文章目錄
無人駕駛技術——無損卡爾曼濾波（UKF）
前面總結了卡爾曼濾波 【無人駕駛技術——初探Kalman濾波器(KF)】和
擴展卡爾曼濾波【無人駕駛技術——擴展Kalman濾波（EKF）】，
今天主要學習下無損卡爾曼濾波（UKF）
無損卡爾曼濾波（UKF）
無損卡爾曼濾波（UKF）是處理非線性過程模型或非線性測量模型的替代方案。但UKF不會對非線性函數進行線性化處理，它使用所謂的sigma點來近似概率分佈。
它有兩個好處：
- sigma點近似非線性轉換的效果比線性化更好；
- 可以省去計算雅可比矩陣的過程。
無損卡爾曼濾波器不需要將非線性函數線性化；相反，無損卡爾曼濾波器從高斯分佈中取代表點。這些點將被插入非線性方程。
無損卡爾曼濾波器隨時間處理測量值的方式和擴展卡爾曼濾波器是一樣的。它們都有預測步驟，這個步驟是獨立於測量模型的，在這一步裏，需要使用上一節學到的CTRV模型。接下來是更新步驟，我們使用雷達測量模型或激光測量模型。
對於UKF 和EKF,我們可以使用同樣的頂層處理鏈，兩者的不同之處是，UKF處理的是非線性過程模型或非線性測量模型的方法，它使用的方法叫無損變換。
UKF Predict
無損濾波預測步驟主要分爲三步：
1.需要知道選擇sigma點的好辦法
2.需要知道如何預測sigma點
3.需要知道如何根據預測的sigma點計算預測、均值和協方差
1.選擇預測點sigma點
將下面各個值代入上面的sigma點矩陣後，計算結果如下：
可見，主要是根據以下公式求得sigma值：
sigma點計算代碼實現
ukf.h
#ifndef UKF_H
#define UKF_H
#include "Dense"
class UKF {
public:
/**
* Constructor
*/
UKF();
/**
* Destructor
*/
virtual ~UKF();
/**
* Init Initializes Unscented Kalman filter
*/
void Init();
/**
* Student assignment functions
*/
void GenerateSigmaPoints(Eigen::MatrixXd* Xsig_out);
void AugmentedSigmaPoints(Eigen::MatrixXd* Xsig_out);
void SigmaPointPrediction(Eigen::MatrixXd* Xsig_out);
void PredictMeanAndCovariance(Eigen::VectorXd* x_pred,
Eigen::MatrixXd* P_pred);
void PredictRadarMeasurement(Eigen::VectorXd* z_out,
Eigen::MatrixXd* S_out);
void UpdateState(Eigen::VectorXd* x_out,
Eigen::MatrixXd* P_out);
};
#endif // UKF_H
ukf.cpp
#include "ukf.h"
#include <iostream>
using Eigen::MatrixXd;
using Eigen::VectorXd;
UKF::UKF() {
Init();
}
UKF::~UKF() {
}
void UKF::Init() {
}
/**
* Programming assignment functions:
*/
void UKF::GenerateSigmaPoints(MatrixXd* Xsig_out) {
// set state dimension
int n_x = 5;
// define spreading parameter
double lambda = 3 - n_x;
// set example state
VectorXd x = VectorXd(n_x);
x << 5.7441,
1.3800,
2.2049,
0.5015,
0.3528;
// set example covariance matrix
MatrixXd P = MatrixXd(n_x, n_x);
P << 0.0043, -0.0013, 0.0030, -0.0022, -0.0020,
-0.0013, 0.0077, 0.0011, 0.0071, 0.0060,
0.0030, 0.0011, 0.0054, 0.0007, 0.0008,
-0.0022, 0.0071, 0.0007, 0.0098, 0.0100,
-0.0020, 0.0060, 0.0008, 0.0100, 0.0123;
// create sigma point matrix
MatrixXd Xsig = MatrixXd(n_x, 2 * n_x + 1);
// calculate square root of P
MatrixXd A = P.llt().matrixL();
/**
* Student part begin
*/
// your code goes here
// calculate sigma points ...
