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Trajectory planning for carrier aircraft on deck using Newton Symplectic pseudo-spectral method

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Abstract

The kinematic models for three dispatch modes of carrier aircraft were established, including individually taxiing, off-axle hitching towing without drawbar, and off-axle hitching towing with drawbar. As the high nonlinearity in the kinematics, a towing system with drawbar was transformed into a simpler virtual on-axle hitching towing system so as to facilitate the trajectory planning. Considering the dispatch efficiency and security, the trajectory planning problems of three dispatch modes were formulated as time-energy hybrid optimal control problems. To solve the nonlinear optimal control problem efficiently, a Symplectic pseudo-spectral method (SPM) was firstly developed based on the third kind of generating function, Symplectic theory and pseudo-spectral discretization. Then the Newton iteration and the SPM were used to determine the optimal terminal time according to the terminal transversality condition. The developed method was applied to solve trajectory planning problems of three dispatch modes, and the direct pseudo-spectral method was implemented for comparison. The simulation results suggest that the developed method can generate smooth dispatch trajectories with higher accuracy and efficiency, where no infeasible solution occurs, leading to better operability and applicability.

Keywords： carrier aircraft ; trajectory planning ; Symplectic pseudo-spectral method (SPM) ; Newton iteration method ; optimal control

LIU Jie, DONG Xian-zhou, HAN Wei, WANG Xin-wei, LIU Chun, JIA Jun. Trajectory planning for carrier aircraft on deck using Newton Symplectic pseudo-spectral method. Journal of Zhejiang University(Engineering Science)[J], 2020, 54(9): 1827-1838 doi:10.3785/j.issn.1008-973X.2020.09.020

1.1. 舰载机滑行运动学模型

$\dot {{X}} = {\left[ {{v_1}\cos \theta_1 ,}\;{{v_1}\sin \theta_1 ,}\;{{{{v_1}{u_1}} / {{L_1}}},}\;{{u_2}} \right]^{\rm{T}}}.$

$\left. \begin{array}{l} 0 \leqslant {v_1} \leqslant {v_1}_{\max },\\ \left| {{u_1}} \right| \leqslant \tan\; {\beta _{1\max} },\\ {a_1}_{{\rm{min}}} \leqslant {u_2} \leqslant {a_1}_{\max }. \end{array} \right\}$

1.2. 无杆牵引系统运动学模型

${v_1} = {v_2}\left( {\cos \;{\beta _1} + {{{L_2}\sin\; {\beta _1}{u_1}} / {{M_0}}}} \right).$

$\begin{split}\dot {{X}} =& \frac{{\rm{d}}}{{{\rm{d}}t}}\left[ {\begin{array}{*{20}{c}} {{x_1}},& {{y_1}},& {{\theta _1}},& {{\theta _2}},& {{v_2}} \end{array}} \right]^{\rm T} =\\ &\left[ {\begin{array}{*{20}{c}} {{{\left( {{L_2}{v_2}\cos\; {\beta _1} + {M_0}{v_2}{u_1}\sin\; {\beta _1}} \right)\cos\; {\theta _1}} / {{L_2}}}} \\ {{{\left( {{L_2}{v_2}\cos\; {\beta _1} + {M_0}{v_2}{u_1}\sin\; {\beta _1}} \right)\sin\; {\theta _1}} / {{L_2}}}} \\ {{{\left( {{L_2}{v_2}\sin\; {\beta _1} - {M_0}{v_2}{u_1}\cos\; {\beta _1}} \right)} / {{L_1}{L_2}}}} \\ {{{{v_2}{u_1}} / {{L_2}}}} \\ {{u_2}} \end{array}} \right].\end{split}$

${v_1} = {{{v_3}\cos \;{\beta _1}\cos\; {\beta _2}\left( {{L_3} + M\tan\; {\beta _2}\tan\; \alpha } \right)} / {{L_3}}}.$

$\begin{split}\dot {{X}} =& \dfrac{{\rm{d}}}{{{\rm{d}}t}}\left[ {\begin{array}{*{20}{c}} {{x_1}}, & {{y_1}} ,& {{\theta _1}} ,& {{\beta _1}} ,& {{\beta _2}} \end{array}} \right]^{\rm T} =\\ &\left[ {\begin{array}{*{20}{c}} {{u_2}\cos \;{\theta _1}} \\ {{u_2}\sin \;{\theta _1}} \\ {{{{u_2}\tan \;{\beta _1}} / {{L_1}}}} \\ {{u_2}\left( {\dfrac{{{L_3}\tan\; {\beta _2} - M{u_1}}}{{{L_2}\cos\; {\beta _1}\left( {{L_3} + M{u_1}\tan\; {\beta _2}} \right)}} - \dfrac{{\tan\; {\beta _1}}}{{{L_1}}}} \right)} \\ {{u_2}\left( {\dfrac{{{L_2}{u_1} - {L_3}\sin \;{\beta _2} + M{u_1}\cos\; {\beta _2}}}{{{L_2}\cos \;{\beta _1}\cos \;{\beta _2}\left( {{L_3} + M{u_1}\tan \;{\beta _2}} \right)}}} \right)} \end{array}} \right].\end{split}$

$J = \int\limits_{{t_0}}^{{t_{\rm f}}} {\left( {{ H} - {{{\lambda }}^{\rm{T}}}{\dot{ X}}} \right)} {\rm d}t.$

$\frac{{\partial { H}}}{{\partial {{U}}}} = {{UR}} + {{{B}}^{\rm{T}}}{{\lambda }} + {{{D}}^{\rm{T}}}{{\beta }} = {{0}}.$

Hamiltonian正则方程为

${\dot{ X}} = \frac{{\partial { H}}}{{\partial {{\lambda }}}},\;{\dot{ \lambda }} = - \frac{{\partial { H}}}{{\partial {{X}}}}.$

2）初始化迭代参数 $k = 0$，设置控制变量、控制变量和协变量的初始猜测解分别为 ${{{X}}^{\left[ 0 \right]}}$${{{U}}^{\left[ 0 \right]}}$${{{\lambda }}^{\left[ 0 \right]}}$.

