This paper presents a scalable online algorithm to generate safe and kinematically feasible trajectories for quadrotor swarms. Existing approaches rely on linearizing Euclidean distance-based collision constraints and on axis-wise decoupling of kinematic constraints to reduce the trajectory optimization problem for each quadrotor to a quadratic program (QP). This conservative approximation often fails to find a solution in cluttered environments. We present a novel alternative that handles collision constraints without linearization and kinematic constraints in their quadratic form while still retaining the QP form. We achieve this by reformulating the constraints in a polar form and applying an Alternating Minimization algorithm to the resulting problem. Through extensive simulation results, we demonstrate that, as compared to Sequential Convex Programming (SCP) baselines, our approach achieves on average a 72% improvement in success rate, a 36% reduction in mission time, and a 42 times faster per-agent computation time. We also show that collision constraints derived from discrete-time barrier functions (BF) can be incorporated, leading to different safety behaviours without significant computational overhead. Moreover, our optimizer outperforms the state-of-the-art optimal control solver ACADO in handling BF constraints with a 31 times faster per-agent computation time and a 44% reduction in mission time on average. We experimentally validated our approach on a Crazyflie quadrotor swarm of up to 12 quadrotors. The code with supplementary material and video are released for reference.

翻译：本文提出了一种可扩展的在线算法，用于生成四旋翼机群的安全且运动学可行轨迹。现有方法依赖于线性化基于欧氏距离的碰撞约束以及运动学约束的轴向解耦，从而将每架四旋翼的轨迹优化问题简化为二次规划（QP）。这种保守近似在杂乱环境中常常无法找到可行解。我们提出了一种新颖的替代方案，无需线性化即可处理碰撞约束，并在保留二次规划形式的同时处理其二次形式的运动学约束。我们通过将约束重新表示为极坐标形式，并对由此产生的问题应用交替最小化算法来实现这一目标。通过大量仿真结果，我们证明与基于序列凸规划（SCP）的基线方法相比，我们的方法在成功率上平均提升了72%，任务时间缩短了36%，且单架四旋翼的计算速度提高了42倍。我们还展示了可以融入基于离散时间障碍函数（BF）的碰撞约束，从而在不显著增加计算开销的情况下实现不同的安全行为。此外，我们的优化器在处理障碍函数约束方面优于最先进的最优控制求解器ACADO，平均单架四旋翼计算速度提高了31倍，任务时间减少了44%。我们通过一个最多包含12架Crazyflie四旋翼的机群实验验证了该方法。代码、补充材料及视频已公开以供参考。