下面我们使用LM法来解决之前用线性最小二乘法难以解决的椭球拟合问题，在磁力计标定中我们的目的是通过磁场测量数据来拟合当前受干扰的地磁场椭球，通过其参数反向缩放和补偿位置，从而将其映射为正球体完成对磁场的校准，这样通过磁场纠正数据计算出的航线角才随机体转动近似线性变化的，通过LM法我们不需要换元处理，同时也可不用把参数方程展开为一般式，直接列出椭球的参数方程：
构建误差方程：
求取每一个参数在第i组测量数据和当前参数估计下的偏导数：
求第i组测量数据对应的雅可比矩阵：
则H和B矩阵为：
因此第j次迭代的参数增量为：
则依据LM算法流程，参数迭代过程如下：
A. 给定参数初始值，最大迭代次数，参数修正停止阈值，正则化参数；
B. 使用dx更新参数，之后计算新参数对应拟合误差；
C. 当新拟合误差小于上一次的误差时修正 正则化参数，并将参数拟合值进行更新：
D. 当新拟合误差大于老误差时，为保证梯度需增大正则化参数并重新计算，因此：
E. 最终当参数增量dx的1范数小于参数修正停止阈值或迭代次数达到最大次数限制时停止拟合；
下面还是基于Matlab进行数值仿真，我使用了网上提供的真实磁力计数据进行拟合，拟合结果如下图所示：
可以看到该磁场数据是采用DJI磁场标定先机体水平旋转一圈，再垂直旋转一圈的方式，最终的拟合结果较为满意，拟合中各参数的变化过程如下图所示：
拟合椭圆参数如下：
p_init =
0.0944
-0.0353
-0.0022
0.2454
0.0628
0.2239
p_out =
-0.0032
0.0107
-0.0012
0.4953
0.7114
0.2578基于该参数可进一步对测量数据进行缩放完成校准，即：
最终将测量数据校准到单位圆的结果如下图所示：
最后附上MATLAB仿真代码，目前当初始值存在较大偏差时还是无法收敛，具体问题希望大家帮忙解答。
function NLSM
close all;
clear all;
clc;
%%拟合椭圆
%(x-a)^2/A^2+(y-b)^2/B^2+(z-c)^2/C^2=1
%
%制造测量
REAL_DATA=1;
if REAL_DATA
data = xlsread('data1.xlsx');
x_real=data(:,1);
y_real=data(:,2)*1.5;
z_real=data(:,3)*0.5;
N=length(x_real);
for i=1:N
x(i)=x_real(i);
y(i)=y_real(i);
z(i)=z_real(i);
end
else
N=20;
w=0.15;
xc = 1;
yc = 2;
zc = -3;
xr = 3.5;
yr = 5;
zr = 4;
p_real=[xc;yc;zc;xr;yr;zr]
[x_real,y_real,z_real] = ellipsoid(xc,yc,zc,xr,yr,zr,N);
k=1;
for i=1:int16(N*0.6)
for j=1:int16(N*0.6)
x(k)=x_real(i,j)+randn(1)*w;
y(k)=y_real(i,j)+randn(1)*w;
z(k)=z_real(i,j)+randn(1)*w;
k=k+1;
end
end
N=k-1;
end
%LM 法不换元
EN_DIR=0;%直接法
MAX_IT=100;
err_thr=1e-5;
dlamada=2;
lamda=100;
%%(x-a)^2/A^2+(y-b)^2/B^2+(z-c)^2/C^2=1
if(1)
% a=0.25;
% b=0.1;
% c=0.15;
% A=0.13;
% B=0.12;
% C=0.3;
ww=0.1;
a=0.1+randn(1)*ww;
b=0.1+randn(1)*ww;
c=0.1+randn(1)*ww;
A=0.1+randn(1)*ww;
B=0.1+randn(1)*ww;
C=0.1+randn(1)*ww;
if REAL_DATA==0
ww=2;
a=xc+randn(1)*ww;% = 1.21;
b=yc+randn(1)*ww;%= 2.32;
c=zc+randn(1)*ww;%= 4.32;
A=xr+randn(1)*ww;% = 2.78;
B=yr+randn(1)*ww;% = 5.76;
C=zr+randn(1)*ww;% = 1.51;
end
p_init=[a;b;c;A;B;C]
else%有挑战的初始值
scale=10;
a=0.25*scale;
b=0.1;
c=0.15*scale;
A=0.13;
B=0.12*scale;
C=0.3;
end
err_all_lm(1)=0;
lamdad(1)=lamda;
temp(1)=1;
a_lm(1)=a;
b_lm(1)=b;
c_lm(1)=c;
A_lm(1)=A;
B_lm(1)=B;
C_lm(1)=C;
f_sum_last=99;
for j=1:MAX_IT
for i=1:N
f=(x(i)-a)^2/A^2+(y(i)-b)^2/B^2+(z(i)-c)^2/C^2-1;
Ja=-2*(x(i)-a)/A^2;
