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adaptive fuzzy sliding mode control

Authors: Seonghyeon Jo([email protected])

Date: Jan 20, 2022

This repository is a MATLAB simulation of adaptive fuzzy sliding mode control for robot manipulator. The robot manipulator uses sawyer 4-dof manipulator and prototype 3-dof manipulator. we compare to simple PID Controller, Sliding Mode Control(SMC), Fuzzy Silding Mode Control(FSMC), Adaptive Fuzzy Sliding Mode Control(AFSMC).

"Adaptive fuzzy sliding mode control using supervisory fuzzy control for 3 DOF planar robot manipulators."

Description of robot manipulator model

In general, the dynamic of a robotic manipulator can be expressed as follows: \M(\q)\ddot{\q} + \C(\q,\dot{\q})\dot{\q}+\G(\q)+\F(\dot{\q}) =\boldsymbol\tau + \boldsymbol\tau_d where \q, \dot{\q}, \ddot{\q}, \boldsymbol\tau, \boldsymbol\tau_d \in \mathcal{R}^{n} denote the joint position vector, velocity vector, acceleration, the vector of control torques, and the vector of disturbance torques, respectively, and \M\in\mathcal{R}^{n\times n} denotes the symmetric positive definite inertia matrix, \C \in \mathcal{R}^{n\times n} denotes the Coriolis matrix, \G \in \mathcal{R}^{n} denotes the gravity vector and \F \in \mathcal{R}^{n} is the friction matrix.

Multiplying $M^{-1}(q)$ by both sides in Equation (1), we have

\ddot{\q}=-\M^{-1}(\q)(\C(\q,\dot{\q})\dot{\q}+\G(\q)+\F(\dot{\q})-\boldsymbol\tau-\boldsymbol\tau_d )

The sliding proportional-integral-derivative PID surface in the space of tracking error can be defined as:

\begin{align*} \s(t)&=K_{p}\e(t)+K_{i}\int{\e}(\tau)d\tau+K_{d}\dot{\e}(t) \end{align*}

where kp is positive proportional gain matrix, ki is positive integral gain matrix, kd is positive derivative gain.

Take the derivative of sliding surface with respect to time, then

\begin{align*} \dot{\s}(t) &= K_p \dot{\e}(t) + K_i \e(t)+ K_{d}\ddot{\e}(t)\\ &= K_p \dot{\e}(t) + K_i \e(t)+K_d (\ddot{\q}_d+\M^{-1}(\q)(\C(\q,\dot{\q})\dot{\q}+\G(\q)+\F(\dot{\q})-\boldsymbol\tau-\boldsymbol\tau_d)\\ \end{align*}

The control effort being derived as the solution of s˙(t) = 0 without considering uncertainty (d(t) = 0) is to achieve the desired performance under nominal model, and it is referred to as equivalent control effort , represented by ueq(t)

\begin{align*} \u_{eq}= (K_d\M^{-1}(\q))^{-1}(K_p \dot{\e} + K_i \e+K_d (\ddot{\q}_d+\M^{-1}(\q)(\C(\q,\dot{\q})\dot{\q}+\G(\q)+\F(\dot{\q}))) \end{align*}

sawyer 4-dof manipulator

RMSE

Method joint 1 joint 2 joint 3 joint 4 sum
PID 0.0394 0.0339 0.0044 0.0083 0.0860
SMC-SIGN 0.0078 0.0349 0.0003 0.0039 0.0469
SMC-SAT 0.0078 0.0358 0.0004 0.0019 0.0460
FSMC 0.0111 0.0146 0.0017 0.0043 0.0317
AFSMC 0.0105 0.0141 0.0013 0.0042 0.0301

Figure

pid controller

joint_input_result_1 joint_input_result_1 joint_input_result_1

smc using sign function

joint_input_result_1 joint_input_result_1 joint_input_result_1

smc using sat function

joint_input_result_1 joint_input_result_1 joint_input_result_1

smc using sat function

joint_input_result_1 joint_input_result_1 joint_input_result_1

fsmc

joint_input_result_1 joint_input_result_1 joint_input_result_1

afsmc

joint_input_result_1 joint_input_result_1 joint_input_result_1