# Motion-Tracking Control of Mobile Manipulation Robotic Systems Using Artificial Neural Networks for Manufacturing Applications

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling of the Mobile Manipulation Robotic System

#### 2.1. System Description

#### 2.2. Kinematic Modeling

#### 2.3. Dynamic Modeling

## 3. A Motion-Control Approach for the Mobile Manipulation Robotic System

## 4. Mobile Manipulation Robotic System Motion Control Using Artificial Neural Networks

## 5. Numeric Simulation Results

#### 5.1. Position Reference Tracking Control in the Joint Space

#### 5.2. Position Reference Tracking Control in the Joint Space Subjected to External Vibratory Torques

#### 5.3. Trajectory Tracking Control in Cartesian Space

#### 5.4. Trajectory Tracking Control in Cartesian Space with Parameter Variation

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANN | Artificial Neural Network |

BS-ANN | B-Spline Artificial Neural Network |

DC | Direct Current |

DOF | Degrees of Freedom |

PD | Proportional Derivative |

PID | Proportional Integral Derivative |

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**Figure 8.**Controlled linear motion of the mobile robot in its local coordinate frame. (

**a**) Controlled linear position ${p}_{{m}_{l}}$. (

**b**) Computed driving force ${F}_{{m}_{l}}$. (

**c**) Linear position error $Er{r}_{{m}_{l}}$. (

**d**) Adaptive ${\omega}_{{m}_{l}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{m}_{l}}$ control parameter. (

**f**) Adaptive ${P}_{{m}_{l}}$ control parameter.

**Figure 9.**Controlled angular motion of the mobile robot in its local coordinate frame. (

**a**) Controlled angular position ${p}_{{m}_{a}}$. (

**b**) Computed driving torque ${\tau}_{{m}_{a}}$. (

**c**) Angular position error $Er{r}_{{m}_{a}}$. (

**d**) Adaptive ${\omega}_{{m}_{a}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{m}_{a}}$ control parameter. (

**f**) Adaptive ${P}_{{m}_{a}}$ control parameter.

**Figure 10.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{1}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{1}}$. (

**c**) Angular position error $Er{r}_{{q}_{1}}$. (

**d**) Adaptive ${\omega}_{{q}_{1}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{q}_{1}}$ control parameter. (

**f**) Adaptive ${P}_{{q}_{1}}$ control parameter.

**Figure 11.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{2}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{2}}$. (

**c**) Angular position error $Er{r}_{{q}_{2}}$. (

**d**) Adaptive ${\omega}_{{q}_{2}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{q}_{2}}$ control parameter. (

**f**) Adaptive ${P}_{{q}_{2}}$ control parameter.

**Figure 12.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{3}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{3}}$. (

**c**) Angular position error $Er{r}_{{q}_{3}}$. (

**d**) Adaptive ${\omega}_{{q}_{3}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{q}_{3}}$ control parameter. (

**f**) Adaptive ${P}_{{q}_{3}}$ control parameter.

**Figure 13.**Induced external vibratory torques. (

**a**) Vibratory force ${F}_{{d}_{L}}$. (

**b**) Vibratory torque ${\tau}_{{d}_{A}}$. (

**c**) Vibratory torque ${\tau}_{{d}_{1}}$. (

**d**) Vibratory torque ${\tau}_{{d}_{2}}$. (

**e**) Vibratory torque ${\tau}_{{d}_{3}}$.

**Figure 14.**Controlled linear motion of the mobile robot in its local coordinate frame subjected to the external force ${F}_{{d}_{L}}$. (

**a**) Controlled linear position ${p}_{{m}_{l}}$. (

**b**) Computed driving force ${F}_{{m}_{l}}$. (

**c**) Linear position error $Er{r}_{{m}_{l}}$. (

**d**) Adaptive ${\omega}_{{m}_{l}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{m}_{l}}$ control parameter. (

**f**) Adaptive ${P}_{{m}_{l}}$ control parameter.

**Figure 15.**Controlled angular motion of the mobile robot in its local coordinate frame subjected to the external torque ${\tau}_{{d}_{A}}$. (

**a**) Controlled angular position ${p}_{{m}_{a}}$. (

**b**) Computed driving torque ${\tau}_{{m}_{a}}$. (

**c**) Angular position error $Er{r}_{{m}_{a}}$. (

**d**) Adaptive ${\omega}_{{m}_{a}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{m}_{a}}$ control parameter. (

**f**) Adaptive ${P}_{{m}_{a}}$ control parameter.

**Figure 16.**Controlled angular motion of the manipulator robot in its joint space subjected to the external torque ${\tau}_{{d}_{1}}$. (

