This document discusses modeling and simulating MEMS accelerometers and gyroscopes using order-reduction methods. It begins with an introduction stating the need for new modeling techniques for MEMS integrated into SoCs. It then outlines experiments to apply SVD and QRD order-reduction to behavioral models of the devices. The document describes creating physical and mathematical Simulink models of the accelerometer and gyroscope. It provides examples of simulation results comparing the original and reduced order models. Analysis shows the reduced models achieve simulation time reductions of 1-26% using SVD and 1-45% using QRD, while decreasing memory usage to 50%. Mathematical models were found to simulate faster than physical ones. In conclusion, SVD provided better time reductions for acceler

1. MODELING AND SIMULATION OF MEMS
ACCELEROMETERS AND GYROSCOPES
USING ORDER-REDUCTION METHODS
RABIE A. EL-ZOGHBI1, VLADIMIR N. LANTSOV1, AND AHMAD J. BAZZI2
1VLADIMIR STATE UNIVERSITY, RUSSIAN FEDERATION
2GRADUATE SCHOOL OF ENGINEERING, GUNMA UNIVERSITY, JAPAN

2. Outline
 Introduction
 Experiments Outline
 MEMS Modeling
 Simulink – Accelerometer
 Simulink – Gyroscope
 MEMS Simulation
 SVD Application
 QRD Application
 Simulation Results Examples (reduced vs. original models)
 Simulation Results Analysis
 Conclusions

3. Introduction
Current research in design methods for integrating micro electromechanical systems
(MEMS) into system-on-chip (SoCs) shows that there is a need for new modeling and
simulation techniques.
We suggest implementing behavioral modeling of MEMS and applying methods of
order-reduction (MOR) to these models to save time and resources in both design
process and simulation.
In this paper, we implemented MEMS accelerometer and gyroscope behavioral models
(both physical and mathematical) and utilized singular values decomposition (SVD)
and orthogonal-triangular decomposition (QRD) methods, which are suitable for
such over determined models.
This resulted in improving the simulation time and system resources, with minimum error
between the reduced and original models.

4. Experiments Outline
MEMS Device Behavioral Model Input Signal MOR
Accelerometer Physical Model
(Mechanical)
Gaussian Signal
(Arbitrary)
SVD
QRD
Sinusoidal Signal
(Synchronous)
SVD
QRD
Mathematical Model Gaussian Signal SVD
QRD
Sinusoidal Signal SVD
QRD
Gyroscope Physical Model Gaussian Signal SVD
QRD
Sinusoidal Signal SVD
QRD
Mathematical Model Gaussian Signal SVD
QRD
Sinusoidal Signal SVD
QRD

5. MODELING OF MEMS ACCELEROMETER
AND GYROSCOPE

6.  Due to the mechanical nature of MEMS devices,
modeling starts with their physical behavior, upon
which a mathematical model is constructed.
 We used mathematical equations describing the
physical model of MEMS from micromechanical
references .
MEMS Modeling

7. Accelerometer Model
 The behavior of MEMS accelerometer is a typical
enforced mass-damper-spring system.
 Represented by the
equation:
 F = m.x”+ k.x + c.x’

8.  System of 2 equation:
 Where:
 MEMS output:
Gyroscope Model

9. Simulink – Accelerometer

10. Simulink – Accelerometer
Physical Model (using Simscape lib.)

11. Simulink – Accelerometer
Mathematical Model

12. Simulink – Gyroscope

13. Simulink – Gyroscope
Physical Model

14. Simulink – Gyroscope
Mathematical Model

15. SIMULATION OF MEMS ACCELEROMETER
AND GYROSCOPE

16. Simulation
 Each model is simulated with 2 input sets
consecutively for 100 seconds:
 Gaussian signal (Arbitrary values)
 Sinusoidal signal
After that, the input matrix A and output matrix B are
constructed with dimension mxn = 5000x10

17. Input sets applied to each model

18. Example of applied inputs

19. Example of MEMS Accelerometer output reading

20. Example of MEMS Gyroscope output reading

21. SVD Application
 The SVD of A, mxn of order r:
 A = UxSxVt
 Where U is orthonormal mxm
matrix, S is diagonal nxn
matrix containing importance
coefficients sorted in
descending order, V is nxn
matrix.
 The reduced matrix is of
order k < r.
 In our experiment, since S1
>> S2, S3, etc.., then the
order of the reduced model is
1.
 For comparison, the reduced
models for order k = 2 to 10
were constructed.
Also, the reduced output
models were compared to the
original one.

22. QRD Application
 QRD of A:
 A = QxR;
 Where Q is orthonormal mxm matrix, and R is
upper triangular nxn matrix.
 In our experiments, order of the reduced models
varied from order k = 1 to 2 in according to the
least simulation time and error.

23. This Simulink model was designed to calculate the error of each
reduced model of order k, and compare the reduced output to
the original one for all models and inputs.

