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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Position sensing with inertial sensors such as accelerometers and gyroscopes usually requires other aided sensors or prior knowledge of motion characteristics to remove position drift resulting from integration of acceleration or velocity so as to obtain accurate position estimation. A method based on analytical integration has previously been developed to obtain accurate position estimate of periodic or quasi-periodic motion from inertial sensors using prior knowledge of the motion but without using aided sensors. In this paper, a new method is proposed which employs linear filtering stage coupled with adaptive filtering stage to remove drift and attenuation. The prior knowledge of the motion the proposed method requires is only approximate band of frequencies of the motion. Existing adaptive filtering methods based on Fourier series such as weighted-frequency Fourier linear combiner (WFLC), and band-limited multiple Fourier linear combiner (BMFLC) are modified to combine with the proposed method. To validate and compare the performance of the proposed method with the method based on analytical integration, simulation study is performed using periodic signals as well as real physiological tremor data, and real-time experiments are conducted using an ADXL-203 accelerometer. Results demonstrate that the performance of the proposed method outperforms the existing analytical integration method.

It is a well-known fact that the use of numerical integration of acceleration/angular rate information from inertial sensors (accelerometers/gyroscopes) to obtain position/orientation information inherently causes position/orientation errors to grow with time, which is commonly known as “integration drift”. For that reason, estimation of position/orientation using inertial sensors is performed with the help of externally-referenced aided sensors or sensing systems [

With the aided sensors or sensing systems, Kalman filters (KF) or extended-Kalman filters (EKF) are commonly used to fuse two sources of information: one coming from the inertial sensors, and the other from aided sensors or sensing systems in an attempt to correct for the drift. For example, correction of orientation drift using EKF and a magnetometer as an aided sensor is described in [

An example application of the use of inertial sensors with prior knowledge of motion is in human-walking studies. The use of prior knowledge of motion of human walking makes it possible to avoid the use of aided sensors or sensing systems for correction of the drift [

These algorithms can estimate desired periodic signals from a mixture of desired periodic signals and undesired signals without altering the phase and magnitude of the desired periodic signal. However, the WFLC and the BMFLC have limitations in that the magnitude of the undesired signals comparing to that of the desired periodic signal cannot be too large in order to achieve satisfactory accuracy of the estimate [

To obtain drift-free position estimates of periodic or quasi-periodic motion using inertial sensors without employing other aided sensors or sensing systems, one possible solution is to employ linear high-pass filtering of drifted position by choosing a cutoff frequency somewhere between the frequencies of low-frequency drift signal and that of the periodic motion which has relatively high frequency. However, linear filtering inherently introduces phase-shift and attenuation [

In this paper, a method is proposed in which a combination of linear filtering and modified-WFLC or modified-BMFLC is employed. The integrated signal will be filtered using a high-pass linear filter. The filtered signal, which is the phase-shifted and attenuated version of the actual desired periodic signal, will then be estimated using WFLC or BMFLC algorithms. Accordingly, the estimate will be the phase-shifted and attenuated version of the actual periodic signal. The estimate of the actual periodic signal is recovered from the phase-shifted and attenuated estimate by compensating for the phase-shift and attenuation introduced by the filter. The compensation is achieved with modification of existing algorithms WFLC and BMFLC.

The main idea behind the proposed method relies on the knowledge of the specification of the linear filter employed in filtering and on that of the frequency content of the desired periodic signal to be estimated. If specification of a filter and frequency of an input signal to the filter are known, the amount of phase-shift and attenuation of the signal at the output of the filter can be known. Using the knowledge of the amount of phase-shift and attenuation introduced by the filter for a particular frequency, compensation for the phase-shift and attenuation of each frequency component in the periodic signal can be performed.

Since WFLC and BMFLC algorithms can provide information on the frequencies in the desired periodic signal together with their respective amplitudes, the algorithms are well suited for the proposed method. In Section 2, the existing algorithms are explained briefly and the modified versions of the algorithms together with the proposed method are presented.

In sub-section 2.1 existing methods for zero-phase estimation of periodic signals are discussed first. It should be clear that the existing methods are not the contribution of this paper, but are described briefly to aid readers clearly understand the proposed method which is the contribution of the paper. In sub-section 2.2, the proposed method of drift-free position estimation using inertial sensors is described. Since the proposed method requires the use of an existing estimation method and its modification is required, modification to the existing methods are proposed and described. It should be noted that analytical integration method for drift-free estimation described in [

Weighted Fourier Linear Combiner (WFLC) [

The WFLC [

The reference input vector to WFLC, _{k} = [_{1k} … _{2Mk}]^{T} is:
_{k}_{k}_{0k}, which is required in the reference input vector, is estimated by modified LMS as follows:
_{0} is adaptive gain parameter.