Jb=-2*(y(i)-b)/B^2;
Jc=-2*(z(i)-c)/C^2;
JA=-2*(x(i)-a)^2/A^3;
JB=-2*(y(i)-b)^2/B^3;
JC=-2*(z(i)-c)^2/C^3;
J=[Ja,Jb,Jc,JA,JB,JC];
if(i==1)%广义法
JJ(1,:)=J;
FF(i)=f;
H= J'*J;
Bb=-J'*f;
f_sum=f*f;
else
JJ=[JJ(:,:);
J];
FF(i)=f;
H=H+ J'*J+lamda*eye(6,6);
Bb=Bb- J'*f;
f_sum=f_sum+f*f;
end
end
if(EN_DIR)%直接法
H= JJ'*JJ+lamda*eye(6,6);
Bb=-JJ'*FF';
clear JJ;
dx=pinv(H)*Bb;
else
dx=pinv(H/N)*Bb/N;
end
%新参数
at=a+dx(1);
bt=b+dx(2);
ct=c+dx(3);
At=A+dx(4);
Bt=B+dx(5);
Ct=C+dx(6);
%计算老的误差
for i=1:N
f_last=(x(i)-a)^2/A^2+(y(i)-b)^2/B^2+(z(i)-c)^2/C^2-1;
if(i==1)
f_sum_last=f_last*f_last;
else
f_sum_last=f_sum_last+f_last*f_last;
end
end
f_sum_last=f_sum_last/N;
%计算新的误差
for i=1:N
f_new=(x(i)-at)^2/At^2+(y(i)-bt)^2/Bt^2+(z(i)-ct)^2/Ct^2-1;
if(i==1)
f_sum_new=f_new*f_new;
else
f_sum_new=f_sum_new+f_new*f_new;
end
end
f_sum_new=f_sum_new/N;
%修正lamada比较误差梯度 简单处理
if f_sum_new<f_sum_last
a=at;%+dx(1);
b=bt;%+dx(2);
c=ct;%+dx(3);
A=At;%+dx(4);
B=Bt;%+dx(5);
C=Ct;%+dx(6);
a_lm(j)=a;
b_lm(j)=b;
c_lm(j)=c;
A_lm(j)=A;
B_lm(j)=B;
C_lm(j)=C;
err_all_lm(j)=f_sum_new;
rho =( f_sum_last-f_sum_new ) / L0_L( dx,H/N,Bb/N);
dlamada=0.33;%
%dlamada=max(0.33,1-(2*rho-1)^3);
lamda=lamda*dlamada;
dlamada=2;
else
lamda=lamda*dlamada;
dlamada=2*dlamada;
end
if lamda>100 && 1
lamda=100;
end
if lamda<0.5 && 1
lamda=0.5;
end
lamdad(j)=lamda;
temp(j)=1;
norm_dx=norm(dx,1);
if(norm_dx<err_thr)
break;
end
end
p_out=[a;b;c;A;B;C]
%标定
for i=1:N
x_fix(i)=(x(i)+a)/A;
y_fix(i)=(y(i)+b)/B;
z_fix(i)=(z(i)+c)/C;
end
figure(5)
plot3(x_real(:),y_real(:),z_real(:),':');
hold on;
plot3(x(:),y(:),z(:),'k+');
hold on;
[x_est,y_est,z_est] = ellipsoid(a,b,c,A,B,C,80);
plot3(x_est(:),y_est(:),z_est(:),'b:');
hold on;
plot3(x_fix(:),y_fix(:),z_fix(:),'r.');
hold on;
grid on;
axis equal;
legend('真实曲线','测量值','LM','校准值')
%
figure(6)
subplot(4,2,1)
plot(err_all_lm,'-k');
grid on;
ylabel('拟合误差');
subplot(4,2,2)
plot(a_lm,'-.b');
grid on;
ylabel('a拟合');
subplot(4,2,3)
plot(b_lm,'-.b');
hold on;
grid on;
ylabel('b拟合');
grid on;
subplot(4,2,4)
plot(c_lm,'-.b');
hold on;
grid on;
ylabel('c拟合');
grid on;
subplot(4,2,5)
plot((lamdad),'-k');
grid on;
ylabel('lamdad阻尼因子');
subplot(4,2,6)
plot(A_lm,'-.b');
grid on;
ylabel('A拟合');
subplot(4,2,7)
plot(B_lm,'-.b');
