Predicting the position of two points with known geometrical constraints

Question

Using the diagram below, I want a systematic way to find the coordinates of L' and R' relative to L and R, given:

1. The distance between L and L'
2. The distance between R and R'
3. The distance between L and R (which is also the distance between L' and R')
4. The angle between AB and the x-axis
5. AB is perpendicular to L'R'

So an ideal solution would give the L' and R' as vectors in terms of L, R, the two distances, and an angle. Otherwise, a solution using the constants in the diagram would be helpful as well.

Purpose: I'm trying to model the dynamic movement of a system in 2d space. L and R represent the last known position of two points on the system, and L' and R' are the predicted location of the points. Two magnetic encoders measure the change in position of those two points, hence why the lengths of LL' and RR' are known. The system also uses a gyroscope to measure its angle relative to its starting position: (0,0). If anyone familiar with control theory and knows of a better approach, I'm all ears.

Thanks!

Answer 1

Being $L'(x_1,y_1)$ and $R'(x_2,y_2)$ the unknown points, and $L(px_1,py_1)$ and $R(px_2,py_2)$ the given points, and some given distances, we know:

$$dist^2 L' to R' = (dL'R')^2 = (x_2-x_1)^2+(y_2-y_1)^2$$ $$dist^2 L to L' = (dLL')^2 = (x_1-px_1)^2+(y_1-py_1)^2$$ $$dist^2 R to R' = (dRR')^2 = (x_2-px_2)^2+(y_2-py_2)^2$$ $$angle L'R' with\; x-axis \implies \; tan(\alpha)=\frac{y_2-y_1}{x_2-x_1}$$

The angle $\alpha$ is angle AB-$\pi/2$. In your case $\alpha=110º-90º=20º$

Solve for $x_1,y_1,x_2,y_2$