CN114996841B - Optimal coplanarity transfer search method based on tangent initial value - Google Patents
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Abstract
Description
技术领域Technical Field
本发明涉及基于正切初值的最优共面转移搜索方法。The invention relates to an optimal coplanar transfer searching method based on tangent initial value.
背景技术Background Art
航天器的轨道动力学是航天工程的基础科学。航天器的轨道设计是保证航天器工作任务顺利执行的关键。自人类进入太空时代以来,轨道设计理论迅速发展,相关方法和技术得到大量实际工程的验证。轨道转移是轨道设计的一种基本问题,是指航天器主动地从原有轨道经过机动过程转移到目标轨道,通常考虑燃料消耗、转移时间、控制精度等设计指标。The orbital dynamics of spacecraft is the basic science of aerospace engineering. The orbital design of spacecraft is the key to ensure the smooth execution of spacecraft missions. Since humans entered the space age, orbital design theory has developed rapidly, and related methods and technologies have been verified by a large number of actual projects. Orbital transfer is a basic problem in orbital design, which refers to the active transfer of a spacecraft from its original orbit to its target orbit through a maneuvering process, usually taking into account design indicators such as fuel consumption, transfer time, and control accuracy.
目前,在轨卫星数量越来越多,转移任务的轨道设计需求随之增长。这对转移任务的求解效率提出了更高要求。在求解燃料最优转移轨道时,通常根据任务所要求的脉冲数,通过遗传算法等全局搜索算法,求得全局最优脉冲转移方案。但是传统方法的受优化变量的限制(3n-4个优化变量),计算时间较长,求解效率较低,难于应对大规模的轨道转移需求。At present, the number of satellites in orbit is increasing, and the demand for orbit design of transfer missions is increasing accordingly. This puts higher requirements on the efficiency of solving transfer missions. When solving the optimal fuel transfer orbit, the global optimal pulse transfer scheme is usually obtained based on the number of pulses required by the mission through global search algorithms such as genetic algorithms. However, the traditional method is limited by the optimization variables (3n-4 optimization variables), the calculation time is long, the solution efficiency is low, and it is difficult to cope with large-scale orbit transfer needs.
发明内容Summary of the invention
本发明的目的是为了解决现有方法受优化变量的限制(3n-4个优化变量),计算时间较长,求解效率较低,难于应对大规模的轨道转移需求的问题,而提出一种基于正切初值的最优共面转移搜索方法。The purpose of the present invention is to solve the problems that the existing methods are limited by optimization variables (3n-4 optimization variables), have long calculation time, low solution efficiency, and are difficult to cope with large-scale orbit transfer needs, and to propose an optimal coplanar transfer search method based on tangent initial values.
一种基于正切初值的最优共面转移搜索方法具体过程为:An optimal coplanar transfer search method based on tangent initial value has the following specific process:
步骤一、完成脉冲优化的初始条件设置:Step 1: Complete the initial condition setting for pulse optimization:
根据航天器初始轨道参数、目标轨道参数以及任务要求的脉冲数n,设置遗传算法的迭代次数、种群数量参数;According to the initial orbit parameters of the spacecraft, the target orbit parameters and the number of pulses n required by the mission, the number of iterations and population size parameters of the genetic algorithm are set;
步骤二、遗传算法求解正切脉冲全局最优解:Step 2: Genetic algorithm is used to find the global optimal solution of tangent pulse:
基于步骤一,将推力方向限制在速度正切方向,设定优化指标为燃料最优,利用遗传算法,对正切脉冲燃料最优解所需要优化的2n-3个变量进行全局搜索,得到正切脉冲条件下的全局燃料最优解;Based on
步骤三、完成fmincon算法的初始化:Step 3: Complete the initialization of the fmincon algorithm:
将遗传算法得到的正切脉冲条件下的全局最优解每次脉冲的径向值设置为0后再代入fmincon算法作为初始猜测;The radial value of each pulse of the global optimal solution under the tangent pulse condition obtained by the genetic algorithm is set to 0 and then substituted into the fmincon algorithm as the initial guess;
步骤四、fmincon算法求解自由方向脉冲的局部最优解:Step 4: Use the fmincon algorithm to find the local optimal solution of the free direction pulse:
由步骤一中给出的初始条件,将脉冲方向限制在轨道平面内,设定优化指标为燃料最优,利用fmincon算法,对3n-4个优化变量进行局部优化,求得自由方向脉冲条件下的局部燃料最优解。Based on the initial conditions given in
本发明的有益效果为:The beneficial effects of the present invention are:
本发明提出了一种基于正切脉冲初始猜测的共面轨道转移燃料最优快速搜索算法。正切脉冲是指推力方向与速度方向共线的脉冲,在工程上有易于实现、安全性高、求解方便、能量近似最优的特点,正切方向是改变轨道能量最快的推力方向。该算法通过搜索正切脉冲的全局最优解,利用正切脉冲优化变量少(2n-3个优化变量)、燃料近似最优的特点,有效提高了计算效率,并以此解作为局部优化算法的初始猜测,进而采用局部优化算法得到最终解,保证燃料最优的同时节省了计算时间。The present invention proposes an optimal rapid search algorithm for coplanar orbit transfer fuel based on the initial guess of tangent pulse. Tangent pulse refers to a pulse whose thrust direction is colinear with the velocity direction. It is easy to implement, highly safe, easy to solve, and has the characteristics of approximately optimal energy in engineering. The tangent direction is the thrust direction that changes the orbit energy the fastest. The algorithm searches for the global optimal solution of the tangent pulse, and utilizes the characteristics of the tangent pulse with few optimization variables (2n-3 optimization variables) and approximately optimal fuel, which effectively improves the calculation efficiency. This solution is used as the initial guess of the local optimization algorithm, and then the local optimization algorithm is used to obtain the final solution, which ensures the optimal fuel while saving calculation time.
