diff --git a/cpp/src/DO/Sara/KalmanFilter/DistributionConcepts.hpp b/cpp/src/DO/Sara/KalmanFilter/DistributionConcepts.hpp
new file mode 100644
index 000000000..43247c75d
--- /dev/null
+++ b/cpp/src/DO/Sara/KalmanFilter/DistributionConcepts.hpp
@@ -0,0 +1,56 @@
+#pragma once
+
+#include <DO/Sara/KalmanFilter/EigenMatrixConcepts.hpp>
+
+#include <concepts>
+#include <limits>
+
+
+namespace DO::Sara::KalmanFilter {
+
+  template <typename T>
+  concept GaussianDistribution = requires(T dist)
+  {
+    typename T::scalar_type;
+    typename T::mean_type;
+    typename T::covariance_matrix_type;
+    // clang-format off
+    // Constructor.
+    { T{typename T::mean_type{}, typename T::covariance_matrix_type{}} } -> std::same_as<T>;
+    // Methods.
+    { dist.mean() } -> std::same_as<const typename T::mean_type&>;
+    { dist.covariance_matrix() } -> std::same_as<const typename T::covariance_matrix_type&>;
+    // clang-format on
+  };
+
+  template <typename T>
+  concept ZeroMeanGaussianDistribution = requires(T dist)
+  {
+    typename T::scalar_type;
+    typename T::mean_type;
+    typename T::covariance_matrix_type;
+    // clang-format off
+    { T{typename T::covariance_matrix_type{}} } -> std::same_as<T>;
+    { dist.covariance_matrix() } -> std::same_as<const typename T::covariance_matrix_type&>;
+    // clang-format on
+  };
+
+  template <typename T>
+  concept StateDistribution = GaussianDistribution<T>;
+
+  template <typename T>
+  concept NoiseDistribution = ZeroMeanGaussianDistribution<T>;
+
+  template <typename T>
+  concept FixedSizeStateDistribution =  //
+      GaussianDistribution<T> &&        //
+      CompileTimeFixedMatrix<decltype(T{}.mean())> &&
+      CompileTimeFixedMatrix<decltype(T{}.covariance_matrix())>;
+
+  template <typename T>
+  concept FixedSizeNoiseDistribution =  //
+      ZeroMeanGaussianDistribution<T> &&
+      CompileTimeFixedMatrix<decltype(T{}.mean())> &&
+      CompileTimeFixedMatrix<decltype(T{}.covariance_matrix())>;
+
+}  // namespace DO::Sara::KalmanFilter
diff --git a/cpp/src/DO/Sara/KalmanFilter/EigenMatrixConcepts.hpp b/cpp/src/DO/Sara/KalmanFilter/EigenMatrixConcepts.hpp
new file mode 100644
index 000000000..6e8e9ab2f
--- /dev/null
+++ b/cpp/src/DO/Sara/KalmanFilter/EigenMatrixConcepts.hpp
@@ -0,0 +1,50 @@
+#pragma once
+
+#include <Eigen/Core>
+
+#include <concepts>
+
+
+namespace DO::Sara::KalmanFilter {
+
+  template <typename T>
+  concept EigenVector = requires(T)
+  {
+    typename T::Scalar;
+    // clang-format off
+    { T{}.rows() } -> std::same_as<Eigen::Index>;
+    { T{}.cols() } -> std::same_as<Eigen::Index>;
+    { T{}(int{}) } -> std::same_as<typename T::Scalar>;
+    // clang-format on
+  };
+
+  template <typename T>
+  concept EigenMatrix = requires(T)
+  {
+    typename T::Scalar;
+    // clang-format off
+    { T{}.rows() } -> std::same_as<Eigen::Index>;
+    { T{}.cols() } -> std::same_as<Eigen::Index>;
+    { T{}(int{}, int{}) } -> std::same_as<typename T::Scalar>;
+    { T{}.transpose() };
+    // clang-format on
+  };
+
+  template <typename T>
+  concept EigenSquareMatrix = EigenMatrix<T> && requires
+  {
+    T::Rows == T::Cols;
+  };
+
+  template <typename T>
+  concept CompileTimeFixedMatrix = requires
+  {
+    typename T::scalar_type;
+    // clang-format off
+    { T::Rows } -> std::same_as<int>;
+    { T::Cols } -> std::same_as<int>;
+    { T{} } -> std::same_as<Eigen::Matrix<typename T::scalar_type, T::Rows, T::Cols>>;
+    // clang-format on
+  };
+
+}  // namespace DO::Sara::KalmanFilter
diff --git a/cpp/src/DO/Sara/KalmanFilter/KalmanFilter.hpp b/cpp/src/DO/Sara/KalmanFilter/KalmanFilter.hpp
deleted file mode 100644
index c2a727862..000000000
--- a/cpp/src/DO/Sara/KalmanFilter/KalmanFilter.hpp
+++ /dev/null
@@ -1,26 +0,0 @@
-#pragma once
-
-
-namespace DO::Sara::KalmanFilter {
-
-  template <typename StateTransitionEquation, typename ObservationEquation>
-  struct KalmanFilter
-  {
