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Gyroscopic drift compensation by using low cost sensors for improved attitude determination
Accepted Manuscript Gyroscopic Drift Compensation by Using Low Cost Sensors for Improved Attitude Determination S.M. Dildar Ali, U. Iqbal Bhatti, K. Munawwar, U. Al-Saggaf, Shoaib Mansoor, Jamshaid Ali PII: DOI: Reference:
S0263-2241(17)30705-4 https://doi.org/10.1016/j.measurement.2017.11.003 MEASUR 5072
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
24 November 2016 26 September 2017 2 November 2017
Please cite this article as: S.M. Dildar Ali, U. Iqbal Bhatti, K. Munawwar, U. Al-Saggaf, S. Mansoor, J. Ali, Gyroscopic Drift Compensation by Using Low Cost Sensors for Improved Attitude Determination, Measurement (2017), doi: https://doi.org/10.1016/j.measurement.2017.11.003
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Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2
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Gyroscopic Drift Compensation by Using Low Cost Sensors for Improved Attitude Determination S. M. Dildar Ali, U. Iqbal Bhatti, K. Munawwar, U. Al-Saggaf, Shoaib Mansoor and Jamshaid Ali
Transformation matrix from e to n
Abstract— Desire of inexpensive Electro-Optic and MicroElectro-Mechanical System (MEMS) inertial sensors has drastically been increased in the recent times for both military and commercial applications. Beside the traditional applications, reduced cost of such sensors has opened new domains in personal navigation. This paper provides a framework for attitude estimation using miniaturized and cost-effective Inertial Measurement Units (IMU). Sensor fusion of gyroscope, accelerometer and True Air Speed (TAS) sensor helps in minimizing the characteristic time growing error present in gyroscopic integrated data. A novel approach of TAS model development is implemented to generate true air speed data in the absence of TAS sensor. The presence of linear acceleration is estimated and eliminated by means of gyroscope and TAS model. Due to the slight difference in two direction vectors, an error function is estimated and constantly compensated by using a Proportional-Integral (PI) block. Coarse tuning of PI gains is applied and simulated results are produced to assess the filter performance by using real vehicle data. It has been presented that the proposed filter can be used to compute a reasonably accurate attitude solution by using low cost inertial sensors when external aiding is unavailable or not useful. Keywords — Attitude, Complementary Filter, True Air Speed Model, Air Turbulence Model, Inertial Sensors, Body Frame.
NOMENCLATURE b frame e frame i frame n frame
Body frame Earth frame Inertial frame Navigation frame (NED frame) Transformation matrix from b to e
frame Transformation matrix from b to i frame Transformation matrix from b to n frame Transformation matrix from e to i frame Syed M. Dildar Ali ([email protected]) and Dr. Umar Iqbal Bhatti ([email protected]) are working with the Aeronautics and Astronautics Engineering Department at Institute of Space Technology (IST), Islamabad 48000, Pakistan. Prof. Dr. Khalid Munawar and Prof. Dr. Ubaid Al-Saggaf are working at Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdul Aziz University, Jeddah, KSA. Shoaib Mansoor is a PhD scholar at CUST Islamabad, Pakistan and is a member of the research team. Dr. Jamshaid Ali is working as a consultant in an Aerospace Industry at Islamabad, Pakistan.