An estimation of the desired periodic or quasi-periodic signal in the input can be calculated as:

However, it should be noted that a good estimation is achieved only when magnitudes of other undesired signal components in _{k}_{k}

One limitation of WFLC is its inability to extract a periodic signal containing more than one dominant frequency. To overcome that, BMFLC [

The reference input to BMFLC is:
_{r}

An estimate of the desired signal can be given by:

Again, as with WFLC, a good estimation is achieved only when magnitudes of other undesired signal components in _{k}_{k}

In this section, the proposed method of drift-free estimation of desired periodic or quasi-periodic signal using one of the algorithms described in the previous section, and compensation for the phase-shift and attenuation introduced by the linear filters is described. The proposed method is described using acceleration as representative inertial sensor output. A block diagram describing the method to obtain the position estimate of desired periodic or quasi-periodic motion which is sensed by an accelerometer is shown in

In the figure, _{k}_{k}_{k}

The input signal to WFLC or BMFLC, _{k}_{k}_{k}_{k}_{k}_{k}

Although employing a single high-pass filer with an appropriate cutoff frequency might be sufficient to remove the drift before filtering with WFLC or BMFLC algorithms, the modifications made to the algorithms are presented in general for any number of linear filters employed. Therefore, the method can handle inherent filters of the sensors.

Let
^{th}^{th}_{0k} at ^{th}

A compensated estimate (an estimate with compensation for phase-shift and attenuation) of the desired signal becomes:

The block diagram of the compensation using the modified reference input in WFLC is shown in

In the case of BMFLC, assuming
^{th}_{r}_{k}

The block diagram of the compensation using the modified reference input in BMFLC is shown in

In this section, simulations with periodic signals and real physiological tremor data are presented. For illustration of the proposed method in implementation, the simulations are performed with the modified BMFLC algorithm.

For the simulation with periodic signals, a synthesized periodic acceleration consisting of sinusoidal components with amplitudes 200 mm/s^{2} and frequencies of 10 Hz, 11 Hz, and 11.5 Hz is used as a desired periodic signal whose position is to be estimated. The desired periodic signal is superimposed with Gaussian white noise having a standard deviation of 40 mm/s^{2} and a DC offset value of 5,000 mm/s^{2} to form the simulated noisy acceleration output from an accelerometer. The DC offset is to simulate a sensor bias voltage while noise represents sensor noise. The simulated noisy acceleration output is shown in

To simulate the hardware filter of an accelerometer, a first-order low-pass filter with the cutoff frequency of 50 Hz is used. Frequency responses of the low-pass filter and the high-pass filter are pre-calculated and the components in the range of 8 to 12 Hz are shown in

The cutoff frequency and the order of the high-pass filter are to be chosen such that the filter removes unwanted low-frequency components of drift significantly. Therefore, it depends on the inertial sensor’s noise level that determines the low-frequency components of drift. For noise level used in the simulation, cutoff frequency of 5 Hz, and fourth order are chosen since it removes the unwanted low-frequency drift significantly. The simulated noisy periodic acceleration output is filtered with the low-pass filter before it is numerically double-integrated to obtain the position. The position obtained from the integration is then used with the BMFLC algorithm.

The following parameters are used for implementation of BMFLC:

The values of
_{1k} and _{6k} are given by:
_{1} = 0.9981, _{2} = 0.9852, _{1} = 77.88 deg, and _{2} = −9.86 deg (refer to

To compare estimation performance of the proposed method with that of analytical integration method, the simulated periodic acceleration is filtered by the simulated filter of the accelerometer, and the filtered signal is then double-integrated analytically using the method described in [

To show that the proposed method also works very well with gyroscopes, a simulated periodic angular velocity consisting of the same frequency content as the simulated periodic acceleration is used. The amplitude of each component is set at 200 mrad/s. Other settings are kept the same, except that the periodic angular velocity is integrated only once to obtain orientation.

The position obtained from the integration, which contains periodic position and drift, is shown in

The phase-shift and attenuation is eliminated using the proposed method as can be seen in

In the previous section, simulation of the proposed method using pure periodic signals was presented. In order to show that the method works well with the quasi-periodic signals such as physiological tremor signals since a tremor is approximately a rhythmic (quasi-periodic) signal [

To simulate a hardware filter [

In this section, real-time experiments using an accelerometer as a representative inertial sensor are described. A periodic motion is generated using a commercially available nanopositioning stage (P-561.3CD from Physik Instrumente, Germany) on which a physiological tremor compensation instrument consisting of accelerometers is mounted. The accuracy of the used positioning stage is better than 100 nm. The accelerometers used in the instrument are ADXL-203 accelerometers from Analog Devices. However, only one accelerometer is used for the experiments to show the effectiveness of the proposed method. The setup for the experiment is shown in

The tremor is defined as roughly sinusoidal, and approximately rhythmic [^{2} although the accelerometer’s measurement range is ±1.7 g. The voltage output from the accelerometer is acquired at 500 Hz using a 16-bit data acquisition (DAQ) card (PD2-MF-150, United Electronic Industries, Inc, USA).