hold on;
grid on;
ylabel('B拟合');
grid on;
subplot(4,2,8)
plot(C_lm,'-.b');
hold on;
grid on;
ylabel('C拟合');
grid on;
end
%Eigen::MatrixXd L = -h.transpose() * J_.transpose() * fx_ - 0.5 * h.transpose() * J_.transpose() * J_ * h;
function out=L0_L( dx,H,B)
L = -dx' * B - 0.5 * dx' * H * dx;
out= L;
end
function out=max(in1,in2)
if in1>in2
out=in1;
else
out=in2;
end
end测试数据需要赋值到Excel中：
-0.3174 0.0249 -0.3932
-0.3078 0.0261 -0.3821
-0.3181 0.0479 -0.3907
-0.3186 0.042 -0.3784
-0.3003 0.0488 -0.4021
-0.3117 0.0588 -0.3937
-0.312 0.0559 -0.3906
-0.3178 0.0667 -0.3958
-0.2968 0.0896 -0.3953
-0.2978 0.0937 -0.3787
-0.2924 0.1088 -0.3742
-0.3033 0.109 -0.3991
-0.2872 0.137 -0.3929
-0.2811 0.145 -0.3944
-0.2606 0.1622 -0.3984
-0.2733 0.1562 -0.3989
-0.2661 0.1772 -0.3838
-0.267 0.1828 -0.3672
-0.2665 0.2032 -0.3875
-0.2542 0.2205 -0.3553
-0.257 0.2346 -0.3649
-0.227 0.2514 -0.3562
-0.2345 0.2591 -0.3522
-0.2315 0.2624 -0.3628
-0.2257 0.266 -0.3565
-0.1996 0.2537 -0.3794
-0.214 0.2604 -0.368
-0.2017 0.2607 -0.3843
-0.1897 0.2737 -0.3779
-0.1823 0.2804 -0.3701
-0.1837 0.2962 -0.3797
-0.1534 0.3001 -0.3784
-0.1603 0.2992 -0.3725
-0.1393 0.3063 -0.3687
-0.1348 0.3112 -0.3732
-0.1189 0.3219 -0.3727
-0.0951 0.3294 -0.3748
-0.0846 0.325 -0.3757
-0.0741 0.3365 -0.3831
-0.0572 0.3271 -0.3775
-0.0288 0.3568 -0.3769
-0.0115 0.3518 -0.3622
0.005 0.354 -0.3691
0.0262 0.3639 -0.3592
0.0401 0.3514 -0.3704
0.0621 0.3543 -0.3771
0.0654 0.3445 -0.3629
0.0796 0.3522 -0.3716
0.1011 0.3492 -0.3537
0.1186 0.3472 -0.3651
0.1302 0.3243 -0.3884
0.1451 0.3234 -0.3861
0.166 0.3131 -0.3773
0.1924 0.2732 -0.3674
0.214 0.2703 -0.3739
0.2356 0.253 -0.3908
0.2357 0.2228 -0.3947
0.2626 0.2191 -0.3994
0.2605 0.1957 -0.3943
0.2647 0.166 -0.3904
0.2904 0.1477 -0.4009
0.2818 0.1119 -0.3837
0.2872 0.0954 -0.3983
0.3004 0.0912 -0.3792
0.2948 0.0733 -0.3898
0.3068 0.0864 -0.3956
0.3032 0.0916 -0.3867
0.3094 0.0853 -0.397
0.3023 0.0814 -0.3865
0.3068 0.0863 -0.3758
0.3324 0.0839 -0.3821
0.2984 0.0679 -0.3895
0.3003 0.0581 -0.3937
0.3358 0.044 -0.3886
0.3099 0.0434 -0.3792
0.3207 0.0434 -0.3802
0.3225 0.0322 -0.3935
0.329 0.0287 -0.384
0.3262 0.0283 -0.3841
0.3226 0.0177 -0.3688
0.3283 -0.0104 -0.377
0.3234 -0.0356 -0.3629
