Kalman Filter

The Kalman filter is a Bayesian filter that uses multivariate Gaussians, a recursive state estimator, a linear quadratic estimator (LQE), and an Infinite Impulse Response (IIR) filter. It is a control theory tool applicable to signal estimation, sensor fusion, or data assimilation problems. The filter is applicable for unimodal and uncorrelated uncertainties. The filter assumes white noise, propagation and measurement functions are differentiable, and that the uncertainty stays centered on the state estimate. The filter is the optimal linear filter under assumptions. The filter updates estimates by multiplying Gaussians rather than integrating differential equations. The filter predicts estimates by adding Gaussians. The filter maintains an estimate of the state and its uncertainty over the sequential estimation process. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory in 1960.

Designing a filter is as much art as science, with the following recipe. Model the real world in state-space notation. Then, compute and select the fundamental matrices, select the states X, P, the processes F, Q, the measurements Z, R, the measurement function H, and if the system has control inputs U, G. Evaluate the performance and iterate.

This library supports various simple and extended filters. The implementation is independent from linear algebra backends. Arbitrary parameters can be added to the prediction and update stages to participate in gain-scheduling or linear parameter varying (LPV) systems. The default filter type is a generalized, customizable, and extended filter. The default type parameters implement a one-state, one-output, and double-precision floating-point type filter. The default update equation uses the Joseph form. Examples illustrate various usages and implementation tradeoffs. A standard formatter specialization is included for representation of the filter states. Filters with state x output x input dimensions as 1x1x1 and 1x1x0 (no input) are supported through vanilla C++. Higher dimension filters require a linear algebra backend. Customization points and type injections allow for implementation tradeoffs.

Examples

1x1 Constant System Dynamic Model Filter

Example from the building height estimation sample. One estimated state and one observed output filter.

```cpp kalman filter{ state{60.}, output, estimateuncertainty{225.}, outputuncertainty{25.} };

filter.update(48.54); ```

full sample code

6x2 Constant Acceleration Dynamic Model Filter

Example from the 2-dimension vehicle location, velocity, and acceleration vehicle estimation sample. Six estimated states and two observed outputs filter.

```cpp kalman filter{ state{0., 0., 0., 0., 0., 0.}, output>, estimateuncertainty{{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.}}, processuncertainty{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.}}}, outputuncertainty{{9., 0.}, {0., 9.}}, outputmodel{{1., 0., 0., 0., 0., 0.}, {0., 0., 0., 1., 0., 0.}}, 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.}}};

filter.predict(); filter.update(-393.66, 300.4); ```

full sample code

4x1 Nonlinear Dynamic Model Extended Filter

Example from the thermal, current of warm air, strength, radius, and location estimation sample. Four estimated states and one observed output extended filter with two additional prediction arguments and two additional update arguments.

```cpp kalman filter{ state{triggerstrength, thermalradius, thermalpositionx, thermalpositiony}, output, estimateuncertainty{{strengthcovariance, 0.F, 0.F, 0.F}, {0.F, radiuscovariance, 0.F, 0.F}, {0.F, 0.F, positioncovariance, 0.F}, {0.F, 0.F, 0.F, positioncovariance}}, processuncertainty{{strengthnoise, 0.F, 0.F, 0.F}, {0.F, distancenoise, 0.F, 0.F}, {0.F, 0.F, distancenoise, 0.F}, {0.F, 0.F, 0.F, distancenoise}}, outputuncertainty{measurenoise}, outputmodel{ -> matrix<1, 4> { const float expon{std::exp(-(std::pow(x[2] - positionx, 2.F) + std::pow(x[3] - positiony, 2.F)) / std::pow(x[1], 2.F))}; const matrix<1, 4> h{ expon, 2 * x(0) * ((std::pow(x(2) - positionx, 2.F) + std::pow(x(3) - positiony, 2.F)) / std::pow(x(1), 3.F)) * expon, -2 * (x(0) * (x(2) - positionx) / std::pow(x(1), 2.F)) * expon, -2 * (x(0) * (x(3) - positiony) / std::pow(x(1), 2.F)) * expon}; return h; }}, transition{ -> vector<4> { const vector<4> drifts{0.F, 0.F, driftx, drifty}; return x + drifts; }}, observation{ -> float { return x(0) * std::exp(-(std::pow(x[2] - positionx, 2.F) + std::pow(x[3] - positiony, 2.F)) / std::pow(x[1], 2.F)); }}, updatetypes, prediction_types};

filter.predict(driftx, drifty); filter.update(positionx, positiony, variometer); ```

full sample code

Other Examples

• 1x1 constant system dynamic model filter of the temperature of a liquid in a tank.
• 1x1x1 constant velocity dynamic model filter of the 1-dimension position of a dog.
• 2x1x1 constant acceleration dynamic model filter of the 1-dimension position and velocity of a rocket altitude.
• 8x4 constant velocity dynamic model filter of the 2-dimension position and velocity of the center, aspect ratio, and height of a bounding box.

Installation

Example of installation commands in Shell:

shell git clone --depth 1 "https://github.com/FrancoisCarouge/kalman" cmake -S "kalman" -B "build" cmake --build "build" --parallel sudo cmake --install "build"

Another variation for your CMake infrastructure via fetch content:

