Altimeter Constant Acceleration Kalman Filter

Author

Jack Johnston

Published

June 24, 2025

1 Overview

The objective of this filter is to estimate the vertical position, velocity, and acceleration of a rocket in real time using altimeter data. These estimates are critical for detecting key flight events such as launch, burnout, apogee, and landing.

The proposed filter operates under a one-dimensional (1D) model of rocket flight and assumes constant acceleration between discrete time steps. A discrete-time Kalman filter is employed to integrate sensor measurements and generate robust state estimates throughout the rocket’s flight.

The system is designed to work with a barometric altimeter, sampling at a variable rate, and accounting for zero mean Gaussian barometer sensor noise.

The filter is designed to run on a resource-constrained microcontroller as part of a rocket’s flight computer, enabling onboard, real-time processing. Its outputs can used to trigger events such as parachute deployment and to provide reliable telemetry for post-flight analysis.

By balancing simplicity with performance, this filter provides a lightweight yet effective solution for real-time rocket flight state estimation.

2 System Model

2.1 State Vector

The state vector, $\mathbf{x}$, contains the key states for which the Kalman Filter generates estimates.

\[\mathbf{x}_k = \begin{bmatrix}h \\ \dot{h}\\ \ddot{h} \end{bmatrix}\]

Where,

$h$ is altitude.
$\dot{h}$ is vertical velocity.
$\ddot{h}$ is vertical acceleration.

2.2 Process Model

This filter eploys the constant acceleration process model, meaning that in between measurement updates the acceleration is assumed to be constant.

The process model is derived from the kinematic equations of motion:

\[\frac{d}{dt} \begin{bmatrix}h(t) \\ \dot{h}(t)\\ \ddot{h}(t)\end{bmatrix} = \begin{bmatrix}\dot{h}(t) \\ \ddot{h}(t) \\ 0\end{bmatrix}\]

The system can be expressed in matrix form as a continuous-time linear differential equation:

\[ \frac{d \mathbf{x}(t)}{dt} = \mathbf{A} \mathbf{x}(t), \quad \text{where} \quad \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \]

To apply this model in a discrete-time Kalman Filter, it must be discretised over a timestep $\Delta t$, resulting in a state transition matrix $\mathbf{F}_k$. This discretisation is derived using the matrix exponential of the continuous-time system matrix $\mathbf{A}$:

\[ \mathbf{F}_k = e^{\mathbf{A} \Delta t} \]

For this system, $\mathbf{A}$ is a nilpotent matrix (i.e., $\mathbf{A}^3 = 0$), so the matrix exponential can be computed exactly using a truncated Taylor series:

\[ e^{\mathbf{A} \Delta t} = \mathbf{I} + \mathbf{A} \Delta t + \frac{1}{2!} \mathbf{A}^2 \Delta t^2 \]

Calculating the terms:

\[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \quad \mathbf{A}^2 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad \mathbf{A}^3 = \mathbf{0} \]

Substituting into the series:

\[ \mathbf{F}_k = \mathbf{I} + \mathbf{A} \Delta t + \frac{1}{2} \mathbf{A}^2 \Delta t^2 = \begin{bmatrix} 1 & \Delta t & \frac{1}{2} \Delta t^2 \\ 0 & 1 & \Delta t \\ 0 & 0 & 1 \end{bmatrix} \]

Thus, the discrete-time process model becomes:

\[ \mathbf{x}_k = \mathbf{F}_k \mathbf{x}_{k-1} + \mathbf{w}_{k-1} \]

Where $\mathbf{w}_{k-1} \sim \mathcal{N}(0, \mathbf{Q}_k)$ is the process noise, which models uncertainty in the acceleration dynamics.

In the discrete-time process model:

\[ \mathbf{x}_k = \mathbf{F}_k \mathbf{x}_{k-1} + \mathbf{w}_{k-1} \]

the process noise $\mathbf{w}_{k-1} \sim \mathcal{N}(0, \mathbf{Q}_k)$ accounts for uncertainty in the system dynamics - primarily unmodelled forces or variability in acceleration.

We assume the noise enters through the acceleration term only, meaning the system is affected by a continuous-time white noise acceleration input with variance $\sigma_a^2$. This leads to a discrete-time process noise covariance matrix $\mathbf{Q}_k$ derived using the following:

\[ \mathbf{Q}_k = \int_0^{\Delta t} e^{\mathbf{A} \tau} \mathbf{G} \mathbf{Q}_c \mathbf{G}^T e^{\mathbf{A}^T \tau} d\tau \]

Where: - $\mathbf{A}$ is the continuous-time system matrix, - $\mathbf{G}$ is the noise input matrix, - $\mathbf{Q}_c = \sigma_a^2$ is the continuous-time noise intensity (scalar), - $\Delta t$ is the time step.

