimal policy, efficiently queried in real time for planning the next footsteps of a humanoid robot. Our empirical results (in simulation and on the Talos platform) demonstrate that, despite the theoretical exponential complexity, optimisations reduce the practical complexity of NAS to a manageable bilinear form, maintaining completeness guarantees and enabling efficient GPU parallelisation. NAS is demonstrated in a variety of scenarios for the Talos robot, both in simulation and on the hardware platform. Future work will focus on further reducing computation times and extending the algorithm's applicability beyond gaited locomotion. Our companion video is available at https://youtu.be/Shkf8PyDg4g

Optimal control (OC) is an effective approach to controlling complex dynamical systems. However, typical approaches to parameterising and learning controllers in optimal control have been ad-hoc, collecting data and fitting it to neural networks. This two-step approach can overlook crucial constraints such as optimality and time-varying conditions. We introduce a unified, function-first framework that simultaneously learns Lyapunov or value functions while implicitly solving OC problems. We propose two mathematical programs based on the Hamilton-Jacobi-Bellman (HJB) constraint and its relaxation to learn time varying value and Lyapunov functions. We show the effectiveness of our approach on linear and nonlinear control-affine problems. The proposed methods are able to generate near optimal trajectories and guarantee Lyapunov condition over a compact set of initial conditions. Furthermore We compare our methods to Soft Actor Critic (SAC) and Proximal Policy Optimisation (PPO). In this comparison, we never underperform in task cost and, in the best cases, outperform SAC and PPO by a factor of 73 and 22, respectively.

State-of-the-art approaches to footstep planning assume reduced-order dynamics when solving the combinatorial problem of selecting contact surfaces in real time. However, in exchange for computational efficiency, these approaches ignore joint torque limits and limb dynamics. In this work, we address these limitations by presenting a topology-based approach that enables model predictive control (MPC) to simultaneously plan full-body motions, torque commands, footstep placements, and contact surfaces in real time. To determine if a robot's foot is inside a contact surface, we borrow the winding number concept from topology. We then use this winding number and potential field to create a contact-surface penalty function. By using this penalty function, MPC can select a contact surface from all candidate surfaces in the vicinity and determine footstep placements within it. We demonstrate the benefits of our approach by showing the impact of considering full-body dynamics, which includes joint torque limits and limb dynamics, on the selection of footstep placements and contact surfaces. Furthermore, we validate the feasibility of deploying our topology-based approach in an MPC scheme and explore its potential capabilities through a series of experimental and simulation trials.

Planning multi-contact motions in a receding horizon fashion requires a value function to guide the planning with respect to the future, e.g., building momentum to traverse large obstacles. Traditionally, the value function is approximated by computing trajectories in a prediction horizon (never executed) that foresees the future beyond the execution horizon. However, given the non-convex dynamics of multi-contact motions, this approach is computationally expensive. To enable online Receding Horizon Planning (RHP) of multi-contact motions, we find efficient approximations of the value function. Specifically, we propose a trajectory-based and a learning-based approach. In the former, namely RHP with Multiple Levels of Model Fidelity, we approximate the value function by computing the prediction horizon with a convex relaxed model. In the latter, namely Locally-Guided RHP, we learn an oracle to predict local objectives for locomotion tasks, and we use these local objectives to construct local value functions for guiding a short-horizon RHP. We evaluate both approaches in simulation by planning centroidal trajectories of a humanoid robot walking on moderate slopes, and on large slopes where the robot cannot maintain static balance. Our results show that locally-guided RHP achieves the best computation efficiency (95\%-98.6\% cycles converge online). This computation advantage enables us to demonstrate online receding horizon planning of our real-world humanoid robot Talos walking in dynamic environments that change on-the-fly.

Abstract--Real-time synthesis of legged locomotion maneuvers in challenging industrial settings is still an open problem, requiring simultaneous determination of footsteps locations several steps ahead while generating whole-body motions close to the robot's limits. State estimation and perception errors impose the practical constraint of fast re-planning motions in a model predictive control (MPC) framework. We first observe that the computational limitation of perceptive locomotion pipelines lies in the combinatorics of contact surface selection. Re-planning contact locations on selected surfaces can be accomplished at MPC frequencies (50-100 Hz). Then, whole-body motion generation typically follows a reference trajectory for the robot base to facilitate convergence. Our contributions are integrated into a complete framework for perceptive locomotion, validated under diverse terrain conditions, and demonstrated in challenging trials that push the robot's actuation limits, as well as in the ICRA 2023 quadruped challenge simulation. ELIABLE and autonomous locomotion for legged robots in arbitrary environments is a longstanding challenge. A. State of the art The hardware maturity of quadruped robots [1], [2], [3], [4] The mathematical complexity of the legged locomotion motivates the development of a motion synthesis framework problem in arbitrary environments is such that an undesired for applications including inspections in industrial areas [5]. Typically, a contact plan describing the contact handling the issues of contact decision (where should the robot locations is first computed, assumed to be feasible, and provided step?) and Whole-Body Model Predictive Control (WB-MPC) as input to a WB-MPC framework to generate wholebody of the robot (what motion creates the contact?). As a result, the contact decision Each contact decision defines high-dimensional, non-linear must be made using an approximated robot model, under the geometric and dynamic constraints on the WB-MPC that uncertainty that results from imperfect perception and state prevent a trivial decoupling of the two issues: How to prove estimation. The complexity of the approximated model has, that a contact plan is valid without finding a feasible wholebody unsurprisingly, a strong correlation with the versatility and motion to achieve it?

Optimal control (OC) using inverse dynamics provides numerical benefits such as coarse optimization, cheaper computation of derivatives, and a high convergence rate. However, to take advantage of these benefits in model predictive control (MPC) for legged robots, it is crucial to handle efficiently its large number of equality constraints. To accomplish this, we first (i) propose a novel approach to handle equality constraints based on nullspace parametrization. Our approach balances optimality, and both dynamics and equality-constraint feasibility appropriately, which increases the basin of attraction to high-quality local minima. To do so, we (ii) modify our feasibility-driven search by incorporating a merit function. Furthermore, we introduce (iii) a condensed formulation of inverse dynamics that considers arbitrary actuator models. We also propose (iv) a novel MPC based on inverse dynamics within a perceptive locomotion framework. Finally, we present (v) a theoretical comparison of optimal control with forward and inverse dynamics and evaluate both numerically. Our approach enables the first application of inverse-dynamics MPC on hardware, resulting in state-of-the-art dynamic climbing on the ANYmal robot. We benchmark it over a wide range of robotics problems and generate agile and complex maneuvers. We show the computational reduction of our nullspace resolution and condensed formulation (up to 47.3%). We provide evidence of the benefits of our approach by solving coarse optimization problems with a high convergence rate (up to 10 Hz of discretization). Our algorithm is publicly available inside CROCODDYL.