Optimal Sensor Fusion for Skew-Redundant Inertial Measurement Units
May 12, 2011 By: Yigiter Yuksel, Naser El-SheimyIntroduction
There are some navigation applications for which the performance of current MEMS inertial units is not sufficient to achieve the desired navigation accuracy. For such applications, one of the foreseeable solutions is to use redundant number of inertial sensors together to generate more accurate inertial data which can lead to better navigation solution.
Unfortunately, the current INS practices do not answer the important questions regarding the optimal use of redundant sensors to achieve an improved navigation performance. For 6Dof IMUs, the optimal solution can be obtained by forming an extended Kalman filter around the nominal INS outputs. However, when an abundant number of sensors exists, the computation of the nominal navigation solution becomes ambiguous. Should only a critical number of sensors (i.e. 3 accelerometers and 3 gyroscopes) be used for a nominal solution or should all the sensors be used? In case of the latter, how will the sensor outputs be combined so that the usual navigation algorithms can be executed? Does the redundancy of sensors provide any additional information other than the redundant raw measurements? If such additional information exists, how can they be represented and processed in an optimal way?
In a recent study [1] we addressed the above questions. In that study we developed a framework for the optimal fusion of redundant inertial sensors. Using this framework, we solved the problem of computing best acceleration/rotation rates (kinematic variables) on the body frame given all (skew) redundant inertial sensor measurements.
Once the optimal kinematic variables are estimated from the redundant sensor measurements, the usual navigation algorithms can be executed with the computed accelerations/rotation rates. The navigation solution obtained in this way is not theoretically optimal in general. However, the difference between the optimal navigation solution and the navigation solution obtained with this method are so small that it can be completely ignored for all practical purposes.
In the following section an overview of the proposed optimal sensor fusion method for the redundant sensors is provided. Some important properties of this proposed method are discussed in the final section.
Overview
An SRIMU can be modeled as follows:
 | (1a) |
 | (1b) |
 | (2) |
where xk is the SRIMU error vector (e.g. stability and repeatability errors) with an initial covariance P0, uk is the real acceleration and rotation rates (which are assumed to be the deterministic unknowns), and yk represents the raw sensor outputs (i.e. accelerometer and gyroscope outputs). wk and vk denote white system disturbance and observation noises with covariance matrices Qk and Rk. In the standard IMU modeling terminology vk is called as angle/velocity random walk. Mk is the SRIMU configuration matrix, each row of which corresponds to the unit vector in the direction of each individual sensor. Ak, Bk, Nk and Ck are matrices with appropriate dimensions.
In [1], we proved that the optimal xn and un given all observations (yk, k=0..n) can be computed with the structure presented in Figure 1. In this figure Mk* represents the following weighted pseudo inverse of the configuration matrix:
 | (3) |
As seen in Figure 1, the optimal accelerations/rotation rates are computed as follows:
 | (4) |
where x̂k is the best error estimate generated by the Kalman filter. This Kalman filter processes the sensor redundancy measurements which are the projection of the raw sensor outputs to the left null space of the configuration matrix Mk. These projections are obtained by multiplying the SRIMU outputs with Tk whose rows are the basis vectors of the left null space of Mk (i.e. TkMk=0).
Figure 1: The optimal sensor fusion structure. yk is the SRIMU output vector. x̂k and ûk are the optimal error and kinematic variables (i.e. acceleration/rotation rate) estimates respectively. (Click to enlarge.)
Under Gaussian noise assumption, it can be proved that x̂k and ûk estimates are MMSE optimal. In addition to this, it is proved in [1] that the generated solution is also optimal in terms of the following deterministic, quadratic cost function definition:
 | (5a) |
| Subject to: | |
 | (5b) |
The association of the optimality with this deterministic cost function is important because such a cost function definition does not require an exact IMU error model knowledge. Regardless of whatever values are used as weight coefficients (e.g. Qk, Rk, P0), the structure in Figure 1 is always optimal in terms of equation (5).
Summary of Key Findings
Starting from the system model (1) and the cost function definition (5), we presented a complete analysis for the INS designs with SRIMUs in [1]. In this section, some key findings of the corresponding study are summarized. Interested readers should refer to [1] for proofs and more detailed elaborations.
- Once best accelerations/rotation rates (i.e. ûk) are computed, the INS algorithms can be directly executed with these computed kinematic variables as shown in Figure 2. In this case, the errors on the computed kinematic variables (i.e. ek = ûk-uk) must be modeled and implemented as a part of the navigation Kalman filter states.
Figure 2: Pseudo optimal configurations for SRIMU based INSs. (Click to enlarge.)
- Although ûk is optimal, the navigation solution generated by the structure in Figure 2 is not optimal. In order to compute the optimal navigation solution the sensor redundancy measurements must be processed by the navigation Kalman filter as shown in Figure 3.
Figure 3: Optimal navigation solution configuration for SRIMU based INSs. (Click to enlarge.)
- The difference between the navigation solutions obtained with Figure 2 and Figure 3 is only marginal. Such a difference can be safely ignored for any practical purpose.
- The error models for the computed kinematic variables (i.e. ek=ûk-uk) can be derived theoretically using the error models of the individual sensors and the transfer function of the Kalman filter. However, when an abundant number of inertial sensors is used in an SRIMU, this method implies too many additional states to be augmented to the states of the navigation Kalman filter. Therefore, instead of using such theoretical models, it is usually better to remodel the optimally fused sensor outputs using a laboratory sensor modeling technique. On the other hand, because of the cross-correlation between the errors on the different axes, the Allan variance method cannot be used for the optimally fused outputs. In [2], we presented a new error modeling method for such redundant sensor systems. This new method uses the correlation of filtered sensor outputs to generate approximate error models for SRIMUs.
- When all the sensors in an SRIMU have identical stochastic error models, the sensor redundancy measurements in Figure 3 becomes orthogonal to the navigation solution. As a result of this, these redundancy measurements can be completely ignored under this condition. Furthermore, the errors on the ûkwls can be remodeled with less number of states using an alternative equivalent (up to 2nd order statistics) representation. Such a model order reduction provides a great simplification when tens of identical sensors are used in an SRIMU.
- In the existing literature, there are some studies which propose use of multiple INSs to process SRIMU outputs. Under certain conditions, multi-INS approaches can also be assumed optimal. However, multi-INS methods are numerically very problematic. Therefore, the single INS approach that is presented in this study should always be preferred over such multi-INS implementations.
References
- Yuksel Y., "Design and Analysis of Inertial Navigation Systems with Skew Redundant Inertial Sensors", UCGE Report 20328, Dept. of Geomatics Engineering, University of Calgary, Calgary, Canada, 2011.
- Yuksel Y.; El-Sheimy N., Aboelmagd N., "Error modeling and characterization of environmental effects for low cost inertial MEMS units", 2010 IEEE/ION Position Location and Navigation Symposium (PLANS), pp.598-612, May 2010.
Yigiter Yuksel and Naser ElSheimy
Department of Geomatics Engineering
University of Calgary, Calgary, Canada
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