3）结合 ${{{X}}^{\left[ k \right]}}$${{{U}}^{\left[ k \right]}}$${{{\lambda }}^{\left[ k \right]}}$，采用基于第三类的保辛伪谱算法计算出式（16）的解： ${{{X}}^{\left[ {k + 1} \right]}}$${{{U}}^{\left[ {k + 1} \right]}}$${{{\lambda }}^{\left[ {k + 1} \right]}}$，具体计算步骤如下.

${{{\upsilon }}^j}\left( \tau \right) = \sum\limits_{l = 0}^{{N^j}} {{{\upsilon }}_l^j\frac{{\left( {{\tau ^2} - 1} \right){{\dot L}^j}\left( \tau \right)}}{{{N^j}\left( {{N^j} + 1} \right)\left( {\tau - \tau _l^j} \right)L_l^j}}} .$

图 2

Fig.2   Taxiing trajectory of aircraft

图 3

Fig.3   Change of control variables of sliding system with time

图 4

Fig.4   Changs in speed and steering angle of taxiing aircraft over time

Tab.1  Comparison results of NSP and pseudospectral method in trajectory planning for taxiing system

 滑行系统 方法 Mayer Lagrange $J$ ${t_{\rm f}}$/s tc/s 舰载机3 NSP 4.76 16.33 21.10 95.27 16.27 伪谱 5.23 15.68 20.91 104.59 91.87 舰载机5 NSP 7.27 1.92 20.91 145.49 7.73 伪谱 7.64 3.81 11.45 152.76 54.40 舰载机11 NSP 10.03 0.63 10.66 200.66 12.72 伪谱 10.11 0.51 10.63 202.28 76.29

图 5

Fig.5   Trajectory of carrier aircraft and tractor of towed carrier aircraft system without drawbar

图 6

Fig.6   Control of towed carrier aircraft system without drawbar

图 7

Fig.7   Velocity and steering angle of aircraft of towed carrier aircraft system without drawbar

Tab.2  Comparison results of NSP and pseudospectral method in trajectory planning for towed carrier aircraft system without drawbar

 无杆牵引系统 方法 Mayer Lagrange $J$ ${t_{\rm f}}$/s tc/s 位置7到10 NSP 8.33 5.30 13.63 83.29 5.33 伪谱 9.53 3.81 13.34 95.27 45.27 位置7到12 NSP 8.47 4.24 12.71 84.69 4.78 伪谱 10.83 65.54 76.37 108.26 68.63 位置4到11 NSP 6.55 5.51 12.06 65.55 2.32 伪谱 6.32 7.80 14.12 63.25 21.03

图 8

Fig.8   Trajectory of carrier aircraft and tractor of towed carrier aircraft system with drawbar by NPS algorithm

NSP算法对在轴无杆虚拟系统进行路径规划的可行性在第4.3节已进行了研究和验证，本节不再赘述，只给出飞机速度和转向角的变化规律. 如图9（a）（b）（c）所示分别为从位置4到位置1、从位置7到位置8和从位置7到位置10的舰载机速度和转向角随时间的变化规律. 分析图9可知，采用本文算法得到的舰载机速度和转向角变化均比较平缓，其中飞机的速度均处于 $\left[ { -1.0,\;1.0} \right]\;{\rm{m/s}}$，即控制变量 ${u_1}$均处于 $\left[ { - 0.9,\;0.9} \right]$，这说明飞机速度和转向角均满足相应的约束，且变化均比较平稳，满足工程实际需求.

图 9

Fig.9   Velocity and steering angle of aircraft of towed carrier aircraft system with drawbar

5. 结　论

1）本研究提出了舰载机单机滑行、离轴无杆牵引系统、离轴有杆牵引系统的运动学模型. 其中，无杆牵引系统的路径规划目前鲜有研究，所建立的运动学模型可以为后续无杆牵引系统的控制等问题提供参考. 提出的有杆牵引系统的运动学模型较现有的模型更加简洁，提出的模型可为有杆牵引系统的研究提供参考. 此外，由于有杆牵引系统最为复杂，所建立的有杆牵引系统运动学模型仍为一个强非线性系统，对初始猜测解以及参数均较为敏感，容易发散. 为此，将复杂的有杆牵引系统转化为一个更加简单的虚拟在轴无杆牵引系统，以实现对舰载机和牵引车的轨迹进行求解，这可为后续的相关研究提供新的思路.

（2）建立了路径规划的最优控制模型，采用最优控制方法进行路径规划，较其他方法而言，该算法所得到的结果不仅可以严格满足终端约束条件，还可以满足最优控制条件，以确保所得到的结果最优.

（3）NSP算法可以以较高的精度和稳定性求解末端时间不定的最优控制问题，对初始猜测解敏感度低、计算效率高、收敛速度快. 此外，本文算法还可以解决时间-能量最优的控制问题，在确保控制平稳的前提下得到使舰载机从起点到终点耗时最短的路径，有利于提高出动和调运效率.

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