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Kawasaki Robot Controller E Series. Reference Manual (2015) - page 17

Appendix 3 Euler’s OAT Angles

Kawasaki Robot AS Language Reference Manual

APPENDIX 3 EULER’S O,A,T ANGLES

Z

z

A

y

T

X

O

O

Y

A

X

T

Y

x

X

The orientation of a coordinate system (x,y,z) with respect to the base coordinate system

(X,Y,Z) is generally expressed using the Euler’s OAT angles. As shown in the figure above,

the three angles can be defined as follows. In the figure above, the two coordinate systems

(x,y,z)and (X,Y,Z)share the same origin.

O: The angle between Zz plane and XZ plane

A: The angle between z axis and Z axis

T: The angle between x axis and X axis

X axis is on the Zz plane and the angle between this axis and the z axis is 90.

These three angles can also be said to represent the angles of rotations necessary for the base

coordinate system (X,Y,Z) to coincide with the coordinate system (x,y,z). The order of

rotation must not be changed, or else will result differently.

1. O rotation of the coordinate system (X,Y,Z) around Z axis.

(This moves (X,Y,Z) to

(X,Y,Z).)

2. A rotation of the coordinate system (X,Y,Z) around Y axis.

(This moves (X,Y,Z) to

(X,Y,z).)

3. T rotation of the coordinate system (X,Y,z) around z axis.

(This moves (X,Y,z) to

(x,y,z).)

Furthermore, this can be considered in terms of polar coordinate values. When point P, which is

on the z axis at the distance of d from the origin, is written as (d, A, O), then O and A in these

coordinate values are equal to O and A described above. The direction of the z axis is expressed

by these two values.

A3-1

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