## 4 Robust Distributed Rendezvous Control
### Visiting UIUVs along the Path to the Root, Based on Signal Strength Measurements  
At the beginning of Algorithm 2, Algorithm 1 in Section 4.1 runs to build a spanning tree \(T\) in a distributed manner. Algorithm 1 has the computational complexity \(O(1)\)[61, 62]. Once \(T\) is generated, the only control applied to each UUV is visiting UUVs along the path, e.g., \(\mathit{PATH}\), to the root in \(T\). This control is a high-level control of every UUV. We introduce a local control to make one UUV visit UUVs along \(\mathit{PATH}\) utilizing signal strength measurements. Consider the case where \(\mathit{PATH}\) consists of a set of UUVs \(p_{1}\to p_{2}\to p_{3}\ldots\to p_{end}\) in this order. Here, \(p_{end}\) is the root. In the inertial reference frame, let \(\mathbf{p}_{j}\in\mathbb{R}^{3}\) indicate the 3D position of \(p_{j}\) for convenience. Suppose that \(u_{i}\) encountered \(\mathbf{p}_{j-1}\in\mathbb{R}^{3}\) and that the next UUV to encounter is \(\mathbf{p}_{j}\in\mathbb{R}^{3}\). Since \(u_{i}\) encountered \(\mathbf{p}_{j-1}\), \(u_{i}\) can detect the signal emitted from \(\mathbf{p}_{j}\) utilizing its intensity sensors. Each intensity sensor on \(u_{i}\) measures the strength of signals emitted from \(\mathbf{p}_{j}\). Signal field intensity is maximized at the signal source. See Equation (5) for RSSL. The gradient direction of a field defines the direction which maximizes the increase in the field. We let a UUV \(u_{i}\) move in the gradient direction of the field. Since the gradient is the direction representing the maximum increase of the field, this maneuver makes the UUV move towards the signal source. The gradient direction is measured in the body\(-\)fixed frame of \(u_{i}\). In the body\(-\)fixed frame of \(u_{i}\), let \(\mathbf{gr}_{i}(P)\in\mathbb{R}^{3}\) present the gradient of \(P\) at the center of \(u_{i}\). Moving in the direction of \(\mathbf{gr}_{i}(P)\) makes \(u_{i}\) move towards the signal source \(\mathbf{p}_{j}\). We next discuss how to derive \(\mathbf{gr}_{i}(P)\) utilizing signal intensity measurements. The local coordinates of every intensity sensor are presented in Figure 3.  
Assume that \(\|\mathbf{c}_{i}^{r}\|\) is sufficiently small for all \(r\in\{1,\ldots,6\}\). Through Taylor expansion up to the first-order derivative, one obtains the following:  
\[P(\mathbf{c}_{i}^{r})=P(u_{i})+\mathbf{c}_{i}^{r}*\mathbf{gr}_{i}(P), \tag{10}\]  
which leads to:  
\[P(\mathbf{c}_{i}^{r})-P(u_{i})=\mathbf{c}_{i}^{r}*\mathbf{gr}_{i}(P). \tag{11}\]  
At each sampling step \(k\), one obtains the following:  
\[\mathbf{P}_{S}=[P_{c}^{1},P_{c}^{2},\ldots,P_{c}^{6}]. \tag{12}\]  
Here, \(P_{c}^{r}=P(\mathbf{c}_{i}^{r})-P(u_{i})\). Furthermore, we utilize the following:  
\[\mathbf{C}_{S}=[\mathbf{c}_{i}^{1};\mathbf{c}_{i}^{2};\ldots\mathbf{c}_{i}^{6 }]. \tag{13}\]  
Utilizing (11)-(13), we derive the following:  
\[\mathbf{P}_{S}^{T}=\mathbf{C}_{S}*\mathbf{gr}_{i}(P). \tag{14}\]  
Thereafter, \(\mathbf{gr}_{i}(P)\in\mathbb{R}^{3}\) is derived using the pseudo-inverse as follows:  
\[\mathbf{gr}_{i}(P)=(\mathbf{C}_{S}^{T}*\mathbf{C}_{S})^{-1}*\mathbf{C}_{S}* \mathbf{P}_{S}^{T}. \tag{15}\]  
where \(\mathbf{gr}_{i}(P)\in\mathbb{R}^{3}\) defines the gradient of \(P\), measured in the the body\(-\)fixed frame of \(u_{i}\). In order to make \(u_{i}\) head towards the signal source, the heading of \(u_{i}\) is set as the gradient direction. Since the gradient is the direction representing the maximum increase of the field, this maneuver makes the UUV move towards the source. In Equation (8), the UUV's heading vector \(\mathbf{h}_{i}(k)\in\mathbb{R}^{3}\) is defined in the inertial reference frame. Since \(\mathbf{gr}_{i}(P)\) is defined in the body\(-\)fixed frame of \(u_{i}\), one changes \(\mathbf{gr}_{i}(P)\) into a gradient vector in the inertial reference frame. Notice that \(\mathbf{R}(\psi(k),\theta(k),\phi(k))*\frac{\mathbf{gr}_{i}(P)}{\|\mathbf{gr} _{i}(P)\|}\) is a gradient vector in the inertial reference frame.