# Input Shaping Control of a Flexible Structure for Rest-to-Rest and Non-Rest-to-Rest Maneuvers
## 4 Rest-to-Rest Problem with Conventional ZVDD Input Shaping  
Applying the conventional ZVDD shaped torque profile in Section 3.2 to the flexible two-body system model, it is possible to achieve a near-rest state and significantly reduce the final vibration. Let the time of the rotational maneuver be \(t_{f}+\epsilon(t)\), where \(\epsilon(t)\) is the additional time due to the use of input shaping. From Equation (24), the time difference between each impulse will be \(\delta t=\pi/c_{3}\). So, we can write \(t_{1110}^{\prime}=t_{1009}^{\prime}=t_{0908}^{\prime}=t_{0706}^{\prime}=t_{060 5}^{\prime}=t_{0504}^{\prime}=t_{0302}^{\prime}=t_{0201}^{\prime}=t_{0100}^{ \prime}=\delta t=\pi/c_{3}\). Note that \(t_{0807}^{\prime}\) and \(t_{0304}^{\prime}\) will have different values.  
Comparing Equations (25) and (26) to the convoluted torque profile, the convoluted profile is three time increments longer than the single-switch torque profile, so  
\[\epsilon(t)=3\delta t=\frac{3\pi}{c_{3}} \tag{27}\]  
The final value of \(\theta_{1}\) at time \(t_{f}+3\delta t\) can be analytically expressed as follows:  
\[\theta_{1}(t_{f}+3\delta t) =(8c_{3}^{4}\theta_{0}+b_{4}c_{1}u_{2}-2b_{4}c_{1}u_{2}\cos(t_{21 }c_{3})+b_{4}c_{1}u_{2}\cos(2t_{21}c_{3})+3b_{4}c_{1}u_{2}\cos(c_{3}\delta t)\] \[+3b_{4}c_{1}u_{2}\cos(2c_{3}\delta t)+b_{4}c_{1}u_{2}\cos(3c_{3} \delta t)-6b_{4}c_{1}u_{2}\cos(c_{3}(t_{21}+2\delta t))+3b_{4}c_{1}u_{2}\cos( c_{3}(2t_{21}+\delta t))\] \[-2b_{4}c_{1}u_{2}\cos(c_{3}(t_{21}+3\delta t))+b_{4}c_{1}u_{2} \cos(c_{3}(2t_{21}+3\delta t))+8t_{21}^{2}b_{2}c_{3}^{4}u_{2}-6b_{4}c_{1}u_{2} \cos(c_{3}(t_{21}+\delta t))\] \[+3b_{4}c_{1}u_{2}\cos(2c_{3}(t_{21}+\delta t))+8t_{21}^{2}b_{4}c _{1}c_{3}^{2}u_{2})/(8c_{3}^{4}) \tag{28}\]  
Equation (28) can be further simplified and written as follows:  
\[\theta_{1}(t_{f}+3\delta t)=\theta_{0}+t_{21}^{2}b_{2}u_{2}+(t_{21}^{2}b_{4}c _{1}u_{2})/c_{3}^{2} \tag{29}\]  
Therefore, if \(t_{21}\) is chosen such that  
\[t_{21}=\sqrt{-\theta_{0}/(u_{2}(b_{2}+b_{4}c_{1}/c_{3}^{2}))} \tag{30}\]  
then \(\theta_{1}(t_{f}+3\delta t)=0\).  
The remaining states at time \(t_{f}+3\delta t\) take the following form:  
\[\dot{\theta}_{1}(t_{f}+3\delta t) = -b_{4}c_{1}u_{2}(\sin(c_{3}(2t_{21}+3\delta t))-6\sin(c_{3}(t_{2 1}+\delta t))+3\sin(2c_{3}(t_{21}+\delta t)) \tag{31}\] \[-2\sin(t_{21}c_{3})+\sin(2t_{21}c_{3})+3\sin(c_{3}\delta t)+3\sin (2c_{3}\delta t)+sin(3c_{3}\delta t)-6\sin(c_{3}(t_{21}+2\delta t))\] \[+3\sin(c_{3}(2t_{21}+\delta t))-2\sin(c_{3}(t_{21}+3\delta t)))/(8 c_{3}^{3})\] \[=0\]\[\theta_{2}(t_{f}+3\delta t) = -b_{4}u_{2}(\cos(c_{3}(2t_{21}+3\delta t))-6\cos(c_{3}(t_{21}+\delta t ))+3\cos(2c_{3}(t_{21}+\delta t))\] \[-2\cos(t_{21}c_{3})+\cos(2t_{21}c_{3})+3\cos(c_{3}\delta t)+3\cos( 2c_{3}\delta t)+\cos(3c_{3}\delta t)-6\cos(c_{3}(t_{21}+2\delta t))\] \[+3\cos(c_{3}(2t_{21}+\delta t))-2\cos(c_{3}(t_{21}+3\delta t))+1)/ (8c_{3}^{2})\] \[=0 \tag{32}\] \[\dot{\theta}_{2}(t_{f}+3\delta t) = b_{4}u_{2}(\sin(c_{3}(2t_{21}+3\delta t))-6\sin(c_{3}(t_{21}+ \delta t))+3\sin(2c_{3}(t_{21}+\delta t))-2\sin(t_{21}c_{3})+\sin(2t_{21}c_{3})\] \[+3\sin(c_{3}\delta t)+3\sin(2c_{3}\delta t)+\sin(3c_{3}\delta t)- 6\sin(c_{3}(t_{21}+2\delta t))+3\sin(c_{3}(2t_{21}+\delta t))\] \[-2\sin(c_{3}(t_{21}+3\delta t)))/(8c_{3})\] \[=0\]  
Note that to calculate Equations (28)-(33), a number of trigonometric formulas are used related to Sine and Cosine. Please see Appendix A for more details.  
In summary, Equations (29)-(33) show that the conventional ZVDD input shaping method can bring the two-body spring-mass to the desired rest state, but the method does not greatly reduce vibrations during the rotational/slew maneuver. When the ZVDD shaped torque profile is applied to the rest-to-rest problem defined in Section 2, the resulting state trajectories are shown in Figure 5.