attitude determination(QEKF,MEKF,UKF).zip_kalman_EKF_虚拟/增强现实-VR/AR下载-pudn.com

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/6251549074bc5c01059a1e1c/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6251549074bc5c01059a1e1c/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Unscented Kalman Filtering for Attitude</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws0">Determination Using MEMS Sensors</div><div class="t m0 x3 h3 y3 ff2 fs1 fc0 sc0 ls2 ws0">Jaw-Kuen Shiau* and I-Chiang Wang</div><div class="t m0 x4 h4 y4 ff3 fs2 fc0 sc0 ls3 ws0">Department of Aerospace Engineering, Tamkang University,</div><div class="t m0 x5 h4 y5 ff3 fs2 fc0 sc0 ls4 ws0">Danshui, New Taipei City 25137, Taiwan, R.O.C.</div><div class="t m0 x6 h5 y6 ff4 fs3 fc0 sc0 ls2 ws0">Abstract</div><div class="t m0 x7 h6 y7 ff2 fs4 fc0 sc0 ls5 ws0">This paper presents the results of a quaternion-based unscented Kalman filtering for attitude</div><div class="t m0 x8 h6 y8 ff2 fs4 fc0 sc0 ls2 ws0">estimation using low cost MEMS sensors. The unscented Kalman filter uses the pitch and roll angles</div><div class="t m0 x8 h6 y9 ff2 fs4 fc0 sc0 ls6 ws0">computed from gravity force decomposition as the measurement for the filter. The immeasurable</div><div class="t m0 x8 h6 ya ff2 fs4 fc0 sc0 ls7 ws0">gravity accelerations are deduced from the outputs of the three axes accelerometers, the relative</div><div class="t m0 x8 h6 yb ff2 fs4 fc0 sc0 ls9 ws0">accelerations, and the accelerations due to body rotation. The constraint of the four elements of the</div><div class="t m0 x8 h6 yc ff2 fs4 fc0 sc0 ls8 ws0">quaternion method is treated as a perfect measurement and is integrated into the system to form a</div><div class="t m0 x8 h6 yd ff2 fs4 fc0 sc0 ls2 ws0">constrained unscented Kalman filter. The heading angle is obtained from a complimentary filter which</div><div class="t m0 x8 h6 ye ff2 fs4 fc0 sc0 lsc ws0">uses the heading signal derived from the magnetic force information from an electronic magnetic</div><div class="t m0 x8 h6 yf ff2 fs4 fc0 sc0 ls2 ws0">sensor and the GPS-derived heading as the inputs. An experiment using an in-house designed motion</div><div class="t m0 x8 h6 y10 ff2 fs4 fc0 sc0 lsc ws0">platform is conducted to evaluate the proposed algorithm. The noise characteristics of the sensor</div><div class="t m0 x8 h6 y11 ff2 fs4 fc0 sc0 ls2 ws0">signals are examined using the laboratory data. Approximations of the time-varying noise variances of</div><div class="t m0 x8 h6 y12 ff2 fs4 fc0 sc0 lsd ws0">the measured signals are obtained through Taylor series expansions. The algorithm is intuitive and</div><div class="t m0 x8 h6 y13 ff2 fs4 fc0 sc0 lse ws0">easy to implement. Moreover, the proposed algorithm and the filter design are successfully</div><div class="t m0 x8 h6 y14 ff2 fs4 fc0 sc0 ls2 ws0">demonstrated through a complete set of flight test data.</div><div class="t m0 x8 h6 y15 ff5 fs4 fc0 sc0 ls2 ws0">Key Words: Flight Information Measurement, Nonlinear Kalman Filter, Quaternion,</div><div class="t m0 x9 h6 y16 ff2 fs4 fc0 sc0 ls2 ws0">Complementary Filter</div><div class="t m0 xa h5 y17 ff4 fs3 fc0 sc0 ls10 ws0">1. Intr oduction</div><div class="t m0 xb h7 y18 ff2 fs5 fc0 sc0 ls2 ws0">Obtaining precise attitude information is essential</div><div class="t m0 xc h7 y19 ff2 fs5 fc0 sc0 ls2 ws0">for aircraft navigation and control applications. With the</div><div class="t m0 xc h7 y1a ff2 fs5 fc0 sc0 ls2 ws0">maturation and advancement of semiconductor