在本发明中,先直接限定脉冲为正切脉冲,从而减少优化变量,得到正切条件下的全局最优解后,以此解作为局部优化算法的初始猜测,进而采用局部优化算法得到最终解。该算法利用了正切脉冲优化变量少的特点,有效提高了计算效率,可与原来的自由脉冲全局搜索方法收敛到同一组燃料最优解,节省了计算时间,对于任意数量的脉冲均可使用。In the present invention, the pulse is first directly limited to a tangent pulse, thereby reducing the optimization variables, and after obtaining the global optimal solution under the tangent condition, this solution is used as the initial guess of the local optimization algorithm, and then the local optimization algorithm is used to obtain the final solution. The algorithm takes advantage of the fact that the tangent pulse has few optimization variables, effectively improves the calculation efficiency, and can converge to the same set of fuel optimal solutions as the original free pulse global search method, saving calculation time, and can be used for any number of pulses.
附图说明BRIEF DESCRIPTION OF THE DRAWINGS
图1为基于正切初值的共面轨道转移燃料最优搜索算法流程图;FIG1 is a flow chart of the optimal search algorithm for coplanar orbit transfer fuel based on tangent initial value;
图2为传统方法求解共面轨道转移燃料最优流程图;FIG2 is a conventional method for solving the optimal flow chart of coplanar orbital fuel transfer;
图3为轨道转移过程示意图;FIG3 is a schematic diagram of the orbit transfer process;
图4为两脉冲燃料对比图;Figure 4 is a comparison of two pulse fuels;
图5为三脉冲燃料对比图。Figure 5 is a comparison chart of three-pulse fuels.
具体实施方式DETAILED DESCRIPTION
具体实施方式一:本实施方式一种基于正切初值的最优共面转移搜索方法具体过程为:Specific implementation method 1: This implementation method is an optimal coplanar transfer search method based on tangent initial value. The specific process is as follows:
步骤一、完成脉冲优化的初始条件设置:Step 1: Complete the initial condition setting for pulse optimization:
根据航天器初始轨道参数、目标轨道参数以及任务要求的脉冲数n,设置遗传算法的迭代次数、种群数量等参数;According to the initial orbit parameters of the spacecraft, the target orbit parameters and the number of pulses n required by the mission, the number of iterations and the number of populations of the genetic algorithm are set;
步骤二、遗传算法求解正切脉冲全局最优解:Step 2: Genetic algorithm is used to find the global optimal solution of tangent pulse:
基于步骤一,将推力方向限制在速度正切方向,设定优化指标为燃料最优,利用遗传算法,对正切脉冲燃料最优解所需要优化的2n-3个变量进行全局搜索,得到正切脉冲条件下的全局燃料最优解;Based on
步骤三、完成fmincon算法的初始化:Step 3: Complete the initialization of the fmincon algorithm:
根据脉冲数等初始条件,合理设置fmincon算法的最大迭代次数,收敛误差限等参数,以保证收敛精度达到仿真要求将遗传算法得到的正切脉冲条件下的全局最优解每次脉冲的径向值设置为0后再代入fmincon算法作为初始猜测;可以调高fmincon算法的最大迭代次数,并合理设置收敛条件,以得到符合收敛条件的最优解;According to the initial conditions such as the number of pulses, the maximum number of iterations of the fmincon algorithm, the convergence error limit and other parameters are reasonably set to ensure that the convergence accuracy meets the simulation requirements. The radial value of each pulse of the global optimal solution under the tangent pulse condition obtained by the genetic algorithm is set to 0 and then substituted into the fmincon algorithm as the initial guess; the maximum number of iterations of the fmincon algorithm can be increased, and the convergence conditions can be reasonably set to obtain the optimal solution that meets the convergence conditions;
步骤四、fmincon算法求解自由方向脉冲的局部最优解:Step 4: Use the fmincon algorithm to find the local optimal solution of the free direction pulse:
