Interpolate Between Axis-Angle Representations

We can interpolate between two orientations that are represented by an axis and an angle either linearly or with slerp (spherical linear interpolation). Here we compare both methods and measure the angular velocity between two successive steps. We can see that linear interpolation results in a non-constant angular velocity. Usually it is a better idea to interpolate with slerp.

Once Loop Reflect

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import axes3d
import matplotlib.animation as animation
from pytransform3d import rotations as pr

velocity = None
last_a = None
rotation_axis = None


def interpolate_linear(start, end, t):
    return (1 - t) * start + t * end


def update_lines(step, start, end, n_frames, rot, profile):
    global velocity
    global last_a

    if step == 0:
        velocity = []
        last_a = start

    if step <= n_frames / 2:
        t = step / float(n_frames / 2 - 1)
        a = pr.axis_angle_slerp(start, end, t)
    else:
        t = (step - n_frames / 2) / float(n_frames / 2 - 1)
        a = interpolate_linear(end, start, t)

    R = pr.matrix_from_axis_angle(a)

    # Draw new frame
    rot[0].set_data(np.array([0, R[0, 0]]), [0, R[1, 0]])
    rot[0].set_3d_properties([0, R[2, 0]])

    rot[1].set_data(np.array([0, R[0, 1]]), [0, R[1, 1]])
    rot[1].set_3d_properties([0, R[2, 1]])

    rot[2].set_data(np.array([0, R[0, 2]]), [0, R[1, 2]])
    rot[2].set_3d_properties([0, R[2, 2]])

    # Update vector in frame
    test = R.dot(np.ones(3) / np.sqrt(3.0))
    rot[3].set_data(
        np.array([test[0] / 2.0, test[0]]), [test[1] / 2.0, test[1]])
    rot[3].set_3d_properties([test[2] / 2.0, test[2]])

    velocity.append(
        pr.angle_between_vectors(a[:3], last_a[:3]) + a[3] - last_a[3])
    last_a = a
    profile.set_data(np.linspace(0, 1, n_frames)[:len(velocity)], velocity)

    return rot


if __name__ == "__main__":
    # Generate random start and goal
    np.random.seed(3)
    start = np.array([0, 0, 0, np.pi])
    start[:3] = pr.norm_vector(np.random.randn(3))
    end = np.array([0, 0, 0, np.pi])
    end[:3] = pr.norm_vector(np.random.randn(3))
    n_frames = 100

    fig = plt.figure(figsize=(12, 5))

    ax = fig.add_subplot(121, projection="3d")
    ax.set_xlim((-1, 1))
    ax.set_ylim((-1, 1))
    ax.set_zlim((-1, 1))
    ax.set_xlabel("X")
    ax.set_ylabel("Y")
    ax.set_zlabel("Z")

    Rs = pr.matrix_from_axis_angle(start)
    Re = pr.matrix_from_axis_angle(end)

    rot = [ax.plot([0, 1], [0, 0], [0, 0], c="r", lw=3)[0],
           ax.plot([0, 0], [0, 1], [0, 0], c="g", lw=3)[0],
           ax.plot([0, 0], [0, 0], [0, 1], c="b", lw=3)[0],
           ax.plot([0, 1], [0, 1], [0, 1], c="gray", lw=3)[0],

           ax.plot([0, Rs[0, 0]], [0, Rs[1, 0]], [0, Rs[2, 0]], c="r", lw=3,
                   alpha=0.5)[0],
           ax.plot([0, Rs[0, 1]], [0, Rs[1, 1]], [0, Rs[2, 1]], c="g", lw=3,
                   alpha=0.5)[0],
           ax.plot([0, Rs[0, 2]], [0, Rs[1, 2]], [0, Rs[2, 2]], c="b", lw=3,
                   alpha=0.5)[0],

           ax.plot([0, Re[0, 0]], [0, Re[1, 0]], [0, Re[2, 0]], c="orange",
                   lw=3, alpha=0.5)[0],
           ax.plot([0, Re[0, 1]], [0, Re[1, 1]], [0, Re[2, 1]], c="turquoise",
                   lw=3, alpha=0.5)[0],
           ax.plot([0, Re[0, 2]], [0, Re[1, 2]], [0, Re[2, 2]], c="violet",
                   lw=3, alpha=0.5)[0]]

    ax = fig.add_subplot(122)
    ax.set_xlim((0, 1))
    ax.set_ylim((0, 1))
    profile = ax.plot(0, 0)[0]

    anim = animation.FuncAnimation(fig, update_lines, n_frames,
                                   fargs=(start, end, n_frames, rot, profile),
                                   interval=50, blit=False)

    plt.show()