Mathematical modeling and analysis of the Delta robot with flexible links

This paper presents a mathematically dynamic model of a Delta robot with flexible links. The mathematical models of the robot cannot be represented by partial differential equations, so this paper utilizes the kineto-elasto-dynamics and the finite element method to perform a mathematical model. Each link of the robot is modeled by multiple beam elements with an axial displacement, an axial torsion, and two transverse displacements. In literature, element assembling usually imposes a set of algebraic constraint equations, which are difficultly solved simultaneously. This paper proposes an alternative approach. A set of global variables based on the D-H method is defined, and the Euler-Lagrange's equation is applied to derive the model without using any constraint equations. The developed model is a set of linear time-varying differential equations, which can describe the flexible motions with respect to the rigid body configuration. Furthermore, the natural frequency analysis and the convergence analysis are performed first, and then two types of paths are designed for the motions of the end-effector. The first path is a constant-speed circular motion in order to demonstrate the numerical simulations of the model at the steady state, and the second path is an inverted-U path, which is commonly used to operate a pick-to-place motion in industry.