由步骤一中给出的初始条件,将脉冲方向限制在轨道平面内,设定优化指标为燃料最优,利用fmincon算法,对3n-4个优化变量进行局部优化,求得自由方向脉冲条件下的局部燃料最优解。Based on the initial conditions given in
具体实施方式二:本实施方式与具体实施方式一不同的是,所述步骤一中完成脉冲优化的初始条件设置:Specific implementation method 2: This implementation method is different from the
根据航天器初始轨道参数、目标轨道参数以及任务要求的脉冲数n,设置遗传算法的迭代次数、种群数量等参数;According to the initial orbit parameters of the spacecraft, the target orbit parameters and the number of pulses n required by the mission, the number of iterations and the number of populations of the genetic algorithm are set;
具体过程为:The specific process is:
初始时刻航天器的轨道参数在J2000地心惯性坐标系下给出,航天器的初始轨道参数包括:半长轴a0、离心率e0、轨道倾角i0、升交点赤经Ω0、近心点角距ω0;The orbital parameters of the spacecraft at the initial moment are given in the J2000 geocentric inertial coordinate system. The initial orbital parameters of the spacecraft include: semi-major axis a 0 , eccentricity e 0 , orbital inclination i 0 , ascending node right ascension Ω 0 , pericenter angular distance ω 0 ;
目标轨道参数包括:半长轴af、离心率ef、轨道倾角if、升交点赤经Ωf、近心点角距ωf;其中,轨道倾角和升交点赤经应与初始时刻航天器的相同,i0=if,Ω0=Ωf,即航天器的初始轨道与目标轨道满足共面条件;The target orbit parameters include: semi-major axis a f , eccentricity e f , orbit inclination i f , ascending node right ascension Ω f , pericenter angular distance ω f ; among which, the orbit inclination and ascending node right ascension should be the same as those of the spacecraft at the initial moment, i 0 = if , Ω 0 = Ω f , that is, the initial orbit of the spacecraft and the target orbit meet the coplanar condition;
初始条件中初始轨道和目标轨道离心率较大时,会出现正切脉冲的全局最优解与自由方向脉冲全局最优解差距较大的情况,即在正切脉冲全局最优解施加脉冲的地点,不在方向自由脉冲全局最优解施加脉冲的地点附近,用本发明中的方法存在一定可能性得到一个局部最优解;遗传算法的种群数量和迭代次数应合理设置以保证收敛性与收敛速度;When the eccentricity of the initial orbit and the target orbit in the initial condition is large, there will be a large gap between the global optimal solution of the tangent pulse and the global optimal solution of the free direction pulse, that is, the location where the tangent pulse global optimal solution applies the pulse is not near the location where the free direction pulse global optimal solution applies the pulse. There is a certain possibility of obtaining a local optimal solution using the method of the present invention; the population size and the number of iterations of the genetic algorithm should be reasonably set to ensure convergence and convergence speed;
在本发明的全流程中所采用的动力学模型均为二体模型,即不考虑地球非球形摄动、太阳光压、大气阻力等摄动因素;The dynamic models used in the whole process of the present invention are all two-body models, that is, the perturbation factors such as the non-spherical perturbation of the earth, the solar light pressure, and the atmospheric resistance are not considered;
其中,μ表示地球的引力常数,r和v分别表示J2000地心惯性坐标系下航天器的位置矢量和速度矢量,|r|表示相应位置矢量的大小;表示r的一阶导数,表示v的一阶导数。Where, μ represents the gravitational constant of the Earth, r and v represent the position vector and velocity vector of the spacecraft in the J2000 geocentric inertial coordinate system, respectively, and |r| represents the magnitude of the corresponding position vector; represents the first-order derivative of r, represents the first derivative of v.
其它步骤及参数与具体实施方式一相同。The other steps and parameters are the same as those in the first embodiment.