For a system with noise entering only through acceleration:

\[ \mathbf{G} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \]

Following the derivation (or by using known results for a constant-acceleration model), the resulting discrete-time process noise covariance matrix is:

\[ \mathbf{Q}_k = \sigma_a^2 \begin{bmatrix} \frac{1}{4} \Delta t^4 & \frac{1}{2} \Delta t^3 & \frac{1}{2} \Delta t^2 \\ \frac{1}{2} \Delta t^3 & \Delta t^2 & \Delta t \\ \frac{1}{2} \Delta t^2 & \Delta t & 1 \end{bmatrix} \]

This matrix captures how acceleration noise propagates into uncertainty in position and velocity over time.

2.3 Measurement Model

Single altimeter measurement (position only)
Altimeter bias removed using a tare procedure before flight

3 Filter Structure

3.1 Initialization

Zero initial state vector
Identity covariance matrix
Altimeter noise and process noise as tunable parameters
Tare buffer logic to determine launch altitude

3.2 Prediction Step

Computes time step dt from system time
Applies constant-acceleration state transition
Updates state estimate and error covariance
Uses time-dependent process noise matrix

3.3 Update Step

Waits until the tare phase completes before accepting updates
Adjusts altimeter reading relative to launch altitude
Updates estimate using Kalman gain
Uses the Joseph form to update the covariance matrix for numerical stability

4 Tare Mechanism

Collects a fixed number of samples before launch
Computes average to establish the zero-altitude reference (ASL)
Filters are only applied after tare is complete

5 Parameters and Tuning

altimeter_noise_std: defines measurement noise
process_noise_cov_acc: variance for acceleration noise
tare_sample_count: number of samples used for tare averaging

6 Example Usage

Description of how to integrate into a loop or system
Requirements for time input and measurement format
Notes on synchronization and timekeeping

7 Limitations

Barometric readings
Vent hole sizing
Supersonic effects

8 Future Improvements

Fuse additional sensors (e.g., IMU) using an Extended Kalman Filter
Automatically detect launch rather than using a fixed tare sample count
Add support for non-uniform sampling or robust noise adaptation

9 Appendix

9.1 Class Summary

AltimeterCAKF: main filter class
- predict(time)
- update(measurement)
- get_process_noise_matrix(dt)

