Study Notes on Multi-Axis Error Modeling Part I and II at Johns Hopkins University

Precision Mechanical Design

Introduction to Multi-Axis Error Modeling

• Location: Gilman Hall, Johns Hopkins Whiting School of Engineering

• Subject Focus: Multi-Axis Error Modeling

Linear Axes in Assemblies

• Assumption: Linear axes are nominally orthogonal to each other; they deviate from orthogonality only by small squareness errors.

• Squareness Representation:

  • Three squareness terms are used to represent the squareness of the system:

  • Squareness of secondary axis to primary axis

  • Squareness of tertiary axis to primary axis

  • Squareness of tertiary axis to secondary axis

• Notation: Primary, secondary, and tertiary axes can be represented by any of the three linear axes. Before modeling, it is necessary to choose the primary and secondary axes.

Two Axis System Example

• Primary Axis: X-axis

• Representation: The relationship is depicted as:

  • $P <br>ightarrow (T)(T)$ where P and T denote the respective positions and orientations along the X and Y axes.

Squareness / Orthogonality of Y to X

• Notation: The relationship is denoted as $Y(ECOY)$ where:

  • ECOY signifies the error in the squareness between Y and X.

  • Various parameters define the transformation accounting for squareness errors.

Y-axis Translation Error in X

• Equation Explanation:

  • The transformation that represents squareness of the Y-axis to X is depicted as:

  • $Z<em>{Y} + Y</em>{tran} + X<em>{ECOY} + Z</em>{OY}E = 0$

Y-axis Transformation

• Transformation Representation:

  • With respect to the X-axis coordinate frame, the transformation includes several parameters:

  • $B<em>Y A</em>Y Z<em>Y C</em>Y + A<em>Y Y</em>Y + C<em>Y B</em>Y X_Y (additional terms) = 0$

Translation of YP in Basis b

• Transformation Representation:

  • $P <br>ightarrow (T)(T)$ with transformation calculations showing various transformations:

  • $$egin{pmatrix} 0 & 0 & 0 & 1 \ 1 & 1 & 1 & 1 \ 0 & 0 & 0 & 1 \ ext{… (additional terms)} \ ext{(..and so on, as per the parameters)}\