Euler angles are a common way of representing rotations in three-dimensional space. However, they suffer from several issues, such as gimbal lock, a phenomenon that makes it impossible to represent certain rotations accurately using Euler angles. Quaternion representations, on the other hand, do not suffer from the same issues and are a popular alternative to Euler angles. In this article, we will cover how to convert from Euler angles to quaternion representations, along with code examples in Python.
Euler angles are a set of three angles that represent the rotations around the three axes of a 3D coordinate system. The three angles, usually represented by ϕ (phi), θ (theta), and ψ (psi), represent rotations around the x, y, and z axes respectively. Euler angles are easy to understand and intuitive for many people, making them a popular way of representing rotations. However, they can quickly become complex to work with, especially when trying to combine multiple rotations.
Quaternions provide a more concise and straightforward way of representing rotations in 3D space. Quaternions are a four-dimensional complex number that consists of a scalar component (w) and three vector components (i, j, k). Quaternion rotations are computed by using quaternion multiplication, rather than matrix multiplication used in Euler angles. Quaternion representations do not suffer from gimbal lock, a situation where changing one of the angles causes the axis of rotation to align with one of the other axes, resulting in the loss of one degree of freedom.
To convert from Euler angles to quaternion representations, we need to apply a series of formulas that depend on the order of rotations (also called the rotation convention). One common convention is the XYZ convention, where rotations are first applied around the x-axis, then the y-axis, and finally, the z-axis. Another is the ZYX convention, where rotations are first applied around the z-axis, then the y-axis, and finally, the x-axis.
Let's look at the formula for converting from Euler angles to quaternion in the XYZ convention:
w = cos(ϕ/2)cos(θ/2)cos(ψ/2) + sin(ϕ/2)sin(θ/2)sin(ψ/2)
i = sin(ϕ/2)cos(θ/2)cos(ψ/2) – cos(ϕ/2)sin(θ/2)sin(ψ/2)
j = cos(ϕ/2)sin(θ/2)cos(ψ/2) + sin(ϕ/2)cos(θ/2)sin(ψ/2)
k = cos(ϕ/2)cos(θ/2)sin(ψ/2) – sin(ϕ/2)sin(θ/2)cos(ψ/2)
In this formula, w represents the scalar component of the quaternion, while i, j, and k represent the vector components. ϕ, θ, and ψ are the Euler angles that represent rotations around the x, y, and z axes respectively.
For example, let's say we want to convert the rotations ϕ = 30°, θ = 45°, and ψ = 60° in the XYZ convention to quaternion representation. We can plug these values into the formula to get the following quaternion:
w = 0.8536, i = 0.1464, j = 0.3536, k = -0.3536
Now let's look at the Python code to convert Euler angles to quaternion in the XYZ convention:
import math
phi = math.radians(30) # rotation around x-axis
theta = math.radians(45) # rotation around y-axis
psi = math.radians(60) # rotation around z-axis
qw = math.cos(phi/2) * math.cos(theta/2) * math.cos(psi/2) + math.sin(phi/2) * math.sin(theta/2) * math.sin(psi/2)
qx = math.sin(phi/2) * math.cos(theta/2) * math.cos(psi/2) – math.cos(phi/2) * math.sin(theta/2) * math.sin(psi/2)
qy = math.cos(phi/2) * math.sin(theta/2) * math.cos(psi/2) + math.sin(phi/2) * math.cos(theta/2) * math.sin(psi/2)
qz = math.cos(phi/2) * math.cos(theta/2) * math.sin(psi/2) – math.sin(phi/2) * math.sin(theta/2) * math.cos(psi/2)
print(f"Quaternion: ({qw:.4f}, {qx:.4f}, {qy:.4f}, {qz:.4f})")
In this code example, we convert the rotations ϕ = 30°, θ = 45°, and ψ = 60° in the XYZ convention to quaternion representation. We first convert the angles to radians using the math.radians function, then apply the formula to compute the quaternion. Finally, we use the print statement to display the resulting quaternion.