24. Example of reduced and original outputs of order 1,2,3,9
Accelerometer, physical model, arbitrary input, SVD
(Optimal is order 1)

25. Example of reduced and original outputs of order 1,2,3,9.
Accelerometer, mathematical model, arbitrary input, QRD
(Optimal is order 2)

26. Example of reduced and original outputs of order 1,2,3,9
Gyroscope, physical model, sinusoidal input, SVD
(Optimal is order 1 – fastest simulation)

27. Example of reduced and original outputs of order 1,2,3,9
Gyroscope, mathematical model, sinusoidal input, QRD
(Optimal is order 2 – minimum error)

28. SIMULATION RESULTS ANALYSIS

29. Accelerometer, physical model,
arbitrary input, SVD & QRD
In this model, using SVD
yields to reduce the original
system to order 1 with
simulation time = 10.059 %
of the original simulation
time and 49.94 % of
memory usage.
Using QRD, the optimal to
use order 2 with simulation
time = 23.53 % of the
original simulation time and
the same memory decrease
as SVD.
0
2
4
6
8
10
12
14
16
18
20
22
24
SimulationTimeins
Simulation Time Decrease
svd
qr
0
0.5
1
1.5
2
2.5
3
3.5
k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10
Errorin%
Reduced model error
svd
qr

30. Accelerometer, physical model,
sinusoidal input, SVD & QRD
 In this model, using SVD
yields to reduce the
original system to order
1 with simulation time =
8.228 % of the original
simulation time and
49.94 % of memory
usage.
 Using QRD, the optimal
to use order 2 with
simulation time = 45.8%
of the original simulation
time and the same
memory decrease as
SVD.
0
10
20
30
40
50
60
70
SimulationTimeins Simulation Time Decrease
qr
svd
0
0.2
0.4
0.6
0.8
1
1.2
k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10
Errorin%
Reduced model error
svd
qr

31. Accelerometer, mathematical model,
arbitrary input, SVD & QRD
In this model, using SVD
yields to reduce the original
system to order 1 with
simulation time = 12.9 % of
the original simulation time
and 49.94 % of memory
usage.
Using QRD, the optimal to
use order 1 with simulation
time = 23.53 % of the
original simulation time and
the same memory decrease
as SVD.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k =
10
Errorin%
Reduced model error
svd
qr
0
2
4
6
8
10
12
14
16
18
20
22
24
SimulationTimeins
Simulation Time Decrease
svd
qr

32. Accelerometer, mathematical model,
sinusoidal input, SVD & QRD
0
2
4
6
8
10
12
14
16
18
20
22
24
SimulationTimeins
Simulation Time Decrease
svd
qr
0.65
0.7
0.75
0.8
0.85
0.9
0.95
k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k =
10
Errorin%
Reduced model error
svd
qr
In this model, using SVD
yields to reduce the original
system to order 1 with
simulation time = 1.064 %
of the original simulation
time and 49.94 % of
memory usage.
Using QRD, the optimal to
use order 1 with simulation
time = 1.001 % of the
original simulation time and
the same memory decrease
as SVD.

33. Gyroscope, physical model,
arbitrary input, SVD & QRD
0
2
4
6
8
10
12
14
16
18
20
22
24
SimulationTimeins
Simulation Time Decrease
svd
qr
0
1
2
3
4
5
6
k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10
Errorin%
Reduced model error
svd
qr
In this model, using SVD
yields to reduce the original
system to order 1 with
simulation time = 16.75 %
of the original simulation
time and 49.94 % of
memory usage.
Using QRD, the optimal to
use order 2 with simulation
time = 16.705 % of the
original simulation time and
the same memory decrease
as SVD.

34. Gyroscope, physical model,
sinusoidal input, SVD & QRD
0
2
4
6
8
10
12
14
16
18
20
22
24
SimulationTimeins
Simulation Time Decrease
svd
qr
0
1
2
3
4
5
6
k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10
Errorin%
Reduced model error
svd
qr
In this model, using SVD
yields to reduce the original
system to order 1 with
simulation time = 26.75 %
of the original simulation
time and 49.94 % of
memory usage.
Using QRD, the optimal to
use order 2 with simulation
time = 16.705 % of the
original simulation time and
the same memory decrease
as SVD.

35. Gyroscope, mathematical model,
arbitrary input, SVD & QRD
0
2
4
6
8
10
12
14
16
18
20
22
24
SimulationTimeins
Simulation Time Decrease
svd
qr
0
1
2
3
4
5
6
k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10
Errorin%
Reduced model error
svd
qr
In this model, using SVD
yields to reduce the original
system to order 1 with
simulation time = 22.8 % of
the original simulation time
and 49.94 % of memory
usage.
Using QRD, the optimal to
use order 2 with simulation
time = 16.919 % of the
original simulation time and
the same memory decrease
as SVD.

36. Gyroscope, mathematical model,
sinusoidal input, SVD & QRD
0
2
4
6
8
10
12
14
16
18
20
22
24
SimulationTimeins
Simulation Time Decrease
svd
qr
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10
Errorin%
Reduced model error
svd
qr
In this model, using SVD
yields to reduce the original
system to order 1 with
simulation time = 12.02 %
of the original simulation
time and 49.94 % of
memory usage.
Using QRD, the optimal to
use order 1 with simulation
time = 11.16 % of the
original simulation time and
the same memory decrease
as SVD.

37. Conclusions
 After analyzing the simulation results from all experiments, we can
conclude the following:
 SVD gives more decreased simulation time for accelerometer than
QRD, with error approximately similar to that of QRD.
 QRD gives better simulation time in the case of the gyroscope, but
with bigger error that of SVD, hence, in accepted limits.
 Using order-reduction methods for MEMS simulation decreases
simulation time in a range from 1 to 26 % of the original time in the
case of SVD, while QRD in a range from 1 to 45 %.
 In both methods, memory usage drops to 50 %.
 Mathematical models simulate faster than physical ones. Besides,
tuning and parameterization of the design is easier and faster.

38. THANK YOU