# Non-Parametric Calibration of the Inverse Kinematic Matrix of a Three-Wheeled Omnidirectional Mobile Robot Based on Genetic Algorithms

^{*}

## Abstract

**:**

## Featured Application

**Odometry calibration of a three-wheeled omnidirectional mobile robot.**

## Abstract

## 1. Introduction

#### New Contribution

## 2. Materials and Methods

#### 2.1. APR-02 Three-Wheeled Omnidirectional Mobile Robot

#### 2.2. Odometry Estimation

#### 2.3. Calibration Trajectories

#### 2.4. Dataset of Training and Validation Trajectories

## 3. Procedure for Genetic Algorithm Calibration of the Inverse Kinematic Matrix

^{−6}, which is the default value used in standard GA searches.

## 4. Results

#### 4.1. Reference Theoretical Value of the Inverse Kinematic Matrix

#### 4.2. Reference Parametric Optimization of the Inverse Kinematic Matrix (from [17])

#### 4.3. Non-Parametic Inverse Kinematic Matrix Calibrated with Genetic Algorithms

^{−17}. Alternatively, the result of the parametric calibration performed in [17] (displayed in Equation (8)) showed ${\mathrm{m}}_{1,2}$ to reach the value of $-0.00056$, which is very difficult to interpret as a parametric inaccuracy originated during the assembling the robot. The differences between the parametric and non-parametric calibration of the inverse kinematic matrices are small but cannot be neglected because the odometry is cumulatively updated ten times per second with the displacement information provided by the encoders.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**APR-02 mobile robot performing a transversal (or lateral) displacement: (

**a**) starting point; (

**b**,

**c**) intermediate points; and (

**d**) destination point.

**Figure 2.**Detail of the omnidirectional motion system of the APR-02 mobile robot. $\left({X}_{R},{Y}_{R}\right)$ is the mobile robot frame in which ${X}_{R}$ represents the front and forward direction of the robot.

**Figure 3.**Parametric definition of the omnidirectional motion system of the APR-02 mobile robot. $\left({X}_{R},{Y}_{R}\right)$ represents the mobile robot frame in which ${X}_{R}$ is the front of the mobile robot.

**Figure 4.**Diagram of the Genetic Algorithm search performed to calibrate the inverse kinematic matrix of the APR-02 mobile robot.

**Figure 5.**Sample trajectory followed by the APR-02 mobile robot for the motion command $\left(v,\alpha ,\omega \ne 0\right)$ = (0.2 m/s, 30°, 0.35 rad/s). The ground truth trajectory was estimated from the information of the LIDAR (red line) while the odometry was estimated with the theoretical value of the inverse kinematic matrix of the mobile robot (blue line).

**Figure 6.**Comparison between the ground truth trajectory followed by the mobile robot (red line) and the odometry estimated with: the theoretical IK (green line), the parametric IK (brown line) and the non-parametric IK (magenta line). Trajectories originated by the following motion commands, $\left(v,\alpha ,\omega \right)$: (

**a**) (0.2 m/s, 30°, 0.35 rad/s); (

**b**) (0.2 m/s, 210°, 0.35 rad/s); (

**c**) (0.2 m/s, 120°, 0.35 rad/s); and (

**d**) (0.2 m/s, 300°, 0.35 rad/s).

**Figure 7.**Detail of the final position of the robot corresponding to the full trajectories displayed in Figure 6. Ground truth trajectory of the mobile robot (red) and odometry of the mobile robot computed using the theoretical IK (green), the parametric IK [17] (brown) and the noon-parametric IK (magenta).

Parameter | Description |
---|---|

$\left(v,\alpha ,\omega ,d\right)$ | Target motion command for the robot $v$: translational velocity of the displacement of the robot, (m/s) $\alpha $: angular orientation of the displacement, (°) $\omega $: angular rotational speed of the robot during the displacement, (rad/s) $d$: linear distance to be achieved during the displacement, (m) |

$\left({\omega}_{Ma},{\omega}_{Mb},{\omega}_{Mc}\right)$ | Target angular velocities for the wheels ${\omega}_{Mi}$: target angular velocity defined for the motor of the wheel $i$, computed when receiving the motion command, (rpm) |

${\omega}_{a,b,c}=\left[\begin{array}{c}t\left(k=1\dots F\right)\\ {\omega}_{a}\left(k=1\dots F\right)\\ {\omega}_{b}\left(k=1\dots F\right)\\ {\omega}_{c}\left(k=1\dots F\right)\end{array}\right]$ | Angular velocities of the wheels provided by the encoders $t\left(k\right)$: time the update $k$ was received, (s) ${\omega}_{i}$: instantaneous angular velocity measured by the encoder of the motor $i$, updated periodically at a frame rate of 10 ms, (rpm) |