In conclusion, converting from Euler angles to quaternion representations can be useful in many applications, especially those involving 3D graphics and animations. While the formulas and conventions might seem daunting at first, with some practice and the help of code examples, anyone can learn to use quaternions to represent 3D rotations accurately and efficiently.
let's dive deeper into some of the topics covered in the article.
Euler angles have been used for a long time to represent rotations in 3D space because they are easy to understand and work with. However, one of their biggest drawbacks is the problem of gimbal lock. Gimbal lock occurs when the rotation axes align, causing a loss of one degree of freedom, leading to inaccurate representations and problematic animations in computer graphics applications. Quaternions, on the other hand, do not suffer from gimbal lock and provide a more concise and robust way of representing 3D rotations.
Quaternions are a four-dimensional complex number that consists of a scalar component (w) and three vector components (i, j, k). Quaternion rotations are carried out using quaternion multiplication, which is similar to ordinary complex multiplication but with an additional cross-product term of the vector components. Quaternion multiplication is also associative, which means that the order of operations does not matter, making it easy to combine multiple rotations. Additionally, quaternions provide interpolations between two orientations that are independent of the path between them, providing exceptionally smooth animations.
In the article, we discussed how to convert from Euler angles to quaternion representation by applying a series of formulas based on the order of rotation convention. The most popular conventions are the XYZ and ZYX conventions, although there are many others. In the XYZ convention, rotations occur first around the x-axis, followed by the y-axis, and finally, the z-axis. In the ZYX convention, rotations occur first around the z-axis, followed by the y-axis, and finally, the x-axis.
The code example provided in the article shows how to convert from Euler angles to a quaternion in the XYZ convention using Python. The example first converts the angles to radians and applies the formula to calculate the quaternion that represents the rotation. The resulting quaternion can then be used for further calculations or to drive 3D animations.
Quaternions find extensive use in computer graphics and robotics applications, ranging from 3D modeling and animation software to drone stabilization systems and robotics control algorithms. In many of these applications, the ability to perform fast and efficient quaternion operations is crucial. Modern computing hardware, including CPUs and GPUs, provide a wide range of libraries and optimizations that make quaternion operations lightning-fast, making them a popular choice for modeling and simulation tasks.
In conclusion, quaternions are a powerful tool for representing 3D rotations and provide an efficient and accurate alternative to Euler angles. With a bit of practice and some code examples, anyone can learn to work with quaternions to perform complex 3D rotations and animations.
Popular questions
Q: What is the problem with using Euler angles to represent 3D rotations?
A: Euler angles suffer from the problem of gimbal lock, a phenomenon that causes a loss of degree of freedom when rotation axes align, leading to inaccurate representations and problematic animations in computer graphics applications.
Q: How are quaternions different from Euler angles in representing 3D rotations?
A: Quaternions provide a more concise and robust way of representing 3D rotations compared to Euler angles. They do not suffer from gimbal lock, use quaternion multiplication for rotation operations, are easy to combine, and provide high-quality interpolation for smooth animations.
Q: What is the most popular rotation convention when converting from Euler angles to quaternion representation?
A: The most popular conventions are the XYZ and ZYX conventions; however, there are many others.
Q: What is the formula for converting Euler angles to quaternion in the XYZ convention?
A: The formula is w = cos(ϕ/2)cos(θ/2)cos(ψ/2) + sin(ϕ/2)sin(θ/2)sin(ψ/2), i = sin(ϕ/2)cos(θ/2)cos(ψ/2) – cos(ϕ/2)sin(θ/2)sin(ψ/2), j = cos(ϕ/2)sin(θ/2)cos(ψ/2) + sin(ϕ/2)cos(θ/2)sin(ψ/2), and k = cos(ϕ/2)cos(θ/2)sin(ψ/2) – sin(ϕ/2)sin(θ/2)cos(ψ/2), where ϕ, θ, and ψ are Euler angles representing rotations around the x, y, and z axes, respectively.
Q: What are some applications of quaternions in computer graphics and robotics?
A: Quaternions find extensive use in computer graphics and robotics applications, spanning 3D modeling and animation software, drone stabilization systems, and robotics control algorithms.
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