• complexanalysis
• conformalmapping
• harmonicplanning
• multirobotplanning
• potentialplanning
• robotplanning
• rootlocus
• schwarzchristoffel

This thesis is focused on the development of advanced analytical tools designed to facilitate the planning of movements for a group of identical robots operating within a two-dimensional environment that contains various polygonal obstacles. The primary objective of this research is to create effective strategies that enable these robots to navigate smoothly and efficiently without colliding with each other or with any obstacles present in their surroundings.
To begin with, the thesis provides a concise introduction to the existing methods used in robotic motion planning. This introduction is followed by a detailed classification of these planning methods, categorizing them based on their approaches and applications. By doing so, the proposed method is contextualized within the broader spectrum of available techniques documented in the current scientific literature, highlighting its unique contributions and advantages.
The thesis then proceeds to introduce the specific notation and terminology that will be used throughout the work. This is essential for establishing a clear and consistent framework for discussing the mathematical and computational aspects of the problem. Alongside the notation, a rigorous mathematical formalization of the motion planning problem is presented.
Following the problem formulation, the thesis outlines the essential mathematical tools and methodologies that are necessary to derive the desired solution. Specifically, it delves into the conformal mapping method and the Generalized Root Locus (GRL) technique. These tools are introduced in a general manner, explaining their relevance and applicability to the problem at hand. The conformal mapping method is particularly important for transforming the physical scenario into a more manageable form, while the GRL technique offers a powerful tool for planning purposes.
The next section of the thesis applies these mathematical methods to a specific instance of the motion planning problem. This involves developing a methodological approach that systematically addresses the challenges posed by the presence of polygonal obstacles. The chosen approach begins with an initial simplification of the problem by employing conformal mapping. This process transforms the original physical environment, which includes polygonal obstacles, into a canonical scenario where the obstacles are represented with circular boundaries. This transformation is not merely theoretical but is implemented numerically using MATLAB's PlgCirToolbox software package, which provides the necessary computational tools to perform the mapping accurately.
Once the environment has been transformed into the canonical domain, the motion planning problem is tackled using root locus-based methods. These methods build upon previous research in the field, expanding the foundational ideas by interpreting the root locus from the perspective of complex potentials. Within this framework, the problem is effectively reduced to solving Laplace's equation, a fundamental partial differential equation, under specific boundary conditions imposed within the canonical domain. Solving Laplace's equation in this context is crucial for determining the potential fields that guide the robots' movements.
Initially, the problem is addressed in an environment devoid of obstacles to emphasize the inherent properties of the root locus curves when no perturbations are present. By doing so, the thesis highlights how these curves behave naturally and lays the groundwork for understanding how obstacles influence their trajectories. To ensure that the resulting motion paths are safe and collision-free, the thesis presents several theorems along with their proofs. These theorems establish conditions under which the root locus curves do not intersect, thereby preventing any potential collisions between the robots as they navigate the environment.
Building upon this foundation, the thesis introduces an iterative approach based on Milne-Thompson's Theorem to incorporate circular obstacles into the analysis. This approach ensures that the streamlines generated by the potential fields do not penetrate the boundaries of the obstacles, maintaining a safe distance between the robots and any obstacles in their path. By iteratively applying this theorem, the research successfully generates sets of paths that are free from collisions both with the obstacles and among the robots themselves within the canonical space.
Once the collision-free paths have been established in the canonical domain, the final step involves mapping these solutions back to the original physical domain. This is achieved by applying the inverse of the conformal mapping that was previously derived. This inverse mapping translates the paths from the simplified canonical scenario back to the actual environment with polygonal obstacles, ensuring that the robots can follow these paths in the real-world setting.
In summary, this thesis demonstrates the effectiveness of using root locus-based analytical tools for multi-robot motion planning in environments with obstacles. It delves deeply into the analytical methodologies that underpin these tools, providing a thorough exploration of their capabilities. Additionally, the research contributes a novel interpretation of the mutual root locus through the lens of Milne-Thompson's Theorem, offering fresh insights into the dynamics of multi-robot systems.