text: #include "kalman_filter.h"
  #include <math.h>
  using Eigen::MatrixXd;
  using Eigen::VectorXd;
  
  /* 
   * Please note that the Eigen library does not initialize 
   *   VectorXd or MatrixXd objects with zeros upon creation.
   */
  
  KalmanFilter::KalmanFilter() {}
  
  KalmanFilter::~KalmanFilter() {}
  
  void KalmanFilter::Init(VectorXd &x_in, MatrixXd &P_in, MatrixXd &F_in,
                          MatrixXd &H_in, MatrixXd &R_in, MatrixXd &Q_in) {
    x_ = x_in;
    P_ = P_in;
    F_ = F_in;
    H_ = H_in;
    R_ = R_in;
    Q_ = Q_in;
  }
  
  void KalmanFilter::Predict() {
    /**
     * TODO: predict the state
     */  
    x_ = F_ * x_;
    MatrixXd Ft = F_.transpose();
    P_ = F_ * P_ * Ft + Q_;
    
  }
  
  // set a new function with the common equations to avoid redundant code
  void KalmanFilter::Update_Using_Error(const VectorXd &y){
    
    MatrixXd Ht = H_.transpose();
    MatrixXd S = H_ * P_ * Ht + R_;
    MatrixXd Si = S.inverse();
    MatrixXd PHt = P_ * Ht;
    MatrixXd K = PHt * Si;
  
    //new estimate
    x_ = x_ + (K * y);
    long x_size = x_.size();
    MatrixXd I = MatrixXd::Identity(x_size, x_size);
    P_ = (I - K * H_) * P_;
  }
  
  void KalmanFilter::Update(const VectorXd &z) {
    /**
     * TODO: update the state by using Kalman Filter equations
     */
    VectorXd z_pred = H_ * x_;
    VectorXd y = z - z_pred;
    Update_Using_Error(y); 
  
  }
  
  void KalmanFilter::UpdateEKF(const VectorXd &z) {
    /**
     * TODO: update the state by using Extended Kalman Filter equations
     */
    
    // To calculate y, we use the equations that map the predicted location x' from Cartesian coordinates to polar coordinates.
    
    double px = x_(0);
    double py = x_(1);
    double vx = x_(2);
    double vy = x_(3);
  
    double rho = sqrt(px*px + py*py);
    double phi = atan2(py, px);
    double rho_dot ;
  
    // check to avoid dividing by rho = 0;
    if (fabs(rho) < 0.0001) { rho_dot = 0; }
    else { rho_dot = (px*vx + py*vy) / rho; }
  
    VectorXd hx = VectorXd(3);
  
    hx << rho, phi, rho_dot;
  
    VectorXd y = z - hx;
    Update_Using_Error(y);
    
  }
  
  
  