0.3216 -0.056 -0.3746
0.3262 -0.0814 -0.3616
0.3121 -0.0876 -0.3682
0.3272 -0.0856 -0.3599
0.3138 -0.0831 -0.3591
0.3316 -0.0822 -0.3644
0.3163 -0.1014 -0.3708
0.3185 -0.0924 -0.3815
0.3176 -0.0868 -0.3787
0.3178 -0.084 -0.3574
0.3166 -0.0826 -0.3849
0.3167 -0.097 -0.3785
0.3199 -0.1227 -0.3609
0.3129 -0.1408 -0.3822
0.2945 -0.149 -0.3691
0.3017 -0.1597 -0.3441
0.3093 -0.1659 -0.3527
0.3012 -0.1655 -0.3562
0.2997 -0.1829 -0.3696
0.2858 -0.1863 -0.3579
0.2714 -0.1952 -0.3889
0.251 -0.2109 -0.3742
0.2421 -0.2193 -0.376
0.234 -0.2348 -0.3714
0.2239 -0.2418 -0.387
0.2168 -0.2601 -0.3655
0.2016 -0.2635 -0.3784
0.1983 -0.2698 -0.3677
0.177 -0.2798 -0.3639
0.1829 -0.2746 -0.3775
0.1644 -0.3002 -0.3731
0.1488 -0.2922 -0.3558
0.14 -0.315 -0.3571
0.1129 -0.3286 -0.3505
0.1053 -0.3383 -0.3369
0.0905 -0.3359 -0.3498
0.0637 -0.3481 -0.3508
0.0503 -0.3455 -0.35
0.0212 -0.3666 -0.3417
0.018 -0.357 -0.3421
0.014 -0.3561 -0.3285
-0.0188 -0.3575 -0.3301
-0.0269 -0.3572 -0.3183
-0.0448 -0.3436 -0.3301
-0.0635 -0.3402 -0.3494
-0.0793 -0.3496 -0.3255
-0.1058 -0.3415 -0.3361
-0.126 -0.3239 -0.3375
-0.1342 -0.3236 -0.3287
-0.1451 -0.3091 -0.3449
-0.1686 -0.2977 -0.3478
-0.183 -0.2909 -0.3471
-0.1963 -0.2869 -0.3417
-0.208 -0.2812 -0.3455
-0.233 -0.2556 -0.3641
-0.2327 -0.2527 -0.3489
-0.2508 -0.2262 -0.3641
-0.2636 -0.2322 -0.3493
-0.2644 -0.2107 -0.359
-0.269 -0.2012 -0.3625
-0.2804 -0.1752 -0.3606
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-0.3143 -0.1277 -0.3664
-0.3016 -0.1217 -0.3629
-0.3012 -0.1333 -0.3565
-0.3062 -0.1122 -0.3664
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-0.3093 -0.0853 -0.3581
-0.3352 -0.0857 -0.3776
-0.3295 -0.0677 -0.3671
-0.34 -0.0475 -0.3543
-0.34 -0.0475 -0.3543
-0.3382 -0.027 -0.3624
-0.3347 -0.0338 -0.3621
-0.3426 -0.0305 -0.3503
-0.338 -0.0241 -0.3594
-0.3417 -0.0203 -0.3612
-0.3392 -0.0229 -0.3457
-0.3383 -0.0126 -0.3612
-0.3411 -0.013 -0.3583
-0.3312 -0.0087 -0.3762
-0.3222 0.0012 -0.3898
-0.3288 0.0031 -0.3703
-0.3273 0.0047 -0.3657
-0.3225 -0.0177 -0.3649
-0.3321 -0.0031 -0.3627
-0.3452 -0.0135 -0.3585
-0.3321 -0.0191 -0.3531
-0.3548 -0.0293 -0.3479
-0.3399 -0.0302 -0.3425
-0.3583 -0.0226 -0.3314
-0.365 -0.0379 -0.3344
-0.3765 -0.0294 -0.3077
