Real-time quadrotor trajectory optimization with time-triggered corridor constraints

One of the challenges for flying quadrotors in cluttered environments is to optimize their trajectories subject to collision-avoidance constraints in real time [1,2]. Along such a trajectory, the position of the quadrotor must stay within a set of collision-free corridors. Each corridor is a bounded convex flight space. The union of all these corridors forms a nonconvex pathway connecting the quadrotor’s current position to its target position [3].

Traditional trajectory optimization methods treat the flight corridor constraints using either mixed-integer programming or sequential convex programming. These methods are either computationally expensive for real-time applications or lack guarantees for constraint satisfaction. For example, the mixed-integer programming uses binary variables to represent the union of all corridors, and then it optimizes quadrotor trajectories together with these binary variables [4,5]. The drawback of mixed-integer programming is that the worst-case computation time of mixed-integer programming increases exponentially as the number of integer variables (which depends on the number of corridors and the trajectory length [4]) increases. An alternative is to replace the corridor constraints with equivalent collision-avoidance constraints, and then approximate the resulting nonconvex trajectory optimization using a sequence of convex ones, such as second-order cone programs [6,7] or semidefinite programs [8,9]. The drawback of sequential convex programming methods is that they either require careful initializations to ensure convergence or can violate collision-avoidance constraints when terminated in finite time. For example, the methods in Refs. [6,7] require a nominal trajectory to initiate the linearization of collision-avoidance constraints. These methods can diverge if the nominal trajectory is far from the optimum. On the other hand, the methods in Refs. [8,9] transform nonconvex collision-avoidance constraints to rank constraints in semidefinite programs. When terminated after a finite number of iterations, the iterates of these methods converge to a local optimal solution that may violate the rank constraints, and consequently the collision-avoidance constraints.

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