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This paper investigates experimental design (DoE) for the calibration of the triaxial accelerometers embedded in a wearable micro Inertial Measurement Unit (μ-IMU). Firstly, a new linearization strategy is proposed for the accelerometer model associated with the so-called autocalibration scheme. Then, an effective Icosahedron design is developed, which can achieve both D-optimality and G-optimality for linearized accelerometer model in ideal experimental settings. However, due to various technical limitations, it is often infeasible for the users of wearable sensors to fully implement the proposed experimental scheme. To assess the efficiency of each individual experiment, an index is given in terms of desired experimental characteristic. The proposed experimental scheme has been applied for the autocalibration of a newly developed μ-IMU.

Wearable health monitoring system is one of the most promising technologies to provide effective solutions to health monitoring for aging populations. Various wearable sensors equipped with artificial intelligence, e.g., neural networks, fuzzy logical, genetic algorithm, particle swarm optimization, and clustering, have already been utilized for specific health monitoring tasks [

With the rapid development of Micro-Electro-Mechanical Systems (MEMS) technology, chip-based wearable sensors are becoming small, inexpensive, lightweight, and low energy-consuming, which stimulate their applications in the development of wearable systems in health monitoring [

Several recent papers [

This paper aims to provide a systematic investigation of Experimental Design (DoE) for autocalibration method. A major focus of DoE is to optimally design suitable input signals to stimulate the system significantly so that the information about the system can be extracted from the experiments. For the identification of a static model of an inertial sensor, a well selected/designed set of experimental observations with desired properties, in terms of DoE, can significantly improve the accuracy of parameter estimation [

Classical accelerometer calibration, normally carried out in a well-controlled laboratory environment, can be formulated as a static linear parameter identification problem, for which DoE theory has been well established [

In this study, a new linearization method for autocalibration [

The paper is structured as follows. The next section introduces the proposed linearization method for the 9- parameter model for autocalibration. In Section 3, an experimental design is proposed and the details of its indices will be analysed. Section 4 shows experimental validation of the designed experiment and Section 5 concludes the paper.

A classical static linear second-order model for an accelerometer can be written as follows:

where

Assume a set of

where

Assume that the random errors

where

which is also known as moment matrix. If

The variance of

In order to compare within different experimental designs, the scaled prediction variance is often defined as follows [

To apply DoE theory for the calibration of MEMS accelerometer, we define uncalibrated acceleration gener-

ated from accelerometer output as

is the real acceleration component on each axis.

A model describing the accelerometer can then be expressed in matrix form as below:

where

The autocalibration method is based on the fact that the overall acceleration which is measured by triaxial accelerometer should equal to the local gravity acceleration “1g” in static condition. The principle of autocalibration is:

By applying the method of autocalibration (see Equation (9)) for the 9-parameter model from Equation (8), we have:

where

where

Equation (11) cannot be written in the form of Equation (1) because of its nonlinearity with the parameters. To estimate the parameters, most existing studies use either nonlinear least square method [

Inspired by these studies, this paper proposes a new linearization scheme to directly linearize Equation (11) and transform it in the form of Equation (1). From this, mature linear DoE theory can be directly applied to handle experimental design and parameter estimation for the autocalibration scheme. In contrast with local linearization (e.g., Taylor expansion around the observation point), the main strategy of the proposed linearization method is based on re-combination of parameters.

Firstly, this approach disregards the items in Equation (11) which have little impact on parameter estimation.

From

We can apply the same simplification method for

where

From Equation (14), let us define new parameters for the re-combined parameters as follows:

Let us use

This can be simplified as:

Since

Parameter | Min | Typ | Max | Unit |
---|---|---|---|---|

Cross-axis | % | |||

Sensitivity (2 g range) | 230 | 256 | 282 | LBS/g |

0 g offset for X, Y | −150 | 0 | 150 | mg |

0 g offset for Z | −250 | 0 | 250 | mg |

Offset vs. temperature X, Y | mg/˚C | |||

Offset vs. temperature Z | mg/˚C |

Applying linear least square estimation (LSE) method for the simplified linear model, we have

where

and

Based on linear least square method, all new unknown parameters from Equation (16) can be estimated. According to the definition of Equation (15), the original 9 independent parameters

In order to estimate 9 unknown parameters, a minimum number of 9 observations are necessary. To balance the cost and accuracy, we propose a 12-observation Icosahedron design, which is a space filling design aiming for the uniformly distribution of experimental observations on experimental domain. This experimental design is for the linearized 9-parameter model derived in Section 2. The idea of Icosahedron design is that all 12 observations uniformly distribute on the surface of sphere.

Due to the constraint of the gravity based calibration, all the experimental observations will be situated uniformly on the surface of a sphere whose radius equals to local gravity “1g”. In another word, these 12 experimental observations will construct an Icosahedron whose circumcircle has radius of “1g”.

