卡尔曼滤波器matlab的输入位置/速度
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【中文标题】卡尔曼滤波器matlab的输入位置/速度【英文标题】：input position/velocity to kalman filter matlab 【发布时间】：2014-06-21 20:17:51 【问题描述】：我正在尝试在 3-D 空间中跟踪对象位置和方向速度。为此，我在 Matlab 中编写了一个类，但是，我的跟踪算法 EKF 的方程/算法运行良好，因为每个当前和以前的状态都可以很好地预测，但是，我想输入一个轨迹Nx3 点，我遇到了这个问题。
但是我也有方向速度的信息，即位置向量的导数。
我在位置估计/预测中仅输入位置或速度感到困惑，如obj.X(:,1) = [6x1] 和obj.Xh(:,1) = [6x1]，这是否意味着[x,y,z,vx,vy,vz]？
如果是这样，我如何INPUT检查它的估计，如果不是，我如何估计POSITION，因为我的目标只是位置估计。
我的 EKF：
classdef EKF <handle
properties (Access=private)
H
K
Z
Q
M
ind
A
X
Xh
P
a
b
end
methods
function obj = EKF(position)
obj.H = [];
obj.K = [];
obj.Z = [];
obj.ind=0; % indicator function. Used for unwrapping of tan
obj.Q =[0 0 0 0 0 0;
0 0 0 0 0 0;
0 0 0 0 0 0;
0 0 0 0.01 0 0;
0 0 0 0 0.01 0;
0 0 0 0 0 0.01];% Covarience matrix of process noise
obj.M=[0.001 0 0;
0 0.001 0;
0 0 0.001]; % Covarience matrix of measurment noise
obj.A=[1 0 0 0.1 0 0;
0 1 0 0 0.1 0;
0 0 1 0 0 0.1;
0 0 0 1 0 0;
0 0 0 0 1 0;
0 0 0 0 0 1]; % System Dynamics
obj.X(:,1)=[position(1,:) position(2,:)];
obj.Xh(:,1)=[position(1,:) position(2,:)];%Assumed initial conditions
obj.Z(:,:,1)=position(1,:)';% initial observation
obj.P(:,:,1)=[0.1 0 0 0 0 0;
0 0.1 0 0 0 0;
0 0 0.1 0 0 0;
0 0 0 0.1 0 0;
0 0 0 0 0.1 0;
0 0 0 0 0 0.1]; %inital value of covarience of estimation error
end
function [obj,predictedS]=EKFpredictor(obj,p,n)
function [ARG]=arctang(a,b,ind)
if b<0 && a>0 % PLACING IN THE RIGHT QUADRANT
ARG=abs(atan(a/b))+pi/2;
elseif b<0 && a<0
ARG=abs(atan(a/b))+pi;
elseif b>0 && a<0
ARG=abs(atan(a/b))+3*pi/2;
else
ARG=atan(a/b);
end
if ind==-1 % UNWARPPING PART
ARG=ARG-2*pi;
else
if ind==1;
ARG=ARG+2*pi;
end
end
end
obj.X(:,n-1)=[obj.X(1:3,n-1)' p]';
%% PROCESS AND OBSERVATION PROCESS WITH GAUSSINA NOISE
% State process % w generating process noise
obj.X(:,n)=obj.A*obj.X(:,n-1)+[0;0;0;sqrt(obj.Q(4,4))*randn(1);sqrt(obj.Q(5,5))*randn(1);sqrt(obj.Q(6,6))*randn(1)];
%generating & observation observation noise
obj.Z(:,:,n)=[sqrt(obj.X(1,n-1)^2+obj.X(2,n-1)^2);arctang(obj.X(2,n-1),obj.X(1,n-1),obj.ind);obj.X(3,n-1)]+[sqrt(obj.M(1,1))*randn(1);sqrt(obj.M(1,1))*randn(1);sqrt(obj.M(1,1))*randn(1)];
%% PREDICTION OF NEXT STATE
% ESTIMATE
obj.Xh(:,n)=obj.A*obj.Xh(:,n-1);
predictedS=obj.Xh(:,n)';
% PRIORY ERROR COVARIENCE
obj.P(:,:,n)=obj.A*obj.P(:,:,n-1)*obj.A'+obj.Q;
%% CORRECTION EQUTIONS
% Jacobian matrix
obj.H(:,:,n-1)=[obj.Xh(1,n)/(sqrt(obj.Xh(1,n)^2+obj.Xh(2,n)^2)), obj.Xh(2,n)/(sqrt(obj.Xh(1,n)^2+obj.Xh(2,n)^2)),0,0,0,0; ...
