## 3 Force-Controlled End-Effector
### Controller Design  
In the process of MRF, the components of MRF fluid change dynamically, mainly temperature and moisture. Generally, under the compensation function of the circulation system during the process, the fluctuation range of moisture is 0.5% and the fluctuation range of temperature is within 1 \({}^{\circ}\)C. Hence, for the specified MRF fluid and material of the workpiece, the stiffness and damping of the constant force end-effector system will be affected by the dynamical variation of MRF fluid parameters, and thus the system cannot be accurately modeled. Therefore, the ideal controller should be insensitive to the changes in  
Figure 5: The system structure of force-controlled end-effector.  
system parameters and has strong anti-interference and robustness. The active disturbance rejection control (ADRC) first proposed by [PERSON] [25] can estimate and compensate the total disturbances in real-time; its adaptive range of parameters is large. Taking the normal polishing force \(F_{n}\) and its differential signal as state variables \(x_{1}\) and \(x_{2}\), the system parameters affected by the change of MRF Fluid state are considered as the parts of the unknown external disturbance \(\omega(t)\). The constant force actuator system model is described as \(f(x_{1},x_{2},\omega(t),t)\), which is a multivariable function of both the states, external disturbances, and time \(t\). Treating \(f(x_{1},x_{2},\omega(t),t)\) as an additional state variable \(x_{3}\), and let \(\dot{x_{3}}=Z(t)\), the system state is described as:  
\[\left\{\begin{array}{c}\dot{x_{1}}=x_{2}\\ \dot{x_{2}}=x_{3}+bu\\ \dot{x_{3}}=Z(t)\\ y=x_{1}\end{array}\right., \tag{6}\]  
where \(y\) is the output, measured and to be controlled, and \(u\) is the input control signal. With \(f(x_{1},x_{2},\omega(t),t)\) unknown, ADRC provides an extended state observer (ESO) to estimate and reject the disturbances of the system. A differential tracker (TD) is used to provide the accurate tracking of the input signal, and the contradiction between overshoot and response speed is solved by arranging a transient process. Different from PID with a linear control law, TD with a nonlinear feedback law is designed to achieve zero error. Combining all of the above, the ADRC takes the form as shown in Equation (7) and the structure of ADRC is shown in Figure 6.  
\[\left\{\begin{array}{c}fh=fhan(x_{1}-x_{0},x_{2},r_{0},h_{0})\\ x_{1}=x_{1}+hx_{2}\\ x_{2}=x_{2}+hfh\\ \overline{x}=x_{1}+h_{1}x_{2}\\ fal(e,a,b)=\left\{\begin{array}{c}|e|^{a}sign(e),|e|>b\\ \frac{e}{p_{r}},|e|\leq b\\ e=z_{1}-y_{0}\\ fe=fal(e,0.5,h)\\ fe_{1}=fal(e,0.25,h)\\ z_{1}=z_{1}+h(z_{2}-\beta_{01}e)\\ z_{2}=z_{2}+h(z_{3}-\beta_{02}fe)\\ z_{3}=z_{3}+h(-\beta_{03}fe_{1})\\ e_{1}=x_{1}-z_{1}\\ e_{2}=x_{2}-z_{2}\\ u_{0}=fhan(e_{1},ce_{2},r_{0},h_{1})\\ u=\frac{u_{0}-z_{3}}{b_{0}}\end{array}\right., \tag{7}\]  
where \(fh\) is a time-optimal solution that guarantees the fastest convergence from the current state to the desired state without any overshoot; \(r_{0}\) is the parameter selected accordingly to speed up or slow down the transient process; \(h\) is the sampling period; \(h_{0}\) is the step size employed to restrain noise amplification of the differential signal, usually it is slightly larger than \(h\); \(h_{1}\) is the precision coefficient that determines the aggressiveness of the control loop and it is usually a multiple of the sampling period \(h\); \(\overline{x}\) is the value of forecast time \(h_{1}\) forward according to differential signal \(x_{2}\) on the basis of value \(x_{1}\); \(e\) is the difference between tracking signal \(z_{1}\) of \(x_{1}\) and the system output \(y_{0}\); \(z_{2}\) is the tracking signal of \(x_{2}\), \(z_{3}\) and \(b_{0}\) are parameters with disturbance compensation; \(\beta_{01},\ \beta_{02},\ \beta_{03}\) are the observer gains; \(c\) is a fine-tuning adjustment parameter; \(x_{0}\) is the input signal; and \(u\) is the control signal.