Before the experiment, static calibration of the accelerometer is performed using a gravity value of 9.81 m/s^{2}. The sensitivity value of the accelerometer obtained from the calibration is 980 mV/g. The time constant and hence the specification of the hardware filter of the accelerometer is measured by giving a step input using ST pin of the accelerometer [

As can be seen from

Error plots in

To achieve a good performance with the proposed method, filter order and cutoff frequency need to be chosen so that the filter removes unwanted low-frequency drift significantly. The choice depends on the sensor noise level and the frequency content of the periodic or quasi-periodic signal. However, a wide range of filter orders and cutoff frequencies are available to be chosen, and hence the choice is not too restrictive. For acceptable reasonable filter order and cutoff frequency, the proposed method outperforms the analytical integration method.

Although the effectiveness of the method is proven using a BMFLC algorithm, it is also valid for the WFLC algorithm. The main reason to employ the WFLC or BMFLC algorithms in the proposed method is because the periodic signals can be modeled by a series of sine and cosine components. The algorithms’ performance is the best for pure periodic motion. If the motion to be estimated is not purely periodic, the algorithms’ performance will be degraded and accuracy of estimation will be affected depending on the degree of non-periodicity.

The performance of the algorithms and hence that of the proposed method also depends on the value of adaptive gain μ. In

The algorithms (and hence the proposed method) work very well as long as the approximate band of frequencies of the periodic motion is known. If the periodic motion consists of more than one dominant frequency, the BMFLC algorithm should be used. If there is only one dominant frequency with its harmonics in the periodic motion and the dominant frequency is varying slowly, WFLC is a better choice since WFLC can track the periodic motion whose frequency is varying slowly such as that of physiological tremors [

This research project is funded partly by the Science and Engineering Research Council (SERC) (SERC Public Funding, Grant #052 101 0017), partly by the Biomedical Research Council (BMRC) (BMRC Grant 07/1/22/19/538) of the Agency for Science, Technology & Research (A*STAR), Singapore and partly supported by Brain Korea 21 (BK21) Project, Korea.

WFLC algorithm.

BMFLC algorithm.

The block diagram showing the steps in estimation of position from accelerometer’s data.

The block diagram of the compensation using the modified reference input in WFLC.

The block diagram of the compensation using the modified reference input in BMFLC.

Simulated acceleration output consisting of the sinusoidal acceleration signals with noise and DC offset.

Frequency response of the first-order low-pass filter with a cutoff frequency of 50 Hz.

Frequency response of the 4th order high-pass Butterworth filter with a cutoff frequency of 5 Hz.

True position and position obtained from double-integration of simulated acceleration.

Plots showing effectiveness of the proposed method. The whole plot of the unfiltered position (shown in green line) can be seen in

Position estimation error with the high-pass filter and the proposed method.

Position estimation error obtained with the proposed method (

Orientation estimation error obtained with the proposed method (

Plots showing the physiological hand tremor (true position) of a subject, estimate of the tremor with and without compensation.

Physiological tremor of a subject (

A picture (Left) and a schematic drawing (right) of the experimental setup.

Experimental determination of time-constant of the accelerometer hardware filter.

(

Estimation errors obtained from (

Position estimation errors with the proposed method and the analytical integration method with and without the simulated hardware filter.

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14.8 | 43.6 | 24.1 | 65.1 | |

6.2 | 23.1 | 6.5 | 23.6 | |

58.2 | 47 | 73 | 64 | |

10.5 | 17.7 | 11.1 | 18.1 |

Orientation estimation errors with the proposed method and the analytical integration method with and without the simulated hardware filter.

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1.00 | 2.92 | 1.64 | 4.45 | |

0.54 | 1.78 | 0.56 | 2.03 | |

45.77 | 39.18 | 66.79 | 60.1 | |

15.01 | 20.58 | 15.68 | 23.49 |

Mean and standard deviation of RMS errors and maximum (peak) errors of estimation of physiological tremor signals from ten subjects with and without proposed compensation method.

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3.46 ± 1.42 | 15.19 ± 6.85 | |

2.31 ± 0.73 | 11.77 ± 7.1 | |

30.17 ± 11.46 | 23.73 ± 17.84 |

Estimation errors obtained with the analytical integration and the proposed method.

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11.5 | 18.9 | |

3 | 8.9 | |

75 | 52 |

RMS error and maximum error of real-time estimation of 10 Hz periodic motion with and without compensation for the phase-shift and attenuation introduced by the filters.

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44 | 66 | |

3.0 | 9 | |

93 | 86 |