# Multi-Layered Interactive Target Guidance with Visual Safety in Convex-Shaped Obstacle Environments
## 2 Problem Formulation
### Environment Model and Occlusion Avoidance  
In this article, the target is assumed to escape in an environment with \(n_{\mathcal{O}}\) convex-shaped obstacles. The \(k\)th obstacle is considered to be a convex set in \(\mathbb{R}^{3}\) and is represented by \(n_{k}\) twice differentiable convex functions \(o_{kj}\) as  
\[\mathcal{O}_{k}\coloneqq\Big{\{}p\in\mathbb{R}^{3}\mid o_{kj}(p)\leq 0,\ \forall j\in\{1,2,\ldots n_{k}\}\Big{\}},\ \ k\in\{1,\ldots,n_{\mathcal{O}}\}. \tag{13}\]  
In this paper, all these functions are assumed to be known and available. In other words, a situation is considered where the environment map has been constructed in advance, and the positions and shapes of the obstacles are known.  
In order to pursue a target objective in a cluttered environment, it is necessary to maintain the absence of obstacles between the camera and the target. We call the property that the camera achieves occlusion avoidance.  
**Definition 2** (Occlusion Avoidance): _The line of sight is defined as a line segment whose endpoints are the origin of the camera-fixed coordinate frame \(\Sigma_{c}\) and a feature point \(p_{ci}\). Suppose that all the line of sights and an obstacle \(\mathcal{O}_{k}\) do not intersect, then occlusion avoidance against the obstacle \(\mathcal{O}_{k}\) is achieved._  
According to this definition, a sufficient condition for occlusion avoidance is presented. First, let \(P_{S}(p)\) be the projected point of a point \(p\in\mathbb{R}^{3}\) onto a closed convex set \(S\subset\mathbb{R}^{3}\). Then, we define \(l_{c,k}=\|p_{wc}-P_{\mathcal{O}_{k}}(p_{wc})\|\) and \(l_{o,k}=\|p_{wo}-P_{\mathcal{O}_{k}}(p_{wo})\|\). The configuration is illustrated in Figure 2. A sufficient condition for occlusion avoidance with an obstacle is provided.  
**Proposition 2**.: _Suppose that \(l_{c,k},l_{o,k},p_{co}\), and \(\Delta_{occ}\in\mathbb{R}\) satisfy_  
\[l_{c,k}+l_{o,k}-\|p_{co}\|-\Delta_{occ}\geq 0, \tag{14}\] \[\Delta_{occ}>2r_{M}, \tag{15}\]  
_for \(k\in\{1,\ldots,n_{\mathcal{O}}\}\). Then, occlusion avoidance against the obstacle \(\mathcal{O}_{k}\) in the sense of Definition 2 is achieved._  
Proof.: It is proven that points whose distances from the origin of \(\Sigma_{o}\) are shorter than \(r_{M}\) are not occluded when (14) and (15) are satisfied. Additionally, \(l_{c,k}:=\|p_{ci}-P_{\mathcal{O}_{k}}(p_{ci})\|(=\|p_{oi}-P_{\mathcal{O}_{k}} (p_{oi})\|)\) is defined. From the triangle inequality, the definition of \(r_{M}\), \(l_{o,k}\leq l_{ci,k}+\|p_{oi}\|\leq l_{ci,k}+r_{M}\) holds. Since \(\|p_{ci}\|\leq\|p_{co}\|+r_{M}\) holds similarly,  
\[l_{c,k}+l_{ci,k}-\|p_{ci}\|>0 \tag{16}\]  
is derived from (14) and (15).  
If occlusion occurs between the feature point \(p_{ci}\) and the camera, there exists a point at which the line of sight and the obstacle \(\mathcal{O}_{k}\) intersect. Define \(\bar{p}_{ci,k}\) as the coordinate of this intersection point with respect to the camera-fixed frame \(\Sigma_{c}\). Then, \(\|\bar{p}_{ci,k}\|+\|p_{ci}-\bar{p}_{ci,k}\|=\|p_{ci}\|\). In addition, \(\bar{p}_{ci,k}\in\mathcal{O}_{k},l_{c,k}\leq\|\bar{p}_{ci,k}\|\) and \(l_{ci,k}\leq\|p_{ci}-\bar{p}_{ci,k}\|\) are derived, resulting in \(l_{c,k}+l_{ci,k}-\|p_{ci}\|\leq 0\). This contradicts (16). Thus, the contraposition is proven.  
Similar to the safe set for field of view maintenance \(\mathcal{D}_{fov}\), the set \(\mathcal{D}_{occ,k}\subset SE(3)^{2}\) is defined as  
\[\mathcal{D}_{occ,k}:=\Big{\{}x_{pe}\in SE(3)^{2}\mid l_{c,k}+l_{o,k}-\|p_{co} \|-\Delta_{occ}\geq 0\Big{\}},\;\;k\in\{1,\ldots,n_{\mathcal{O}}\}, \tag{17}\]  
where \(\Delta_{occ}>2r_{M}\). We call this the _safe set of occlusion avoidance_ for the \(k\)-th obstacle. Furthermore, we define the safety set of occlusion avoidance for all obstacles as  
\[\mathcal{D}_{occ}:=\bigcap_{k\in\{1,\ldots,n_{\mathcal{O}}\}}\mathcal{D}_{occ,k}. \tag{18}\]