## 4 Optimal Design  
### Optimization Objective  
It is assumed that the motorized spindle has \(L\) stress levels under stress \(S_{i}(i=1,\ldots,I)\), the \(k\)-th performance test time of the \(j\) sample under group \(l\) stress is \(t_{ijk}(l=1,2,\ldots,L\), \(L\) is the number of stress levels; \(j=1,2,\ldots,n_{l}\), \(n_{l}\) is the sample size under the stress level \(l\); \(k=1,2,\ldots,n_{l}\), \(m_{l}\) is the inspection times under the stress level \(l\)), the detected performance value is \(y_{ljk}\). Brownian motion is a Gaussian process. Therefore, the performance increment \(\Delta y=y_{ljk}-y_{l|j(k-1)}\) is independent and follows normal distribution. According to Equation (13), the probability density function of independent increment is as follows:  
\[f(x)=\frac{1}{\sigma\sqrt{2\pi\Delta t}}\exp\left\{-\frac{[x-\mu\times\Delta t ]^{2}}{2\sigma^{2}\times\Delta t}\right\} \tag{14}\]  
The logarithmic likelihood function of Equation (14) is shown in Equation (15):  
\[\ln p(x|\theta)\propto-\frac{1}{2}\sum_{l=1}^{L}\sum_{j=1}^{n_{l}}\sum_{k=1} ^{m_{l}}\left\{[\ln(2\pi\Delta t)+\ln(\sigma^{2})]+\frac{[\chi_{ljk}-\mu_{k} \times\Delta t]^{2}}{\sigma^{2}\Delta t}\right\} \tag{15}\]  
where \(\theta=\gamma_{0},\gamma_{1},\ldots,\gamma_{I}\). For SSADT of the motorized spindle, the sample size under each stress combination is the same, \(n_{l}=(1/l)\times n\) (\(n\) is the total number of samples). \(D_{s}\)-optimality is the generalization of \(D\)-optimality, which is an information-based optimization method. For the model with more parameters, it focuses on the evaluation accuracy of important parameters. In other words, only some important parameters in \(\theta\) are estimated. Therefore, Equation (6) is divided into two parts:  
\[\ln(\mu)=g_{1}(x)^{\prime}\theta_{1}+g_{2}(x)^{\prime}\theta_{2} \tag{16}\]  
where \(\theta_{1}\) contains \(s\) important parameters (\(\theta_{1}=\gamma_{1},\ldots,\gamma_{s}\)), \(\theta_{2}\) contains \((L-s)\) unimportant parameters (\(\theta_{2}=\gamma_{s+1},\ldots,\gamma_{t},\gamma_{0}\)). Therefore, \(g_{1}(x)^{\prime}=(\ln S_{11},\ldots,\ln S_{sl})\), \(g_{2}(x)^{\prime}=(\ln S_{s+1,1},\ldots,\ln S_{IL},1)\). Based on \(D_{s}\)-optimality, the information matrix \(M_{\xi}(M_{\xi}=\det[F(\theta)])\) is divided into four parts  
\[M_{\xi}=\left[ \begin{array}{cc}M_{11}(\xi)&M_{12}(\xi)\\ M_{21}(\xi)&M_{22}(\xi)\end{array} \right] \tag{17}\]  
where \(\xi\) is the selected test scheme. According to the concept of the information matrix [36], we can get the following conclusions:  
\[M_{\xi}=G^{\prime}G/L \tag{18}\]  
where \(G=\left[ \begin{array}{ccc}\varphi_{11}&\ldots&\varphi_{\rm IL}\\ \vdots&&\vdots\\ \varphi_{\rm II}&\ldots&\varphi_{\rm IL}\end{array} \right]\). Therefore:  
\[\mathrm{M}_{\xi}=\left[ \begin{array}{ccc}\mathrm{E}\Big{(}-\frac{\partial^ {2}lnp(x|\theta)}{\partial^{2}\gamma_{1}}\Big{)}&\ldots&\mathrm{E}\Big{(}- \frac{\partial^{2}lnp(x|\theta)}{\partial\gamma_{1}\partial\gamma_{0}}\Big{)} &\mathrm{E}\Big{(}-\frac{\partial^{2}lnp(x|\theta)}{\partial\gamma_{1} \partial\sigma^{2}}\Big{)}\\ \mathrm{E}\Big{(}-\frac{\partial^{2}lnp(x|\theta)}{\partial\gamma_{2}\partial \gamma_{1}}\Big{)}&\mathrm{E}\Big{(}-\frac{\partial^{2}lnp(x|\theta)}{ \partial\gamma_{2}\partial\gamma_{0}}\Big{)}&\mathrm{E}\Big{(}-\frac{\partial ^{2}lnp(x|\theta)}{\partial\gamma_{2}\partial\sigma^{2}}\Big{)}\\ \vdots&\ddots&\vdots\\ \mathrm{E}\Big{(}-\frac{\partial^{2}lnp(x|\theta)}{\partial\gamma_{0}\partial \gamma_{1}}\Big{)}&\vdots&\mathrm{E}\Big{(}-\frac{\partial^{2}lnp(x|\theta)}{ \partial\gamma_{0}\partial\sigma^{2}}\Big{)}&\mathrm{E}\Big{(}-\frac{\partial ^{2}lnp(x|\theta)}{\partial\sigma^{2}}\Big{)}\\ \mathrm{E}\Big{(}-\frac{\partial^{2}lnp(x|\theta)}{\partial\sigma^{2}\partial \gamma_{1}}\Big{)}&\mathrm{E}\Big{(}-\frac{\partial^{2}lnp(x|\theta)}{ \partial\sigma^{2}\partial\gamma_{0}}\Big{)}&\mathrm{E}\Big{(}-\frac{\partial ^{2}lnp(x|\theta)}{(\partial\sigma^{2})^{2}}\Big{)}\end{array}