Concurrent design and motion planning in robotics using differentiable optimal control

Abstract

Robot design optimization (what the robot is) and motion planning (how the robot moves) are two problems that are connected. Robots are limited by their design in terms of what motions they can execute – for instance a robot with a heavy base has less payload capacity compared to the same robot with a lighter base. On the other hand, the motions that the robot executes guide which design is best for the task. Concurrent design (co-design) is the process of performing robot design and motion planning together. Although traditionally co-design has been viewed as an offline process that can take hours or days, we view interactive co-design tools as the next step as they enable quick prototyping and evaluation of designs across different tasks and environments.

In this thesis we adopt a gradient-based approach to co-design. Our baseline approach embeds the motion planning into bi-level optimization and uses gradient information via finite differences from the lower motion planning level to optimize the design in the upper level. Our approach uses the full rigid-body dynamics of the robot and allows for arbitrary upper-level design constraints, which is key for finding physically realizable designs. Our approach is also between 1.8 and 8.4 times faster on a quadruped trotting and jumping co-design task as compared to the popular genetic algorithm covariance matrix adaptation evolutionary strategy (CMA-ES). We further demonstrate the speed of our approach by building an interactive co-design tool that allows for optimization over uneven terrain with varying height.

Furthermore, we propose an algorithm to analytically take the derivative of nonlinear optimal control problems via differential dynamic programming (DDP). Analytical derivatives are a step towards addressing the scalability and accuracy issues of finite differences. We further compared with a simultaneous approach for co-design that optimizes both motion and design in one nonlinear program. On a co-design task for the Kinova robotic arm we observed a 54-times improvement in computational speed. We additionally carry out hardware validation experiments on the quadruped robot Solo. We designed longer lower legs for the robot, which minimize the peak torque used during trotting. Although we always observed an improvement in peak torque, it was less than in simulation (7.609% versus 28.271%). We discuss some of the sim-toreal issues including the structural stability of joints and slipping of feet that need to be considered and how they can be addressed using our framework.

In the second part of this thesis we propose solutions to some open problems in motion planning. Firstly, in our co-design approach we assumed fixed contact locations and timings. Ideally we would like the motion planner to choose the contacts instead. We solve a related, but simpler problem, which is the control of satellite thrusters, which are similar to robot feet but do not have the constraint of having to be in contact with the ground to exert force on the robot. We introduce a sparse, L1 cost on control inputs (thrusters) and implement optimization via DDP-style solvers. We use full rigid-body dynamics and achieve bang-bang control via optimization, which is a difficult problem due to the discrete switching nature of the thrusters.

Lastly, we present a method for planning and control of a hybrid, wheel-legged robot. This is a difficult problem, as the robot needs to always actively balance on the wheel even when not driving or jumping forward. We propose the variablelength wheeled inverted pendulum (VL-WIP) template model that captures only the necessary dynamic interactions between wheels and base. We embedded this into a model-predictive controller (MPC) and demonstrated highly dynamic behaviors, including swinging-up and jumping over a gap.

Both of these motion planning problems expand the ability of our motion planning tools to new domains, which is an integral part also of the co-design algorithms, as co-design aims to optimize both design, and motion, together.