# The Application of Quaternions to Strap-Down MEMS Sensor Data

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## Abstract

**:**

## 1. Introduction

#### 1.1. Strap-Down Microelectromechanical Sensors

#### 1.2. Introduction to Quaternions

**u**, the rotation can be represented as a combination of the scalar θ and vector

**u**. The quaternion,

**q**, appears in the unusual form of the sum of a scalar, s, and vector,

**v**

**i**,

**j**, and

**k**are equivalent to the Cartesian unit vectors but are the basic quaternion units that have the unusual, non-commutative, multiplicative properties, known as the Hamilton products

**p**and

**q**, may be written as

**q**followed by

**p**.

**q**is a unit quaternion its inverse is the conjugate quaternion,

**q***, and is found simply by changing the sign of the imaginary part

**q**, representing a rotation of θ about the axis,

**u**, is constructed as

**v**, is treated as a quaternion with real part ${v}_{0}$= 0, giving

**q**is the orientation quaternion for an object in Cartesian space, then Equations (6)–(9) also represent the transformation of vectors from the object’s FOR into Cartesian coordinates.

#### 1.3. Euler Angles

## 2. Application to Object for Inertial Data

#### Other Practical Considerations

## 3. An Illustrative Example

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The roll, pitch, and yaw of the blast loaded object as a function of time. The object had an initial nose down pitch of 10.5° in the earth’s FOR.

**Figure 4.**Selected animation frames showing the motion of the test object being thrown into the soft-catch pit. Time of 0.01 s shows the initial location prior to movement and 2.01 s shows the final resting location.

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Dickson, P.M.; Rae, P.J.
The Application of Quaternions to Strap-Down MEMS Sensor Data. *Sensors* **2021**, *21*, 7658.
https://doi.org/10.3390/s21227658

**AMA Style**

Dickson PM, Rae PJ.
The Application of Quaternions to Strap-Down MEMS Sensor Data. *Sensors*. 2021; 21(22):7658.
https://doi.org/10.3390/s21227658

**Chicago/Turabian Style**

Dickson, Peter M., and Philip J. Rae.
2021. "The Application of Quaternions to Strap-Down MEMS Sensor Data" *Sensors* 21, no. 22: 7658.
https://doi.org/10.3390/s21227658