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
-0.2935 -0.1697 -0.3629
-0.3041 -0.1668 -0.3711
-0.2995 -0.1446 -0.3546
-0.3143 -0.1277 -0.3664
-0.3016 -0.1217 -0.3629
-0.3012 -0.1333 -0.3565
-0.3062 -0.1122 -0.3664
-0.3207 -0.1069 -0.3764
-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
-0.414 0.1818 -0.2566
-0.3994 0.1779 -0.2482
-0.4016 0.1832 -0.2224
-0.4053 0.1727 -0.233
-0.4115 0.1791 -0.2335
-0.4049 0.1613 -0.2617
-0.4126 0.1661 -0.2531
-0.391 0.1488 -0.2699
-0.4134 0.1415 -0.2693
-0.4029 0.1372 -0.2809
-0.3977 0.1032 -0.2783
-0.3914 0.0968 -0.2748
-0.4133 0.0796 -0.28
-0.4202 0.0787 -0.2925
-0.4111 0.0741 -0.2874
-0.3973 0.06 -0.2849
-0.4174 0.079 -0.2848
-0.4013 0.0927 -0.2859
-0.4003 0.0726 -0.2853
-0.3953 0.0835 -0.3006
-0.4047 0.0534 -0.308
-0.4043 0.042 -0.3138
-0.3902 0.0164 -0.3232
-0.3979 -0.0091 -0.3214
-0.4095 -0.0336 -0.3366
-0.3984 -0.0625 -0.3354
-0.4047 -0.0719 -0.3324
-0.393 -0.0934 -0.339
-0.3991 -0.1015 -0.3268
-0.3907 -0.0989 -0.3463
-0.3804 -0.1207 -0.3194
-0.3839 -0.1456 -0.351
-0.3707 -0.1814 -0.3449
-0.3723 -0.1687 -0.3194
-0.3816 -0.1655 -0.3351
-0.3752 -0.1546 -0.3275
-0.3774 -0.1492 -0.3246
-0.3755 -0.159 -0.3396
-0.3702 -0.1597 -0.3332
-0.3794 -0.1725 -0.3272
-0.3718 -0.1945 -0.334
-0.3646 -0.2052 -0.3242
-0.3825 -0.2393 -0.3105
-0.3645 -0.2356 -0.3052
-0.3774 -0.2589 -0.2991
-0.3702 -0.2695 -0.297
-0.363 -0.2802 -0.2903
-0.38 -0.2882 -0.2787
-0.3539 -0.3007 -0.2649
-0.3478 -0.3087 -0.2415
-0.3407 -0.3208 -0.2348
-0.3367 -0.3217 -0.2254
-0.3147 -0.3348 -0.2057
-0.3197 -0.3456 -0.1981
-0.307 -0.3398 -0.1656
-0.3117 -0.3463 -0.1473
-0.2954 -0.3471 -0.1267
-0.2946 -0.3528 -0.1143
-0.2874 -0.3491 -0.0988
-0.2674 -0.3537 -0.0947
-0.2492 -0.3471 -0.0848
-0.2471 -0.37 -0.0582
-0.1978 -0.3507 -0.0304
-0.1972 -0.374 0.0313
-0.1748 -0.3511 0.073
-0.1652 -0.3502 0.1375
-0.1804 -0.3366 0.2082
-0.1987 -0.3435 0.2395
-0.2343 -0.2834 0.2639
-0.2756 -0.2888 0.2773
-0.2988 -0.2746 0.2743
-0.3273 -0.2739 0.2653
-0.341 -0.2757 0.2617
-0.3318 -0.263 0.2664
-0.346 -0.2706 0.2537
-0.3455 -0.2489 0.2547
-0.3509 -0.2481 0.2605
-0.3501 -0.2552 0.247
-0.3559 -0.243 0.2541
-0.3646 -0.234 0.2382
-0.3801 -0.2403 0.23
-0.3833 -0.2305 0.2067
-0.385 -0.2178 0.2216
-0.3827 -0.2232 0.2035
-0.3759 -0.2079 0.2202
-0.3795 -0.2329 0.2191
-0.3756 -0.2194 0.219
-0.3768 -0.2182 0.2204
-0.3624 -0.1932 0.2236
-0.3638 -0.2093 0.2285
-0.3592 -0.203 0.2468
-0.3624 -0.1933 0.2388
-0.3597 -0.2088 0.2439
-0.3714 -0.1873 0.2505
-0.3548 -0.1996 0.2485
-0.3483 -0.2031 0.2611
-0.3552 -0.2041 0.2715
-0.354 -0.1895 0.2864
-0.3361 -0.1714 0.3098
-0.3535 -0.1838 0.3016
-0.3694 -0.1787 0.3114
-0.3505 -0.1647 0.315
-0.3643 -0.168 0.3236
-0.3492 -0.166 0.3243
-0.3677 -0.1598 0.3156
-0.3598 -0.1631 0.3222
-0.3607 -0.1257 0.3151
-0.3512 -0.1417 0.3175
-0.3534 -0.1363 0.3249
-0.3649 -0.1435 0.3138
-0.3546 -0.1494 0.3144
-0.3584 -0.1311 0.3153
-0.369 -0.1427 0.3258
-0.3591 -0.1384 0.3033
-0.3605 -0.14 0.2941
-0.3649 -0.1435 0.3062
-0.3761 -0.1464 0.3073
-0.3672 -0.1221 0.2995
-0.3603 -0.1372 0.3169
-0.3525 -0.126 0.317
-0.3483 -0.1082 0.3244
-0.3522 -0.1216 0.3093
-0.3588 -0.1197 0.3242
-0.3669 -0.1193 0.3223
-0.3651 -0.1146 0.3009
-0.356 -0.1193 0.3243
-0.3551 -0.1091 0.3165
-0.3453 -0.105 0.3352
-0.3489 -0.0997 0.3379
-0.353 -0.1003 0.3377
-0.3489 -0.0997 0.3379
-0.3495 -0.107 0.3381
-0.3546 -0.0874 0.3145