This thesis aims to address the real-time planning and nonlinear control of quadrupedal locomotion such that the resulting gaits are robust to various kinds of disturbances. Specifically, this work addresses two scenarios. Namely, a quasi-static formulation in which an inertial appendage (i.e., a tail) is used to assist the quadruped in negating external push disturbances, and an agile formulation which is derived in a manner such that an appendage could easily be added in future work to examine the affect of tails on agile and high-speed motions.

Initially, this work presents a unified method in which bio-inspired articulated serpentine robotic tails may be integrated with walking robots, specifically quadrupeds, in order to produce stable and highly robust locomotion. The design and analysis of a holonomically constrained 2 degree of freedom (DOF) tail is shown and its accompanying nonlinear dynamic model is presented. The model created is used to develop a hierarchical control scheme which consists of a high-level path planner and a full-order nonlinear low-level controller. The high-level controller is based on model predictive control (MPC) and acts on a linear inverted pendulum (LIP) model which has been extended to include the forces produced by the tail by augmenting the LIP model with linearized tail dynamics. The MPC is used to generate center of mass (COM) and tail trajectories and is subject to the net ground reaction forces of the system, tail shape, and torque saturation of the tail in order to ensure overall feasibility of locomotion. At the lower level, a full-order nonlinear controller is implemented to track the generated trajectories using quadratic program (QP) based input-output (I-O) feedback linearization which acts on virtual constraints. The analytical results of the proposed approach are verified numerically through simulations using a full-order nonlinear model for the quadrupedal robot, Vision60, augmented with a tail, totaling at 20 DOF. The simulations include a variety of disturbances to show the robustness of the presented hierarchical control scheme.

The aforementioned control scheme is then extended in the latter portion of this thesis to achieve more dynamic, agile, and robust locomotion. In particular, we examine the use of a single rigid body model as the template model for the real-time high-level MPC, which is linearized using variational based linearization (VBL) and is solved at 200 Hz as opposed to an event-based manner. The previously defined virtual constraints controller is also extended so as to include a control Lyapunov function (CLF) which contributes to both numerical stability of the QP and aids in stability of the output dynamics. This new hierarchical scheme is validated on the A1 robot, with a total of 18 DOF, through extensive simulations to display agility and robustness to ground height variations and external disturbances. The low-level controller is then further validated through a series of experiments displaying the ability for this algorithm to be readily transferred to hardware platforms.