# Toward a More Complete, Flexible, and Safer Speed Planning for Autonomous Driving via Convex Optimization

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## Abstract

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## 1. Introduction

**safety-guaranteed speed planner**should be able to generate a solution satisfying at least all the hard constraints (safety) in Table 1. A mature speed planner should cover all the constraints that include soft and hard ones.

- We summarize the most common constraints raised in various autonomous driving scenarios as the requirements for speed planner design and metrics to measure the capacity of the existing speed planners roughly for autonomous driving. We clarify which constraints need to be addressed by speed planners to guarantee safety in general.
- In light of these requirements and metrics, we present a more general, flexible and complete speed planning mathematical model including friction circle, dynamics, smoothness, time efficiency, time window, ride comfort, IoD, path and boundary conditions constraints compared to similar methods explained in [3,11]. We addressed the limitations of the method of Lipp et al. [3] by introducing a pseudo jerk objective in longitudinal dimension to improve smoothness, adding time window constraints at certain point of the path to avoid dynamics obstacles, capping a path constraint (most-likely non-smooth) on speed decision variables to deal with task constraints like speed limits, imposing a boundary condition at the end point of the path to guarantee safety for precise stop or merging scenarios. Compared to the approach of Liu et al. [11], our formulation optimizes the time efficiency directly while still staying inside of the friction circle, which ensures our method exploits the full acceleration capacity of the vehicle when necessary.
- We introduce a semi-hard constraint concept to describe unique characters of the comfort box constraints and implement this kind of constraints using slack variables and penalty functions, which emphasizes comfort while guaranteeing fundamental motion safety without sacrificing the mobility of cars. To the best of our knowledge, none of the existing methods handle these constraints like ours. In contrast, Refs. [7,8,9,10,11] regarded comfort box constraints as hard constraints, which dramatically reduces the solution space and in consequence limits the mobility of cars.
- We demonstrate that our problem still preserves convexity with the added constraints, and hence, that the global optimality is guaranteed. This means our problem can be solved using state-of-the-art convex optimization solvers efficiently as well. We also provide some evidence to prove that our solution is able to keep consistent when the boundary conditions encounter some disturbances, which means only the part of results needed to be adjusted will be regulated due to the global optimality. This may benefit the track performance of speed controllers by providing a relative stable reference. It is not the case for these methods that solve the speed planning problem using local optimization techniques like [11]. A small change of boundary conditions or initial guess may result in a totally different solution due to local minimas in their problem.
- We showcase how our formulation can be used in various autonomous driving scenarios by providing several challenging case studies solved in our framework, such as safe stop on a curvy road with different entry speeds, dealing with jaywalking in two different ways and merging from a freeway entrance ramp to expressways with safety guaranteed.

## 2. Related Work

## 3. Problem Formulation

#### 3.1. Path Representation

#### 3.2. Vehicle Model and Vehicle Dynamics Constraints

#### 3.3. Friction Circle Constraints

#### 3.4. Time Efficiency Objective

#### 3.5. IoD Objective

#### 3.6. Smoothness Objective

#### 3.7. Path Constraints

- Speed limits on certain segments of roads happen to be common driving scenarios in urban environments. The speed limits cannot be exceeded by autonomous driving systems, or the driving system will violate the traffic regulations and be fined. The restrictions may happen along the whole path or just segments of the path, which is a little different from an overall speed threshold constraint and the IoD objective.
- A high-level planning system (i.e., behavior planning system, task planning system) may provide the upper boundary or lower boundary of the speed profile to a speed planner to make it behave well or satisfy certain task requirements. A speed planner has to plan a speed profile that stays in the prescribed region or below the envelope.

#### 3.8. Boundary Condition Constraints

#### 3.9. Time Window Constraints

#### 3.10. Comfort Box Constraints

#### 3.11. Overall Convex Optimization Problem Formulation

- For the objectives, ${J}_{T}$ is an integral of a negative power function and is therefore convex. ${J}_{S}$ is an integral of a squared power of absolute value and is therefore convex. ${J}_{V}$ is an integral of an identity power of absolute value and is therefore convex. So are $\parallel {\sigma}^{\tau}\parallel $ and $\parallel {\sigma}^{\eta}\parallel $. As ${\omega}_{1}$, ${\omega}_{2}$, ${\omega}_{3}$, ${\lambda}_{1}$, ${\lambda}_{2}$ are all nonnegative, J as a nonnegative weighted sum of convex functions, is convex.
- For (6), the dynamics equality constraint is affine in $\alpha $, $\beta $, u and is therefore convex. For equality constraints about decision variables (8), since the derivative is a linear operator, the relation between $\alpha $ and $\beta $ is convex. For the inequality path constraint (17), $\beta \left({s}_{i}\right)$ is a sublevel set of convex set in the interval $[{s}_{m},{s}_{n}]$ and is thus convex. The equality and inequality constraints about boundary conditions (19) are linear constraints, thus convex. As the ${T}_{i}$ is an integral of a negative power function, therefore convex and ${T}_{U}$ is a fixed upper boundary, the time window inequality constraint (20) is a convex constraint.
- For the convex set constraint about the friction circle (11), the norm of u is convex, upper bounds are fixed and ${v}_{max}^{2}$ is fixed, so the control set constraint is the intersection of three convex sets and is therefore convex.
- The comfort box constraints with slack variables ${\sigma}^{\tau}$ and ${\sigma}^{\eta}$ are second-order cone constraints and convex.