frame
L l
Specific force in b frame Local gravity vector Angular rate of the Earth Gyroscope provided angular rates Angular rate of body w.r.t to e frame expressed in b frame Angular rate of Earth w.r.t to i frame expressed in e frame Roll angle (p) Pitch angle (q) Yaw angle (r) Latitude Longitude I. INTRODUCTION
P
recise attitude measurement remains the foremost requirement in all aerial applications. It plays a vital role in maneuvering of an aircraft/Unmanned Aerial Vehicle (UAV) or a guided missile to follow a trajectory to achieve the set targets. Moreover, with the rapid growth of industrial automation, attitude determination has become very critical in the field of robotics and humanoids [1]. Due to recent increased demand of cheaper and reliable systems, it has become the prime concern to utilize cheaper sensors i.e., low cost IFOG (Interferometic Fiber Optic Gyroscope) and MEMS (Micro Electro-Mechanical System) based inertial sensors, with more sophisticated state estimation algorithms [2]. However, the complexity of such algorithm increases as the quality of the measuring sensor decreases. Without relying on any external reference, data fusion of available sensors can minimize the attitude errors with an intelligent and robust attitude algorithm. Over the past decades, major research has been carried out to minimize the time growing attitude error with minimum computational requirements. Earlier, Shuster and Oh [3] have used TRIAD algorithm for the deterministic attitude while the QUEST (Quaternion-Estimator) algorithm has been implemented to measure the optimal attitude of spacecraft in weighted configuration [4]. Markely [5] has also presented a comparison of different attitude representations using two vector measurement approaches. A similar work has also been
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 presented by Shuster [6], which provides different relationships between twelve attitude representations. The Kalman Filter (KF) and its variants such as the Extended Kalman Filter (EKF) are well studied techniques for error estimation and gyroscope bias elimination [7]. However, such filters cannot always guarantee the stability and optimality of the solution in case of nonlinearity or drastic change in noise model [8]. Likewise, EKF can also fail, when vehicle dynamics is highly nonlinear or having poor a-priori state estimates [7]. Furthermore, the KF and EKF are difficult to apply robustly where computational resources are limited making them unsuitable for limited scale applications. [23, 9, 4]. Other than KF variants, the most commonly class of filtration technique is the complimentary filters. Such filters have good frequency filtering properties for linear systems [24, 10]. Complementary filter combines measurements from two or more available sources with different characteristics and having minimal distortion in the signal output [11]. Recently, a lot of development has taken place on nonlinear analogous of Single Input Single Output (SISO) filters for attitude determination [12, 13]. Implementation of such schemes on UAVs uses the accelerometer provided information to estimate the gravitational direction. Such filters, however, fail when the vehicle dynamics are large and accelerometer measurements can no longer provide suitable estimates of gravitational direction [14]. Rehbindera and Hub have also proposed a solution by fusing IMU based sensor information to offer stable attitude estimates for robotic applications [15]. Euston et al. [14] work allows full estimation of vehicle attitude as well as gyroscope biases based just on the accelerometer and gyroscope outputs. The non-linear complementary filter provides an excellent attitude estimate suitable for small UAVs [25]. Furthermore, a more recent work on sensor fusion by Yamato et al. [16] provides a new framework for robotic applications, by using angular velocity vectors and direction vectors. The stated approach contains two-step measurement update methodology, namely, attitude estimation by simple rate integration and attitude determination via vector matching by using Wahba problem technique [12]. The first function is for regulation of error accumulation while the second functional block estimates the reference attitude from the bias removing feedback. Such filtering techniques are used for attitude estimation for swarm UAVs [26] and aerial robots [27]. A MARG (Magnetic, Angular Rate and Gravity) sensor is employed with a
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complementary filter to integrate with an IMU to provide the attitude solution for MAV (Micro-Air vehicles) [28]. Similar configuration has been tested with a sensor array for improved attitude estimation [29]. The current study proposes a continuation of above work by developing an attitude estimation algorithm for miniaturized and cost-effective systems. The gyroscope provided attitude solution typically suffers from a time growing error due to the presence of certain bias levels in gyroscope signal. The current sensor fusion configuration scheme utilizes a triad of low cost inertial sensor (MEMS/IFOG) along with a True Air Speed (TAS) sensor model to independently aid the attitude solution. A similar kind of work has already been proposed by Euston et al. but the scheme fails to address a problem when TAS sensor data is not available. The situation may also arise in the applications where TAS sensor has no use and cannot provide sufficient results e.g. robots, land rovers and platform stabilizers. A complete mathematical description for the TAS model along with certain air disturbance, is devised. For ease and available data set limitations, complete algorithm has been applied for body frame attitude computation without any known reference. The obtained results clearly exhibit a potential of the proposed scheme to estimate a more accurate attitude solution. II. PROPOSED ATTITUDE DETERMINATION TECHNIQUE The problem stated above, can be addressed either by providing an external aiding or implementing a filtration scheme to compensate and minimize growing drifts in gyroscopes. The presented work demonstrates a methodology of estimating an improved attitude solution as compared to conventional schemes through a nonlinear complementary filter, also augmented by vehicle dynamics. The attitude determination technique consists of two computational blocks. The first block computes and simulates the true air speed sensor data which later combines with the gyroscopic output to provide the estimates of direction vector. The gyroscope and TAS model fusion provide the best estimates of gravitational or direction vector. A non-linear and non-inertial acceleration model is used to compensate the accelerometer output to obtain a zero bias estimate of the gravitational direction. The model is based on a simple centripetal force model derived from the airspeed and the rate of turn of the vehicle [14].Abbreviations and Acronyms
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2
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Complementary filter performs a low pass filtering on the low frequency attitude which is obtained from accelerometers and operates as a high pass filter on biased high frequency attitude estimates, obtained from the direct integration of gyroscopes. Both the estimates are then fused together to achieve all-pass estimate of the attitude. The schematic of the proposed filter is shown in Figure1. The performance of the algorithm is examined using the body frame attitude solution in order to maintain simplicity for technique validation. Its implementation in NED frame can also be applied, however, that analysis is unnecessary. The combined system is relatively simple to implement and achieves better performance as compared to an unaided system or loosely coupled system. The algorithm is implemented and tested on a data set of a ground vehicle [17] and favorable results are obtained.
