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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/).

In this paper, we summarize the results of using dynamic models borrowed from tracking theory in describing the time evolution of the state vector to have an estimate of the angular motion in a gyro-free inertial measurement unit (GF-IMU). The GF-IMU is a special type inertial measurement unit (IMU) that uses only a set of accelerometers in inferring the angular motion. Using distributed accelerometers, we get an angular information vector (AIV) composed of angular acceleration and quadratic angular velocity terms. We use a Kalman filter approach to estimate the angular velocity vector since it is not expressed explicitly within the AIV. The bias parameters inherent in the accelerometers measurements' produce a biased AIV and hence the AIV bias parameters are estimated within an augmented state vector. Using dynamic models, the appended bias parameters of the AIV become observable and hence we can have unbiased angular motion estimate. Moreover, a good model is required to extract the maximum amount of information from the observation. Observability analysis is done to determine the conditions for having an observable state space model. For higher grades of accelerometers and under relatively higher sampling frequency, the error of accelerometer measurements is dominated by the noise error. Consequently, simulations are conducted on two models, one has bias parameters appended in the state space model and the other is a reduced model without bias parameters.

A conventional IMU is composed of three accelerometers and three gyroscopes mounted in a strap-down configuration. Accelerometers are sensors that measure acceleration and gyroscopes are sensors that measure the angular rate of rotation. Gyros are usually corrupted by various sources of errors such as bias instability, noise, scale factor errors…

The use of distributed accelerometers as an alternative to conventional gyros to infer the angular motion has been a subject of intensive research. Unlike the standard IMU, the GF-IMU uses only accelerometers to infer the acceleration and the angular velocity. It is possible to get the Coriolis acceleration vector, which contains a direct expression of the angular velocity vector, through configurations contain rotating accelerometers. However, the focus of this work is on fixed accelerometer configurations because they are simpler to implement. There are several reasons to use accelerometers for inferring the angular motion. Generally, accelerometers are less costly, less heavy and less power consuming than comparable gyros, which have typically the disadvantage of complicated manufacturing techniques, high cost, high power consumption, high weight, large volume, and limited dynamic range [

The rest of this paper is organized as follows: Section 2 gives a background about the angular motion estimation in a GF-IMU and describes the configuration used in this work. Section 3 lists the dynamic models which can be used for the Kalman filter process update. Section 4 gives a sensor error model with a review of the calibration procedure. Section 5 presents an extended Kalman filter (EKF) solution using a Singer model with appended bias parameters. Section 6 presents the observability analysis for the augmented state space model. Section 7 presents an EKF solution without appending bias parameters. Section 8 gives simulation results for the augmented model and Section 9 gives simulation results for the reduced model. Finally, Section 10 presents our conclusions.

Using certain fixed GF-IMU configurations of accelerometers, we get an angular acceleration vector and quadratic terms of the angular velocity. Quadratic angular velocity terms do not have an accumulative error as in the case when the angular acceleration is integrated. Proper filter setup combining the angular acceleration and the quadratic terms can assist in the convergence to the right sign as the quadratic terms have undetermined sign solutions. The integration of the different types of data coming from the GF-IMU has been a subject of intensive research. In [

The configuration shown in

Mainly, we consider this configuration because a minimum of twelve accelerometers are needed to determine angular velocity magnitude and direction (algebraic sign cannot be determined uniquely). The most amount of the angular motion information, which is the AIV composed of the nine angular terms shown in

In general, a good model is important to extract the maximum amount of information from the observation. We will utilize the proper dynamic models in the angular motion estimation in the GF-IMU. We focus on the dynamic models used for maneuvering target tracking surveyed in [

This model assumes that the angular acceleration is a Wiener process, or more generally and precisely, the angular acceleration is a process with independent increments, which is not necessarily a Wiener process. This model is referred as a constant angular acceleration model (CAA) or a nearly constant angular acceleration model. It can be considered as a special case of a Gauss-Markov process. This model makes the angular acceleration a process with an increasing variance:

The discrete-time form is given as:

Since we have time uncorrelated noise, the corresponding state space representation of the Wiener sequence of angular acceleration vector combined with the angular velocity vector is given as:

This model was initially used for modeling linear acceleration [_{i}_{a}

Considering a first-order linearization for the exponential term in

The autocorrelation function Ψ_{α}

The corresponding state space representation of the Wiener sequence angular acceleration model in 3D motion including angular velocity vector is given as:

The variance is selected according to the ternary-uniform mixture as suggested in [

Angular jerk, which is the derivative of the angular acceleration, can be used in the same way as that of the angular acceleration based models. Using angular jerk based models increases the dimension of the state space vector which increases the computational load.

In this section, we give a simple error model of the accelerometer which considers the bias only. The section ends with a review of simple calibration procedure which fits the GF-IMU.

Each accelerometer measurement is assumed to be corrupted by a bias _{a}_{acc}

The variance of the discrete-time noise component _{disc.}

Every drifting accelerometer bias has the unit of

In discrete-time the random walk bias model is given as:

Using the accelerometer's error model shown in _{1}…_{9} represent the new bias parameters and _{1}…_{9} represent the noise errors.

