# Modeling and Analysis of Dynamics of Rigid-Flexible Coupled Parallel Robots
## 1 Introduction  
Traditional robots are connected by rigid connecting rods with large motion inertia, which can only realize low motion speed and acceleration, thus affecting work efficiency. With the continuous development of modern machinery in the direction of lightweight, low energy consumption, and high efficiency, rigid-flexible coupled robots with increased member flexibility or flexible members replacing rigid members are becoming increasingly widely used [1]. However, the vibration and elastic deformation caused by component flexibility can affect the motion accuracy and dynamics of the system, which has become the focus of the current research on this type of rigid-flexible coupled robot [2, 3, 4, 5].  
Rigid-flexible coupled robots are mainly divided into tandem and parallel robots, among which the dynamics modeling problem of rigid-flexible coupled parallel robots has been a hot research topic in recent years [6, 7]. [PERSON] et al. [8, 9] established the dynamics model of planar flexible parallel robots using the hypothetical modal method and illustrated the effect of the vibration of the flexible rod on the dynamics of the whole flexible robot system. [PERSON] et al. [10] constructed a dynamic model of a flexible planar parallel robot by combining [PERSON]'s principle with the principle of virtual work. [PERSON] et al. [11] used [PERSON] principle to build a dynamics model of a spatial Delta-S rigid-flexible coupled parallel robot and performed a dynamic static analysis. [PERSON] et al. [12, 13] constructed a dynamic model of a planar 3-RRR flexible parallel robot and flexible robot containing flexible hinges. Other modeling methods include the Kineto-Elastodynamics (KED) method, the finite element method, and the concentrated mass method [14, 15, 16]. The above literature shows that most of the modeling and experimental studies have been conducted for planarflexible parallel robots, while the studies for spatial flexible parallel robots are fewer and limited to systems with a single energy domain.  
Power bond graph is a system dynamics modeling method [17]. [PERSON] et al. [18] proposed a method for modeling the dynamics of flexible multibody systems based on the hypothetical modal method and bond graph theory and constructed a Euler-Bernoulli beam bond graph model. [PERSON] et al. [19] pointed out that flexible robotic systems are both rigid-flexible coupled and multi-energy-domain physical systems and established a bond graph model of a 2-dimensional finite element beam structure, and analyzed the dynamic response. These studies provide a reasonable paradigm for developing models for such multi-energy domain systems. However, the power bond graph is only applicable to simple systems and is not yet satisfactory when dealing with multi-dimensional and complex spatial motions.  
The vector bond graph has the ability to deal with multi-dimensional and multi-energy domain systems, which is suitable for spatial multi-rod flexible parallel modeling, and the state equations are easier to derive. Therefore, this paper proposes a dynamics modeling method based on the vector bond graph, establishes a multi-energy domain global dynamics model of Delta-type rigid-flexible coupled parallel robot considering rod flexibility, and investigates the coupled vibration of the control subsystem and mechanical subsystem of the system. The model is simulated and analyzed, and the comparison of the results of different simulation software verifies the validity and reasonableness of the model. This method provides a reference for the modeling of other spatially parallel systems with rigid-flexible coupling, and lays a foundation for the subsequent design of the control system.