manu</div><div class="t m0 xd h7 y1b ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y1c ff2 fs5 fc0 sc0 ls2 ws0">facturing technology, MEMS sensors are increasingly</div><div class="t m0 xc h7 y1d ff2 fs5 fc0 sc0 ls2 ws0">used in flight attitude calculations [1-9]. However, accu</div><div class="t m0 xd h7 y1e ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y1f ff2 fs5 fc0 sc0 ls2 ws0">racy requirements usually cannot be satisfied by using</div><div class="t m0 xc h7 y20 ff2 fs5 fc0 sc0 ls2 ws0">the inexpensive MEMS sensors. Therefore, some forms</div><div class="t m0 xc h7 y21 ff2 fs5 fc0 sc0 ls2 ws0">of Kalman filtering or complementary fusion algorithms</div><div class="t m0 xc h7 y22 ff2 fs5 fc0 sc0 ls11 ws0">are normally employed to provide more accurate and</div><div class="t m0 xc h7 y23 ff2 fs5 fc0 sc0 ls11 ws0">reliable attitude information in the MEMS attitude de</div><div class="t m0 xd h7 y24 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y25 ff2 fs5 fc0 sc0 ls2 ws0">termination systems [2-6]. A fuzzy logic based closed-</div><div class="t m0 xc h7 y26 ff2 fs5 fc0 sc0 ls2 ws0">loop strapdown attitude system for UAV is presented in</div><div class="t m0 xc h7 y27 ff2 fs5 fc0 sc0 ls2 ws0">[7]. A miniature MEMS-based attitude and heading re</div><div class="t m0 xd h7 y28 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y29 ff2 fs5 fc0 sc0 ls12 ws0">ference system, which makes use of an extended Kalman</div><div class="t m0 xe h7 y2a ff2 fs5 fc0 sc0 ls2 ws0">filter with adaptive gain, and with accelerometers and</div><div class="t m0 xe h7 y2b ff2 fs5 fc0 sc0 ls2 ws0">magnetic sensor as the measurements, was presented in</div><div class="t m0 xe h7 y2c ff2 fs5 fc0 sc0 ls2 ws0">[8]. A precision attitude determination employing a mul</div><div class="t m0 xf h7 y2d ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y2e ff2 fs5 fc0 sc0 ls2 ws0">tiple model adaptive estimation scheme was reported in</div><div class="t m0 xe h7 y2f ff2 fs5 fc0 sc0 ls2 ws0">[9]. For highly nonlinear systems, the extended Kalman</div><div class="t m0 xe h7 y30 ff2 fs5 fc0 sc0 ls2 ws0">filter may not provide enough accuracy for air vehicle</div><div class="t m0 xe h7 y31 ff2 fs5 fc0 sc0 ls13 ws0">attitude estimation. The unscented Kalman filter will</div><div class="t m0 xe h7 y32 ff2 fs5 fc0 sc0 ls2 ws0">achieve greater attitude estimation performance than the</div><div class="t m0 xe h7 y33 ff2 fs5 fc0 sc0 ls13 ws0">extended Kalman filter through the use of unscented</div><div class="t m0 xe h7 y34 ff2 fs5 fc0 sc0 ls2 ws0">transformation [10-13]. In particular, [10] and [12] de</div><div class="t m0 xf h7 y35 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y36 ff2 fs5 fc0 sc0 ls12 ws0">monstrated the success of the unscented Kalman filter</div><div class="t m0 xe h7 y37 ff2 fs5 fc0 sc0 ls14 ws0">design for spacecraft attitude estimation through com</div><div class="t m0 xf h7 y38 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y39 ff2 fs5 fc0 sc0 ls2 ws0">puter simulation. In [11], an unscented Kalman filter for</div><div class="t m0 xe h7 y3a ff2 fs5 fc0 sc0 ls2 ws0">in-motion alignment of low-cost MEMS-based inertial</div><div class="t m0 xe h7 y3b ff2 fs5 fc0 sc0 ls2 ws0">measurement unit was discussed. It demonstrated that</div><div class="t m0 xe h7 y3c ff2 fs5 fc0 sc0 ls2 ws0">the unscented Kalman filter has the capability to deal</div><div class="t m0 xc h8 y3d ff7 fs6 fc0 sc0 ls15 ws0">Journal of Applied Science and Engineering, Vol. 16, No. 