```cmake include(FetchContent)

FetchContentDeclare( fcarouge-kalman GITREPOSITORY "https://github.com/FrancoisCarouge/kalman" FINDPACKAGEARGS NAMES fcarouge-kalman) FetchContent_MakeAvailable(fcarouge-kalman)

targetlinklibraries(your_target PRIVATE fcarouge-kalman::kalman) ```

For more, see installation instructions.

Reference

Class kalman

Also documented in the fcarouge/kalman.hpp header.

Declaration

cpp template <typename Filter> class kalman final : public kalman_internal::conditional_member_types<Filter>

Template Parameters

| Template Parameter | Definition | | --- | --- | | Filter | Exposition only. The deduced internal filter template parameter. Class template argument deduction (CTAD) figures out the filter type based on the declared configuration. See deduction guide. |

Member Types

| Member Type | Dimensions | Definition | Also Known As | | --- | --- | --- | --- | | estimate_uncertainty | x by x | Type of the estimated, hidden covariance matrix p. | P, Σ | | gain | x by z | Type of the gain matrix k. | K, L | | innovation_uncertainty | z by z | Type of the innovation uncertainty matrix s. | S | | innovation | z by 1 | Type of the innovation column vector y. | Y | | input_control | x by u | Type of the control transition matrix g. This member type is defined only if the filter supports input control. | G, B | | input | u by 1 | Type of the control column vector u. This member type is defined only if the filter supports input. | U | | output_model | z by x | Type of the observation transition matrix h. This member type is defined only if the filter supports output model. | H, C | | output_uncertainty | z by z | Type of the observation, measurement noise covariance matrix r. | R | | output | z by 1 | Type of the observation column vector z. | Z, Y, O | | process_uncertainty | x by x | Type of the process noise covariance matrix q. | Q | | state_transition | x by x | Type of the state transition matrix f. | F, Φ, A | | state | x by 1 | Type of the state estimate, hidden column vector x. | X |

The member types are optionally present according to the filter configuration.

Member Functions

| Member Function | Definition | | --- | --- | | (constructor) | Constructs a Kalman filter from its declared configuration. | | (move constructor) | Constructs the filter, default. | | (move assignment operator) | Assigns values to the filter, default. | | (destructor) | Destructs the filter. |

