# Multi-Layered Interactive Target Guidance with Visual Safety in Convex-Shaped Obstacle Environments
## 3 Safety Layer for Visual Safety
### Visual Safety
Then, the left-hand side of (45) is expanded as  
\[2p_{co}^{T}T(e^{\xi\theta_{ox}}v_{wo}^{b}-v_{wc}^{b}-\hat{\omega}_{wc }^{b}p_{co})\] \[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]0 \[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]1 \[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]2  
Thus, substituting  
\[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]3  
into (48), (45) is transformed into  
\[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]4  
In addition, since the inequality \(\mathbf{e}_{co}^{T}e^{\xi\theta_{ow}}v_{wo}^{b}\geq-\frac{\|p_{co}\|}{2}+ \sqrt{\Delta_{fov}}\) is derived from the condition (43), and \(p_{co}^{T}Te^{-\xi\theta_{ox}}\mathbf{e}_{co}\) is larger than \(\sqrt{\Delta_{fov}}\) in \(\mathcal{D}_{fov}\), \(p_{co}^{T}Te^{-\xi\theta_{ox}}\mathbf{e}_{co}\sqrt{\Delta_{fov}}\geq\Delta_{fov}\) holds. Therefore,  
\[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]5  
is a sufficient condition for (50) in \(\mathcal{D}_{fov}\). Next, let \(\theta_{c,k}\) and \(\theta_{o,k}\) be angles formed by \(\mathbf{e}_{c,k}\), \(\mathbf{e}_{o,k}\), and \(\mathbf{e}_{co}\), respectively. Then, there exist the unit vectors \(\mathbf{u}_{c,k}\) and \(\mathbf{u}_{o,k}\) so that they are orthogonal to \(\mathbf{e}_{co}\) and decompose \(\mathbf{e}_{c,k}\) and \(\mathbf{e}_{o,k}\) into  
\[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]6 \[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]7  
With this decomposition, (46) can be transformed into  
\[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]8 \[V^{b}_{wc}\in K_{fov}\cap K_{occ,k},\;\;\forall k=1,\dots,n_{ \mathcal{O}}. \tag{42b}\]9  
From the above inequality, the discussion proceeds with three cases: (i) \(|\cos\theta_{c,k}|\leq 1-\frac{\gamma}{\|p_{co}\|^{2}}\), (ii) \(\cos\theta_{c,k}\leq-\left(1-\frac{\gamma}{\|p_{co}\|^{2}}\right)\), and (iii) \(\cos\theta_{c,k}\geq 1-\frac{\gamma}{\|p_{co}\|^{2}}\). Note that \(1-\frac{\gamma}{\|p_{co}\|^{2}}>0\) holds due to assumption (44). First, in the case of (i), \(|\sin\theta_{c,k}|\geq\frac{\gamma}{\|p_{co}\|^{2}}\) holds. Therefore,  
Figure 5.0  
achieves (51) and (54). Next, in the case of (ii), the controller  
Figure 5.1achieves (51) and (54). Finally, in the case of (iii), when \(\theta_{c,k}\
eq 0\), the controller  
Figure 5.2 Figure 5.3  
satisfies (51) and (54). Additionally, when \(\theta_{c,k}=0\), the controller  
Figure 5.4  
does, due to the fact \(\theta_{c,k}\geq\theta_{o,k}\), which is derived from the assumption of convexity of the obstacle. From the above discussion, it is proven that there always exists an input \(V_{wc}^{b}\) satisfying (45) and (46). Therefore, \(K_{fov}(x_{pe})\cap K_{occ,k}(x_{pe})\) is not empty for all \(x_{pe}\) in \(\mathcal{D}_{fov}\cap\mathcal{D}_{occ,k}\). Condition (43) implies that the target has sufficiently small velocity in the direction of the camera robot. Depending on \(v_{o}\) in (6), the movement of the target object may result in it approaching the camera robot. Such movement can be considered unfavorable for visual safety. The controllers in (55)-(58) give inputs that achieve (45) and (46) and whose norm value is finite in each case. Note that from (55)-(58), the existence of input \(V_{wc}^{b}\in K_{fov}\cap K_{occ,k}\) can be guaranteed, with the norm value bounded above by an affine function of \(\|v_{uw}^{b}\|\) due to the fact that \(\|p_{co}\|\geq\Delta_{fov}\) when \(x_{pe}\in\mathcal{D}_{fov}\cap\mathcal{D}_{occ,k}^{T}\). From Proposition 3, when there is only one obstacle, the existence of the optimal solution for (42) is guaranteed, thus ensuring visual safety.