9.1.5.9. Exercise: Expressing a series of frames

In Fig. 9.6, the position of the elbow in global coordinates is \((0.34, 0.371, 0)\). Therefore, the frame the expresses the position and incline of the forearm is:

\[\begin{split} ~^\text{global}_\text{forearm}T = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 & 0.34 \\ \sin(\theta) & \cos(\theta) & 0 & 0.371 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{split}\]

Using Kinetics Toolkit’s conventions, create a NumPy array that expresses the series of frames for a movement consisting of three inclines: 0°, 15° and 25°.

Show code cell content Hide code cell content

import numpy as np


frames = np.array(
    [
        [
            [np.cos(0.0), -np.sin(0.0), 0.0, 0.34],
            [np.sin(0.0), np.cos(0.0), 0.0, 0.371],
            [0.0, 0.0, 1.0, 0.0],
            [0.0, 0.0, 0.0, 1.0],
        ],
        [
            [np.cos(np.deg2rad(15.0)), -np.sin(np.deg2rad(15.0)), 0.0, 0.34],
            [np.sin(np.deg2rad(15.0)), np.cos(np.deg2rad(15.0)), 0.0, 0.371],
            [0.0, 0.0, 1.0, 0.0],
            [0.0, 0.0, 0.0, 1.0],
        ],
        [
            [np.cos(np.deg2rad(25.0)), -np.sin(np.deg2rad(25.0)), 0.0, 0.34],
            [np.sin(np.deg2rad(25.0)), np.cos(np.deg2rad(25.0)), 0.0, 0.371],
            [0.0, 0.0, 1.0, 0.0],
            [0.0, 0.0, 0.0, 1.0],
        ],
    ]
)

frames

array([[[ 1.        , -0.        ,  0.        ,  0.34      ],
        [ 0.        ,  1.        ,  0.        ,  0.371     ],
        [ 0.        ,  0.        ,  1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ]],

       [[ 0.96592583, -0.25881905,  0.        ,  0.34      ],
        [ 0.25881905,  0.96592583,  0.        ,  0.371     ],
        [ 0.        ,  0.        ,  1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ]],

       [[ 0.90630779, -0.42261826,  0.        ,  0.34      ],
        [ 0.42261826,  0.90630779,  0.        ,  0.371     ],
        [ 0.        ,  0.        ,  1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ]]])