# Analysis and Optimization Based on Factors Affecting the Spiral Climbing Locomotion of Snake-like Robot

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## Abstract

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## 1. Introduction

## 2. Analysis of Locomotion Patterns

#### 2.1. Method for Locomotion

#### 2.2. Control of Spiral Climbing Locomotion

#### 2.3. Influencing Factors

#### 2.3.1. Radius of Spiral Climbing Gaits

#### 2.3.2. Contact Point

## 3. Analysis of Optimization

#### 3.1. Cost Function Based on Factors

#### 3.1.1. The Number of Joint Modules

#### 3.1.2. The Forward Velocity of Spiral Climbing

#### 3.1.3. The Output Torque of Joints

#### 3.2. Optimization Design

_{max}= 0.5 means that we choose the central contact model for calculation. Moreover, we define $l\in \left[0.2,2\right]$ and ${r}_{\mathrm{m}}\in \left[0.2,2\right]$, both of which limit the joint module’s size of the design to no larger than the cylinder’s radius. Furthermore, $\epsilon \in \left[0,\mathsf{\pi}/2\right]$ requires that the snake-like robot can’t be vertical relative to the ground. We calculate the sensitivity factor as $\sigma =\left[\begin{array}{ccc}0.0819& 0.3788& 1.01\end{array}\right]$. Each weight in the multi-objective optimization function has to be positive, representing the importance of the corresponding influencing factor compared to the other two factors. Also, the three need to be content with ${w}_{1}+{w}_{2}+{w}_{3}=1$. Consequently, we can obtain the optimization parameters of a single joint module according to different cylinder and spiral climbing. Table 3 shows the optimization input parameters and the optimized outputs.

## 4. Simulation

#### 4.1. Modeling of the Snake-like Robot

#### 4.2. Simulations and Results

#### 4.2.1. Vibration at Startup

#### 4.2.2. Periodic Variation

#### 4.2.3. Contact Force

#### 4.2.4. Non-Contact Zone

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Different types of spiral climbing movement by a snake: (

**A**) “Folds and Stretches”—wrapping helically and moving forward; and (

**B**) “Spirals and Laterals”—Gripping during lateral undulation.

**Figure 3.**Different spiral climbing gaits when changing the parameters: (

**A**,

**D**) the effect of $A$ on spiral climbing gait; (

**B**,

**E**) the effect of $\beta $ on spiral climbing gait; (

**C**,

**F**) the effect of $\lambda $ on spiral climbing gait.

**Figure 5.**The contact models of a snake-like robot with the cylinder: (

**A**) Central point contact model. (

**B**) Non-central point contact model.

**Figure 9.**Comparison of modules with different radii: (

**A**) joint module with radius of ${r}_{m1}$; and (

**B**) joint module with radius of ${r}_{m2}$.

**Figure 11.**The snake-like robot whose joint module is designed based on the optimization cases: (

**A**) case 1; (

**B**) case 2; (

**C**) case 3; and (

**D**) case 4.

**Figure 12.**The snake-like robot with orthogonal joints: (

**A**)the joint modules are connected orthogonally; and (

**B**) the snake-like robot is modeled by 20 modules which can be divided into head, tail, and central modules.

**Figure 15.**The mechanic characteristics of adjacent modules: (

**A**) velocity and acceleration relative to time; and (

**B**) joint torque and consumption relative to time.

**Figure 16.**The output angle and contact force of two adjacent joints: (

**A**) the output angle and contact force of joint modules; and (

**B**) the contact force is applied to modules at different time points.

**Figure 17.**The output angle and contact force of four adjacent joints: (

**A**) the output angle and contact force of joint modules; and (

**B**) the contact force is applied to modules at different time points.

**Figure 19.**Non-contact zone: (

**A**) JCF of the 9th module; (

**B**) JCF of the 10th module; (

**C**) JCF of the 11th module; and (

**D**) JCF of the 12th module.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|

$A$ | 0.40 | 0.40 | 0.50 | 0.50 | 0.60 | 0.66 | 0.80 | 0.80 | 1.00 | 1.00 |

$\lambda $ | 1.00 | 1.30 | 0.90 | 1.35 | 1.30 | 0.70 | 1.00 | 1.20 | 1.20 | 1.35 |

