UKF跟踪算法是一种常用的非线性滤波算法，用于在不确定度较高的情况下估计系统的状态。其原理是通过对观测到的状态信息进行加权处理来计算系统的实际状态值。以下是UKF跟踪算法的原理和cpp代码实现的详细说明。
UKF跟踪算法原理
UKF跟踪算法是一种基于卡尔曼滤波算法的改进版，它通过引入非线性变换来适应非线性系统，并通过使用无迹卡尔曼滤波的方法来更新状态估计。其主要步骤如下：
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初始化状态估计。根据现有的观测数据和先验知识，利用卡尔曼滤波器对状态变量进行初始化估计。
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构造sigma点。利用系统状态的均值和协方差矩阵来构造一组sigma点，包括系统状态及其方差范围内的其他可能状态。
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通过非线性变换来计算sigma点的预测状态。
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计算预测状态的均值和协方差矩阵。根据预测状态的sigma点和权重，计算预测状态的均值和协方差矩阵。
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计算观测的sigma点。利用预测状态的均值和协方差矩阵，通过观测方程来计算观测值的sigma点。
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计算观测值的均值和协方差矩阵。根据观测值的sigma点和权重，计算观测值的均值和协方差矩阵。
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计算卡尔曼增益。根据预测状态的协方差矩阵和观测值的协方差矩阵，计算卡尔曼增益。
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更新状态估计。利用卡尔曼增益和观测值的差异来更新状态估计。
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重复步骤3-8，直到状态估计收敛。
UKF跟踪算法cpp代码实现
以下是UKF跟踪算法的C++代码实现。假设我们有一个非线性系统，其状态变量为x，观测变量为z。
#include <iostream>
#include <cmath>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
// 定义状态变量和观测变量的维度
const int nx = 4; // 状态变量的维度
const int nz = 2; // 观测变量的维度
// 定义sigma点的个数及其权重
const int n_sigma = 2 * nx + 1;
double lambda = 3 - nx;
// 定义系统的状态变量和协方差矩阵
VectorXd x(nx);
MatrixXd P(nx, nx);
// 定义观测变量和协方差矩阵
VectorXd z(nz);
MatrixXd R(nz, nz);
// 参数设置
double delta_t = 0.1;
// 定义非线性状态转移函数
VectorXd f(VectorXd x, double delta_t) {
VectorXd x_pred(nx);
x_pred(0) = x(0) + x(2) * delta_t;
x_pred(1) = x(1) + x(3) * delta_t;
x_pred(2) = x(2);
x_pred(3) = x(3);
return x_pred;
}
// 定义非线性观测函数
VectorXd h(VectorXd x) {
VectorXd z_pred(nz);
z_pred(0) = sqrt(x(0) * x(0) + x(1) * x(1));
z_pred(1) = atan2(x(1), x(0));
return z_pred;
}
// 定义非线性函数Jacobian矩阵
MatrixXd Jacobian(VectorXd x) {
double px = x(0);
double py = x(1);
double vx = x(2);
double vy = x(3);
double c1 = px * px + py * py;
double c2 = sqrt(c1);
double c3 = (c1 * c2);
MatrixXd Hj = MatrixXd::Zero(nz, nx);
// check division by zero
if (fabs(c1) < 0.0001) {
cout << "CalculateJacobian () - Error - Division by Zero" << endl;
return Hj;
}
// compute the Jacobian matrix
Hj(0, 0) = px / c2;
Hj(0, 1) = py / c2;
Hj(1, 0) = -py / c1;
Hj(1, 1) = px / c1;
Hj(1, 2) = py * (vx * py - vy * px) / c3;
Hj(1, 3) = px * (vy * px - vx * py) / c3;
return Hj;
}
// 定义计算sigma点的函数
MatrixXd compute_sigma_points(VectorXd x, MatrixXd P, double lambda) {
// compute square root of P
MatrixXd A = P.llt().matrixL();
// compute sigma points ...
