# Dynamic Analysis of a Delta Parallel Robot with Flexible Links and Joint Clearances
## 1 Introduction  
The Delta parallel robot (DPR) is a three-degree-of-freedom (3-DOF) translational manipulator that consists of a fixed base linked to a mobile platform by three arms. The first model of the Delta parallel robot was invented in 1987 by [PERSON] [1] as a suitable structure for high-speed and high-acceleration tasks, specially used for pick and place operations, but also for packaging, sorting, precision positioning, and other applications. In the industry, parallel robots have light structures and usually operate at high speeds and accelerations with heavy payload; as a result, negative kinematic and dynamic effects may intervene in the operation due to joint clearances and frictions or link flexibility. Preliminary knowledge of the behavior of these robots represents a critical asset for their optimal design. Various studies related to the analytical modeling (both kinematic and dynamic approaches) of parallel robots but also their CAD modelling and simulation can be found in the literature. To the best of our knowledge, no relevant works have been identified that address the idea of analyzing the cumulative effect of the flexibility of elastic elements in combination with the clearances and friction from the spherical joints. These three parameters can have a major role in the dynamic behavior of parallel robots. The analysis of the effects of these factors is exemplified in the paper on the case study of a Delta parallel robot (DPR). A new DPR is proposed and developed in [2], along with its dynamic optimization. A direct and inverse pose modeling method for a DPR is addressed in [3] based on ADAMS, and a DPR kinematic model is presented in [4], completed by a closed-form inverse dynamic model using the Newton laws, a formulation called \"in two spaces\". An analytical approach for the dimensional synthesis of a Delta parallel robot is presented in [5]. The analytical solution presented, with dimensional optimization for the link length, aims to find theDPR workspace. A dynamic dimensional synthesis using the pressure/transmission angle constraints of a DPR is described in [6]. Two types of pressure angle are defined, and thus direct and indirect singularities can be identified. The kinematic calibration and sensitivity analysis for a DPR is described in [7], a method which can be used successfully for other parallel robots. An experimental verification of a newly developed DPR, based on the dynamic model derived by using the Hamilton's principle, is presented in [8]. An inverse dynamic model is developed using the Euler-Lagrange approach, a model validated with real torques data obtained from of a model of the Delta parallel robot developed by SIPRO [9]. Other approaches to dynamic DPR modeling are presented in [10], for example, the Lagrange equations, Newton-Euler formulation and the Principle of Virtual Work. An analysis of an analytical dynamic model of the Delta robot was presented in [11] and another analytical model was validated by experimental data [12]. Different methods for dynamic modeling of parallel robots are presented in [13; 14; 15; 16; 17], with numerical simulation and experimental validation. In addition, the kinematic modeling of parallel robots, especially the Delta robot, has been extensively addressed in the literature [18; 19; 20; 21; 22] including forward and inverse modeling and kinematic optimization. The numerical simulation of parallel robots is an attractive topic, mainly using MATLAB software ([[https://en.wikipedia.org/wiki/MATLAB](https://en.wikipedia.org/wiki/MATLAB)]([https://en.wikipedia.org/wiki/MATLAB](https://en.wikipedia.org/wiki/MATLAB))) [23; 24] or MATLAB simulation validated by experimental research [25; 26]. In dynamic studies, various assumptions can be considered, such as rigid vs. flexible links (analytical or numerical), ideal condition vs. friction and clearance in joints.