// set sigma points as columns of matrix Xsig
Xsig.col(0) = x;
// set remaining sigma points
for (int i = 0; i < n_x; ++i)
{
Xsig.col(i+1) = x + sqrt(lambda+n_x) * A.col(i);
Xsig.col(i+1+n_x) = x - sqrt(lambda+n_x) * A.col(i);
}
/**
* Student part end
*/
// print result
// std::cout << "Xsig = " << std::endl << Xsig << std::endl;
// write result
*Xsig_out = Xsig;
}
/**
* expected result:
* Xsig =
* 5.7441 5.85768 5.7441 5.7441 5.7441 5.7441 5.63052 5.7441 5.7441 5.7441 5.7441
* 1.38 1.34566 1.52806 1.38 1.38 1.38 1.41434 1.23194 1.38 1.38 1.38
* 2.2049 2.28414 2.24557 2.29582 2.2049 2.2049 2.12566 2.16423 2.11398 2.2049 2.2049
* 0.5015 0.44339 0.631886 0.516923 0.595227 0.5015 0.55961 0.371114 0.486077 0.407773 0.5015
* 0.3528 0.299973 0.462123 0.376339 0.48417 0.418721 0.405627 0.243477 0.329261 0.22143 0.286879
*/
main.cpp
#include <iostream>
#include "Dense"
#include "ukf.h"
using Eigen::MatrixXd;
int main() {
// Create a UKF instance
UKF ukf;
/**
* Programming assignment calls
*/
MatrixXd Xsig = MatrixXd(5, 11);
ukf.GenerateSigmaPoints(&Xsig);
// print result
std::cout << "Xsig = " << std::endl << Xsig << std::endl;
return 0;
}
添加過程噪聲處理
爲什麼需要添加噪聲進行擴充？
因爲過程噪聲對狀態有非線性影響。
構建噪聲擴充矩陣 c++
ukf.h
#ifndef UKF_H
#define UKF_H
#include "Dense"
class UKF {
public:
/**
* Constructor
*/
UKF();
/**
* Destructor
*/
virtual ~UKF();
/**
* Init Initializes Unscented Kalman filter
*/
void Init();
/**
* Student assignment functions
*/
void GenerateSigmaPoints(Eigen::MatrixXd* Xsig_out);
void AugmentedSigmaPoints(Eigen::MatrixXd* Xsig_out);
void SigmaPointPrediction(Eigen::MatrixXd* Xsig_out);
void PredictMeanAndCovariance(Eigen::VectorXd* x_pred,
Eigen::MatrixXd* P_pred);
void PredictRadarMeasurement(Eigen::VectorXd* z_out,
Eigen::MatrixXd* S_out);
void UpdateState(Eigen::VectorXd* x_out,
Eigen::MatrixXd* P_out);
};
#endif // UKF_H
ukf.cpp
#include <iostream>
#include "ukf.h"
using Eigen::MatrixXd;
using Eigen::VectorXd;
UKF::UKF() {
Init();
}
UKF::~UKF() {
}
void UKF::Init() {
}
/**
* Programming assignment functions:
*/
void UKF::AugmentedSigmaPoints(MatrixXd* Xsig_out) {
// set state dimension
int n_x = 5;
// set augmented dimension
int n_aug = 7;
// Process noise standard deviation longitudinal acceleration in m/s^2
double std_a = 0.2;
// Process noise standard deviation yaw acceleration in rad/s^2
double std_yawdd = 0.2;
// define spreading parameter
double lambda = 3 - n_aug;
// set example state
VectorXd x = VectorXd(n_x);
x << 5.7441,
1.3800,
2.2049,
0.5015,
0.3528;
// create example covariance matrix
MatrixXd P = MatrixXd(n_x, n_x);
P << 0.0043, -0.0013, 0.0030, -0.0022, -0.0020,
-0.0013, 0.0077, 0.0011, 0.0071, 0.0060,
0.0030, 0.0011, 0.0054, 0.0007, 0.0008,
-0.0022, 0.0071, 0.0007, 0.0098, 0.0100,
-0.0020, 0.0060, 0.0008, 0.0100, 0.0123;
// create augmented mean vector
VectorXd x_aug = VectorXd(7);
// create augmented state covariance
MatrixXd P_aug = MatrixXd(7, 7);
// create sigma point matrix
MatrixXd Xsig_aug = MatrixXd(n_aug, 2 * n_aug + 1);
/**
