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Volume 13
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Article

Workspace Definition in Parallelogram Manipulators: A Theoretical Framework Based on Boundary Functions

by
Luis F. Luque-Vega
1,
Jorge A. Lizarraga
2,
Dulce M. Navarro
2,
Jose R. Navarro
2,
Rocío Carrasco-Navarro
3,
Emmanuel Lopez-Neri
4,
Jesús Antonio Nava-Pintor
5,
Fabián García-Vázquez
5 and
Héctor A. Guerrero-Osuna
5,*
1
Department of Technological and Industrial Processes, ITESO, Tlaquepaque 45604, Jalisco, Mexico
2
Departamento de Investigación, Centro de Enseñanza Técnica Industrial, Guadalajara 44638, Jalisco, Mexico
3
Department of Mathematics and Physics, ITESO, Tlaquepaque 45604, Jalisco, Mexico
4
Centro de Investigación, Innovación y Desarrollo Tecnológico CIIDETEC-UVM, Universidad del Valle de México, Tlaquepaque 45601, Jalisco, Mexico
5
Unidad Académica de Ingeniería Eléctrica, Universidad Autónoma de Zacatecas, Zacatecas 98000, Zacatecas, Mexico
*
Author to whom correspondence should be addressed.
Technologies 2025, 13(9), 404; https://doi.org/10.3390/technologies13090404
Submission received: 31 May 2025 / Revised: 31 July 2025 / Accepted: 15 August 2025 / Published: 5 September 2025
(This article belongs to the Special Issue Collaborative Robotics and Human–AI Interactions)
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Abstract

Robots with parallelogram mechanisms are widely employed in industrial applications due to their mechanical rigidity and precise motion control. However, the analytical definition of feasible workspace regions free from self-collisions remains an open challenge, especially considering the nonlinear and composite nature of such regions. This work introduces a mathematical model grounded in a collision theorem that formalizes boundary functions based on joint variables and geometric constraints. These functions explicitly define the envelope of safe configurations by evaluating relative positions between critical structural components. Using the MinervaBotV3 as a case study, the symbolic joint-space boundaries and their corresponding geometric regions in both 2D and 3D are computed and visualized. The feasible region is refined through centroid-based scaling to introduce safety margins and avoid singularities. The results show that this framework enables analytically continuous workspace representations, improving trajectory planning and reliability in constrained environments. Future work will extend this method to spatial mechanisms and real-time implementations in hybrid robotic systems.
Keywords: parallelogram mechanisms; feasible workspace; self-collision constraints; boundary functions; symbolic modeling; MinervaBotV3; trajectory planning; geometric workspace parallelogram mechanisms; feasible workspace; self-collision constraints; boundary functions; symbolic modeling; MinervaBotV3; trajectory planning; geometric workspace

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MDPI and ACS Style

Luque-Vega, L.F.; Lizarraga, J.A.; Navarro, D.M.; Navarro, J.R.; Carrasco-Navarro, R.; Lopez-Neri, E.; Nava-Pintor, J.A.; García-Vázquez, F.; Guerrero-Osuna, H.A. Workspace Definition in Parallelogram Manipulators: A Theoretical Framework Based on Boundary Functions. Technologies 2025, 13, 404. https://doi.org/10.3390/technologies13090404

AMA Style

Luque-Vega LF, Lizarraga JA, Navarro DM, Navarro JR, Carrasco-Navarro R, Lopez-Neri E, Nava-Pintor JA, García-Vázquez F, Guerrero-Osuna HA. Workspace Definition in Parallelogram Manipulators: A Theoretical Framework Based on Boundary Functions. Technologies. 2025; 13(9):404. https://doi.org/10.3390/technologies13090404

Chicago/Turabian Style

Luque-Vega, Luis F., Jorge A. Lizarraga, Dulce M. Navarro, Jose R. Navarro, Rocío Carrasco-Navarro, Emmanuel Lopez-Neri, Jesús Antonio Nava-Pintor, Fabián García-Vázquez, and Héctor A. Guerrero-Osuna. 2025. "Workspace Definition in Parallelogram Manipulators: A Theoretical Framework Based on Boundary Functions" Technologies 13, no. 9: 404. https://doi.org/10.3390/technologies13090404

APA Style

Luque-Vega, L. F., Lizarraga, J. A., Navarro, D. M., Navarro, J. R., Carrasco-Navarro, R., Lopez-Neri, E., Nava-Pintor, J. A., García-Vázquez, F., & Guerrero-Osuna, H. A. (2025). Workspace Definition in Parallelogram Manipulators: A Theoretical Framework Based on Boundary Functions. Technologies, 13(9), 404. https://doi.org/10.3390/technologies13090404

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