## 4 Predict the Evader Position Two Steps Forward in Time Considering Noisy Environments
### The Relationship Between the Estimate Error and the Controller Performance  
We next show the relationship between the estimate error and the performance of our chasing controller. Usually, locations can be measured rather accurately. Hence, we assume that \(\mathbf{r}_{k}^{\varepsilon}\approx\tilde{\mathbf{r}}_{k}^{\varepsilon}\) in this subsection.  
In this subsection, we assume that \(q=\inf\). This implies that we analyze the relationship between the estimate error and the performance of our chasing controller, not considering the maximum turn rate of the pursuer.  
Due to the estimate error, motion camouflage state (\(\angle(\mathbf{r}_{k}^{p}-\mathbf{r}_{k}^{\varepsilon},\mathbf{r}_{k+1}^{p} -\mathbf{r}_{k+1}^{\varepsilon})\) is zero) cannot be achieved at every time index. We derive the equation for \(\angle(\mathbf{r}_{k}^{p}-\mathbf{r}_{k}^{\varepsilon},\mathbf{r}_{k+1}^{p} -\mathbf{r}_{k+1}^{\varepsilon})\) under our chasing controller.  
Theorem 3.: _The pursuer moves by applying \(\mathbf{u}_{k}\) at every time index \(k\). Consider the case where the pursuer at time index \(k\) cannot access \(\mathbf{r}_{k+1}^{\varepsilon}\) accurately. Let \(\tilde{\mathbf{r}}_{k+1}^{\varepsilon}\) denote an estimate of \(\mathbf{r}_{k+1}^{\varepsilon}\). \(\mathbf{n}_{k+1}=\mathbf{r}_{k+1}^{\varepsilon}-\tilde{\mathbf{r}}_{k+1}^{\varepsilon}\) is the error in the estimate. Let \(\mu_{k+1}\) denote \(\angle(\tilde{\mathbf{r}}_{k+1}^{\varepsilon}-\mathbf{r}_{k+1}^{\varepsilon},\mathbf{r}_{k+1}^{\varepsilon}-\tilde{\mathbf{r}}_{k+1}^{\varepsilon})\) for convenience. \(\sin(\angle(\mathbf{r}_{k}^{p}-\mathbf{r}_{k}^{\varepsilon},\mathbf{r}_{k+1 }^{\varepsilon}-\mathbf{r}_{k+1}^{\varepsilon}))=\frac{\sin(\mu_{k+1})\| \mathbf{n}_{k+1}\|}{d_{k+1}}\). Here, \(d_{k+1}\) is \(\|\mathbf{r}_{k+1}^{p}-\mathbf{r}_{k+1}^{\varepsilon}\|\)._  
Proof.: See Figure 3 for an illustration of \(\mu_{k+1}\). Utilizing the geometry in this figure, we derive  
\[\sin(\angle(\mathbf{r}_{k+1}^{p}-\tilde{\mathbf{r}}_{k+1}^{\varepsilon}, \mathbf{r}_{k+1}^{p}-\mathbf{r}_{k+1}^{\varepsilon}))=\frac{\sin(\mu_{k+1}) \|\mathbf{n}_{k+1}\|}{d_{k+1}}. \tag{35}\]  
Since the pursuer at time index \(k\) can only access \(\tilde{\mathbf{r}}_{k+1}^{\varepsilon}\) instead of \(\mathbf{r}_{k+1}^{\varepsilon}\), we draw \(\bar{L}_{k+1}\) intersecting \(\tilde{\mathbf{r}}_{k+1}^{\varepsilon}\) such that \(\bar{L}_{k+1}\) is parallel to \(L_{k}\). See Figure 3 for an illustration. Utilizing our chasing controller \(\mathbf{u}_{k}\), \(\mathbf{r}_{k+1}^{p}\) is the heading point on \(\bar{L}_{k+1}\). Utilizing the same argument as in Theorem 1, we obtain  
\[\angle(\mathbf{r}_{k+1}^{p}-\tilde{\mathbf{r}}_{k+1}^{\varepsilon},\mathbf{r} _{k}^{p}-\mathbf{r}_{k}^{\varepsilon})=0. \tag{36}\]  
Utilizing (35) and (36), we derive  
\[\sin(\angle(\mathbf{r}_{k}^{p}-\mathbf{r}_{k}^{\varepsilon},\mathbf{r}_{k+1 }^{p}-\mathbf{r}_{k+1}^{\varepsilon}))=\frac{\sin(\mu_{k+1})\|\mathbf{n}_{k+1 }\|}{d_{k+1}}. \tag{37}\]  
Theorem 3 depicts that in the case where \(d_{k+1}\) is too small compared to \(\sin(\mu_{k+1})\|\mathbf{n}_{k+1}\|\), \(\angle(\mathbf{r}_{k+1}^{p}-\mathbf{r}_{k+1}^{\varepsilon},\mathbf{r}_{k}^{p} -\mathbf{r}_{k}^{\varepsilon})\) is large. In other words, the rotation rate of LOS is large. This implies that the pursuer may leave the motion camouflage state as the distance between the evader and the pursuer is too small.