Finite Time Formation Control for Multiple Vehicles Based on Pontryagin's Minimum Principle
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摘要: 研究基于庞特里亚金极小值原理的多运载体有限时间编队问题.运载体刻画为欧氏群切丛上演化的全驱动刚体动力学模型.编队机动时间以及队形的几何结构是由编队任务指定的.对于期望的队形，首先利用庞特里亚金最小值原理给出了开环最优控制.为了克服开环控制对扰动的敏感性并增加针对初始条件不确定性摄动的鲁棒性，在假定运载体间通讯为全联通的模式下，通过反馈将系统当前状态作为初始状态，当前时刻作为初始时刻，进一步将开环控制律转化为闭环形式.为了验证所得结果，给出了平面及空间运载体编队的仿真算例.Abstract: The paper studies the problem of finite time formation control for multiple vehicles based on Pontryagin's minimum principle. The vehicle is modeled as a fully actuated rigid body with the dynamics evolving on the tangent bundle of Euclidean group. Both the formation maneuver time and the geometric structure of the formation are specified by the formation task. For the required formation, an open loop optimal control law is derived by using Pontryagin's minimum principle. In order to overcome the sensitivity of the open-loop control to the disturbance and increase the robustness of the control law to the initial perturbation, the open loop control law is converted to the closed loop form. This is done by feeding the current state back and initializing the control law at the current time, under the assumption that the mode of communication between the vehicles is all-to-all. For demonstration of the result, some numerical examples of formations for both planar and spacial vehicles are included.
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Key words:
- Finite time formation control /
- consensus /
- multiple vehicles /
- minimum principle
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图 1 两运载体的质心位置、姿态演化轨迹(第3图为$t =t_f$时的质心位置、姿态演化轨迹.
Fig. 1 The evolution trajectories of position and attitude of the two vehicles (The 3rd figure are the evolution trajectories of position and attitude at $t = t_f$.
图 2 取得一致性过程中两运载体的时间行为(从上到下: 空间坐标系下的位形(质心位置与姿态角);相对于第一个运载体刚体坐标系下的相对位形; 刚体坐标系下的速度;刚体坐标系下的控制. 从左到右: 相应变量的 $x$-轴坐标;$y$-轴坐标; 姿态角$\theta$轴坐标.
Fig. 2 The time behaviors of the two vehicles during the process of achieving consensus (From top to down: configuration(position and attitude angle) in the space frame; relative configuration with respect to the body frame of first vehicle;velocity in the body frame; control in the body frame. From left to right: the coordinates of the corresponding quantity in $x$-axis,$y$-axis,and attitude angle $\theta$.
图 3 两运载体的质心位置、姿态演化轨迹(左下角为$t = t_f$时的质心位置、姿态演化轨迹.
Fig. 3 The evolution trajectories of position and attitude of the two vehicles (The figure on the bottom left are the evolution trajectories of position and attitude at $t = t_f$.)
图 4 取得一致性过程中两运载体的时间行为(从上到下:空间坐标系下的位形(质心位置与姿态角);相对于第一个运载体刚体坐标系下的相对位形;刚体坐标系下的速度; 刚体坐标系下的控制. 从左到右: 相应变量的$x$-轴坐标; $y$-轴坐标; 姿态角$\theta$轴坐标
Fig. 4 The time behaviors of the two vehicles during the process of achieving consensus (From the top down: configuration(position and attitude angle) in the space frame; relative configuration with respect to the body frame of first vehicle;velocity in the body frame; control in the body frame. From left to right: the coordinates of the corresponding quantity in $x$-axis,$y$-axis,and attitude angle $\theta$.
图 5 两运载体的质心位置、姿态演化轨迹(左下角为$t = t_f$时的质心位置、姿态演化轨迹.
Fig. 5 The evolution trajectories of position and attitude of the two vehicles (The figure on the bottom left are the evolution trajectories of position and attitude at $t = t_f$.
图 6 取得一致性过程中两运载体的时间行为(从上到下:空间坐标系下的位形(质心位置与姿态角);相对于第一个运载体刚体坐标系下的相对位形;刚体坐标系下的速度; 刚体坐标系下的控制. 从左到右: 相应变量的$x$-轴坐标; $y$-轴坐标; 姿态角$\theta$轴坐标.
Fig. 6 The time behaviors of the two vehicles during the process of achieving consensus (From top to down: configuration(position and attitude angle) in the space frame; relative configuration with respect to the body frame of first vehicle;velocity in the body frame; control in the body frame. From left to right: the coordinates of the corresponding quantity in $x$-axis,$y$-axis,and attitude angle $\theta$.
图 7 两运载体的质心位置、姿态演化轨迹(左上角为$t = t_f$时的质心位置、姿态演化轨迹
Fig. 7 The evolution trajectories of position and attitude of the two vehicles (The figure on the top left are the evolution trajectories of position and attitude at $t = t_f$.)
图 8 取得一致性过程中两运载体的时间行为(从上到下: 空间坐标系下的位形(质心位置与姿态角);相对于第一个运载体刚体坐标系下的相对位形; 刚体坐标系下的速度;刚体坐标系下的控制. 从左到右: 相应变量的 $x$-轴坐标;$y$-轴坐标; 姿态角$\theta$轴坐标.
Fig. 8 The time behaviors of the two vehicles during the process of achieving consensus (From the top down: configuration(position and attitude angle) in the space frame; relative configuration with respect to the body frame of first vehicle;velocity in the body frame; control in the body frame. From left to right: the coordinates of the corresponding quantity in$x$-axis,$y$-axis,and attitude angle $\theta$.)
