# Quaternions and their Use in the ZeroKey Quantum RTLS

What is a quaternion, why is it used for expressing the orientation of devices, and how is it used in the ZeroKey Quantum RTLS?

**What is a Quaternion?**

A quaternion is a general mathematical entity with four components. A *unit* quaternion is a mathematical object with a magnitude or length equal to 1. In other words, its norm, represented as *||q||*, is 1.

A unit quaternion is normalized, and often denoted as:

**q = (w, x, y, z), where w ^{2}+x^{2}+y^{2}+z^{2}=1**

In the context of orientation representation, unit quaternions are particularly useful because they can represent rotations without scaling. This means that applying a unit quaternion to a vector will only rotate it without changing its magnitude. ZeroKey follows the XYZ convention.

**Numerical Stability and Global Representation**

Unit quaternions offer numerical stability, particularly when performing multiple rotations or complex transformations. Unlike Euler angles, which a prone to gimbal lock – a phenomenon where certain orientations cause a loss of one degree of freedom – quaternions avoid this problem. This stability ensures a global and reliable orientation representation, even in scenarios involving rapid or complex motion, which is essential for accurate tracking in the ZeroKey Quantum RTLS.

**Smooth Interpolation**

In applications such as animation, robot control, or motion tracking, it’s often necessary to seamlessly transition between different orientations. Unit quaternions facilitate this process by enabling smooth interpolation. Linearly interpolating between two-unit quaternions ensures that the interpolated orientations follow the shortest part on the unit sphere, maintaining a smooth and visually pleasing transition.

**How are Quaternions Used in the ZeroKey Quantum RTLS?**

In the context of ZeroKey Quantum RTLS, which tracks the location and orientation of Mobiles in real-time, quaternions are used to precisely describe the orientation of these devices in three-dimensional space. This is important because traditional methods often suffer from singularities and ambiguities, particularly when dealing with multiple rotations. The Quantum RTLS presents a quaternion for the orientation of a device with each position update that the device gives. This can be observed in the console logs and the ZeroKey API.

The following is an example of a console log output:

`rot: 0.99999, 0.00040, -0.00143, -0.00470`

**Inertial Measurement Unit (IMU)**

Orientation information is only available on ZeroKey Mobiles that contain a 6 DOF IMU. Currently, the QTM-UMR-10, QTM-DWR10, QTM-SMR10, and QTM-HMC10 are all enabled with this hardware feature.

By using quaternions, ZeroKey Quantum RTLS can accurately represent the orientation of Mobiles without encountering singularities and discontinuities, making them ideal for real-time applications where precise orientation tracking is paramount. Additionally, they are computationally efficient and can be easily manipulated using quaternion algebra, facilitating complex calculations needed for orientation tracking and motion analysis.