# Simple Reflex Controller for Decentralized Motor Coordination Based on Resonant Oscillation

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## Abstract

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## 1. Introduction

## 2. Reflex Controller for Resonance Mode Excitation

## 3. Linear Spring–Mass–Damper System

#### 3.1. Resonance Frequency Analysis

#### 3.2. Simulations of Resonance Mode Excitation

#### 3.3. Discussion of the Simulation

## 4. Gait Generation in Legged Robot Model

#### 4.1. Model Formulation

#### 4.2. Resonance Frequency Analysis

#### 4.3. Simulations of Resonance Mode Excitation

#### 4.4. Modification to the Controller

#### 4.5. Discussion of the Simulation

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Concept of the proposed reflex controller. The simple and fast-response reflex controllers in each part of the robot allows the robot to sync itself by only physical interaction through the whole body dynamics.

**Figure 2.**Overview of the proposed controller module. Each module is composed of an actuator part, which is modeled as a linear actuator with a spring–damper, and a controller part, which provides oscillation and a reflex function. If the controller module is subjected to an external force from the robot body and environment, then the controller senses an internal force in the spring–damper and adjusts the natural length of the actuator part.

**Figure 3.**Feedback effect in the proposed controller. When the controller does not sense forces, it oscillates the body in a constant frequency by driving the linear actuator; however, if the controller senses a non-zero internal force in the spring–damper, then it slows down the oscillation.

**Figure 4.**Simulation setting of spring–mass–damper system. Two control modules (red spring– dampers) are in the left and right. Passive springs–dampers (black spring–dampers) are in the middle that transmit the physical interaction.

**Figure 5.**Simulation result of the spring–mass system. Figure 5a shows a result with $k=1$, and Figure 5b shows a result with $k=2$. The blue bullet denotes that the solutions of the two controllers are in an in-phase manner, and the red bullet are the solutions in an anti-phase manner. The triangles at the top of the graph indicates the resonant frequencies ${\omega}_{1},\phantom{\rule{3.33333pt}{0ex}}{\omega}_{2},\text{}\mathrm{and}\text{}{\omega}_{3}$ that is computed in advance by mathematical analysis.

**Figure 6.**The simplified sagittal plane model [27] of a quadruped robot. The model assumes that the motions of the left side of the body is mirrored to right. The proposed controller are applied to the fore and hind legs.

**Figure 7.**Simulation result of the legged robot model. Figure 7a shows the simulation result with the spring-damper force feedback, and Figure 7b shows the simulation result with the modified controller using only spring force feedback. The blue bullet denotes the solutions that the two legs are in an in-phase manner, and the red bullet denotes that the solutions are in an anti-phase manner. The triangles at the top of the graph indicates the resonant frequencies ${\omega}_{1}$ and ${\omega}_{2}$, which are analytically derived.

$\mathit{k}=1$ | |||

1st mode | 2nd mode | 3rd mode | |

Analysis | ${\omega}_{1}=0.77$ | ${\omega}_{2}=1.41$ | ${\omega}_{3}=1.85$ |

Simulation | ${\omega}_{1}=[0.6,\phantom{\rule{3.33333pt}{0ex}}0.8]$ | ${\omega}_{2}=[1.4,\phantom{\rule{3.33333pt}{0ex}}1.6]$ | ${\omega}_{3}=[2,\phantom{\rule{3.33333pt}{0ex}}2.6]$ |

$\mathbf{k}=\mathbf{2}$ | |||

1st Mode | 2nd Mode | 3rd Mode | |

Analysis | ${\omega}_{1}=1.08$ | ${\omega}_{2}=2$ | ${\omega}_{3}=2.61$ |

Simulation | ${\omega}_{1}=[0.4,\phantom{\rule{3.33333pt}{0ex}}1.4]$ | ${\omega}_{2}=[2.2,\phantom{\rule{3.33333pt}{0ex}}2.4]$ | ${\omega}_{3}=[3,\phantom{\rule{3.33333pt}{0ex}}3.6]$ |

1st Mode | 2nd Mode | |
---|---|---|

Analysis | ${\omega}_{1}=44.7$ | ${\omega}_{2}=77.5$ |

Simulation | ${\omega}_{1}=[20,\phantom{\rule{3.33333pt}{0ex}}24]$ | ${\omega}_{2}=[36,\phantom{\rule{3.33333pt}{0ex}}44]$ |

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**MDPI and ACS Style**

Masuda, Y.; Ishikawa, M.
Simple Reflex Controller for Decentralized Motor Coordination Based on Resonant Oscillation. *Robotics* **2018**, *7*, 23.
https://doi.org/10.3390/robotics7020023

**AMA Style**

Masuda Y, Ishikawa M.
Simple Reflex Controller for Decentralized Motor Coordination Based on Resonant Oscillation. *Robotics*. 2018; 7(2):23.
https://doi.org/10.3390/robotics7020023

**Chicago/Turabian Style**

Masuda, Yoichi, and Masato Ishikawa.
2018. "Simple Reflex Controller for Decentralized Motor Coordination Based on Resonant Oscillation" *Robotics* 7, no. 2: 23.
https://doi.org/10.3390/robotics7020023