MatrixXd Xsig = MatrixXd::Zero(nx, n_sigma);
Xsig.col(0) = x;
for (int i = 0; i < nx; i++) {
Xsig.col(i + 1) = x + sqrt(lambda + nx) * A.col(i);
Xsig.col(i + 1 + nx) = x - sqrt(lambda + nx) * A.col(i);
}
return Xsig;
}
// 定义计算扩展sigma点的函数
MatrixXd compute_augmented_sigma_points(VectorXd x, MatrixXd P, double lambda, double std_a, double std_yawdd) {
// create augmented mean vector
VectorXd x_aug = VectorXd::Zero(7);
x_aug.head(5) = x;
x_aug(5) = 0;
x_aug(6) = 0;
// create augmented state covariance
MatrixXd P_aug = MatrixXd::Zero(7, 7);
P_aug.topLeftCorner(5, 5) = P;
P_aug(5, 5) = std_a * std_a;
P_aug(6, 6) = std_yawdd * std_yawdd;
// create sigma point matrix
MatrixXd Xsig_aug = MatrixXd::Zero(7, n_sigma);
Xsig_aug.col(0) = x_aug;
// create square root matrix
MatrixXd L = P_aug.llt().matrixL();
// create augmented sigma points
for (int i = 0; i < 7; i++) {
Xsig_aug.col(i + 1) = x_aug + sqrt(lambda + 7) * L.col(i);
Xsig_aug.col(i + 1 + 7) = x_aug - sqrt(lambda + 7) * L.col(i);
}
return Xsig_aug;
}
// 定义sigma点预测值函数
MatrixXd predict_sigma_points(MatrixXd Xsig_aug, double delta_t) {
MatrixXd Xsig_pred = MatrixXd::Zero(nx, n_sigma);
// predict sigma points
for (int i = 0; i < n_sigma; 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 yaw_rate = Xsig_aug(4, i);
double nu_a = Xsig_aug(5, i);
double nu_yawdd = Xsig_aug(6, i);
// predicted state values
double px_p, py_p;
// avoid division by zero
if (fabs(yaw_rate) > 0.001) {
px_p = p_x + v / yaw_rate * (sin(yaw + yaw_rate * delta_t) - sin(yaw));
py_p = p_y + v / yaw_rate * (cos(yaw) - cos(yaw + yaw_rate * 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 + yaw_rate * delta_t;
double yawd_p = yaw_rate;
// add noise
px_p = px_p + 0.5 * nu_a * delta_t * delta_t * cos(yaw);
py_p = py_p + 0.5 * nu_a * delta_t * delta_t * sin(yaw);
v_p = v_p + nu_a * delta_t;
yaw_p = yaw_p + 0.5 * nu_yawdd * delta_t * delta_t;
yawd_p = yawd_p + nu_yawdd * 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;
}
return Xsig_pred;
}
// 定义计算预测状态均值和协方差矩阵函数
void predict_mean_covariance(VectorXd& x_pred, MatrixXd& P_pred, MatrixXd Xsig_pred) {
// create vector for weights
VectorXd weights = VectorXd::Zero(n_sigma);
weights(0) = lambda / (lambda + nx);
for (int i = 1; i < n_sigma; i++) {
weights(i) = 0.5 / (lambda + nx);
}
// predict state mean
x_pred = Xsig_pred * weights;
// predict state covariance matrix
P_pred.fill(0.0);
for (int i = 0; i < n_sigma; i++) {
// state difference
VectorXd x_diff = Xsig_pred.col(i) - x_pred;
// 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_pred = P_pred + weights(i) * x_diff * x_diff.transpose();
}
}
// 定义计算预测观测均值和协方差矩阵函数
void predict_measure_mean_covariance(VectorXd& z_pred, MatrixXd& S_pred, MatrixXd Xsig_pred) {
// transform sigma points into measurement space
MatrixXd Zsig = MatrixXd::Zero(nz, n_sigma);
for (int i = 0; i < n_sigma; i++) {