For Icosahedron design, if the radius of its circumcircle is “1g”, then all 12 observations can be pinpointed on rectangular coordinate system (see

For each individual observation, the relationship of

・

・

・

G-optimal design is seeking to minimize the maximum value of the scaled prediction variance in Equation (7) over the experimental region [

Observation | A | B | C | |||
---|---|---|---|---|---|---|

1 | 0 | −a | −b | |||

2 | 0 | a | −b | |||

3 | 0 | −a | b | |||

4 | 0 | a | b | |||

5 | −a | −b | 0 | |||

6 | a | −b | 0 | |||

7 | −a | b | 0 | |||

8 | a | b | 0 | |||

9 | −b | 0 | −a | |||

10 | b | 0 | −a | |||

11 | −b | 0 | a | |||

12 | b | 0 | a |

G-optimal is an important measurement of performance which indicates satisfactory prediction of output throughout the design region.

We propose the following theorem to show the proposed Icosahedron design is G-optimal.

Theorem 1. The proposed Icosahedron design for the linearized 9-parameter accelerometer model

is G-optimal.

Proof. The variance equation of predicted

Recall scaled prediction variance from Equation (7):

A G-optimal design

Regarding Equation (21), G-optimal is equivalent to

where

According to [

where

That is, for a specific experimental design, if

then this experimental design is G-optimal design [

Consider the Icosahedron design proposed above for 9-parameter model:

Recall Icosahedron design, matrix

Substituting the value of

Considering that

we have

Under the constrain

According to the theorem of G-optimal in [

Another desired design characteristic of experimental design is D-optimality. The criterion of D-optimality is maximizing the determinant of the information matrix for continuous design or moment matrix for exact design [

which leads to minimize the size of the confidence ellipsoid for the estimator

Kiefer and Wolfowitz [

Based on KWT theorem, we show the proposed Icosahedron design is also D-optimal.

Theorem 2. The proposed Icosahedron design is D-optimal for the linearized 9-parameter model

Proof. Let us consider a continuous experimental design

To convert exact design

Based on KWT Equivalence Theorem [

The

Based on the described experimental design in Section 3, we tried to implement the Icosahedron design. It is not supervising that the proposed plan cannot be fully implemented by using the “non-professional” calibration devices. To access the quality of a specific experiment, we adopt the following index, D-efficiency [

where

To evaluate the efficiency of a specific experiment, ideally, we need the exact input value of each observation for a specific design

However, for auto-calibration, the input value on each axis of a particular observation, denoted by a vector A, cannot be directly measured from the calibration device. We therefore have to use the output of the accelerometer, which is under calibration, to estimate the real input acceleration A. Equation (8) describes the relationship between the uncalibrated acceleration output V and the real acceleration input A if assuming the scale factor and offset are accurate. Let us recall Equation (8) and simplify it as:

In order to obtain real acceleration A, we need to compute scale factor S and offset O first. Towards the end of Section 2, we mentioned the scale factor

where Y is local gravity “1 g”, V_{1} represents uncalibrated acceleration from accelerometer, B_{1} is vector of re- combined parameters defined in Equation (15),

where Y is local gravity “1 g”, V_{1} is the known quantity from accelerometer and B_{1} is re-combined parameter by scale factor S and offset O. S_{1} and O_{1} can then be solved by using LSE as shown in Equation (18).

From Equation (29), we have

Due to the fact that we neglected some little impact items during LSE, A_{1} will not be exactly the same as real acceleration A, but A_{1} is closer to real acceleration A comparing to V_{1}. In this case, when A_{1} is closer to A, the value of offset O will be reduced. Recall from Section 2 that all disregarded items contain offset O, it means the mean of the summation of all disregarded items _{1} with A_{1} (A_{1} is marked as

From Equation (15) and Equation (18), the new scale factor S_{2} and offset O_{2} can then be solved. Recall Equation (29):

In this case,

Let us repeat this procedure, A_{i} is approaching to real acceleration A while offset O is reducing to 0. The accuracy of LSE will increase because all disregarded items contain offset O will drop to 0. Eventually, offset O and cross-axis factors

The overall equation of Equation (29) is:

Now, as

We performed the calibration experiment in the Center of Health Technologies (CHT), University of Technology, Sydney (UTS), without using a turntable. In contrast with the ideal setting, the posterior type D-effi- ciency for our experiment is around 99.7% which is slighter smaller than 100%. It indicates our experiment achieved desired results.

Experimental results also showed the Mean Square Error (MSE) has been reduced from 0.00344 g^{2} to 0.000255 g^{2} by the proposed experimental design/calibration method.

This study investigates the DoE for autocalibration of triaxial accelerometer in a wearable micro Inertial Measurement Unit (μ-IMU), and our contribution is two-fold. Firstly, a new model linearization strategy is proposed to linearize the nonlinear model associated with the autocalibration of triaxial accelerometer. The major technique of the proposed linearization strategy is based on recombination of parameters rather than local linearization around observation point (e.g. Taylor expansion). With such a linearized model, the classical linear model identification and DoE approaches can be applied to calibrate the triaxial accelerometer in a non-experimental environment. The second contribution is that this paper introduces a new experimental scheme, Icosahedron design. We have proved that this scheme is both G-optimal and D-optimal for the linearized 9-parameter triaxial accelerometer model. Experimental results also demonstrate that the proposed DoE scheme can significantly decrease the MSE of triaxial accelerometer after calibration. This indicates that the proposed linearization method is reliable and efficient. We believe that the proposed experimental design approach can provide an efficient tool for the users of wearable sensors to efficiently calibrate the sensors in free living condition.