## 4. Implementation

#### 4.1. Discretization of ${J}_{T}$, ${J}_{S}$, and ${J}_{V}$

#### 4.2. Discretization of ${r}^{\prime}\left(s\right)$ and ${r}^{\u2033}\left(s\right)$

## 5. Numerical Results

#### 5.1. Smoothness

#### 5.2. Boundary Condition Constraint

**A**,

**B**in Figure 8 showed that our method is able to satisfy the final speed boundary condition while optimizing time efficiency (

**A**) with a sharp slow-down slope or optimizing time efficiency and smoothness at the same time (

**B**) with a flatter slow-down slope at the end. We conducted the second set of experiments with both time efficiency and smoothness objectives considered using same coefficients but with different type of boundary conditions,

**E**is generated, which is the optimal shape under the given objectives. By adding an equality constraint (

**D**) and an inequality constraint (

**C**) to the final speed, we observed notable differences of the last portion of the speed profile among these results. The last segments of the speed profile are adapted by the optimization to satisfy the given constraints. The other parts almost stay the same for case

**C**,

**D**,

**E**due to global optimality. A similar phenomenon is observed between the results of MTSOS and case

**A**in Figure 8. Only the part that needs to be adjusted is regulated. This is an appealing feature for speed tracking regarding temporal consistency of references and control stability. Since time efficiency is one of the objectives, it makes sense that the final speed of the case

**C**reached the upper boundary at the end when given a feasible range.

#### 5.3. Path Constraint

#### 5.4. IoD Task Constraints

**A**and the dash-dash line

**B**in Figure 10) to show the behaviors of our planner. We first ran the MTSOS planner to generate the upper boundary of the speed profile for reference. For the desired speed profile

**A**in Figure 10, we consider the time efficiency objective and IoD objective only by ${\omega}_{1}=1$, ${\omega}_{2}=0$, ${\omega}_{3}=10$ and relaxed all the other constraints to generate the speed profile, shown as the orange curve in Figure 10. The orange curve aligned well with the desired speed profile except for the part that the desired speed exceeds the limit of the friction circle. For the exceeding part, the orange curve stayed as close as possible to the desired speed but limited by the speed upper boundary constrained by the friction circle. This result uncovers the strong safety feature of our method. Moreover, taking the smoothness objective into consideration by making ${\omega}_{2}=0.1$, the quality of the speed profile is further improved (see the green curve in Figure 10). We also tested the IoD constraint against the totally feasible desired speed profile B using the same parameters setting with the previous experiment. The blue curve in Figure 10 depicted the planning result without considering smoothness. The resulting speed almost perfectly aligned with desired speed

**B**. Similarly, the quality of the speed profile was significantly improved by add the smoothness objective (see light red curve in Figure 10).

#### 5.5. Time Window Constraint

#### 5.6. Semi-Hard Comfort Box Constraint

## 6. Case Study

#### 6.1. Speed Planning for Safe Stop

#### 6.2. Speed Planning Dealing with Jaywalking on a Curvy Road

#### 6.3. Speed Planning for Freeway Entrance Ramp Merging

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The comparison of solution spaces of the normal friction circle, the comfort box and the shrinked friction circle constraints.

**Figure 4.**A cross scenario with moving vehicles. (

**a**) A cross scenario without traffic lights. The blue car is the autonomous car. The orange cars (${C}_{1}$, ${C}_{2}$) are the oncoming vehicles with prescribed speed profiles. (

**b**) A S-T graph that describes different types of time windows and possible solutions to avoid moving vehicles. The S is the arc-length along the path of the autonomous car.

