# Improving Energy Efficiency of an Autonomous Bicycle with Adaptive Controller Design

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## Abstract

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## 1. Introduction

## 2. Modelling

## 3. Controllers

## 4. Energy Calculation

## 5. Simulations

#### 5.1. Choice of the Plant Parameters Values

#### 5.2. Controller Gains Calculation

#### 5.3. Energy Consumption

#### 5.4. Energy Saving

- Relative to the energy consumption of the lateral stability control system by using a non-adaptive control (${E}_{\theta}^{NA}$):$$E{S}_{\theta}(\%)=\frac{{E}_{\theta}^{NA}-{E}_{\theta}^{A}}{{E}_{\theta}^{NA}}\times 100$$
- Relative to the energy consumption of the forward velocity control system (${E}_{v}$):$$E{S}_{v}(\%)=\frac{{E}_{\theta}^{NA}-{E}_{\theta}^{A}}{{E}_{v}}\times 100$$
- Relative to the total energy consumption of the system using a non-adaptive control (${E}^{NA}$):$$ES(\%)=\frac{{E}_{\theta}^{NA}-{E}_{\theta}^{A}}{{E}^{NA}}\times 100$$

#### 5.5. Trajectory Tracking

**a**) change of the mass centre position [33]; (

**b**) change of the reference inclination [34]; (

**c**) gyroscopic stabilization [21]. We have used as an illustrative example the change of the reference inclination strategy.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

Bicycle model | |

d | Distance of the bicycle centre of mass from the rear-wheel. |

g | Gravitational acceleration. |

h | Height of the bicycle centre of mass. |

v | Forward velocity. |

w | Wheelbase. |

x | Displacement in the x coordinate. |

y | Displacement in the y coordinate. |

G | Transfer function of the stability dynamic model of the bicycle. |

${P}^{A}$ | Trajectory followed by the bicycle with the adaptive controller. |

${P}^{NA}$ | Trajectory followed by the bicycle with the non-adaptive controller. |

R | Turning radius. |

Z | System disturbances. |

$\alpha $ | Front-wheel angle. |

$\theta $ | Roll angle. |

$\sigma $ | Path curvature. |

$\psi $ | Yaw angle. |

${\mathrm{\Lambda}}^{F}\left(\theta \right)$ | Relation of the yaw angle $\psi $ with the roll angle $\theta $ of the bicycle. |

${\mathrm{\Lambda}}^{I}\left(\psi \right)$ | Relation of the roll angle $\theta $ with the yaw angle $\psi $ of the bicycle. |

Controllers | |

${k}_{R}$ | Controller Gains. |

$uE$ | Dimensionless energy unit. |

${x}^{*}$ | Reference displacement in the x coordinate. |

${y}^{*}$ | Reference displacement in the y coordinate. |

${J}_{u}$ | Energy functional. |

${K}_{p}$ | Proportional gain of the PI controller. |

${K}_{i}$ | Integral gain of the PI controller. |

${P}^{*}$ | Desired trajectory. |

$PI$ | Proportional-Integral controller. |

${\theta}^{*}$ | Roll angle reference. |

${\psi}^{*}$ | Yaw angle reference. |

Model of the current electric motor | |

${A}^{v}$ | Transfer function parameter of the current electric motor controlled in velocity. |

${A}^{\theta}$ | Transfer function parameter of the current electric motor controlled in position. |

${B}^{v}$ | Transfer function parameter of the current electric motor controlled in velocity. |

${B}^{\theta}$ | Transfer function parameter of the current electric motor controlled in position. |

${G}_{M}^{v}$ | Transfer function of the current electric motor controlled in velocity. |

${G}_{M}^{\theta}$ | Transfer function of the current electric motor controlled in position. |

${V}_{c}^{v}$ | Control voltage of the motor controlled in velocity. |

${V}_{c}^{\theta}$ | Control voltage of the motor controlled in position. |

Energies and energy savings | |

E | Energy consumed by the bicycle. |

${E}_{v}$ | Energy used for the forward motion control. |

${E}_{\theta}$ | Energy used for the lateral stability control. |

$ES$ | Energy saving relative to the total energy consumption of the system by using a non-adaptive control. |

$E{S}_{v}$ | Energy saving relative to the energy consumption of the forward velocity control system. |

$E{S}_{\theta}$ | Energy saving relative to the energy consumption of the lateral stability control system by using a non-adaptive control. |

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**Figure 3.**(

**a**) Control loop with a classic $PI$ controller; (

**b**) Control loop with an adaptive $PI$ controller.

**Figure 6.**(

**a**) Value of the ${K}_{p}$ controller gain according to the bicycle forward velocity. (

**b**) Value of the ${K}_{i}$ controller gain according to the bicycle forward velocity.

**Figure 7.**(

**a**) Energy consumption of the lateral stability control system with adaptive controller and non-adaptive controller; (

**b**) Energy consumption of the forward velocity control system; (

**c**) Total energy consumption of the bicycle.

**Figure 9.**Energy saving relative to the energy consumption of the lateral stability control system of the bicycle by using a non-adaptive control.

**Figure 11.**Energy saving relative to the total energy consumption of the system by using a non-adaptive control.

**Figure 13.**(

**a**) Forward velocity profile; (

**b**) Desired trajectory; (

**c**) Desired yaw angle; (

**d**) Desired roll angle.

**Figure 14.**Desired trajectory (${P}^{*}$), trajectory followed by the bicycle with the adaptive controller (${P}^{A}$) and trajectory followed by the bicycle with the non-adaptive controller (${P}^{NA}$).

**Figure 15.**Total energy consumption by the bicycle with the adaptive (${E}^{A}$) and non-adaptive controller (${E}^{NA}$).

Bicycle Model | Forward Motor Model | Rotation Motor Model |
---|---|---|

h = 0.39 m | ||

d = 0.34 m | ${A}^{v}$ = 13.69 A/(m${}^{2}$ V Kg ) | ${A}^{\theta}$ = 9.12 A/(m${}^{2}V$ Kg) |

w = 0.71 m | ${B}^{v}$ = 15.24 s${}^{-1}$ m${}^{-2}$ | ${B}^{\theta}$ = 15.24 s${}^{-1}$ m${}^{-2}$ |

g = 9.81 m/s${}^{2}$ |

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**MDPI and ACS Style**

Rodriguez-Rosa, D.; Payo-Gutierrez, I.; Castillo-Garcia, F.J.; Gonzalez-Rodriguez, A.; Perez-Juarez, S. Improving Energy Efficiency of an Autonomous Bicycle with Adaptive Controller Design. *Sustainability* **2017**, *9*, 866.
https://doi.org/10.3390/su9050866

**AMA Style**

Rodriguez-Rosa D, Payo-Gutierrez I, Castillo-Garcia FJ, Gonzalez-Rodriguez A, Perez-Juarez S. Improving Energy Efficiency of an Autonomous Bicycle with Adaptive Controller Design. *Sustainability*. 2017; 9(5):866.
https://doi.org/10.3390/su9050866

**Chicago/Turabian Style**

Rodriguez-Rosa, David, Ismael Payo-Gutierrez, Fernando J. Castillo-Garcia, Antonio Gonzalez-Rodriguez, and Sergio Perez-Juarez. 2017. "Improving Energy Efficiency of an Autonomous Bicycle with Adaptive Controller Design" *Sustainability* 9, no. 5: 866.
https://doi.org/10.3390/su9050866