Exploring Trajectory Planning: Concepts, Decision‑Making, and Challenges in Autonomous Driving

Hello everyone, today we will introduce the exploration and challenges of trajectory planning, covering four aspects: the concept of trajectory planning, decision making, lateral planning, and longitudinal planning.

Concept of Trajectory Planning:

Trajectory planning aims to answer the question of how a vehicle should move, for example deciding how to turn left when pedestrians, cyclists, and a truck are nearby.

The inputs to trajectory planning include a topological map, obstacles and their predicted trajectories, traffic‑signal states, localization, destination, and vehicle state; the output is a trajectory, a time‑to‑position function (t → (x, y, z)), where z is omitted because current vehicles do not fly.

Trajectory planning is essentially an optimization problem. Constraints include traffic rules, collision avoidance, and controllability (the planned path must be feasible for the vehicle). The objective is to make the motion feel human‑like and comfortable, with variations such as more stable or more aggressive driving styles.

Mathematically we first ask whether the problem is convex, because convex problems are easier to solve. Trajectory planning is non‑convex. To handle this, it is split into lateral planning (s → (x, y)) that determines the shape of the trajectory, and longitudinal planning (t → s) that determines the speed profile along that shape. Lateral planning is non‑convex, as illustrated by the example where the linear combination of two feasible lane‑change trajectories yields an infeasible, overly slow path.

Decision:

Because trajectory planning is non‑convex, a decision module is needed to restrict the solution space and transform the problem into a convex one. Decision making selects among alternatives such as yielding or proceeding, which then allows the use of convex optimization techniques like linear programming, quadratic programming, or sequential quadratic programming. Decision making itself is NP‑hard, and typical approaches include rule‑based heuristics, dynamic programming, and machine‑learning‑based policies.

Decision challenges include the NP‑hard nature of the problem, difficulty of modeling human experience in cost functions, and the need for robust perception and prediction. Machine‑learning methods are often employed to handle complex scenarios such as sudden lane changes or ambiguous pedestrian behavior.

Lateral Planning:

Lateral planning determines the shape of the trajectory (the steering plan). In lane‑based scenarios, a reference line is generated offline, and the problem becomes solving for the lateral offset L relative to that line (s → L). Constraints include staying within lane boundaries and avoiding collisions; objectives aim to stay close to the lane center, keep curvature low, and ensure smooth curvature variation.

After discretizing the reference line, the problem reduces to a quadratic programming (QP) formulation where the decision variables are the lateral offsets of future points, subject to boundary constraints and an objective that penalizes deviation from the center, large curvature, and rapid curvature change.

Challenges for lateral planning arise in unstructured environments (e.g., intersections without clear lane markings) and in dynamic situations where obstacles appear suddenly, requiring rapid re‑planning while maintaining smooth, controllable trajectories.

Environmental prediction is also difficult; for example, a fast‑approaching motorbike may force a vehicle to cancel a lane change and return to its original lane, demanding a smooth and feasible path.

Longitudinal Planning:

Longitudinal planning determines the speed profile along the previously generated path, subject to traffic‑rule compliance and collision avoidance. The objective is comfort (low acceleration and jerk) while maintaining reasonable travel speed.

Examples include handling pedestrians crossing the road, where different decisions (yield or proceed) lead to distinct constraints in the s‑t diagram.

Additional scenarios such as yellow‑light decisions are illustrated with s‑t diagrams, showing how aggressive crossing requires a steeper s‑t slope.

Following a decelerating lead vehicle can be modeled in the s‑t space to generate an optimal speed curve.

Longitudinal challenges also stem from perception and prediction uncertainties, such as sudden “ghost” pedestrians or ambiguous vehicle intentions, which make safe speed planning difficult.

Extreme cases require coordinated lateral‑longitudinal responses, for example a high‑speed vehicle inserting into traffic, where maintaining safety distance and avoiding collisions become the primary goals.

Author Introduction:

Liang Yaxiong, Pony.ai Tech Lead, holds a master's degree in Computer Science from Tsinghua University and is an ACM‑ICPC Asia Regional Gold Medalist. He previously worked at Hulu and now leads autonomous‑driving path planning and control research at Pony.ai.

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