2, pp. 165-176 (2013) DOI: 10.6180/jase.2013.16.2.08</div><div class="t m0 xc h9 y3e ff2 fs6 fc0 sc0 ls2 ws0">*Corresponding author. E-mail:shiauj@mail.tku.edu.tw</div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6251549074bc5c01059a1e1c/bg2.jpg"><div class="t m0 xc h7 y3f ff2 fs5 fc0 sc0 ls2 ws0">with large and small attitude errors of an inertial naviga</div><div class="t m0 xd h7 y40 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y41 ff2 fs5 fc0 sc0 ls2 ws0">tion system. In [13], an unscented Kalman filter design</div><div class="t m0 xc h7 y42 ff2 fs5 fc0 sc0 ls2 ws0">for attitude estimation of the strapdown inertial naviga</div><div class="t m0 xd h7 y43 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y44 ff2 fs5 fc0 sc0 ls2 ws0">tion system was reported. The simulation results indi</div><div class="t m0 xd h7 y45 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y46 ff2 fs5 fc0 sc0 ls2 ws0">cated that the performance of the unscented Kalman fil</div><div class="t m0 xd h7 y47 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y48 ff2 fs5 fc0 sc0 ls2 ws0">ter far exceeds the standard extended Kalman filter.</div><div class="t m0 xb h7 y49 ff2 fs5 fc0 sc0 ls2 ws0">This study considers the development of a quater</div><div class="t m0 xd h7 y4a ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y4b ff2 fs5 fc0 sc0 ls2 ws0">nion-based unscented Kalman filter for attitude estima</div><div class="t m0 xd h7 y4c ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y4d ff2 fs5 fc0 sc0 ls2 ws0">tion using low-cost MEMS sensors. The filter utilizes the</div><div class="t m0 xc h7 y4e ff2 fs5 fc0 sc0 ls2 ws0">evolution of the four elements in the quaternion method</div><div class="t m0 xc h7 y4f ff2 fs5 fc0 sc0 ls2 ws0">for attitude determination as the dynamic model, with</div><div class="t m0 xc h7 y50 ff2 fs5 fc0 sc0 ls2 ws0">the four elements as the states of the filter. The pitch and</div><div class="t m0 xc h7 y51 ff2 fs5 fc0 sc0 ls2 ws0">roll angles computed from gravity force decomposition</div><div class="t m0 xc h7 y52 ff2 fs5 fc0 sc0 ls2 ws0">are considered as the measurement for the filter. In addi</div><div class="t m0 xd h7 y53 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y54 ff2 fs5 fc0 sc0 ls2 ws0">tion, the constraint of the four elements of the quaternion</div><div class="t m0 xc h7 y55 ff2 fs5 fc0 sc0 ls2 ws0">method is treated as a perfect measurement and is inte</div><div class="t m0 xd h7 y56 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y57 ff2 fs5 fc0 sc0 ls2 ws0">grated into the filter computation. The immeasurable</div><div class="t m0 xc h7 y58 ff2 fs5 fc0 sc0 ls2 ws0">gravity accelerations are deduced from the outputs of the</div><div class="t m0 xc h7 y59 ff2 fs5 fc0 sc0 ls2 ws0">three axes accelerometers, the relative accelerations, and</div><div class="t m0 xc h7 y5a ff2 fs5 fc0 sc0 ls2 ws0">the accelerations due to body rotation. The source for the</div><div class="t m0 xc h7 y5b ff2 fs5 fc0 sc0 ls14 ws0">heading is different from that for the pitch and roll an-</div><div class="t m0 xc h7 y5c ff2 fs5 fc0 sc0 ls2 ws0">gles. Therefore we use a complimentary filter to compute</div><div class="t m0 xc h7 y5d ff2 fs5 fc0 sc0 ls2 ws0">the heading angle. The complimentary filter uses the</div><div class="t m0 xc h7 y5e ff2 fs5 fc0 sc0 ls2 ws0">heading signal derived from the magnetic force informa-</div><div class="t m0 xc h7 y5f ff2 fs5 fc0 sc0 ls2 ws0">tion from an electronic magnetic sensor and the