Mathematical Problems in Engineering

Volume 2018, Article ID 6910468, 16 pages

https://doi.org/10.1155/2018/6910468

## Research on Stiffness of Multibackbone Continuum Robot Based on Screw Theory and Euler-Bernoulli Beam

Department of Control Science and Engineering, Tongji University, Shanghai 201804, China

Correspondence should be addressed to Bin He; nc.ude.ijgnot@nibeh

Received 8 November 2017; Revised 10 April 2018; Accepted 15 April 2018; Published 16 May 2018

Academic Editor: John D. Clayton

Copyright © 2018 Bin He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Continuum robots have become a focus for extensive research, since they can work well in complex and confined environments. The main contribution of this paper is to establish a stiffness model of a single section multibackbone continuum robot and analyze the effect of the structural parameters of continuum robot on the overall rotation and translation stiffness. First, a stiffness model which indicates the end configuration of continuum robot under external load is deduced by the screw theory and Euler-Bernoulli beam. Then, the stiffness elements are fully analyzed, therefore, obtaining the influence of the structural parameters of continuum robot on the stiffness elements. Meanwhile, a numerical analysis of stiffness elements is given. Furthermore, the minimum and maximum rotation/translation stiffness are introduced to analyze the effect of the structural parameters of continuum robot on the overall rotation and translation stiffness. Finally, the experiments are used to validate the proposed stiffness model. The experimental results show that the proposed stiffness model of continuum robot is correct and the errors are less than 7%.

#### 1. Introduction

Continuum robot is a new kind of bionic robot, inspired by elephant trunks and octopus tentacles. Continuum robot does not have its own joints, which can produce flexible deformation in any part, so it has a strong ability to avoid obstacles and better adapt to the complex unstructured environments. Continuum robots offer a number of potential advantages over the traditional rigid-link robots in applications involving disaster relief [1], industrial applications [2], and medical aid [3].

The main types of continuum robots include the rod-driven continuum robot [4], cable-driven continuum robot [5], pneumatic continuum robot [6], and concentric tube continuum robot [7]. Du et al. [8] developed an optimization notched continuum manipulator based on performance evaluation indices. Tian et al. [9] proposed the kinematic analysis of a continuum bionic robot with three flexible actuation rods. Moreover, Bergeles et al. [10] presented optimization framework based on anatomical and surgical task constraints. Li et al. [11] developed a new constrained wire-driven flexible mechanism which has a larger workspace and is more dexterous compared to the existing surgical arms. He et al. [12] proposed a multibackbone continuum robot which is driven by NiTi alloy wire. Trivedi et al. [13] discussed the novel capabilities of soft robots, described examples from nature that provide biological inspiration, surveyed the state of the art, and outlined existing challenges in soft robot design, modeling, fabrication, and control. Webster III and Jones [14] reviewed several modeling approaches in a common frame and notational convention, illustrating that, for piecewise constant curvature, they produce identical results.

The analysis of stiffness is an important step in the design and control of continuum robots, since it determines the relationship between the deformation and the force of continuum robots. Selig and Ding [15] applied the screw theory [16] to analyze compliance and stiffness matrices of a beam. Pei et al. [17] studied the compliance of cartwheel flexural hinges. Ding and Dai [18] investigated spatial continuous compliance on both serial and parallel mechanisms based on screw theory and Lie groups, applying eigenvectors and eigenvalues to identify principal screws in the mechanisms. Awtar and Sen [19] proposed a generalized constraint model for compliance and stiffness analysis of 2D beam flexures. Krishnan et al. [20] studied serial and parallel concatenation of building blocks. Tunay [21] proposed a concept of equivalent bending stiffness to establish the kinematic model of continuum robot. Gao et al. [22] established a mathematical model for predicting the loaded posture of a single section continuum manipulator. Qi et al. [23] analyzed the compliance characteristics of a new planar spring continuum robot. Gravagne et al. [24] discussed the dynamics of a planar continuum backbone section, incorporating a large-deflection dynamic model. Trivedi et al. [25] presented a new approach for modeling soft robotic manipulators that incorporates the effect of material nonlinearities and distributed weight and payload. Camarillo et al. [26] proposed a new linear model for transforming desired beam configuration to tendon displacements and vice versa. Fras et al. [27] described the design and implementation of a static model used for position estimation of a flexible modular medical manipulator equipped with optic-fiber based sensors. Till and Rucker [28] exhibited low numerical damping, handled arbitrarily large time steps, and provided an accurate, high-order representation of the rod shape in steady state. Hadi Sadati et al. [29] introduced a novel series solution for variable-curvature Cosserat rod static and Lagrangian dynamic method.

Several different performance indices have been proposed for stiffness evaluation, including determinant of stiffness matrix, average stiffness, and minimum and maximum stiffness [30, 31]. The determinant of stiffness matrix does not distinguish specific stiffness values, and the average stiffness cannot give enough information of stiffness values [32]. The minimum and maximum stiffness are minimum and maximum eigenvalue of stiffness matrix, respectively, which can indicate variation range of stiffness values, and the corresponding eigenvector directions represent the minimum and maximum stiffness directions, respectively. However, it is well known that the conventional minimum and maximum stiffness cannot be applied to a 6 × 6 symmetric stiffness/compliance matrix. This is due to the fact that the eigenvalues of a stiffness/compliance matrix are not consistent unit and no way to compare the sizes of eigenvalues. As a consequence, we define the minimum and maximum rotation/translation stiffness, to evaluate the influence of the structural parameters of continuum robot on the rotation/translation stiffness of continuum robot.

In the paper, a general loading is represented by a wrench, while a general deformation is represented by a twist. The main contribution is to present a method to establish the stiffness model of a single section multibackbone continuum robot based on screw theory and Euler-Bernoulli beam, as well as analyzing the effect of the structural parameters of continuum robot on the overall rotation and translation stiffness by the minimum and maximum rotation/translation stiffness. The remainder of the paper is organized as follows: In Section 2, we give structure overview of a single section multibackbone continuum robot. In Section 3, based on the screw theory and Euler-Bernoulli beam, a brief stiffness model which illustrates the end configuration of the continuum robot under external load is deduced. In Section 4, the stiffness elements are analyzed to fully indicate relationship between wrench and twist of continuum robot and obtain the influence of the structural parameters of continuum robot on the stiffness elements. In addition, the minimum and maximum rotation/translation stiffness are introduced to evaluate the effect of the structural parameters of continuum robot on the overall rotation and translation stiffness. In Section 5, the proposed stiffness model is validated by the experiments. The experimental results indicate that proposed stiffness model of continuum robot is correct. In Section 6, some discussions are given. In Section 7, conclusions are provided.

#### 2. Overview of a Single Section Multibackbone Continuum Robot

The simplified structure of a single section multibackbone continuum robot is shown in Figure 1. The continuum robot is composed of a base disk, several spacer disks, an end disk, and four super elastic NiTi wires as its backbones. The central NiTi wire is the primary backbone and the remaining NiTi wires are the secondary backbones. The primary backbone is attached to all disks, and all disks are distributed in equal distance. The secondary backbones are equidistant from primary backbone and from each other. The secondary backbones are only attached to the end disk and slide through holes in base disk and spacer disks, which have double effect: driving robot to achieve two degrees of freedom bending motion and providing auxiliary support to increase the stiffness of continuum robot.