# Estimation of accelerating target using position measurements only

I am currently thinking about approaches to estimating the position and velocity of an accelerating target.

At this time, I have tried a few approaches that work alright. I have tried two variations of Kalman filters, the first where it estimates position and velocity given the following:

• Dynamics: \begin{align} \dot{x} &= v + \eta_1\\ \dot{v} &= \eta_2 \end{align}

• Measurement: $$y = x + \eta_3$$ where $\eta_i$ are noise from a Gaussian distribution.

The other Kalman Filter approach was based on the following:

• Dynamics: \begin{align} \dot{a_1} &= \eta_1\\ \dot{a_2} &= \eta_2 \end{align}

• Measurement: $$y_{i+1} = y_{i}+a_1 \Delta t + a_2 \frac{\Delta t^2}{2}$$

I also implemented a filter using local regression based on the last $N$ position samples in the time series.

All of these filters do fairly well, but I still question if there might be something better. I have tried looking for any papers or lectures on the subject but struggle to find anything worthwhile.

One other thing that coworker and myself discussed was potentially using Euler's Equation for Rigid Body Dynamics to add some constraints into a target's ability to turn too rapidly based on using a ballpark inertia matrix (assuming we're going after a small car).

This obviously makes the dynamics nonlinear and would push us to using something like an Unscented Kalman filter. Does anyone know whether this would be worth it? We suspect it could help our estimate not make any unrealistic jumps that it might otherwise make with the purely linear model, but don't know if this would make a big enough difference.

I don't have a really good answer for you, but I can see a possible problem with your second attempt: the value of $a_3$ has the potential to completely blow up the dynamics because it is multiplying $t^3$.
In cases like this, you are better off defining an interval over which you wish to do the estimation, $[T_1,T_2]$ and then choosing well-behaved polynomials on this interval (e.g. the Legendre polynomials, $P_k(x)$) and multiplying them.
That way your measurement becomes $$y(t) = \alpha_1 P_0(x;T_1, T_2) + \alpha_2 P_1(x;T_1, T_2) + \alpha_3 P_2(x;T_1, T_2)$$ where $P_k(\cdot;T_1, T_2)$ is the (scaled) $k^{\rm th}$ order Legendre polynomial: $$P_k(\cdot;T_1, T_2) = \frac{1}{T_2 - T_1} P_k\left( \frac{2t -T_1 - T_2}{T_2 - T_1} \right)$$ where $P_k(\cdot)$ is the standard Legendre polynomial.