In this work we designed a QP-based controller combining real-time walking pattern generation and constrained nonlinear control to achieve robotic walking under Zero-Moment Point (ZMP) and torque constraints.
This method utilize the fact that existing solutions to both walking pattern generation and constrained nonlinear control (summarized in the following table) have been independently constructed as Quadratic Programs (QPs, as shown below) and that these constructions can be related through an equality constraint on the instantaneous acceleration of the center of mass (COM).
The QP of constrained nonlinear control.
The QP of COM planning (in the form of MPC) with linear inverted pendulum (LIP).
| Controller | Walking Pattern Generation | QP controller with RES-CLF |
|---|---|---|
| Purpose | COM planning for LIP | Tracking control for nonlinear system |
| Pros | ZMP constraints over the horizon | instantaneous ZMP, torque, and CLF constraints for nonlinear dynamics |
| Cons | The LIP might not reflect the nonlinear dyanmics | The ZMP constraints over the horizon cannot be guaranteed |
To summarize, this unified controller (as shown below) solves a single Quadratic Program which incorporates elements from Model Predictive Control (MPC) based center of mass planning and from rapidly exponentially stabilizing control Lyapunov function (RES-CLF). The resulting QP-based controller simultaneously solves for a COM trajectory that satisfies ZMP constraints over a future horizon while also producing joint torques consistent with instantaneous acceleration, torque, ZMP and RES-CLF constraints.
