Fuzzy-backstepping control of quadruped robots

Abstract

In this paper, fuzzy-backstepping control method is utilized to control quadruped robots. The backstepping control method is inherently a stable control method since its design procedure is based on the Lyapunov stability theory. However, for an acceptable performance of this controller, the feedback gains must be appropriately selected. Using fuzzy systems, these gains are adaptively tuned for best performance of the robot against uncertainties in system parameters, external disturbances, and measurement noises. Next, the stability of the robot is investigated using the Poincaré map stability criterion. This is because the Lyapunov method does not guarantee stability for the periodic systems such as legged robots. The proposed method is simulated on the highly nonlinear equations of a quadruped robot. The performance of the proposed method is compared with the backstepping control to show the importance of fine-tuning the parameters using fuzzy system. The simulating results for a nine-degree-of-freedom quadruped robot show very good performance of the proposed control method, especially against uncertainties in system parameters, external disturbances, and measurement noises, which are common in robots.

Appendix

The kinematic equations of the quadruped robot are given below:

$$ \begin{aligned} & x_{{C_{1} }} = d_{1} \sin \theta_{1} ,\,\,y_{{C_{1} }} = d_{1} \cos \theta_{1} \\ & x_{{C_{2} }} = l_{1} \sin \theta_{1} + d_{2} \sin \theta_{2} ,\,\,y_{{C_{2} }} = l_{1} \cos \theta_{1} + d_{2} \cos \theta_{2} \\ & x_{{C_{3} }} = l_{1} \sin \theta_{1} + l_{2} \sin \theta_{2} + (l_{3} - d_{3} )\sin \theta_{3} ,\,y_{{C_{3} }} = l_{1} \cos \theta_{1} + l_{2} \cos \theta_{2} - (l_{3} - d_{3} )\cos \theta_{3} \\ & x_{C4} = l_{1} \sin \theta_{1} + l_{2} \sin \theta_{2} + l_{3} \sin \theta_{3} + (l_{4} - d_{4} )\sin \theta_{4} ,\,y_{{C_{4} }} = l_{1} \cos \theta_{1} + l_{2} \cos \theta_{2} - l_{3} \cos \theta_{3} - (l_{4} - d_{4} )\cos \theta_{4} \\ & x_{{C_{5} }} = d_{5} \sin \theta_{5} ,\,y_{{C_{5} }} = d_{5} \cos \theta_{5} \\ \end{aligned} $$

$$ \begin{aligned} & x_{{C_{6} }} = l_{5} \sin \theta_{5} + d_{6} \sin \theta_{6} ,\,\,\,\,y_{{C_{6} }} = l_{5} \cos \theta_{5} + d_{6} \cos \theta_{6} \\ & x_{{C_{7} }} = l_{5} \sin \theta_{5} + l_{6} \sin \theta_{6} + (l_{7} - d_{7} )\sin \theta_{7} ,\,\,\,\,\,y_{{C_{7} }} = l_{5} \cos \theta_{5} + l_{6} \cos \theta_{6} - (l_{7} - d_{7} )\cos \theta_{7} \\ & x_{{C_{8} }} = l_{5} \sin \theta_{5} + l_{6} \sin \theta_{6} + l_{7} \sin \theta_{7} + (l_{8} - d_{8} )\sin \theta_{8} ,\,\,\,\,\,y_{{C_{8} }} = l_{5} \cos \theta_{5} + l_{6} \cos \theta_{6} - l_{7} \cos \theta_{7} - (l_{8} - d_{8} )\cos \theta_{8} \\ & x_{{C_{9} }} = l_{1} \sin \theta_{1} + l_{2} \sin \theta_{2} + l_{9} \cos \theta_{9} ,\,\,\,\,\,y_{{C_{9} }} = l_{1} \cos \theta_{1} + l_{2} \cos \theta_{2} + l_{9} \sin \theta_{9} \\ \end{aligned} $$

where $l_{i}$ and $d_{i}$($i = 1, \ldots ,9$) are the length and the half of the length of the ith link, respectively, and $x_{{c_{i} }}$ and $y_{{c_{i} }}$ are the x–y coordinate of the center of mass of the ith link, respectively.