Hi Dave,
I have a tiny bit of experience with Kalman filtering for state
estimation, and attached a toy example that might help get you
started. I have written it to work with Octave which is open source so
you can install it if you don't already have it. The idea as I
understand it is that you have a model of how the state updates, like
Newton's laws. You predict what you'll measure, compare that to your
actual measurement and the apply the Kalman gain matrix K to the
difference to update your estimate of the state.
You can run the example in Octave with
octave:1> kalman_example
In the example, I have an object moving in a parabolic arc, and in a
commeted-out line, an object moving in a straight line. The
measurements that are the input the filter are stored in y. You can
plot the measurements out with
plot(y(1,:), y(2,:), 'o')
At the end of the script the estimate state positions are plotted with
plot(x_hat_plus(1,:), x_hat_plus(2,:), 'o')
The tricky parts seem to be what sort values to put in the initial P,
Q, and R covariance matrices.
Hope this helps.
Cheers,
Keith
On Sun, Aug 21, 2011 at 12:08 PM, <kth.cur...@yahoo.com> wrote:
>
>
> ---------- Forwarded message ----------
> Date: Wed, 17 Aug 2011 16:18:51 -0400
> From: Dave <e...@dc9.tzo.com>
> Reply-To: "Enhanced Machine Controller (EMC)"
> <emc-users@lists.sourceforge.net>
> To: emc-users@lists.sourceforge.net
> Subject: [Emc-users] OT ? Kalman Filter
>
>
> I have an application that I believe could greatly benefit from a Kalman
> Filter. Kalman filters are oftentimes used for guidance systems and
> that is what I need it for - except that I am guiding a boat!
> However the same filter I understand can also be used to predict
> (estimate) velocity and position based upon what would normally be
> considered incomplete information. One example I heard what that a
> Kalman filter can
> be used to predict velocity when used with low count encoders - as in
> low count encoders used on servo motors.
>
> If you do a look up of "Kalman filter" on Youtube you can get an idea of
> their flexibility.
>
> Does anyone have any experience with Kalman filters? I have searched
> the web for a simple explanation and John Crenshaw of embedded magazine
> started a series of articles in the March 2011 edition (sort of a
> Kalman filter for dummies) but he stopped short of publishing the second
> article which gets into the actual filter software explanation.
>
> I found a "primer" on the web that I downloaded. It is a PDF document
> that is 189 pages long!
>
> Where do I start with this??
>
> I get the feeling that if I just grab some source code and implement
> something it won't work since the filter has to be "setup" to allow the
> software to properly predict or estimate a value.
>
> Thanks,
>
> Dave
>
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### example to filter location measurements of a moving object
### build measurement data to filter
delta_t = 0.1; # 1 second updates
# x state vector x-position, y-position, x-velocity, y-velocity
N = 1000;
t = (1:N)*delta_t;
# parabolic arc
x = [ 10*t ; 15*t - 3*t.^2 ; 10 + 0*t; 15 - 6*t];
#straight line
#x = [ 10*t ; -15*t ; 10 + 0*t; 15 + 0*t];
# H is measurement matrix, only measures position for this example
H = [ 1 0 0 0;
0 1 0 0 ];
#build measurement data
y = H*x + 30*randn(2,N);
plot(y(1,:), y(2,:), 'o')
### set up initial matrices for Kalman filter
# F is state update matrix, assuming Newton's Laws for model
F = [ 1 0 delta_t 0;
0 1 0 delta_t;
0 0 1 0;
0 0 0 1 ] ;
# P is error covariance, set initial matrix
P = diag([1000 1000 100 100]);
# R measurement noise covariance, set initial matrix,
R = diag([1000 1000]);
# Q control noise covariance, set matrix for duration of filtering, for this example, assume only velocity changes
#try different control noise to allow for curves
#Q = diag([0 0 1000 1000]);
Q = diag([0 0 10 10]);
#Q = diag([0 0 1 1]);
#Q = diag([0 0 0 0]); # seems pretty idealistic
### apply Kalman filter to measurements
x_hat_plus = zeros(4,N);
x_hat_plus(:,1) = [y(1,1) y(2,1) 0 0]; # initial estimate
for i = 2:N
x_hat_minus = F * x_hat_plus(:,i-1) ; # model extrapolation
# x_hat_minus += G * u for control terms if known
y_hat = H * x_hat_minus; # estimated measurement
yvec = y(:,i); # real, noisy, measurement
P_minus = F * P * F' + Q;
K = P_minus * H' * inv(H * P_minus * H' + R);
# assimilate the new measurement
x_hat_plus(:,i) = x_hat_minus + K*(yvec - y_hat);
#update covariance
A = eye(4,4) - K*H;
P = A*P_minus*A' + K*R*K';
endfor
plot(x_hat_plus(1,:), x_hat_plus(2,:), 'o')
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