-obj.Xh(2,n)/(sqrt(obj.Xh(1,n)^2+obj.Xh(2,n)^2)), obj.Xh(1,n)/(sqrt(obj.Xh(1,n)^2+obj.Xh(2,n)^2)),0,0,0,0; ...
0,0,1,0,0,0];
% Kalman Gain
obj.K(:,:,n)=obj.P(:,:,n)*obj.H(:,:,n-1)'*(obj.M+obj.H(:,:,n-1)*obj.P(:,:,n)*obj.H(:,:,n-1)')^(-1);
% INNOVATION
Inov=obj.Z(:,:,n)-[sqrt(obj.Xh(1,n)^2+obj.Xh(2,n)^2);arctang(obj.Xh(2,n),obj.Xh(1,n),obj.ind);obj.Xh(3,n)];
%computes final estimate
obj.Xh(:,n)=obj.Xh(:,n)+ obj.K(:,:,n)*Inov;
%computes covarience of estimation error
obj.P(:,:,n)=(eye(6)-obj.K(:,:,n)*obj.H(:,:,n-1))*obj.P(:,:,n);
%% unwrapping the tan function
if abs(arctang(obj.Xh(1,n),obj.Xh(2,n),0)-arctang(obj.Xh(1,n-1),obj.Xh(2,n-1),0))>=pi
if obj.ind==1
obj.ind=0;
else
obj.ind=1;
end
end
end
end
end
检查脚本：
predictedS = EKF(POSITION);
for n = 2:length(POSITION)
[predictedS,S]=predictedS.EKFpredictor(POSITION(ii,:),n);
S1 = S(:,1:3);
plot3(S1(:,1),S1(:,2),S1(:,3),'g');
hold on
end
hold on
plot3(POSITION(:,1),POSITION(:,2),POSITION(:,3),'b')
位置矩阵
-188.1651 187.7193 34.1940
-185.6452 185.0441 33.8262
-183.4172 182.3138 33.5098
-181.4431 179.5418 33.2382
-179.6895 176.7406 33.0055
-178.1260 173.9217 32.8063
-176.7259 171.0961 32.6359
-175.4649 168.2737 32.4900
-174.3218 165.4639 32.3650
-173.2774 162.6754 32.2573
-172.3147 159.9165 32.1640
-171.4185 157.1948 32.0825
-170.5753 154.5171 32.0103
-169.7732 151.8902 31.9453
-169.0016 149.3201 31.8858
-168.2509 146.8122 31.8299
-167.5129 144.3717 31.7762
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-165.3053 137.4984 31.6164
-164.5524 135.3690 31.5602
-163.7832 133.3256 31.5010
-162.9939 131.3705 31.4383
-162.1811 129.5057 31.3715
-161.3420 127.7328 31.3000
-160.4744 126.0528 31.2235
-159.5762 124.4667 31.1416
-158.6458 122.9747 31.0540
-157.6819 121.5767 30.9606
-156.6837 120.2724 30.8610
-155.6503 119.0611 30.7553
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3.0788 34.4572 4.4439
4.0386 33.1311 4.2370
5.0041 31.7819 4.0305
5.9760 30.4120 3.8244
6.9551 29.0239 3.6185
7.9419 27.6201 3.4129
8.9370 26.2033 3.2074
9.9409 24.7760 3.0020
10.9540 23.3409 2.7966
11.9766 21.9007 2.5911
13.0089 20.4580 2.3856
14.0511 19.0157 2.1800
15.1030 17.5763 1.9744
16.1647 16.1425 1.7687
17.2359 14.7171 1.5630
18.3161 13.3025 1.3573
19.4050 11.9015 1.1518
20.5018 10.5166 0.9464
21.6057 9.1502 0.7414
22.7157 7.8049 0.5369
23.8307 6.4831 0.3330
24.9495 5.1870 0.1299
26.0705 3.9191 -0.0722
27.1922 2.6814 -0.2731
28.3126 1.4762 -0.4724
29.4298 0.3055 -0.6699
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31.6458 -1.9246 -1.0583
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35.9320 -5.8892 -1.7982
