# Should the input of a Kalman filter always be a signal and its derivative?

I always see the Kalman filter used with such input data. For example, the inputs are commonly a position and the correspondent velocity:

$$(x, \dfrac{dx}{dt})$$

In my case, I only have 2D positions and angles at each sample time:

$$P_i(x_i, y_i) \qquad \text{and} \qquad (\alpha_1, \alpha_2, \alpha_3)$$

Should I compute velocities for each point and for each angle to be able to fit the Kalman framework?

• I am never an expert of Kalman filter, but I think some answers against next questions may be requied to make some model by yourself. In your case, 2D position of what do you have ? and what are angles you have? Are there any relations among 2D position and the angles? And, what do you want to get by using Kalman filter? Smoothed locus of 2D position or what? – fumio ueda Sep 23 '11 at 17:01
• The positions I have are 3D points projected on the screen of a device. The angles are the gyroscope-mesured Euler angles of the device. The relation between them is kinda complex. What I want is a stabilization of the projected points, reflecting the absence or the low movement of the camera. Hope it can help. – Stéphane Péchard Sep 26 '11 at 9:04

There is a good example of this on the Wikipedia page. In that example, 1-dimensional linear motion of an object is considered. The object's state variables consist of its position versus time and its velocity on the one-dimensional line of movement. The example assumes that the only observable is the object's position versus time; its velocity is not observed directly. Therefore, the filter structure "infers" the velocity estimate based on the position measurements and the known relationship between velocity and position (i.e. $\dot{x_k} \approx \frac{(x_k - x_{k-1})}{\Delta t}$ if acceleration is assumed to be slowly-varying).
That depends on your system model, if you can model the system with only positions and angles, that is $x=[x_i, y_i, \alpha_1, \alpha_2, \alpha_3]^T$, it's okay you did not compute the velocity, if it's not possible, you may consider the other way.