$\left(x,y,\theta \right)=\left[\begin{array}{c}x\left(k=1\dots F\right)\\ y\left(k=1\dots F\right)\\ \theta \left(k=1\dots F\right)\end{array}\right]$ | Trajectory of the mobile robot estimated with the odometry $x,y$: location of the robot, (m) $\theta $: angular orientation of the robot, (°) |

$LIDAR=\left[\begin{array}{ccc}{t}_{LIDAR}\left(p=1\right)& & {t}_{LIDAR}\left(p=N\right)\\ {\left[\begin{array}{c}d\left(1\right)\\ \dots \\ d\left(1080\right)\end{array}\right]}^{1}& \dots & {\left[\begin{array}{c}d\left(1\right)\\ \dots \\ d\left(1080\right)\end{array}\right]}^{N}\end{array}\right]$ | Scans provided by the onboard LIDAR ${t}_{LIDAR}\left(p\right)$: time the scan $p$ was received, (s) $d\left(r\right)$: distance scan corresponding to the angular orientation $r$, updated periodically at a frame rate from 200 to 300 ms, (mm) |

${\left(x,y,\theta \right)}_{LIDAR}=\left[\begin{array}{c}{x}_{LIDAR}\left(p=1\dots N\right)\\ {y}_{LIDAR}\left(p=1\dots N\right)\\ {\theta}_{LIDAR}\left(p=1\dots N\right)\end{array}\right]$ | Ground Truth trajectory of the mobile robot estimated with SLAM ${x}_{LIDAR},{y}_{LIDAR}$: location of the robot, (m) ${\theta}_{LIDAR}$: angular orientation of the robot, (°) |

**Table 2.**Values of the cost function obtained when computing the odometry of the 180 trajectories included in the training dataset with the theoretical and calibrated inverse kinematic matrices.

${\mathit{M}}^{-1}$ | $\mathit{C}{\mathit{F}}_{\mathit{T}\mathit{h}\mathit{e}\mathit{o}\mathit{r}\mathit{e}\mathit{t}\mathit{i}\mathit{c}\mathit{a}\mathit{l}\mathit{I}\mathit{K}}$ | $\mathit{C}{\mathit{F}}_{\mathit{T}\mathit{R}\mathit{A}\mathit{I}\mathit{N}\mathit{I}\mathit{N}\mathit{G}}$ | Improvement |
---|---|---|---|

Theoretical IK | 0.1234 | - | - |

Parametric IK [17] | 0.1234 | 0.0215 | 82.61% |

Non-parametric IK * | 0.1234 | 0.0215 * | 82.60% * |

**Table 3.**Values of the cost function obtained when computing the odometry of the 180 trajectories included in the validation dataset with the theoretical and calibrated inverse kinematic matrices.

${\mathit{M}}^{-1}$ | $\mathit{C}{\mathit{F}}_{\mathit{T}\mathit{h}\mathit{e}\mathit{o}\mathit{r}\mathit{e}\mathit{t}\mathit{i}\mathit{c}\mathit{a}\mathit{l}\mathit{I}\mathit{K}}$ | $\mathit{C}{\mathit{F}}_{\mathit{V}\mathit{A}\mathit{L}\mathit{I}\mathit{D}\mathit{A}\mathit{T}\mathit{I}\mathit{O}\mathit{N}}$ | Improvement |
---|---|---|---|

Theoretical IK | 0.1251 | - | - |

Parametric IK [17] | 0.1251 | 0.0229 | 81.68% |

Non-parametric IK * | 0.1251 | 0.0227 * | 81.81% * |

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**MDPI and ACS Style**

Palacín, J.; Rubies, E.; Bitrià, R.; Clotet, E. Non-Parametric Calibration of the Inverse Kinematic Matrix of a Three-Wheeled Omnidirectional Mobile Robot Based on Genetic Algorithms. *Appl. Sci.* **2023**, *13*, 1053.
https://doi.org/10.3390/app13021053

**AMA Style**

Palacín J, Rubies E, Bitrià R, Clotet E. Non-Parametric Calibration of the Inverse Kinematic Matrix of a Three-Wheeled Omnidirectional Mobile Robot Based on Genetic Algorithms. *Applied Sciences*. 2023; 13(2):1053.
https://doi.org/10.3390/app13021053

**Chicago/Turabian Style**

Palacín, Jordi, Elena Rubies, Ricard Bitrià, and Eduard Clotet. 2023. "Non-Parametric Calibration of the Inverse Kinematic Matrix of a Three-Wheeled Omnidirectional Mobile Robot Based on Genetic Algorithms" *Applied Sciences* 13, no. 2: 1053.
https://doi.org/10.3390/app13021053