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42.4382 -10.9875 -2.8990
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43.9201 -11.9690 -3.1473
44.5777 -12.3765 -3.2574
45.1739 -12.7276 -3.3574
45.7044 -13.0220 -3.4465
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46.8252 -13.1076 -3.6554
46.4728 -12.8385 -3.6015
46.0132 -12.5166 -3.5302
45.4440 -12.1435 -3.4412
44.7635 -11.7207 -3.3339
43.9705 -11.2500 -3.2081
43.0642 -10.7335 -3.0636
42.0447 -10.1736 -2.9001
40.9125 -9.5729 -2.7177
39.6694 -8.9344 -2.5163
38.3176 -8.2612 -2.2962
36.8606 -7.5569 -2.0575
35.3031 -6.8254 -1.8007
33.6507 -6.0708 -1.5265
31.9107 -5.2976 -1.2355
30.0915 -4.5108 -0.9289
28.2036 -3.7154 -0.6076
26.2588 -2.9170 -0.2733
24.2713 -2.1214 0.0723
22.2570 -1.3346 0.4272
20.2347 -0.5630 0.7891
18.2252 0.1868 1.1551
16.2524 0.9082 1.5222
14.3433 1.5943 1.8866
12.5280 2.2383 2.2443
10.8405 2.8333 2.5905
9.3185 3.3728 2.9199
8.0041 3.8501 3.2266
6.9442 4.2595 3.5039
6.1905 4.5954 3.7442
5.8005 4.8530 3.9391
速度矢量：
747.0176 -736.8417 -110.3954
660.0126 -754.1758 -95.0541
584.1147 -767.6202 -81.6712
518.1547 -777.4587 -70.0407
461.0804 -783.9474 -59.9769
411.9453 -787.3191 -51.3131
369.8994 -787.7867 -43.8993
334.1793 -785.5465 -37.6009
304.1001 -780.7806 -32.2971
279.0476 -773.6596 -27.8797
258.4712 -764.3446 -24.2515
241.8777 -752.9889 -21.3256
228.8250 -739.7391 -19.0241
218.9174 -724.7368 -17.2771
211.8001 -708.1194 -16.0221
207.1551 -690.0205 -15.2031
204.6971 -670.5712 -14.7697
204.1703 -649.8998 -14.6768
205.3444 -628.1329 -14.8839
208.0123 -605.3948 -15.3543
211.9872 -581.8082 -16.0553
217.1002 -557.4940 -16.9569
223.1984 -532.5713 -18.0323
230.1428 -507.1572 -19.2570
237.8068 -481.3667 -20.6086
246.0746 -455.3126 -22.0667
254.8402 -429.1053 -23.6127
264.0056 -402.8525 -25.2293
273.4808 -376.6587 -26.9007
283.1821 -350.6256 -28.6122
293.0316 -324.8513 -30.3500
302.9568 -299.4301 -32.1014
312.8900 -274.4525 -33.8545
322.7675 -250.0051 -35.5982
332.5296 -226.1698 -37.3220
342.1204 -203.0241 -39.0161
351.4872 -180.6409 -40.6713
360.5803 -159.0881 -42.2789
369.3532 -138.4289 -43.8309
377.7625 -118.7209 -45.3196
385.7672 -100.0170 -46.7381
393.3294 -82.3645 -48.0796
400.4140 -65.8056 -49.3383
406.9884 -50.3772 -50.5085
413.0230 -36.1107 -51.5851
418.4909 -23.0322 -52.5637
423.3679 -11.1627 -53.4400
427.6328 -0.5178 -54.2107
431.2672 8.8919 -54.8725
434.2555 17.0610 -55.4232
436.5850 23.9889 -55.8605
438.2461 29.6799 -56.1832
439.2320 34.1430 -56.3902
439.5386 37.3917 -56.4811