Characteristics

| Characteristic | Definition | | --- | --- | | f | Manages the state transition matrix F. Gets or sets the value. Configures the callable object of expression state_transition(const state &, const input &, const PredictionTypes &...) to compute the value. The default value is the matrix with all its diagonal elements equal to ones, and zeroes everywhere else. | | g | Manages the control transition matrix G. Gets or sets the value. Configures the callable object of expression input_control(const PredictionTypes &...) to compute the value. The default value is the matrix with all its diagonal elements equal to ones, and zeroes everywhere else. This member function is defined only if the filter supports input control. | | h | Manages the observation transition matrix H. Gets or sets the value. Configures the callable object of expression output_model(const state &, const UpdateTypes &...) to compute the value. The default value is the matrix with all its diagonal elements equal to ones, and zeroes everywhere else. This member function is defined only if the filter supports output model. | | k | Manages the gain matrix K. Gets the value last computed during the update. The default value is the matrix with all its diagonal elements equal to ones, and zeroes everywhere else. | | p | Manages the estimated covariance matrix P. Gets or sets the value. The default value is the matrix with all its diagonal elements equal to ones, and zeroes everywhere else. | | q | Manages the process noise covariance matrix Q from the process noise w expected value E[wwᵀ] and its variance σ² found by measuring, tuning, educated guesses of the noise. Gets or sets the value. Configures the callable object of expression process_uncertainty(const state &, const PredictionTypes &...) to compute the value. The default value is the null matrix. | | r | Manages the observation, measurement noise covariance matrix R from the measurement noise v expected value E[vvᵀ] and its variance σ² found by measuring, tuning, educated guesses of the noise. Gets or sets the value. Configures the callable object of expression output_uncertainty(const state &, const output &, const UpdateTypes &...) to compute the value. The default value is the null matrix. | | s | Manages the innovation uncertainty matrix S. Gets the value last computed during the update. The default value is the matrix with all its diagonal elements equal to ones, and zeroes everywhere else. | | u | Manages the control column vector U. Gets the value last used in prediction. This member function is defined only if the filter supports input. | | x | Manages the state estimate column vector X. Gets or sets the value. The default value is the null column vector. | | y | Manages the innovation column vector Y. Gets the value last computed during the update. The default value is the null column vector. | | z | Manages the observation column vector Z. Gets the value last used during the update. The default value is the null column vector. |

The characteristics are optionally present according to the filter configuration.

Modifiers

| Modifier | Definition | | --- | --- | | predict | Produces estimates of the state variables and uncertainties. | | update | Updates the estimates with the outcome of a measurement. |