**a**) Controlled angular position ${p}_{{q}_{1}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{1}}$. (

**c**) Angular position error $Er{r}_{{q}_{1}}$. (

**d**) Adaptive ${\omega}_{{q}_{1}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{q}_{1}}$ control parameter. (

**f**) Adaptive ${P}_{{q}_{1}}$ control parameter.

**Figure 17.**Controlled angular motion of the manipulator robot in its joint space subjected to the external torque ${\tau}_{{d}_{2}}$. (

**a**) Controlled angular position ${p}_{{q}_{2}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{2}}$. (

**c**) Angular position error $Er{r}_{{q}_{2}}$. (

**d**) Adaptive ${\omega}_{{q}_{2}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{q}_{2}}$ control parameter. (

**f**) Adaptive ${P}_{{q}_{2}}$ control parameter.

**Figure 18.**Controlled angular motion of the manipulator robot in its joint space subjected to the external torque ${\tau}_{{d}_{3}}$. (

**a**) Controlled angular position ${p}_{{q}_{3}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{3}}$. (

**c**) Angular position error $Er{r}_{{q}_{3}}$. (

**d**) Adaptive ${\omega}_{{q}_{3}}$ control parameter. (

**e**) Adaptive ${\zeta}_{{q}_{3}}$ control parameter. (

**f**) Adaptive ${P}_{{q}_{3}}$ control parameter.

**Figure 19.**Controlled linear motion of the mobile robot in its local coordinate frame subjected to the external force ${F}_{{d}_{L}}$ with nonlinear PD-like control. (

**a**) Controlled linear position ${p}_{{m}_{l}}$. (

**b**) Computed driving force ${F}_{{m}_{l}}$. (

**c**) Linear position error $Er{r}_{{m}_{l}}$.

**Figure 20.**Controlled angular motion of the mobile robot in its local coordinate frame subjected to the external torque ${\tau}_{{d}_{A}}$ with nonlinear PD-like control. (

**a**) Controlled angular position ${p}_{{m}_{a}}$. (

**b**) Computed driving torque ${\tau}_{{m}_{a}}$. (

**c**) Angular position error $Er{r}_{{m}_{a}}$.

**Figure 21.**Controlled angular motion of the manipulator robot in its joint space subjected to the external torque ${\tau}_{{d}_{1}}$ with nonlinear PD-like control. (

**a**) Controlled angular position ${p}_{{q}_{1}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{1}}$. (

**c**) Angular position error $Er{r}_{{q}_{1}}$.

**Figure 22.**Controlled angular motion of the manipulator robot in its joint space subjected to the external torque ${\tau}_{{d}_{2}}$ with nonlinear PD-like control. (

**a**) Controlled angular position ${p}_{{q}_{2}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{2}}$. (

**c**) Angular position error $Er{r}_{{q}_{2}}$.

**Figure 23.**Controlled angular motion of the manipulator robot in its joint space subjected to the external torque ${\tau}_{{d}_{3}}$ with nonlinear PD-like control. (

