# Path Planning and Guidance Laws of a Formula Student Driverless Car

^{*}

## Abstract

**:**

## 1. Introduction

- Develop an algorithm for path planning under the assumptions that there is an a priori knowledge of the track layout and an algorithm for online obstacle avoidance considering static obstacles;
- Design different control strategies to effectively steer the vehicle, as well as a low-level controller, in order to obtain more accurate results, ensure vehicle stability, and avoid wheel lock or spin;
- Test, evaluate, and compare the different algorithms, using a developed, realistic model of the vehicle.

- Use the centerline to place the attractive force, to cut the corners while keeping within track;
- Implement decoupled planning (not just decoupled control);
- Deal with obstacle avoidance using transverse forces, to ensure that they are overcome without unduly decelerating the movement and, thus, wasting energy;
- Values for the observation, warning, and danger radius should vary continuously with velocity (using a spline).

## 2. Vehicle Modelling

#### 2.1. Realistic Model

- The linear $\mathbf{v}$ and angular $\mathrm{\Omega}$ velocities of the centre of gravity (CG) expressed in the local frame (6 variables);
- The CG position ${\mathbf{p}}_{CG}$ expressed in the global frame (3 variables);
- The Euler angles $\mathrm{\Phi}$ associated with the rotations from global to local frame (3 variables);
- The angular speeds of each of the four wheels $\mathrm{\omega}$ (4 variables).

- The four wheel torques ${\mathbf{t}}_{w}$;
- The steering angle $\mathrm{\delta}$.

- The suspension deformations ${\mathbf{\Delta}}_{\mathit{z}}$;
- The slip ratios $\kappa $;
- The slip angles $\alpha $;
- The forces ${\mathbf{f}}_{x},{\mathbf{f}}_{y},{\mathbf{f}}_{z}$ and moments ${\mathbf{m}}_{z}$ resulting from the tire–ground interaction.

**Powertrain**

**Steering**

**Suspension**

**Tires**

#### 2.2. Simplified Models

**Lateral bicycle dynamic model**

**Bicycle dynamic model in terms of tracking errors**

## 3. Planning Algorithms

#### 3.1. Path Planning

**Reference path**

**Reference speed**

- →
- Backward pass

- A user-defined limit ${v}_{{x}_{\mathrm{lim}}}$;

- →
- Forward pass

- →
- Powertrain constraint

#### 3.2. Obstacle Avoidance

**Reference path**

- Track limits and obstacles are differentiated, meaning that they are defined by different repulsive fields;
- The obstacles’ repulsive force is forced to have the direction of the vehicle’s closest edge, instead of being perpendicular to the contour lines from the different danger levels;
- The ability to check if a given obstacle was already overtaken is included (when the CG is ahead of all the edges of the rectangle representing that same obstacle);
- The ability to change the repulsive and attractive gains if a collision is predicted (by projecting the current trajectory a fixed distance ahead) is incorporated.

**Reference speed**

- Since a negative velocity is not allowed in FS competitions, the profile is changed to take this constraint into account;
- If no possible passage is detected, the reference speed is set to zero;
- Due to possible chattering caused by a linear piecewise profile, a cubic spline interpolation is performed between the different radii, allowing a smooth transition between regions;
- To avoid an “overshoot” in the observation zone, the velocity associated with this outer radius is slightly decreased from the maximum velocity;
- ${R}_{O},\phantom{\rule{4pt}{0ex}}{R}_{W}$ and ${R}_{D}$ are parameterised as a function of the velocity, with a linear relation, and not established as fixed values. ${R}_{C}$ is fixed, encompassing the vehicle with an extra safety distance.

## 4. Decoupled Control Approach

#### 4.1. Available Sensors and Required Estimations

#### 4.2. Longitudinal Control

#### 4.3. Lateral Control

**Cross-track and heading errors**

**Control strategies:**

- →
- Pure pursuit (PP)

- →
- Linear quadratic Gaussian (LQG)

- →
- Kinematics lateral speed (KLS)

- →
- Modified sliding mode (MSM)

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SAE | Society of Automotive Engineers |

GPS | Global Positioning System |

FS | Formula Student |

FST | Formula Student team from Instituto Superior Técnico—Universidade de Lisboa |

LiDAR | Light detection and ranging |

MPC | Model predictive control |

DOF | Degree of freedom |

4WD | Four-wheel driven |

CG | Centre of gravity |

RWD | Rear-wheel driven |

PP | Pure pursuit |

LQG | Linear quadratic Gaussian |

LQR | Linear quadratic regulator |

ARE | Algebraic Riccati equation |

KLS | Kinematics lateral speed |

MSM | Modified sliding mode |

FSG | Formula Student Germany |

FSI | Formula Student Italy |

RMS | Root-mean-square |

DNF | Did not finish |

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**Figure 5.**Position error ${e}_{y}$ and heading error ${e}_{\psi}$ in a path-following vehicle. The vehicle reference point is represented as the CG, but another location could also be considered.

**Figure 6.**Evolution of U with d for several values of $\gamma $: the higher the value of $\gamma $, the lower the danger of the boundary. In this figure, two different potential fields are represented, one for each limit. As such, U increases from bottom to top $(\uparrow )$ for both limits, but d increases from left to right $(\to )$ for the left limit and from right to left $(\leftarrow )$ for the right limit.

