Nonlinear swing-down control of the Acrobot: Analysis and optimal gain design

Area of Science:

• Robotics
• Control Theory
• Mechanical Engineering

Background:

• The Acrobot, a two-link planar robot with a single actuated joint, presents challenges in stabilizing its downward equilibrium.
• Controlling the Acrobot requires addressing its nonlinear dynamics and achieving rapid stabilization from diverse initial conditions.

Purpose of the Study:

• To develop and analyze a novel sinusoidal-derivative (SD) controller for the Acrobot's swing-down control.
• To ensure rapid stabilization of the Acrobot at its downward equilibrium point from nearly all initial states.

Main Methods:

• Design of a sinusoidal-derivative (SD) controller using linear feedback of the actuated joint's angle (sinusoidal function) and angular velocity.
• Analytical determination of optimal control gains that minimize the real parts of the dominant closed-loop poles.
• Stability analysis under conditions of no friction and limited sensor measurements (actuated joint angle and angular velocity).

Main Results:

• The SD controller achieves the swing-down control objective when the sinusoidal gain exceeds a negative constant and the derivative gain is positive.
• Optimal control gains were analytically derived, optimizing the closed-loop system's stability.
• The closed-loop poles can exhibit various configurations (double complex conjugate, quadruple real, or triple real) based on the Acrobot's physical parameters.

Conclusions:

• The proposed sinusoidal-derivative (SD) controller effectively stabilizes the Acrobot at its downward equilibrium.
• Simulations confirm the SD controller's superior performance over the traditional derivative (D) controller in terms of stabilization speed.