Abstract
The deterministic errors of an accelerometer comprise the prevailing i) bias, ii) scale factor, and iii) non-orthogonality. Together, these errors result in a nonlinear measurement model, which is conventionally solved via an iterative nonlinear least-squares method. In contrast to the conventional approach, we propose a novel method to transform the above nonlinear model into a system of linear equations, resulting in an exact, closed-form solution of the deterministic errors. The developed mathematical formulations are first verified in a simulation setting, followed by a real-time implementation using Robot Operating System for small micro-electromechanical inertial measurement units.
1 PROPOSED METHODOLOGY
To the best of our knowledge, the existing methodologies (Clausen & Skaloud, 2020; Syed et al., 2007; Tedaldi et al., 2014) for estimating the deterministic errors of an accelerometer make use of iterative nonlinear least-squares techniques. The primary reason for taking such an approach seems to be the evident nonlinear observation model presented below (Syed et al., 2007):
1
where z = [zx zy zz]T is the measurement and x = [xx xy xz]T is the corresponding true accelerometer value, which is unknown. 
denotes white noise, and 
represents the bias. The scaling factors (sx, sy, and sz) and non-orthogonality (θxy, θzy, θzx) are encapsulated by a lower triangular matrix, M:
The model represented by Equation (1) comprises three equations with nine unknowns representing the deterministic errors and three unknown true accelerometer readings. However, under static conditions, the norm of the true accelerometer reading is assumed to be constant and, in the absence of errors, corresponds to gravity, i.e., x⊤x = g2, where 
denotes acceleration due to gravity, thereby limiting the number of unknowns to nine. Therefore, with sufficient accelerometer measurements in different orientations (poses), the nine unknowns can be determined. We first solve this system as it is in Section 1.1. We then assume M = I3 and present a simplified solution in Section 1.2.
Contributions: In sum, our method provides the scientific community with a single-step linear estimation approach for computing deterministic accelerometer errors, in contrast to commonly used iterative and multi-step methods (Haupt et al., 1996). Furthermore, in addition to solving a nonlinear problem using a linear estimator, our methodology facilitates bias estimation without a knowledge of local gravity, distinguishing our approach from existing methods. Importantly, the method presented herein also enables attitude-free calibration of accelerometer triads, similar to existing methods.
1.1 Full Linear Least Squares (Full LLS)
By rewriting Equation (1) as x = M−1 (z – b) and substituting this equation in the static constraint (x⊤x = g2), we obtain the following system:
2
where S = ATA with A = M−1; r = ATq with q = M−1b; d = q⊤q – g2. Equation (2) is then rewritten for n measurements:
3
The linear system in Equation (3) is a homogeneous system of the form XΘ = 0 with n rows (equations) and 10 columns (unknowns). Although the columns of X appear to be independent, they are not, as they are obtained through the static constraint, with the assumption of {M, b} being constants. As there is only one constraint, the rank of X, with a sufficient number of measurements, cannot exceed nine, leading to a one-dimensional null space (or kernel space) in which Θ exists. Intuitively speaking, the columns representing 
, and 1 are not independent as 
. A nontrivial solution to this system is obtained via singular value decomposition (SVD) such that ||Θ|| = 1, yielding S = λŜ, 
, 
with Ŝ, 
, and 
being the components of Θ found using SVD and 
. We first compute the bias by using the equations 
and λŜ = ATA to obtain the following:
4
This result proves that the bias computation does not depend on a knowledge of g or λ (which is yet unknown). Secondly, we compute λ by using 
and λŜ = ATA to obtain the following:
5
It should be noted that the computation of λ depends on a knowledge of g. Finally, to compute M, we use S = λŜ = ATA. Because M is a lower triangular matrix, its inverse, A = M−1, is also a lower triangular matrix. The matrix A is determined as follows:
This results in M as follows:
6
Note that the diagonal of M cannot be negative; therefore, a6, a3, a1 are required to be positive.
1.2 Bias-Only Linear Least Squares (Bias-Only LLS)
By substituting M = I3 in Equation (1) and rewriting the static constraint (x⊤x = g2) for n measurements, we obtain the following:
7
This system can be solved by a linear least-squares method, and the bias computation does not depend on a knowledge of g, which is consistent with the result obtained by the full LLS method. The formulations presented here and in Section 1.1 result in closed-form formulae for estimating deterministic errors without reliance on iterative methods or initial guesses, thereby precluding the likelihood of numerical divergence. Moreover, both proposed methods require at least q +1 static measurements for q unknowns.