具体实施方式三:本实施方式与具体实施方式一或二不同的是,所述步骤二中遗传算法求解正切脉冲全局燃料最优解:Specific implementation method three: This implementation method is different from specific implementation methods one or two in that the genetic algorithm in step two solves the tangent pulse global fuel optimal solution:
基于步骤一,将推力方向限制在正切方向,设定优化指标为燃料最优,利用遗传算法,对正切脉冲燃料最优解所需要优化的2n-3个变量进行全局搜索,得到正切脉冲条件下的全局燃料最优解;Based on
具体过程为:The specific process is:
在步骤二中,燃料消耗量的计算和优化过程如下:In step 2, the calculation and optimization process of fuel consumption is as follows:
步骤二一、假设脉冲数量为n,得到燃料最优解所需要优化的状态变量数量至少为2n-3个,选取待优化状态变量为X=[θ1,θ2,...,θn-1,Δv1,Δv2,...,Δvn-2];Step 21: Assuming that the number of pulses is n, the number of state variables to be optimized to obtain the optimal fuel solution is at least 2n-3, and the state variables to be optimized are selected as X=[θ 1 ,θ 2 ,...,θ n-1 ,Δv 1 ,Δv 2 ,...,Δv n-2 ];
其中,θi表示第i次脉冲时航天器所在位置与初始轨道升交点矢径方向OX之间的夹角,取值范围在[0,2π];Δvi为第i次脉冲的大小,考虑到地球卫星椭圆轨道的速度大小,脉冲不必过大,取值范围设置为[-10,10],以与原速度方向同向为正,单位为km/s;Among them, θ i represents the angle between the position of the spacecraft at the time of the i-th pulse and the radial direction OX of the ascending node of the initial orbit, and the value range is [0,2π]; Δvi is the size of the i-th pulse. Considering the speed of the earth satellite's elliptical orbit, the pulse does not need to be too large. The value range is set to [-10,10], with the same direction as the original speed direction as positive, and the unit is km/s;
步骤二二、根据初始轨道参数[a0,e0,i0,Ω0,ω0]和待优化状态变量可进行轨道递推,以第一次脉冲过程为例,轨道递推过程表示如下:Step 22: According to the initial orbital parameters [a 0 , e 0 , i 0 , Ω 0 , ω 0 ] and the state variables to be optimized, orbital recursion can be performed. Taking the first pulse process as an example, the orbital recursion process is expressed as follows:
由初始轨道参数和θ1可以得到第一次施加脉冲前航天器在θ1的轨道六根数[a1,e1,i1,Ω1,ω1,f1],其中前五个轨道参数与初始轨道参数相同,即满足a1=a0、e1=e0、i1=i0、Ω1=Ω0、ω1=ω0,真近点角f1=θ1-ω1;通过在θ1处施加脉冲前的轨道六根数和脉冲大小Δv1求解第一次施加脉冲后在θ1处的轨道六根数[a1′,e1′,i1′,Ω1′,ω1′,f1′];The six orbital elements [a 1 ,e 1 ,i 1 ,Ω 1 ,ω 1 ,f 1 ] of the spacecraft at θ 1 before the first pulse is applied can be obtained from the initial orbital parameters and θ 1 , where the first five orbital parameters are the same as the initial orbital parameters, that is, they satisfy a 1 =a 0 , e 1 =e 0 , i 1 =i 0 , Ω 1 =Ω 0 , ω 1 =ω 0 , and the true anomaly f 1 =θ 1 -ω 1 ; the six orbital elements [a 1 ′,e 1 ′,i 1 ′,
步骤二三、易知对于轨道六根数的前五项,有a1′=a2、e1′=e2、i1′=i2、Ω1′=Ω2、ω1′=ω2;通过θ2可求得f2=θ2-ω2;由此可得施加第二次脉冲Δv2前的轨道六根数[a2,e2,i2,Ω2,ω2,f2],通过在θ2处施加脉冲前的轨道六根数和脉冲Δv2求解第二次施加脉冲后在θ2处的轨道六根数[a2′,e2′,i2′,Ω2′,ω2′,f2′];Step 23: It is easy to know that for the first five items of the six orbital numbers, a 1 ′=a 2 , e 1 ′=e 2 , i 1 ′=i 2 , Ω 1 ′=Ω 2 , ω 1 ′=ω 2 ; f 2 =θ 2 -ω 2 can be obtained through θ 2 ; thus, the six orbital numbers [a 2 ,e 2 ,i 2 ,Ω 2 ,ω 2 ,f 2 ] before the second pulse Δv 2 is applied can be obtained, and the six orbital numbers [a 2 ′,e 2 ′,i 2 ′,Ω 2 ′,ω 2 ′,f 2 ′] at θ 2 after the second pulse is applied can be solved by the six orbital numbers before the pulse is applied at θ 2 and the pulse Δv 2 ;
按此方式进行轨道递推直到得到在θn-2处施加脉冲Δvn-2后的轨道参数[an-2′,en-2′,in-2′,Ωn-2′,ωn-2′,fn-2′],通过已知的目标轨道的轨道参数[af,ef,if,Ωf,ωf],以及待优化的状态变量中的θn-1,采用已有方法可求得最后两次正切脉冲的大小Δvn-1,Δvn。The orbital recursion is performed in this way until the orbital parameters [a n - 2 ′,e n-2 ′,i n-2 ′,Ω n-2 ′,ω n -2 ′,f n-2 ′ ] are obtained after the pulse Δv n-2 is applied at θ n-2. Through the known orbital parameters of the target orbit [a f ,e f , if ,Ω f ,ω f ] and θ n-1 among the state variables to be optimized, the sizes of the last two tangent pulses Δv n-1 and Δv n can be obtained by using existing methods.
