# Multi-Robot Formation Control Based on CVT Algorithm and Health Optimization Management
## 3 Health Optimization Management  
### Problem Description  
To improve the health of each robot and enhance the robustness and task completion of the formation, health degradation should be considered. Ensuring the survivability of a single robot will often have a significant impact on the task performance and future reliability of the overall formation, as factors such as low battery, high power consumption rate or unforeseen damage to physical structures can cause the robot to slow down in speed or task processing. Although these health problems caused by degraded operating conditions or malfunctions do not need to be dealt with immediately, continued operation will have a significant impact on the mission performance and future reliability of the robot itself and even on the formation. In response to the above problems, the formation health optimization management algorithm was introduced to restrict the speed of the unhealthy robot's Voronoi cell, in order to make it move more conservatively, maximizing the endurance of the unhealthy robot. This makes the formation system better able to deal with unforeseen damage such as mechanical failures and emergencies, making it more robust when performing tasks in different scenarios. In this algorithm, \(u_{i}\) is the control input of robot \(i\). To better describe the health status of the robot, in response to the above problems, \(h_{i}(t)\) is specially introduced, where \(h_{i}(t)\) at time \(t\) can be expressed as:  
\[\left\{ \begin{array}{l}h_{i}(t)=h_{i}(0)-d_{i}(t)\\ d_{i}(t)=\int_{0}^{t}k_{d,i}u_{i}^{2}dt+f_{d,i}(t)\\ \dot{h}_{i}(t)\leq 0,\dot{f}_{d,i}(t)\geq 0,h_{i}(t),f_{d,i}(t)\in[0,1]\end{array} \right. \tag{23}\]  
where \(h_{i}(0)\) is the initial health of the robot, \(d_{i}(t)\) is the health loss of the robot at each moment, \(k_{d,i}\) is the usage gain coefficient, and \(f_{d,i}(t)\) is the negative impact of the external environment on the robot, such as in damage caused by external collisions. Assuming that the health of each robot decreases with the progress of its motion control and possible external damage, Equation (23) is a function of its control accumulation and any external damage that may be encountered. Controlling the accumulation means that fast-moving and turning robots lose their health faster than slow-moving or stationary robots, which can be simplified to a combination of remaining power and structural health in a physical sense. The current health status of a multi-robot formation can be calculated using the average health level of all robots:  
\[\left\{ \begin{array}{c}h^{*}(t)=\frac{1}{n}\sum\limits_{i=1}^{n}h_{i}(t)\\ \dot{h}^{*}(t)<0,h^{*}(t)\in[0,1]\end{array} \right. \tag{24}\]Assume that each robot can obtain the overall average health level \(h^{*}(t)\) of all robots in the entire task area by communicating with neighboring robots within a limited range. The movement of each robot is controlled by a proportional controller, which is controlled by the proportional derivative controller of the formation health optimization management algorithm:  
\[\left\{ \begin{array}{l}u_{i}(t)=k_{p}\cdot[1+k_{h,i}(t)]\cdot(p_{i}-p_{i}{}^{*} )\\ e_{i}(t)=h_{i}(t)-h^{*}(t)\\ k_{h,i}(t)=k_{h_{p}}e_{i}(t)+k_{h_{d}}\dot{e}_{i}(t)\end{array} \right. \tag{25}\]  
where \(k_{h,i}(t)\) represents the gain obtained by the robot changing the mobile formation health optimization management system, \(e_{i}(t)\) represents the difference between a robot and the overall average health level \(k_{h_{p}}\), and \(k_{h_{d}}\) represents the gain coefficient of the formation health optimization management algorithm. Finally, combining the optimization of robot health management and the flexibility of the CVT algorithm, the remaining robots re-divide the original area after the robot whose health value is reduced to zero stops working (the robot cannot continue to work normally due to energy exhaustion, failure, or the deterioration of the external environment), to minimize the impact of this type of robot on the formation task.