2 RESULTS
Simulations: Our simulation environment is created in Python, where ground truth values are sampled from a uniform distribution: bias in [0.9 mg, 1.3 mg], scale factor in [300 ppm, 600 ppm], and non-orthogonality in [0.02°, 0.04°]. Most of these values are taken from the datasheet of ADIS-16475 to simulate accelerometer measurements based on Equation (1) with g = 9.8055 m/s2. We first analyzed the accuracy of the implementation by simulating 24 accelerometer measurements (or poses) and corrupting them with Gaussian noise with standard deviation in [10−7, 10−3] m/s2. In addition to implementing our methods, we also use an iterative nonlinear least-squares approach (Clausen & Skaloud, 2020), in which we either assume M = I3 (bias-only nonlinear least squares [NLLS]) or not (full NLLS). The bias estimation obtained by following this strategy is presented in Figure 1(a), where the performances of both full NLLS and full LLS are found to be comparable and of high quality. Meanwhile, the performance of the proposed LLS is better than that of the NLLS for bias-only determination. The maximum estimation errors (across the noise range) are ~ 0.5% for the scale factor and ~ 1% for non-orthogonality; because these error values are comparatively small and the same for both methods, they not plotted. Secondly, when analyzing the effect of the number of accelerometer measurements, we found that the estimation error decreases for all entities with an increasing number of measurements; for the bias, this is shown in Figure 1(b). Thirdly, we analyzed the effect of insufficient knowledge of g. The bias estimation error is plotted in Figure 1(c), where diverging trends are observed for bias-only NLLS, whereas the proposed bias-only LLS has a much lower and consistent error. The scale-factor estimation error is plotted in Figure 1(d), showing high sensitivity to a knowledge of g for both methods. In contrast, for non-orthogonality, the error has a range of 1% – 2% and hence is not plotted. Notably, when M = I3 is used to generate inertial measurement unit (IMU) measurements, the bias estimation, shown in Figure 1(e), confirms the superiority of the proposed bias-only LLS over bias-only NLLS in terms of knowledge of g. Moreover, the computation times of the proposed full and bias-only LLS are reduced by factors of ~ 35 and 20, respectively, compared with NLLS computation times.
Simulation results
Field tests using bias-only LLS and full LLS: In laboratory tests (L1 and L2), 4x ADIS 16475 micro-electromechanical IMUs (dahu 0–3) were used to process data for 35 poses in L1 and 15 poses in L2. Subsequent tests (F1 and F2) involve these IMUs in addition to STIM318 on a custom drone, TP-2, focusing on six static poses for practical reasons; see Figure 2 for the experimental setup. We use Robot Operating System in C++ for implementing the proposed methods in real time; the obtained deterministic errors are presented in Table 1. Note that the accelerometer data from the ADIS IMUs in F1 and F2 are preprocessed based on the scale factor and non-orthogonality values from L1 prior to the bias computation; M = I3 is assumed for STIM318. The estimated values of deterministic errors are consistent among themselves and not far from those in the datasheet.
(a) Lab test scenario (L1 and L2) at Ecole Polytechnique Federale de Lausanne; (b) field test scenario (F1 and F2) at Vufflens-la-ville, Switzerland
Field Test Results Highlighting the i) Norm of Bias, ii) Mean Scale Factor, and iii) Mean Non-Orthogonality
3 CONCLUSION
We have derived closed-form expressions of deterministic accelerometer errors utilizing static measurements. Our full LLS method demonstrates performance on par with the prevailing full NLLS technique. Meanwhile, the proposed bias-only LLS method exhibits superiority, in terms of accuracy, over the existing bias-only NLLS approach. Theoretically, our methodologies necessitate only a single additional observation while obviating the iterative and initial guess requirements of the NLLS method. Moreover, the proposed methods are at least one order of magnitude faster than the existing approaches. Therefore, the developed methods hold promise for practical estimation of turn-on systematic effects, such as accelerometer biases, before flight. This capability, in turn, can enhance attitude initialization for drones within an integrated navigation system.
HOW TO CITE THIS ARTICLE
Burkhard, J., Sharma, A., & Skaloud, J. (2024). Linear estimation of deterministic accelerometer errors. NAVIGATION, 71(3). https://doi.org/10.33012/navi.656
ACKNOWLEDGMENTS
This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 754354.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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