29. UKF Update Assignment 1

Assignment UKF Update

UKF Update Assignment

Task Description:

You will be completing the missing code in ukf.cpp, UpdateState() function.

Task List:

Task Feedback:

Nice! Have you tested your code below?

Helpful Equations

Cross-correlation Matrix

T_{k+1|k} = \sum_{i=1}^{n_\sigma} w_i (X_{k+1|k,i} - x_{k+1|k})\ (Z_{k+1|k,i} - z_{k+1|k})^T

Kalman gain K

K_{k+1|k} = T_{k+1|k}S^{-1}_{k+1|k}

Update State

x_{k+1|k+1} = x_{k+1|k}+K_{k+1|k}(z_{k+1}-z_{k+1|k})

Covariance Matrix Update

P_{k+1|k+1} = P_{k+1|k}-K_{k+1|k}S_{k+1|k}K^T_{k+1|k}

Start Quiz: 

#include "Dense"
#include "ukf.h"

using Eigen::MatrixXd;
using Eigen::VectorXd;

int main() {

  // Create a UKF instance
  UKF ukf;

  /**
   * Programming assignment calls
   */
  VectorXd x_out = VectorXd(5);
  MatrixXd P_out = MatrixXd(5, 5);
  ukf.UpdateState(&x_out, &P_out);

  return 0;
}
#include <iostream>
#include "ukf.h"

using Eigen::MatrixXd;
using Eigen::VectorXd;

UKF::UKF() {
  Init();
}

UKF::~UKF() {
}

void UKF::Init() {

}

/**
 * Programming assignment functions: 
 */

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

  // calculate Kalman gain K;

  // update state mean and covariance matrix

  /**
   * 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;
}
#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

expected result x:

x =

5.92276

1.41823

2.15593

0.489274

0.321338

expected result P:

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

Next Concept