# What is the mathematical relationship between parent and child frame in tf?

I am very confused by the coordinate frame in this tutorial.

In this command line syntax

```
rosrun tf static_transform_publisher x y z yaw pitch roll frame_id child_frame_id period_in_ms
```

- Does the value
`x y z yaw pitch roll`

represents the transform from`child_frame_id`

to`frame_id`

? Or the other way around? And what is the relationship between

`frame_id`

and`child_frame_id`

? What is the order of the yaw pitch row in the transformation? Which one is the correct relationship?*Extrinsic*rotation (occurs about the axes of the fixed coordinate system):`child_frame_id = R(yaw) * R(pitch) * R(roll) * frame_id + [x,y,z]`

`child_frame_id = R(roll) * R(pitch) * R(yaw) * frame_id + [x,y,z]`

*Intrinsic*rotation (occurs about the axes of a coordinate system XYZ attached to a moving body):`child_frame_id = frame_id * R(yaw) * R(pitch) * R(roll) + [x,y,z]`

`child_frame_id = frame_id * R(roll) * R(pitch) * R(yaw) + [x,y,z]`