Optimization-based motion planning for legged robots

Legged machines have the potential to traverse terrain that wheeled robots cannot. These capabilities are useful in scenarios such as stairs in homes or debris-filled disaster scenes, such as earthquake areas. This thesis develops one of the algorithms necessary to achieve such a task, a motion-planner, which translates a high-level task into a desired motion-plan for a legged robot to execute. A core difficulty in legged locomotion is that a forward body movement cannot be directly generated, but results from contact forces between the end-effectors and the environment. Furthermore, there are various physical restrictions on the contact forces that can be generated. These can make hand-crafting valid motions for all the interdependent quantities such as body motion, end-effector motion and contact forces tedious and even infeasible for complex tasks. Alternatively, Trajectory Optimization can be utilized to generate motions in a more general, automated way. Only a high-level task is specified, while the algorithm determines the motions and forces taking into account the physical restrictions of legged locomotion. This approach is attractive because once the problem has been properly modeled, the algorithm would, in an ideal case, produce motions for any high-level task, solving legged locomotion planning on a general level. This thesis presents approaches to transcribe, using e.g. collocation, the legged locomotion problem into a mathematical optimization problem, then solvable by off-the-shelf software. The three developed motion-planning algorithms generate motion-plans for increasingly complex terrains and tasks. Simulation and experiments on the quadruped robots HyQ and ANYmal verify the physical correctness of the motion-plans. The final formulation and main contribution of this thesis holistically determines the gait- sequence, step-timings, footholds, contact forces, swing-leg motions and 6-dimensional body motion, given a desired goal state and 2.5D height map of the non-flat terrain. We model the robot as a single rigid body (SRBD) controlled by the contact forces at the end- effectors. Our novel phase-based parameterization of end-effector motion and forces allows optimizing over the discrete gait sequence using only continuous decision variables. The algorithm efficiently generates highly dynamic motion-plans with flight-phases for legged systems, such as monopeds, bipeds, and quadrupeds.

Time-Optimal Path-Following Operation in the Presence of Uncertainty

Dominique Bonvin, Timm Faulwasser, Alejandro Gabriel Marchetti, Predrag Milosavljevic

Path-following tasks, which refer to dynamic motion planning along pre-specified geometric references, are frequently encountered in applications such as milling, robot-supported measurements, and trajectory planning for autonomous vehicles. Different convex and non-convex optimal control formulations have been proposed to tackle these problems for the case of perfect models. This paper analyzes path-following problems in the presence of plant-model mismatch. The proposed adaptation strategies rely on concepts that are well known in the field of real-time optimization. We present conditions guaranteeing that, upon convergence, a minimum-time solution is attained despite the presence of plant-model mismatch. We draw upon a simulated robotic example to illustrate our results.

Ieee2016

Bimanual Skill Learning with Pose and Joint Space Constraints

Sylvain Calinon

As humanoid robots become commonplace, learning and control algorithms must take into account the new challenges imposed by this morphology, if we aim to fully exploit their potential. One of the most prominent characteristics of such robots is their bimanual structure. Most research on learning bimanual skills has focused on the coordination between end-effectors, exploiting operational space formulations. However, motion patterns in bimanual scenarios are not exclusive to operational space, also occurring at joint level. Moreover, in addition to position, the end-effector orientation is also essential for bimanual operation. Here, we propose a framework for simultaneously learning constraints in configuration and operational spaces, while considering end-effector orientations, overlooked in previous works. In particular, we extend the Task-Parameterized Gaussian Mixture Model (TP-GMM) with novel, Jacobian-based operators that address the foregoing problem. The proposed framework is evaluated in a bimanual task with the COMAN humanoid that requires the consideration of operational and configuration space movements.

2018

Frequency-Aware Model Predictive Control

Farbod Farshidian

Transferring solutions found by trajectory optimization to robotic hardware remains a challenging task. When the optimization fully exploits the provided model to perform dynamic tasks, the presence of unmodeled dynamics renders the motion infeasible on the real system. Model errors cannot be only a result of model simplifications, but also naturally arise when deploying the robot in unstructured and nondeterministic environments. Predominantly, compliant contacts and actuator dynamics lead to bandwidth limitations. While classical control methods provide tools to synthesize controllers that are robust to a class of model errors, such a notion is missing in modern trajectory optimization, which is solved in the time domain. We propose frequency-shaped cost functions to achieve robust solutions in the context of optimal control for legged robots. Through simulation and hardware experiments we show that motion plans can be made compatible with bandwidth limits set by actuators and contact dynamics. The smoothness of the model predictive solutions can be continuously tuned without compromising the feasibility of the problem. Experiments with the quadrupedal robot ANYmal, which is driven by highly compliant series elastic actuators, showed significantly improved tracking performance of the planned motion, torque, and force trajectories and enabled the machine to walk robustly on terrain with unmodeled compliance.

2019