* Student part begin
*/
// create augmented mean state
x_aug.head(5) = x;
x_aug(5) = 0;
x_aug(6) = 0;
// create augmented covariance matrix
P_aug.fill(0.0);
P_aug.topLeftCorner(5,5) = P;
P_aug(5,5) = std_a*std_a;
P_aug(6,6) = std_yawdd*std_yawdd;
// create square root matrix
MatrixXd L = P_aug.llt().matrixL();
// create augmented sigma points
Xsig_aug.col(0) = x_aug;
for (int i = 0; i< n_aug; ++i) {
Xsig_aug.col(i+1) = x_aug + sqrt(lambda+n_aug) * L.col(i);
Xsig_aug.col(i+1+n_aug) = x_aug - sqrt(lambda+n_aug) * L.col(i);
}
/**
* Student part end
*/
// print result
std::cout << "Xsig_aug = " << std::endl << Xsig_aug << std::endl;
// write result
*Xsig_out = Xsig_aug;
}
/**
* expected result:
* Xsig_aug =
* 5.7441 5.85768 5.7441 5.7441 5.7441 5.7441 5.7441 5.7441 5.63052 5.7441 5.7441 5.7441 5.7441 5.7441 5.7441
* 1.38 1.34566 1.52806 1.38 1.38 1.38 1.38 1.38 1.41434 1.23194 1.38 1.38 1.38 1.38 1.38
* 2.2049 2.28414 2.24557 2.29582 2.2049 2.2049 2.2049 2.2049 2.12566 2.16423 2.11398 2.2049 2.2049 2.2049 2.2049
* 0.5015 0.44339 0.631886 0.516923 0.595227 0.5015 0.5015 0.5015 0.55961 0.371114 0.486077 0.407773 0.5015 0.5015 0.5015
* 0.3528 0.299973 0.462123 0.376339 0.48417 0.418721 0.3528 0.3528 0.405627 0.243477 0.329261 0.22143 0.286879 0.3528 0.3528
* 0 0 0 0 0 0 0.34641 0 0 0 0 0 0 -0.34641 0
* 0 0 0 0 0 0 0 0.34641 0 0 0 0 0 0 -0.34641
*/
main.cpp
#include "Dense"
#include "ukf.h"
using Eigen::MatrixXd;
int main() {
// Create a UKF instance
UKF ukf;
/**
* Programming assignment calls
*/
MatrixXd Xsig_aug = MatrixXd(7, 15);
ukf.AugmentedSigmaPoints(&Xsig_aug);
return 0;
}
2.預測sigma點
預測sigma點方法
預測sigma點 c++代碼實現
ukf.h
#ifndef UKF_H
#define UKF_H
#include "Dense"
class UKF {
public:
/**
* Constructor
*/
UKF();
/**
* Destructor
*/
virtual ~UKF();
/**
* Init Initializes Unscented Kalman filter
*/
void Init();
/**
* Student assignment functions
*/
void GenerateSigmaPoints(Eigen::MatrixXd* Xsig_out);
void AugmentedSigmaPoints(Eigen::MatrixXd* Xsig_out);
void SigmaPointPrediction(Eigen::MatrixXd* Xsig_out);
void PredictMeanAndCovariance(Eigen::VectorXd* x_pred,
Eigen::MatrixXd* P_pred);
void PredictRadarMeasurement(Eigen::VectorXd* z_out,
Eigen::MatrixXd* S_out);
void UpdateState(Eigen::VectorXd* x_out,
Eigen::MatrixXd* P_out);
};
#endif // UKF_H
ukf.cpp
#include <iostream>
#include "ukf.h"
using Eigen::MatrixXd;
using Eigen::VectorXd;
UKF::UKF() {
Init();
}
UKF::~UKF() {
}
void UKF::Init() {
}
/**
* Programming assignment functions:
*/
void UKF::SigmaPointPrediction(MatrixXd* Xsig_out) {
// set state dimension
int n_x = 5;
// set augmented dimension
int n_aug = 7;
// create example sigma point matrix
MatrixXd Xsig_aug = MatrixXd(n_aug, 2 * n_aug + 1);
Xsig_aug <<
5.7441, 5.85768, 5.7441, 5.7441, 5.7441, 5.7441, 5.7441, 5.7441, 5.63052, 5.7441, 5.7441, 5.7441, 5.7441, 5.7441, 5.7441,
1.38, 1.34566, 1.52806, 1.38, 1.38, 1.38, 1.38, 1.38, 1.41434, 1.23194, 1.38, 1.38, 1.38, 1.38, 1.38,
2.2049, 2.28414, 2.24557, 2.29582, 2.2049, 2.2049, 2.2049, 2.2049, 2.12566, 2.16423, 2.11398, 2.2049, 2.2049, 2.2049, 2.2049,