**a**) Controlled angular position ${p}_{{q}_{3}}$. (

**b**) Computed driving torque ${\tau}_{{q}_{3}}$. (

**c**) Angular position error $Er{r}_{{q}_{3}}$.

**Figure 25.**Trajectory tracking control of the mobile manipulation robotic system in Cartesian space. (

**a**) Three-dimensional trajectory tracking. (

**b**) Trajectory tracking on the x-axis; (

**c**) x-axis position error. (

**d**) Trajectory tracking on the y-axis; (

**e**) y-axis position error. (

**f**) Trajectory tracking on the z-axis; (

**g**) z-axis position error.

**Figure 26.**Controlled linear motion of the mobile robot in its local coordinate frame. (

**a**) Controlled linear position ${p}_{{m}_{l}}$. (

**b**) Performed linear velocity u. (

**c**) Computed driving force ${F}_{{m}_{l}}$. (

**d**) Linear position error $Er{r}_{{m}_{l}}$. (

**e**) Adaptive ${\omega}_{{m}_{l}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{m}_{l}}$ control parameter. (

**g**) Adaptive ${P}_{{m}_{l}}$ control parameter.

**Figure 27.**Controlled angular motion of the mobile robot in its local coordinate frame. (

**a**) Controlled angular position ${p}_{{m}_{a}}$. (

**b**) Performed angular velocity $\dot{\varphi}$. (

**c**) Computed driving torque ${\tau}_{{m}_{a}}$. (

**d**) Angular position error $Er{r}_{{m}_{a}}$. (

**e**) Adaptive ${\omega}_{{m}_{a}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{m}_{a}}$ control parameter. (

**g**) Adaptive ${P}_{{m}_{a}}$ control parameter.

**Figure 28.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{1}}$. (

**b**) Performed angular velocity ${\dot{q}}_{1}$. (

**c**) Computed driving torque ${\tau}_{{q}_{1}}$. (

**d**) Angular position error $Er{r}_{{q}_{1}}$. (

**e**) Adaptive ${\omega}_{{q}_{1}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{q}_{1}}$ control parameter. (

**g**) Adaptive ${P}_{{q}_{1}}$ control parameter.

**Figure 29.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{2}}$. (

**b**) Performed angular velocity ${\dot{q}}_{2}$. (

**c**) Computed driving torque ${\tau}_{{q}_{2}}$. (

**d**) Angular position error $Er{r}_{{q}_{2}}$. (

**e**) Adaptive ${\omega}_{{q}_{2}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{q}_{2}}$ control parameter. (

**g**) Adaptive ${P}_{{q}_{2}}$ control parameter.

**Figure 30.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{3}}$. (

**b**) Performed angular velocity ${\dot{q}}_{3}$. (

**c**) Computed driving torque ${\tau}_{{q}_{3}}$. (

**d**) Angular position error $Er{r}_{{q}_{3}}$. (

**e**) Adaptive ${\omega}_{{q}_{3}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{q}_{3}}$ control parameter. (

**g**) Adaptive ${P}_{{q}_{3}}$ control parameter.

**Figure 31.**Trajectory tracking control of the mobile manipulation robotic system in the Cartesian space. (

**a**) Three-dimensional trajectory tracking. (

**b**) Trajectory tracking on the x-axis; (

**c**) x-axis position error. (

**d**) Trajectory tracking on the y-axis; (

**e**) y-axis position error. (

**f**) Trajectory tracking on the z-axis; (

**g**) z-axis position error.

**Figure 32.**Controlled linear motion of the mobile robot in its local coordinate frame. (

**a**) Controlled linear position ${p}_{{m}_{l}}$. (

**b**) Performed linear velocity u. (

**c**) Computed driving force ${F}_{{m}_{l}}$. (

**d**) Linear position error $Er{r}_{{m}_{l}}$. (

**e**) Adaptive ${\omega}_{{m}_{l}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{m}_{l}}$ control parameter. (

**g**) Adaptive ${P}_{{m}_{l}}$ control parameter.

**Figure 33.**Controlled angular motion of the mobile robot in its local coordinate frame. (

**a**) Controlled angular position ${p}_{{m}_{a}}$. (

**b**) Performed angular velocity $\dot{\varphi}$. (

**c**) Computed driving torque ${\tau}_{{m}_{a}}$. (

**d**) Angular position error $Er{r}_{{m}_{a}}$. (

**e**) Adaptive ${\omega}_{{m}_{a}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{m}_{a}}$ control parameter. (

**g**) Adaptive ${P}_{{m}_{a}}$ control parameter.

**Figure 34.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{1}}$. (

**b**) Performed angular velocity ${\dot{q}}_{1}$. (

**c**) Computed driving torque ${\tau}_{{q}_{1}}$. (

**d**) Angular position error $Er{r}_{{q}_{1}}$. (

**e**) Adaptive ${\omega}_{{q}_{1}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{q}_{1}}$ control parameter. (

**g**) Adaptive ${P}_{{q}_{1}}$ control parameter.

