# Defining Feasible Joint and Geometric Workspaces Through Boundary Functions  
[PERSON]  
1 Departamento de Investigacion, Centro de Ensenazza Tecnica Industrial, Guadalajara 44638, Jalisco, Mexico; 1  
Several numerical methods have been extensively studied for workspace analysis, including algorithms designed to map manipulator workspace boundaries, probabilistic numerical analysis approaches [1], and Monte Carlo-based techniques [2]. These methodologies have been widely used to approximate workspace limits and singularity regions, demonstrating effectiveness in various robotic configurations. For instance, the Monte Carlo method proposes to enhance precision by leveraging Gaussian growth to refine workspace estimations. However, despite improving accuracy, its reliance on iterative random sampling limits its ability to produce continuous boundary trajectories. Similarly, the numerical algorithms developed by [PERSON] et al. [3] employ a continuation method to systematically trace workspace boundaries, yet they require significant computational resources, particularly for kinematically redundant manipulators. Alternative strategies have been proposed to overcome these limitations, such as algebraic approaches for kinematic analysis [4] and probabilistic mapping methods for workspace determination [5]. To overcome these limitations, [PERSON]. [6] proposed a mathematical formulation to define feasible, collision-free workspaces through boundary functions that account for both geometric constraints and internal link interactions. This approach is particularly relevant for mechanisms with parallelogram structures, where geometric modeling plays a crucial role in improving workspace representation and ensuring the accurate detection of feasible regions, thereby mitigating the inefficiencies associated with numerical sampling methods. In addition, hybrid techniques have been explored to address workspace characterization. [PERSON] et al. [1] introduced a probabilistic approximation to generate boundary curves in workspaces; however, its dependence on randomization introduces potential inaccuracies, especially for manipulators with complex nonlinear constraints. Meanwhile, workspace determination methods for redundant manipulators, such as those proposed by [PERSON] et al. [7], have been developed to reduce computational redundancy, but they often struggle to precisely define workspace boundaries when joint configurations exceed a certain degree of redundancy. Structural optimization techniques have also been investigated to enhance workspace determination. [PERSON] et al. [8] formulated workspace generation as a direct kinematic problem, reducing computational complexity while maintaining accuracy in identifying singularity curves. Similarly, topology optimization methods [9] have been utilized to improve workspace efficiency, though they introduce additional complexities in parametric system optimization. Algebraic formulations [10] further support workspace evaluation, providing efficient solutions for robotic design in constrained environments. Apart from geometric and numerical considerations, the feasibility of a workspace is influenced by control strategies. The workspace observer method introduced by [PERSON] et al. [11] presents a robust control strategy for redundant manipulators, focusing on real-time workspace regulation. However, such control-based approaches do not inherently define feasible workspaces and must be supplemented with mathematical workspace constraints, as examined in this study. Alternative geometric transformations have also been proposed to tackle workspace determination challenges. The coverings transformation concept, explored by [PERSON] et al. [12], introduces an optimization-based method for approximating solutions to nonlinear inequalities that describe workspace constraints. Likewise, numerical approaches developed for parallel mechanisms [13] demonstrate that direct mapping techniques provide reliable workspace estimations but often involve significant computational loads.