**Figure 5.**An example path from [3].

Category | Constraint Name | Description | Property |
---|---|---|---|

Soft Constraints | Smoothness (S) | continuity of speed, acceleration and jerk over the path | performance |

Time Efficiency (TE) | time used by travelling along the path | performance | |

IoD | integral of speed deviations | performance | |

Hard Constraints | Friction Circle (FC) | total force should be within the friction circle | safety |

Path Constraints (PC) | speed limits on path segments | safety | |

Time Window (TW) | time window to reach a certain point on path | safety | |

Boundary Condition (BC) | speed at the end of the path | safety & performance | |

Semi-hard Constraints | Comfort Box (CB) | comfort acceleration and deceleration bounds | performance |

Method | S | TE | IoD | FC | PC | TW | BC | CB | Optimality | Safety | Mobility | Flexibility |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Li et al. [7] | ✓ | ✗ | ✗ | ✗ | ✓ | ✗ | ✓ | ✓ | ✗ | low | low | low |

Gu et al. [8,9,10] | ✓ | ✗ | ✗ | ✗ | ✓ | ✗ | ✓ | ✓ | ✗ | medium | medium | medium |

Dakibay et al. [4] | ✗ | ✗ | ✗ | ✓ | ✓ | ✗ | ✓ | ✗ | ✗ | medium | high | low |

Liu et al. [11] | ✓ | ✓ | ✗ | ✗ | ✓ | ✓ | local | medium | medium | medium | ||

Lipp et al. [3] | ✗ | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | global | low | high | low |

Ours | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | global | high | high | high |

**Mobility:**determined by how much mobility capacity of the vehicle the planner is able to leverage;

**Optimality:**determined by whether the planner is able to identify an optimal solution in terms of its objective;

**Flexibility:**determined by how many type of scenarios the planner is able to handle by only adjusting parameters without changing underlying problem formulation or problem structures;

**Safety:**determined by four aspects, ability to stop in front of obstacles (BC) precisely, ability to deal with emergencies (FC), ability to impose task constraints like speed limits, and ability to handle dynamic obstacles (TW).

Parameter | Description | Values | Unit |
---|---|---|---|

m | Mass of the car | 0.1453 | kg |

$\mu $ | Friction coefficient | 0.70 | 1 |

g | Acceleration of gravity | 9.83 | m/s${}^{2}$ |

${a}_{c}^{\eta}$ | Longitudinal acceleration threshold for comfort | 0.4 $\mathsf{\mu}$g | m/s${}^{2}$ |

${a}_{c}^{\tau}$ | Lateral acceleration threshold for comfort | 0.4 $\mathsf{\mu}$g | m/s${}^{2}$ |

${a}_{max}^{\tau}$ | Max. longitudinal acceleration of the car. | 0.5 $\mathsf{\mu}$g | m/s${}^{2}$ |

${v}_{max}$ | Max. speed of the car. | 1.8 | m/s |

Profile Figure 11 | Coefficients | Time Window (s) | Travel Time at ${\mathit{s}}_{\mathit{f}}$ (s) |
---|---|---|---|

blue | ${\omega}_{1}=1,{\omega}_{2}=0.5$ | free | 6.626 |

green | ${\omega}_{1}=1,{\omega}_{2}=0.5$ | ${t}_{{s}_{f}}\in (0,5]$ | 4.999 |

red | ${\omega}_{1}=1,{\omega}_{2}=0.5$ | ${t}_{{s}_{f}}\in (0,4]$ | 4.000 |

Parameter | Description | Values | Unit |
---|---|---|---|

w | Car width | 2.45 | m |

l | Car length | 4.9 | m |

$wb$ | Car wheelbase | 2.8448 | m |

$tr$ | Car track | 1.5748 | m |

m | Mass of the car. | 1500.0 | kg |

$\mu $ | Friction coefficient | 0.7 | 1 |

g | Acceleration of gravity | 9.83 | m/s${}^{2}$ |

${a}_{c}^{\eta}$ | Longitudinal acceleration threshold for comfort | 0.4 $\mathsf{\mu}$g | m/s${}^{2}$ |

${a}_{c}^{\tau}$ | Lateral acceleration threshold for comfort | 0.4 $\mathsf{\mu}$g | m/s${}^{2}$ |

${a}_{max}^{\tau}$ | Max. longitudinal acceleration of the car | 0.5 $\mathsf{\mu}$g | m/s${}^{2}$ |

${v}_{max}$ | Max. speed of the car | 30 | m/s |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Chen, H.; Waslander, S.L.; Yang, T.; Zhang, S.; Xiong, G.; Liu, K. Toward a More Complete, Flexible, and Safer Speed Planning for Autonomous Driving via Convex Optimization. *Sensors* **2018**, *18*, 2185.
https://doi.org/10.3390/s18072185

**AMA Style**

Zhang Y, Chen H, Waslander SL, Yang T, Zhang S, Xiong G, Liu K. Toward a More Complete, Flexible, and Safer Speed Planning for Autonomous Driving via Convex Optimization. *Sensors*. 2018; 18(7):2185.
https://doi.org/10.3390/s18072185

**Chicago/Turabian Style**

Zhang, Yu, Huiyan Chen, Steven L. Waslander, Tian Yang, Sheng Zhang, Guangming Xiong, and Kai Liu. 2018. "Toward a More Complete, Flexible, and Safer Speed Planning for Autonomous Driving via Convex Optimization" *Sensors* 18, no. 7: 2185.
https://doi.org/10.3390/s18072185