in the body frame and the answer is also expressed in the body frame. (4)
III. TRUE AIR SPEED (TAS) MODEL
The total angular rate of the body with respect to inertial frame can be expressed as the sum of two rates such that:
True Air speed sensor provides the speed of the aircraft relative to air mass in which it is flying [18]. In the absence of TAS data, the desired data set is generated using the equipped INS through a TAS model. The proposed TAS model under this scheme has brought a novelty in previous cited work in the introduction section. Although body frame mechanization is a well-known formulation for example described in Titterton and Weston [19]. However, it is typically used for control purposes and not for propagation of navigation variables. In this way it has become possible to use a conventional model for air disturbance named as Dryden Model to be incorporated in the TAS model, for example as in Maaz et al. [20]. The output of a conventional Dryden Model are typically provided in the body frame and hence are directly utilized for providing better imitation of original TAS. The TAS model requires two additional computations i.e. velocity computation in the body frame and an air turbulence model.
where;
and
The update equation, from e-frame to b-frame is (5)
(6) At time t = 0;
and
A. Body Frame Mechanization Body frame velocities are computed by using the INS data. The following mathematical derivation provides a clear understanding of body frame mechanization along with necessary initial conditions. The Coriolis law for the body frame mechanization (subscript and superscript b) can be expressed as (1) Substituting the
from inertial frame mechanization as
formulated in appendix Eq. (A.9). (2) (3) In body frame,
Dot representation is used when the derivative is calculated
Fig. 1. Basic Schematic of Proposed Algorithm with Complementary Filter (BF denotes Body Frame) Fig. 2. RSS Comparison of Velocities in Body Frame and NED Frame
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2
(8)
The above derivation provides encouraging results for body frame velocity computation. The root sum square (RSS) of the velocities in the body frame is identical to NED frame velocities. Fig. 2 demonstrates the validation of body frame velocity computation by comparing RSS of velocities in different frames for a ground vehicle. This figure is of tutorial value and acts as a cross checking measure for validity of the body frame based TAS model. Furthermore this can be used as a benchmark for the case when disturbances are added for simulation of a more realistic TAS model. It is acknowledged, that the work done for body frame mechanization is not completely novel however its use in the particular context of navigation and for ease of disturbance model shows its worthiness. B. Effect of Air Turbulence Model The velocities computed through INS data in the body frame are the ideal velocities. However, in actual, velocities may differ due to the presence of certain disturbances in environment i.e., wind turbulences (side slip, wind gusts etc.). To further enhance the validity of the INS computed velocities, velocities are passed from another model i.e. Simulink programmed Dryden Wind Turbulence Model to measure and introduce turbulence intensities as a function of altitude [21]. The Dryden model adds turbulence to the aerospace model by passing band-limited white noise through appropriate forming filters and provides output as stochastic processes with the Dryden gusts power spectral densities. Wind turbulences varies with altitude and vehicle speed. As the altitude increases, the magnitude of air gusts decreases and
vehicle experience much less disturbances. The example is of aircraft cruise flight, which typically experiences much smaller air turbulences in a level flight while it may suffers from heavy turbulence during landing and takeoff. The model can be validated by varying the input parameters and analyzing the resultant turbulences in each axis. The vehicle in this case, is travelling with 40 to 50 km/hr at an altitude of 35 meters, the turbulence disturbance comparison is shown in two parts of Figure 3. The results clearly shows that
Fig. 4. Effects of Turbulence Velocities on INS Computed XY Velocities
difference in turbulence velocity is quite small for these two speeds. Hence for the complete simulation work, the speed of the vehicle has been finalized as 50 km/hr. C. TAS Simulation Results As described earlier, due to the absence of air speed sensor data, TAS model is used to provide input to the proposed filter. Once the disturbances are known, they can be added in the computed velocities to get the final velocities in the body frame. TAS model results are presented in Figure 4 which provides the simulated true air speed of the vehicle in body frame. IV.
