Trajectory and Foothold Optimization using Low-Dimensional Models for Rough Terrain Locomotion

We present a trajectory optimization framework for legged locomotion on rough terrain. We jointly optimize the center of mass motion and the foothold locations, while considering terrain conditions. We use a terrain costmap to quantify the desirability of a foothold location. We increase the gait's adaptability to the terrain by optimizing the step phase duration and modulating the trunk attitude, resulting in motions with guaranteed stability. We show that the combination of parametric models, stochastic-based exploration and receding horizon planning allows us to handle the many local minima associated with different terrain conditions and walking patterns. This combination delivers robust motion plans without the need for warm-starting. Moreover, we use soft-constraints to allow for increased flexibility when searching in the cost landscape of our problem. We showcase the performance of our trajectory optimization framework on multiple terrain conditions and validate our method in realistic simulation scenarios and experimental trials on a hydraulic, torque controlled quadruped robot

Legged Robots

International audienc

Legged Robots

International audienc

Hierarchical planning of dynamic movements without scheduled contact sequences

Most animal and human locomotion behaviors for solving complex tasks involve dynamic motions and rich contact interaction. In fact, complex maneuvers need to consider dynamic movement and contact events at the same time. We present a hierarchical trajectory optimization approach for planning dynamic movements with unscheduled contact sequences. We compute whole-body motions that achieve goals that cannot be reached in a kinematic fashion. First, we find a feasible CoM motion according to the centroidal dynamics of the robot. Then, we refine the solution by applying the robot's full-dynamics model, where the feasible CoM trajectory is used as a warm-start point. To accomplish the unscheduled contact behavior, we use complementarity constraints to describe the contact model, i.e. environment geometry and non-sliding active contacts. Both optimization phases are posed as Mathematical Program with Complementarity Constraints (MPCC). Experimental trials demonstrate the performance of our planning approach in a set of challenging tasks

Preview optimization for learning locomotion policies on rough terrain

Legged robots promise a clear advantage in unstructured and challenging terrain, scenarios such as disaster relief, search and rescue, forestry and construction site. Dynamic locomotion on rough terrain has to guarantee stability and maximizing the cross-ability of a local set of candidate footholds. Trajectory optimization improves such performance metric while satisfying locomotion stability. Terrain conditions increase significantly the dimensionality of the optimization problem. Moreover, decoupling footstep selection and Center of Mass (CoM) motion generation may limit the success of the task. We are inspired by the observation that humans solve complex problems through intensive reasoning in the initial phases, which allows them to solve faster and naturally similar problems. In the same vein, the preview optimization allows the robot to infer the locomotion skills required on challenging terrain, and then use the data to build a locomotion policy that can be the used in real-time. A set of preview model allows us to reduce the dimensionality of the problem, which is desirable for trajectory optimization and policy reconstruction

Preview optimization for learning locomotion policies on rough terrain

Legged robots promise a clear advantage in unstructured and challenging terrain, scenarios such as disaster relief, search and rescue, forestry and construction site. Dynamic locomotion on rough terrain has to guarantee stability and maximizing the cross-ability of a local set of candidate footholds. Trajectory optimization improves such performance metric while satisfying locomotion stability. Terrain conditions increase significantly the dimensionality of the optimization problem. Moreover, decoupling footstep selection and Center of Mass (CoM) motion generation may limit the success of the task. We are inspired by the observation that humans solve complex problems through intensive reasoning in the initial phases, which allows them to solve faster and naturally similar problems. In the same vein, the preview optimization allows the robot to infer the locomotion skills required on challenging terrain, and then use the data to build a locomotion policy that can be the used in real-time. A set of preview model allows us to reduce the dimensionality of the problem, which is desirable for trajectory optimization and policy reconstruction

A feasibility-driven approach to control-limited DDP

Differential dynamic programming (DDP) is a direct single shooting method for trajectory optimization. Its efficiency derives from the exploitation of temporal structure (inherent to optimal control problems) and explicit roll-out/integration of the system dynamics. However, it suffers from numerical instability and, when compared to direct multiple shooting methods, it has limited initialization options (allows initialization of controls, but not of states) and lacks proper handling of control constraints. In this work, we tackle these issues with a feasibility-driven approach that regulates the dynamic feasibility during the numerical optimization and ensures control limits. Our feasibility search emulates the numerical resolution of a direct multiple shooting problem with only dynamics constraints. We show that our approach (named BOX-FDDP) has better numerical convergence than BOX-DDP+ (a single shooting method), and that its convergence rate and runtime performance are competitive with state-of-the-art direct transcription formulations solved using the interior point and active set algorithms available in KNITRO. We further show that BOX-FDDP decreases the dynamic feasibility error monotonically—as in state-of-the-art nonlinear programming algorithms. We demonstrate the benefits of our approach by generating complex and athletic motions for quadruped and humanoid robots. Finally, we highlight that BOX-FDDP is suitable for model predictive control in legged robots

Simultaneous Contact, Gait and Motion Planning for Robust Multi-Legged Locomotion via Mixed-Integer Convex Optimization

Traditional motion planning approaches for multi-legged locomotion divide the problem into several stages, such as contact search and trajectory generation. However, reasoning about contacts and motions simultaneously is crucial for the generation of complex whole-body behaviors. Currently, coupling theses problems has required either the assumption of a fixed gait sequence and flat terrain condition, or non-convex optimization with intractable computation time. In this paper, we propose a mixed-integer convex formulation to plan simultaneously contact locations, gait transitions and motion, in a computationally efficient fashion. In contrast to previous works, our approach is not limited to flat terrain nor to a pre-specified gait sequence. Instead, we incorporate the friction cone stability margin, approximate the robot's torque limits, and plan the gait using mixed-integer convex constraints. We experimentally validated our approach on the HyQ robot by traversing different challenging terrains, where non-convexity and flat terrain assumptions might lead to sub-optimal or unstable plans. Our method increases the motion generality while keeping a low computation time.Comment: 8 pages, IEEE Robotics and Automation Letter