Awesome

adaptive fuzzy sliding mode control

Authors: Seonghyeon Jo(cpsc.seonghyeon@gmail.com)

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: <img src= "https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5CM%28%5Cq%29%5Cddot%7B%5Cq%7D+%2B+%5CC%28%5Cq%2C%5Cdot%7B%5Cq%7D%29%5Cdot%7B%5Cq%7D%2B%5CG%28%5Cq%29%2B%5CF%28%5Cdot%7B%5Cq%7D%29+%3D%5Cboldsymbol%5Ctau+%2B+%5Cboldsymbol%5Ctau_d" alt="\M(\q)\ddot{\q} + \C(\q,\dot{\q})\dot{\q}+\G(\q)+\F(\dot{\q}) =\boldsymbol\tau + \boldsymbol\tau_d"> where <img src= "https://render.githubusercontent.com/render/math?math=%5Ctextstyle++%5Cq%2C+%5Cdot%7B%5Cq%7D%2C+%5Cddot%7B%5Cq%7D%2C+%5Cboldsymbol%5Ctau%2C+%5Cboldsymbol%5Ctau_d+%5Cin+%5Cmathcal%7BR%7D%5E%7Bn%7D" alt=" \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 <img src= "https://render.githubusercontent.com/render/math?math=%5Ctextstyle+%5CM%5Cin%5Cmathcal%7BR%7D%5E%7Bn%5Ctimes+n%7D" alt="\M\in\mathcal{R}^{n\times n}"> denotes the symmetric positive definite inertia matrix, <img src= "https://render.githubusercontent.com/render/math?math=%5Ctextstyle+%5CC+%5Cin+%5Cmathcal%7BR%7D%5E%7Bn%5Ctimes+n%7D" alt="\C \in \mathcal{R}^{n\times n}"> denotes the Coriolis matrix, <img src= "https://render.githubusercontent.com/render/math?math=%5Ctextstyle+%5CG+%5Cin+%5Cmathcal%7BR%7D%5E%7Bn%7D" alt="\G \in \mathcal{R}^{n}"> denotes the gravity vector and <img src= "https://render.githubusercontent.com/render/math?math=%5Ctextstyle+%5CF+%5Cin+%5Cmathcal%7BR%7D%5E%7Bn%7D" alt="\F \in \mathcal{R}^{n}"> is the friction matrix.

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

<img src= "https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cddot%7B%5Cq%7D%3D-%5CM%5E%7B-1%7D%28%5Cq%29%28%5CC%28%5Cq%2C%5Cdot%7B%5Cq%7D%29%5Cdot%7B%5Cq%7D%2B%5CG%28%5Cq%29%2B%5CF%28%5Cdot%7B%5Cq%7D%29-%5Cboldsymbol%5Ctau-%5Cboldsymbol%5Ctau_d+%29" alt="\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:

<img src= "https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cs%28t%29%26%3DK_%7Bp%7D%5Ce%28t%29%2BK_%7Bi%7D%5Cint%7B%5Ce%7D%28%5Ctau%29d%5Ctau%2BK_%7Bd%7D%5Cdot%7B%5Ce%7D%28t%29%0A%5Cend%7Balign%2A%7D" alt="\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

<img src= "https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cdot%7B%5Cs%7D%28t%29+%26%3D+K_p+%5Cdot%7B%5Ce%7D%28t%29+%2B+K_i+%5Ce%28t%29%2B+K_%7Bd%7D%5Cddot%7B%5Ce%7D%28t%29%5C%5C%0A%26%3D+K_p+%5Cdot%7B%5Ce%7D%28t%29+%2B+K_i+%5Ce%28t%29%2BK_d+%28%5Cddot%7B%5Cq%7D_d%2B%5CM%5E%7B-1%7D%28%5Cq%29%28%5CC%28%5Cq%2C%5Cdot%7B%5Cq%7D%29%5Cdot%7B%5Cq%7D%2B%5CG%28%5Cq%29%2B%5CF%28%5Cdot%7B%5Cq%7D%29-%5Cboldsymbol%5Ctau-%5Cboldsymbol%5Ctau_d%29%5C%5C%0A%5Cend%7Balign%2A%7D" alt="\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)

<img src= "https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cu_%7Beq%7D%3D+%28K_d%5CM%5E%7B-1%7D%28%5Cq%29%29%5E%7B-1%7D%28K_p+%5Cdot%7B%5Ce%7D+%2B+K_i+%5Ce%2BK_d+%28%5Cddot%7B%5Cq%7D_d%2B%5CM%5E%7B-1%7D%28%5Cq%29%28%5CC%28%5Cq%2C%5Cdot%7B%5Cq%7D%29%5Cdot%7B%5Cq%7D%2B%5CG%28%5Cq%29%2B%5CF%28%5Cdot%7B%5Cq%7D%29%29%29%0A%5Cend%7Balign%2A%7D" alt="\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

Figure

pid controller

<img src="./sawyer-4dof-manipulator/fig/joint_position_result_1.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/joint_input_result_1.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/position_error_result_1.png" alt="joint_input_result_1" width="220" />

smc using sign function

<img src="./sawyer-4dof-manipulator/fig/joint_position_result_2.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/joint_input_result_2.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/position_error_result_2.png" alt="joint_input_result_1" width="220" />

smc using sat function

<img src="./sawyer-4dof-manipulator/fig/joint_position_result_2.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/joint_input_result_2.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/position_error_result_2.png" alt="joint_input_result_1" width="220" />

smc using sat function

<img src="./sawyer-4dof-manipulator/fig/joint_position_result_3.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/joint_input_result_3.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/position_error_result_3.png" alt="joint_input_result_1" width="220" />

fsmc

<img src="./sawyer-4dof-manipulator/fig/joint_position_result_4.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/joint_input_result_4.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/position_error_result_4.png" alt="joint_input_result_1" width="220" />

afsmc

<img src="./sawyer-4dof-manipulator/fig/joint_position_result_5.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/joint_input_result_5.png" alt="joint_input_result_1" width="220" /> <img src="./sawyer-4dof-manipulator/fig/position_error_result_5.png" alt="joint_input_result_1" width="220" />