A Kinematic Model and Trajectory Tracking Control of Tractor-Aircraft System Based on Front Wheel Angle Compensation
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摘要: 传统的牵引车-飞机系统运动学模型在低速滑行牵引工况下精度不足，导致轨迹跟踪控制误差大、响应慢，难以满足新型离港方式对轨迹精度和安全性的高要求。为提升运动学模型精度与轨迹跟踪性能，研究了基于牵引车前轮转向角补偿函数的运动学模型补偿方法。以威海广泰AM210无杆牵引车和B737-800飞机为研究对象，先建立传统的牵引车-飞机系统运动学模型，然后将传统运动学模型与Trucksim车辆模型进行开环联合仿真对比分析，通过引入补偿函数来补偿2种模型之间的轨迹偏差，同时设计了基于非线性模型预测控制（nonlinear model predictive control，NMPC）的牵引车-飞机系统轨迹跟踪控制器。以双移线工况作为参考轨迹，搭建MATLAB/Simulink与Trucksim闭环联合仿真模型，并将NMPC控制器与传统比例积分微分控制（proportional integral derivative，PID）的轨迹跟踪控制器进行轨迹仿真对比分析，验证NMPC控制器的优越性。进一步在2 m/s和4 m/s牵引车速度下分别对基于传统、补偿后的运动学模型的NMPC控制器跟踪性能进行评估并且分析存在不同初始偏差下对牵引车-飞机系统轨迹跟踪性能的影响。仿真结果显示：在2 m/s和4 m/s牵引车速度下基于补偿后的运动学模型的NMPC控制器可以使跟踪峰值误差分别降低61.93%和41.63%，均方根误差分别降低56.14%和37.69%。在存在的初始偏差情况下，基于NMPC的轨迹跟踪控制器能够使系统在30 s内完成对初始偏差（横向偏差0.5~1 m、航向角偏差0.05~0.1 rad）的修正，无超调现象。Abstract: The traditional kinematic model of the tractor-aircraft system exhibits insufficient accuracy under low-speed taxiing towing conditions, resulting in significant trajectory tracking control errors and slow responses, which is difficult to meet the stringent requirements for trajectory accuracy and safety for the new departure mode. To improve the accuracy of the kinematic model and the performance of trajectory tracking, this study proposes a kinematic model compensation method based on a front wheel steering angle compensation function of the tractor. Taking the Weihai Guangtai AM210 rodless tractor and the B737-800 aircraft as research objects, a traditional kinematic model of the tractor-aircraft system is first established. Then, the traditional kinematic model and the Trucksim vehicle model are subjected to open-loop joint simulation and comparative analysis. The trajectory deviations between these two models are compensated by introducing a compensation function. Meanwhile, a trajectory tracking controller for the tractor-aircraft system based on nonlinear model predictive control (NMPC) is designed. Taking the double-shifting line condition as the reference trajectory, a closed-loop joint simulation model utilizing MATLAB/Simulink and Trucksim is built, and the trajectory simulation comparison and analysis between the NMPC controller and the traditional proportional-integral-derivative control trajectory tracking controller are conducted to verify the superiority of the NMPC controller. The tracking performance of the NMPC controller based on the traditional and compensated kinematic models is further evaluated at tractor speeds of 2 m/s and 4 m/s respectively, and the influence of different initial deviations on the trajectory tracking performance of the tractor-aircraft system is analyzed. The simulation results show that the NMPC controller based on the compensated kinematic model can reduce the peak tracking error by 61.93% and 41.63%, and the root mean square errors by 56.14% and 37.69%, at tractor speeds of 2 m/s and 4 m/s, respectively. Under the condition of existing initial deviations, the trajectory tracking controller based on NMPC can enable the system to correct the initial deviations (lateral deviation of 0.5 to 1 m and heading angle deviation of 0.05 to 0.1 rad) within 30 s without overshoot.
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图 2 传统运动学模型与Trucksim车辆模型铰接点的轨迹比较
Figure 2. A comparison of the trajectories of hinge points between the traditional kinematic model and the Trucksim vehicle model
图 4 传统运动学模型与Trucksim车辆模型在初始0.2 s内铰接点的轨迹比较
Figure 4. A comparison of the trajectories of hinge points within the initial 0.2 seconds between the traditional kinematic model and the Trucksim vehicle model
图 5 补偿后运动学模型与Trucksim车辆模型在初始0.2 s内铰接点的轨迹比较
Figure 5. A comparison of the trajectories of the hinge points between the compensated kinematic model and the Trucksim vehicle model within the initial 0.2 seconds
图 8 不同控制器下牵引车-飞机系统的轨迹跟踪对比
Figure 8. Trajectory tracking comparison of tractor-aircraft systems under different controllers
图 9 不同控制器下牵引车-飞机系统轨迹跟踪误差结果
Figure 9. Trajectory tracking error results of the tractor-aircraft system under different controllers
图 10 基于传统运动学模型控制器的轨迹跟踪和距离误差结果
Figure 10. Trajectory tracking and distance error results based on the traditional kinematic model controller
图 11 基于传统、补偿后运动学模型控制器的轨迹跟踪和距离误差结果
Figure 11. Trajectory tracking and distance error results based on the traditional, compensated kinematic model controller
图 12 飞机与牵引车的相对夹角变化结果
Figure 12. The change of the relativeAngle between the aircraft and the tractor
图 13 不同速度下2种运动学仿真模型的轨迹跟踪误差对比
Figure 13. Comparison of trajectory tracking errors of two kinematic simulation models at different speeds
图 14 不同初始偏差下牵引车-飞机系统的轨迹跟踪对比
Figure 14. Trajectory tracking comparison of tractor-aircraft systems under different initial deviations
图 15 不同初始偏差下牵引车-飞机系统轨迹跟踪误差结果
Figure 15. The trajectory tracking error results of the tractor-aircraft system under different initial deviations
表 1 牵引车-飞机系统参数
Table 1. Tractor-Aircraft system parameters
参数/m 数值 AM210牵引车轴距L 4.5 B737-800飞机轴距L1 15.6 牵引车前轴几何中心与铰接点距离L2 2.96 牵引车后轴几何中心与铰接点距离L3 1.54 
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表 2 控制器参数
Table 2. Controller parameter
参数 值 T/ms 50 NP 15 Nc 2 Q Diag{100, 100, 100, 50} R Diag{50, 50}
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表 3 均方根误差值
Table 3. Root mean square error value(RMSE)
控制器 横向误差/m 飞机航向角误差/rad 牵引车航向角误差/rad NMPC 0.041 5 0.001 9 0.003 9 PID 0.092 4 0.012 7 0.032 7
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表 4 不同速度下2种控制器仿真模型的峰值跟踪误差和均方根跟踪误差
Table 4. The peak tracking error and root mean square tracking error of the two controller simulation models at different speeds
速度/（m/s） 运动学模型 峰值误差/m 均方根误差/m 2 传统模型 0.190 7 0.041 5 2 补偿后模型 0.072 6 0.018 2 4 传统模型 0.123 0 0.026 8 4 补偿后模型 0.071 8 0.016 7
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表 5 不同初始偏差参数
Table 5. Different initial deviation parameters
序号 $\Delta x_{1} / \mathrm{m}$ $\Delta y_{1} / \mathrm{m}$ $\Delta \varphi_{1} / \mathrm{rad}$ $\Delta \varphi_{2} / \mathrm{rad}$ 案例1 -0.5 -0.5 0 0 案例2 -0.6 -0.8 0 0 案例3 -0.7 -0.7 0 0.05 案例4 -1 -1 0 0.1
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