439.1652 39.4439 -56.4561
438.1136 40.3216 -56.3157
436.3887 40.0510 -56.0612
433.9980 38.6619 -55.6941
430.9519 36.1879 -55.2167
427.2637 32.6660 -54.6315
422.9491 28.1362 -53.9417
418.0263 22.6418 -53.1508
412.5162 16.2286 -52.2629
406.4418 8.9450 -51.2822
399.8286 0.8419 -50.2137
392.7040 -8.0280 -49.0624
385.0975 -17.6100 -47.8340
377.0402 -27.8478 -46.5342
368.5651 -38.6837 -45.1692
359.7067 -50.0589 -43.7454
350.5007 -61.9136 -42.2693
340.9841 -74.1874 -40.7478
331.1949 -86.8191 -39.1879
321.1718 -99.7478 -37.5966
310.9543 -112.9119 -35.9812
300.5823 -126.2505 -34.3490
290.0961 -139.7028 -32.7073
279.5359 -153.2086 -31.0634
268.9420 -166.7085 -29.4246
258.3544 -180.1440 -27.7981
247.8127 -193.4578 -26.1911
237.3559 -206.5940 -24.6106
227.0223 -219.4979 -23.0633
216.8492 -232.1168 -21.5561
206.8730 -244.3996 -20.0953
197.1286 -256.2972 -18.6872
187.6500 -267.7626 -17.3378
178.4695 -278.7510 -16.0527
169.6176 -289.2200 -14.8372
161.1234 -299.1297 -13.6966
153.0142 -308.4427 -12.6353
145.3153 -317.1243 -11.6579
138.0499 -325.1425 -10.7682
131.2395 -332.4684 -9.9699
124.9031 -339.0756 -9.2661
119.0578 -344.9411 -8.6595
113.7184 -350.0445 -8.1526
108.8976 -354.3688 -7.7473
104.6055 -357.9000 -7.4450
100.8503 -360.6271 -7.2470
97.6377 -362.5424 -7.1537
94.9714 -363.6413 -7.1656
92.8526 -363.9221 -7.2824
91.2804 -363.3865 -7.5035
90.2517 -362.0391 -7.8279
89.7614 -359.8877 -8.2544
89.8022 -356.9429 -8.7810
90.3649 -353.2184 -9.4058
91.4382 -348.7306 -10.1262
93.0092 -343.4990 -10.9393
95.0630 -337.5454 -11.8421
97.5833 -330.8946 -12.8311
100.5519 -323.5737 -13.9025
103.9493 -315.6124 -15.0523
107.7547 -307.0426 -16.2764
111.9460 -297.8982 -17.5701
116.4998 -288.2156 -18.9288
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编辑 1：
我已经按照您的建议实施了 LKF，但是我仍然没有跟踪任何点。 请看我的实现。
classdef EKF <handle
properties (Access=private)
H
K
Z
Q
M
ind
A
X
Xh
P
a
b
end
methods
function obj = EKF(position)
obj.Q =[0 0 0 0 0 0;
0 0 0 0 0 0;
0 0 0 0 0 0;
0 0 0 0.01 0 0;
0 0 0 0 0.01 0;
0 0 0 0 0 0.01];% Covarience matrix of process noise
obj.M=[0.001 0 0;
0 0.001 0;
0 0 0.001]; % Covarience matrix of measurment noise
obj.A=[1 0 0 0.1 0 0;
0 1 0 0 0.1 0;
0 0 1 0 0 0.1;
0 0 0 1 0 0;
0 0 0 0 1 0;
0 0 0 0 0 1]; % System Dynamics
obj.X = zeros(6,1);
obj.X(1:3,1) = position(1,:);
obj.P = [0.1 0 0 0 0 0;
0 0.1 0 0 0 0;
0 0 0.1 0 0 0;
0 0 0 0.1 0 0;
0 0 0 0 0.1 0;
0 0 0 0 0 0.1];
end
function [obj,predictedS]=EKFpredictor(obj,p)
%% PROCESS AND OBSERVATION PROCESS WITH GAUSSINA NOISE
% State process % w generating process noise
obj.X = obj.A*obj.X+ ...