0.5015, 0.44339, 0.631886, 0.516923, 0.595227, 0.5015, 0.5015, 0.5015, 0.55961, 0.371114, 0.486077, 0.407773, 0.5015, 0.5015, 0.5015,
0.3528, 0.299973, 0.462123, 0.376339, 0.48417, 0.418721, 0.3528, 0.3528, 0.405627, 0.243477, 0.329261, 0.22143, 0.286879, 0.3528, 0.3528,
0, 0, 0, 0, 0, 0, 0.34641, 0, 0, 0, 0, 0, 0, -0.34641, 0,
0, 0, 0, 0, 0, 0, 0, 0.34641, 0, 0, 0, 0, 0, 0, -0.34641;
// create matrix with predicted sigma points as columns
MatrixXd Xsig_pred = MatrixXd(n_x, 2 * n_aug + 1);
double delta_t = 0.1; // time diff in sec
/**
* Student part begin
*/
// predict sigma points
// avoid division by zero
// write predicted sigma points into right column
// predict sigma points
for (int i = 0; i< 2*n_aug+1; ++i) {
// extract values for better readability
double p_x = Xsig_aug(0,i);
double p_y = Xsig_aug(1,i);
double v = Xsig_aug(2,i);
double yaw = Xsig_aug(3,i);
double yawd = Xsig_aug(4,i);
double v_a = Xsig_aug(5,i);
double v_b = Xsig_aug(6,i);
// predicted state values
double px_p, py_p;
// avoid division by zero
if (fabs(yawd) > 0.001) {
px_p = p_x + v/yawd * ( sin (yaw + yawd*delta_t) - sin(yaw));
py_p = p_y + v/yawd * ( cos(yaw) - cos(yaw+yawd*delta_t) );
} else {
px_p = p_x + v*delta_t*cos(yaw);
py_p = p_y + v*delta_t*sin(yaw);
}
double v_p = v;
double yaw_p = yaw + yawd*delta_t;
double yawd_p = yawd;
// add noise
px_p = px_p + 0.5*v_a*delta_t*delta_t * cos(yaw);
py_p = py_p + 0.5*v_a*delta_t*delta_t * sin(yaw);
v_p = v_p + v_a*delta_t;
yaw_p = yaw_p + 0.5*v_b*delta_t*delta_t;
yawd_p = yawd_p + v_b*delta_t;
// write predicted sigma point into right column
Xsig_pred(0,i) = px_p;
Xsig_pred(1,i) = py_p;
Xsig_pred(2,i) = v_p;
Xsig_pred(3,i) = yaw_p;
Xsig_pred(4,i) = yawd_p;
}
/**
* Student part end
*/
// print result
std::cout << "Xsig_pred = " << std::endl << Xsig_pred << std::endl;
// write result
*Xsig_out = Xsig_pred;
}
main.cpp
#include "Dense"
#include "ukf.h"
using Eigen::MatrixXd;
int main() {
// Create a UKF instance
UKF ukf;
/**
* Programming assignment calls
*/
MatrixXd Xsig_pred = MatrixXd(15, 5);
ukf.SigmaPointPrediction(&Xsig_pred);
return 0;
}
3.根據預測的sigma點預測狀態均值和協方差矩陣
預測均值和協方差c++代碼實現
ukf.h
#ifndef UKF_H
#define UKF_H
#include "Dense"
class UKF {
public:
/**
* Constructor
*/
UKF();
/**
* Destructor
*/
virtual ~UKF();
/**
* Init Initializes Unscented Kalman filter
*/
void Init();
/**
* Student assignment functions
*/
void GenerateSigmaPoints(Eigen::MatrixXd* Xsig_out);
void AugmentedSigmaPoints(Eigen::MatrixXd* Xsig_out);
void SigmaPointPrediction(Eigen::MatrixXd* Xsig_out);
void PredictMeanAndCovariance(Eigen::VectorXd* x_pred,
Eigen::MatrixXd* P_pred);
void PredictRadarMeasurement(Eigen::VectorXd* z_out,
Eigen::MatrixXd* S_out);
void UpdateState(Eigen::VectorXd* x_out,
Eigen::MatrixXd* P_out);
};
#endif // UKF_H
ukf.cpp
#include <iostream>
#include "ukf.h"
using Eigen::MatrixXd;
using Eigen::VectorXd;
UKF::UKF() {
Init();
}
UKF::~UKF() {
}
void UKF::Init() {
}
/**
