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Engineer here, I have an indoor tracking system where I have to subsystems that calculate the angle (angles a and b) towards an object of interest. The angles are calculated to the nearest degree.

If I want to calculate the angle and distance from say A to the object I use the known distance between A and B, say C. and I calculate the distance from A to the object as:

$$A = \frac{C}{\sin c} \sin a$$

Where c can be calculated as: $180^{\circ} - a - b$

My issue however, is say consider a distance between the two subsystems C equal to $0.5m$, and the furthest distance I want to be able to track is $5m$, what is the uncertainty of the location determined by the system?

What route can I go to work this out?

Makoto
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1 Answers1

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One remark before all my engineer nonsense : what you call $A$ is the distance from $B$ to $C$. But it doesn't change the problem.

You can use differential calculus : considering the uncertainty on $C$ is negligible, you have (with $a$ and $b$ in radians) : $$dA=\frac{C}{\sin^2(a+b)}\left[(\sin b)da-(\sin a\cos(a+b))db\right]$$ so the uncertainty comes from the denominator $\sin^2(a+b)$. But as $$\sin(a+b)=\sin(\pi-a-b)=\sin c$$ and $$\sin c=\frac{C\sin a}{A} \ge \frac{\sin a}{10}$$ and same from $b$, and $a$ and $b$ are roughly right angles for not too small distances, the worst you can have is $$\frac{1}{\sin^2(a+b)}\le 100$$ So the uncertainty on $A$ is roughly $$\delta A \approx 100C(\delta a+\delta b) \approx 100\frac{\pi}{180}$$ so an error of $1.75$ meters :-)

You could obtain better precision with a more thorough calculus, I guess... This is one worst case, and I think the presence of the term $\cos(a+b)$ means you can at least divide this error by $2$...

Nicolas FRANCOIS
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