# Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control

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## Abstract

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## 1. Introduction

- Establish the dynamic stability of the system and make it self-balanced in the linear direction (forward and backward movement), with variations in rider input velocity, using the method of LQR control, and for a wide range of rider masses;
- Establish an LQG control mechanism to account for wind disturbances and perform a comparison with regular LQR controlled output by considering external disturbances such as wind and friction as noise to maintain dynamic stability in such conditions;
- Estimate position state output through Kalman Filter estimation;
- Develop a nonlinear reference model for the monowheel system to deal with higher-order uncertain dynamics in terms of the model parameters (rider mass, rider upper body height);

## 2. Problem Statements

#### 2.1. Control Objectives and Design Objectives

- Vehicle dynamics: This is the analysis of the reaction of the vehicle to driver inputs on a given solid surface. This part of engineering is predominantly based on classical mechanics. Aerodynamics, drive train and braking, mass distribution, suspension and steering, and tires are some of the factors that govern vehicle dynamics.
- Chassis designing: The outer frame supporting the driver should be robust and carry heavy loads without failure.
- Wheel mechanics: The calculation of forces acting on the wheel during the response using sensors such as the gyro sensor and accelerometer is required.
- Wheel hub design: This is where the real hardware comes together, requiring free movement between the wheel and driving systems.

#### 2.2. Selection of Parameters and Calculations

#### 2.3. Selection of Motor

## 3. Kinematics and Dynamics

#### 3.1. Linearization

## 4. Control Analysis

#### 4.1. Pole Placement Control

#### 4.2. LQR Control

## 5. Kalman Filter Estimation and LQG Control

#### 5.1. Factoring in Velocity

#### 5.2. Disturbance Adaptation

## 6. Reference Model Assisted Adaptive Control

#### 6.1. Lyapunov Stability Analysis

#### 6.2. Building the Adaptive Controlled System

#### 6.3. Simulation Results and Discussion

## 7. Concluding Remarks

- Self-balancing and stability have not been analyzed for cases with sudden braking;
- All analysis was conducted under the assumption that the frame’s mass is 35 kg and the maximum velocity to be attained is 20 km/h;
- A standard incline gradient ratio of 1:4000 feet is assumed. For inclinations greater than that, there will be a change in the calculations and analysis;
- After introducing a load disturbance, the system takes around 20 s to stabilize at the desired value.;
- The system is adaptively controllable only within the parameter ranges given in Table 4.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Step responses at various masses. (

**a**) Weight = 10 kg. (

**b**) Weight = 40 kg. (

**c**) Weight = 120 kg. (

**d**) Weight = 180 kg.

**Figure 7.**State responses, where x is position, $\theta $ is angle of pendulum from cart, ${x}^{\prime}$ is velocity, and ${\theta}^{\prime}$ is angular velocity.

**Figure 12.**Velocity responses with LQR and LQG control. (

**a**) Velocity response with LQR control. (

**b**) Velocity estimation during LQR control. (

**c**) Velocity response with LQG control. (

**d**) Velocity estimation during LQG control.

**Figure 16.**Velocity response of the system at different masses. (

**a**) Weight = 30 kg. (

**b**) Weight = 180 kg.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Gross weight of vehicle | ${W}_{g}$ | 110 | kg |

Maximum speed of vehicle | v | 18 | km/h |

Wheel diameter with tyre | D | 0.56 | m |

Maximum acceleration | a | $1.5$ | m/s${}^{2}$ |

Parameter | Formula/Symbol | Value | Unit |
---|---|---|---|

Coefficient of rolling resistance | ${C}_{r}$ | 0.02 | – |

Density of air | $\rho $ | $1.22556$ | kg/m${}^{3}$ |

Coefficient of air resistance | ${C}_{a}$ | 0.9 | – |

Exposed area to air friction | ${A}_{f}$ | $0.7$ | m${}^{2}$ |

Gravity constant | g | $9.81$ | m/s${}^{2}$ |

Gradient of surface | – | 1 in 4000 | feet |

Rolling force | ${F}_{r}={C}_{r}Wg$ | 21.58 | N |

Aerodynamic force | ${F}_{a}=\frac{1}{2}{C}_{a}{A}_{f}\rho {v}^{2}$ | 6.89 | N |

Gradient force | ${F}_{g}=Wg\phantom{\rule{2.84526pt}{0ex}}sin\left(\theta \right)$ | 0.27 | N |

Total force | $F={F}_{r}+{F}_{a}+{F}_{g}$ | 28.75 | N |

Tractive force | ${F}_{t}rac=W{a}_{max}$ | 45.83 | N |

Nominal power | $P=Fv$ | 143.73 | Watts |

Peak power | $P={F}_{trac}{v}_{max}$ | 229.167 | Watts |

RPM | $\frac{speed\phantom{\rule{2.84526pt}{0ex}}(m/min)}{Circum.\phantom{\rule{2.84526pt}{0ex}}of\phantom{\rule{2.84526pt}{0ex}}wheel}$ | 170 | – |

Nominal torque | $\tau =\frac{P\phantom{\rule{2.84526pt}{0ex}}\times \phantom{\rule{2.84526pt}{0ex}}95488}{RPM}$ | 8.05 | Nm |

Peak torque | ${\tau}_{peak}$ | 12.83 | Nm |

Parameter | Symbol | Unit |
---|---|---|

m | Mass of the pendulum rod | 75 kg |

M | Mass of the cart | 35 kg |

l | Pendulum length up to COG | 0.6 m |

J${}_{m}$ | Motor rotor moment of inertia | $3.26\times {10}^{-2}$ kg·m${}^{2}$ |

R${}_{m}$ | Motor armature resistance | $6.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

k${}_{b}$ | Motor back EMF constant | 0.013 V/rad/s |

k${}_{t}$ | Motor torque constant | 0.58 N·m/A |

R | Motor pinion radius | 0.04 m |

C | Friction coefficient for cart | 0.04 N/m/s |

I${}_{p}$ | Inertia of pendulum rod | 6.75 kg·m${}^{2}$ |

G | Gravitational constant | 9.81 m/s${}^{2}$ |

Parameter | Nominal Value | Bounds |
---|---|---|

${p}_{1}$ | 80 kg | 50–120 kg |

${p}_{2}$ | 1.67 m${}^{-1}$ | 1.25–2.5 m${}^{-1}$ |

${K}_{1}$ | [0.0221 63.223 0.0159] | - |

${K}_{2}$ | [0; 1; 0] | - |

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

${\alpha}_{1}$ | 0.23 |

${\alpha}_{2}$ | 2.8 |

${\alpha}_{3}$ | ${10}^{-8}$ |

${\beta}_{1}$ | 0.1 |

${\beta}_{2}$ | 2 |

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**MDPI and ACS Style**

Sengupta, I.; Gupta, S.; Deb, D.; Ozana, S.
Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control. *Appl. Sci.* **2021**, *11*, 9766.
https://doi.org/10.3390/app11209766

**AMA Style**

Sengupta I, Gupta S, Deb D, Ozana S.
Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control. *Applied Sciences*. 2021; 11(20):9766.
https://doi.org/10.3390/app11209766

**Chicago/Turabian Style**

Sengupta, Ipsita, Sagar Gupta, Dipankar Deb, and Stepan Ozana.
2021. "Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control" *Applied Sciences* 11, no. 20: 9766.
https://doi.org/10.3390/app11209766