# Vibration Suppression of Collaborative Robot Servo with Dual Encoders
## 7 Control Principle of Dual Encoder Vibration Suppression
### Closed-Loop Control with Dual Encoder
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Symbol** & **Default Value** & **Unit** & **Adjustment Rang** \\ \hline Position loop proportional gain & K\({}_{\text{pp}}\) & 0.8 & - & 0.5–1.3 \\ Speed loop proportional gain & K\({}_{\text{vp}}\) & 20 & - & 8–40 \\ Speed loop integral time constant & T\({}_{\text{vi}}\) & 0.05 & s & 0.01–2 \\ Current loop proportional gain & K\({}_{\text{fp}}\) & 0.7 & - & 0.5–1.5 \\ Current loop integral time constant & T\({}_{\text{1}}\) & 0.002 & - & 0.0005–0.5 \\ Power device gain coefficient & K\({}_{\text{pwm}}\) & 1 & - & - \\ Switch period & T\({}_{\text{pwm}}\) & \(1\times 10^{-4}\) & s & \(0.6\times 10^{-4}\)–\(1.6\times 10^{-4}\) \\ Deadband time & T\({}_{\text{dt}}\) & \(2\times 10^{-6}\) & s & \(1\times 10^{-6}\)–\(3\times 10^{-6}\) \\ Equivalent resistance of motor & R\({}_{\text{s}}\) & 0.00125 & ohm & - \\ Equivalent inductance of motor & L\({}_{\text{q}}\) & 0.0002 & H & - \\ Number of motor poles & N\({}_{\text{p}}\) & 8 & Pair & - \\ Motor torque constant & K\({}_{\text{t}}\) & 100 & 0 & 0 \\ Torsional stiffness & K\({}_{\text{s}}\) & 394 & Nm/rad & - \\ Motor inertia & J\({}_{1}\) & \(2.46\times 10^{-4}\) & kg\(\cdot\)m\({}^{2}\) & - \\ Load inertia & J\({}_{2}\) & \(1.15\times 10^{-3}\) & kg\(\cdot\)m\({}^{2}\) & - \\ Current sampling gain coefficient & Q\({}_{\text{l}}\) & 1 & - & - \\ Current sampling delay & T\({}_{\text{si}}\) & \(1\times 10^{-4}\) & s & - \\ Current filtering coefficient & T\({}_{\text{fi}}\) & \(6\times 10^{-4}\) & s & - \\ Speed sampling gain coefficient & Q\({}_{\text{v}}\) & 1 & - & - \\ Speed sampling delay time & T\({}_{\text{sv}}\) & \(1\times\text{T}_{\text{pwm}}\) & cycle & - \\ Speed filtering coefficient & T\({}_{\text{fv}}\) & \(5\times\text{T}_{\text{pwm}}\) & cycle & - \\ Position sampling gain coefficient & Q\({}_{\text{p}}\) & 1 & - & - \\ Position sampling delay time & T\({}_{\text{sp}}\) & \(1\times\text{T}_{\text{pwm}}\) & cycle & - \\ Position filtering coefficient & T\({}_{\text{fp}}\) & 0 & cycle & - \\ Sampling ratio coefficient of encoder on & k1 & 0.7 & - & 0–1 \\ the motor side & & & & \\ Sampling ratio coefficient of encoder in & k2 & 0.3 & - & 0–1 \\ the reducer side & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Transfer function-related parameters. and the load-side speed sampling weight k2 are automatically matched, and once identified, these values are automatically saved. The specific optimization procedure implemented in an actual collaborative robot is described as follows: Joint 2 was programmed to execute a trajectory ranging from \(-180^{\circ}\) to \(+180^{\circ}\). Considering the pronounced vibration at low speeds, the maximum rotational speed was constrained to 500 RPM, with both acceleration and deceleration times set to a standard value of 500 ms. The optimization process focused on determining the optimal motor-side speed sampling weight (k1) and load-side speed sampling weight (k2), with both parameters varying within the range of [0, 1.0]. The adjustment resolution was set to 0.05 for both k1 and k2. A systematic parameter space exploration was conducted by scanning through all possible combinations of k1 and k2 values. For each parameter pair, the absolute deviation between the reducer-side speed and the target speed was recorded and accumulated to calculate the total deviation metric (\(\Sigma\)). Through comparative analysis of all obtained \(\Sigma\) values, the parameter corresponding to the minimum \(\Sigma\) was identified as the optimal weight configuration.