# Model Predictive Control of a Novel Wheeled–Legged Planetary Rover for Trajectory Tracking

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hardware and Kinematics of the Rover

#### 2.1. Mechanical Structure

#### 2.2. Perception and Control System

#### 2.3. Kinemactics of the Rover

_{B}—X

_{B}Y

_{B}Z

_{B}} of the whole robot is located at the center of the plane and is composed of the centers of four hip joints. We established the D-H coordinate systems in Figure 4 for each leg. Because each leg has the same kinematic structure, the D-H parameters of the four legs are also the same. {O

_{0}—x

_{0}y

_{0}z

_{0}} is the base frame of each leg (i.e., the leg frame), which is located at the center of the hip joint. {O

_{4}—x

_{4}y

_{4}z

_{4}} is the wheel frame of each leg. The D-H parameters are shown in Table 1. Therefore, the transformation matrix from frame i − 1 to i for the ith limb can be written as

_{34}= sin(θ

_{3}+θ

_{4}), c

_{34}= cos(θ

_{3}+θ

_{4}), s

_{2}= sinθ

_{2}, c

_{2}= cosθ

_{2}, s

_{3}= sinθ

_{3}, and c

_{3}= cosθ

_{3}. Here, θ

_{1}= $\pi /2$ and $\mathit{P}={({P}_{x},\text{}{P}_{y}\text{}{P}_{z})}^{T}$. are the positions of the wheel center with respect to the leg frame.

**P**is given, the rotational angle of each joint, θ

_{2}, θ

_{3}, and θ

_{4}, can be obtained as

## 3. Control Strategy

#### 3.1. Locomotive Equations

**I**is the identity matrix.

#### 3.2. Trajectory Tracking Model Based on MPC

#### 3.2.1. Objective Function

_{p}and N

_{c}are the prediction and control horizons, respectively. Here,

**Q**and

**R**are the weighting matrices; $\rho $ is the weight coefficient, and $\epsilon $ is the relaxation factor.

#### 3.2.2. Constraints

**u**and control input increment Δ

**u**satisfy

#### 3.3. Streering Strategy

#### 3.4. Wheel Speed Allocation (WSA)

## 4. Results and Discussion

#### 4.1. Simulations

_{w}is the radius of the wheel; ${\omega}_{id}$ is the practical angular velocity of the wheel, which can be measured by the wheels’ encoders; and ${v}_{i}$ is the practical linear velocity of the wheel center. Figure 7 shows the comparison of the slippage percentages with and without the WSA module. Without the WSA module, the slippage reached up to 0.25, while with the WSA, the maximum of the slippage was less than 0.13. It was found that wheel slip was obviously decreased by the WSA component.

_{p}= 6; N

_{c}= 3; $\rho =10;$ $\epsilon =0$ for the lower limit and $\epsilon =10$ for the upper limit. The PID parameters for the linear velocity were set as: ${k}_{p1}=2,\text{}{k}_{i1}=1,\text{}{k}_{d1}=0$. Since there was only a linear velocity in the body frame, the PID module for the control of angular velocities did not work. Figure 8, Figure 9 and Figure 10 show the trajectory tracking results for three speeds. It was found that the robot could track the corresponding target values within a short period of time under the three different speeds. There was a large increase in trajectory errors at the beginning of the tracking process. The reason for this is that the initial direction of the target speed was the same as the x axis in the global coordinate system. There was an obvious delay before the actual speed reached the target value, and the speed error was relatively larger at the beginning. Furthermore, it was found that there were obvious overshoots in the velocity responses from the MPC method in the beginning. These overshoots facilitated trajectory tracking, and thus, the forward velocity of the robot could approximate the desired value quickly. In the meantime, the overshoot increased as the desired speed increases. The overshoot at 0.4 m/s was the largest one among the three forward speeds. It should also be noted that the final velocity response errors increased as the target speed increased. However, as a whole, the trajectory errors for the three speeds were all relatively small, validating the MPC module and the whole control strategy.

_{p}= 6; N

_{c}= 3; $\rho =10;$ $\epsilon =0$ for the lower limit and $\epsilon =10$ for the upper limit. The PID parameters for the linear velocity were set as: ${k}_{p1}=0.7,\text{}{k}_{i1}=0,\text{}{k}_{d1}=5$. Figure 11 gives the comparisons of the theoretical and real trajectories and velocities. It was found that the robot could track well the reference trajectory and the reference velocity when running along the S-type trajectory. Furthermore, the errors in the x and y coordinates were very small, the relative error of which were less than 2% and 1.75%, respectively, as seen in Figure 12.