因此,整个求解过程中的待优化状态变量共有2n-3个,即通过优化X=[θ1,θ2,...,θn-1,Δv1,Δv2,...,Δvn-2]可求解得到正切脉冲条件下的燃料最优转移轨道;通过遗传算法对得到正切脉冲条件下的燃料最优转移轨道所需要优化的2n-3个变量进行全局搜索,得到正切脉冲条件下的全局燃料最优解。Therefore, there are a total of 2n-3 state variables to be optimized in the entire solution process, that is, by optimizing X=[θ 1 ,θ 2 ,...,θ n-1 ,Δv 1 ,Δv 2 ,...,Δv n-2 ], the optimal fuel transfer orbit under tangent pulse conditions can be solved; through the genetic algorithm, a global search is performed on the 2n-3 variables that need to be optimized to obtain the optimal fuel transfer orbit under tangent pulse conditions, and the global fuel optimal solution under tangent pulse conditions is obtained.
其它步骤及参数与具体实施方式一或二相同。The other steps and parameters are the same as those in the first or second embodiment.
具体实施方式四:本实施方式与具体实施方式一至三之一不同的是,所述步骤二二中根据初始轨道参数[a0,e0,i0,Ω0,ω0]和待优化状态变量可进行轨道递推,以第一次脉冲过程为例,轨道递推过程表示如下:Specific implementation mode 4: This implementation mode is different from the
由初始轨道参数和θ1可以得到第一次施加脉冲前航天器在θ1的轨道六根数[a1,e1,i1,Ω1,ω1,f1],其中前五个轨道参数与初始轨道参数相同,即满足a1=a0、e1=e0、i1=i0、Ω1=Ω0、ω1=ω0,真近点角f1=θ1-ω1;通过在θ1处施加脉冲前的轨道六根数和脉冲大小Δv1求解第一次施加脉冲后在θ1处的轨道六根数[a1′,e1′,i1′,Ω1′,ω1′,f1′];The six orbital elements [a 1 ,e 1 ,i 1 ,Ω 1 ,ω 1 ,f 1 ] of the spacecraft at θ 1 before the first pulse is applied can be obtained from the initial orbital parameters and θ 1 , where the first five orbital parameters are the same as the initial orbital parameters, that is, they satisfy a 1 =a 0 , e 1 =e 0 , i 1 =i 0 , Ω 1 =Ω 0 , ω 1 =ω 0 , and the true anomaly f 1 =θ 1 -ω 1 ; the six orbital elements [a 1 ′,e 1 ′,i 1 ′,
具体过程为:The specific process is:
第一次施加脉冲时航天器的位置矢量和速度矢量[r1,v1]可表示为The position vector and velocity vector [r 1 ,v 1 ] of the spacecraft when the first pulse is applied can be expressed as
r1=C3(Ω1)C1(i1)C3(ω1)[|r1|sinf1;|r1|cosf1;0]r 1 =C 3 (Ω 1 )C 1 (i 1 )C 3 (ω 1 )[|r 1 |sinf 1 ; |r 1 |cosf 1 ; 0]
式中,C1、C3表示旋转矩阵,旋转矩阵的形式是:In the formula, C 1 and C 3 represent rotation matrices, and the form of the rotation matrix is:
其中α表示Ω1、i1或ω1;where α represents Ω 1 , i 1 or ω 1 ;
在θ1处施加脉冲Δv1后,位置矢量不变,速度矢量改变,此时航天器的位置矢量和速度矢量[r1′,v1′]可表示为After applying pulse Δv 1 at θ 1 , the position vector remains unchanged, but the velocity vector changes. At this time, the position vector and velocity vector of the spacecraft [r 1 ′,v 1 ′] can be expressed as
r1′=r1 r 1 ′=r 1
从而可以用施加脉冲后的位置速度矢量r1′,v1′求解出在θ1处施加脉冲Δv1后的轨道六根数[a1′,e1′,i1′,Ω1′,ω1′,f1′]。Therefore, the position and velocity vectors r 1 ′ and v 1 ′ after the pulse is applied can be used to solve the six orbital elements [a 1 ′, e 1 ′, i 1 ′, Ω 1 ′, ω 1 ′, f 1 ′] after the
