What's the best way of modeling 3d target motion with only 2d angle observations?

Question

A maneuvering target is flying in 3d cartesian space, but a sensor (passive infrared or mic array, etc.) can only observe it in polar coordinate with 2d orientations (azimuth, elevation). For simplicity, there is only one target and one stationary observer fixed in origin coordinates. The question is what's the best way of modeling its states and motions for Kalman tracking?

From my investigations so far, there could be three approaches:

1. All in 3d format. represent location of target states and observations all onto a unit sphere ($X= [x, y, z], |X|=1$), and normalize the location each time after state correcting/predicting (as was done in a sound source tracking system).

2. All in 2d format (azimuth, elevation). Simplest but may not well simulate the linear motions in real 3d cartesian space, but with circular curves instead.

3. remain 3d location states and 2d angle observations, use EKF/UKF to tackle nonlinear transformation with constraint that $|X|=1$.

Answer 1

So, let's see how to start this. Let's make the states the 3D location and the 3D velocities: $$ \mathbf{x}_k = \left [ x_k\ \dot{x}_k\ y_k\ \dot{y}_k\ z_k\ \dot{z}_k \right ]^T $$ Then, following the Mathworks page I referenced in my comment, let's just assume a random walk for the state update equation: $$ \mathbf{x}_{k+1} = \mathbf{A} \mathbf{x}_{k} + \epsilon_k $$ where $$\mathbf{A} = \left [\begin{array}{cccccc} 1 & \Delta t & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 &1 & \Delta t & 0 & 0 \\ 0 & 0 &0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & \Delta t \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right] $$ and $\epsilon_k$ is the independent, identically distributed process noise.

Then the output equation is just $$ \mathbf{z}_k = \left [ \begin{array}{c} \arcsin\left(\frac{z_k}{ \sqrt{x_k^2 + y_k^2 + z_k^2}}\right)\\ \arctan\left(\frac{y_k}{ x_k}\right)\\ \end{array} \right ] + \eta_k $$ where $\eta_k$ is the measurement noise and the measurements are the elevation $$\phi = \arcsin\left(\frac{z_k}{ \sqrt{x_k^2 + y_k^2 + z_k^2}}\right)$$ and the azimuth $$\psi = \arctan \left(\frac{y_k}{ x_k}\right)$$

And I'd just then apply your EKF equations to that. I can't see a good reason to normalize $|\mathbf{x}_k | = 1$, just with what I've written.

It may be that the model doesn't work well without some sort of correction, but I'd try it without the normalization first.