Robotics: Theory and Examples

4. Understanding Quaternions in Robotics

In the world of robotics, controlling and representing the orientation and rotation of a robot’s components is crucial. Quaternions, a mathematical tool, have become increasingly popular for this purpose due to their numerous advantages over traditional representations like Euler angles. In this article, we will explore the role of quaternions in robotics, their benefits, and how they can be applied in a 2R pick-and-place robot simulation.

4.1. Understanding Quaternions in Robotics

Quaternions are a type of complex number with four components, usually represented as \(q = a + bi + cj + dk\), where \(a\), \(b\), \(c\), and \(d\) are real numbers, and \(i\), \(j\), and \(k\) are imaginary units. They are particularly useful in robotics for representing 3D orientations and rotations, and they facilitate a more accurate and efficient means of controlling a robot’s movements.

4.2. Advantages of Quaternions over Euler Angles

Avoiding Gimbal Lock: Euler angles use a sequence of three rotations about different axes to represent orientations. However, this representation can suffer from a phenomenon called gimbal lock, where two of the axes become aligned, causing the loss of one degree of freedom. Quaternions, on the other hand, avoid gimbal lock due to their unique mathematical properties.
More Efficient Interpolation: When generating smooth trajectories between orientations, quaternions enable a more efficient interpolation technique called Spherical Linear Interpolation (SLERP). This method results in smoother and more natural movements compared to interpolating Euler angles.

4.3. Applying Quaternions in a 2R Pick-and-Place Robot Simulation

In a 2R pick-and-place robot simulation, we can use quaternions to control the orientation of the robot arm and end-effector. Here’s how:

Representing Orientation: Convert the robot arm’s current orientation to a quaternion representation. This can be done using a function that takes the current Euler angles or rotation matrix and returns the corresponding quaternion.

import numpy as np
from scipy.spatial.transform import Rotation

def euler_to_quaternion(roll, pitch, yaw):
    rotation = Rotation.from_euler('xyz', [roll, pitch, yaw], degrees=True)
    return rotation.as_quat()

roll, pitch, yaw = 30, 45, 60
quaternion = euler_to_quaternion(roll, pitch, yaw)

Interpolating Between Orientations: If we need the robot arm to move between two orientations, we can use SLERP to interpolate between the corresponding quaternions. This technique will generate a smooth trajectory for the robot arm’s movement, avoiding abrupt changes in orientation.

from scipy.spatial.transform import Slerp

def interpolate_quaternions(q1, q2, t):
    slerp = Slerp([0, 1], Rotation.from_quat([q1, q2]))
    return slerp([t]).as_quat()[0]

q1 = euler_to_quaternion(0, 0, 0)
q2 = euler_to_quaternion(30, 45, 60)
t = 0.5  # Interpolation factor between 0 and 1
resulting_quaternion = interpolate_quaternions(q1, q2, t)

Updating the Robot Arm’s Orientation: Once the new quaternion is determined using SLERP, we can convert it back to Euler angles or a rotation matrix to update the robot arm’s orientation in the simulation.

def quaternion_to_euler(q):
    rotation = Rotation.from_quat(q)
    return rotation.as_euler('xyz', degrees=True)

# Convert the resulting quaternion back to Euler angles
resulting_euler_angles = quaternion_to_euler(resulting_quaternion)

# Update the robot arm's orientation using the new Euler angles

Incorporating quaternions into a (2R pick-and-place) robot simulation can significantly improve the control and representation of the robot arm’s orientation. Quaternions offer several advantages over traditional Euler angles. By understanding and applying quaternions in robotics, engineers and researchers can create more accurate, efficient, and natural movements for advanced robotic systems.

4.4. Exercise

Example 1

What is the primary advantage of using quaternions over Euler angles for representing 3D rotations in robotics?

Solution:

The primary advantage of using quaternions over Euler angles is that quaternions avoid gimbal lock, which is a loss of a degree of freedom caused by the alignment of two rotational axes. Quaternions also enable more efficient interpolation of orientations using Spherical Linear Interpolation (SLERP), resulting in smoother and more natural motions for robotic systems.

Example 2

How can quaternions be used to interpolate between two orientations in 3D space?

Solution:

Quaternions can be used to interpolate between two orientations using Spherical Linear Interpolation (SLERP). SLERP is a method that smoothly interpolates between two unit quaternions along the shortest great circle arc on the 4D hypersphere, ensuring minimal rotational change between the two orientations. This method is computationally efficient and produces a more natural and continuous rotation than linear interpolation of Euler angles.

Example 3

How do you convert between quaternion and Euler angle representations in a robotic application? Use an example in Python.

Solution:

In a robotic application, you can convert between quaternion and Euler angle representations using dedicated conversion functions or available libraries, such as SciPy in Python. For example, you can use scipy.spatial.transform.Rotation class to convert between quaternions and Euler angles:

from scipy.spatial.transform import Rotation

def euler_to_quaternion(euler_angles):
    rotation = Rotation.from_euler('xyz', euler_angles, degrees=True)
    return rotation.as_quat()

def quaternion_to_euler(quaternion):
    rotation = Rotation.from_quat(quaternion)
    return rotation.as_euler('xyz', degrees=True)

These functions can be used to convert between the two representations, allowing you to take advantage of the benefits of quaternions while still working with familiar Euler angle representations in your robotic application.

Robotics: Theory and Examples

4. Understanding Quaternions in Robotics

4.1. Understanding Quaternions in Robotics

4.2. Advantages of Quaternions over Euler Angles

4.3. Applying Quaternions in a 2R Pick-and-Place Robot Simulation

4.4. Exercise

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