As a fundamental component of autonomous driving systems, motion planning has garnered significant attention from both academia and industry. This paper focuses on efficient and spatial-temporal optimal trajectory optimization in unstructured environments using compact convex approximations of vehicle shapes. Conventional approaches typically model the task as an optimal control problem by discretizing the motion process in state configuration space. However, this often results in a tradeoff between optimality and efficiency since generating high-quality motion trajectories often requires high-precision discretization of the dynamic process, which imposes a substantial computational burden. To address this issue, we leverage the differential flatness property of car-like robots to simplify the trajectory representation and analytically formulate the spatial-temporal joint optimization problem with flat outputs in a compact manner, while ensuring the feasibility of nonholonomic dynamics. Moreover, we achieve efficient obstacle avoidance with a collision-free driving corridor for unmodelled obstacles and signed distance approximations for dynamic moving objects. We present comprehensive benchmarks with State-of-the-Art methods, demonstrating the significance of the proposed method in terms of efficiency and trajectory quality. Real-world experiments verify the practicality of our algorithm. We will release our codes for the research community.

Collision evaluation is of essential importance in various applications. However, existing methods are either cumbersome to calculate or not exact. Therefore, considering the cost of implementation, most whole-body planning works, which require evaluating collision between robots and environments, struggle to tradeoff between accuracy and computationally efficiency. In this paper, we propose a zero-gap whole-body collision evaluation that can be formulated as a low-dimensional linear programming. This evaluation can be solved analytically in linear complexity. Moreover, the method provides gradient efficiently, making it accessible to optimization-based applications. Additionally, this method provides support for obstacles represented by either points or hyperplanes. Experiments on the widely used aerial and car-like robots validate the versatility and practicality of our method.