9.2.3.5. Exercise: A fixed point in a moving coordinate system

In Fig. 9.6, the position of the elbow in the arm coordinate system is \((0, -0.5, 0)\).

Using Kinetics Toolkit’s geometry module, express the position of the elbow in global coordinates for a movement of shoulder flexion of 0°, 5°, 10° and 15°.

a) Do this exercise by creating the corresponding homogeneous transform using ktk.geometry.create_transforms, then by multiplying this transform by the point coordinates using ktk.geometry.matmul.

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import kineticstoolkit.lab as ktk
import numpy as np


p_elbow_ref_arm = np.array([[0.0, -0.5, 0.0, 1.0]])

T_arm = ktk.geometry.create_transforms(
    angles=[
        np.deg2rad(0.0),
        np.deg2rad(5.0),
        np.deg2rad(10.0),
        np.deg2rad(15.0),
    ],
    seq="z",
)

ktk.geometry.matmul(T_arm, p_elbow_ref_arm)

array([[ 0.        , -0.5       ,  0.        ,  1.        ],
       [ 0.04357787, -0.49809735,  0.        ,  1.        ],
       [ 0.08682409, -0.49240388,  0.        ,  1.        ],
       [ 0.12940952, -0.48296291,  0.        ,  1.        ]])

b) Do this exercise using the ktk.geometry.rotate and ktk.geometry.translate function.

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p_elbow_ref_arm = np.array([[0.0, -0.5, 0.0, 1.0]])

ktk.geometry.rotate(
    p_elbow_ref_arm,
    angles=[
        np.deg2rad(0.0),
        np.deg2rad(5.0),
        np.deg2rad(10.0),
        np.deg2rad(15.0),
    ],
    seq="z",
)

array([[ 0.        , -0.5       ,  0.        ,  1.        ],
       [ 0.04357787, -0.49809735,  0.        ,  1.        ],
       [ 0.08682409, -0.49240388,  0.        ,  1.        ],
       [ 0.12940952, -0.48296291,  0.        ,  1.        ]])