其它步骤及参数与具体实施方式一至三之一相同。The other steps and parameters are the same as those in
具体实施方式五:本实施方式与具体实施方式一至四之一不同的是,所述步骤二三中易知对于轨道六根数的前五项,有a1′=a2、e1′=e2、i1′=i2、Ω1′=Ω2、ω1′=ω2;通过θ2可求得f2=θ2-ω2;由此可得施加第二次脉冲Δv2前的轨道六根数[a2,e2,i2,Ω2,ω2,f2],通过在θ2处施加脉冲前的轨道六根数和脉冲Δv2求解第二次施加脉冲后在θ2处的轨道六根数[a2′,e2′,i2′,Ω2′,ω2′,f2′];Specific implementation mode five: This implementation mode is different from specific implementation modes one to four in that, in steps two and three, it is easy to know that for the first five items of the six orbital numbers, a 1 ′=a 2 , e 1 ′=e 2 , i 1 ′=i 2 , Ω 1 ′=Ω 2 , ω 1 ′=ω 2 ; f 2 =θ 2 -ω 2 can be obtained through θ 2 ; thus, the six orbital numbers [a 2 ,e 2 ,i 2 ,Ω 2 ,ω 2 ,f 2 ] before the second pulse Δv 2 is applied can be obtained, and the six orbital numbers [a 2 ′,e 2 ′,i 2 ′,Ω 2 ′,ω 2 ′,f 2 ′ ] at θ 2 after the second pulse is applied can be solved by the six orbital numbers before the pulse is applied at θ 2 and the pulse Δv 2 ;
按此方式进行轨道递推直到得到在θn-2处施加脉冲Δvn-2后的轨道参数[an-2′,en-2′,in-2′,Ωn-2′,ωn-2′,fn-2′],通过已知的目标轨道的轨道参数[af,ef,if,Ωf,ωf],以及待优化的状态变量中的θn-1,采用已有方法可求得最后两次正切脉冲的大小Δvn-1,Δvn;The orbital recursion is performed in this way until the orbital parameters [a n - 2 ′,e n-2 ′,i n-2 ′,Ω n-2 ′,ω n-2 ′,f n- 2 ′] after applying the pulse Δv n-2 at θ n-2 are obtained. The sizes Δv n-1 , Δv n of the last two tangent pulses can be obtained by using the known orbital parameters [a f ,e f , if ,Ω f ,ω f ] of the target orbit and θ n -1 among the state variables to be optimized using the existing method.
过程如下:The process is as follows:
θ=θn-θn-1 θ=θ n -θ n-1
式中θ表示θn夹角与θn-1夹角的差值;λ表示中间变量;Where θ represents the difference between the angle θ n and the angle θ n-1 ; λ represents the intermediate variable;
利用飞行方向角γ与真近点角的关系可得Using the relationship between the flight direction angle γ and the true anomaly angle, we can get
式中γj表示施加第j次脉冲Δvj前的飞行方向角,|rj|表示施加第j次脉冲Δvj时的位置矢量大小,pj表示轨道的半通径;Where γ j represents the flight direction angle before applying the jth pulse Δv j , |r j | represents the position vector size when applying the jth pulse Δv j , and p j represents the semi-path of the orbit;
两边同时乘以pnsinθ,Multiply both sides by p n sinθ,
整理可得Arrangement available
令则 make but
式中k1、k2表示中间变量,表示中间变量;Where k 1 and k 2 represent intermediate variables, represents an intermediate variable;
过程中的转移轨道半通径和角动量分别为The semi-diameter and angular momentum of the transfer orbit in the process are
从而可求得最后两次脉冲的大小Δvn-1,Δvn Thus, the magnitudes of the last two pulses Δv n-1 and Δv n can be obtained.
