## 1 Introduction  
A parafoil delivery system is a special precision air conveying system [1, 2, 3, 4]. By controlling the shape of the parafoil canopy, this system can change the flight direction and achieve precise landing, which is hard to realize by the traditional parachute system. Based on this advantage, parafoil has broad prospects in the fields of military and civilian airdrop supply and aircraft recovery. For example, NASA applied a 689 m\({}^{2}\) parafoil to recovery X-38 aircraft [5, 6]. The German Aerospace Center (DLR) has also developed a small and widely instrumented aircraft Alex (a parafoil delivery system), and proposed the application of the GNC algorithm in autonomous landing [7, 8]. Other research, such as [9, 10], explored the modeling approach for ram-air parachutes. Ref. [11] proposed pods that provide distributed sensors in the whole parachute canopy and a fusion algorithm to merge the pod data into useful canopy state estimation, which improves the accuracy of canopy state estimation. Ref. [12] presented a feasibility study on the engine and engine frame recovery system of an existing expendable heavy launch vehicle. By analyzing the dynamic features of parafoil, it can be observed that its vertical velocity is nearly uncontrollable for the unpowered parafoil. Due to this feature, the flight time of parafoil trajectory planning depends mainly on the initial launching altitude. The initial altitude of the parafoil determines that the flight time and the adjustment time for trajectory change in the flight process is very limited. Parafoil has two research directions of trajectory planning: the optimal trajectory planning and the multiple trajectory planning. Based on the above two research directions, the trajectory planning of the parafoil delivery system is studied by many scholars. The reference flight trajectory is usually composed of standard straight lines and arcs. In [7], a GNC algorithm based on T-Approach is applied to multi-trajectory planning. Meanwhile, the trajectory of optimal trajectory planning is an irregular curve. In the early research on the multiple planning method, [PERSON]. [13] proposed a simplified 6-DOF model predictive control strategy for autonomous control of parafoil. The trajectory of this method is composed by many short straight lines. Then, [PERSON]. [14; 15; 16] proposed the terminal guidance strategy of autonomous parafoil, which has good robustness. The trajectory of this method is the arc in the terminal planning strategy. [PERSON]. [17; 18] proposed the robust trajectory planning of parafoil in uncertain wind environments. [PERSON]. [19] proposed a hybrid trajectory planning strategy, which is suitable for a class of parafoils. Using this method, the generated trajectory consists of multiple straight lines and arcs. [PERSON]. [20] proposed a terminal trajectory planning method in the form of a Bezier curve. The path of this method is composed of straight lines and circles. [PERSON]. [21] explored the direct multiple shooting method to obtain the optimal trajectories. In [22], parafoil guidance used the Line of Sight guidance algorithm. In [23], multiple parafoils coordinate and navigate autonomously according to the trajectory, which is a Dubins path composed of straight lines and arcs. For the multi system planning method, [PERSON]. [24; 25] explored the coordinated trajectory tracking method of multi parafoils to realize the accurate airdrop of multiple parafoils. Different from the traditional multi trajectory planning, the flight trajectory of the optimal trajectory planning is an irregular curve. Therefore, the optimal trajectory planning method can satisfy more complex constraints, such as terrain avoidance and multi-objective global optimization of control quantities [26; 27; 28]. For example, [PERSON]. [29; 30] applied multivariable control to the autonomous attitude optimal control system of parafoil. Meanwhile, unlike the fully autonomous airdrops, [PERSON]. [31] explored a semi-autonomous Human-in-the-loop control method for precise landing.