**Figure 35.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{2}}$. (

**b**) Performed angular velocity ${\dot{q}}_{2}$. (

**c**) Computed driving torque ${\tau}_{{q}_{2}}$. (

**d**) Angular position error $Er{r}_{{q}_{2}}$. (

**e**) Adaptive ${\omega}_{{q}_{2}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{q}_{2}}$ control parameter. (

**g**) Adaptive ${P}_{{q}_{2}}$ control parameter.

**Figure 36.**Controlled angular motion of the manipulator robot in its joint space. (

**a**) Controlled angular position ${p}_{{q}_{3}}$. (

**b**) Performed angular velocity ${\dot{q}}_{3}$. (

**c**) Computed driving torque ${\tau}_{{q}_{3}}$. (

**d**) Angular position error $Er{r}_{{q}_{3}}$. (

**e**) Adaptive ${\omega}_{{q}_{3}}$ control parameter. (

**f**) Adaptive ${\zeta}_{{q}_{3}}$ control parameter. (

**g**) Adaptive ${P}_{{q}_{3}}$ control parameter.

Variable | Definition |
---|---|

O | Location of the mobile robot’s center of mass |

a | Distance from the center of mass to the front axis of the mobile robot |

b | Distance from the center of mass to the rear axis of the mobile robot |

$2c$ | Distance between the wheels of the mobile robot |

r | Radius of each wheel of the mobile robot |

l | Height of the mobile robot |

${l}_{p}$ | Distance from the center of mass to the base of the manipulator robot |

${l}_{1}$ | First link length of the manipulator robot |

${l}_{{c}_{2}}$ | Second link length to the center of mass |

${l}_{2}$ | Second link length of the manipulator robot |

${l}_{{c}_{3}}$ | Third link length to the center of mass |

${l}_{3}$ | Third link length of the manipulator robot |

${m}_{p}$ | Mobile robot chassis mass |

${m}_{w}$ | Mass of each wheel of the mobile robot |

${m}_{1}$ | First link mass of the manipulator robot |

${m}_{2}$ | Second link mass of the manipulator robot |

${m}_{3}$ | Third link mass of the manipulator robot |

${I}_{{z}_{P}}$ | Moment of inertia around the Z-axis of the mobile robot |

${I}_{{y}_{w}}$ | Moment of inertia around the Y-axis of each wheel of the system |

${I}_{{z}_{w}}$ | Moment of inertia around the Z-axis of each wheel of the system |

${I}_{{z}_{1}}$ | Moment of inertia around the Z-axis of the first link |

${I}_{{y}_{2}}$ | Moment of inertia around the Y-axis of the second link |

${I}_{{z}_{2}}$ | Moment of inertia around the Z-axis of the second link |

${I}_{{y}_{3}}$ | Moment of inertia around the Y-axis of the third link |

${I}_{{z}_{3}}$ | Moment of inertia around the Z-axis of the third link |

u | Linear velocity of the mobile robot |

$\omega $ | Angular velocity of the mobile robot |

$\varphi $ | Angle of rotation in the Z-axis of the mobile robot |

${q}_{1}$ | Angle of rotation in the Z-axis of the first link |

${q}_{2}$ | Angle of rotation in the Y-axis of the second link |

${q}_{3}$ | Angle of rotation in the Y-axis of the third link |

${h}_{p}$ | XYZ position of the coupling point on the mobile robot |

${h}_{{q}_{3}}$ | XYZ position of the mobile manipulator robot’s end-effector |

g | Gravitational acceleration constant |

R | Inertial coordinate system |

**Table 2.**Parameters and constants of the mobile manipulation robotic system used in simulation scenarios.

Parameter | Quantity | Units | Description |
---|---|---|---|

c | 0.1920 | m | Distance from O to each side’s wheels of the robot |

r | 0.06 | m | Radius of each wheel of the mobile robot |

l | 0.2272 | m | Height of the mobile robot |

${l}_{p}$ | 0.1 | m | Distance from O to the base of the manipulator robot |

${l}_{1}$ | 0.0645 | m | First link length of the manipulator robot |

${l}_{2}$ | 0.2031 | m | Second link length of the manipulator robot |

${l}_{3}$ | 0.3018 | m | Third link length of the manipulator robot |

${m}_{p}$ | 7.1368 | Kg | Mobile robot chassis mass |

${m}_{w}$ | 0.18 | Kg | Mass of each wheel of the mobile robot |

${m}_{1}$ | 0.7238 | Kg | First link mass of the manipulator robot |

${m}_{2}$ | 0.8524 | Kg | Second link mass of the manipulator robot |

${m}_{3}$ | 0.5085 | Kg | Third link mass of the manipulator robot |

${I}_{{z}_{P}}$ | 0.2308 | Kg m^{2} | Inertia moment in the Z-axis of the mobile robot |

${I}_{{y}_{w}}$ | 0.0003 | Kg m^{2} | Inertia moment in the Y-axis of the wheels |

