Fast UAV Trajectory Optimization Using Bilevel Optimization With Analytical Gradients

Abstract

In the article, we present an efficient optimization framework that solves trajectory optimization problems by decoupling state variables from timing variables, thereby decomposing a challenging nonlinear programming (NLP) problem into two easier subproblems. With timing fixed, the state variables can be optimized efficiently using convex optimization, and the timing variables can be optimized in a separate NLP, which forms a bilevel optimization problem. The challenge of obtaining the gradient of the timing variables is solved by sensitivity analysis of parametric NLPs. The exact analytic gradient is computed from the dual solution as a by-product, whereas existing finite-difference techniques require additional optimization. The bilevel optimization framework efficiently optimizes both timing and state variables which is demonstrated on generating trajectories for an UAV. Numerical experiments demonstrate that bilevel optimization converges significantly more reliably than a standard NLP solver, and analytical gradients outperform finite differences in terms of computation speed and accuracy. Physical experiments demonstrate its real-time applicability for reactive target tracking tasks.

Robust & Asymptotically Locally Optimal UAV-Trajectory Generation Based on Spline Subdivision

“…The trajectories computed using numerical integration by [26] are locally optimal in either shape variables or time variables but not both at the same time. The latest works [30], [31] achieve shapetime joint optimality via bilevel optimization or weighted combination. We following [31] and optimize a weighted combination of shape and time cost functions.…”

“…For a trajectory with N Bézier curve pieces each having order M , we need (M − 2)N + 3 control points organized as in Figure 2, so w ∈ R 3((M −2)N +3) . Unlike [30], [31], [35], this natural parameterization does not require additional linear constraints between adjacent curve pieces to ensure continuity.…”

“…Time efficacy can be formulated as a terminal cost such as R(p, T ) = T , combining the semi-infinite velocity and acceleration bounds. Unlike [30], [10], that warp each Bézier piece of p(t, w) using a separate time parameter, we use a single, global time rescaling. In other words, the UAV flies through each polynomial piece for T N seconds.…”

“…Infeasible Recovery: Nonlinear constrained optimizers, e.g. Sequential Quadratic Programming (SQP), have been used in [40], [35], [30] to directly handle nonconvex constraints. These methods rely on the constraints' directional derivatives and penalty functions to pull infeasible solutions back to the feasible domain.…”

“…3) Our formulation achieves simultaneous optimality in trajectory shapes and time allocations. As compared with [30] where a separate decision variable is needed for time-allocation of each spline, we only need a global time re-scaling.…”