Roll with the Pitches

Euler Rotations
Whether it’s a roadheader, rockbolter, excavator or TBM, one of the fundamental components on any navigation/guidance system is the dual-axis level sensor located on the main body of the machine. The purpose is obvious: to correct the calibrated coordinate system of the machine to the gravity system. But you need to be careful about a few things…

There are a variety of methods for undertaking the calculations required to correct for pitch and roll. As surveyors and engineers we’re all quite familiar thinking in terms of orthogonal, three-axis coordinate systems (x, y, z). Euler rotations are a way of applying rotations to a set of coordinates. A 3x3 matrix rotates a 3x1 vector by the required amounts. Do this once for rotations about the pitch axis (Rx), then again about the roll axis (Ry), and the resulting set of coordinates has been corrected for pitch and roll…well, sort of! Here’s where some limitations start to emerge.

Order Dependency
Depending on the order of operations, the magnitude of the rotations and the geometry of the object you are rotating, you will not always get the results you want. The effect of the second rotation is smaller by proportion of its true value because of the change in geometry caused by the first rotation. For 0° of pitch there is no effect. For 5° of pitch there is .4% error in roll. For 15° of pitch there is 3.5% error in roll.

RollModified = Asin(Sin(MachineRollRads) / Cos(Abs(MachinePitchRads)))

Errors in Monitoring?
Monitoring of structures often uses orthogonally mounted tilt meters upon which similar calculations are required. For a 10m x 10m slab which has undergone tilts of 10mm/m in both axes you can expect an error of 0.5mm in the second axis – too much if you consider that a digital level, let alone a good total station, could easily measure that.

Final Word
Most navigation systems operate in relatively level environments, say within 10 degrees of level – so the error is minimized. As tempting as it is to say that the error is negligible the calculations to correct for this are pretty straight-forward, and should not be ignored. With all the other error sources, environmental effects and mechanical magnifications why not get the pitch and roll perfect…? KODA does.

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Until next time, KODA