# Searching for a Cheap Robust Steering Controller

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Controller Selection

- Nonlinear Feedback and Feed-forward control (FDBK + FFW) [23];
- Target and Control Driver Model (TandC) [41];
- Discrete-Time Infinite-Horizon Linear Quadratic Regulator (LQR). (Formally, this acronym should be DTLQR or dLQR, but since there is no other LQR formulation to differentiate from, LQR is used instead) [42].

#### 2.1.1. Nonlinear Feedback and Feed-Forward Control

Algorithm 1 FDBK + FFW Gain Optimization Routine |

for each ${U}_{x}$ do |

$\zeta \leftarrow 0$ |

${x}_{LA}\leftarrow 0$ |

while $min\left(\zeta \right)\le {\zeta}_{thresh}$ do |

${x}_{LA}\leftarrow {x}_{LA}+{\delta}_{{x}_{LA}}$ |

${k}_{p}\leftarrow 0$ |

while $min\left({\omega}_{n}\right)\le {\omega}_{thresh}$ do |

${k}_{p}\leftarrow {k}_{p}+{\delta}_{{k}_{p}}$ |

Compute $min\left(\zeta \right)$ |

Compute $min\left({\omega}_{n}\right)$ |

end while |

end while |

end for |

#### 2.1.2. Target and Control Driver Model

Algorithm 2 T&C Gain Optimization Routine |

for each ${U}_{x}$ do |

Compute $\zeta $ for all ${k}_{p}$ and ${k}_{LA}$ |

Remove all ${k}_{p}$ and ${k}_{LA}$ where $\zeta \le {\zeta}_{thresh}$ |

${c}_{{k}_{p}}\leftarrow \frac{{k}_{p}-\mathrm{min}\left({k}_{p}\right)}{\mathrm{max}\left({k}_{p}\right)-\mathrm{min}\left({k}_{p}\right)}$ |

${c}_{{k}_{LA}}\leftarrow \frac{{k}_{LA}-\mathrm{min}\left({k}_{LA}\right)}{\mathrm{max}\left({k}_{LA}\right)-\mathrm{min}\left({k}_{LA}\right)}$ |

${c}_{DM}\leftarrow $ compute disk margin (skew parameter set to 0) |

$({k}_{p},{k}_{LA})\leftarrow \mathrm{argmin}({W}_{{k}_{p}}{c}_{{k}_{p}}+{W}_{{k}_{LA}}{c}_{{k}_{LA}}+{W}_{DM}{c}_{DM})$ |

end for |

#### 2.1.3. Linear Quadratic Regulator

#### 2.1.4. Linear Parameter Varying Youla Controller

Algorithm 3 LPV Youla Tuning Routine |

Require: $|Q(s,{U}_{x}){|}_{\infty}={\left|\frac{u}{r}\right|}_{\infty}<-10$ dB |

for each ${U}_{x}$ do |

Compute $P(s,{U}_{x})$ |

Compute ${T}_{{H}_{\infty}}(s,{U}_{x})$ as solution to H-infinity loop shaping problem |

$({\omega}_{n1},{\zeta}_{1},{\omega}_{n2},{\zeta}_{2},{\tau}_{2})\leftarrow T\left(s\right)\approx {T}_{{H}_{\infty}}(s,{U}_{x})$ |

end for |

- The bandwidth of $T\left(s\right)$ increases as velocity increases.
- ${\left|S\left(s\right)\right|}_{\infty}$ remains less than 5 dB.
- ${\left|Q\left(s\right)\right|}_{\infty}$ remains close to −10 dB until higher velocities are reached (≈20 m/s). After that, as velocity increases, ${\left|Q\left(s\right)\right|}_{\infty}$ decreases. This is a direct result of setting $T\left(s\right)$ constant for velocities above 20 m/s.
- $PS\left(s\right)$ has a rather large DC gain that increases with velocity.
- ${\left|PS\left(s\right)\right|}_{\infty}$ increases as velocity increases.