[0;0;0;sqrt(obj.Q(4,4))*randn(1);sqrt(obj.Q(5,5))*randn(1); ...
sqrt(obj.Q(6,6))*randn(1)];
% predictedX=obj.Xp;
predictedS=obj.X';
%% PREDICTION OF NEXT STATE
obj.P=obj.A*obj.P*obj.A'+obj.Q;
%% CORRECTION EQUTIONS
% Jacobian
obj.Z = p';
obj.H = zeros(3,6);
obj.H(1,1) = 1;
obj.H(2,2) = 1;
obj.H(3,3) = 1;
% Kalman Gain
S = obj.H*obj.P*obj.H' + obj.M;
obj.K = obj.P*obj.H'*inv(S);
% INNOVATION
Y = obj.Z - obj.H*obj.X;
obj.X = obj.X + obj.K*Y;
obj.P = (eye(6)-obj.K*obj.H)*obj.P; % alternatives exist for this calculation
end
end
end
检查 LKF：
predictedS = EKF(POSITION);
for n = 2:length(POSITION)
[predictedS,S]=predictedS.EKFpredictor(POSITION(ii,:));
S1 = S(:,1:3);
plot3(S1(:,1),S1(:,2),S1(:,3),'g');
hold on
end
【问题讨论】：
【参考方案1】：出于好奇，您为什么要使用扩展卡尔曼滤波器 (EKF)？由于您在 3D 空间中跟踪对象，每个位置（测量或观察）输入由 (x,y,z) 三元组给出），并且（输出）状态向量 (X) 是 3D 位置（具有速度分量)，为什么不直接使用更简单的线性卡尔曼滤波器 (LKF)？这样可以避免从 (x,y,z) 坐标空间到范围和方位的转换，避免雅可比的一阶导数等。
由于您的目标是估计位置，因此我建议您使用 LKF，我将在下面逐步介绍您的代码时对其进行描述。
初始化
您的POSITION 矩阵为 279x3（VELOCITY 相同），这意味着我们有 279 个观察值将用于纠正（或更新）对象。对于初始化，我们只需要一个位置（我现在将省略速度）所以而不是
predictedS = EKF(POSITION);
我们可以的
predictedS = EKF(POSITION(1,:));
您的类的构造函数使用一些默认值实例化一些矩阵（Q、A、M），然后分别初始化状态向量和协方差矩阵X 和P：
obj.X(:,1)=[position(1,:) position(2,:)];
obj.P(:,:,1)=[0.1 0 0 0 0 0;
0 0.1 0 0 0 0;
0 0 0.1 0 0 0;
0 0 0 0.1 0 0;
0 0 0 0 0.1 0;
0 0 0 0 0 0.1];
我忽略了Z 和Xh 的初始化。为简单起见，我还将忽略输入参数n，因此不跟踪历史信息。
在X 的初始化中，它变成了一个 6x1 向量，其中前三个元素对应于第一个观察值 - 这是有道理的，因为它们是 (x,y,z) 坐标 - 以及最后三个元素设置为第二个观察值（这是您传递所有 279 个观察值的时候）。这是不正确的，因为X 的速度 (vx,vy,vz) 被赋予了不正确的值 - 位置而不是速度。如果您不知道物体的初始速度，那么卡尔曼滤波器将随着时间的推移估算它们。所以我们可以简单地将上面的状态初始化替换为
obj.X = zeros(6,1);
obj.X(1:3,1) = position(1,:);
协方差初始化的更简单的初始化变成了
obj.P = [0.1 0 0 0 0 0;
0 0.1 0 0 0 0;
0 0 0.1 0 0 0;
0 0 0 0.1 0 0;
0 0 0 0 0.1 0;
0 0 0 0 0 0.1];
您已为每个分量 (x,y,z,vx,vy,vz) 分配了相同的误差不确定性（方差）。这可能是不现实的，因为通常位置不确定性具有较大的方差。您是在猜测这些值还是基于其他知识？
顺便说一句，状态向量中元素的单位是什么？米和米每秒，或类似的东西？我问的部分原因是您的速度（VELOCITIES）与您的位置变化相比很大，部分原因是初始化将转换矩阵定义为
obj.A=[1 0 0 0.1 0 0;
0 1 0 0 0.1 0;
0 0 1 0 0 0.1;
0 0 0 1 0 0;
0 0 0 0 1 0;
0 0 0 0 0 1];
0.1 是指每次更新之间的时间单位。在这种情况下是什么？十分之一秒？如果是这样，那似乎与速度和位置变化无关（除非我错过了什么）。
预测
代码调用预测（以及随后的校正）如下
[predictedS,S]=predictedS.EKFpredictor(POSITION(ii,:),n);
ii 未定义，我不清楚为什么要传入n（除非您想保留所有预测和更新的历史记录）。我暂时忽略它并将其更改为
[predictedS,S]=tracker.EKFpredictor(POSITION(n,:));
第 n 个位置（从 n==2 开始）传入预测器的位置。
您的预测代码（用于状态向量）类似于
obj.X(:,n-1)=[obj.X(1:3,n-1)' p]';
obj.X(:,n)=obj.A*obj.X(:,n-1)+ ...
