Despite the great progress in quadrupedal robotics during the last decade, selecting good contacts (footholds) in highly uneven and cluttered environments still remains an open challenge. This paper builds upon a state-of-the-art approach, already successfully used for humanoid robots, and applies it to our robotic platform; the quadruped robot ANY-mal. The proposed algorithm decouples the problem into two subprob-lems: first a guide trajectory for the robot is generated, then contacts are created along this trajectory. Both subproblems rely on approximations and heuristics that need to be tuned. The main contribution of this work is to explain how this algorithm has been retuned to work with ANY-mal and to show the relevance of the approach with a variety of tests in realistic dynamic simulations.
For simple cases like flat terrains, optimization techniques are able to correctly solve, simultaneously, the motion of the main body of the robot and its footstep placements @cite_3 @cite_14 . Using more complex models and solvers, the problem can be reformulated to solve motion on other terrains @cite_17 @cite_12 . However, this algorithm is doomed to fall into local minima, e.g. ignoring intermediate steps on stairs or trying to jump over impassable obstacles. Such behaviors can be reduced by first relaxing the complementary constraint of contacts then slowly converging back to the initial problem @cite_16 @cite_10 @cite_21 . Alternatively, one can decompose the non-smooth terrain into different convex and even patches, and rely on Mixed-Integer Programming to find the best patches for each footstep @cite_23 @cite_18 . However the computation time of those approaches make them difficult to use on a real robot. Overall, optimization techniques are not well suited for collisions and all of the presented approaches ignore this problem.
@cite_18 @cite_14 @cite_21 @cite_3 @cite_23 @cite_16 @cite_10 @cite_12 @cite_17
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Traditional motion planning approaches for multilegged 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 nonconvex optimization with intractable computation time. In this letter, 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 prespecified 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 nonconvexity and flat terrain assumptions might lead to suboptimal or unstable plans. Our method increases the motion robustness while keeping a low computation time. We present a new method for planning footstep placements for a robot walking on uneven terrain with obstacles, using a mixed-integer quadratically-constrained quadratic program (MIQCQP). Our approach is unique in that it handles obstacle avoidance, kinematic reachability, and rotation of footstep placements, which typically have required non-convex constraints, in a single mixed-integer optimization that can be efficiently solved to its global optimum. Reachability is enforced through a convex inner approximation of the reachable space for the robot's feet. Rotation of the footsteps is handled by a piecewise linear approximation of sine and cosine, designed to ensure that the approximation never overestimates the robot's reachability. Obstacle avoidance is ensured by decomposing the environment into convex regions of obstacle-free configuration space and assigning each footstep to one such safe region. We demonstrate this technique in simple 2D and 3D environments and with real environments sensed by a humanoid robot. We also discuss computational performance of the algorithm, which is currently capable of planning short sequences of a few steps in under one second or longer sequences of 10–30 footsteps in tens of seconds to minutes on common laptop computer hardware. Our implementation is available within the Drake MATLAB toolbox [1]. This paper investigates the computational complexity of planning the motion of a body B in 2-D or 3-D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3-D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACE-hard if B is given a velocity modulus bound on its movements and is NP-hard even if B has no velocity modulus bound, where, in both cases, B has 6 degrees of freedom. To prove these results, we use a unique method of simulation of a Turing machine that uses time to encode configurations (whereas previous lower bound proofs in robotic motion planning used the system position to encode configurations and so required unbounded number of degrees of freedom). We also investigate a natural class of dynamic problems that we call asteroid avoidance problems : B, the object we wish to move, is a convex polyhedron that is free to move by translation with bounded velocity modulus, and the polyhedral obstacles have known translational trajectories but cannot rotate. This problem has many applications to robot, automobile, and aircraft collision avoidance. Our main positive results are polynomial time algorithms for the 2-D asteroid avoidance problem, where B is a moving polygon and we assume a constant number of obstacles, as well as single exponential time or polynomial space algorithms for the 3-D asteroid avoidance problem, where B is a convex polyhedron and there are arbitrarily many obstacles. Our techniques for solving these asteroid avoidance problems use “normal path” arguments, which are an intereting generalization of techniques previously used to solve static shortest path problems. We also give some additional positive results for various other dynamic movers problems, and in particular give polynomial time algorithms for the case in which B has no velocity bounds and the movements of obstacles are algebraic in space-time. 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.