# On the Intersection of Computational Geometry Algorithms with Mobile Robot Path Planning
## 2 Path Planning
### Transitioning from a Single-Robot to Multi-Robot Path Planning  
As delineated in the single-robot path-planning problem, the objective revolves around the generation of an optimal and feasible path in a given configuration space. The essence of this approach lies in the computation of paths in the presence of obstacles, ensuring the absence of any intersection with the obstacle space \(C_{\text{obs}}\). With a clear understanding of the single-robot paradigm, transitioning to the realm of multi-robot path planning necessitates the recognition and accommodation of additional complexities. **Constraints and Challenges:** While the foundational principles of single-robot path planning can be employed in the multi-robot scenario, several key constraints emerge:  
1. **Inter-Robot Collisions:** Unlike the single-robot scenario, the risk of collisions is not limited to static obstacles. Dynamic collisions between robots present a substantial challenge. The movement of one robot might obstruct the path of another, necessitating continuous reevaluation and adaptation;
2. **Coordinated Movement:** Ensuring that all robots reach their respective destinations within optimal time frames demands a high degree of coordination. This often involves sacrificing individual robot efficiency for collective efficiency;
3. **Increased Configuration Complexity:** The configuration space exponentially grows with the addition of each robot. This expansion augments the computational overhead and complicates the search for optimal paths. **A Bridge:** CG serves as a bridge to extend single-robot path-planning solutions to cater to multiple robots. Several techniques come into play:  
* **Minkowski Sum [25]:** This can be employed to compute the configuration space obstacles when dealing with moving robots. By considering one robot as the primary agent and treating other robots as moving obstacles, the problem can be reduced to a single-robot scenario in an augmented environment;
* **Voronoi Diagrams [26]:** These can assist in partitioning the workspace, providing each robot with a distinct region to navigate. This ensures a degree of separation between the robots, minimizing potential collisions;
* **Visibility Graphs [27]:** For environments with multiple robots and obstacles, visibility graphs can be extended to incorporate dynamic inter-robot constraints. This ensures that, while navigating from source to destination, the robot considers both static and dynamic elements in its path. **Role of Kinematics and Dynamics:** In multi-robot systems, purely geometric considerations, although fundamental, may not suffice for effective path planning, especially in scenarios involving intricate robotic systems such as autonomous cars. The kinematic and dynamic parameters, including velocities and accelerations, play a pivotal role. Ignoring these parameters can lead to unrealistic path solutions that a physical robot cannot execute due to constraints in acceleration, turning radius, or speed limitations. For instance, two autonomous cars approaching an intersection may need to adjust their speeds to avoid collision, even if their geometric paths do not intersect. This emphasizes the need for path-planning algorithms to be kinodynamically aware, accounting for both the geometric constraints of the environment and the kinematic/dynamic constraints of the robots. Integrating these considerations ensures the generated paths are not only collision-free but also feasible, given the robots' physical characteristics and capabilities. Thus, a more holistic path-planning approach for multi-robot systems incorporates both geometric algorithms, as provided by computational geometry, and kinodynamic considerations to cater to the real-world complexities and constraints of robotic systems. The solutions discussed below consider both CG and kinodynamic systems for multi-robot path planning.