Adaptive Control Lyapunov Function-Based Controller Using Fixed Point Iteration

The Proportional Integral Derivative (PID), the Computed Torque Control (CTC), the Fuzzy Logic (FL), and the Artificial Neural Network (ANN) controllers are some examples of methods widely used in the industrial sectors for control purposes. All of them have been largely developed and investigated, but besides their advantages, they have some drawbacks which have not been completely solved yet. Other types of approaches, especially used in Robotics, are classical adaptive methods such as the Adaptive Inverse Dynamics Control (AIDC), the Adaptive Slotine-Li Control (ASCL), and the Model Reference Adaptive Controller (MRAC). They need a formally accurate robot model and they fail in the presence of unknown external disturbances or friction. This work aims to propose a new Fixed Point Iteration-based (FPI) adaptive controller that applies Control Lyapunov Function for kinematic design, and a multi-dimensional generalization of the Rodrigues formula for adaptation. The first one is derived using the error of the second derivative of the generalized coordinates and tries to move the error to zero exponentially. The second one is used to develop an Adaptive Deformation function to compensate for the inaccurate dynamic model where the new output for the vector of the second derivative of the generalized coordinates is obtained by smooth rotations. The concept was simulated by using Julia code which demonstrated the feasibility of the idea providing after the transient an oscillating error which ranges between -0.005 and 0.004 rad.