## 4 Pre-Filter Acceleration Feedback Enhanced PID Controller  
### Pre-Filter Acceleration Feedback Enhanced Control Strategy  
The modules of the reconfigurable SCARA robot are connected by bolt contacts, which will inevitably generate a slight gap at the joint. When the robot operates at high speed, the gap will greatly amplify the disturbance, which makes the control of the robot have high nonlinearity and strong hysteresis. Meanwhile, this system is a multi-input single-output complex system, the input of which includes stimulus signals, disturbance signal and so on. Therefore, a new pre-filter AFE control strategy was proposed to achieve high robustness.  
According to the Euler-Lagrange equation, the dynamic equation of a single joint of the reconfigurable SCARA robot can be expressed as Equation (2).  
\[\dot{q}=\tau+\triangle-C\big{(}q,\dot{q}\big{)}.\dot{q}-G(q)-F\big{(}q,\dot{q} \big{)} \tag{2}\]  
where \(J\) is the inertia coefficient, \(q\) is the position vector, \(\tau\) is the input torque, \(\Delta\) is the disturbance torque, \(C\big{(}q,\dot{q}\big{)}.\dot{q}\) is the Coriolis force, \(G\) (\(q\)) is the static torque, and \(F\big{(}q,\dot{q}\big{)}\) is the friction torque.  
Then the pre-filter is integrated into the classic AFE control strategy, and pre-filter AFE control strategy is shown in Figure 5.  
High-gain acceleration feedback can be expressed as Equation (3). According to Equations (2)-(4) can be obtained.  
\[\tau=k\big{(}\dot{v}-\ddot{q}\big{)} \tag{3}\]  
\[\left\{\begin{array}{c}\tau=\frac{k}{1+\dot{k}}\dot{v}+\frac{k}{J+k}M_{q}- \frac{k}{J+k}\triangle\\ M_{q}=C\big{(}q,\dot{q}\big{)}.\dot{q}+G(q)+F\big{(}q,\dot{q}\big{)}\end{array}\right. \tag{4}\]  
where \(k\) is the gain constant and \(\dot{v}\) is the output of the upper controller. The system structure Equation (5) can be obtained by introducing the pre-filter \(B(s)/A(s)\),  
\[\tau=\frac{k}{J+k}\dot{v}+\frac{k}{J+k}\frac{B(s)}{A(s)}M_{q}-\frac{k}{J+k} \frac{B(s)}{A(s)}\triangle \tag{5}\]  
Figure 5: Pre-filter acceleration feedback enhanced control strategy.  
where \(B(s)/A(s)\) is the transfer function of the pre-filter. \(B(s)\) and \(A(s)\) are the Laplace transform of output and input respectively. According to Equations (5) and (2), the dynamic equation of a single rotational joint can be re-expressed as Equation (6),  
\[\bar{Jq}=\frac{1}{\frac{1}{k}+\frac{1}{l}}\dot{v}+\frac{B(s)-\left(\frac{1}{k}+1 \right)A(s)}{\left(\frac{1}{k}+1\right)A(s)}M_{q}-\frac{B(s)-\left(\frac{1}{k}+ 1\right)A(s)}{\left(\frac{l}{k}+1\right)A(s)}\triangle \tag{6}\]  
It can be seen that the gain constant \(k\) exists in the reciprocal. When the gain constant \(k\) is larger, the anti-disturbance ability of the control system is stronger. Let \(k\rightarrow+\infty\), Equation (7) can be approximately obtained,  
\[\bar{Jq}=\bar{Jv}+\frac{B(s)-A(s)}{A(s)}M_{q}-\frac{B(s)-A(s)}{A(s)}\triangle \tag{7}\]  
According to the system Equation (6), it can be seen that the disturbance torque needs to pass through a filter before disturbing the control system. Therefore, the pre-filters \(A(s)/B(s)\) play an important role in suppressing the system disturbance, while the gain constant \(k\) is transformed into an intermediate parameter, which does not appear in other aspects of the controller. Moreover, according to the swing experiment, the disturbance frequency range of the load (hanging plate weight) of the reconfigurable SCARA robot is 0-0.3 Hz, which is the low frequency range. Therefore, the high-pass filter \(L(s)\) was used to suppress low-frequency disturbances. The pre-filters can be expressed as Equation (8),  
\[\frac{B(s)}{A(s)}=\frac{c}{s+c},L(s)=\frac{B(s)-A(s)}{A(s)}=\frac{s}{s+c} \tag{8}\]  
where \(c\) is the positive constant. According to Equations (7) and (8), the control system can be expressed as Equation (9),  
\[\bar{Jq}=\bar{Jv}+\frac{s}{s+c}M_{q}-\frac{s}{s+c}\triangle \tag{9}\]