A parametric model for tetra joints applied to compliant delta robot analysis

International Journal of Intelligent Robotics and Applications https://doi.org/10.1007/s41315-025-00431-9 REGULAR PAPER A parametric model for tetra joints applied to compliant delta robot analysis Seyyed Masoud Kargar1 · Giovanni Berselli1,2 Received: 16 December 2024 / Accepted: 7 February 2025 © The Author(s) 2025 Abstract Unlike rigid mechanisms, compliant mechanisms offer frictionless motion with enhanced precision in compact designs by deriving their motion from the deflection of flexible elements. The Tetra joint, a class of compliant spherical joints, has garnered significant attention in recent years. However, its unique design-featuring a trapezoidal cross-section and trapezoidal blade flexures-poses significant modeling challenges. This work extends our previous study, where we developed a geometric model for streamlined Tetra joint modeling suitable for implementation in Computer-Aided Engineering (CAE) platforms. In this study, we expand the applicability of this model to the analysis and simulation of complex robotic systems, specifically compliant Delta robots. A compliant Delta robot constructed with Tetra joints, termed DeltaFlex, is numerically and parametrically modeled using the developed geometric model of the Tetra joint. To identify the primary contributors to kinematic errors, a systematic analysis is performed. This methodology combines numerical simulations with statistical correlation analysis to evaluate the relationships between design parameters and system errors. The results reveal that the angle between the two sides of the triangle in the Tetra joint has a dominant effect on system errors. Specifically, smaller angles lead to significantly higher kinematic errors. Keywords Compliant mechanism · Tetra joint · Parametric modeling · Parallel robots · DeltaFlex robot · FEA · CAE 1 Introduction Compliant mechanisms differ from rigid-body mechanisms by utilizing the bending of thin, flexible regions within their structure to transfer and transmit motion. This type of mechanical design offers notable advantages such as eliminating backlash, enabling frictionless motion, and reducing the number of components, leading to a more compact and efficient final design (Howell et al. 2013; Berselli et al. 2011; Kargar and Berselli 2024). Spherical compliant mechanisms (Bilancia et al. 2021; Hao et al. 2024; Baggetta et al. 2024), flexible counterparts to traditional ball-and-socket joints (Fig. 1a), enable rotational motion across three perpendicular axes by utilizing * Seyyed Masoud Kargar 1 Department of Mechanical, Energy, Management and Transportation Engineering (DIME), University of Genova, Genova, Italy 2 Advanced Robotics Department (ADVR), Fondazione Istituto Italiano di Tecnologia (IIT), Genova, Italy slender structural elements for precise spherical motion transmission. While ball-and-socket joints are integral to systems like the human body (Ikemoto et al. 2015), spherical compliant mechanisms are preferred in advanced applications such as robotics, parallel kinematic devices (Brecher et al. 2010; Dong et al. 2008), nanopositioners (Wu et al. 2008), and other precision engineering systems (Moon and Kota 2002). These mechanisms typically achieve motion through spherical notch joints or wire flexures (Howell et al. 2013; Naves et al. 2019; Lobontiu and Paine 2002). Spherical notch joints offer a compact design but are limited to small angular displacements, whereas wire flexures provide a broader range of motion at the expense of reduced support stiffness and increased susceptibility to buckling under load (Howell et al. 2013; Hogervorst et al. 2022). Other alternatives for achieving three rotational degrees of freedom include mechanisms like Tetra joints (Rommers et al. 2021; Hogervorst et al. 2022; Kargar et al. 2024), which stand out for their compactness and self-supporting design. These attributes make them particularly advantageous for Vol.:(0123456789) S. M. Kargar, G. Berselli Fig. 1  a Ball and socket joint; b tetra joint; and c deflected tetra joint Kargar and Berselli (2024) applications such as three-dimensional printing (Parmiggiani et al. 2023). Tetra joints consist of serially connected blade flexures without the inclusion of an intermediate body. These blade flexures, constructed from tetrahedral elements, are nested in a configuration that constrains the end-effector (point e, see Fig. 1 (b)) to rotate exclusively around a remote center of rotation (point CR ) located in space. When a horizontal force is applied to the end-effector, the joint rotates about point CR , enabling spherical motion centered at this remote rotation point (Fig. 1c). To ensure precise functionality, it is crucial that the planes of all three blade flexures forming the tetrahedral elements converge accurately at point CR. Modeling and analyzing Tetra joints present significant challenges due to: – The trapezoidal geometry of the blade flexures; and – The trapezoidal profile of the blade flexures’ cross-sections. Analytical models for these joints have been explored in the literature. For instance, in Rommers et al. (2021), an analytical model was developed to determine the stiffness of the joint in various translational and rotational directions. This study considered both tetrahedron and prism configurations, with the prism element being a degenerate form of the tetrahedron, where the remote center of rotation is assumed to be at infinity. While the analytical model for the prism element successfully captures the joint’s behavior and aligns with finite element analysis (FEA) results, the tetrahedron element’s analytical predictions show significant discrepancies compared to FEA. This error is attributed to the approximation of the trapezoidal shape of the blade flexures and their cross-sectional geometry. For the numerical modeling of these joints, we developed a CAE platform to streamline their design and optimization (Kargar et al. 2024), leveraging the robust capabilities of CAE tools in modeling and optimizing diverse structural systems Kargar et al. (2024). After defining the geometric model, we employed software tools to enhance the joint’s mobility while minimizing any shift in its center. While we were able to analyze the joint and propose a new design, Fig. 2  DeltaFlex robot (Parmiggiani et al. 2024, 2023) Tetra III, the model is constrained by the use of shell elements, which overlook the variation in thickness along the height of the joint, particularly due to its trapezoidal crosssectional shape. To effectively design these joints within intricate structures and robotic systems, it is crucial to develop a three-dimensional (3D) geometric and parametric model. Such a model can accurately represent the joint’s overall behavior, enabling designers to analyze and optimize it according to specific design criteria. Key parameters in a Tetra joint include the beam lengths, height, thickness, the angle between the sides of the to (...truncated)