* Programming assignment functions:
*/
void UKF::PredictMeanAndCovariance(VectorXd* x_out, MatrixXd* P_out) {
// set state dimension
int n_x = 5;
// set augmented dimension
int n_aug = 7;
// define spreading parameter
double lambda = 3 - n_aug;
// create example matrix with predicted sigma points
MatrixXd Xsig_pred = MatrixXd(n_x, 2 * n_aug + 1);
Xsig_pred <<
5.9374, 6.0640, 5.925, 5.9436, 5.9266, 5.9374, 5.9389, 5.9374, 5.8106, 5.9457, 5.9310, 5.9465, 5.9374, 5.9359, 5.93744,
1.48, 1.4436, 1.660, 1.4934, 1.5036, 1.48, 1.4868, 1.48, 1.5271, 1.3104, 1.4787, 1.4674, 1.48, 1.4851, 1.486,
2.204, 2.2841, 2.2455, 2.2958, 2.204, 2.204, 2.2395, 2.204, 2.1256, 2.1642, 2.1139, 2.204, 2.204, 2.1702, 2.2049,
0.5367, 0.47338, 0.67809, 0.55455, 0.64364, 0.54337, 0.5367, 0.53851, 0.60017, 0.39546, 0.51900, 0.42991, 0.530188, 0.5367, 0.535048,
0.352, 0.29997, 0.46212, 0.37633, 0.4841, 0.41872, 0.352, 0.38744, 0.40562, 0.24347, 0.32926, 0.2214, 0.28687, 0.352, 0.318159;
// create vector for weights
VectorXd weights = VectorXd(2*n_aug+1);
// create vector for predicted state
VectorXd x = VectorXd(n_x);
// create covariance matrix for prediction
MatrixXd P = MatrixXd(n_x, n_x);
/**
* Student part begin
*/
// set weights
weights(0) = lambda / (lambda + n_aug);
double weight = 0.5 / (lambda + n_aug);
for(int i = 1; i < 2 * n_aug + 1; ++i)
{
weights(i) = weight;
}
// predict state mean
x.fill(0.0);
for (int i = 0; i < 2 * n_aug + 1; ++i)
{ // iterate over sigma points
x = x + weights(i) * Xsig_pred.col(i);
}
// predict state covariance matrix
P.fill(0.0);
for (int i = 0; i < 2 * n_aug + 1; ++i)
{ // iterate over sigma points
// state difference
VectorXd x_diff = Xsig_pred.col(i) - x;
//angle normalization
while (x_diff(3)> M_PI) x_diff(3)-=2.*M_PI;
while (x_diff(3)<-M_PI) x_diff(3)+=2.*M_PI;
P = P + weights(i) * x_diff * x_diff.transpose() ;
}
/**
* Student part end
*/
// print result
std::cout << "Predicted state" << std::endl;
std::cout << x << std::endl;
std::cout << "Predicted covariance matrix" << std::endl;
std::cout << P << std::endl;
// write result
*x_out = x;
*P_out = P;
}
main.cpp
#include "Dense"
#include "ukf.h"
using Eigen::MatrixXd;
using Eigen::VectorXd;
int main() {
// Create a UKF instance
UKF ukf;
/**
* Programming assignment calls
*/
VectorXd x_pred = VectorXd(5);
MatrixXd P_pred = MatrixXd(5, 5);
ukf.PredictMeanAndCovariance(&x_pred, &P_pred);
return 0;
}
UKF Update
測量預測
將預測狀態轉換爲測量空間，定義轉換的模型叫測量模型（Measurement model）。
測量模型是一個非線性模型，測量噪音具有單純的累加效果，因此這裏我們不需要擴充噪音，有更好的處理方法。
1.首先，將預測到的sigma點轉換到測量空間
2.利用這些轉換後的點計算預測測量值的均值和協方差矩陣S
同樣，轉換後的預測sigma點存儲爲矩陣的列（col）。
將預測到的sigma點轉換到測量空間：
這裏，我們不使用擴充噪聲矩陣了，故等式中的w_k+1 設爲0。
計算均值和協方差：
這裏，用測量噪音矩陣R來替代擴充噪音矩陣。
測量預測c++代碼實現
ukf.h
#ifndef UKF_H
#define UKF_H
#include "Dense"
class UKF {
public:
/**
* Constructor
*/
UKF();
/**
* Destructor
*/
virtual ~UKF();
/**
* Init Initializes Unscented Kalman filter
*/
void Init();
/**
* Student assignment functions
*/
void GenerateSigmaPoints(Eigen::MatrixXd* Xsig_out);
