Nonlinear model predictive control on SO(3) for dynamic legged locomotion

Abstract

the exponential map is used as a retraction for the optimization variable evolving on the SO(3) manifold as a composition of a nominal point on SO(3) with the variations in the tangent space at that point. By doing so, the 9-element variable can be concisely represented as a 3-element variable living in Euclidean space without additional constraints. Based on that, analytic Jacobians required for constructing the gradient and Gauss-Newton Hessian approximation matrix of the cost function are derived. Having formulated the NMPC problem in terms of a constrained nonlinear least-squares problem, an efficient Gauss-Newton algorithm is introduced to enable real-time calculation of optimal solutions. In particular, fast and efficient algorithms are developed to speed up the calculation of the gradient and Gauss-Newton Hessian approximation matrix by establishing an equivalent relationship between the solution of linear quadratic optimal control and the solution of the forward dynamics problem of articulated rigid body systems. Finally, to enable the robot to move on not only horizontal surfaces but also vertical and inverted surfaces, contact constraints such as friction and tipping over constraints, which are imperative to reliably perform agile climbing locomotion against gravity, are derived to explicitly considered in the controller for providing admissible ground reaction forces without foot slipping and tipping over. The performance of the overall framework is investigated by applying the proposed control framework to two different-sized quadrupedal hardware platforms, namely KAIST Hound and MARVEL, including agile and versatile locomotion on horizontal, vertical, and inverted surfaces.

Legged robotic systems have the potential to perform various maneuvers in complex and unstructured environments. However, conventional control approaches of legged robots still lag far behind compared to the motor skills of legged animals such as geckos and lizards that can agilely and robustly move on arbitrary inclined surfaces. Towards reducing the gap of motor skills between legged robotic systems and legged animals, this thesis addresses a geometry-aware constrained nonlinear model predictive control (NMPC) framework and efficient algorithms to solve the nonlinear optimization problem for agile and versatile legged locomotion in 3-dimensional space, including horizontal, vertical, and inverted surfaces. This work first formulates the legged locomotion control problem in terms of the NMPC problem. The NMPC framework assumes a legged robot as a floating-base single rigid body with ground reaction forces applied to the body as control inputs. Specifically, the state of the body orientation is represented using a rotation matrix evolving on the 3D special orthogonal group SO(3) manifold, thereby liberating from the issues such as Gimbal lock related to Euler angles as an orientation representation. Also, the distance metric of orientation used in the NMPC problem is represented as exponential coordinates, which express the rotation error along the (shortest) geodesic path on the SO(3) manifold in a geometrically meaningful way. The underlying difficulty posed by directly using the rotation matrix (9-variable) as an optimization variable is that additional constraints (e.g., determinant and orthonormal constraints) should be imposed on the variable to enforce it lie on the SO(3) manifold. However, the additional constraints would unnecessarily increase the computation time, which hinders real-time implementation and thus affects the overall performance of the controller. To address this issue, in this context, a well-established Lie theory is introduced into the NMPC framework