图 9 系统状态的时间行为 (从上到下:空间坐标系下的欧拉角; 质心位置以及刚体坐标系下的旋转速度和平移速度.从左到右: 相应变量的$x$-轴坐标; $y$-轴坐标;$z$-轴坐标.
Fig. 9 Time behaviors of system$'$s states (From the top down: Euler angle; position in the space frame; rotation velocity in the body frame; translation velocity in the body frame. From left to right: the coordinates of the corresponding quantity in$x$-axis,$y$-axis,and $z$-axis.
图 10 系统相对于第一个运载体刚体坐标系下的相对状态 (从上到下: 相对欧拉角;相对位置; 相对旋转速度; 相对平移速度. 从左到右:相应变量的$x$-轴坐标; $y$-轴坐标; $z$-轴坐标.
Fig. 10 Time behaviors of system$'$s relative states with respect to the body coordinate of the first agent (From top to down: relative Euler angle; relative position; relative rotation velocity; relative translation velocity. From left to right: the coordinates of the corresponding quantity in $x$-axis,$y$-axis,and$z$-axis.)
图 11 在刚体坐标系下的控制时间行为 (从上到下:广义力矩; 广义力. 从左到右: 相应变量的$x$-轴坐标;$y$-轴坐标; $z$-轴坐标.)
Fig. 11 Time behaviors of system$'$s control in the body coordinate (From top to down: generalized torque; and generalized force. From left to right: the coordinates of the corresponding quantity in $x$-axis,$y$-axis,and $z$-axis.)
图 12 空间四运载体质心位置、姿态演化轨迹
Fig. 12 The evolution trajectories of position and attitude of the four vehicles
表 1 运载体的初始位形
Table 1 Initial configurations of agents
序号 1 2 $x(0) $ 80 $-80$ $y(0) $ 100 $-100$ $\theta(0) $ $\dfrac{\pi}{2}$ $-\dfrac{\pi}{2}$ 初始位形是由空间坐标系下的质心位置坐标$(x,y)$ 和姿态角$\theta$给出. 
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表 2 运载体的初始位形
Table 2 Initial configurations of agents
序号 1 2 3 4 $x(0) $ 100 -100 -100 100 $y(0) $ 100 -100 100 -100 $\theta(0) $ 0 $-\pi$ ${\pi}/{2}$ $-{\pi}/{2}$ 初始位形是由空间坐标系下的质心位置坐标$(x,y)$ 和姿态角$\theta$给出.
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表 3 终端时刻相对位形
Table 3 Relative configurations at final time
相对指标$(1i)$ (11) (12) (13) (14) $x_{1i}(t_f)$ 0 -40 -40 -80 $y_{1i}(t_f)$ 0 -40 40 0 $\theta_{1i}(t_f)$ 0 0 0 0 队形由相对于第一运载体刚体坐标系的相对质心位置坐标$(x_{1i},y_{1i})$和相对姿态角$\theta_{1i}$给出.
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表 4 运载体的初始位形和初始速度
Table 4 Initial configurations and initial velocities of vehicles
序号 1 2 3 4 $x(0) $ 100 -100 -100 100 $y(0) $ 100 -100 100 -100 $\theta(0) $ 0 $-\pi$ ${\pi}/{2}$ $-{\pi}/{2}$ $v_x(0) $ 15 10 7 5 $v_y(0) $ 0 0 0 0 $v_{\theta}(0) $ 0.05 0.02 0.08 0.05 初始位形由空间坐标系下的质心位置坐标$(x,y)$和姿态角$\theta$给出. 初始速度由刚体坐标系下平移速度$(v_x,v_y)$和角速度$v_{\theta}$ 给出.
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表 5 运载体的初始位形和初始速度
Table 5 Initial configurations and initial velocities of vehicles
序号 1 2 3 4 $\theta_r(0) $ 0 0 0 0 $\theta_p(0) $ 0 0 0 0 $\theta_y(0) $ ${\pi}/{4}$ $-{3\pi}/{4}$ ${\pi}/{2}$ $-{\pi}/{2}$ $x(0) $ 100 -100 -100 100 $y(0) $ 100 -100 100 -100 $z(0) $ 0 0 0 0 $\omega_x(0) $ 0 0 0 0 $\omega_y(0) $ -0.15 0 0 0 $\omega_z(0) $ -0.08 -0.05 0.08 0.05 $v_x(0) $ 15 10 7 5 $v_y(0) $ 0 0 0 0 $v_z(0) $ 0 0 0 0 初始位形是由包括滚转,俯仰,偏航三个欧拉角$(\theta_r,\theta_p,\theta_y)$以及空间坐标系下质心的位置坐标$(x,y,z)$给定的.初始速度是在刚体坐标系下给定的,它们分别是绕$x$-轴,$y$-轴,$z$-轴的旋转速度$(\omega_x,\omega_y,\omega_z)$,以及沿$x$-轴,$y$-轴,$z$-轴的平移速度$(v_x,v_y,v_z)$.
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表 6 终端时刻相对位形
Table 6 Relative configurations at final time
相对指标$(1i)$ (11) (12) (13) (14) $\theta_{r,1i}(t_f)$ 0 0 0 0 $\theta_{p,1i}(t_f)$ 0 0 0 0 $\theta_{y,1i}(t_f)$ 0 0 0 0 $x_{1i}(t_f)$ 0 $-100$ -50 -50 $y_{1i}(t_f)$ 0 0 50 -50 $z_{1i}(t_f)$ 0 0 0 0 相对位形是由相对于第一运载体刚体坐标系下的相对欧拉角$(\theta_{r,1i},\theta_{p,1i},\theta_{y,1i})$及相对位置坐标$(x_{1i},y_{1i},z_{1i})$给出的.
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