This paper describes a collection of optimization algorithms for achieving dynamic planning, control, and state estimation for a bipedal robot designed to operate reliably in complex environments. To make challenging locomotion tasks tractable, we describe several novel applications of convex, mixed-integer, and sparse nonlinear optimization to problems ranging from footstep placement to whole-body planning and control. We also present a state estimator formulation that, when combined with our walking controller, permits highly precise execution of extended walking plans over non-flat terrain. We describe our complete system integration and experiments carried out on Atlas, a full-size hydraulic humanoid robot built by Boston Dynamics, Inc.

Abstract-To plan dynamic, whole-body motions for robots, one conventionally faces the choice between a complex, fullbody dynamic model containing every link and actuator of the robot, or a highly simplified model of the robot as a point mass. In this paper we explore a powerful middle ground between these extremes. We exploit the fact that while the full dynamics of humanoid robots are complicated, their centroidal dynamics (the evolution of the angular momentum and the center of mass (COM) position) are much simpler. By treating the dynamics of the robot in centroidal form and directly optimizing the joint trajectories for the actuated degrees of freedom, we arrive at a method that enjoys simpler dynamics, while still having the expressiveness required to handle kinematic constraints such as collision avoidance or reaching to a target. We further require that the robot's COM and angular momentum as computed from the joint trajectories match those given by the centroidal dynamics. This ensures that the dynamics considered by our optimization are equivalent to the full dynamics of the robot, provided that the robot's actuators can supply sufficient torque. We demonstrate that this algorithm is capable of generating highly-dynamic motion plans with examples of a humanoid robot negotiating obstacle course elements and gait optimization for a quadrupedal robot. Additionally, we show that we can plan without pre-specifying the contact sequence by exploiting the complementarity conditions between contact forces and contact distance.

Abstract-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 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 robustness while keeping a low computation time.

The DARPA Robotics Challenge Trials held in December 2013 provided a landmark demonstration of dexterous mobile robots executing a variety of tasks aided by a remote human operator using only data from the robot's sensor suite transmitted over a constrained, fieldrealistic communications link. We describe the design considerations, architecture, implementation and performance of the software that Team MIT developed to command and control an Atlas humanoid robot. Our design emphasized human interaction with an efficient motion planner, where operators expressed desired robot actions in terms of affordances fit using perception and manipulated in a custom user interface. We highlight several important lessons we learned while developing our system on a highly compressed schedule.

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In this paper, we present a novel formulation of the inverse kinematics (IK) problem with generic constraints as a mixed-integer convex optimization program. The proposed approach can solve the IK problem globally with generic task space constraints: a major improvement over existing approaches, which either solve the problem in only a local neighborhood of the user initial guess through nonlinear non-convex optimization, or address only a limited set of kinematics constraints. Specifically, we propose a mixed-integer convex relaxation of non-convex [Formula: see text] rotation constraints, and apply this relaxation on the IK problem. Our formulation can detect if an instance of the IK problem is globally infeasible, or produce an approximate solution when it is feasible. We show results on a seven-joint arm grasping objects in a cluttered environment, an 18-degree-of-freedom quadruped standing on stepping stones, and a parallel Stewart platform. Moreover, we show that our approach can find a collision free path for a gripper in a cluttered environment, or certify such a path does not exist. We also compare our approach against the analytical approach for a six-joint manipulator. The open-source code is available at http://drake.mit.edu .

Abstract-While legged animals are adept at traversing rough landscapes, it remains a very challenging task for a legged robot to negotiate unknown terrain. Control systems for legged robots are plagued by dynamic constraints from underactuation, actuator power limits, and frictional ground contact; rather than relying purely on disturbance rejection, considerable advantage can be obtained by planning nominal trajectories which are more easily stabilized. In this paper, we present an approach for designing nominal periodic trajectories for legged robots that maximize a measure of robustness against uncertainty in the geometry of the terrain. We propose a direct collocation method which solves simultaneously for a nominal periodic control input, for many possible one-step solution trajectories (using ground profiles drawn from a distribution over terrain), and for the periodic solution to a jump Riccati equation which provides an expected infinite-horizon cost-to-go for each of these samples. We demonstrate that this trajectory optimization scheme can recover the known deadbeat openloop control solution for the Spring Loaded Inverted Pendulum (SLIP) on unknown terrain. Moreover, we demonstrate that it generalizes to other models like the bipedal compass gait walker, resulting in a dramatic increase in the number of steps taken over moderate terrain when compared against a limit cycle optimized for efficiency only.