-0.3968 -0.0279 -0.2736
-0.43 -0.0498 -0.2303
-0.4298 -0.047 -0.209
-0.454 -0.059 -0.1733
-0.4533 -0.0663 -0.1334
-0.4788 -0.0786 -0.0839
-0.4892 -0.0571 -0.0391
-0.4825 -0.0579 0.0223
-0.4949 -0.0596 0.0339
-0.4836 -0.0541 0.0969
-0.4782 -0.0549 0.1018
-0.46 -0.0484 0.1544
-0.4489 -0.0298 0.1911
-0.4531 -0.0477 0.1989
-0.4573 -0.0325 0.2045
-0.4252 -0.0385 0.2397
-0.4209 -0.0366 0.2536
-0.4191 -0.0465 0.257
-0.429 -0.0362 0.2517
-0.4281 -0.0405 0.2686
-0.4126 -0.0342 0.2814
-0.4197 -0.038 0.2812
-0.4318 -0.0366 0.2577
-0.4158 -0.0404 0.2738
-0.4231 -0.0457 0.2934
-0.4051 -0.042 0.2988
-0.3987 -0.0469 0.2992
-0.3842 -0.0365 0.321
-0.3957 -0.0438 0.3298
-0.3936 -0.0348 0.3205
-0.3861 -0.0425 0.3302
-0.397 -0.0425 0.3282
-0.3985 -0.0442 0.3342
-0.3942 -0.0422 0.3435
-0.3851 -0.0468 0.3517
-0.3842 -0.0524 0.3458
-0.3757 -0.0643 0.3465
-0.3864 -0.0455 0.3531
-0.377 -0.0472 0.3536
-0.4124 -0.0634 0.3539
-0.3807 -0.0433 0.3396
-0.381 -0.0462 0.3443
-0.387 -0.0369 0.3468
-0.3921 -0.0333 0.3556
-0.3949 -0.0337 0.3555
-0.3817 -0.0074 0.354
-0.3962 0.0138 0.3345
-0.3893 0.0306 0.3329
-0.3809 0.0331 0.3378
-0.3904 0.0493 0.3171
-0.3983 0.0512 0.2999
-0.3928 0.052 0.2955
-0.3851 0.063 0.3109
-0.3918 0.0795 0.2888
-0.3891 0.1101 0.3004
-0.3708 0.1328 0.2778
-0.3763 0.1481 0.2619
-0.3659 0.1899 0.2431
-0.3835 0.2065 0.222
-0.3924 0.2125 0.2108
-0.3888 0.2232 0.2
-0.3813 0.2155 0.1944
-0.3904 0.2361 0.1721
-0.3883 0.2595 0.1503
-0.3781 0.2682 0.1353
-0.3848 0.2846 0.1315
-0.3652 0.2915 0.1323
-0.376 0.2916 0.1119
-0.3715 0.2966 0.1029
-0.3755 0.3135 0.0824
-0.377 0.312 0.0626
-0.3697 0.3173 0.049
-0.3736 0.3039 0.0309
-0.3827 0.3244 0.0284
-0.3713 0.3302 0.0227
-0.3752 0.3168 0.0076
-0.3661 0.3267 -0.0167
-0.3694 0.3206 -0.0304
-0.3841 0.3073 -0.0628
-0.3843 0.3044 -0.0735
-0.3828 0.3062 -0.0902
-0.3713 0.2977 -0.123
-0.37 0.2808 -0.1592
-0.3862 0.2672 -0.1657
-0.3964 0.2587 -0.1798
-0.3955 0.2386 -0.1716
-0.414 0.2291 -0.1982
-0.399 0.2137 -0.1925
-0.4035 0.2088 -0.1911
-0.4122 0.202 -0.2112
-0.4026 0.2033 -0.2184
-0.3868 0.181 -0.2537
-0.3934 0.1989 -0.256
-0.4052 0.2203 -0.2266
-0.4076 0.1926 -0.226
-0.4162 0.1871 -0.2293
-0.3836 0.1871 -0.2339
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