## 3 Double-Layer Corridor
### Static-Layer Corridor  
A series of partially overlapping convex sets are often used to model the safe region of robots to minimize the number of collision avoidance constraints [11]. These convex sets are defined as static-layer corridors in this paper. The collision avoidance between robot and obstacle is guaranteed by static-layer corridors.  
To construct static-layer corridors, the minimum safe distance between the robot and the obstacle needs to be obtained. Assume that the position of the robot \(m\) at time \(k\) is a multivariate Gaussian random variable \(\mathbf{p}_{m,k}\sim\mathcal{N}(\widetilde{\mathbf{p}}_{m,k},\Sigma_{m,k})\) and obstacle \(o\) is located at \(\mathbf{q}_{o}\). Then, \(\mathbf{p}_{m,k}-\mathbf{q}_{o}\) is also a Gaussian random variable, i.e., \(\mathbf{p}_{m,k}-\mathbf{q}_{o}\sim\mathcal{N}(\widetilde{\mathbf{p}}_{m,k}-\mathbf{q}_{o}, \Sigma_{m,k})\). Thus, the collision probability is given by Equation (8).  
\[\mathcal{P}(\mathbf{x}_{m,k}\in C_{m,k})=\int_{\|p_{m}-q_{o}\|<d_{m}}p\Big{(}\mathbf{p}_ {m,k}-\mathbf{q}_{o}\Big{)}d\Big{(}\mathbf{p}_{m,k}-\mathbf{q}_{o}\Big{)} \tag{8}\]  
Equation (8) is the integral over the circle of the multivariate Gaussian probability density function. Referring to [25], the circle collision region \(C_{m0,k}\) is enlarged into a half space \(C_{m0,k}\). \(C_{m0,k}\) is defined as Equation (9).  
\[C_{m0,k}\,^{t}:=\Big{\{}\eta_{m0,k}^{T}\Big{(}\mathbf{p}_{m,k}-\mathbf{q}_{o}\Big{)}<d _{m}\Big{\}} \tag{9}\]  
In Equation (9), \(\eta_{m0,k}=\big{(}\widetilde{\mathbf{p}}_{m,k}-\mathbf{q}_{o}\big{)}/\|\widetilde{ \mathbf{p}}_{m,k}-\mathbf{q}_{o}\|\). Since \(C_{m0,k}\subset C_{m0,k}\,^{t}\), Equation (10) can be obtained  
For the \(k\)th static-layer corridor \(S^{s}_{m,k}\) of robot \(m\), it satisfies Equation (17) and Equation (18).  
\[S^{s}_{m,k}C_{mo,k} \tag{17}\]  
\[S^{s}_{m,k}\cap S^{s}_{m,k+1}\
eq\varnothing \tag{18}\]  
Equation (17) guarantees that the intersection of the \(k\)th static-layer corridor and the obstacle region is empty. Moreover, Equation (18) enforces that the \(k\)th static-layer corridor and the \((k+1)\)th static-layer corridor are sequentially connected.  
The distance between the boundary of a static-layer corridor and the obstacle is set as \(d^{s}_{m}\), which is shown in Figure 2. \(d^{s}_{m}\) is set to an initial value at the beginning and \(d^{s}_{m}\) updates during the robot's operation.  
\[d^{s}_{m}=inv(\eta_{m,k}{}^{\prime})(d_{m}+\Psi^{-1}(1-2\delta_{o})\sqrt{2\eta _{m,k}^{\prime}\Sigma_{m,k}\eta_{m,k}{}^{\prime}}) \tag{19}\]  
\[\eta_{m,k}{}^{\prime}=(\mathbf{p}_{M}-\mathbf{p}_{N})/\|\mathbf{p}_{M}-\mathbf{p}_{N}\| \tag{20}\]  
In Equation (19), \(inv(\eta_{m,k}{}^{\prime})\) represents the generalized inverse matrix of \(\eta_{m,k}{}^{\prime}\). In Equation (20), \(\mathbf{p}_{M}\) is the position of point M and \(\mathbf{p}_{N}\) is the position of point N. M is a point on the boundary of static-layer corridor and N is its vertical point on the obstacle area. If robot \(m\) stays inside the static-layer corridor, then Equation (16) satisfies. When the robot satisfies Equation (16), the robot cannot collide with the obstacle.  
Thus, it can be seen that if the state of robot \(m\) satisfies Equation (21), there is no obstacle collision.  
\[x_{m,k}\in S^{s}_{m} \tag{21}\]  
Figure 1: Static-layer corridors.