# Rapid and Precise Zoom Lens Design Based on Voice Coil Motors with Tunnel Magnetoresistance Sensors
## 3 Rapid and Precise Zoom System Design Example
### Movement Trajectory Planning  
For the optical system shown in Figure 4, the maximum travel distances of the two zoom lens groups are 38 mm and 26 mm, respectively, with weights of 30 g and 16 g. In this context, it is necessary to plan the trajectories for both zoom lens groups individually. The trajectory planning design principles emphasize smooth acceleration and deceleration throughout the motion process, with no sudden changes in acceleration and maintaining a certain stabilization time.  
The zoom lens group with the maximum travel distance and the heaviest load is presented as an example to illustrate the movement trajectory planning design process. The current travel distance of Zoom Group 1 is 38 mm, and it is required to stabilize within 0.3 s. To ensure system stability, a third-order symmetrical S-curve is used for the trajectory planning of the zoom group. It controls the rate of change of the acceleration and velocity, achieving smooth acceleration transitions [22]. S-curve trajectory planning avoids sudden changes in acceleration, reducing mechanical shocks and vibrations, thereby enhancing the system's stability.  
The design involves the following:  
1. Increasing acceleration phase for 0.05 s;
2. Decreasing acceleration phase for 0.05 s;
3. Increasing deceleration phase for 0.05 s;
4. Decreasing deceleration phase for 0.05 s.  
During the motion process, the rate of change of the acceleration remains constant. Let the absolute value be \(J_{0}\); then, in each motion phase, the rates of change of acceleration \(J(t)\), acceleration \(a(t)\), velocity \(v(t)\), and displacement \(s(t)\) over time are as given by Equations (1)-(4).  
Increasing acceleration phase, \(0\leq t\leq 0.05\) s:  
\[\left\{\begin{array}{l}J(t)=J_{0}\\ a(t)=J_{0}t\\ v(t)=\frac{1}{2}J_{0}t^{2}\\ s(t)=\frac{1}{6}J_{0}t^{3}\end{array}\right. \tag{1}\]  
Figure 4: The zoom optical system layout. Different colored lines represent light rays at different field angles.  
Decreasing acceleration phase, 0.05 s \(\leq t\leq\) 0.1 s:  
\[\left\{\begin{array}{c}J(t)=-J_{0}\\ a(t)=0.05J_{0}-J_{0}(t-0.05)\\ v(t)=\frac{1}{2}J_{0}\times\ 0.05^{2}+0.05\times J_{0}(t-0.05)-\frac{1}{2}J_{0}(t-0.05) ^{2}\\ s(t)=\frac{1}{6}J_{0}\times\ 0.05^{3}+\frac{1}{2}J_{0}\times\ 0.05^{2}\times(t-0.05) +\frac{1}{2}J_{0}\times 0.05\times(t-0.05)^{2}-\frac{1}{6}J_{0}(t-0.05)^{3} \end{array}\right. \tag{2}\]  
Increasing deceleration phase, 0.1 s \(\leq t\leq\) 0.15 s:  
\[\left\{\begin{array}{c}J(t)=-J_{0}\\ a(t)=-J_{0}(t-0.1)\\ v(t)=J_{0}\times\ 0.05^{2}-\frac{1}{2}J_{0}(t-0.1)\,^{2}\\ s(t)=J_{0}\times\ 0.05^{3}+J_{0}\times\ 0.05^{2}\times(t-0.1)-\frac{1}{6}J_{0}(t-0. 1)\,^{3}\end{array}\right. \tag{3}\]  
Decreasing deceleration phase, 0.15 s \(\leq t\leq\) 0.2 s:  
\[\left\{\begin{array}{c}J(t)=J_{0}\\ a(t)=-J_{0}\times 0.05+J_{0}(t-0.15)\\ v(t)=\frac{1}{2}J_{0}\times\ 0.05^{2}-0.05\times J_{0}\times(t-0.15)+\frac{1}{2}J_{0}(t-0. 15)\,^{2}\\ s(t)=\frac{5}{6}J_{0}\times\ 0.05^{3}+\frac{1}{2}J_{0}\times\ 0.05^{2}\times(t-0.15)- \frac{1}{2}J_{0}\times 0.05\times(t-0.15)^{2}+\frac{1}{6}J_{0}(t-0.15)\,^{3} \end{array}\right. \tag{4}\]  
In each motion phase, the parameters change over time, as shown in Figure 5. From Figure 5, it can be seen that the acceleration reaches its maximum absolute value of 7.6 m/s\({}^{2}\) at 0.05 s and 0.15 s, and the velocity reaches its maximum value of 0.38 m/s at 0.1 s. The travel distance for Zoom Group 2 is 26 mm. Using the same method for trajectory planning for Zoom Group 2, the maximum acceleration and velocity are found to be 5.2 m/s\({}^{2}\) and 0.26 m/s, respectively.