GPS-</div><div class="t m0 xc h7 y60 ff2 fs5 fc0 sc0 ls17 ws0">derived heading as the inputs to achieve a long-term</div><div class="t m0 xc h7 y61 ff2 fs5 fc0 sc0 ls18 ws0">accurate, reliable and less noisy heading information. An</div><div class="t m0 xc h7 y62 ff2 fs5 fc0 sc0 ls2 ws0">experiment using an in-house designed motion platform</div><div class="t m0 xc h7 y63 ff2 fs5 fc0 sc0 ls2 ws0">is conducted to evaluate the proposed attitude estimation</div><div class="t m0 xc h7 y64 ff2 fs5 fc0 sc0 ls2 ws0">algorithm. The noise characteristics of the sensor signals</div><div class="t m0 xc h7 y65 ff2 fs5 fc0 sc0 ls2 ws0">are examined using the laboratory data. Approximations</div><div class="t m0 xc h7 y66 ff2 fs5 fc0 sc0 ls2 ws0">of the time-varying noise variances of the measured sig</div><div class="t m0 xd h7 y67 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y68 ff2 fs5 fc0 sc0 ls2 ws0">nals are obtained through Taylor series expansions.</div><div class="t m0 xb h7 y69 ff2 fs5 fc0 sc0 ls19 ws0">The proposed attitude estimation algorithm is demon</div><div class="t m0 xd h7 y6a ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y6b ff2 fs5 fc0 sc0 ls2 ws0">strated through a set of flight test data collected from the</div><div class="t m0 xc h7 y6c ff2 fs5 fc0 sc0 ls2 ws0">in-house designed MEMS-based attitude determination</div><div class="t m0 xc h7 y6d ff2 fs5 fc0 sc0 ls2 ws0">system. The duration of the flight covers a total of 3,620</div><div class="t m0 xc h7 y6e ff2 fs5 fc0 sc0 ls1a ws0">seconds. Details of the evolutions of the attitude angles and</div><div class="t m0 xc h7 y6f ff2 fs5 fc0 sc0 ls1b ws0">some related information such as body axes velocities, an</div><div class="t m0 xd h7 y70 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 y71 ff2 fs5 fc0 sc0 ls1c ws0">gle of attack, and side slip angles are examined and explored</div><div class="t m0 xc h7 y72 ff2 fs5 fc0 sc0 ls1b ws0">in the paper. The results show the long-term stability and</div><div class="t m0 xc h7 y73 ff2 fs5 fc0 sc0 ls1d ws0">practicality of the proposed attitude estimation algorithm.</div><div class="t m0 x10 h5 y74 ff4 fs3 fc0 sc0 ls1e ws0">2. Attitude Determination</div><div class="t m0 xb h7 y75 ff2 fs5 fc0 sc0 ls1f ws0">The determination of attitude considered in this</div><div class="t m0 xe h7 y76 ff2 fs5 fc0 sc0 ls2 ws0">study is based on a set of signals measured from an in</div><div class="t m0 xe h7 y77 ff2 fs5 fc0 sc0 ls2 ws0">house-designed Flight Information Measurement Unit</div><div class="t m0 xe h7 y78 ff2 fs5 fc0 sc0 ls2 ws0">(FIMU) and a GPS receiver. The basic system structure</div><div class="t m0 xe h7 y79 ff2 fs5 fc0 sc0 ls2 ws0">is shown in Figure 1. The main function of the FIMU</div><div class="t m0 xe h7 y7a ff2 fs5 fc0 sc0 ls2 ws0">system is to measure the essential data required for atti</div><div class="t m0 xf h7 y7b ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y7c ff2 fs5 fc0 sc0 ls20 ws0">tude estimation including the accelerations, angular rates,</div><div class="t m0 xe h7 y7d ff2 fs5 fc0 sc0 ls2 ws0">magnetic fields, and information from the GPS in a real-</div><div class="t m0 xe h7 y7e ff2 fs5 fc0 sc0 ls2 ws0">time and continuous manner. Hence, the navigation com</div><div class="t m0 xf h7 y7f ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y80 ff2 fs5 fc0 sc0 ls2 ws0">puter can use these information to determine the vehi</div><div class="t m0 xf h7 y81 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y82 ff2 fs5 fc0 sc0 ls2 ws0">cle’s attitude. Details of the design and description of the</div><div class="t m0 xe h7 y83 ff2 fs5 fc0 sc0 ls2 ws0">MEMS-based FIMU system can be found in [14].