Landing Trajectory Design for UAV Considering Control Restrictions and Landing Speed

The article presents a method for designing the trajectory of the UAV in space, taking into account the restriction on control. The chosen optimal controls are namely normal overload with restrictions, tangential overload with restrictions and lateral overload. The Pontryagin maximum principle allows the transition of the optimal control problem to a boundary value problem. The parameter continuation method is applied to solve the boundary problem. The article results reveal reference trajectories in different cases of UAV landing. This result allows the design of reference trajectories for the UAV to attain the highest landing efficiency.


Introduction
During the process of landing, the value of UAV landing speed is critically significant in case of landing on short runway or emergency landing… UAV is expected to land with a small landing speed; otherwise, the large landing speed may lead to unsafety circumstances such as the UAV going off the runway, the UAV may flip or change direction when landing. In addition to landing speed, control restrictions also have a significant effect on landing quality. During the landing process, the situation is diverse, and the drones should closely abide by some reference trajectories to achieve efficient landing with respect to some performance indices [1,2]. Therefore, in this article, the authors establish a reference trajectory for UAV with consideration of values of different landing speed and optimal controls namely normal overload with restrictions, tangential overload with restrictions and lateral overload. This problem can be handled by 2 methods: analytical and numerical one. The analytical method offers feedback control, however, depending on the boundary conditions coupling with the restricted control during flight, seeking for an optimal control would be of arduousness. With the aim to establish a reference trajectory in the service of landing cases, the authors select the numerical method to solve the bespoken problem. This method burgeons results in a quick manner in case of restricted control and variable boundaries. For better application of the numerical method, the authors convert the optimal control problem to the boundary problem, the parameter continuation method [3][4][5][6][7] is used to successfully handle the boundary problem. The simulation results show that the UAV lands with different speed values and the control is within the allowable range.

Optimal Landing Trajectory
The system of equations of UAV movement in space includes the following differential equations [3]:  (1) in which: Vvelocity of UAV  -flight path angle  -heading angle x, y, z -UAV coordinates g -gravity acceleration (g = 9,80665 m/s²) ,, x y z n n n -respectively corresponding tangential overload, normal overload, lateral overload.
The authors find the optimal control at each time that makes Hamilton function H reach the maximum Due to the fact that z n is within the allowable range when conducting the survey or at the beginning of the landing phase, the movement direction of the UAV is asymptotical to or coincided with the runway direction, so . Accordingly, the system of equations for the UAV full movement includes the combination of the system of equations (1) and (2).
Then, there goes an essence to find the initial condition To solve the boundary problem, the authors use the method of parameter continuation.

Parameter Continuation Method
The essence of the parameter continuation method lies at the formal reduction of the considered boundary value problem to the Cauchy problem [4][5][6][7]. The boundary problem for a dynamic system with boundary conditions can be represented as an equation for the residuals at the right end of the trajectory: In which: Considering the immersion of equation (4) in a one-parameter family: (5) in which:  is the continuation parameter, and the writers represent the vector z as a function of this parameter: z = z (), moreover z (0) = z 0 from equation (4). They require equality (5) for any 0    1. Obviously, for  = 0, equation (5) coincides with (4), and for  = 1the equation for residuals for the desired boundary value problem (3).
Differentiating equation (5) with respect to the continuation parameter  and solving the resulting expression for the derivative dz/d, we obtain a formal reduction of equation ( Figure 1 illustrates the UAV trajectory in space corresponding to various landing speeds. It can be apparently seen that the higher the landing speed is the more tension the trajectory has.
The results shown indicate that the value of the Hamilton function at the end f t is close to 0 in all cases, which demonstrates that the landing time has been optimized ( Figure 2

Conclusion
Via surveying the flight trajectory, the authors may state that in case 1, landing with low speed holds more advantages in emergency circumstances in which the aircraft encounters alarming problems and must land on a short runway, but the decreasing landing speed comes with the increase of tangential overload. Hence, the actualization of flight will be of difficulty if the UAV fails to meet the tangential overload as calculated during flight. In case 2, with different beginning positions, the landing trajectory would be variable, the more the landing direction deviates from the runway direction, the greater the control energy consumes. The research results claims that different flight trajectories can be designed and the feasibility may be evaluated when realizing flight trajectory, thereby offering reference trajectories. In this article, there remains a point mis-considering the noise influence during flight since using numerical method instead of analytical method would reveal several disadvantages with noise involvement. Therefore, the authors are expected to consider the impact of noise in flight in the coming studies.