// extract values for better readibility
double p_x = Xsig_pred(0, i);
double p_y = Xsig_pred(1, i);
double v = Xsig_pred(2, i);
double yaw = Xsig_pred(3, i);
// measurement model
Zsig(0, i) = sqrt(p_x * p_x + p_y * p_y);
Zsig(1, i) = atan2(p_y, p_x);
}
// mean predicted measurement
z_pred = Zsig * weights;
// innovation covariance matrix S
S_pred.fill(0.0);
for (int i = 0; i < n_sigma; i++) {
// residual
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_pred = S_pred + weights(i) * z_diff * z_diff.transpose();
}
// add measurement noise covariance matrix
S_pred = S_pred + R;
}
// 定义计算卡尔曼增益函数
void calculate_Kalman_gain(MatrixXd& K, MatrixXd& Tc, MatrixXd S_pred, MatrixXd Xsig_pred, VectorXd x_pred, VectorXd z_pred) {
// calculate cross correlation matrix
Tc.fill(0.0);
for (int i = 0; i < n_sigma; i++) {
// residual
VectorXd z_diff = Xsig_pred.col(i) - x_pred;
// angle normalization
while (z_diff(3) > M_PI) z_diff(3) -= 2. * M_PI;
while (z_diff(3) < -M_PI) z_diff(3) += 2. * M_PI;
// state difference
VectorXd x_diff = Xsig_pred.col(i) - x_pred;
// 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;
Tc = Tc + weights(i) * x_diff * z_diff.transpose();
}
// calculate Kalman gain K;
K = Tc * S_pred.inverse();
}
// 定义更新状态估计函数
void update_state(VectorXd& x_pred, MatrixXd& P_pred, VectorXd z, MatrixXd K, VectorXd z_pred) {
// calculate residual
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_pred = x_pred + K * z_diff;
P_pred = P_pred - K * S_pred * K.transpose();
}
int main() {
// 定义初始状态变量和协方差矩阵
x << 0, 0, 0, 0;
P << 1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 10, 0,
0, 0, 0, 10;
// 定义观测变量和噪声协方差矩阵
z << 0, 0;
R << 0.1, 0,
0, 0.1;
// 定义噪声参数
double std_a = 0.2;
double std_yawdd = 0.2;
// 计算sigma点的权重
VectorXd weights = VectorXd::Zero(n_sigma);
weights(0) = lambda / (lambda + nx);
for (int i = 1; i < n_sigma; i++) {
weights(i) = 0.5 / (lambda + nx);
}
// 仿真时间
double time = 0;
// 执行UKF跟踪算法
while (time < 30) {
// 定义噪声向量
VectorXd nu = VectorXd::Zero(2);
nu(0) = std_a * sqrt(delta_t);
nu(1) = std_yawdd * sqrt(delta_t);
// 计算扩展sigma点
MatrixXd Xsig_aug = compute_augmented_sigma_points(x, P, lambda, std_a, std_yawdd);
// 预测sigma点的状态均值和协方差矩阵
MatrixXd Xsig_pred = predict_sigma_points(Xsig_aug, delta_t);
VectorXd x_pred(nx);
MatrixXd P_pred(nx, nx);
predict_mean_covariance(x_pred, P_pred, Xsig_pred);
// 计算预测观测的均值和协方差矩阵
VectorXd z_pred(nz);
MatrixXd S_pred(nz, nz);
predict_measure_mean_covariance(z_pred, S_pred, Xsig_pred);
// 计算卡尔曼增益
MatrixXd K(nx, nz);
MatrixXd Tc(nx, nz);
calculate_Kalman_gain(K, Tc, S_pred, Xsig_pred, x_pred, z_pred);
// 更新状态变量
VectorXd z_true(nz);
z_true << sqrt(x(0) * x(0) + x(1) * x(1)),
atan2(x(1), x(0));
update_state(x_pred, P_pred, z_true + nu, K, z_pred);
// 更新时间
time += delta_t;
}
return 0;
}
在上面的代码中，我们首先定义了状态变量和观测变量的维度，以及sigma点的个数和权重。然后，我们定义了系统的状态变量和协方差矩