<< Chapter < Page Chapter >> Page >
Explanation of the Kalman Filter implementation for digital rocket apogee detection.

Kalman Filter

The Kalman filter is a time domain method of incorporating knowledge of the physical model of the system and of the reliability of the sensors to accurately estimate the state of the system. Implementation of the Kalman filter first requires the creation of an accurate physical model of the system. The two equations which are used to determine the estimate of the current state from that of the previous state are:

x k Ax k 1
x k x k K m H x k
x s v a

Near apogee, the physical equations governing the rocket’s flight are simple, which makes A simple. The only force acting on the rocket is gravity only (because drag forces vary with the square of the velocity they can be neglected near apogee, where the velocity is close to zero).

s v t 1 2 a t 2
v a t
a g
1 Δt Δt 2 2 0 1 Δt 0 0 1

Where ∆t is the time between x k and x k+1 .

m is a vector of the measured values from the sensors. Position is measured with the barometer and acceleration by the accelerometer.

s m a m

H is a matrix which maps x k to m:

1 0 0 0 0 1

Finally, K , the Kalman gain matrix, weights the difference between the measured values and the estimated values. K is typically computed in real time as the system changes. However, the formula for K is rather complicated and therefore difficult to implement on a microcontroller in real time. Luckily, because the rocket’s flight can be approximated over the whole flight by the system and because the sensor variances do not change, K can be precomputed via the following recursive process:

K P H T H P H T R 1
P I K P P
P A P A T Q

In a small number of repetitions, K will converge. In these equations, R is the measurement noise covariance matrix which holds the variances for each sensor:

σ p 2 0 0 σ a 2

P is called the error covariance matrix, and it is first approximated with a guess, and then recursively defined like the K matrix. Finally, Q is the process noise covariance matrix, and is associated with the amount of noise added to the estimate in each time step. The code for calculating the K matrix is shown below:

.% Calculates the Kalman gain H = [1 0 0; 0 0 1]; % maps x (state variables) to z (sensor data) R = [35.8229 0; 0 .0012]; % measurement noise covariance Q = [0 0 0; 0 0 0; 0 0 1]; % process noise covariance matrix T = .05; % time stepA = [1 T 1/2 * T^2; 0 1 T; 0 0 1]; % maps previous state to next state% these three equations recursively define k (matrix of kalman gains) % and P (error covariance matrix)P = eye(3); % initial guess for p for i = 1:20K = P*H'/(H*P*H' + R); % Kalman gainsP = (eye(3) - K *H)*P; P = A*P*A' + Q;end display(K)display(H) display(P)

The last piece of code demonstrates the actual implementation of the Kalman filter in Matlab.

.% implements Kalman filter on altitude and accelerometer data. Required vectors are alt and accel, which are vectors cointaining the altitude and accelerometer data at times corresponding to the time vector t. t = .05:.05:15;estimate = zeros(3,length(t)); estimate(:,1) = [alt(1); 0; accel(1)]; for i = 2:length(t)estimate(:,i) = A*estimate(:,i-1); estimate(:,i) = estimate(:,i) + K*([alt(i);accel(i)]- H *estimate(:,i));end

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Digital detection of rocket apogee. OpenStax CNX. Dec 18, 2013 Download for free at http://cnx.org/content/col11599/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital detection of rocket apogee' conversation and receive update notifications?

Ask