Geometrically Constrained Trajectory Optimization for Multicopters

Abstract

We present an optimization-based framework for multicopter trajectory planning subject to geometrical spatial constraints and user-defined dynamic constraints. The basis of the framework is a novel trajectory representation built upon our novel optimality conditions for unconstrained control effort minimization. We design linear-complexity operations on this representation to conduct spatial-temporal deformation under various planning requirements. Smooth maps are utilized to exactly eliminate geometrical constraints in a lightweight fashion. A wide range of state-input constraints are supported by the decoupling of dense constraint evaluation from sparse parameterization, and backward differentiation of flatness map. As a result, the proposed framework transforms a generally constrained multicopter planning problem into an unconstrained optimization that can be solved reliably and efficiently. Our framework bridges the gaps among solution quality, planning frequency and constraint fidelity for a multicopter with limited resources and maneuvering capability. Its generality and robustness are both demonstrated by applications and experiments for different tasks. Extensive simulations and benchmarks are also conducted to show its capability of generating highquality solutions while retaining the computation speed against other specialized methods by orders of magnitudes.

DIRECT: A Differential Dynamic Programming Based Framework for Trajectory Generation

“…For trajectory generation with time optimization as well as safety and dynamical feasibility constraints, we compare our algorithm with the alternating spatial-temporal optimization [6] and bi-level optimization with analytic gradient [9]. 4 In [6] the energy and time costs are separately optimized while in [11] the overall cost function takes a similar form as ours, but without the terminal cost and the time cost takes the form of w k t k . The benchmark is conducted using an i7-8550U CPU.…”

“…, where t ini k is the initial allocated time for segment k and t * k is the optimized allocated time computed by each algorithm) and 4 According to Fig. 9 of [11], these two algorithms are the state-of-the-art methods for generating a safe and dynamical feasible trajectory with time optimization. The code for [11] is currently unavailable.…”

“…An alternating minimization scheme was proposed in [10] where complex safety constraints were not considered. Recently in [11], the authors formulate an unconstrained nonlinear optimization that can be solved by quasi-Newton methods efficiently, but the safety and dynamical feasibility are enforced as penalty functions on discretized points instead of hard constraints. In terms of large scale trajectory (≥ 10 3 ), the authors in [12] proved that the minimum snap problem with fixed time allocation can be solved in linear complexity by exploiting the structure of optimality conditions.…”

“…Following [4], [11], [13], we consider the problem of enforcing the smoothness of trajectory up to the (m − 1)-th (m = n+1…”

Robust Multi-Robot Trajectory Optimization Using Alternating Direction Method of Multiplier

“…Although the convergence speed is comparable in terms of number of iterations, our two methods (AM and ADMM) clearly outperform in terms of computational time. By not restricting the trajectory to precomputed corridors as done in [24,42,43,40,44], our method allows a larger solution space and returns a shorter trajectory. We summarize the quality of trajectory as computed by three methods in Table I.…”