1. Paige Thielen, ME535 Spring 2018
Least Squares Accelerometer Calibration in Precision Measurement Equipment
Paige Thielen, ME535 Spring 2018

2. Abstract
Various methods of accelerometer calibration can be used to increase the precision of acceleration measurements. The methods tested are two 12-parameter linear least squares optimizations, one using four calibration orientations, one using eight orientations, and two 15-parameter least squares optimizations using eight and 19 calibration orientations. Based on the data gathered, while it is not necessary to change the calibration method currently in use, good results could be obtained from applying a 12-parameter, 8-orientation least squares calibration without significant increase in time required for calibration.

3. Introduction
The system being analyzed results from a project that I worked on for almost two years at my previous job
I was tasked with investigating inconsistencies encountered during calibration which would cause two subsequent test sequences to yield different results
Intention was to characterize the amount of error between measurements taken during identical test profiles with the same DUT
I use measurements taken during that investigation, along with the accelerometer manufacturer’s guidelines as a basis for this study

4. Introduction
Compare various least squares calibrations to determine which method would be sufficient for the level of accuracy desired in the equipment
The best possible method is one that requires the least number of calibration positions to compute a model that will provide the desired level of accuracy
The company does not currently use a least squares method for calibrating the equipment and my previous analysis proved that the method of calibration currently in use is sufficiently accurate to provide a measurement with a relatively low calibration time (~4 minutes)

5. Device Under Test (DUT)
Level sensor designed to measure roll and pitch of specialized equipment
Each device contains two accelerometers which measure the position of three axes: x, y, and z
During assembly, accelerometers are inserted into the body of the transmitter at an unknown orientation
x-axis is approximately aligned with the axis of the (cylindrical) measurement device
y- and z-axes can be rotated at any angle with respect to the 12 o’clock position

6. System Model 𝑝𝑖𝑡𝑐ℎ= sin −1 𝑥 𝑐 𝑟𝑜𝑙𝑙= tan −1 𝑦 𝑐 𝑧 𝑐
xc, yc, and zc are corrected accelerometer outputs
Need to determine xc, yc, and zc using least squares

7. Methods 12 parameter least squares at 4 orientations
21 orientations are combination of suggested orientations for 4 points, 8 points, 6 points, and 3 points in manufacturer’s manual
Current calibration method for comparison

8. Methods
4 Points
8 Points

9. Methods
4 Points

10. Methods
8 Points

11. 12 Parameter Model
W and V are the parameters to be estimated from least squares
𝐺 𝑓𝑥 𝐺 𝑓𝑦 𝐺 𝑓𝑧 = raw accelerometer readings in x, y, and z, normalized to g
𝐺 12𝑥 𝐺 12𝑦 𝐺 12𝑧 ≈ −𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙 , 𝜃 is pitch angle and 𝜙 is roll angle

12. 12 Parameter Model
Elements of G12 represent the true x, y, and z components of the applied gravitational field at each measurement orientation
The matrix X of measurements of the independent variables is
𝑿= 𝑥 1 𝑦 1 𝑧 𝑥 2 𝑦 2 𝑧 2 1 ⋮ ⋮ ⋮ 1 𝑥 𝑚 𝑦 𝑚 𝑧 𝑚 1

13. 12 Parameter Model
𝛽 𝑥 = 𝑊 𝒙𝒙 𝑊 𝒙𝒚 𝑊 𝒙𝒛 𝑉 𝑥 , and its y and z versions, are calculated using least squares:
𝛽 𝑥 = 𝑿 𝑇 𝑿 −1 𝑿 𝑇 𝒀 𝑥
𝛽 𝑦 = 𝑿 𝑇 𝑿 −1 𝑿 𝑇 𝒀 𝑦
𝛽 𝑧 = 𝑿 𝑇 𝑿 −1 𝑿 𝑇 𝒀 𝑧

14. 15 Parameter Model
W, V, and 𝚪 are the parameters to be estimated from least squares
𝐺 𝑓𝑥 𝐺 𝑓𝑦 𝐺 𝑓𝑧 = raw accelerometer readings in x, y, and z, normalized to g
𝐺 15𝑥 𝐺 15𝑦 𝐺 15𝑧 ≈ −𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙 , 𝜃 is pitch angle and 𝜙 is roll angle

15. 15 Parameter Model
Elements of G15 represent the true x, y, and z components of the applied gravitational field at each measurement orientation
The matrix X of measurements of the independent variables is
𝑿 𝒙 = 𝑥 1 𝑦 1 𝑧 𝑥 𝑥 2 𝑦 2 𝑧 𝑥 ⋮ ⋮ ⋮ ⋮ ⋮ 𝑥 𝑚 𝑦 𝑚 𝑧 𝑚 1 𝑥 𝑚 3

16. 15 Parameter Model
𝛽 𝑥 = 𝑊 𝒙𝒙 𝑊 𝒙𝒚 𝑊 𝒙𝒛 𝑉 𝑥 Γ 𝑥𝑥 , and its y and z versions, are calculated using least squares:
𝛽 𝑥 = 𝑿 𝒙 𝑇 𝑿 𝒙 −1 𝑿 𝒙 𝑇 𝒀 𝑥
𝛽 𝑦 = 𝑿 𝒚 𝑇 𝑿 𝒚 −1 𝑿 𝒚 𝑇 𝒀 𝑦
𝛽 𝑧 = 𝑿 𝒛 𝑇 𝑿 𝒛 −1 𝑿 𝒛 𝑇 𝒀 𝑧

17. Results

18. Results

19. Conclusion
Using 12 parameters at four orientations and 12 parameters at 8 orientations outperformed the 15-parameter calibration at 21 orientations
Maximum reasonable number of measurements that can be taken during each calibration cycle is eight
Current number of calibration positions is six
Current number of parameters is twelve, it would be more difficult to implement 15
Based on these results, there is no compelling reason to change the calibration model unless significant time could be saved to increase production output