# Output Function Optimization for Faster Convergence Rate in Underactuated Bipedal Walking Control

#### Abstract

Bipedal walking control has been studied for decades. Because of the variant degrees of freedom of the robot during walking, underactuation does occur in some phases of the walking process. Underactuation is defined as the situation in which the number of the actuators is less than the degrees of freedom of the robot. In general, underactuated cases are harder to be handled than fully-actuated cases due to the lack of control effort at the unactuated joints. As a result, underactuation in bipedal walking control has always been one of the main issues in achieving walking stabilization. Previously, underactuated bipedal walking stabilization has been achieved based on the nonlinear control theories. The concept of Hybrid-Zero-Dynamics (HZD) has been introduced, and the stability of walking has been formally evaluated using the method of Poincar\'e map. The output functions are defined based on the desired walking pattern. By applying feedback control law through input-output feedback linearization, asymptotically stable walking can be achieved. More recently, time-dependent orbital stabilization of underactuated bipedal walking has also been realized by including time-dependent desired gait in the output functions. In addition, a stability condition is established to check the stability of the full-order system instead of the internal dynamics only. In both cases, though the converging rate of the output functions to zero is assumed by the input-output feedback linearization design, actual convergence rate of the robot to the desired trajectories sill depends on the behavior of the resulting internal dynamics and as such, could still be quite slow. In the above work, input-output feedback linearization has been utilized in the controller design of underactuated bipedal walking control. With this design methodology, although the output functions can be driven to zero exponentially fast, the overall rate of convergence of the robot to the desired trajectories still depends on the behavior of the internal dynamics. Since internal dynamics only depend on the output function definitions, it is thus interesting to see how different definitions of output functions affect the behavior of the resulting internal dynamics, which is the focus of this thesis. Specifically, the output functions in those previous studies are defined as some combinations of the tracking errors between the actual joint positions and the desired positions only, which leads to relative degrees of to be two for each output. In this thesis, the output functions are broadened to a larger class that could include the velocity tracking errors as well, resulting in outputs of relative degree one only. Though for the embedded velocity terms leading to the higher dimensions of internal dynamics and making the analysis of the biped system harder, the more general form of the output function definitions allows the resulting internal dynamics to be further optimized for a faster overall rate of convergence of the robot to the desired trajectories. This strategy is shown to yield control solutions that have a faster rate of convergence to the desired walking pattern and better robustness to external perturbations. Finally, the proposed control strategy is simulated on a planar robot with five revolute joints, point feet, and a torso. Simulation results show the validity of the proposed controller design and realization of the orbital stabilization of bipedal walking. Improvements in the overall rate of convergence and the robustness to external forces will be shown in the comparison with the controller design in previous studies.

#### Degree

M.S.M.E.

#### Advisors

Yao, Purdue University.

#### Subject Area

Mechanical engineering|Robotics

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