</div><div class="t m0 x11 h7 y84 ff2 fs5 fc0 sc0 ls2 ws0">Determination of flight attitude involves the com</div><div class="t m0 xf h7 y85 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y86 ff2 fs5 fc0 sc0 ls2 ws0">putation of aircraft pitch angle, roll angle, and heading</div><div class="t m0 xe h7 y87 ff2 fs5 fc0 sc0 ls2 ws0">angle. Pitch angle and roll angle can be computed th</div><div class="t m0 xf h7 y88 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y89 ff2 fs5 fc0 sc0 ls2 ws0">rough the measured aircraft accelerations and body rates</div><div class="t m0 xe h7 y8a ff2 fs5 fc0 sc0 ls2 ws0">from accelerometers and rate gyros. The heading angle is</div><div class="t m0 xe h7 y8b ff2 fs5 fc0 sc0 ls22 ws0">determined by computing the magnetic heading. With</div><div class="t m0 xe h7 y8c ff2 fs5 fc0 sc0 ls2 ws0">the measured three axes acceleration signals and pitch,</div><div class="t m0 xe h7 y8d ff2 fs5 fc0 sc0 ls2 ws0">roll, and yaw rate information – the pitch and roll angles</div><div class="t m0 xe h7 y8e ff2 fs5 fc0 sc0 ls2 ws0">can be determined either by computing the gravitational</div><div class="t m0 xe h7 y8f ff2 fs5 fc0 sc0 ls2 ws0">acceleration components on the body axes, or by using</div><div class="t m0 xe h7 y90 ff2 fs5 fc0 sc0 ls2 ws0">the Euler quaternion method [15]. Computation of the</div><div class="t m0 xe h7 y91 ff2 fs5 fc0 sc0 ls23 ws0">gravitational acceleration components provides long-term</div><div class="t m0 xe h7 y92 ff2 fs5 fc0 sc0 ls2 ws0">accuracy, although it is accompanied by high noise con-</div><div class="t m0 xe h7 y93 ff2 fs5 fc0 sc0 ls2 ws0">tents. The quaternion method, however, provides low</div><div class="t m0 xe h7 y94 ff2 fs5 fc0 sc0 ls2 ws0">noise contents and fast response to changes in the input</div><div class="t m0 xe h7 y95 ff2 fs5 fc0 sc0 ls2 ws0">signals, but tends to drift with time due to gyro bias er</div><div class="t m0 xf h7 y96 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y97 ff2 fs5 fc0 sc0 ls2 ws0">rors.</div><div class="t m0 x11 h7 y98 ff2 fs5 fc0 sc0 ls2 ws0">Assume that at a certain instant, the pitch angle of</div><div class="t m0 xe h7 y99 ff2 fs5 fc0 sc0 ls2 ws0">the aircraft is q, roll angle is f, and the gravitational force</div><div class="t m0 xe h7 y9a ff2 fs5 fc0 sc0 ls2 ws0">components along the body axes (X-, Y-, and Z-axis) are</div><div class="t m0 xe ha y9b ff3 fs5 fc0 sc0 ls2 ws0">g</div><div class="t m0 x12 hb y9c ff3 fs7 fc0 sc0 ls2 ws0">x</div><div class="t m0 x13 h7 y9d ff2 fs5 fc0 sc0 ls2 ws0">, g</div><div class="t m0 x14 hb y9c ff3 fs7 fc0 sc0 ls2 ws0">y</div><div class="t m0 x15 h7 y9d ff2 fs5 fc0 sc0 ls2 ws0">, and g</div><div class="t m0 x16 hb y9c ff3 fs7 fc0 sc0 ls2 ws0">z</div><div class="t m0 x17 h7 y9d ff2 fs5 fc0 sc0 ls2 ws0">respectively. The relationship between the</div><div class="t m0 xe h7 y9e ff2 fs5 fc0 sc0 ls2 ws0">gravitational acceleration components and the attitude</div><div class="t m0 xe h7 y9f ff2 fs5 fc0 sc0 ls2 ws0">angle are</div><div class="t m0 xc h8 ya0 ff7 fs6 fc0 sc0 ls2 ws0">166 Jaw-Kuen Shiau and I-Chiang Wang</div><div class="t m0 xe hc ya1 ff4 fs2 fc0 sc0 ls2 ws0">Figure 1. System Structure of the Attitude Estimation Sys</div><div class="t m0 xf hc ya2 ff2 fs2 fc0 sc0 ls2 ws0">-</div><div class="t m0 x18 hc ya3 ff2 fs2 fc0 sc0 ls2 ws0">tem.