基于凸优化的自动驾驶安全速度规划方法

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本文由张等人在2018年发表，标题为《通过凸优化实现自动驾驶中固定路径的速度规划：向更完整、灵活与安全的解决方案》（Toward a More Complete, Flexible, and Safer Speed Planning for Autonomous Driving via Convex Optimization）。该研究主要关注的是自动驾驶汽车在静态和动态环境中，如何设计一个全面、灵活且安全的速度规划方法来解决沿着预设路径的行驶问题。论文的核心贡献主要体现在五个方面： 1. **约束总结**：作者首先梳理了自动驾驶在不同场景中常见的速度规划约束，这些约束包括但不限于车速限制、避障要求、路径跟随精度、行驶舒适性以及动态环境中的预测性行为等。这些约束对于确保速度规划算法的适用性和有效性至关重要。 2. **方法框架**：基于凸优化理论，作者提出了一种新颖的方法，这种方法能够有效地整合这些复杂的约束条件，转化为求解速度控制参数的优化问题。凸优化的优势在于它保证了解决问题的全局最优解，并且优化过程易于求解。 3. **灵活性**：相比于传统的方法，新提出的算法更加灵活，可以根据实时路况和车辆状态动态调整速度规划策略，提高驾驶的适应性和响应性。这使得车辆能够在复杂多变的道路环境中进行高效而安全的驾驶。 4. **安全性保障**：在速度规划过程中，作者强调了安全性考虑，通过严格的数学模型和稳健的决策机制，确保在遇到突发情况时，车辆能够迅速做出安全反应，避免潜在的风险。 5. **实证验证**：文章还提供了详尽的实验验证，展示了新方法在实际道路测试中的性能，证明了其在静态和动态环境中的有效性和鲁棒性，为自动驾驶领域的速度规划提供了一个实用的参考。这篇论文不仅提出了一个完整的速度规划框架，还展示了其在实际应用中的潜力，对于推动自动驾驶技术的发展，尤其是在安全性和灵活性方面的提升具有重要意义。

Sensors 2018, 18, 2185 6 of 29

Lipp et al. [

] presented a convex-optimization-based general minimum time speed planning

method over the ﬁxed path based on the approach proposed by [

]. The friction circle constraint is well

considered as a convex set constraint acting on the problem formulation, which leads to an elegant

solution. Not only the capacity of mobility of cars are fully explored, but also the total time travelling

along the path is explicitly and analytically represented as a soft constraint to achieve time efﬁciency.

The problem is solved by a customized interior point method using log barrier functions efﬁciently.

Thanks to the preserving convexity of the problem formulation, the global optimality of solutions

is guaranteed. However, smoothness of the speed proﬁle is not consider, which most likely results

in the same issues that we mentioned about Dakibay’s work regarding tracking performance and

safety concerns. In addition, the use of customized Newton-based solver requires that constraints

and objective functions are all at least twice differentiable, which seems very restrictive on the type of

constraints that users can impose in convex optimization. Convex problems with non-differentiable

constraint terms cannot be solved by their framework.

Liu et al. [

] recently introduced a temporal optimization approach, optimizing time stamps

for all waypoints along a ﬁxed path with respect to time window constraints at each point, and then

using a slack convex feasible set algorithm to solve it iteratively. Smoothness of the speed proﬁle and

time efﬁciency are taken into account in the problem formulation. However, the time efﬁciency is

considered in an indirect way that optimizes IoD with respect to a reference speed over the path. Their

formulation leads to a highly nonlinear and non-convex problem and is solved by a local optimization

method, thus only local optimality is guaranteed. They addressed some important constraints in

Table 1 such as smoothness, time window and comfort box constraints in their formulation but left

out the friction circle constraint, which does not fully exploit the acceleration capacity of the vehicle.

In addition, since they optimized timestamps directly, we do not see a quick way to impose a path

constraint or a point constraint as a hard one to manipulate speed proﬁles.

3. Problem Formulation

Assuming a curvature continuous path has been generated by a hierarchical motion planning

framework like [

], the speed planning is to ﬁnd a time-efﬁcient, safe, and smooth speed proﬁle

travelling along the ﬁxed path with respect to both safety and performance constraints.

To solve the proposed problem, we optimize the performance criterions from three aspects,

smoothness

, time efﬁciency

, and speed deviation

from a desired speed, with others left as

hard constraints or semi-hard constraints. We ﬁrst introduce the path representation and explain

the relationship of an arc-length parametrized path and a time parametrized path, then present

mathematical expressions of all the constraints, and pose the optimization problem at the end.

3.1. Path Representation

The goal of speed planning is to ﬁnd a speed proﬁle along a ﬁxed path. Since the path is known,

we need to reconstruct the mapping between the known path and the speed proﬁle, then represent the

speed proﬁle with parameters determined by the prescribed path. A rich set of parameterized path

representations has been proposed in the literature, including B-spline [

], Bezier curve [

clothoid [

], polynomial curve [

] and polynomial spiral [

]. It is trivial to convert all the

listed curve models to a simple waypoints representation, but not vice versa. To avoid the non-trivial

converting between curve models, we use the general waypoints parametrization to represent a

ﬁxed path, with the orientation and curvature encoded implicitly by the path. Formally, we deﬁne a

waypoints parametrized curve as a workspace path. A workspace path, r, of the body point, b, at the

center of the rear axle with footprint,

, is deﬁned as

r : [

] → R

. More speciﬁcally, we consider

the following arc-length parametric form in Cartesian coordinate system,

r(s) = (x(s), y(s)), s ∈ [0, s

(1)

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