void AugmentedSigmaPoints(Eigen::MatrixXd* Xsig_out);
void SigmaPointPrediction(Eigen::MatrixXd* Xsig_out);
void PredictMeanAndCovariance(Eigen::VectorXd* x_pred,
Eigen::MatrixXd* P_pred);
void PredictRadarMeasurement(Eigen::VectorXd* z_out,
Eigen::MatrixXd* S_out);
void UpdateState(Eigen::VectorXd* x_out,
Eigen::MatrixXd* P_out);
};
#endif // UKF_H
ukf.cpp
#include <iostream>
#include "ukf.h"
using Eigen::MatrixXd;
using Eigen::VectorXd;
UKF::UKF() {
Init();
}
UKF::~UKF() {
}
void UKF::Init() {
}
/**
* Programming assignment functions:
*/
void UKF::PredictRadarMeasurement(VectorXd* z_out, MatrixXd* S_out) {
// set state dimension
int n_x = 5;
// set augmented dimension
int n_aug = 7;
// set measurement dimension, radar can measure r, phi, and r_dot
int n_z = 3;
// define spreading parameter
double lambda = 3 - n_aug;
// set vector for weights
VectorXd weights = VectorXd(2*n_aug+1);
double weight_0 = lambda/(lambda+n_aug);
double weight = 0.5/(lambda+n_aug);
weights(0) = weight_0;
for (int i=1; i<2*n_aug+1; ++i) {
weights(i) = weight;
}
// radar measurement noise standard deviation radius in m
double std_radr = 0.3;
// radar measurement noise standard deviation angle in rad
double std_radphi = 0.0175;
// radar measurement noise standard deviation radius change in m/s
double std_radrd = 0.1;
// create example matrix with predicted sigma points
MatrixXd Xsig_pred = MatrixXd(n_x, 2 * n_aug + 1);
Xsig_pred <<
5.9374, 6.0640, 5.925, 5.9436, 5.9266, 5.9374, 5.9389, 5.9374, 5.8106, 5.9457, 5.9310, 5.9465, 5.9374, 5.9359, 5.93744,
1.48, 1.4436, 1.660, 1.4934, 1.5036, 1.48, 1.4868, 1.48, 1.5271, 1.3104, 1.4787, 1.4674, 1.48, 1.4851, 1.486,
2.204, 2.2841, 2.2455, 2.2958, 2.204, 2.204, 2.2395, 2.204, 2.1256, 2.1642, 2.1139, 2.204, 2.204, 2.1702, 2.2049,
0.5367, 0.47338, 0.67809, 0.55455, 0.64364, 0.54337, 0.5367, 0.53851, 0.60017, 0.39546, 0.51900, 0.42991, 0.530188, 0.5367, 0.535048,
0.352, 0.29997, 0.46212, 0.37633, 0.4841, 0.41872, 0.352, 0.38744, 0.40562, 0.24347, 0.32926, 0.2214, 0.28687, 0.352, 0.318159;
// create matrix for sigma points in measurement space
MatrixXd Zsig = MatrixXd(n_z, 2 * n_aug + 1);
// mean predicted measurement
VectorXd z_pred = VectorXd(n_z);
// measurement covariance matrix S
MatrixXd S = MatrixXd(n_z,n_z);
/**
* Student part begin
*/
// transform sigma points into measurement space
for(int i=0; i < 2 * n_aug + 1; ++i)
{
float p_x = Xsig_pred(0,i);
float p_y = Xsig_pred(1,i);
float v = Xsig_pred(2,i);
float yaw = Xsig_pred(3,i);
float yawd = Xsig_pred(4,i);
float th_2 = sqrt(p_x*p_x + p_y*p_y);
float rho_z = th_2;
float yaw_z = atan2(p_y,p_x);
float rhod_z = (p_x*cos(yaw)*v + p_y*sin(yaw)*v) / th_2;
Zsig(0,i) = rho_z;
Zsig(1,i) = yaw_z;
Zsig(2,i) = rhod_z;
}
// calculate mean predicted measurement
z_pred.fill(0.0);
for(int i=0; i < 2 * n_aug + 1; ++i)
{
z_pred = z_pred + weights(i) *Zsig.col(i);
}
// calculate innovation covariance matrix S
S.fill(0.0);
for (int i=0; i < 2*n_aug+1; ++i)
{
VectorXd z_diff = Zsig.col(i) - z_pred;
// angle normalization