</div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6251549074bc5c01059a1e1c/bg3.jpg"><div class="t m0 x19 h7 ya4 ff2 fs5 fc0 sc0 ls2 ws0">(1)</div><div class="t m0 xc h7 ya5 ff2 fs5 fc0 sc0 ls13 ws0">With g</div><div class="t m0 x1a hb ya6 ff3 fs7 fc0 sc0 ls2 ws0">x</div><div class="t m0 x1b h7 ya7 ff2 fs5 fc0 sc0 ls2 ws0">, g</div><div class="t m0 x1c hb ya6 ff3 fs7 fc0 sc0 ls2 ws0">y</div><div class="t m0 x1d h7 ya7 ff2 fs5 fc0 sc0 ls25 ws0">,andg</div><div class="t m0 x1e hb ya6 ff3 fs7 fc0 sc0 ls2 ws0">z</div><div class="t m0 x1f h7 ya7 ff2 fs5 fc0 sc0 ls17 ws0">information, the roll angle f and pitch</div><div class="t m0 xc h7 ya8 ff2 fs5 fc0 sc0 ls26 ws0">angle q can be computed from</div><div class="t m0 x19 h7 ya9 ff2 fs5 fc0 sc0 ls2 ws0">(2)</div><div class="t m0 xc h7 yaa ff2 fs5 fc0 sc0 ls27 ws0">But g</div><div class="t m0 x20 hb yab ff3 fs7 fc0 sc0 ls2 ws0">x</div><div class="t m0 x1a h7 yac ff2 fs5 fc0 sc0 ls2 ws0">, g</div><div class="t m0 x21 hb yab ff3 fs7 fc0 sc0 ls2 ws0">y</div><div class="t m0 x8 h7 yac ff2 fs5 fc0 sc0 ls28 ws0">,andg</div><div class="t m0 x22 hb yab ff3 fs7 fc0 sc0 ls2 ws0">z</div><div class="t m0 xa h7 yac ff2 fs5 fc0 sc0 ls18 ws0">cannot be measured directly during</div><div class="t m0 xc h7 yad ff2 fs5 fc0 sc0 ls17 ws0">flight. The immeasurable gravity accelerations can be</div><div class="t m0 xc h7 yae ff2 fs5 fc0 sc0 ls17 ws0">computed from the outputs of the three axes accelero</div><div class="t m0 xd h7 yaf ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 yb0 ff2 fs5 fc0 sc0 ls29 ws0">meters, the relative accelerations, and the accelerations</div><div class="t m0 xc h7 yb1 ff2 fs5 fc0 sc0 ls29 ws0">due to body rotation. The accelerations measured from</div><div class="t m0 xc h7 yb2 ff2 fs5 fc0 sc0 ls29 ws0">the accelerometers are the total accelerations along the</div><div class="t m0 xc h7 yb3 ff2 fs5 fc0 sc0 ls2a ws0">X-, Y-, and Z-axis (a</div><div class="t m0 x23 hb yb4 ff3 fs7 fc0 sc0 ls2 ws0">x</div><div class="t m0 x24 h7 yb5 ff2 fs5 fc0 sc0 ls2 ws0">, a</div><div class="t m0 x25 hb yb4 ff3 fs7 fc0 sc0 ls2 ws0">y</div><div class="t m0 x26 h7 yb5 ff2 fs5 fc0 sc0 ls2 ws0">, a</div><div class="t m0 x27 hb yb4 ff3 fs7 fc0 sc0 ls2 ws0">z</div><div class="t m0 x28 h7 yb5 ff2 fs5 fc0 sc0 ls2b ws0">). The relationship between</div><div class="t m0 xc h7 yb6 ff2 fs5 fc0 sc0 ls29 ws0">the measured accelerations (a</div><div class="t m0 x29 hb yb7 ff3 fs7 fc0 sc0 ls2 ws0">x</div><div class="t m0 x2a h7 yb8 ff2 fs5 fc0 sc0 ls2 ws0">, a</div><div class="t m0 x2b hb yb7 ff3 fs7 fc0 sc0 ls2 ws0">y</div><div class="t m0 x2c h7 yb8 ff2 fs5 fc0 sc0 ls2 ws0">, a</div><div class="t m0 x2d hb yb7 ff3 fs7 fc0 sc0 ls2 ws0">z</div><div class="t m0 x3 h7 yb8 ff2 fs5 fc0 sc0 ls2c ws0">) and the gravita-</div><div class="t m0 xc h7 yb9 ff2 fs5 fc0 sc0 ls2c ws0">tional force components are [16]</div><div class="t m0 x19 h7 yba ff2 fs5 fc0 sc0 ls2 ws0">(3)</div><div class="t m0 xc h7 ybb ff2 fs5 fc0 sc0 ls2a ws0">where (U, V, W) are the velocities along the X-, Y-, and</div><div class="t m0 xc h7 ybc ff2 fs5 fc0 sc0 ls2d ws0">Z-axis, whereas (p, q, r) are the pitch rate, roll rate, and</div><div class="t m0 xc h7 ybd ff2 fs5 fc0 sc0 ls2e ws0">yaw rate, respectively.