9.2 Code Snippet
Link or reference to full implementation file (e.g., GitHub repo or altimeter_cakf.py)

--- title: "Altimeter Constant Acceleration Kalman Filter" author: "Jack Johnston" date: today format: html: toc: true toc-depth: 3 number-sections: true code-fold: true code-tools: true theme: cosmo # Or another clean theme bibliography: ../references.bib # csl: ../ieee.csl # Optional if you want specific citation styling freeze: auto --- ## Overview The objective of this filter is to estimate the vertical position, velocity, and acceleration of a rocket in real time using altimeter data. These estimates are critical for detecting key flight events such as launch, burnout, apogee, and landing. The proposed filter operates under a one-dimensional (1D) model of rocket flight and assumes constant acceleration between discrete time steps. A discrete-time Kalman filter is employed to integrate sensor measurements and generate robust state estimates throughout the rocket’s flight. The system is designed to work with a barometric altimeter, sampling at a variable rate, and accounting for zero mean Gaussian barometer sensor noise. The filter is designed to run on a resource-constrained microcontroller as part of a rocket's flight computer, enabling onboard, real-time processing. Its outputs can used to trigger events such as parachute deployment and to provide reliable telemetry for post-flight analysis. By balancing simplicity with performance, this filter provides a lightweight yet effective solution for real-time rocket flight state estimation. ## System Model ### State Vector The state vector, $\mathbf{x}$, contains the key states for which the Kalman Filter generates estimates. $$\mathbf{x}_k = \begin{bmatrix}h \\ \dot{h}\\ \ddot{h} \end{bmatrix}$$ Where, - $h$ is altitude. - $\dot{h}$ is vertical velocity. - $\ddot{h}$ is vertical acceleration. ### Process Model This filter eploys the constant acceleration process model, meaning that in between measurement updates the acceleration is assumed to be constant. The process model is derived from the kinematic equations of motion: $$\frac{d}{dt} \begin{bmatrix}h(t) \\ \dot{h}(t)\\ \ddot{h}(t)\end{bmatrix} = \begin{bmatrix}\dot{h}(t) \\ \ddot{h}(t) \\ 0\end{bmatrix}$$ The system can be expressed in matrix form as a continuous-time linear differential equation: $$ \frac{d \mathbf{x}(t)}{dt} = \mathbf{A} \mathbf{x}(t), \quad \text{where} \quad \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} $$ To apply this model in a discrete-time Kalman Filter, it must be discretised over a timestep $\Delta t$, resulting in a state transition matrix $\mathbf{F}_k$. This discretisation is derived using the **matrix exponential** of the continuous-time system matrix $\mathbf{A}$: $$ \mathbf{F}_k = e^{\mathbf{A} \Delta t} $$ For this system, $\mathbf{A}$ is a **nilpotent matrix** (i.e., $\mathbf{A}^3 = 0$), so the matrix exponential can be computed exactly using a truncated Taylor series: $$ e^{\mathbf{A} \Delta t} = \mathbf{I} + \mathbf{A} \Delta t + \frac{1}{2!} \mathbf{A}^2 \Delta t^2 $$ Calculating the terms: $$ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \quad \mathbf{A}^2 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad \mathbf{A}^3 = \mathbf{0} $$ Substituting into the series: $$ \mathbf{F}_k = \mathbf{I} + \mathbf{A} \Delta t + \frac{1}{2} \mathbf{A}^2 \Delta t^2 = \begin{bmatrix} 1 & \Delta t & \frac{1}{2} \Delta t^2 \\ 0 & 1 & \Delta t \\ 0 & 0 & 1 \end{bmatrix} $$ Thus, the discrete-time process model becomes: $$ \mathbf{x}_k = \mathbf{F}_k \mathbf{x}_{k-1} + \mathbf{w}_{k-1} $$ Where $\mathbf{w}_{k-1} \sim \mathcal{N}(0, \mathbf{Q}_k)$ is the process noise, which models uncertainty in the acceleration dynamics. In the discrete-time process model: $$ \mathbf{x}_k = \mathbf{F}_k \mathbf{x}_{k-1} + \mathbf{w}_{k-1} $$ the process noise $\mathbf{w}_{k-1} \sim \mathcal{N}(0, \mathbf{Q}_k)$ accounts for uncertainty in the system dynamics - primarily unmodelled forces or variability in acceleration. We assume the noise enters through the acceleration term only, meaning the system is affected by a continuous-time white noise acceleration input with variance $\sigma_a^2$. This leads to a **discrete-time process noise covariance matrix** $\mathbf{Q}_k$ derived using the following: $$ \mathbf{Q}_k = \int_0^{\Delta t} e^{\mathbf{A} \tau} \mathbf{G} \mathbf{Q}_c \mathbf{G}^T e^{\mathbf{A}^T \tau} d\tau $$ Where: - $\mathbf{A}$ is the continuous-time system matrix, - $\mathbf{G}$ is the noise input matrix, - $\mathbf{Q}_c = \sigma_a^2$ is the continuous-time noise intensity (scalar), - $\Delta t$ is the time step. For a system with noise entering only through acceleration: $$ \mathbf{G} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} $$ Following the derivation (or by using known results for a constant-acceleration model), the resulting discrete-time process noise covariance matrix is: $$ \mathbf{Q}_k = \sigma_a^2 \begin{bmatrix} \frac{1}{4} \Delta t^4 & \frac{1}{2} \Delta t^3 & \frac{1}{2} \Delta t^2 \\ \frac{1}{2} \Delta t^3 & \Delta t^2 & \Delta t \\ \frac{1}{2} \Delta t^2 & \Delta t & 1 \end{bmatrix} $$ This matrix captures how acceleration noise propagates into uncertainty in position and velocity over time. ### Measurement Model - Single altimeter measurement (position only) - Altimeter bias removed using a tare procedure before flight ## Filter Structure ### Initialization - Zero initial state vector - Identity covariance matrix - Altimeter noise and process noise as tunable parameters - Tare buffer logic to determine launch altitude ### Prediction Step - Computes time step `dt` from system time - Applies constant-acceleration state transition - Updates state estimate and error covariance - Uses time-dependent process noise matrix ### Update Step - Waits until the tare phase completes before accepting updates - Adjusts altimeter reading relative to launch altitude - Updates estimate using Kalman gain - Uses the Joseph form to update the covariance matrix for numerical stability ## Tare Mechanism - Collects a fixed number of samples before launch - Computes average to establish the zero-altitude reference (ASL) - Filters are only applied after tare is complete ## Parameters and Tuning - `altimeter_noise_std`: defines measurement noise - `process_noise_cov_acc`: variance for acceleration noise - `tare_sample_count`: number of samples used for tare averaging ## Example Usage - Description of how to integrate into a loop or system - Requirements for time input and measurement format - Notes on synchronization and timekeeping ## Limitations - Barometric readings - Vent hole sizing - Supersonic effects ## Future Improvements - Fuse additional sensors (e.g., IMU) using an Extended Kalman Filter - Automatically detect launch rather than using a fixed tare sample count - Add support for non-uniform sampling or robust noise adaptation ## Appendix ### Class Summary - `AltimeterCAKF`: main filter class - `predict(time)` - `update(measurement)` - `get_process_noise_matrix(dt)` ### Code Snippet - Link or reference to full implementation file (e.g., GitHub repo or `altimeter_cakf.py`)