Frame defined with respect to another frame (Person in a train)

In this example, we will see how to define a frame with respect to another frame.

Lets consider a person walking in a train. The person is represented by a frame \(F\) and the train is represented by a frame \(E\). It is possible to represent the position of the person in the train by defining the frame \(F\) in the frame \(E\) coordinates.

Define the frames

The frame \(F\) is defined with respect to the frame \(E\) by setting the parent of the frame \(F\) to the frame \(E\).

from py3dframe import Frame, Rotation
import numpy as np

rotation = Rotation.from_euler('xyz', [0, 0, 0], degrees=True)
translation = np.array([1, 2, 3]).reshape(3, 1)

frame_E = Frame.from_rotation(translation=translation, rotation=rotation, convention=0)

rotation = Rotation.from_euler('xyz', [0, 0, 0], degrees=True)
translation = np.array([0, 0, 0]).reshape(3, 1)

frame_F = Frame.from_rotation(translation=translation, rotation=rotation, convention=0, parent=frame_E)

When the frame \(E\) (train) moves, the frame \(F\) (person) moves with it. However the position of the person in the train does not change, since it is defined with respect to the train.

• origin is the position of the frame in the local coordinates of the parent frame.

• global_origin is the position of the frame in the global coordinates.

print(f"Before train moves :")
print(f"Global train position: {frame_E.global_origin}")
print(f"Global person position: {frame_F.global_origin}")

frame_E.origin = np.array([4, 5, 6]).reshape(3, 1)

print(f"After train moves: ")
print(f"Global train position: {frame_E.global_origin}")
print(f"Global person position: {frame_F.global_origin}")

frame_F.origin = np.array([1, 2, 3]).reshape(3, 1)

print(f"After person moves in the train: ")
print(f"Global train position: {frame_E.global_origin}")
print(f"Global person position: {frame_F.global_origin}")

Before train moves :
Global train position: [[1.]
 [2.]
 [3.]]
Global person position: [[1.]
 [2.]
 [3.]]
After train moves:
Global train position: [[4.]
 [5.]
 [6.]]
Global person position: [[4.]
 [5.]
 [6.]]
After person moves in the train:
Global train position: [[4.]
 [5.]
 [6.]]
Global person position: [[5.]
 [7.]
 [9.]]

Perform the transformation between the frames

The transformation from the frame \(E\) to the frame \(F\) can be perform using the class py3dframe.FrameTransform as follows.

This object will follow the movement of the train and the person, and will always give the correct transformation from the train to the person.

from py3dframe import FrameTransform

transform = FrameTransform(input_frame=frame_E, output_frame=frame_F)

print("Transformation from train to person (before person moves): ")
print(transform.translation)
print(transform.rotation.as_euler('xyz', degrees=True))

frame_F.euler_angles = [0, 0, 90]

print("Transformation from train to person (after person moves): ")
print(transform.translation)
print(transform.rotation.as_euler('xyz', degrees=True))

Transformation from train to person (before person moves):
[[1.]
 [2.]
 [3.]]
[0. 0. 0.]
Transformation from train to person (after person moves):
[[1.]
 [2.]
 [3.]]
[  0.           0.         116.62015618]

This object can be used to transform points or vector from one frame to another. The points and vectors can be transformed in both directions, from the train to the person and from the person to the train.

The points and vectors must be \((3, N)\) numpy arrays, where \(N\) is the number of points or vectors to transform.

point_E = np.array([1, 2, 3]).reshape(3, 1)
point_F = transform.transform(point=point_E) # In convention 0 : pE = R * pF + T
point_E_bis = transform.inverse_transform(point=point_F)
assert np.allclose(point_E, point_E_bis)

print(f"Point in train coordinates: {point_E.flatten()}")
print(f"Point in person coordinates: {point_F.flatten()}")

vector_E = np.array([1, 2, 3]).reshape(3, 1)
vector_F = transform.transform(vector=vector_E) # In convention 0 : vE = R * vF
vector_E_bis = transform.inverse_transform(vector=vector_F)
assert np.allclose(vector_E, vector_E_bis)

print(f"Vector in train coordinates: {vector_E.flatten()}")
print(f"Vector in person coordinates: {vector_F.flatten()}")

Point in train coordinates: [1 2 3]
Point in person coordinates: [0. 0. 0.]
Vector in train coordinates: [1 2 3]
Vector in person coordinates: [ 1.33991971 -1.7901439   3.        ]