A triangular number-based reward function for trajectory planning of wing-in-ground crafts towards fast-response and low-deviation altitude changes

Abstract Altitude adjustment for wing-in-ground (WIG) crafts during obstacle avoidance or mission execution necessitates a rapid response followed by precise stabilisation at the target altitude. Conventional approaches typically treat the minimisation of response time and the minimisation of deviation from the target altitude as two separate objectives. These are assigned weights, and their performance is balanced through iterative trial-and-error adjustment of these weights, ultimately finding a target trajectory after multiple optimisation runs. This paper proposes a novel reward function that yields, via a single deep reinforcement learning pass, a trajectory minimising both the response time and deviation for WIGs’ altitude changes. Formally, this function bounds the reward value based on the deviation magnitude and incorporates triangular numbers to assess the response time. Its underlying principle ensures that the reward gain from reducing the response time outweighs the reward loss incurred by the potential increase in accumulated deviation resulting from that same reduction. The validity of this reward function is verified through mathematical derivation. A numerical example involving a WIG descending to a lower altitude is presented, using proximal policy optimisation. Comparative results against traditional multi-objective functions requiring extensive parameter tuning demonstrate that the proposed reward function directly produces the desired trajectory. This reward function provides a valuable reference for trajectory planning in second-order motion systems aiming to achieve minimised rise time and subsequent minimised integrated absolute error.