# Observation Model Jacobian for Fixed Transforms

Let's say I have a hypothetical sensor that provides, for example, velocity estimates, and I affix that sensor at some non-zero rotational offset from the robot's base. I also have an EKF that is estimating the robot's velocity.

Normally, the innovation calculation for an EKF looks like this:

$$y_k = z_k - h(x_k)$$

In this case, $h$ would just be the rotation matrix of the rotational offset. What are the ramifications if instead, I pre-process the sensor measurement by rotating $z_k$ by the inverse rotation, which will put its coordinates in the frame of the robot? Can I then safely just make $h$ the identity matrix $I$?

• The KF is a closed-form estimator. Without seeing equations, you can't really say if it's the same. – Josh Vander Hook Jun 18 '14 at 18:48

In your example, if "rotating" the velocity sensor meant that it detected: $v+b$ for velcoity vector $v$ and some fixed bias $b$, then yes, simply removing the bias before incorporating the data is fine. Your innovation has the same value, since $h(x)$ is already the operation you described (applying the rotation).
But if the velocity sensor now measured $v+f(x)$, as in it measured greater velocity if the robot was turning right than left (as is the case of optical flow calculations from a camera facing a little to the right instead of directly forward), then you'd have to be much more careful. (Update jacobians or have near-perfect pre-processing).