</div><div class="t m0 xb h7 ybe ff2 fs5 fc0 sc0 ls2 ws0">For magnetic heading computation, assume that the</div><div class="t m0 xc h7 ybf ff2 fs5 fc0 sc0 ls2 ws0">components of the Earth’s magnetic field along the X-,</div><div class="t m0 xc h7 yc0 ff2 fs5 fc0 sc0 ls2 ws0">Y-, and Z-axis are H</div><div class="t m0 x2e hb yc1 ff3 fs7 fc0 sc0 ls2 ws0">x</div><div class="t m0 x2f h7 yc2 ff2 fs5 fc0 sc0 ls2 ws0">, H</div><div class="t m0 x30 hb yc1 ff3 fs7 fc0 sc0 ls2 ws0">v</div><div class="t m0 x31 h7 yc2 ff2 fs5 fc0 sc0 ls2 ws0">and H</div><div class="t m0 x32 hb yc1 ff3 fs7 fc0 sc0 ls2 ws0">z</div><div class="t m0 x33 h7 yc2 ff2 fs5 fc0 sc0 ls2 ws0">, respectively. Further</div><div class="t m0 xd h7 yc3 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xc h7 yc4 ff2 fs5 fc0 sc0 ls2 ws0">more, the resolved components of H</div><div class="t m0 x34 hb yc5 ff3 fs7 fc0 sc0 ls2 ws0">x</div><div class="t m0 x2d h7 yc6 ff2 fs5 fc0 sc0 ls2 ws0">, H</div><div class="t m0 x35 hb yc5 ff3 fs7 fc0 sc0 ls2 ws0">v</div><div class="t m0 x36 h7 yc6 ff2 fs5 fc0 sc0 ls2 ws0">and H</div><div class="t m0 x37 hb yc5 ff3 fs7 fc0 sc0 ls2 ws0">z</div><div class="t m0 x38 h7 yc6 ff2 fs5 fc0 sc0 ls2 ws0">, in the</div><div class="t m0 xc h7 yc7 ff2 fs5 fc0 sc0 ls2 ws0">horizontal plane along the heading axis H</div><div class="t m0 x39 hd y9c ff2 fs7 fc0 sc0 ls2 ws0">1</div><div class="t m0 x3a h7 y9d ff2 fs5 fc0 sc0 ls2 ws0">and at right</div><div class="t m0 xc h7 y9e ff2 fs5 fc0 sc0 ls2 ws0">angles to the heading axis H</div><div class="t m0 x28 hd yc8 ff2 fs7 fc0 sc0 ls2 ws0">2</div><div class="t m0 x3b h7 yc9 ff2 fs5 fc0 sc0 ls2 ws0">are given by [16]</div><div class="t m0 x19 h7 yca ff2 fs5 fc0 sc0 ls2f ws0">(4)</div><div class="t m0 x19 h7 ycb ff2 fs5 fc0 sc0 ls2f ws0">(5)</div><div class="t m0 xc h7 ycc ff2 fs5 fc0 sc0 ls29 ws0">Thus, the magnetic heading of the aircraft y</div><div class="t m0 x3c hb ycd ff3 fs7 fc0 sc0 ls2 ws0">M</div><div class="t m0 x3d h7 yce ff2 fs5 fc0 sc0 ls13 ws0">is</div><div class="t m0 x19 h7 ycf ff2 fs5 fc0 sc0 ls30 ws0">(6)</div><div class="t m0 xc h7 yd0 ff2 fs5 fc0 sc0 ls2c ws0">If the local magnetic declination is y</div><div class="t m0 x3e hb yd1 ff3 fs7 fc0 sc0 ls31 ws0">md</div><div class="t m0 x35 h7 y74 ff2 fs5 fc0 sc0 ls11 ws0">, the measured</div><div class="t m0 xc h7 yd2 ff2 fs5 fc0 sc0 ls2b ws0">aircraft heading, y</div><div class="t m0 xa hb yd3 ff3 fs7 fc0 sc0 ls2 ws0">m</div><div class="t m0 x3f h7 yd4 ff2 fs5 fc0 sc0 ls2b ws0">, can be determined from y</div><div class="t m0 x40 hb yd3 ff3 fs7 fc0 sc0 ls2 ws0">m</div><div class="t m0 x38 h7 yd4 ff2 fs5 fc0 sc0 ls2 ws0">= y</div><div class="t m0 x6 hb yd3 ff3 fs7 fc0 sc0 ls2 ws0">M</div><div class="t m0 x41 h7 yd4 ff2 fs5 fc0 sc0 ls2 ws0">+</div><div class="t m0 xc ha yd5 ff6 fs5 fc0 sc0 ls2 ws0">y</div><div class="t m0 x42 hb yd6 ff3 fs7 fc0 sc0 ls31 ws0">md</div><div class="t m0 x43 h7 y75 ff2 fs5 fc0 sc0 ls26 ws0">. In this computation, the pitch angle q and roll an</div><div class="t m0 xd h7 yd7 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 y76 ff2 fs5 fc0 sc0 ls26 ws0">gle f are assumed to be known. That is, we have to esti</div><div class="t m0 xf h7 yd8 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 yd9 ff2 fs5 fc0 sc0 ls2 ws0">mate q and f first before computing the aircraft heading.