# Ground Simulation Tests in Two-Dimensional Dynamic Acceleration Environment
## 4 Synchronous Loading of 2-D Dynamic AE
### Vector Axial Angular Displacement Motion Trajectory Planning  
Hermite interpolation can be used to eliminate the issues that arise when the interpolation conditions contain derivatives. Therefore, the motions of the axial positions of the unit being tested are fitted using Hermite interpolation polynomials, which can guarantee that the displacement and velocity at the interpolation points are equal to those of the ideal motion. Each interpolation segment of the PVT motion mode is fitted with a two-point cubic Hermite interpolation polynomial. Only by  
Figure 12: Position compound controller based on feed-forward proportional-integral-derivative (PID).  
specifying the position, velocity and running time of the segment and calculating the motion trajectory that meets the constraint conditions, the more compact control of the trajectory can be realized.  
The angular displacement motions of the unit being tested shown in Figure 13 are divided into six segments. Three cubic Hermite interpolations in the different intervals are used to ensure the following point: the function values (displacement) and the first derivative values (velocity) at the interpolation points are exactly the same as the values of the motion trajectory. The velocity curve can be obtained by taking the derivative of the displacement curve. The cubic Hermite interpolation function corresponding to the first segment of the curve is calculated as follows:  
\[\left\{\begin{array}{l}P(x)=f(x_{0})h_{0}(x)+f(x_{1})h_{1}(x)+f^{\prime}(x_ {0})H_{0}(x)+f^{\prime}(x_{1})H_{1}(x)\\ h_{0}(x)=(1+2\frac{x-x_{0}}{x_{1}-x_{0}})(\frac{x-x_{1}}{x_{0}-x_{1}})^{2}\\ h_{1}(x)=(1+2\frac{x-x_{1}}{x_{0}-x_{1}})(\frac{x-x_{0}}{x_{0}-x_{1}})^{2}\\ H_{0}(x)=(x-x_{0})(\frac{x-x_{1}}{x_{0}-x_{1}})^{2}\\ H_{1}(x)=(x-x_{1})(\frac{x-x_{0}}{x_{1}-x_{0}})^{2}\\ x_{0}=0,f(x_{0})=0,f^{\prime}(x_{0})=0\\ x_{1}=1,f(x_{1})=6,f^{\prime}(x_{1})=-5.2\end{array}\right. \tag{16}\]  
where h\({}_{0}\)(x), h\({}_{1}\)(x), H\({}_{0}\)(x), and H\({}_{1}\)(x) are the interpolation basis functions.  
The cubic Hermitian interpolation polynomial is as follows:  
\[p(x)=6x^{2}(3-2x)-5.2x^{2}(x-1)=-17.2x^{3}+23.2x^{2},\ x\in[0,1] \tag{17}\]  
Similarly, the cubic Hermitian interpolation polynomials corresponding to the other five segments can be obtained. The displacement curve after interpolation is depicted in Figure 14.  
Figure 13: Angular displacement curve of the unit being tested.  
The cubic Hermitian interpolation must extract the first-order derivative of the motion curve at the interpolation point, which is smoother than the motion trajectory of the cubic spline interpolation and is suitable for S-shaped acceleration/deceleration controls of high-speed machining. As shown in Figure 11, the impact load is achieved in 0.12 s. The launch AE requires that the axial acceleration of the unit to be increased from zero \(g\) to 20 \(g\) in 0.12 s, and the radial acceleration is zero \(g\). To simulate this AE, the initial position of the unit is vertical with the arm. During the acceleration, the servo motor controls the vector shaft rotation angle so that the axial direction of the unit is along the resultant acceleration direction of the arm. The shaft angular displacement-tracking curves based on the PVT mode are plotted in Figure 15. As can be observed, the unit is rotated \(90^{\circ}\) from the initial position in a pre-planned trajectory within 0.12 s. When the revolution speed of the arm reaches a steady state, the axial direction of the unit is along the normal direction of the arm, the axial acceleration is \(\mathrm{r}\omega^{2}\), and the radial acceleration of the unit is maintained at \(\mathrm{r}\mathrm{d}\omega/\mathrm{d}t=0\)\(g\).