kf_6x2x0_2d_vehicle_position.cpp

Estimating the vehicle location.

Estimating the vehicle location.

Copyright

This example is transcribed from KalmanFilter.NET copyright Alex Becker.

See also

https://www.kalmanfilter.net/multiExamples.html#ex9

In this example, we would like to estimate the location of the vehicle in the XY plane. The vehicle has an onboard location sensor that reports X and Y coordinates of the system. We assume constant acceleration dynamics. In this example we don't have a control variable u since we don't have control input. Let us assume a vehicle moving in a straight line in the X direction with a constant velocity. After traveling 400 meters the vehicle turns right, with a turning radius of 300 meters. During the turning maneuver, the vehicle experiences acceleration due to the circular motion (an angular acceleration). The measurements period: Δt = 1s (constant).

The initial guess, at the origin, lies some 500 m from the vehicle, far beyond its 22 m standard deviation: the first innovations are inconsistent until the estimate converges. The turn, from the seventeenth measurement, surprises the constant acceleration model again. The acceleration is never measured, yet through the turn its estimate tracks the centripetal acceleration v²/R of a 25 m/s vehicle on a 300 m radius.

/*  __          _      __  __          _   _
| |/ /    /\   | |    |  \/  |   /\   | \ | |
| ' /    /  \  | |    | \  / |  /  \  |  \| |
|  <    / /\ \ | |    | |\/| | / /\ \ | . ` |
| . \  / ____ \| |____| |  | |/ ____ \| |\  |
|_|\_\/_/    \_\______|_|  |_/_/    \_\_| \_|
 
Kalman Filter
Version 0.5.4
https://github.com/FrancoisCarouge/Kalman
 
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#include "fcarouge/kalman.hpp"
#include "fcarouge/linalg.hpp"
 
#include <cassert>
#include <cmath>
 
namespace fcarouge::sample {
namespace {
template <auto Size> using vector = column_vector<double, Size>;
template <auto Row, auto Column> using matrix = matrix<double, Row, Column>;
using state = fcarouge::state<vector<6>>;
 
[[maybe_unused]] const auto sample{[] -> int {
  // A 6x2x0 filter, constant acceleration dynamic model, no control.
  kalman filter{
      // The state X is chosen to be the position, velocity, acceleration in the
      // XY plane: [px, vx, ax, py, vy, ay]. We don't know the vehicle location;
      // we will set initial position, velocity and acceleration to 0.
      state{0., 0., 0., 0., 0., 0.},
      // The vehicle has an onboard location sensor that reports output Z as X
      // and Y coordinates of the system.
      output<vector<2>>,
      // The estimate uncertainty matrix P.
      // Since our initial state vector is a guess, we will set a very high
      // estimate uncertainty. The high estimate uncertainty results in a high
      // Kalman Gain, giving a high weight to the measurement.
      estimate_uncertainty{{500., 0., 0., 0., 0., 0.},
                           {0., 500., 0., 0., 0., 0.},
                           {0., 0., 500., 0., 0., 0.},
                           {0., 0., 0., 500., 0., 0.},
                           {0., 0., 0., 0., 500., 0.},
                           {0., 0., 0., 0., 0., 500.}},
      // The process uncertainty noise matrix Q, constant, computed in place,
      // with  random acceleration standard deviation: σa = 0.2 m.s^-2.
      process_uncertainty{0.2 * 0.2 *
                          matrix<6, 6>{{0.25, 0.5, 0.5, 0., 0., 0.},
                                       {0.5, 1., 1., 0., 0., 0.},
                                       {0.5, 1., 1., 0., 0., 0.},
                                       {0., 0., 0., 0.25, 0.5, 0.5},
                                       {0., 0., 0., 0.5, 1., 1.},
                                       {0., 0., 0., 0.5, 1., 1.}}},
      // The output uncertainty matrix R. Assume that the x and y measurements
      // are uncorrelated, i.e. error in the x coordinate measurement doesn't
      // depend on the error in the y coordinate measurement. In real-life
      // applications, the measurement uncertainty can differ between
      // measurements. In many systems the measurement uncertainty depends on
      // the measurement SNR (signal-to-noise ratio), angle between sensor (or
      // sensors) and target, signal frequency and many other parameters.
      // For the sake of the example simplicity, we will assume a constant
      // measurement uncertainty: R1 = R2...Rn-1 = Rn = R The measurement error
      // standard deviation: σxm = σym = 3m. The variance 9.
      output_uncertainty{{9., 0.}, {0., 9.}},
      // The output model matrix H. The dimension of zn is 2x1 and the dimension
      // of xn is 6x1. Therefore the dimension of the observation matrix H shall
      // be 2x6.
      output_model{{1., 0., 0., 0., 0., 0.}, {0., 0., 0., 1., 0., 0.}},
      // The state transition matrix F would be:
      state_transition{{1., 1., 0.5, 0., 0., 0.},
                       {0., 1., 1., 0., 0., 0.},
                       {0., 0., 1., 0., 0., 0.},
                       {0., 0., 0., 1., 1., 0.5},
                       {0., 0., 0., 0., 1., 1.},
                       {0., 0., 0., 0., 0., 1.}}};
 
