Kalman 0.4.0
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 statespace 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 gainscheduling or linear parameter varying (LPV) systems. The default filter type is a generalized, customizable, and extended filter. The default type parameters implement a onestate, oneoutput, and doubleprecision floatingpoint 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.
Example from the building height estimation sample. One estimated state and one observed output filter.
Example from the 2dimension vehicle location, velocity, and acceleration vehicle estimation sample. Six estimated states and two observed outputs 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.
Example of installation commands in Shell:
Another variation for your CMake infrastructure via fetch content:
For more, see installation instructions.
Also documented in the fcarouge/kalman.hpp header.
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 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 Function  Definition 

(constructor)  Constructs the filter. Configures the filter the deduction guides. 
(move constructor)  Constructs the filter, default. 
(move assignment operator)  Assigns values to the filter, default. 
(destructor)  Destructs the filter. 
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 identity matrix. 
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 identity matrix. 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 identity matrix. 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 identity matrix. 
p  Manages the estimated covariance matrix P. Gets or sets the value. The default value is the identity matrix. 
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 identity matrix. 
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.
Modifier  Definition 

predict  Produces estimates of the state variables and uncertainties. 
update  Updates the estimates with the outcome of a measurement. 
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.
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:
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:
double
with about 16 significant digits to reduce loss of information compared to float
.Design, development, and testing uncovered unexpected facets of the projects:
float
data type has about seven significant digits. Floating point error is a loss of information to account for in design.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.
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. 
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, EKFIMM, EnKF, EulerKF, FadingMemory, Finite/FixedMemory, ForwardBackward, FKF, IEKF, Joseph, KF, Linearized, MEKF, MRPEKF, MRPUKF, MSCKF, SKF, Smoother, UKFGSF, UKFIMM, USQUE, UDU, and UT.
Resources to learn about Kalman filters:
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