24. Predicted Mean and Covariance Assignment 2

Solution

The first part of my solution is setting a vector of weights as we discussed in the last video.

The predicted state mean is just the implementation of the corresponding equation.

When calculating the predicted state covariance matrix, I did something you might not have done. In the equation we always need the difference between the mean predicted state and a sigma points. The problem here is that the state contains an angle. As you have learned before, subtracting angles is a problem for Kalman filters, because the result might be 2\pi plus a small angle, instead of just a small angle. That’s why I normalize the angle here.

Make sure you always normalize when you calculate the difference between angles.

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_pred = VectorXd(5);
  MatrixXd P_pred = MatrixXd(5, 5);
  ukf.PredictMeanAndCovariance(&x_pred, &P_pred);

  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::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
  double weight_0 = lambda/(lambda+n_aug);
  weights(0) = weight_0;
  for (int i=1; i<2*n_aug+1; ++i) {  // 2n+1 weights
    double weight = 0.5/(n_aug+lambda);
    weights(i) = weight;
  }

  // predicted 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);
  }

  // predicted 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;
}
#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 =

5.93637

1.49035

2.20528

0.536853

0.353577

expected result p:

P =

0.00543425 -0.0024053 0.00341576 -0.00348196 -0.00299378

-0.0024053 0.010845 0.0014923 0.00980182 0.00791091

0.00341576 0.0014923 0.00580129 0.000778632 0.000792973

-0.00348196 0.00980182 0.000778632 0.0119238 0.0112491

-0.00299378 0.00791091 0.000792973 0.0112491 0.0126972

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