# Multi-Layered Interactive Target Guidance with Visual Safety in Convex-Shaped Obstacle Environments
## 2 Problem Formulation  
### Pursuer and Evader Model  
The Pursuer is a mobile robot equipped with a camera and the Evader is a moving target object. The camera robot and the target are both modeled as 3-D rigid bodies. They have the body-fixed coordinate frames \(\Sigma_{c}\) and \(\Sigma_{o}\) on their center of gravity. The position of a coordinate frame is represented as \(p\in\mathbb{R}^{3}\). Let \(\xi\in\mathbb{R}^{3}\) and \(\theta\in\mathbb{R}\) be a rotational axis and angle of the coordinate frame, respectively. Then, the orientation of the coordinateframe is represented by a rotational matrix \(e^{\delta\theta}\in SO(3)\coloneqq\{R\in\mathbb{R}^{3\times 3}\mid R^{T}R=RR^{T}=I_{3},\ \det(R)=I_{3}\}\), where the operator \"\(\wedge\)\" gives  
\[\hat{a}= \begin{bmatrix}0&-a_{3}&a_{2}\\ a_{3}&0&-a_{1}\\ -a_{2}&a_{1}&0\end{bmatrix}  \tag{1}\]  
for a vector \(a=[a_{1}\ a_{2}\ a_{3}]^{T}\in\mathbb{R}^{3}\). The inverse operator \"\(\vee\)\" gives \((\hat{a})^{\vee}=a\). Here, SO(3) represents the special orthogonal group in three dimensions. The poses of the camera robot and the target with respect to \(\Sigma_{uv}\) are denoted by \(g_{uv}=(p_{uv},e^{\delta_{uv}\theta_{uv}})\), \(g_{uv}=(p_{uv},e^{\delta_{uv}\theta_{uv}})\in SE(3)\coloneqq\mathbb{R}^{3} \times SO(3)\), respectively. Hereafter, a rotational matrix \(e^{\delta_{u}\theta_{u}}\) is simply represented as \(e^{\delta_{u}\theta_{u}}\). We also use a matrix representation for an element of \(SE(3)\), where \(SE(3)\) represents the special Euclidean group in three dimensions. The matrix and vector representation of a pose \(g=(p,e^{\delta\theta})\in SE(3)\) are given by  
\[g= \begin{bmatrix}e^{\delta\theta}&p\\ 0&1\end{bmatrix} ,\ \ \ \text{vec}(g)\coloneqq \begin{bmatrix}p\\ \frac{1}{2}(e^{\delta\theta}-e^{-\delta\theta})^{\vee}\end{bmatrix} , \tag{2}\]  
respectively. To reduce notational complexity, the same notation is used for the direct product representation \(g=(p,e^{\delta\theta})\) and the matrix representation in (2). The relative pose from \(\Sigma_{c}\) to \(\Sigma_{o}\) is represented by \(g_{co}=(p_{co},e^{\delta\theta_{co}})\), and its matrix form is \(g_{co}=g_{wc}^{-1}g_{wo}\). In this paper, the system state consists of \(g_{wo}\) and \(g_{wc}\). The system state is represented as follows:  
\[x_{pe}=(g_{wo},g_{wc})\in SE(3)\times SE(3)\eqqcolon SE(3)^{2}. \tag{3}\]  
Let \(v_{wc}^{b}\) and \(v_{wo}^{b}\in\mathbb{R}^{3}\) denote translational velocities and \(\omega_{wc}^{b}\) and \(\omega_{wo}^{b}\in\mathbb{R}^{3}\) denote the angular velocities of the camera robot and the target object, respectively. Then, the rigid body motions are given by  
\[ \begin{split}\dot{g}_{wc}=g_{wc}\dot{V}_{wc}^{b},\ \ \dot{V}_{wc}^{b}\coloneqq\begin{bmatrix}\dot{\omega}_{wc}^{b}&v_{wc}^{b}\\ 0&0\end{bmatrix},\\ \dot{g}_{wo}=g_{wo}\dot{V}_{wo}^{b},\ \ \dot{V}_{wo}^{b}\coloneqq \begin{bmatrix}\dot{\omega}_{wo}^{b}&v_{wo}^{b}\\ 0&0\end{bmatrix}.\end{split}  \tag{4}\]  
The stacked vectors \(V_{wc}^{b}:[(v_{wc}^{b})^{T}(\omega_{wc}^{b})^{T}]^{T}\in\mathbb{R}^{6}\) and \(V_{wo}^{b}\coloneqq[(v_{wo}^{b})^{T}(\omega_{wo}^{b})^{T}]^{T}\in\mathbb{R}^{6}\) are called body velocities. The relative rigid body motion is given by  
\[\dot{g}_{co}=-\dot{V}_{wc}^{b}g_{co}+g_{co}\dot{V}_{wo}^{b}. \tag{5}\]  
In this article, it is assumed that the body velocity of the target \(V_{wo}^{b}\) depends on \(g_{wo}\) and \(g_{wc}\). Additionally, the translational body velocity \(v_{wo}^{b}\) is assumed to be  
\[v_{wo}^{b}=v_{o}+v_{e}, \tag{6}\]  
where \(v_{o}(g_{wo},g_{wc})\) is the input determined by the target object according to its objective, which satisfies \(\|v_{o}(t)\|\leq v_{max}\) for all time. The velocity \(v_{e}(g_{wo},g_{wc})\) denotes the evasive response to the camera robot. We model the translational velocity \(v_{e}\) as  
\[v_{e}=k_{e}\frac{e^{-\delta_{wo}}(p_{wo}-p_{wc})}{\|p_{wo}-p_{wc}\|^{3}},\ \ k_{e}>0, \tag{7}\]  
similarly to previous works [5; 10; 14].