[0;0;0;sqrt(obj.Q(4,4))*randn(1);sqrt(obj.Q(5,5))*randn(1); ...
sqrt(obj.Q(6,6))*randn(1)];
我不太清楚为什么第一行从X 获取位置元素并将其与新位置p 连接起来。这似乎是不正确的，因为状态向量中有两倍的位置信息，并且速度信息消失了。我认为第一行可以（应该？）被忽略并替换为只是
obj.X = obj.A*obj.X+ …
[0;0;0;sqrt(obj.Q(4,4))*randn(1);sqrt(obj.Q(5,5))*randn(1); ...
sqrt(obj.Q(6,6))*randn(1)];
predictedX=obj.Xp;
将预测状态向量设置为上述（我假设这就是 predictedS 所指的）
predictedS=obj.X(:,n)';
此时无需保留Xh。
可以预测协方差，如您所展示的（我刚刚删除了n）
obj.P=obj.A*obj.P*obj.A'+obj.Q;
更正
使用 LKF，校正要简单得多。观察或测量矩阵Z 很简单
obj.Z = p';
雅可比只是
obj.H = zeros(3,6);
obj.H(1,1) = 1;
obj.H(2,2) = 1;
obj.H(3,3) = 1;
卡尔曼增益是
S = obj.H*obj.P*obj.H' + obj.M;
obj.K = obj.P*obj.H'*inv(S);
请注意，在上面使用 M 时，您的测量/观察不确定性默认为
obj.M=[0.001 0 0;
0 0.001 0;
0 0 0.001];
再一次，单位很重要，并且考虑到非常小的值，这意味着测量/观察/新位置在用于校正轨迹时将被赋予更重的权重 - 因此新位置将极大地影响校正后的状态向量。
在您的代码中，您在计算 S 时做了类似的事情
(obj.M+obj.H(:,:,n-1)*obj.P(:,:,n)*obj.H(:,:,n-1)')^(-1)
^(-1) 的使用不适用于矩阵的逆矩阵。 inv(A) 是一种在这种情况下可能就足够的替代方案。
创新和修正如下
Y = obj.Z - obj.H*obj.X;
obj.X = obj.X + obj.K*Y;
obj.P = (eye(6)-obj.K*obj.H)*obj.P; % alternatives exist for this calculation
考虑使用 LKF - 它应该可以更好地估计位置，因为您可以忽略范围和方位的转换，然后再返回。还要重新检查速度的计算。考虑到A 矩阵中定义的位置变化和转换时间，它们是否有意义。
【讨论】：
请参阅我对 LKF 的实现的 EDIT 1，另外，predictedX=obj.Xp; 是什么，因为它未定义且未在代码中的任何位置使用。
这是一个错字 - 应该是 predictedX=obj.X。 （我不确定为什么你输出预测的状态向量而不是校正的状态向量。）注意当我运行你的 LKF 版本时，（预测的）轨道位置在每次迭代时都以绿色绘制，并且最后与输入数据几乎相同（蓝色图覆盖了它）。这是意料之中的，因为输入位置误差很小。我将对象实例化为tracker = LKF(POSITION(1,:));，并在每次迭代中预测为[tracker,Xp]=tracker.LKFpredictor(POSITION(n,:));。 （我将您的类定义从 EKF 重命名为 LKF。）
是的，蓝色的图会覆盖它，因为那是一条连续的完整轨迹路径，但绿色的只是一些未连接的点。另外，我输出预测的一个，因为我需要预测，另一个实际是自动更新，因为类的 obj 也在输入争论中。
+1 表示努力。这超出了 OP 的要求，而且您超出了要求。杰夫干得好！
@rayryeng 从我这边，它的 0，而不是 -1 因为努力已经完成，但是在预测下一个状态时，等于原始状态是不可接受的，因此它们不使用过滤器。引用obj.Z=p' 其中obj.Z 是观察矩阵必须从p 计算不等于p以上是关于卡尔曼滤波器matlab的输入位置/速度的主要内容，如果未能解决你的问题，请参考以下文章
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