while (z_diff(1)> M_PI) z_diff(1)-=2.*M_PI;
while (z_diff(1)<-M_PI) z_diff(1)+=2.*M_PI;
S = S + weights(i) * z_diff * z_diff.transpose();
}
// add measurement noise covariance matrix
MatrixXd R = MatrixXd(n_z,n_z);
R << std_radr*std_radr, 0, 0,
0, std_radphi*std_radphi, 0,
0, 0,std_radrd*std_radrd;
S = S + R;
/**
* Student part end
*/
// print result
std::cout << "z_pred: " << std::endl << z_pred << std::endl;
std::cout << "S: " << std::endl << S << std::endl;
// write result
*z_out = z_pred;
*S_out = S;
}
main.cpp
#include "Dense"
#include "ukf.h"
using Eigen::MatrixXd;
using Eigen::VectorXd;
int main() {
// Create a UKF instance
UKF ukf;
/**
* Programming assignment calls
*/
VectorXd z_out = VectorXd(3);
MatrixXd S_out = MatrixXd(3, 3);
ukf.PredictRadarMeasurement(&z_out, &S_out);
return 0;
}
預測結果如下：
z_pred:
6.12155
0.245993
2.10313
S:
0.0946171 -0.000139447 0.00407016
-0.000139447 0.000617548 -0.000770652
0.00407016 -0.000770652 0.0180917
Update之更新狀態
更新狀態c++代碼實現
void UKF::UpdateState(VectorXd* x_out, MatrixXd* P_out) {
// set state dimension
int n_x = 5;
// set augmented dimension
int n_aug = 7;
// set measurement dimension, radar can measure r, phi, and r_dot
int n_z = 3;
// define spreading parameter
double lambda = 3 - n_aug;
// set vector for weights
VectorXd weights = VectorXd(2*n_aug+1);
double weight_0 = lambda/(lambda+n_aug);
double weight = 0.5/(lambda+n_aug);
weights(0) = weight_0;
for (int i=1; i<2*n_aug+1; ++i) {
weights(i) = weight;
}
// create example matrix with predicted sigma points in state space
MatrixXd Xsig_pred = MatrixXd(n_x, 2 * n_aug + 1);
Xsig_pred <<
5.9374, 6.0640, 5.925, 5.9436, 5.9266, 5.9374, 5.9389, 5.9374, 5.8106, 5.9457, 5.9310, 5.9465, 5.9374, 5.9359, 5.93744,
1.48, 1.4436, 1.660, 1.4934, 1.5036, 1.48, 1.4868, 1.48, 1.5271, 1.3104, 1.4787, 1.4674, 1.48, 1.4851, 1.486,
2.204, 2.2841, 2.2455, 2.2958, 2.204, 2.204, 2.2395, 2.204, 2.1256, 2.1642, 2.1139, 2.204, 2.204, 2.1702, 2.2049,
0.5367, 0.47338, 0.67809, 0.55455, 0.64364, 0.54337, 0.5367, 0.53851, 0.60017, 0.39546, 0.51900, 0.42991, 0.530188, 0.5367, 0.535048,
0.352, 0.29997, 0.46212, 0.37633, 0.4841, 0.41872, 0.352, 0.38744, 0.40562, 0.24347, 0.32926, 0.2214, 0.28687, 0.352, 0.318159;
// create example vector for predicted state mean
VectorXd x = VectorXd(n_x);
x <<
5.93637,
1.49035,
2.20528,
0.536853,
0.353577;
// create example matrix for predicted state covariance
MatrixXd P = MatrixXd(n_x,n_x);
P <<
0.0054342, -0.002405, 0.0034157, -0.0034819, -0.00299378,
-0.002405, 0.01084, 0.001492, 0.0098018, 0.00791091,
0.0034157, 0.001492, 0.0058012, 0.00077863, 0.000792973,
-0.0034819, 0.0098018, 0.00077863, 0.011923, 0.0112491,
-0.0029937, 0.0079109, 0.00079297, 0.011249, 0.0126972;
// create example matrix with sigma points in measurement space
MatrixXd Zsig = MatrixXd(n_z, 2 * n_aug + 1);
Zsig <<
6.1190, 6.2334, 6.1531, 6.1283, 6.1143, 6.1190, 6.1221, 6.1190, 6.0079, 6.0883, 6.1125, 6.1248, 6.1190, 6.1188, 6.12057,