  // Now we can predict the next state based on the initialization values.
  filter.predict();
 
  // The measurement values: z1 = [-393.66 m, 300.4 m]
  filter.update(-393.66, 300.4);
  filter.predict();
 
  // And so on, run a step of the filter, predicting and updating, every
  // measurements period: Δt = 1s (constant, built-in).
  const auto step{[&filter](double position_x, double position_y) -> void {
    filter.update(position_x, position_y);
    filter.predict();
  }};
 
  step(-375.93, 301.78);
 
  // Verify the example estimated state at 0.1% accuracy.
  assert(std::abs(1 - (filter.x()[0] / -277.8)) < 0.001 &&
         std::abs(1 - (filter.x()[1] / 148.3)) < 0.001 &&
         std::abs(1 - (filter.x()[2] / 94.5)) < 0.001 &&
         std::abs(1 - (filter.x()[3] / 249.8)) < 0.001 &&
         std::abs(1 - (filter.x()[4] / -85.9)) < 0.001 &&
         std::abs(1 - (filter.x()[5] / -63.6)) < 0.001 &&
         "The state estimates expected at 0.1% accuracy.");
 
  step(-351.04, 295.1);
  step(-328.96, 305.19);
  step(-299.35, 301.06);
  step(-273.36, 302.05);
  step(-245.89, 300);
  step(-222.58, 303.57);
  step(-198.03, 296.33);
  step(-174.17, 297.65);
  step(-146.32, 297.41);
  step(-123.72, 299.61);
  step(-103.47, 299.6);
  step(-78.23, 302.39);
  step(-52.63, 295.04);
  step(-23.34, 300.09);
  step(25.96, 294.72);
  step(49.72, 298.61);
  step(76.94, 294.64);
  step(95.38, 284.88);
  step(119.83, 272.82);
  step(144.01, 264.93);
  step(161.84, 251.46);
  step(180.56, 241.27);
  step(201.42, 222.98);
  step(222.62, 203.73);
  step(239.4, 184.1);
  step(252.51, 166.12);
  step(266.26, 138.71);
  step(271.75, 119.71);
  step(277.4, 100.41);
  step(294.12, 79.76);
  step(301.23, 50.62);
  step(291.8, 32.99);
  step(299.89, 2.14);
 
  assert(std::abs(1 - (filter.x()[0] / 298.5)) < 0.006 &&
         std::abs(1 - (filter.x()[1] / -1.65)) < 0.006 &&
         std::abs(1 - (filter.x()[2] / -1.9)) < 0.006 &&
         std::abs(1 - (filter.x()[3] / -22.5)) < 0.006 &&
         std::abs(1 - (filter.x()[4] / -26.1)) < 0.006 &&
         std::abs(1 - (filter.x()[5] / -0.64)) < 0.006 &&
         "The state estimates expected at 0.6% accuracy.");
  assert(std::abs(1 - (filter.p()(0, 0) / 11.25)) < 0.001 &&
         std::abs(1 - (filter.p()(0, 1) / 4.5)) < 0.001 &&
         std::abs(1 - (filter.p()(0, 2) / 0.9)) < 0.001 &&
         std::abs(1 - (filter.p()(1, 1) / 2.4)) < 0.001 &&
         std::abs(1 - (filter.p()(2, 2) / 0.2)) < 0.001 &&
         std::abs(1 - (filter.p()(3, 3) / 11.25)) < 0.001 &&
         "The estimate uncertainty expected at 0.1% accuracy."
         "At this point, the position uncertainty px = py = 5, which means "
         "that the standard deviation of the prediction is square root of 5m.");
 
  // As you can see, the Kalman Filter tracks the vehicle quite well. However,
  // when the vehicle starts the turning maneuver, the estimates are not so
  // accurate. After a while, the Kalman Filter accuracy improves. While the
  // vehicle travels along the straight line, the acceleration is constant and
  // equal to zero. However, during the turn maneuver, the vehicle experiences
  // acceleration due to the circular motion - the angular acceleration.
  // Although the angular acceleration is constant, the angular acceleration
  // projection on the x and y axes is not constant, therefore ax and ay are not
  // constant. Our Kalman Filter is designed for a constant acceleration model.
  // Nevertheless, it succeeds in tracking maneuvering vehicle due to a properly
  // chosen σa parameter. I would like to encourage the readers to implement
  // this example in software and see how different values of σa of R influence
  // the actual Kalman Filter accuracy, Kalman Gain convergence, and estimation
  // uncertainty.
 
  return 0;
}()};
} // namespace
} // namespace fcarouge::sample