Trajectory Control of Quadrotor With Cable-Suspended Load via Dynamic Feedback Linearization
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摘要: 三维空间下的四旋翼吊挂运输系统是一种欠驱动、强耦合、多变量的非线性系统. 根据系统的动力学特点, 将系统分解为双质点系绳连接子系统和四旋翼姿态控制子系统. 选择与系统自由度维数相同的广义坐标并基于虚位移原理计算对应的广义力, 从而建立系统的拉格朗日动力学方程. 利用微分平滑特性证明了运输系统存在平凡零动态, 因此可通过动态反馈转化为线性和能控系统. 经过2次动态扩展和变量代换, 原系统扩展为总相对阶等于系统状态维度的线性能控系统. 基于赫尔维茨稳定性判据, 设计了跟踪误差指数收敛的动态反馈控制律. 该方法可作为一类非线性系统控制器设计的标准方法. 最后以三维空间的螺旋曲线及水平面内频率变化的圆周曲线为参考轨迹进行仿真, 仿真结果验证了控制系统的有效性.Abstract: A quadrotor with cable-suspended load in 3-D space is considered, which is underactuated, strongly coupling and nonlinear. According to the dynamic feature, the system is decoupled into quadrotor attitude control subsystem and double points link subsystem. The generalized coordinate with the same dimension of the system freedom is selected and the generalized force is calculated based on the principle of virtual displacement, then the Lagrange dynamic equation of the system is established. The quadrotor-load system is proved to have ordinary zero dynamics based on differentially flat property, so it can be transformed into linear and controllable system by dynamic feedback. After 2 dynamic expansions and variable substitutions, the original system is extended to a linear controllable system whose total relative orders are equal to the system state dimensions. Based on Hurwitz stability criterion, a dynamic feedback controller with exponential convergence of position error is designed. This method can be used as a standard method for a class of nonlinear systems. Finally, the spiral curve in 3-D space and the circular curve in the horizontal plane with varying frequency are taken as the reference trajectory for simulation. Simulation results demonstrate the effectiveness of the proposed method.
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Key words:
- Quadrotor /
- transport /
- zero dynamics /
- dynamic feedback
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图 3 两种控制方法下螺旋曲线跟踪轨迹对比((a)动态反馈控制方法; (b)几何控制方法)
Fig. 3 Track a spiral curve via two control methods. ((a) Dynamic feedback control; (b) Geometry control)
图 4 两种控制方法下跟踪螺旋曲线误差收敛情况对比((a)动态反馈控制方法; (b)几何控制方法)
Fig. 4 Position errors convergence when tracking a spiral curve via two control methods. ((a) Dynamic feedback control; (b) Geometry control)
图 8 螺旋曲线跟踪过程中系绳振荡角曲线
Fig. 8 Curve of swing angel on the cable when tracking a spiral curve
图 9 两种控制方法跟踪圆周曲线轨迹对比(a)动态反馈控制方法; (b)分段控制方法
Fig. 9 Track a circle via two control methods. (a) Dynamic feedback control; (b) Two-time-scale control
图 10 两种控制方法跟踪圆周曲线误差收敛情况对比((a)动态反馈方法; (b)分段控制方法)
Fig. 10 Position errors convergence when tracking a circle curve via two control methods. ((a) Dynamic feedback control; (b) Two-time-scale control)
表 1 仿真中使用的模型参数
Table 1 Model parameters in the simulations
变量 参数 单位 mq 0.4 kg ml 0.1 kg l1 0.8 m l2 0.2 m g −9.8 m·s−2 
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