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Zhu Y, Liu J, Yu R, Mu Z, Huang L, Chen J, Chen J. Attitude Solving Algorithm and FPGA Implementation of Four-Rotor UAV Based on Improved Mahony Complementary Filter. SENSORS (BASEL, SWITZERLAND) 2022; 22:6411. [PMID: 36080870 PMCID: PMC9460884 DOI: 10.3390/s22176411] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/01/2022] [Revised: 08/21/2022] [Accepted: 08/22/2022] [Indexed: 06/15/2023]
Abstract
With the development of modern industry, small UAVs have been widely used in agriculture, mapping, meteorology, and other fields. There is an increasing demand for the core attitude-solving algorithm of UAV flight control. In this paper, at first, a novel attitude solving algorithm is proposed by using quaternions to represent the attitude matrix and using Allan variance to analyze the gyroscope error and to quantify the trend of the error over time, so as to improve the traditional Mahony complementary filtering. Simulation results show that the six-axis data from the initial sensors (gyroscope and accelerometer) agree well with the measured nine-axis data with an extra magnetometer, which reduces the complexity of the system hardware. Second, based on the hardware platform, the six-axis data collected from MPU6050 are sent to FPGA for floating-point operation, transcendental function operation, and attitude solution module for processing through IIC communication, which effectively validates the attitude solution by using the proposed method. Finally, the proposed algorithm is applied to a practical scenario of a quadrotor UAV, and the test results show that the RMSE does not exceed 2° compared with the extended Kalman filter method. The proposed system simplifies the hardware but keeps the accuracy and speed of the solution, which may result in application in UAV flight control.
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Affiliation(s)
- Yanping Zhu
- School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
| | - Jing Liu
- School of Networks and Telecommunications Engineering, Jinling Institute of Technology, Nanjing 211199, China
| | - Ran Yu
- School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
| | - Zijian Mu
- School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
| | - Lei Huang
- School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
| | - Jinli Chen
- School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
| | - Jianan Chen
- School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
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Xiang M, Scalzo Dees B, Mandic DP. Multiple-Model Adaptive Estimation for 3-D and 4-D Signals: A Widely Linear Quaternion Approach. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2019; 30:72-84. [PMID: 29993725 DOI: 10.1109/tnnls.2018.2829526] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Quaternion state estimation techniques have been used in various applications, yet they are only suitable for dynamical systems represented by a single known model. In order to deal with model uncertainty, this paper proposes a class of widely linear quaternion multiple-model adaptive estimation (WL-QMMAE) algorithms based on widely linear quaternion Kalman filters and Bayesian inference. The augmented second-order quaternion statistics is employed to capture complete second-order statistical information in improper quaternion signals. Within the WL-QMMAE framework, a widely linear quaternion interacting multiple-model algorithm is proposed to track time-variant model uncertainty, while a widely linear quaternion static multiple-model algorithm is proposed for time-invariant model uncertainty. A performance analysis of the proposed algorithms shows that, as expected, the WL-QMMAE reduces to semiwidely linear QMMAE for [Formula: see text]-improper signals and further reduces to strictly linear QMMAE for proper signals. Simulation results indicate that for improper signals, the proposed WL-QMMAE algorithms exhibit an enhanced performance over their strictly linear counterparts. The effectiveness of the proposed recursive performance analysis algorithm is also validated.
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Valle ME, de Castro FZ. On the Dynamics of Hopfield Neural Networks on Unit Quaternions. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2018; 29:2464-2471. [PMID: 28489551 DOI: 10.1109/tnnls.2017.2691462] [Citation(s) in RCA: 25] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this paper, we first address the dynamics of the elegant multivalued quaternionic Hopfield neural network (MV-QHNN) proposed by Minemoto et al. Contrary to what was expected, we show that the MV-QHNN, as well as one of its variation, does not always come to rest at an equilibrium state under the usual conditions. In fact, we provide simple examples in which the network yields a periodic sequence of quaternionic state vectors. Afterward, we turn our attention to the continuous-valued quaternionic Hopfield neural network (CV-QHNN), which can be derived from the MV-QHNN by means of a limit process. The CV-QHNN can be implemented more easily than the MV-QHNN model. Furthermore, the asynchronous CV-QHNN always settles down into an equilibrium state under the usual conditions. Theoretical issues are all illustrated by examples in this paper.
