An Optimized Linear Model Predictive Control Solver for Online Walking Motion Generation

International audienceThis article addresses the fast solution of a Quadratic Program underlying a Linear Model Predictive Control scheme that generates walking motions. We introduce an algorithm which is tailored to the particular requirements of this problem, and therefore able to solve it efficiently. Different aspects of the algorithm are examined, its computational complexity is presented, and a numerical comparison with an existing state of the art solver is made. The approach presented here, extends to other general problems in a straightforward way

Real-Time Motion Planning of Legged Robots: A Model Predictive Control Approach

We introduce a real-time, constrained, nonlinear Model Predictive Control for the motion planning of legged robots. The proposed approach uses a constrained optimal control algorithm known as SLQ. We improve the efficiency of this algorithm by introducing a multi-processing scheme for estimating value function in its backward pass. This pass has been often calculated as a single process. This parallel SLQ algorithm can optimize longer time horizons without proportional increase in its computation time. Thus, our MPC algorithm can generate optimized trajectories for the next few phases of the motion within only a few milliseconds. This outperforms the state of the art by at least one order of magnitude. The performance of the approach is validated on a quadruped robot for generating dynamic gaits such as trotting.Comment: 8 page

Frequency-Aware Model Predictive Control

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 can be 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

Trajectory generation for multi-contact momentum-control

Simplified models of the dynamics such as the linear inverted pendulum model (LIPM) have proven to perform well for biped walking on flat ground. However, for more complex tasks the assumptions of these models can become limiting. For example, the LIPM does not allow for the control of contact forces independently, is limited to co-planar contacts and assumes that the angular momentum is zero. In this paper, we propose to use the full momentum equations of a humanoid robot in a trajectory optimization framework to plan its center of mass, linear and angular momentum trajectories. The model also allows for planning desired contact forces for each end-effector in arbitrary contact locations. We extend our previous results on LQR design for momentum control by computing the (linearized) optimal momentum feedback law in a receding horizon fashion. The resulting desired momentum and the associated feedback law are then used in a hierarchical whole body control approach. Simulation experiments show that the approach is computationally fast and is able to generate plans for locomotion on complex terrains while demonstrating good tracking performance for the full humanoid control

Walking Stabilization Using Step Timing and Location Adjustment on the Humanoid Robot, Atlas

While humans are highly capable of recovering from external disturbances and uncertainties that result in large tracking errors, humanoid robots have yet to reliably mimic this level of robustness. Essential to this is the ability to combine traditional "ankle strategy" balancing with step timing and location adjustment techniques. In doing so, the robot is able to step quickly to the necessary location to continue walking. In this work, we present both a new swing speed up algorithm to adjust the step timing, allowing the robot to set the foot down more quickly to recover from errors in the direction of the current capture point dynamics, and a new algorithm to adjust the desired footstep, expanding the base of support to utilize the center of pressure (CoP)-based ankle strategy for balance. We then utilize the desired centroidal moment pivot (CMP) to calculate the momentum rate of change for our inverse-dynamics based whole-body controller. We present simulation and experimental results using this work, and discuss performance limitations and potential improvements

Online Optimization-based Gait Adaptation of Quadruped Robot Locomotion

Quadruped robots demonstrated extensive capabilities of traversing complex and unstructured environments. Optimization-based techniques gave a relevant impulse to the research on legged locomotion. Indeed, by designing the cost function and the constraints, we can guarantee the feasibility of a motion and impose high-level locomotion tasks, e.g., tracking of a reference velocity. This allows one to have a generic planning approach without the need to tailor a specific motion for each terrain, as in the heuristic case. In this context, Model Predictive Control (MPC) can compensate for model inaccuracies and external disturbances, thanks to the high-frequency replanning. The main objective of this dissertation is to develop a Nonlinear MPC (NMPC)-based locomotion framework for quadruped robots. The aim is to obtain an algorithm which can be extended to different robots and gaits; in addition, I sought to remove some assumptions generally done in the literature, e.g., heuristic reference generator and user-defined gait sequence. The starting point of my work is the definition of the Optimal Control Problem to generate feasible trajectories for the Center of Mass. It is descriptive enough to capture the linear and angular dynamics of the robot as a whole. A simplified model (Single Rigid Body Dynamics model) is used for the system dynamics, while a novel cost term maximizes leg mobility to improve robustness in the presence of nonflat terrain. In addition, to test the approach on the real robot, I dedicated particular effort to implementing both a heuristic reference generator and an interface for the controller, and integrating them into the controller framework developed previously by other team members. As a second contribution of my work, I extended the locomotion framework to deal with a trot gait. In particular, I generalized the reference generator to be based on optimization. Exploiting the Linear Inverted Pendulum model, this new module can deal with the underactuation of the trot when only two legs are in contact with the ground, endowing the NMPC with physically informed reference trajectories to be tracked. In addition, the reference velocities are used to correct the heuristic footholds, obtaining contact locations coherent with the motion of the base, even though they are not directly optimized. The model used by the NMPC receives as input the gait sequence, thus with the last part of my work I developed an online multi-contact planner and integrated it into the MPC framework. Using a machine learning approach, the planner computes the best feasible option, even in complex environments, in a few milliseconds, by ranking online a set of discrete options for footholds, i.e., which leg to move and where to step. To train the network, I designed a novel function, evaluated offline, which considers the value of the cost of the NMPC and robustness/stability metrics for each option. These methods have been validated with simulations and experiments over the three years. I tested the NMPC on the Hydraulically actuated Quadruped robot (HyQ) of the IIT’s Dynamic Legged Systems lab, performing omni-directional motions on flat terrain and stepping on a pallet (both static and relocated during the motion) with a crawl gait. The trajectory replanning is performed at high-frequency, and visual information of the terrain is included to traverse uneven terrain. A Unitree Aliengo quadruped robot is used to execute experiments with the trot gait. The optimization-based reference generator allows the robot to reach a fixed goal and recover from external pushes without modifying the structure of the NMPC. Finally, simulations with the Solo robot are performed to validate the neural network-based contact planning. The robot successfully traverses complex scenarios, e.g., stepping stones, with both walk and trot gaits, choosing the footholds online. The achieved results improved the robustness and the performance of the quadruped locomotion. High-frequency replanning, dealing with a fixed goal, recovering after a push, and the automatic selection of footholds could help the robots to accomplish important tasks for the humans, for example, providing support in a disaster response scenario or inspecting an unknown environment. In the future, the contact planning will be transferred to the real hardware. Possible developments foresee the optimization of the gait timings, i.e., stance and swing duration, and a framework which allows the automatic transition between gaits