#### 2.2. Computational Comparison

#### 2.3. Simulation

#### 2.4. Reference Generation

#### 2.4.1. Dynamic Feasibility

#### 2.4.2. Reference Extraction

#### 2.5. ODD Definitions

#### 2.6. Metrics

## 3. Results and Discussion

#### 3.1. Relative Performance

#### 3.2. Relative Robustness

## 4. Conclusions

- Vehicle model and performance requirements: This work used a very simple method to derive lateral performance requirements for controller development. However, there exist several factors that this simple method neglects [71]. Specifically, the choice of vehicle significantly impacts these results. This work uses a full-size SUV, whose track width is 1.7 m, which is a rather wide passenger vehicle. If a smaller vehicle, such as a sedan (typically around 1.5 to 1.6 m track width), were used in this study, the required maximum lateral error would have increased. Conversely, if a larger vehicle, such as a commercial truck, were used, the maximum lateral error would have decreased.
- Stochastic disturbances: One simulation was performed per controller, route, and ODD combination. The underlying assumption is that this simulation represents the worst case in that condition, which is a rather significant oversimplification. In the future, the simulator’s real-time factor can be improved and a Monte Carlo experiment can be used to evaluate the worst-case, average, and best-case performance of each controller. In this sense, it makes sense to redefine the probability of failure accordingly.
- Longitudinal control: The longitudinal control performance is not considered in this work. However, the longitudinal controller must also meet pre-defined performance requirements.
- Reliable sensors: Section 2.3 described a system architecture that uses an EKF to fuse information from a GPS, IMU, and Wheel Speed Sensor. Therefore, the results presented in this work rely on the assumption that each of these sensors are available. For IMUs and Wheel Speed Sensors, this is not too great an assumption. However, assuming GPS is available is more significant. This is partly the motivation for introducing the Rural ODD, in which the RTK-GPS is assumed unavailable, but DGPS is available. More investigation is needed by the research community to develop robust, multi-modal localization techniques that can provide quality feedback to the controller in GPS-denied areas.
- Simplified road disturbances: The vertical road disturbances are modeled as colored noise according to ISO 8608 [13]. This neglects potholes and other sudden road jumps. These types of disturbances might present destabilizing conditions that are not captured in this work.
- Adverse weather conditions assumed to be uniform: The adverse weather conditions (Rainstorm and Blizzard ODDs) assume the road friction is uniform (with the addition of colored noise). In reality, puddles and ice patches can exist that cause large and sudden changes in friction. Furthermore, the simulation is incapable of simulating hydroplaning. All of these disturbances greatly complicate simulating adverse weather conditions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**FDBK + FFW gains with respect to velocity. (

**a**) FDBK + FFW ${k}_{p}$. (

**b**) FDBK + FFW ${x}_{LA}$.

**Figure 7.**Collision avoidance paths. (

**a**) Top-down view of the single lane change path. (

**b**) Top-down view of the double lane change path.

Variable | Name | Unit |
---|---|---|

${\delta}_{t}$ | Steering angle at time t | radian |

${\psi}_{veh}$ | Vehicle yaw angle | radian |

${\psi}_{ref}$ | Reference yaw angle | radian |

${Y}_{ref}$ | Global Y coordinate of reference point | m |

${Y}_{veh}$ | Global Y coordinate of the vehicle’s c.g. | m |

${X}_{ref}$ | Global X coordinate of the reference point | m |

${X}_{veh}$ | Global X coordinate of the vehicle’s c.g. | m |

${U}_{x}$ | Vehicle’s longitudinal velocity | m/s |

${\dot{\psi}}_{veh}$ | Vehicle’s yaw rate | rad/s |

${x}_{LA}$ | Look-ahead distance | m |

R | Vehicle’s instantaneous radius of curvature | m |

${k}_{LA}$ | Look-ahead gain | |

${k}_{p}$ | Controller gain |

Name | Maximum Absolute Curvature [1/m] | Average Absolute Curvature [1/m] | Path Length [m] |
---|---|---|---|

Single lane change (SLC) | 0.033 | 0.006 | 210 * |

Double lane change (DLC) | 0.015 | 0.002 | 424 * |

CA-17 | 0.007 | 0.001 | 13,825 |

I-15 | 0.003 | $6\times {10}^{-4}$ | 9924 |

S Road | 0.008 | 0.003 | 1609 |

Nominal | Realistic | Rural | Rainstorm | Blizzard | |
---|---|---|---|---|---|

Noise level | A | A | C | A | D |

Wind speed [m/s] | N/A | 0 | 5 | 13.4 | 13.4 |

Friction | 1.0 | 1.0 | 1.0 | 0.7 | 0.4 |

Feedback | Perfect | RTK | DGPS | RTK | RTK |

Speed adjustment [%] | 0 | 0 | 0 | −16 | −37 |

ODD Name | CA17 | S | I15 | SLC | DLC |
---|---|---|---|---|---|

Nominal | 3.91 | 6.91 | 3.19 | 3.62 | 5.05 |

Realistic | 4.10 | 8.60 | 3.59 | 5.80 | 5.35 |

Rural | 5.92 | 8.29 | 4.57 | 5.78 | 6.06 |

Rainstorm | 2.95 | 5.22 | 2.54 | 3.24 | 3.94 |

Blizzard | 2.88 | 4.08 | 2.75 | 3.21 | 2.99 |

Controller(s) | Route(s) | ODD(s) | Cause |
---|---|---|---|

T&C | I15 | Rural | Lateral error oscillations exceed threshold |

FDBK + FFW | S | Nominal, Realistic, Rural | Lateral error exceeds threshold |

FDBK + FFW | I15 | Rural | Lateral error exceeds threshold |

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**MDPI and ACS Style**

Vidano, T.; Assadian, F.
Searching for a Cheap Robust Steering Controller. *Electronics* **2024**, *13*, 1908.
https://doi.org/10.3390/electronics13101908

**AMA Style**

Vidano T, Assadian F.
Searching for a Cheap Robust Steering Controller. *Electronics*. 2024; 13(10):1908.
https://doi.org/10.3390/electronics13101908

**Chicago/Turabian Style**

Vidano, Trevor, and Francis Assadian.
2024. "Searching for a Cheap Robust Steering Controller" *Electronics* 13, no. 10: 1908.
https://doi.org/10.3390/electronics13101908