0.24428, 0.2337, 0.27316, 0.24616, 0.24846, 0.24428, 0.24530, 0.24428, 0.25700, 0.21692, 0.24433, 0.24193, 0.24428, 0.24515, 0.245239,
2.1104, 2.2188, 2.0639, 2.187, 2.0341, 2.1061, 2.1450, 2.1092, 2.0016, 2.129, 2.0346, 2.1651, 2.1145, 2.0786, 2.11295;
// create example vector for mean predicted measurement
VectorXd z_pred = VectorXd(n_z);
z_pred <<
6.12155,
0.245993,
2.10313;
// create example matrix for predicted measurement covariance
MatrixXd S = MatrixXd(n_z,n_z);
S <<
0.0946171, -0.000139448, 0.00407016,
-0.000139448, 0.000617548, -0.000770652,
0.00407016, -0.000770652, 0.0180917;
// create example vector for incoming radar measurement
VectorXd z = VectorXd(n_z);
z <<
5.9214, // rho in m
0.2187, // phi in rad
2.0062; // rho_dot in m/s
// create matrix for cross correlation Tc
MatrixXd Tc = MatrixXd(n_x, n_z);
/**
* Student part begin
*/
// calculate cross correlation matrix
for(int i=0; i < 2 * n_aug + 1; i++)
{
VectorXd x_diff = Xsig_pred.col(i) - x;
// angle normalization
while (x_diff(3)> M_PI) x_diff(3)-=2.*M_PI;
while (x_diff(3)<-M_PI) x_diff(3)+=2.*M_PI;
VectorXd z_diff = Zsig.col(i) - z_pred;
// angle normalization
while (z_diff(1)> M_PI) z_diff(1)-=2.*M_PI;
while (z_diff(1)<-M_PI) z_diff(1)+=2.*M_PI;
Tc = Tc + weights(i)*x_diff *z_diff.transpose();
}
// calculate Kalman gain K;
MatrixXd K = Tc * S.inverse();
// update state mean and covariance matrix
VectorXd z_diff = z - z_pred;
// angle normalization
while (z_diff(1)> M_PI) z_diff(1)-=2.*M_PI;
while (z_diff(1)<-M_PI) z_diff(1)+=2.*M_PI;
// update state mean and covariance matrix
x = x + K * z_diff;
P = P - K*S*K.transpose();
/**
* Student part end
*/
// print result
std::cout << "Updated state x: " << std::endl << x << std::endl;
std::cout << "Updated state covariance P: " << std::endl << P << std::endl;
// write result
*x_out = x;
*P_out = P;
}
預測結果如下：
Updated state x:
5.92276
1.41823
2.15593
0.489274
0.321338
Updated state covariance P:
0.00361579 -0.000357881 0.00208316 -0.000937196 -0.00071727
-0.000357881 0.00539867 0.00156846 0.00455342 0.00358885
0.00208316 0.00156846 0.00410651 0.00160333 0.00171811
-0.000937196 0.00455342 0.00160333 0.00652634 0.00669436
-0.00071719 0.00358884 0.00171811 0.00669426 0.00881797
噪音參數和評估選擇
對於CTRV模型，有兩個參數定義了過程噪聲：
分別代表了縱向加速度噪聲（您可能會看到這稱爲線性加速度） 和 偏航加速度噪聲（也稱爲角加速度）
在項目中，這兩個值都需要調整。爲了得到有效的解決方案，您必須測試不同的值。在視頻中，dominik提到使用
作爲跟蹤車輛的起點。在UKF項目中，您將跟蹤自行車而不是車輛。所以9可能不是一個合適的加速度噪聲參數。調諧將涉及：
猜測適當的參數值
運行UKF過濾器
決定結果是否足夠好
調整參數並重復該過程
如何選擇噪音參數和評估選擇，這不僅適用於無損卡爾曼濾波，同樣適用於貝葉斯濾波器。
如何選擇噪音參數
直線加速度噪聲參數直觀
偏航加速度噪聲參數直觀
測量噪聲參數
測量噪聲參數表示傳感器測量的不確定性。一般來說，製造商將在傳感器手冊中提供這些值。在UKF項目中，不需要調整這些參數。
怎麼評估選擇
濾波器的一致性檢查
NIS,Normalized Innovation Squared ,歸一化新息平方
新息是預測測量值和實際測量值之間的差，
歸一化是指相對於矩陣S的協方差而言，因此我們這裏有矩陣S的逆。
NIS只是一個標量數字，計算非常簡單，如下圖公式。NIS的值分佈符合卡方分佈，
df指自由度，這裏指測量空間的維度。
舉個例子，假設 第三行
第一列0.95代表在統計意義上，所有情況下有95%的概率，你的NIS會超過0.352；
最後一列0.5代表在統計意義上，所有情況下有5%的概率，你的NIS會超過0.7815；
假設本例中，所有情況下有5%的概率，你的NIS會超過0.7815，若是如下圖所示，則表明正常，
如果得到這樣一張圖，則說明系統低估了不確定性。
如果得到下面這樣一張圖，則說明系統高估了不確定性。你估算的精確度比你
想的要高。
不幸的是，NIS測試不會告訴你錯誤原因。但至少它提供了一些反饋信息。
例如，在本例中，你可以嘗試降低過程噪音，然後再試。