Notation | Description | Value | Units |
---|---|---|---|

m | Mass | 256 | kg |

${I}_{xx}$ | Moment of inertia around x | 39.00 | kg$\xb7{\mathrm{m}}^{2}$ |

${I}_{yy}$ | Moment of inertia around y | 141.61 | kg$\xb7{\mathrm{m}}^{2}$ |

${I}_{zz}$ | Moment of inertia around z | 160.62 | kg$\xb7{\mathrm{m}}^{2}$ |

R | Wheels radius | 0.228 | m |

${L}_{F}$ | Distance of front axis to the CG | 0.816 | m |

${L}_{R}$ | Distance of rear axis to the CG | 0.724 | m |

L | Wheelbase | 1.54 | m |

${L}_{W}$ | Track width | 1.20 | m |

${L}_{H}$ | CG height | 0.265 | m |

${A}_{P}$ | Projected frontal area | 1.05 | ${\mathrm{m}}^{2}$ |

${J}_{w}$ | Wheels’ rotational inertia | 0.24 | kg$\xb7{\mathrm{m}}^{2}$ |

${a}_{w}$ | Half-length of contact patch | 0.06 | m |

For the realistic model only | |||

${C}_{t}$ | CG translation coefficient | 0.8 | kg/m |

${C}_{d}$ | CG downforce coefficient | 1.96 | kg/m |

${C}_{r}$ | CG rotation coefficient | 0.001 | kg$\xb7{\mathrm{s}}^{2}$ |

${C}_{{r}_{\omega}}$ | Wheels’ rotation coefficient | 0.003 | kg$\xb7{\mathrm{s}}^{2}$ |

c | Damping coefficient for each suspension quarter | 2000 | N·s/m |

${k}_{F}$ | Spring constant for front suspension quarter | 52490 | N/m |

${k}_{R}$ | Spring constant for rear suspension quarter | 45000 | N/m |

${r}_{{\mathrm{motion}}_{F}}$ | Front suspension quarter motion ratio | 1.11 | – |

${r}_{{\mathrm{motion}}_{R}}$ | Rear suspension quarter motion ratio | 1.14 | – |

Notation | Value for FSG | Value for FSI | Units |
---|---|---|---|

${K}_{att}$ | 1 | 1 | N/m |

${K}_{rep}$ | 2 | 1 | N |

$\gamma $ | 10 | 2.5 | - |

$\mathrm{offset}$ | 4 | 6 | - |

${d}_{\mathrm{offset}}$ | 6.16 | 6.54 | m |

${d}_{\mathrm{min}}$ | 0.75 | 0.75 | m |

$\Delta s$ | 1.50 | 1.04 | m |

${\eta}_{\mathrm{trans}}$ | 0.70 | 0.70 | - |

${v}_{{x}_{\mathrm{lim}}}$ | 26.5 | 26.5 | m/s |

FSG | FSI | |||||
---|---|---|---|---|---|---|

Reference Path | t(s) | Improvement | t(s) | Improvement | ||

Centerline | 30.86 s | – | – | 24.43 s | – | – |

Potential Field | 27.00 s | 3.86 s | 12.50% | 21.59 s | 2.84 s | 11.63% |

FSG Track | FSI Track | ||||||
---|---|---|---|---|---|---|---|

Controller | ${\mathit{L}}_{\mathbf{l}\mathbf{a}\mathbf{d}}$ | RMS(${\mathit{e}}_{\mathit{y}}$) (m) | Time (s) | Penalty | RMS(${\mathit{e}}_{\mathit{y}}$) (m) | Time (s) | Penalty |

PP | 1 | 0.05 | 27.62 | – | 0.07 | 22.15 | – |

2 | 0.07 | 27.59 | – | 0.12 | 22.19 | – | |

LQG | 1 | 27.69 | 0.04 | – | 0.05 | 22.26 | – |

2 | – | – | DNF | – | – | DNF | |

KLS | – | 0.05 | 27.67 | – | 0.05 | 22.13 | – |

MSM | – | 0.10 | 27.62 | – | 0.11 | 22.13 | – |

${\mathit{R}}_{\mathit{O}}$ | ${\mathit{R}}_{\mathit{W}}$ | ${\mathit{R}}_{\mathit{D}}$ | ${\mathit{R}}_{\mathit{C}}$ | |
---|---|---|---|---|

Distance (m) | $1.5\phantom{\rule{2.84544pt}{0ex}}{v}_{x}$ | $1.2\phantom{\rule{2.84544pt}{0ex}}{v}_{x}$ | $0.9\phantom{\rule{2.84544pt}{0ex}}{v}_{x}$ | $1.5\phantom{\rule{2.84544pt}{0ex}}L$ |

Velocity (m/s) | ${v}_{{x}_{\mathrm{max}}}$ | ${v}_{{x}_{\mathrm{max}}}-3$ | ${v}_{{x}_{\mathrm{max}}}/2-1$ | 0 |

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## Share and Cite

**MDPI and ACS Style**

Santos, S.D.R.; Azinheira, J.R.; Botto, M.A.; Valério, D.
Path Planning and Guidance Laws of a Formula Student Driverless Car. *World Electr. Veh. J.* **2022**, *13*, 100.
https://doi.org/10.3390/wevj13060100

**AMA Style**

Santos SDR, Azinheira JR, Botto MA, Valério D.
Path Planning and Guidance Laws of a Formula Student Driverless Car. *World Electric Vehicle Journal*. 2022; 13(6):100.
https://doi.org/10.3390/wevj13060100

**Chicago/Turabian Style**

Santos, Solange D. R., José Raul Azinheira, Miguel Ayala Botto, and Duarte Valério.
2022. "Path Planning and Guidance Laws of a Formula Student Driverless Car" *World Electric Vehicle Journal* 13, no. 6: 100.
https://doi.org/10.3390/wevj13060100