则总的燃料消耗量可表示为The total fuel consumption can be expressed as
其它步骤及参数与具体实施方式一至四之一相同。The other steps and parameters are the same as those in
具体实施方式六:本实施方式与具体实施方式一至五之一不同的是,所述步骤四中fmincon算法求解自由方向脉冲的局部燃料最优解:Specific implementation method 6: This implementation method is different from any one of
由步骤一中给出的初始条件,将脉冲方向限制在轨道平面内,设定优化指标为燃料最优,利用fmincon算法,对3n-4个优化变量进行局部优化,求得自由方向脉冲条件下的局部燃料最优解;Based on the initial conditions given in
在步骤四中,脉冲方向均为自由方向的脉冲时,燃料消耗量的计算和优化过程如下:In step 4, when the pulse directions are all in the free direction, the calculation and optimization process of fuel consumption is as follows:
假设脉冲数量为n,得到最优解所需要的待优化的状态变量数量至少为3n-4个,选取状态变量为X=[θ1,θ2,...,θn,Δvi1,Δvi2,...,Δvi(n-2),Δvj1,Δvj2,...,Δvj(n-2)],其中,θk表示第k次脉冲时航天器所在位置与初始轨道升交点矢径方向OX之间的夹角,取值范围在[0,2π];Δvik为第k次脉冲径向分量的大小,取值范围为[-10,10],Δvjk为第k次脉冲周向分量的大小,取值范围为[-10,10],以与原速度方向同向为正,单位为km/s;Assuming that the number of pulses is n, the number of state variables to be optimized required to obtain the optimal solution is at least 3n-4, and the state variable is selected as X=[ θ1 , θ2 ,..., θn , Δvi1 , Δvi2 ,...,Δvi (n-2) , Δvj1 , Δvj2 ,..., Δvj(n-2) ], where θk represents the angle between the position of the spacecraft at the kth pulse and the radial direction OX of the ascending node of the initial orbit, and its value range is [0,2π]; Δvik is the magnitude of the radial component of the kth pulse, and its value range is [-10,10]; Δvjk is the magnitude of the circumferential component of the kth pulse, and its value range is [-10,10], with the same direction as the original velocity direction being positive, and the unit is km/s;
与正切轨道递推过程类似,以第一次脉冲过程为例,可通过初始轨道参数以及θ1,Δvi1,Δvj1递推得到施加脉冲后的位置速度矢量r1′,v1′,从而求解出施加脉冲后的轨道六根数,再通过θ2得到f2,通过在θ2处施加脉冲前的轨道六根数和脉冲Δvi2,Δvj2求解第二次施加脉冲后在θ2处的轨道六根数[a2′,e2′,i2′,Ω2′,ω2′,f2′],即可进行下一步的轨道递推;Similar to the tangent orbit recursion process, taking the first pulse process as an example, the position velocity vector r 1 ′, v 1 ′ after the pulse is applied can be recursively obtained through the initial orbit parameters and θ 1 , Δv i1 , Δv j1 , so as to solve the six orbital numbers after the pulse is applied, and then f 2 is obtained through θ 2. The six orbital numbers [a 2 ′, e 2 ′, i 2 ′, Ω 2 ′ , ω 2 ′, f 2 ′ ] at θ 2 after the second pulse is applied can be solved through the six orbital numbers before the pulse is applied at θ 2 and the pulse Δv i2 , Δv j2, and the next step of orbital recursion can be carried out;
最后求解得到第n-2次施加脉冲后的轨道六根数,已知目标轨道的轨道参数,并考虑待优化的变量θn-1,θn,问题被转化为传统的最优双脉冲转移问题,可利用最优双脉冲转移得到Δvi(n-1),Δvj(n-1),Δvin,Δvjn的解析解,因此,全过程需要3n-4个待优化的状态变量,即可以得到整个转移过程总的燃料消耗量Finally, the six orbit numbers after the n-2th pulse is applied are obtained. Given the orbital parameters of the target orbit and considering the variables to be optimized θ n-1 , θ n , the problem is transformed into the traditional optimal double-pulse transfer problem. The optimal double-pulse transfer can be used to obtain the analytical solutions of Δvi (n-1) , Δvj(n-1) , Δv in , Δv jn . Therefore, the whole process requires 3n-4 state variables to be optimized, that is, the total fuel consumption of the whole transfer process can be obtained.
其它步骤及参数与具体实施方式一至五之一相同。The other steps and parameters are the same as those in
采用以下实施例验证本发明的有益效果:The following examples are used to verify the beneficial effects of the present invention:
实施例一:Embodiment 1:
两脉冲算例Two-pulse example
以下为一组两脉冲的具体算例,算例的轨道参数如表1所示:The following is a specific example of two pulses. The orbital parameters of the example are shown in Table 1:
表1算例一的轨道参数Table 1 Orbital parameters of Example 1
遗传算法设置PopulationSize为150,Generations为100。fmincon算法设置MaxFunEvals为6000,MaxIter为5000,TolCon为1e-6。The genetic algorithm sets PopulationSize to 150 and Generations to 100. The fmincon algorithm sets MaxFunEvals to 6000, MaxIter to 5000, and TolCon to 1e-6.
