In a world designed for legs, quadrupeds, bipeds, and humanoids have the opportunity to impact emerging robotics applications from logistics, to agriculture, to home assistance. The goal of this survey is to cover the recent progress toward these applications that have been driven by model-based optimization for the real-time generation and control of movement. The majority of the research community has converged on the idea of generating locomotion control laws by solving an optimal control problem (OCP) in either a model-based or data-driven manner. However, solving the most general of these problems online remains intractable due to complexities from intermittent unidirectional contacts with the environment, and from the many degrees of freedom of legged robots. This survey covers methods that have been pursued to make these OCPs computationally tractable, with a specific focus on how environmental contacts are treated, how the model can be simplified, and how these choices affect the numerical solution methods employed. The survey focuses on model-based optimization while paving its way for broader combination with learning-based formulations to accelerate progress in this growing field.

We present a contact planner for complex legged locomotion tasks: standing up, climbing stairs using a handrail, crossing rubble, and getting out of a car. The need for such a planner was shown at the DARPA Robotics Challenge, where such behaviors could not be demonstrated (except for egress). Current planners suffer from their prohibitive algorithmic complexity because they deploy a tree of robot configurations projected in contact with the environment. We tackle this issue by introducing a reduction property: the reachability condition. This condition defines a geometric approximation of the contact manifold, which is of low dimension, presents a Cartesian topology, and can be efficiently sampled and explored. The hard contact planning problem can then be decomposed into two subproblems: first, we plan a path for the root without considering the whole-body configuration, using a sampling-based algorithm; then, we generate a discrete sequence of whole-body configurations in static equilibrium along this path, using a deterministic contact-selection algorithm. The reduction breaks the algorithm complexity encountered in previous works, resulting in the first interactive implementation of a contact planner (open source). While no contact planner has yet been proposed with theoretical completeness, we empirically show the interest of our framework: in a few seconds, with high success rates, we generate complex contact plans for various scenarios and two robots: HRP-2 and HyQ. These plans are validated in dynamic simulations or on the real HRP-2 robot.

Controlling the robot with a permanently-updated optimal trajectory, also known as model predictive control, is the Holy Grail of whole-body motion generation. Before obtaining it, several challenges should be faced: computation cost, non-linear local minima, algorithm stability, etc. In this paper, we address the problem of applying the updated optimal control in real-time on the physical robot. In particular, we focus on the problems raised by the delays due to computation and by the differences between the real robot and the simulated model. Based on the optimal-control solver MuJoCo, we implemented a complete model-predictive controller and we applied it in real-time on the physical HRP-2 robot. It is the first time that such a whole-body model predictive controller is applied in real-time on a complex dynamic robot. Aside from the technical contributions cited above, the main contribution of this paper is to report the experimental results of this première implementation.

Task-space inverse dynamics (TSID) is a well-known optimization-based technique for the control of highly redundant mechanical systems, such as humanoid robots. One of its main flaws is that it does not take into account any of the uncertainties affecting these systems: poor torque tracking, sensor noises, delays, and model uncertainties. As a consequence, the resulting control-state trajectories may be feasible for the ideal system, but not for the real one. We propose to improve the robustness of TSID by modeling uncertainties in the joint torques, either as Gaussian random variables or as bounded deterministic variables. Then we try to immunize the constraints of the system to any—or at least most—of the realizations of these uncertainties. When the resulting optimization problem is computationally too expensive for online control, we propose ways to approximate it that lead to computation times below 1 ms. Extensive simulations in a realistic environment show that the proposed robust controllers greatly outperform the classic one, even when other unmodeled uncertainties affect the system (e.g., errors in the inertial parameters, delays in the velocity estimates).