# Joint Communication-Motion Planning in Networked Robotic Systems
## 2 Conventional Relay-Assisted Robot System
### Joint Communication-Motion Planning: Decide
#### 2.3.2 JCMP Solutions  
Equation (6) involves the joint optimization of the motion (position of the relay robot) and communication (transmit powers of the relay robot and sensing robot), where the variables are \(x_{\rm r,k}\), \(P_{\rm s,k}\), and \(P_{\rm r,k}\). Solving Equation (6) directly may incur a relative high computation complexity. Therefore, we first consider a simpler and classical _communication planning problem_, namely, _given the positions of the sensing robots, the relay robot and the base station, how do we minimize the total communication energy consumption given the PER constraint?_ As will be shown below, we may obtain the closed-form expression of the optimal transmit power of the sensing robots and the relay robot. Based on these closed-form optimal power expressions, the original problem in Equation (6) is transformed into a _motion planning problem_, _namely, given the optimal communication energy consumption for each possible position of the relay robot, how do we optimize the motion of the relay robot for minimizing the total energy consumption_? Hence, during the \(k\)-th time step, if the position of the relay robot \(x_{\rm r,k}\) is given, the motion energy is fixed, and we may formulate the communication planning problem according to Equation (6) as  
\[ \begin{array}{l}\min_{P_{\rm s,k},P_{\rm r,k}}J_{k}=P_{\rm s,k}t_{\rm s}+P_{ \rm r,k}t_{\rm r}\\ {\rm s.\ t.\ }\ p_{\rm srd}^{\rm 2e}(P_{\rm s,k},P_{\rm r,k})\leq p_{\rm ub}. \end{array} \ \ \forall k\in[1,K] \tag{7}\]  
By analyzing Equation (7), we may find that the objective function is linear, which is convex. For the constraint, we define the function \(h=p_{\rm srd}^{\rm 2e}(P_{\rm s,k},P_{\rm r,k})-p_{\rm ub}\), and calculate its Hessian matrix. The results show that the Hessian matrix \(H\) is positive definite, which confirms the communication planning problem of Equation (7) is convex, and we may obtain an optimal solution with Karush-Kuhn-Tucker (KKT) conditions. Using the Lagrange function and by applying KKT conditions, we may derive the expressions of the optimal transmit powers in closed-form:  
\[ \begin{array}{l}P_{\rm s,k}^{*}(x_{\rm r,k})=\frac{\sqrt{t_{\rm r}t_{\rm s}t_ {\rm s}t_{\rm r}}+t_{\rm s}t_{\rm s}}{t_{\rm s}P_{\rm ub}}\\ P_{\rm r,k}^{*}(x_{\rm r,k})=\frac{\sqrt{t_{\rm r}t_{\rm s}t_{\rm s}t_{\rm s}t_ {\rm s}}+t_{\rm r}t_{\rm s}}{t_{\rm r}P_{\rm ub}}\end{array}  \tag{8}\]  
where we have \(u_{k}=\alpha_{n}/G_{sr,k}\), \(v_{k}=\alpha_{n}/G_{rd,k}\), \(G_{sr,k}=\|x_{\rm s,k}-x_{\rm r,k}\|^{-\beta}/N_{0}B\), and \(G_{rd,k}=\|x_{\rm r,k}-x_{\rm d,k}\|^{-\beta}/N_{0}B\) with \(x_{\rm s,k}\) being the position of the sensing robot during the \(k\)-th time step. By substituting Equation (8) into Equation (6), we may transform the original problem into a motion planning problem, which is only relative to the positions \(x_{r}=\{x_{r,1},x_{r,2},\ldots,x_{r,K}\}\) of the relay robot. \[ \begin{array}{l}\min_{x_{r}}\ J=\sum\limits_{k=1}^{K}P_{\rm s,k}^{*}(x_{\rm r,k})t_{s}+P_{\rm r,k}^{*}(x_{\rm r,k})t_{r}+\omega l_{\rm R}(x_{\rm r,k},x_{\rm r,k-1})\\ {\rm s.\ t.\ }l_{\rm R}(x_{\rm r,k},x_{\rm r,k-1})\leq v_{\rm max}T,\,\forall k \in[1,K]\end{array} . \tag{9}\]Letting \(\xi_{k}(x_{r,k})=P^{*}_{s,k}(x_{r,k})t_{s}+P^{*}_{r,k}(x_{r,k})t_{r}\), we have  
\[ \begin{split}&\xi_{k}(x_{r,k})=P^{*}_{s,k}(x_{r,k})t_{s}+P^{*}_{r,k}(x_ {r,k})t_{r}\\ &=\left(\kappa_{s}\|x_{r,k}-x_{s,k}\|^{\frac{\delta}{2}}+\kappa_{ r}\|x_{r,k}-x_{\rm bs}\|^{\frac{\delta}{2}}\right)^{2}\\ &=\left(\kappa_{s}\|\text{A}_{k}x_{r}+(-x_{s,k})\|^{\frac{\delta} {2}}+\kappa_{r}\|\text{A}_{k}x_{r}+(-x_{\rm bs})\|^{\frac{\delta}{2}}\right)^{ 2}\\ &=[f_{1}(x_{r})+f_{2}(x_{r})]^{2}=[g(x_{r})]^{2}.\end{split}  \tag{10}\]  
Both \(\text{A}_{k}x_{r}+(-x_{s,k})\) and \(\text{A}_{k}x_{r}+(-x_{\rm bs})\) are affine mappings of \(x_{r}\), which guarantees the convexity of \(f_{1}(x_{r})\), \(f_{2}(x_{r})\) and \(g(x_{r})\). Considering \(g(x_{r})\) is a nonnegative convex function, \([g(x_{r})]^{2}\) is also convex.