# If a Kalman filter can only receive information on $(x, y)$ position, is there any reason to have acceleration as part of the model?

I'm new to working with Kalman filters. I have a Kalman filter that's working well for modelling the 2D position of an object. The model at the moment holds information on position ($x$ and $y$) and velocity ($x$ and $y$). I only have position information available for measurement. I was wondering if adding acceleration to the model would improves things in any way, where the only measurement information available is still just the $x$ and $y$ position. Would adding acceleration to the model make it so sudden jerks in movement are handled better or would there be no difference at all?

• Does the object have an accelerated motion in reality? If so, what sort of acceleration is it? Commented Jun 13, 2017 at 12:03
• It's following the position of a person. I was thinking that the person could stop walking, or walk and then start running. Do you think this a good situation for acceleration or would it be a waste of time? Commented Jun 13, 2017 at 13:00
• As @PeterK has already described, that kind of a jerky random walk motion (motion with short bursts of accelerations and then almost constant velocity until next burst) may not benefit from adding a deterministic acceleration state. But instead can use a process noise injection in that state to account for the randomness in the acceleration. Commented Jun 13, 2017 at 14:08

Whether to include a particular parameter in the model depends on how much you know about the parameter. If you're already including 2D velocity in the state, then the acceleration will be coming in through the random noise assumptions you are making.

Whether tracking a person will improve if you include the 2D acceleration in the state depends on what assumptions you can make about how the person's acceleration changes.

For some models, we might know that the jerk$^1$ is completely random but not the acceleration. In that case, it would make sense to include the acceleration and feed the randomness in as the jerk (change to the acceleration).

${\tiny^1\sf \mbox{The derivative of acceleration, I'm not casting aspersions. }}$

You add an acceleration state or any state for that matter if you expect that state have dependence from measurement to measurement. The state should have memory. If the perturbations are statistically independent for each time evolution of the model, it should be a noise term. States have memory, noise does not.

In the event that noise is correlated, we use and augmented state for noise with memory and a independent perturbation. Again a state has memory, perturbations do not.

There are many situations where states have memory for some sequential time evolutions and then no longer do. This is where and Interacting Multiple Model (IMM) is useful. You only carry the extra state when you need to.

As an example you would have 2 process models, one where acceleration is a noise term and another, where acceleration is an included state. Both models run at the same time but are updated by the best model for that time evolution. The key to the best model is the innovations sequence of each model. You can have 2 models 3 models or what is practical.

There are a few different forms of a Kalman Filter. IMM requires the explicit innovations form.