A method for determining Dual Axis Offset $17x

[phil-barrett]

Dual axis offset for auto-squaring is a really nice feature but I have struggled with how to measure super small differences to make it work. In some ways, having adjustable sensor targets is easier but requires trial and error to dial in precisely. But, I believed there should be a way to determine the offset from a single test cut and set out to do that. I'd love any thoughts or comments on this method.

While the machine is told to cut a square, gantry misalignment will cause a parallelogram to be cut. See fig 1.

The idea is to cut a square and measure the diagonals. If the diagonals are not equal, you have cut a parallelogram. The error is due to the gantry not being square. One can determine the dual axis offset via the measurements by using the Pythagorean Theorem. See figure 2 for the overall layout.

The offset error (d) in figure 2 is exaggerated to show intent. The actual offset error will likely be quite small. Figure 3a and 3b show 2 right triangles.

Using the Pythagorean Theorem, the diagonals $r_1$ and $r_2$ are defined as

$$r_1^2 = h^2 + \left(h+d\right)^2$$

$$r_2^2 = h^2 + \left(h-d\right)^2$$

Rearranging,

$$\left(h+d\right)^2 = r_1^2 - h^2$$

$$\left(h-d\right)^2 = r_2^2 - h^2$$

Taking the square root of both sides

$$h+d = \sqrt{r_1^2 - h^2}$$

$$h-d = \sqrt{r_2^2 - h^2}$$

a bit more rearranging and you get

$$d = \sqrt{r_1^2 - h^2} - h$$

$$d = h - \sqrt{r_2^2 - h^2}$$

The values of $h$, $r_1$ and $r_2$ are measured and the equations evaluated to get two values of $d$. These are likely to be slightly different because of measurement error so averaging them will get a better result. If they are significantly different, measure the block again.

Here is the basic process:
Before you start, make sure that your machine's travel resolution for each axis (Steps/mm, $1x0) is dialed in.

1. Create gcode to cut a square of size $h$ X $h$. 80 mm is a good size to be able to measure with 6”/150mm calipers.
2. Cut the $h$ X $h$ square on your machine.
3. Measure the diagonals of the square. Make sure your calipers measure parallel to the diagonal line to avoid parallax error. Call these $r_1$ and $r_2$. $r_1$ should run from the rear primary motor side to the secondary motor side in the front.
4. Use $h$, $r_1$ and $r_2$ to evaluate the above formulas.
5. Average the two values of $d$ to get a more accurate result. Plug that into the $17x Grbl setting.
6. Scale $d$ by the ratio of h and distance between your primary and secondary homing sensor.

This spreadsheet makes it easy. dual axis offset calculator.xls Be sure to rehome the machine and then repeat the process to verify squareness.