Format

A specialization of the standard formatter is provided for the filter. Use std::format to store a formatted representation of all of the characteristics of the filter in a new string. Standard format parameters to be supported.

```cpp kalman filter;

std::println("{}", filter); // {"f": 1, "k": 1, "p": 1, "r": 0, "s": 1, "x": 0, "y": 0, "z": 0} // The characteristics are optionally present according to the filter configuration. ```

Considerations

Motivations

Kalman filters can be difficult to learn, use, and implement. Users often need fair algebra, domain, and software knowledge. Inadequacy leads to incorrectness, underperformance, and a big ball of mud.

This package explores what could be a Kalman filter implementation a la standard library. The following concerns are considered: - Separation of the application domain and integration needs. - Separation of the mathematical concepts and linear algebra implementation. - Generalization and specialization of modern language and library support.

Selected Tradeoffs

In theory there is no difference between theory and practice, while in practice there is. The following engineering tradeoffs have been selected for this library implementation:

• Update and prediction additional arguments are stored in the filter at the costs of memory and performance for the benefits of consistent data access and records.
• The default floating point data type for the filter is double with about 16 significant digits to reduce loss of information compared to float.
• The ergonomics and precision of the default filter takes precedence over performance.

Lessons Learned

Design, development, and testing uncovered unexpected facets of the projects:

• The filter's state, output, and input column vectors should be strongly typed parameters to allow the filter to participate in full compile-time safeties verification.
• There exist Kalman filters with hundreds of state variables.
• The float data type has about seven significant digits. Floating point error is a loss of information to account for in design.
• The units of useful matrices are factorizable, i.e. the unit of an element is expressed as the product of the row and column indexed units. The deduced units result type of a matrix product collapses and folds the inner indexed units merely returning the outer units.
• Safe physical linear algebra not only includes types and units safety, but also coordinate axes and frames reference.

Performance

The benchmarks share some performance information. Custom specializations and implementations can outperform this library. Custom optimizations may include: using a different covariance estimation update formula; removing symmetry support; using a different matrix inversion formula; removing unused or identity model dynamics supports; implementing a generated, unrolled filter algebra expressions; or running on accelerator hardware.

Resources

Definitions

| Term | Definition | | --- | --- | | EKF | The Extended Kalman Filter is the nonlinear version of the Kalman filter. Useful for nonlinear dynamics systems. This filter linearizes the model about an estimate working point of the current mean and covariance. | | ESKF | The Error State Kalman Filter is the error estimation version of the Kalman filter. Useful for linear error state dynamics systems. This filter estimates the errors rather than the states. | MSCKF | The Multi-State Constraint Kalman Filter is an EKF-based approach that leverages constraints between multiple and varied measurements to improve pose estimation accuracy and robustness. | | UKF | The Unscented Kalman Filter is the sampled version of the Extended Kalman Filter. Useful for highly nonlinear dynamics systems. This filter samples sigma points about an estimate working point of the current mean using an Unscented Transformation technique. |

Further terms should be defined and demonstrated for completeness: CKF, EKF-IMM, EnKF, Euler-KF, Fading-Memory, Finite/Fixed-Memory, Forward-Backward, FKF, IEKF, Joseph, KF, Linearized, MEKF, MRP-EKF, MRP-UKF, SKF, Smoother, UKF-GSF, UKF-IMM, USQUE, UDU, and UT.

Related Resources

• A New Approach to Linear Filtering and Prediction Problems by Kalman, Rudolph Emil in Transactions of the ASME - Journal of Basic Engineering, Volume 82, Series D, pp 35-45, 1960 - Transcription by John Lukesh.
• KalmanFilter.NET by Alex Becker.
• Kalman and Bayesian Filters in Python by Roger Labbe.
• How Kalman Filters Work by Tucker McClure of An Uncommon Lab.
• Wikipedia Kalman filter by Wikipedia, the free encyclopedia.
• Applications of Kalman Filtering in Aerospace 1960 to the Present by Mohinder S. Grewal and Angus P. Andrews. IEEE 2010.
• Taking Static Type-Safety to the Next Level - Physical Units for Matrices by Daniel Withopf at CppCon 2022 [slides].
• Units Libraries and Autonomous Vehicles: Lessons from the Trenches by Chip Hogg at CppCon 2021.

Projects

The library is used in projects:

• GstKalman: A GStreamer Kalman filter video plugin.

Your project link here!

Third Party Acknowledgement

The library is designed, developed, and tested with the help of third-party tools and services acknowledged and thanked here:

• actions-gh-pages to upload the documentation to GitHub pages.
• Clang for compilation and code sanitizers.
• CMake for build automation.
• cmakelang for pretty CMake list files.
• cppcheck for static analysis.
• Doxygen for documentation generation.
• Doxygen Awesome for pretty documentation.
• Eigen for linear algebra.
• GCC for compilation and code sanitizers.
• Google Benchmark to implement the benchmarks.
• lcov to process coverage information.
• mp-units the quantities and units library for C++.
• MSVC for compilation and code sanitizers.
• Valgrind to check for correct memory management.

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Continuous Integration & Deployment Actions

License

Kalman Filter is public domain:

This is free and unencumbered software released into the public domain.

Anyone is free to copy, modify, publish, use, compile, sell, or distribute this software, either in source code form or as a compiled binary, for any purpose, commercial or non-commercial, and by any means.

In jurisdictions that recognize copyright laws, the author or authors of this software dedicate any and all copyright interest in the software to the public domain. We make this dedication for the benefit of the public at large and to the detriment of our heirs and successors. We intend this dedication to be an overt act of relinquishment in perpetuity of all present and future rights to this software under copyright law.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

For more information, please refer to https://unlicense.org