-    using State = typename StateTransitionEquation::State;
-    using Observation = typename StateTransitionEquation::Observation;
-
-    auto predict(const State& x) -> State
-    {
-      return _state_transition_equation.predict(x);
-    }
-
-    auto update(const State& x_predicted, const Observation& z) -> State
-    {
-      return _observation_equation.update(x_predicted, z);
-    }
-
-    StateTransitionEquation _state_transition_equation;
-    ObservationEquation _observation_equation;
-  };
-
-}  // namespace DO::Sara::KalmanFilter
diff --git a/cpp/src/DO/Sara/KalmanFilter/ObservationEquation.hpp b/cpp/src/DO/Sara/KalmanFilter/ObservationEquation.hpp
new file mode 100644
index 000000000..bbd4cfd93
--- /dev/null
+++ b/cpp/src/DO/Sara/KalmanFilter/ObservationEquation.hpp
@@ -0,0 +1,71 @@
+#pragma once
+
+#include <DO/Sara/KalmanFilter/DistributionConcepts.hpp>
+
+
+namespace DO::Sara::KalmanFilter {
+
+  template <GaussianDistribution State,                     //
+            GaussianDistribution Observation,               //
+            ZeroMeanGaussianDistribution ObservationNoise,  //
+            EigenMatrix ObservationModelMatrix>
+  struct ObservationEquation
+  {
+    using T = typename ObservationModelMatrix::Scalar;
+    using Innovation = Observation;
+    using KalmanGain = typename Observation::covariance_matrix_type;
+
+    inline static auto observation_model_matrix() -> ObservationModelMatrix
+    {
+      auto H = ObservationModelMatrix{};
+
+      static const auto I = Eigen::Matrix4<T>::Identity();
+      static const auto O = Eigen::Matrix<T, 4, 8>::Zero();
+      H << I, O;
+
+      return H;
+    }
+
+    inline auto innovation(const State& x_a_priori,
+                           const Observation& z) const  //
+        -> Innovation
+    {
+      return {
+          z - H * x_a_priori,  //
+          H * x_a_priori.covariance_matrix() * H.transpose() +
+              v.covariance_matrix()  //
+      };
+    }
+
+    inline auto kalman_gain_matrix(const Observation& x_predicted,
+                                   const Innovation& S) const  //
+        -> KalmanGain
+    {
+      return x_predicted.covariance_matrix() * H.transpose() *
+             S.covariance_matrix().inverse();
+    }
+
+    inline auto update(const State& x_predicted, const Observation& z) -> State
+    {
+      const auto y = innovation(x_predicted, z);
+      const auto K = kalman_gain_matrix(x_predicted, y);
+
+      static const auto I = State::CovarianceMatrix::Identity();
+      return {
+          x_predicted.mean() + K * y.mean(),             //
+          (I - K * H) * x_predicted.covariance_matrix()  //
+      };
+    }
+
+    inline auto residual(const Observation& z,
+                         const State& x) const  //
+        -> typename Observation::mean_type
+    {
+      return z.mean() - H * x.covariance_matrix();
+    }
+
+    const ObservationModelMatrix H;
+    ObservationNoise v;
+  };
+
+}  // namespace DO::Sara::KalmanFilter
diff --git a/cpp/src/DO/Sara/KalmanFilter/StateTransitionModel.hpp b/cpp/src/DO/Sara/KalmanFilter/StateTransitionModel.hpp
new file mode 100644
index 000000000..cbe44afac
--- /dev/null
+++ b/cpp/src/DO/Sara/KalmanFilter/StateTransitionModel.hpp
@@ -0,0 +1,25 @@
+#pragma once
+
+#include <DO/Sara/KalmanFilter/DistributionConcepts.hpp>
+
+
+namespace DO::Sara::KalmanFilter {
+
+  template <EigenMatrix StateTransitionMatrix,
+            GaussianDistribution StateDistribution,
+            ZeroMeanGaussianDistribution ProcessNoiseDistribution>
+  struct StateTransitionEquation
+  {
+    auto predict(const StateDistribution& x) -> StateDistribution
+    {
+      return {
+          F * x.mean(),                                                      //
+          F * x.covariance_matrix() * F.transpose() + w.covariance_matrix()  //
+      };
+    };
+
+    StateTransitionMatrix F;
+    ProcessNoiseDistribution w;
+  };
+
+}  // namespace DO::Sara::KalmanFilter