仿真结果如表2所示:The simulation results are shown in Table 2:
表2算例一仿真结果Table 2 Simulation results of example 1
可见本发明方法与传统方法收敛到了同一组解,并且在计算效率上有着巨大优势。It can be seen that the method of the present invention and the traditional method converge to the same set of solutions and have a huge advantage in computational efficiency.
三脉冲算例Three-pulse example
以下为一组三脉冲的具体算例,算例的轨道参数如表3所示:The following is a specific example of a set of three pulses. The orbital parameters of the example are shown in Table 3:
表3算例二的轨道参数Table 3 Orbital parameters of Example 2
算法的参数设置与算例一相同。仿真结果如表4所示:The algorithm parameter settings are the same as in Example 1. The simulation results are shown in Table 4:
表4算例二仿真结果Table 4 Simulation results of example 2
可见本发明方法与传统方法收敛到了同一组解,并且在计算效率上有着巨大优势。It can be seen that the method of the present invention and the traditional method converge to the same set of solutions and have a huge advantage in computational efficiency.
方法总体效果Overall effect of the method
为得到可靠结论进行大量算例的仿真:To obtain reliable conclusions, a large number of simulation examples are carried out:
初始轨道近地点高度选取300~2000km的随机数,目标轨道近地点高度为初始轨道近地点高度的0.3~3倍,倍数为随机数。初始轨道离心率e0、目标轨道离心率ef为0~1的随机数。ω0、ωf为0~2π的随机数。以上随机数均为均匀分布,两脉冲取600组进行仿真,三脉冲取250组进行仿真。算法的参数设置同上。The initial orbit perigee height is a random number between 300 and 2000 km, and the target orbit perigee height is 0.3 to 3 times the initial orbit perigee height, and the multiple is a random number. The initial orbit eccentricity e 0 and the target orbit eccentricity e f are random numbers between 0 and 1. ω 0 and ω f are random numbers between 0 and 2π. The above random numbers are uniformly distributed. 600 groups of two pulses are taken for simulation, and 250 groups of three pulses are taken for simulation. The algorithm parameter settings are the same as above.
得到的传统方法与计算时间的对比如表5所示。The comparison between the traditional method and the calculation time is shown in Table 5.
表5计算时间对比表Table 5 Comparison of calculation time
将本发明中所用方法求得的燃料最优解与传统方法得到的燃料最优解的燃料进行对比,对比结果如图4和图5所示。图例的百分比为本方法相比传统方法多消耗的燃料百分比,由于两个轨道之间相互转移所消耗的燃料相同,为绘图方便,默认e0小于ef。纵轴高度表示符合e0、ef分类的仿真组数。The fuel optimal solution obtained by the method used in the present invention is compared with the fuel optimal solution obtained by the traditional method, and the comparison results are shown in Figures 4 and 5. The percentage in the legend is the percentage of fuel consumed by the present method compared with the traditional method. Since the fuel consumed by the transfer between the two tracks is the same, for the convenience of drawing, it is assumed that e 0 is less than e f . The height of the vertical axis represents the number of simulation groups that meet the classification of e 0 and e f .
由图4和图5结果可知,在离心率较小情况下,本发明所用方法与传统方法求得的燃料指标几乎没有差别。由于地球轨道卫星一般不会设计大离心率轨道,本发明所用方法是具有普遍性的。As shown in Figures 4 and 5, when the eccentricity is small, the fuel index obtained by the method used in the present invention and the traditional method is almost the same. Since earth orbit satellites are generally not designed with large eccentricity orbits, the method used in the present invention is universal.
可见,大量算例表明,本发明所提出的方法通常能与传统方法收敛到同一组解,并且在计算时间上有着1~2个数量级的优势。It can be seen that a large number of examples show that the method proposed in the present invention can usually converge to the same set of solutions as the traditional method, and has an advantage of 1 to 2 orders of magnitude in calculation time.
本发明还可有其它多种实施例,在不背离本发明精神及其实质的情况下,本领域技术人员当可根据本发明作出各种相应的改变和变形,但这些相应的改变和变形都应属于本发明所附的权利要求的保护范围。The present invention may also have many other embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art may make various corresponding changes and modifications based on the present invention, but these corresponding changes and modifications should all fall within the scope of protection of the claims attached to the present invention.
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