${I}_{{z}_{w}}$ | 0.0002 | Kg m^{2} | Inertia moment in the Z-axis of the wheels |

${I}_{{z}_{1}}$ | 0.0015 | Kg m^{2} | Inertia moment in the Z-axis of the first link |

${I}_{{y}_{2}}$ | 0.0054 | Kg m^{2} | Inertia moment in the Y-axis of the second link |

${I}_{{z}_{2}}$ | 0.0013 | Kg m^{2} | Inertia moment in the Z-axis of the second link |

${I}_{{y}_{3}}$ | 0.0031 | Kg m^{2} | Inertia moment in the Y-axis of the third link |

${I}_{{z}_{3}}$ | 0.0032 | Kg m^{2} | Inertia moment in the Z-axis of the third link |

g | 9.81 | m/s^{2} | Gravitational acceleration constant |

**Table 3.**Parameters and constants of the mobile manipulation robotic system used for the fourth simulation scenario.

Parameter | Quantity | Units | Description |
---|---|---|---|

c | 0.25 | m | Distance from O to each side’s wheels of the robot |

r | 0.075 | m | Radius of each wheel of the mobile robot |

l | 0.4 | m | Height of the mobile robot |

${l}_{p}$ | 0.27 | m | Distance from O to the base of the manipulator robot |

${l}_{1}$ | 0.2 | m | First link length of the manipulator robot |

${l}_{2}$ | 0.3 | m | Second link length of the manipulator robot |

${l}_{3}$ | 0.3 | m | Third link length of the manipulator robot |

${m}_{p}$ | 19.2372 | Kg | Mobile robot chassis mass |

${m}_{w}$ | 0.901 | Kg | Mass of each wheel of the mobile robot |

${m}_{1}$ | 3.293 | Kg | First link mass of the manipulator robot |

${m}_{2}$ | 3.436 | Kg | Second link mass of the manipulator robot |

${m}_{3}$ | 1.229 | Kg | Third link mass of the manipulator robot |

${I}_{{z}_{P}}$ | 0.7004 | Kg m^{2} | Inertia moment in the Z-axis of the mobile robot |

${I}_{{y}_{w}}$ | 0.0015 | Kg m^{2} | Inertia moment in the Y-axis of the wheels |

${I}_{{z}_{w}}$ | 0.0030 | Kg m^{2} | Inertia moment in the Z-axis of the wheels |

${I}_{{z}_{1}}$ | 0.0230 | Kg m^{2} | Inertia moment in the Z-axis of the first link |

${I}_{{y}_{2}}$ | 0.0457 | Kg m^{2} | Inertia moment in the Y-axis of the second link |

${I}_{{z}_{2}}$ | 0.0461 | Kg m^{2} | Inertia moment in the Z-axis of the second link |

${I}_{{y}_{3}}$ | 0.0242 | Kg m^{2} | Inertia moment in the Y-axis of the third link |

${I}_{{z}_{3}}$ | 0.0255 | Kg m^{2} | Inertia moment in the Z-axis of the third link |

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## Share and Cite

**MDPI and ACS Style**

Galvan-Perez, D.; Beltran-Carbajal, F.; Rivas-Cambero, I.; Yañez-Badillo, H.; Favela-Contreras, A.; Tapia-Olvera, R.
Motion-Tracking Control of Mobile Manipulation Robotic Systems Using Artificial Neural Networks for Manufacturing Applications. *Mathematics* **2023**, *11*, 3489.
https://doi.org/10.3390/math11163489

**AMA Style**

Galvan-Perez D, Beltran-Carbajal F, Rivas-Cambero I, Yañez-Badillo H, Favela-Contreras A, Tapia-Olvera R.
Motion-Tracking Control of Mobile Manipulation Robotic Systems Using Artificial Neural Networks for Manufacturing Applications. *Mathematics*. 2023; 11(16):3489.
https://doi.org/10.3390/math11163489

**Chicago/Turabian Style**

Galvan-Perez, Daniel, Francisco Beltran-Carbajal, Ivan Rivas-Cambero, Hugo Yañez-Badillo, Antonio Favela-Contreras, and Ruben Tapia-Olvera.
2023. "Motion-Tracking Control of Mobile Manipulation Robotic Systems Using Artificial Neural Networks for Manufacturing Applications" *Mathematics* 11, no. 16: 3489.
https://doi.org/10.3390/math11163489