Abstract
Autonomous estimation of the state is of key importance in UAVs, as the measurement systems may experience faults and failures. Thus estimation techniques must provide estimates of the most important variables used in the control algorithms for safe, autonomous, unmanned flights. In this paper, a filter with low computational complexity for attitude estimation of a quadrotor UAV is introduced, with a model suitable for Fault-Tolerant Observation. The new filtration method, called the Square Root Unscented Complementary Kalman Filter (SRUCKF), is based on the commonly-known Kalman Filter (KF) in its nonlinear version, namely the Square Root Unscented Kalman Filter (SRUKF). The fundamental equation of the KF is modified so that the complementary feature of the filter is exalted. The new filter introduces characteristics that are analyzed on the basis of its application in quadrotor state estimation. Finally, the results are compared to an ordinary filter of the same type (using the Unscented Transformation). The presented studies indicate that the newly derived filter (SRUCKF) handles strong nonlinearities and gives results similar to those obtained from the SRUKF. Furthermore, it introduces lower computational burden, as the undergoing process uses diagonal matrices in its crucial places. In the paper, the estimation algorithms are tailored to a quadrotor UAV (Crazyflie 2.0), for which a quaternion-based model is proposed. The contribution of the paper lies in a Kalman-based novel state observer and its application in Fault-Tolerant Observation (FTO).
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Acknowledgments
The authors would like to express their gratitude to Mr. Xiang He from the University of Utah, for successful cooperation in the field of robotics and for providing the necessary flight data for a number of studies.
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Appendices
Appendix A: Algorithms of Kalman Filter, Using UT
1.1 A.1 Unscented Kalman Filter
The Unscented Kalman Filter is as a multistep, recursive algorithm. In the first step the whole process is initialized. It is done by calculating the a posteriori state vector x0 and covariance matrix P0:
where i is in the range < 1; k > and stands for the measurement sample number (in general: k < 10). This applies only when the state is measurable, otherwise the unobservable elements are set to zero. The UKF procedure can be presented in the following steps:
for i ∈< 1; 2n >. Note that sign(...) has an exception, i.e. sign(0) = 0.
Parameters are given as follows:
where α, β, and κ are tuning parameters: α is responsible for sigma points spread, κ is a scaling parameter, and β is noise distribution parameter (set to 2 for Gaussian distributions).
The UKF has some significant differences when compared to the EKF or the KF. First of all, it does not approximate the covariance matrix \(\mathbf {P}_{k}^{-}\) or Pk, instead it transfers the so-called ”sigma points” (\(\mathbf {\mathcal {Y}}_{k}^{i}\) and \(\mathbf {\chi }_{k}^{i}\)) through nonlinear functions (f(...) and h(...)) and calculates weighted covariances from these points. It is a simple and fast method, provided that the sigma points are properly calculated. A single sigma point is a sum of the state vector (a priori or a posteriori) and a column or a row of the square root of a given covariance matrix (a priori or a posteriori, respectively).
1.2 A.2 Square Root Unscented Kalman Filter
The process is initialized with the a posteriori estimate and the square root of the a posteriori covariance matrix:
The algorithm can have the following layout:
Appendix B: Lemmas and Proofs
Lemma 1
Diagonalizable matrix.
If matrix A is diagonalizable, then there exists matrix P , such as:
where D is diagonal.
Lemma 2
Diagonalizable symmetric matrix.
If matrix A is symmetric, then it is diagonalizable with orthogonal matrices, such as:
where Q is orthogonal matrix and D is diagonal.
Proof
Square root of a symmetric matrix.
If matrix A is decomposed to diagonal and orthogonal matrix, i.e.
then its square root B (A = BB) is equal to the following:
which is proven by the following:
□
Lemma 3
Derivative of a scalar with respect to a matrix.
For any matrix\(\mathbf {X}\in \mathbb {R}^{(n\times m)}\)andvariable y (which is equal to a function ofX)it:
On th basis of Eq. 145, matrix trace derivative proprieties can be proven. For any matricesX,Yand symmetricmatrixZitcan be written:
Lemma 4
Kronecker product of quaternions in a matrix-based notation.
Let us assume that three quaternions:α,β,andγaregiven in the following relation:
By introducing the matrix forms of α and β it can be written that:
where:
In Eq. 150, \(\tilde {\mathbf {a}}\) and \(\tilde {\mathbf {b}}\) are skew symmetric matrices:
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Gośliński, J., Giernacki, W. & Królikowski, A. A Nonlinear Filter for Efficient Attitude Estimation of Unmanned Aerial Vehicle (UAV). J Intell Robot Syst 95, 1079–1095 (2019). https://doi.org/10.1007/s10846-018-0949-7
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DOI: https://doi.org/10.1007/s10846-018-0949-7