</div><div class="t m0 x11 h7 yda ff2 fs5 fc0 sc0 ls2 ws0">A common method used to compute the attitude an</div><div class="t m0 xf h7 ydb ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 ydc ff2 fs5 fc0 sc0 ls2 ws0">gle in inertial navigation system is the Euler quaternion</div><div class="t m0 xe h7 ydd ff2 fs5 fc0 sc0 ls2 ws0">method [16], which uses the Euler four symmetrical pa</div><div class="t m0 xf h7 yde ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 ydf ff2 fs5 fc0 sc0 ls2 ws0">rameters (e</div><div class="t m0 x44 hd ye0 ff2 fs7 fc0 sc0 ls2 ws0">0</div><div class="t m0 x45 h7 ye1 ff2 fs5 fc0 sc0 ls2 ws0">, e</div><div class="t m0 x46 hd ye0 ff2 fs7 fc0 sc0 ls2 ws0">1</div><div class="t m0 x47 h7 ye1 ff2 fs5 fc0 sc0 ls2 ws0">, e</div><div class="t m0 x48 hd ye0 ff2 fs7 fc0 sc0 ls2 ws0">2</div><div class="t m0 x49 h7 ye1 ff2 fs5 fc0 sc0 ls2 ws0">, e</div><div class="t m0 x4a hd ye0 ff2 fs7 fc0 sc0 ls2 ws0">3</div><div class="t m0 x4b h7 ye1 ff2 fs5 fc0 sc0 ls2 ws0">) to define the aircraft attitude. The</div><div class="t m0 xe h7 ye2 ff2 fs5 fc0 sc0 ls2 ws0">relationship between the attitude angle and the four pa</div><div class="t m0 xf h7 ye3 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 ye4 ff2 fs5 fc0 sc0 ls2 ws0">rameters are</div><div class="t m0 x4c h7 ye5 ff2 fs5 fc0 sc0 ls2 ws0">(7)</div><div class="t m0 xe h7 ye6 ff2 fs5 fc0 sc0 ls18 ws0">The four parameters are subjected to the following con</div><div class="t m0 xf h7 ye7 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 ye8 ff2 fs5 fc0 sc0 ls18 ws0">straint equation</div><div class="t m0 x4c h7 ye9 ff2 fs5 fc0 sc0 ls2 ws0">(8)</div><div class="t m0 xe h7 yea ff2 fs5 fc0 sc0 ls2b ws0">The dynamics of the four parameters regarding the air-</div><div class="t m0 xe h7 yeb ff2 fs5 fc0 sc0 ls2b ws0">craft body rate (p, q, r) are characterized by the follow-</div><div class="t m0 xe h7 yec ff2 fs5 fc0 sc0 ls11 ws0">ing form:</div><div class="t m0 x4c h7 yed ff2 fs5 fc0 sc0 ls2 ws0">(9)</div><div class="t m0 xe h7 yee ff2 fs5 fc0 sc0 ls17 ws0">Therefore, the attitude angles in (7) may be calculated</div><div class="t m0 xe h7 yef ff2 fs5 fc0 sc0 ls13 ws0">by solving Equation (9). Only body information - (p, q,</div><div class="t m0 xe h7 yf0 ff3 fs5 fc0 sc0 ls2 ws0">r) directly measured from rate gyros - is required to</div><div class="t m0 xe h7 yf1 ff2 fs5 fc0 sc0 ls11 ws0">solve Equation (9). However, when MEMS gyros are</div><div class="t m0 xe h7 yf2 ff2 fs5 fc0 sc0 ls11 ws0">used for body rate measurement, long-term drift usually</div><div class="t m0 xe h7 yf3 ff2 fs5 fc0 sc0 ls2d ws0">encountered due to gyro bias errors and integral opera</div><div class="t m0 xf h7 yf4 ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 yf5 ff2 fs5 fc0 sc0 ls22 ws0">tion for solving Equation (9).</div><div class="t m0 x4d h5 yf6 ff4 fs3 fc0 sc0 ls34 ws0">3. Unscented Kalman Filter Design</div><div class="t m0 x11 h7 yf7 ff2 fs5 fc0 sc0 ls2 ws0">In this study, the measured pitch and roll angles are</div><div class="t m0 xe h7 yf8 ff2 fs5 fc0 sc0 ls2 ws0">deduced from the outputs of the accelerometers, gyros,</div><div class="t m0 xe h7 yf9 ff2 fs5 fc0 sc0 ls2 ws0">and the GPS receiver while the heading angle is from the</div><div class="t m0 xe h7 yfa ff2 fs5 fc0 sc0 ls2 ws0">electronic compass unit. To simplify the design, two fil</div><div class="t m0 xf h7 yfb ff2 fs5 fc0 sc0 ls2 ws0">-</div><div class="t m0 xe h7 yfc ff2 fs5 fc0 sc0 ls2 ws0">ters are designed for the attitude estimation. The pitch</div><div class="t m0 xe h7 yfd ff2 fs5 fc0 sc0 ls2 ws0">and roll angles are determined based on a constrained</div><div class="t m0 xe h7 yfe ff2 fs5 fc0 sc0 ls2 ws0">unscented Kalman filter. The heading angle estimation is</div><div class="t m0 x4e h8 ya0 ff7 fs6 fc0 sc0 ls2 ws0">Unscented Kalman Filtering for Attitude Determination Using MEMS Sensors 167</div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>