Vibration reduction of delta robot based on trajectory planning

https://doi.org/10.1016/j.mechmachtheory.2020.104004Get rights and content

Highlights

  • The elastodynamics model is established based on KED method.

  • Elastodynamics, rigid body dynamics and trajectory planning of the robot are integrated.

  • A universal method to improve the precision of flexible robot is proposed.

Abstract

In this paper, trajectory planning is used to suppress vibration of the Delta robot. Firstly, elastodynamic model of the Delta robot based on KED (Kineto-Elastic-Dynamic) method, substructure synthesis technology and Lagrange equation while considering the flexibility of the links, joints and reducers is built. Secondly, the trajectory planning is performed using polynomial function in the operating space, and its influence on output torque of three motors and elastic deformation of the moving platform is analyzed. Based on these, a comprehensive optimization objective is proposed for trajectory optimization, which is to minimize the maximum absolute elastic deformation and the mean absolute elastic deformation of the moving platform in two directions. Meanwhile, output torque of three motors is constrained. Finally, an optimal trajectory with minimum vibration is solved in the operating space. Results show that the optimal trajectory can effectively reduce elastic deformation of the moving platform.

Introduction

With the development of parallel robots toward high speed, high precision and light weight, main moving parts of which are made of lightweight materials. The typical example is that active limbs and passive limbs in the Delta robot are made of carbon fiber. Compared with the links made of metal materials, the lightweight flexible links will cause greater vibration which can effect reduction in the end-effector's working accuracy. In serious cases, the robot will lose its ability to work due to the vibration caused by flexible links, especially when the robot works in a high speed. Therefore, vibration suppression of the Delta robot with flexible links is an important research direction.

Elastodynamic modeling of the Delta parallel robot is an important prerequisite for vibration suppression. Compared with rigid body dynamic models [1], [2], [3], elastodynamic models of parallel robots are more complicated, which reflect the large-scale rigid body movement and small-scale elastic deformation. There are mainly the following methods for elastodynamic modeling, flexible multi-body system dynamics, KED method and dynamic substructure method [4,5]. Zhang et al. [6] established an elastodynamic model of a 3-DOF parallel robot with flexible links for high-speed pick-and-place operation based on substructure displacement method. Piras G et al. [7] analyzed the elastic vibration and natural frequencies of a planar fully parallel robot with flexible links based on finite element method, which shows that the configuration of the mechanism has a significant influence on the nature of the resultant elastic vibrations. Sun et al. [8] formulate elastodynamic model of 5-DoF PKMs (parallel kinematic machines) by a step-by-step strategy. On this basis, dynamic performances, including natural frequency, elastic deformation, and maximum stress, are analyzed. Wang et al. [9] propose a substructuring dynamic modeling procedure for closed-loop flexible-link mechanisms, which extracts Craig–Bampton mode sets from FE (the lagrange finite element) models using component mode synthesis theory. Dwivedy et al. [10] analyzed various modeling methods for elastic vibration of parallel manipulators considering flexible limbs and joints. In recent years, the screw theory has also been used to build the dynamic model of parallel robots [11], [12]. The KED method neglects the effect of elastic deformation on the motion of rigid body, which makes the modeling process relatively simple. Therefore, the KED method is used to establish the elastodynamic model of Delta robot in this paper.

As to vibration suppression of parallel robots with flexible limbs, there are mainly two methods. The first is based on active control. Wang et al. [13] restrained the vibration of a parallel robot by strain rate feedback control, which is caused by rigid body motion and electromechanical coupling of transduction devices and the host linkage. Chou et al. [14] proposed a robust control methodology based on the state-space model of a flexible linkage mechanism. Mohamed et al. [15] applied a command shaping techniques based on input shaping, low-pass and band-stop filtering to the feedforward control of flexible robots, thus restraining the vibration of robots. Lee et al. [16] used an improved input shaping technology to study vibration suppression of double flexible rods. The second is based on trajectory planning. Park et al. [17] used Fourier series and polynomial functions to design the optimal suppression trajectory based on a mountain climbing search algorithm. Korayem et al. [18] solved the optimal vibration suppression trajectory of a flexible limb system in point-to-point control. Using cubic spline function in joint space and particle limb optimization algorithm, Abe [19] proposed an optimal trajectory planning technique for suppressing residual vibration of two-link rigid-flexible manipulators, which takes the axial displacement and nonlinear curvature arising from large bending deformation into consideration. Trajectory planning for vibration suppression is effective and universal. Besides, there is no higher requirement for the controller. Therefore, it has higher feasibility in practical application. Based on discussions above, this paper adopts trajectory planning to suppress vibration.

Inspired by the previously mentioned vibration reduction method, this paper reduces the elastic vibration of the Delta robot by trajectory planning in the operating space. The specific process is as follows. Section 2 establishes the kinematic model and rigid body dynamic model of the robot. And the transformation relationship between different coordinate systems is given for subsequent elastodynamics modeling. In Section 3, the elastodynamic model is established according to KED method, substructure synthesis technology and Lagrange equation. In Section 4, the ANSYS workbench simulation results of the robot are given to verify the accuracy of the built elastodynamic model and the reliability of the modeling method. And then trajectory planning is carried out in the operating space to suppress vibration. Conclusion is drawn in Section 5.

Section snippets

Kinematic modeling

The 3D model of the Delta robot is shown in Fig. 1. The robot consists of a moving platform, a fixed base, and three identical kinematic chains which are connected to the moving platform and fixed base by spherical joints and revolute joints, respectively. Each kinematic chain contains an active limb and a passive limb, which are connected to each other by a spherical joint. Motors and reducers are installed on the fixed base. Drove by three motors, the robot can achieve translational motion in

Elastodynamic modeling

In this section, the elastodynamic model of Delta will be established through KED method and substructure synthesis technology. Based on the KED method, the real motion of the system is assumed to be the superposition of rigid body motion and instantaneous structural elastic deformation. First, according to the Euler-Bernoulli spatial beam element, differential equations of motion of subcomponents will be built. Then, by introducing deformation compatibility conditions, the elastodynamic model

Vibration reduction analysis

4.1. Model validation In order to validate the accuracy of the proposed elastodynamic model, ANSYS workbench is used to evaluate the first three mode shapes at a typical configuration where x = 0 m, y = 0 m, z = −0.8 m. Main parameters of the Delta robot are listed in Table 1, Table 2, Table 3, Table 4. The first three mode shapes are shown in Fig. 9, and analytic and FEA (Finite Element Analysis) results of the lower three natural frequencies are listed in Table 5.

As can be seen, error of the

Conclusion

In this paper, KED method, substructure synthesis technology and Lagrange equation were used to establish the elastodynamic model of the Delta robot that considered the flexibility of the links, joints and reducers. Treating joints as equivalent springs facilitates the establishment of models. Compared with finite element models, the analytical model proposed in this paper is much more succinct and can be numerically solved with high computational efficiency. The accuracy of the model is

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Acknowledgment

This work was supported by the National Natural Science Foundation of China [grant number 51474320].

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