Fig. 3. Turbulence Velocity as generated by the used model at an input speed of 40 km/hr (upper figure) and 50 km/hr (lower figure) respectively
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SIMULATION RESULTS
The IMU used for the implementation of proposed attitude estimation algorithm is the Honeywell Commercial Inertial Measurement Unit (CIMU), which contains optical gyroscopes and quartz based accelerometers. CIMU is a navigation grade IMU with gyros of accuracy range of 0.01 degree per hour. By implementing the proposed TAS model, air speed data limitation has been resolved and the availability of required input parameters for the proposed scheme is ensured. The gravitational direction vectors can now be computed from the system and quaternion gravitational estimates. The error, hence arisen due to the difference between two gravitational vectors, is compensated by using PI filter. The PI filter is conventional Proportional and Integration filter traditionally used in the realm of feedback control systems [22]. In this type of filtering algorithm, the error signal is fed to the filter and the whole loop is operated
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 in a feedback configuration. By tuning the gains of the filter which are known as Proportional and Integration gains, the closed loop operate to minimize the error. The proportional gain is simply a multiplying factor and the Integration gain is a discrete (or continuous) integration routine operated upon the input error. This small routine involves current error value and error history. The output of both of these operations are added together. It has been prove in control systems theory that this sort of filter is very successful in minimizing the input error even if it contains a bias [22]. The tuning of PI block is critical and needs to be addressed during the calibration process of IMU/INS. The gains of each block affect the final attitude solution and the effect of each block is discussed separately.
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B. Effect of Integral Term (KI) Upon the optimization of KP gain, the effect of integral term on filter performance needs to be investigated further. The integral gain (KI) is applied, once the solution is in an acceptable range after KP tuning i.e., KP = 0.001. KI is time
A. Effect of Proportional Term (KP) Initially, the performance of the filter is restricted only to the proportional term (KP), by nulling the integral gain i.e., KI=0. This helps in understanding the direct effect of KP on attitude solution. KP is kept small primarily, which may later be tuned with appropriate gain value for desirable performance. The Fig. 6. Roll Angle at Final PI Tuning
Fig. 5. Attitude Solution with Only Proportional Block (KP= 0.001)
results are compared with the unaided attitude solution i.e., attitude solution from raw gyroscope rates. Starting with a unity magnitude for proportional gain (KP=1) and further decreasing the KP gain up to 0.001, a reasonable roll, pitch and yaw angles are achieved, as shown in Figure 5.The attitude computed directly from the raw gyroscope rates is termed as ‘Raw Gyro Rate’ with red legend and will be treated as the reference for current study. Whereas, the proposed filter provided attitude solution after the error/bias drift compensation, has been termed as ‘Drift Compensated Rate’, with a green legend. From the series of simulations, it is observed that filter solution becomes divergent at higher values of KP and hence the value for KP should be small. The filter provided attitude solution follows a similar trend of rotations as experienced by the body mounted IMU. The obtained result are much similar to that of the raw gyro solution but maintain a slight difference as we expect to observe since the raw gyro attitude solution is prone to suffer from time integrated bias error.
Fig. 7. Pitch Angle at Final PI Tuning
Fig. 8. Yaw Angle at Final PI Tuning
integrated gain, therefore, magnitude of integral gain should be much smaller than KP (about 10 to 100 times). Initially, with only the proportional gain, solution followed a similar trend to raw gyro attitude solution. But with the
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 addition of larger integral gain the solution deviates with time. Yaw angle is also disturbed at higher value of KI. Keeping the KP gain constant and lowering the integral gain, the resultant solution becomes less divergent. The results obtained with the addition of integral term in previous solution are shown in Figures 6, 7 and 8. These results provide a clear picture for gain range of the integral term. KI is much smaller than KP (KI=0.00001, 100 times smaller in this case). The obtained results are better as compared to previously simulated results. The estimated roll, pitch and yaw angles are largely improved and have become similar to raw gyro rate solution. A slight difference in roll angle may further be minimized with fine tuning. However, no significant improvement in yaw angle is observed. This also validates the proposed attitude estimation algorithm as gravitational direction vectors cannot compensate the azimuth error. A further tuning of the filter is necessary to fully optimize the attitude solution which may be achieved by change in PI gains. Furthermore, the integral term in proposed mechanism allow to eliminate the gyro bias and helps in providing a more precise solution.
