飞动轴系转换推导作业MATLAB推算对比结果

--- title: FeiDong.m：机体↔航迹 L_bk 双路线推导 date: 2026-09-22 tags: [飞行力学, MATLAB, 坐标变换] --- # FeiDong.m：机体↔航迹 L_bk 双路线推导  关键在于分解步骤，轴系转动的时候一个轴一个轴来更好理解，航迹轴->（μ）风轴->（α、β）机体轴 ## 源代码 FeiDong.m ```matlab clc; clear; syms chi gamma psi theta Phi real syms alpha beta mu real syms V real Lx = @(a) [1 0 0; 0 cos(a) sin(a); 0 -sin(a) cos(a)]; Ly = @(a) [cos(a) 0 -sin(a); 0 1 0; sin(a) 0 cos(a)]; Lz = @(a) [cos(a) sin(a) 0; -sin(a) cos(a) 0; 0 0 1]; L_gk = Ly(gamma)*Lz(chi); L_gb = Ly(theta)*Lx(Phi)*Lz(psi); L_bk_A = simplify(L_gb.' * L_gk); L_bw = Ly(alpha)*Lz(-beta); L_wk = Lx(mu); L_bk_B = simplify(L_bw*L_wk); disp('=== L_bk（路线A：χ,γ,ψ,θ,Φ）==='); disp(L_bk_A) disp('=== L_bk（路线B：α,β,μ）==='); disp(L_bk_B) disp('--- 检查1：L_bk_A * L_bk_A'' - I ---'); disp(simplify(L_bk_A*L_bk_A.' - eye(3))) disp('--- det(L_bk_A) ---'); disp(simplify(det(L_bk_A))) disp('--- det(L_bk_B) ---'); disp(simplify(det(L_bk_B))) v_b = simplify(L_bk_B*[V;0;0]); disp('--- 检查2：v_k=[V;0;0] 时 v_b（应为 Vcosαcosβ, Vsinβ, Vsinαcosβ）---'); disp(v_b) ang_A = [30 5 20 4 10]; A = double(subs(L_bk_A, {chi,gamma,psi,theta,Phi}, deg2rad(ang_A))); disp('--- 检查3a：det(A)、正交误差（应 =1 / 0）---'); disp(['det(A) = ', num2str(det(A)), ' 正交误差 = ', num2str(norm(A*A.'-eye(3)))]); v_b = A*[1;0;0]; beta0 = asin(v_b(2)); alpha0 = atan2(v_b(3), v_b(1)); Lbw_r = Ly(alpha0)*Lz(-beta0); R = Lbw_r.'*A; mu0 = atan2(R(2,3), R(2,2)); B = Lbw_r*Lx(mu0); disp('--- 检查3b：反解结果（度数）---'); disp(['alpha = ', num2str(rad2deg(alpha0)), ... ' beta = ', num2str(rad2deg(beta0)), ... ' mu = ', num2str(rad2deg(mu0))]); disp(['两条路线矩阵最大偏差 = ', num2str(max(abs(A(:)-B(:))))]); L_special = simplify(subs(L_bk_B, {alpha,beta,mu}, {0,0,0})); disp('--- 检查4：α=β=μ=0 时 L_bk_B（应为单位阵）---'); disp(L_special) ``` ## 运行输出 ```text === L_bk（路线A：χ,γ,ψ,θ,Φ）=== [cos(chi)*cos(gamma)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)) + cos(chi)*sin(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) + cos(Phi)*sin(chi)*sin(psi), cos(gamma)*sin(chi)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)) + sin(chi)*sin(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) - cos(Phi)*cos(chi)*sin(psi), cos(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) - sin(gamma)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta))] [cos(chi)*cos(gamma)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)) + cos(chi)*sin(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) - cos(Phi)*cos(psi)*sin(chi), cos(gamma)*sin(chi)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)) + sin(chi)*sin(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) + cos(Phi)*cos(chi)*cos(psi), cos(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) - sin(gamma)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta))] [ cos(Phi)*cos(chi)*cos(theta)*sin(gamma) - cos(Phi)*cos(chi)*cos(gamma)*sin(theta) - sin(Phi)*sin(chi), sin(Phi)*cos(chi) - cos(Phi)*cos(gamma)*sin(chi)*sin(theta) + cos(Phi)*sin(chi)*cos(theta)*sin(gamma), cos(Phi)*cos(gamma - theta)] === L_bk（路线B：α,β,μ）=== [cos(alpha)*cos(beta), sin(alpha)*sin(mu) - cos(alpha)*cos(mu)*sin(beta), - cos(mu)*sin(alpha) - cos(alpha)*sin(beta)*sin(mu)] [ sin(beta), cos(beta)*cos(mu), cos(beta)*sin(mu)] [cos(beta)*sin(alpha), - cos(alpha)*sin(mu) - cos(mu)*sin(alpha)*sin(beta), cos(alpha)*cos(mu) - sin(alpha)*sin(beta)*sin(mu)] --- 检查1：L_bk_A * L_bk_A' - I --- [ (cos(chi)*cos(gamma)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)) + cos(chi)*sin(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) + cos(Phi)*sin(chi)*sin(psi))^2 + (cos(gamma)*sin(chi)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)) + sin(chi)*sin(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) - cos(Phi)*cos(chi)*sin(psi))^2 + (cos(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) - sin(gamma)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)))^2 - 1, (cos(gamma)*sin(chi)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)) + sin(chi)*sin(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) + cos(Phi)*cos(chi)*cos(psi))*(cos(gamma)*sin(chi)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)) + sin(chi)*sin(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) - cos(Phi)*cos(chi)*sin(psi)) + (cos(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) - sin(gamma)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)))*(cos(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) - sin(gamma)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta))) + (cos(chi)*cos(gamma)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)) + cos(chi)*sin(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) - cos(Phi)*cos(psi)*sin(chi))*(cos(chi)*cos(gamma)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)) + cos(chi)*sin(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) + cos(Phi)*sin(chi)*sin(psi)), 0] [(cos(gamma)*sin(chi)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)) + sin(chi)*sin(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) + cos(Phi)*cos(chi)*cos(psi))*(cos(gamma)*sin(chi)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)) + sin(chi)*sin(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) - cos(Phi)*cos(chi)*sin(psi)) + (cos(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) - sin(gamma)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)))*(cos(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) - sin(gamma)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta))) + (cos(chi)*cos(gamma)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)) + cos(chi)*sin(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) - cos(Phi)*cos(psi)*sin(chi))*(cos(chi)*cos(gamma)*(cos(psi)*cos(theta) - sin(Phi)*sin(psi)*sin(theta)) + cos(chi)*sin(gamma)*(cos(psi)*sin(theta) + sin(Phi)*cos(theta)*sin(psi)) + cos(Phi)*sin(chi)*sin(psi)), (cos(chi)*cos(gamma)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)) + cos(chi)*sin(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) - cos(Phi)*cos(psi)*sin(chi))^2 + (cos(gamma)*sin(chi)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)) + sin(chi)*sin(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) + cos(Phi)*cos(chi)*cos(psi))^2 + (cos(gamma)*(sin(psi)*sin(theta) - sin(Phi)*cos(psi)*cos(theta)) - sin(gamma)*(cos(theta)*sin(psi) + sin(Phi)*cos(psi)*sin(theta)))^2 - 1, 0] [ 0, 0, 0] --- det(L_bk_A) --- 1 --- det(L_bk_B) --- 1 --- 检查2：v_k=[V;0;0] 时 v_b（应为 Vcosαcosβ, Vsinβ, Vsinαcosβ）--- V*cos(alpha)*cos(beta) V*sin(beta) V*cos(beta)*sin(alpha) --- 检查3a：det(A)、正交误差（应 =1 / 0）--- det(A) = 1 正交误差 = 1.1105e-16 --- 检查3b：反解结果（度数）--- alpha = -4.1857 beta = -9.7309 mu = -9.8802 两条路线矩阵最大偏差 = 1.1102e-16 --- 检查4：α=β=μ=0 时 L_bk_B（应为单位阵）--- [1, 0, 0] [0, 1, 0] [0, 0, 1] ```