${R}_{d}/\mathrm{cm}$ | 5.19 | 17.27 | 4.53 | 21.86 | 9.78 | 3.34 | 7.40 | 12.49 | 6.46 | 8.78 |

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{c}}_{5}$ | ${\mathit{c}}_{6}$ | ${\mathit{c}}_{7}$ | ${\mathit{c}}_{8}$ | |
---|---|---|---|---|---|---|---|---|

Optimal value | −0.0812 | 2.2446 | −0.5158 | 0.0149 | 0.0187 | 0.0756 | −1.2878 | 0.4128 |

Symbol | Meaning | Case 1 | Case 2 | Case 3 | Case 4 |
---|---|---|---|---|---|

${r}_{\mathrm{p}}$ | Radius of Cylinder | 20 | 20 | 20 | 20 |

$\mu $ | Friction Coefficient | 0.4 | 0.4 | 0.4 | 0.4 |

$\epsilon $ | Helical Pitch | 10 | 10 | 10 | 10 |

${\phi}_{\mathrm{max}}/\xb0$ | Maximum Rotation Angle | 75 | 75 | 75 | 75 |

${\delta}_{i}$ | Coefficient of Contact Point | 0.5 | 0.5 | 0.5 | 0.5 |

${l}_{\mathrm{min}}/\mathrm{cm}$ | Minimum Length of Module | 5 | 5 | 5 | 5 |

${{r}_{\mathrm{m}}|}_{\mathrm{max}}/\mathrm{cm}$ | Maximum Length of Module | 5 | 5 | 5 | 5 |

${w}_{1}$ | Weight of $cos{t}_{n}$ | 0.15 | 0.15 | 0.70 | 0.33 |

${w}_{2}$ | Weight of $cos{t}_{v}$ | 0.15 | 0.70 | 0.15 | 0.33 |

${w}_{3}$ | Weight of $cos{t}_{\tau}$ | 0.70 | 0.15 | 0.15 | 0.33 |

$l/\mathrm{cm}$ | Length of Module | 9.2326 | 11.2376 | 8.0152 | 12.1638 |

${r}_{m}/\mathrm{cm}$ | Radius of Module | 4.2781 | 4.5013 | 3.0201 | 4.1032 |

Parameter | Value | Parameter | Value |
---|---|---|---|

Stiffness $(N/\mathrm{mm})$ | 2855.00 | Dynamic Friction Coeff. | 0.25 |

Damping $((N\cdot \mathrm{s})/\mathrm{mm})$ | 0.57 | Static Friction Vel. $(\mathrm{mm}/\mathrm{s})$ | 0.10 |

Exponent | 1.10 | Dynamic Friction Vel. $(\mathrm{mm}/\mathrm{s})$ | 10.00 |

Penetration Depth $\left(\mathrm{mm}\right)$ | 0.10 | Coefficient of Restitution | 0.80 |

Static Friction Coeff. | 0.30 | - | - |

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**MDPI and ACS Style**

Zhang, P.; Zang, Y.; Guan, B.; Wu, Z.; Gao, Z.
Analysis and Optimization Based on Factors Affecting the Spiral Climbing Locomotion of Snake-like Robot. *Electronics* **2022**, *11*, 4002.
https://doi.org/10.3390/electronics11234002

**AMA Style**

Zhang P, Zang Y, Guan B, Wu Z, Gao Z.
Analysis and Optimization Based on Factors Affecting the Spiral Climbing Locomotion of Snake-like Robot. *Electronics*. 2022; 11(23):4002.
https://doi.org/10.3390/electronics11234002

**Chicago/Turabian Style**

Zhang, Peng, Yong Zang, Ben Guan, Zhaolin Wu, and Zhiying Gao.
2022. "Analysis and Optimization Based on Factors Affecting the Spiral Climbing Locomotion of Snake-like Robot" *Electronics* 11, no. 23: 4002.
https://doi.org/10.3390/electronics11234002