_{p}= 6; N

_{c}= 3; $\rho =10;$ for the lower limit and $\epsilon =10$ for the upper limit. The PID parameters for the linear velocity were set as: ${k}_{p1}=2,\text{}{k}_{i1}=1,\text{}{k}_{d1}=0$. Figure 13 gives the changes of the real velocities. It was found that the robot could still track the reference velocity after a relatively short time. Moreover, with the control, the robot could track the reference trajectories of both the x coordinate and the y coordinate, depicted in Figure 14 and Figure 15. Compared to the lower speed, the position errors increased. However, the relative errors of the position points were small. The maxima of the relative position errors at 2 m/s and 4 m/s were less than 3% and 8.5%, respectively.

#### 4.2. Experiments

_{p}= 6; N

_{c}= 3; $\rho =10;$ $\epsilon =0$ for the lower limit and $\epsilon =10$ for the upper limit. The PID parameters for the control of linear velocities were set as: ${k}_{p1}=8,\text{}{k}_{i1}=0,\text{}{k}_{d1}=0.2$. Figure 18 shows the slippage in the experiments. Note that the slippages of Leg 1 and 4 (Leg 2 and 3) are almost the same because they suffer the same terrain condition. It was found that the average slippage of all of the legs demonstrated an obvious decrease after WSA control, up to 20%. Figure 19 shows the experimental results of trajectory tracking. As it can be seen, the rover could strictly track the reference trajectory. The deflection with respect to the reference trajectory was less than 2% F.S., which is a relatively small error. Accordingly, the control strategy was validated by the experiments.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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θ_{i} | d_{i} | α_{i} | a_{i} | |
---|---|---|---|---|

1 | θ_{1} | 0 | π/2 | 0 |

2 | θ_{2} | 0 | −π/2 | L_{1} = 87.5 |

3 | θ_{3} | 0 | 0 | L_{2} = 350 |

4 | θ_{4} | 0 | 0 | L_{3} = 310 |

Categories | Terminology | Definition |
---|---|---|

Input parameters | ${x}_{cr}^{w}$$,\text{}{y}_{cr}^{w}$ | Points on reference path |

${\gamma}_{r}^{w}$ | Desired yaw angle of body | |

${v}_{xr}^{B}$$,{\omega}_{zr}^{B}$ | Desired linear and angular velocities of body | |

Control parameters | N_{p}, N_{c} | Prediction and control horizons |

$\rho ,\text{}\epsilon $ | Weight coefficient and the relaxation factor | |

k_{p}_{1}, k_{i}_{1}, k_{d}_{1} | PID parameters for control of the linear velocity of body | |

k_{p}_{2}, k_{i}_{2}, k_{d}_{2} | PID parameters for control of the angular velocity of body | |

Output parameters | ${u}^{*}\left(t|t\right)=\left({v}_{x}^{B},{\omega}_{z}^{B}\right)$ | The linear and angular velocities of the body |

${\delta}_{rf}$$,{\delta}_{lf}$$,,{\delta}_{lr}$ | Steering angles | |

${\omega}_{id}$. | Rotational speeds of the wheel motors |

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**MDPI and ACS Style**

He, J.; Sun, Y.; Yang, L.; Gao, F. Model Predictive Control of a Novel Wheeled–Legged Planetary Rover for Trajectory Tracking. *Sensors* **2022**, *22*, 4164.
https://doi.org/10.3390/s22114164

**AMA Style**

He J, Sun Y, Yang L, Gao F. Model Predictive Control of a Novel Wheeled–Legged Planetary Rover for Trajectory Tracking. *Sensors*. 2022; 22(11):4164.
https://doi.org/10.3390/s22114164

**Chicago/Turabian Style**

He, Jun, Yanlong Sun, Limin Yang, and Feng Gao. 2022. "Model Predictive Control of a Novel Wheeled–Legged Planetary Rover for Trajectory Tracking" *Sensors* 22, no. 11: 4164.
https://doi.org/10.3390/s22114164