Bio-Medical Materials and Engineering 24 (2014) 529–537 DOI 10.3233/BME-130839 IOS Press

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A 3D undulatory locomotion system inspired by nematode C. elegans Deng Xina,b,* and XuJian-Xinb a

College of Computer Science and Technology, Chongqing University of Post and Telecommunications, Chongqing, 400065, China bDepartment of Electrical and Computer Engineering, National University of Singapore, 117576

Abstract. This paper provides an undulatory locomotion model inspired by C. elegans, whose nervous system and muscular structure are well studied. C. elegans is divided into 11 muscle segments according to its anatomical structure, and represented as a multi-joint rigid link model in this work. In each muscle segment, there are four pieces of muscles located in four quadrants. The muscles change their lengths according to the outputs of nervous system. In this work, the dynamic neural networks (DNN) are adopted to represent the nervous system. The DNN are divided into the head DNN and the body DNN. The head DNN produces the sinusoid waves to generate the forward and backward undulatory movements. The body DNN with 11 segments is responsible for passing the sinusoid wave and creating the phase lag. The 3D locomotion of this system are implemented by using the DNN to control the muscle lengths, and then using the muscle lengths to control the angles between two consecutive links on both horizontal and vertical planes. The test results show good performances of this model in both forward and backward locomotion in 3D, which could serve as a prototype of the micro-robot for clinical use. Keywords: C. elegans, dynamic neural networks, locomotion, 3D

1. Introduction Undulatory locomotion is one of the fundamental behaviors of the footless animals, for example, reptiles such as snakes, and worms such as C. elegans. By mimicking undulatory locomotion, the snakelike or worm-like machines can accomplish many difficult tasks such as rescuing survivors in complex areas, and checking the stomach, blood vessels, or intestine for clinical use [1]. Snake has been widely studied to disclose the mechanism of undulatory motion. However, due to the huge amount of neurons and muscle-bone structures of the snake, it is difficult to study the motion mechanism from the viewpoint of neural circuit. Fortunately, C. elegans offers us an idea model to study the undulatory locomotion behaviors and mechanism. Its nervous system contains 302 neurons and all the connections between these neurons and muscles are visibly known [2]. The undulatory locomotion shape of C. elegans is very similar to the snake, whereas the neural circuit of C. elegans is much simpler than that of snake, so it can be modeled easily. The study on undulatory locomotion behavior of C. elegans begins in recent years. Suzuki etc. explored the locomotion behaviors of C. elegans in forward movement, backward movement, and turn*

Corresponding author. E-mail: [email protected].

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X. Deng and J.-X. Xu / A 3D undulatory locomotion system inspired by nematode C. elegans

ing [3–6]. Boyle etc. investigated how the muscles and neurons of C. elegans created the S wave and how it was propagated from the head to tail [7, 8]. In these research work, C. elegans crawls in 2D. The work in [9] claims that the snake lifts up the most bent parts during full speed movement. Through analyzing the image record of C. elegans [10], we find that C. elegans also lifts parts of its body during high speed locomotion. Inspired by this fact, we construct a 3D locomotion model of C. elegans in this work. The novelty of this work involves four aspects. First, the model is represented by ͳʹ multi-joint rigid links according to the biological muscle structure of C. elegans. Second, each joint represents the center of its corresponding muscle segment, and is controlled by four quadrants muscles in the same muscle segment. The joint angles on the x-y plane and the x-z plane are determined by the muscle lengths in four quadrants. The relationship between joint angles and the lengths of muscles are determined. Third, muscle models are constructed to make the muscle lengths vary according to the outputs of DNN. Forth, the neural circuit of C. elegans is depicted by DNN, which involve the head DNN and the body DNN. The head DNN produce the sinusoid waves, and the body DNN passes the wave and make the outputs to muscles. The phase lag produced by the DNN yields the sinusoid shape of the body. The 3D locomotion model of C. elegans performs well in computer test for both forward and backward movements. Furthermore, analysis of actual C. elegans is carried out to verify our model’s effectiveness. 2. Methods 2.1. The structure of the locomotion model of C. elegans 2.1.1. Modeling the body of C. elegans C. elegans with a simply cylindrical body is about 1 millimeter in length. The body wall muscles can be classified into 4 quadrants on the transverse plane, as shown in Fig. 1 (a). These four quadrants are dorsal-left (DL), ventral-left (VL), ventral-right (VR), and dorsal-right (DR). Its 95 body wall muscle cells are arranged as pairs located in four quadrants along the body, as shown in Fig. 1 (b). During locomotion, C. elegans lies aside and moves over a surface by propagating dorsal/ventral flexures along its body [12]. Based on the muscles structure, the body can be divided into 11 muscle segments. The center of each muscle segment in Fig. 1 (b) is depicted as a joint in Fig. 1 (c). As shown in Fig. 1 (c), the whole body is represented by 13 joints, which are connected by 12 links. Thus, in this work the body of C. elegans is depicted as a multi-joint rigid link system.

Fig. 1. Muscle and body structure of C. elegans.

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X. Deng and J.-X. Xu / A 3D undulatory locomotion system inspired by nematode C. elegans

2.1.2. Modeling the nervous system of C. elegans Head DNN The head DNN functions as the central pattern generator (CPG) to create the sinusoid wave. The head DNN contains two dynamic neurons C1 and C2 to generate the sinusoid waves,

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A 3D undulatory locomotion system inspired by nematode C. elegans.

This paper provides an undulatory locomotion model inspired by C. elegans, whose nervous system and muscular structure are well studied. C. elegans is...
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