Why do we need quaternions?

Why do we need quaternions?

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis.

How do you rotate with quaternions?

Rotating smoothly and directly from one set of Euler angles to another is a pain. With quaternions, it’s as simple as multiplication. Typically you will take the orientation you have (as a quaternion) and just multiply by the rotation (another quaternion) you want to apply.

Why do quaternions represent rotations?

It uses a Quaternion object to represent rotation of various geometries. For rotations, quaternions are superior to using Euler angles. The reason is that quaternions avoid a problem known as gimbal lock. That happens if two of the three rotational axes happen to align.

Are quaternions a ring?

The ring of real quaternions is a division ring. (Recall that a division ring is a unital ring in which every element has a multiplicative inverse. It is not necessarily also a commutative ring. A division ring that is commutative is simply a field.)

How do you convert Euler angles to quaternion?

1 Answer

1. First, roll around the world x axis. The quaternion for this is qx,α=cosα2+(sinα2)i.
2. Second, pitch around the world y axis. The quaternion is qy,β=cosβ2+(sinβ2)j.
3. Third, yaw around the world z axis. The quaternion is qz,γ=cosγ2+(sinγ2)k.

Why is gimbal lock a problem?

The gimbal lock problem can be countered by adding a redundant degree of freedom in the form of an extra gimbal, an extra joint in a robotic arm, etc. As you pointed out, it’s the singularity at the poles of the representation that’s the problem.

How do gimbals work?

A gimbal is a pivoting support that allows you to rotate an object along a single axis. A common 3 axis gimbal allows a camera mounted on it to be independent of the movement of the one holding the gimbal. These 3 axes are identified as pitch, yaw, and roll.

Why does gimbal lock happen?

Gimbal lock occurs when the orientation of the sensor cannot be uniquely represented using Euler Angles. Intuitively, the cause of gimbal lock is that when the pitch angle is 90 degrees, yaw and roll cause the sensor to move in exactly the same fashion.