In our setup, we adjust the separation distance manually to be unique for the three distributed triads with common orientation for all triads. Every accelerometer triad needs to be calibrated for three types of errors which are misalignment, scale factor and bias errors. Examples for the accelerometer triad calibration procedures can be found in [

Though we have twelve accelerometers and this means we have twelve unknown bias parameters, we are interested in estimating the resulting nine bias parameters _{1}…_{9} in the AIV given in

The initial state vector can be set as:

The initial estimation error covariance is given as:

Based on the previously described motion dynamic

The process covariance is computed as:

The

The

The measurement vector is composed of the angular acceleration vector and six quadratic terms of angular velocity combined with the nine bias parameters is given as:

The measurement Jacobian matrix is computed as:

The measurement error covariance matrix is computed as:

The Kalman gain is updated as documented in literature e.g., Simon [

The

The

Using the dynamic model gives us the possibility to have all the bias parameters in the resulting angular terms observable under some conditions. In this section, we determine under which conditions is the state space observable. First, we remove the noise in this observability analysis which leaves us with the simpler homogeneous state-space system given in continuous-time as:
_{i}

The zero-order Lie derivative of the measurement is the measurement itself,

Higher order Lie derivatives are computed as:

For our model, which has

The system is observable if the observability matrix

Without appending bias parameters, the state vector is reduced to the angular velocity and the angular acceleration vectors and it is given as:

The reduced state vector is clearly observable because the quadratic angular terms can solve for the angular velocity as shown in [

The initial state vector and initial estimation error covariance are assigned in a similar way as given in

Based on the previously described dynamic model we can write the discrete-time space model as:

The

The AIV, which is composed of the angular acceleration vector and six quadratic terms of angular velocity, is given as:

The measurement Jacobian matrix is computed as:

In this section, we give simulation results for a sinusoidal trajectory which is considered often in literature [

This scenario of motion is a two-dimensional harmonic angular oscillation. Three-dimensional harmonic angular oscillation is considered as a coning motion, however, the GF-IMU system responds better in this case because of having all the AIV terms as non-zero, which increases observability. The mathematical description of the angular motion is given as:

For a practical value of the accelerometer's noise and bias levels, we consider the specifications of the accelerometers manufactured by Analog Devices. We want to have a portable IMU so we choose the separation distance to be 0.4

The state vector is initialized properly around the true state with initial state error of 5% of the true value. The bias values were selected randomly from a distribution with one standard deviation of 2400 μg as shown in

Errors in the estimated angular velocity and angular acceleration vectors are plotted in

Clearly, we see a good convergence to the true angular velocity and angular acceleration components. However, there is a small oscillation in the

The plots show the convergence of all estimated bias parameters in AIV to their exact value with small steady state error. Moreover, from extensive simulations we find that reducing the noise error level of accelerometers gives a smoother and a faster convergence of bias parameters.

It is well known that proper initialization of the state vector for the EKF, which implies linearization, is important to avoid filter divergence. However, the filter can tolerate a limited level of initialization error if the nonlinearity is not high. Since there is no

In this part of the simulations, we consider the same trajectory described previously, but with bias parameters non-appended to the state vector which means we simply ignored them. For this reduced model, we can find the criteria for ignoring bias parameters based on accelerometers specifications of bias and noise errors given in

At relatively high sampling rates (e.g., 0.01 s or more), the magnitude of the error due to white noise is about 10 times the magnitude of error due to remaining bias for a tactical grade accelerometer. Consequently, for this sampling rate, the noise error dominates the bias error in this accelerometer category and for this scenario the EKF model works without a big difference from the one without bias. Hence, ignoring the bias and approximating the error as white noise error model can be justified considering that the Kalman filter will tolerate such a small remaining bias error.

We repeat the previously used trajectory profile with the same settings except for the noise and bias levels which are set to _{acc.}

First, the execution time for this model was much smaller than that of the previous model, which has bias parameters appended, because we have a much simpler state model. Using a reduced model implies reducing computational load remarkably. The plots of errors in the estimated angular velocity and the estimated angular accelerations are shown in

To see the effect of ignoring higher bias values in the AIV, we repeated the simulation with the same parameter values as used in subsection 8.1 and created a plot of the estimated angular velocity vector. Such values of parameters do not meet the criteria that accelerometer's error is dominated by white noise error and hence this results in a biased estimate of the angular velocity vector as shown in

We have presented a novel solution for estimating the spatial angular motion and bias parameters in a GF-IMU utilizing the dynamic models. The integration scheme is performed using an EKF. Observability analysis for the augmented model shows that the state space model is observable whenever the angular acceleration vector has non-zero magnitude. Simulation results shows that the filter can estimate the angular motion and bias parameters in the AIV for proper and improper initialization. Moreover in case of using tactical grade accelerometers or better, the error is dominated by noise error and hence the model can possibly be reduced to include only angular motion terms without degrading performance. Further research can be done to estimate the remaining bias parameters in the accelerometers.

We greatly appreciate the support of the German Academic Exchange Service (DAAD) for the doctoral work of Ezzaldeen Edwan within the International Postgraduate Programme (IPP) Multi Sensorics. The German Research Foundation (DFG) has funded part of the work reported herein; grant number KN 876/2-1, which is gratefully acknowledged.

A configuration of multiple distributed tri-axial accelerometers.

Angular velocity trajectory profile.

Angular acceleration trajectory profile.

Angular velocity vector estimation errors.

Angular acceleration vector estimation errors.

Estimated and reference bias parameters in the AIV.

Angular velocity vector estimation errors for improper initialization.

Angular acceleration vector estimation errors for improper initialization.

Angular velocity vector estimation errors for the reduced model.

Angular acceleration vector estimation errors for the reduced model.

Angular velocity vector estimation errors for the reduced model with large AIV bias values.

Numerical values of the simulation parameters.

_{m} |
_{acc} |
_{d}_{(m)} | |||
---|---|---|---|---|---|

Value | 0.4112 | 0.5 | 200 |
2400 |
0.4 |

Accelerometer Categories.

Noise Floor VRW (μg/✓Hz) | 2000 | 1000 | 100–400 | 5–10 |

Bias Stability (μg) | 2400 | 1200 | 50–500 | 5–10 |