  This manuscript primarily aims to enhance the performance of whole-body
controllers(WBC) for underactuated legged locomotion. We introduce a systematic
parameter design mechanism for the floating-base feedback control within the
WBC. The proposed approach involves utilizing the linearized model of
unactuated dynamics to formulate a Linear Quadratic Regulator(LQR) and solving
a Riccati gain while accounting for potential physical constraints through a
second-order approximation of the log-barrier function. And then the user-tuned
feedback gain for the floating base task is replaced by a new one constructed
from the solved Riccati gain. Extensive simulations conducted in MuJoCo with a
point bipedal robot, as well as real-world experiments performed on a quadruped
robot, demonstrate the effectiveness of the proposed method. In the different
bipedal locomotion tasks, compared with the user-tuned method, the proposed
approach is at least 12% better and up to 50% better at linear velocity
tracking, and at least 7% better and up to 47% better at angular velocity
tracking. In the quadruped experiment, linear velocity tracking is improved by
at least 3% and angular velocity tracking is improved by at least 23% using the
proposed method.
