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A distributed set of accelerometers based on the minimum number of 12 accelerometers allows for computation of the magnitude of angular rate without using the integration operation. However, it is not easy to extract the magnitude of angular rate in the presence of the accelerometer noises, and even worse, it is difficult to determine the direction of a rotation because the angular rate is present in its quadratic form within the inertial measurement system equations. In this paper, an extended Kalman filter scheme to correctly estimate both the direction and magnitude of the angular rate through fusion of the angular acceleration and quadratic form of the angular rate is proposed. We also provide observability analysis for the general distributed accelerometers-based inertial measurement unit, and show that the angular rate can be correctly estimated by general nonlinear state estimators such as an extended Kalman filter, except under certain extreme conditions.

A distributed accelerometers-based inertial measurement unit (IMU) uses only accelerometers to compute a specific force and angular rate. It has several advantages over the conventional gyroscope-equipped IMU, including robustness, easy calibration, low cost, and less power consumption. Furthermore, it can also function as an angular acceleration sensor, and thus it has applications in highly dynamic systems area, as well as low-cost, medium performance inertial navigation systems.

Development efforts for a distributed accelerometers-based IMU have been going on for over 30 years [

The minimum number of accelerometers necessary to directly calculate the angular rate for a rigid body motion is 12 [

In contrast to those solutions, we propose in this paper an extended Kalman filter scheme to aid the integration of angular acceleration by six quadratic terms of angular rate in order to correctly estimate both the direction and magnitude of the angular rate. Compared to the previous works, we apply a Kalman filter to estimate angular rate only, since the specific force and angular acceleration can be computed algebraically. For an extended Kalman filter setup, the 4th–6th variables among the computed 12 kinematic variables in the system equation are designated as a state propagation equation, and the last six variables are designated as a measurement equation, which is the approach originally introduced in [

Consider the inertial frame {^{i}

If an accelerometer is rigidly attached at point _{k}

The output of the accelerometer is directly related with the specific force at the center of the body frame {

Through simple algebraic manipulations, it is easily shown that _{i}_{i}_{i}

Now suppose that

If the configuration matrix ^{+} = (^{T}J^{−1}^{T}

For a configuration of 12 accelerometers, the left inverse matrix ^{+} becomes the inverse matrix ^{−1}, and the main requirement for the placement of the accelerometers is that the configuration matrix is invertible. Several accelerometer array configurations have been proposed in the literature, and

Additional requirements can be imposed to determine the positions and the sensing directions of the accelerometers, since we have 60 parameters for an array of 12 accelerometers to be determined. For example, we can try to find an accelerometer array configuration yielding the minimum influence of the accelerometer noise on the variances of the kinematic variables, since it depends on the accelerometer configuration as follows:
_{y}_{A}

The 12 kinematic variables can be computed from

Several methods have been proposed to resolve this sign ambiguity. Some methods resort to additional sensors such as the conventional low-cost gyroscopes or GPS to determine the correct sign of the angular rate. However, the common approaches compare the estimates with those obtained from the integration of the angular acceleration. In this paper, we use an extended Kalman filter to fuse two sensed sources, the angular acceleration and the quadratic form of the angular rate, and thus to estimate the sign and magnitude of the angular rate at the same time.

For an extended Kalman filter setup, the angular accelerations, which are the 4th–6th terms of

Then, the angular rate propagation and measurement equations are written as:

Observability denotes the ability to uniquely determine the states of a dynamic system from its measurements and it is the indication for the feasibility of design of a state estimator such as a Kalman filter [_{0} is a 18 × 3 matrix, and ∇ stands for a gradient operator with respect to

As seen from

In the first case, wherein at least one of angular acceleration terms is not zero, the rank of observability matrix is three and thus the system is observable for every angular rate including the origin of angular rate vector space,

In the second case, in which all of angular rate terms are constant, the system becomes static but it is still observable if at least one of angular rate terms is not zero since the rank of observability matrix is three. However, the observability condition fails at the origin of angular rate vector space, which means that the system is locally observable in the angular rate vector space. Indeed, it is not possible to determine the direction of angular rate using only quadratic terms of the angular rate, since the quadratic terms do not give a unique solution. Instead, it yields two possible solutions, each of which having equal magnitude but opposite direction in the angular rate vector space, as shown in

In the general application there are certain instants where the second case may occur. However, this is not a problem because the recursive predictor-corrector scheme of Kalman filter makes it possible to continuously incorporate an

The last case is a special case in which all of angular rate and angular acceleration terms are zero. Since all elements of observability matrix are zero, the system becomes completely unobservable. However, although the system is completely unobservable, this case can be treated easily since we can be informed that all angular accelerations and rates are zero from the designated measurement equation.

The angular rate propagation _{k}_{k}_{k}_{k}

An extended Kalman filter can be designed using these two equations. The time update of the angular rate estimate and its covariance matrix are given by:
_{k}_{+1}|_{k}_{k}_{+1}, along with the associated error covariance

The measurement update provides a correction based on the measurement _{k}_{+1} to yield the

The thresholds for angular acceleration and quadratic angular rate terms can be chosen by several methods. One method may be using the standard deviation of their measurement noises, and another method may be using a moving average of the measurements to obtain smoother transition.

To evaluate the performance of the proposed scheme for the angular rate estimation, computer simulations are performed. We use the 12 accelerometer-based IMU shown in

The noise incorporated in the simulation is white, and the noise associated with each accelerometer measurement is assumed to have the same density of 100 μg/

The arbitrary trajectory shown in

_{3}, used for the simulations. Other angular rate and angular acceleration terms are assumed to be zero, as shown in

The extremely severe situation for distributed an accelerometer-based IMU is the case in which all of the angular rates are constant from the initial time when IMU is on, because there is no available information on the direction of rotation. According to the observability analysis, the extended Kalman filter cannot guarantee that the angular rate estimate will converge toward the correct values, unless the initial angular rate estimate is closer to the true angular rate than to the false angular rate.

The simulation was performed with two different initial estimates within the extended Kalman filter.

The situation shown in _{3} follows a sinusoidal pattern at 65 s. As shown in _{k}

Although a typical 12 distributed accelerometer-based IMU allows for the computing of the magnitude of an angular rate without using the integration of the outputs of accelerometers, it is difficult to determine the direction of a rotation because the angular rate is present in its quadratic form within the IMU system equations. To tackle this inherent disadvantage, we proposed an extended Kalman filter scheme to aid the integration of angular acceleration by six quadratic terms of angular rate in order to correctly estimate both the direction and magnitude of the angular rate. We also provided observability analysis for the general 12 accelerometer-based IMU, and showed that the angular rate can be correctly estimated by general nonlinear state estimators such as an extended Kalman filter, except under certain extreme conditions.

The performance of the proposed scheme for the angular rate estimation was evaluated using computer simulations under three special cases. The simulation results agree well with our analyses, and show that the proposed scheme correctly estimates specific force, angular acceleration, and the direction and magnitude of the angular rate as well.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2007-331-D00084).

Inertial frame and body frame.

Two possible configurations (

Two possible solutions in angular rate space.

Reference trajectories of angular acceleration and specific force: Case 1.

Angular rate estimation errors: Case 1.

Angular rate estimation error of _{3}: Case 2.

Angular acceleration: Case 2.

Angular rate estimation with close initial conditions: Case 3-1.

Angular rate estimation with far initial conditions: Case 3-1.

Angular acceleration: Case 3-2.

Angular rate estimation: Case 3-2.

Response of measurement prediction covariance: (