Bias Instability Figures That Decide Whether Dead Reckoning Holds
Inertial dead reckoning holds only while gyroscope bias instability bounds cubic tilt divergence within allowable spatial position tolerance thresholds.
Cubic Positional Divergence
Error Component
Volumetric error in positioning systems where the displacement deviation increases as the third power of the distance from an origin. High precision stages exhibit cubic positional divergence when non-linear structural flexure or bearing misalignments compound over long travel lengths. It establishes the upper boundary for volumetric accuracy in large format metrology equipment.
Geometric Growth
Mechanical linkages and lead screw assemblies introduce small angular errors that translate into significant tip deviations at the end of an arm. In systems suffering from cubic positional divergence, the error remains negligible near the home position but accelerates rapidly as the axis extends. Calibration routines using laser interferometers map these errors to create compensation tables for the controller.
Calibration Strategy
Standardized verification involves measuring the tool center point at various coordinates throughout the entire working envelope. This mapping for cubic positional divergence requires a stable thermal environment to ensure that expansion does not mask the underlying mechanical signature. Precise laser trackers provide the ground truth against which the machine internal encoders are compared.
Software models then integrate these results to create a full error map for the controller.
Correction Limit
Software algorithms can counteract predictable non-linearities by applying inverse offsets to the motion commands. However, cubic positional divergence becomes difficult to manage when the mechanical structure lacks the stiffness to maintain repeatability. The residual error after compensation defines the actual precision of the system in the field.