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Martín López LM, Aguilar de Soto N, Miller P, Johnson M. Tracking the kinematics of caudal-oscillatory swimming: a comparison of two on-animal sensing methods. ACTA ACUST UNITED AC 2016; 219:2103-9. [PMID: 27207638 DOI: 10.1242/jeb.136242] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2015] [Accepted: 05/11/2016] [Indexed: 11/20/2022]
Abstract
Studies of locomotion kinematics require high-resolution information about body movements and the specific acceleration (SA) that these generate. On-animal accelerometers measure both orientation and SA but an additional orientation sensor is needed to accurately separate these. Although gyroscopes can perform this function, their power consumption, drift and complex data processing make them unattractive for biologging. Lower power magnetometers can also be used with some limitations. Here, we present an integrated and simplified method for estimating body rotations and SA applicable to both gyroscopes and magnetometers, enabling a direct comparison of these two sensors. We use a tag with both sensors to demonstrate how caudal-oscillation rate and SA are adjusted by a diving whale in response to rapidly changing buoyancy forces as the lungs compress while descending. The two sensors gave similar estimates of the dynamic forces, demonstrating that magnetometers may offer a simpler low-power alternative for miniature tags in some applications.
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Affiliation(s)
- Lucía Martina Martín López
- SMRU (Sea Mammal Research Unit), University of St Andrews, St Andrews, Fife KY16 8LB, UK Asociación Ipar Perspective, C/Karabiondo 17, 48600 Sopela, Bizkaia, Spain
| | - Natacha Aguilar de Soto
- CREEM (Centre for Research in Ecological and Environmental Modelling), University of St Andrews, Fife KY16 9LZ, UK BIOECOMAC (Biodiversidad, Ecología Marina y Conservación), Universidad de La Laguna, 38200 La Laguna, Tenerife, Spain
| | - Patrick Miller
- SMRU (Sea Mammal Research Unit), University of St Andrews, St Andrews, Fife KY16 8LB, UK
| | - Mark Johnson
- SMRU (Sea Mammal Research Unit), University of St Andrews, St Andrews, Fife KY16 8LB, UK Zoophysiology, Department of Bioscience, Aarhus University, Building 1131, C. F. Moellers Alle 3, Aarhus C DK-8000, Denmark
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Xu D, Xia Y, Mandic DP. Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2016; 27:249-261. [PMID: 26087504 DOI: 10.1109/tnnls.2015.2440473] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions typically require the calculation of the gradient and Hessian. However, real functions of quaternion variables are essentially nonanalytic, which are prohibitive to the development of quaternion-valued learning systems. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized Hamilton-real (GHR) calculus, thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field, rather than using the isomorphism with the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the novel quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are also introduced, opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms. The proposed GHR calculus is shown to yield the same generic algorithm forms as the corresponding real- and complex-valued algorithms. Advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks.
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Xia Y, Jahanchahi C, Nitta T, Mandic DP. Performance Bounds of Quaternion Estimators. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2015; 26:3287-3292. [PMID: 25643416 DOI: 10.1109/tnnls.2015.2388782] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The quaternion widely linear (WL) estimator has been recently introduced for optimal second-order modeling of the generality of quaternion data, both second-order circular (proper) and second-order noncircular (improper). Experimental evidence exists of its performance advantage over the conventional strictly linear (SL) as well as the semi-WL (SWL) estimators for improper data. However, rigorous theoretical and practical performance bounds are still missing in the literature, yet this is crucial for the development of quaternion valued learning systems for 3-D and 4-D data. To this end, based on the orthogonality principle, we introduce a rigorous closed-form solution to quantify the degree of performance benefits, in terms of the mean square error, obtained when using the WL models. The cases when the optimal WL estimation can simplify into the SWL or the SL estimation are also discussed.
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Xia Y, Jahanchahi C, Mandic DP. Quaternion-valued echo state networks. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2015; 26:663-673. [PMID: 25794374 DOI: 10.1109/tnnls.2014.2320715] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Quaternion-valued echo state networks (QESNs) are introduced to cater for 3-D and 4-D processes, such as those observed in the context of renewable energy (3-D wind modeling) and human centered computing (3-D inertial body sensors). The introduction of QESNs is made possible by the recent emergence of quaternion nonlinear activation functions with local analytic properties, required by nonlinear gradient descent training algorithms. To make QENSs second-order optimal for the generality of quaternion signals (both circular and noncircular), we employ augmented quaternion statistics to introduce widely linear QESNs. To that end, the standard widely linear model is modified so as to suit the properties of dynamical reservoir, typically realized by recurrent neural networks. This allows for a full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances, and a rigorous account of second-order noncircularity (improperness), and the corresponding power mismatch and coupling between the data components. Simulations in the prediction setting on both benchmark circular and noncircular signals and on noncircular real-world 3-D body motion data support the analysis.
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