Fig. 9. Drift Compensation in Roll, Pitch and Yaw Channels for KP=0.001 and KI=0.00001
V.
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SOLUTION COMPARISON
The technique is also applied to data set obtained from commercially available system known as CMIGITS. This system is developed by Systron Donner and lies in the
Fig. 11. Attitude Difference between Compensated and Raw Output
medium accuracy range of 3.0 deg/hr [30]. It must also be mentioned here that the technique is also tested for very lower grade MEMS based systems without success due to relatively large amount of noise. The systems tried are developed by VectorNav (VN-200) and Shimmer (Shimmer 3) [31, 32]. The results for drift compensation obtained are presented as below in Figure 9 and Figure 10. It can be seen that compensation has indeed been achieved in the roll and pitch channels as expected. Based on this compensation, it is further studied how much is the effect of these corrections on the performance of the system. The attitude difference between the corrected and the raw system outputs is plotted in Figure 11. It is claimed that this difference in accuracy is achieved by use of the proposed technique. Since the data is limited to 200 seconds, typical performance indicator of deg per hour cannot be compensated. However, by an approximate extrapolation of results reveal that the technique is effective in curtailing the error growth around one deg per hour in a medium accuracy system. It is pertinent to mention here that further verification of the algorithm requires use of much higher quality system to act as reference purpose which is quite expensive and not accessible. VI. CONCLUSION
Fig. 10. Drift Compensation in Roll, Pitch and Yaw Channels for KP=0.0001 and KI=0.000001
The main purpose of this study is to develop a better attitude solution algorithm for UAVs, robots and other aerial applications by utilizing cheaper and low grade sensors. A filtering technique is implemented via data fusion of inertial sensors and true air speed model. TAS model encompasses air turbulence effects which adds effectiveness in proposed work for real world applications. The proposed algorithm is implemented in MATLAB and results are produced using ground vehicle data. Simulation results clearly illustrates that the proposed technique solution is adaptable and flexible which can be tuned with an available
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 reference attitude profile. The resultant solution exhibits relatively small effect of gyroscopic drift i.e., the drift in gyroscope integrated data is minimized. The error growth is continuously monitored and measured by taking a feedback input from gravitational direction vectors. Further, the instantaneous error is compensated by adjusting appropriate gains of the PI block. Coarse tuning PI gains may bring solution in an acceptable range and consequently, fine tuning may further optimize the final solution. The presented filtering methodology is simple and easy to implement. Furthermore, it requires lesser computational power as compared to other filters (particle filter, Kalman filter etc.) making it more suitable for medium range applicants.
ACKNOWLEDGMENT The authors duly acknowledge the technical and research support of Aeronautics and Astronautics Department, Institute of Space Technology (IST), Islamabad. Authors would also like to thank Dr. M. Aleem Mirza for his consistent encouragement. The authors also acknowledge the technical support of Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdul Aziz University, Jeddah, KSA. REFERENCES [1]
[2]
APPENDIX In order to obtain a relation for velocity computation in inertial frame, we start from the basic definitions of acceleration ‘a’ and accelerometer output ‘f’ i.e. the specific force: (A.1)
[3]
[4]
[5]
(A.2) [6]
(A.3)
[7]
By Coriolis Law [6] (A.4)
[8]
where [9]
Differentiating Eq. (A.4) [10]
(A.5)
[11]
Since [12]
and [13]
Therefore Eq. (A.5) becomes [14]
(A.6) [15]
Substituting Eq. (A.6) in Eq. (A.3) and rearranging
[16]
(A.7) [17]
Also, (A.8)
[18]
Comparing Eq. (A.7) and Eq. (A.8) results as:
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(A.9) [19]
E. North, J. Georgy, M. Tarbouchi, U. Iqbal, and A. Noureldin, "Enhanced Mobile Robot Outdoor Localization using INS/GPS Integration", in The IEEE International Conference on Computer Engineering and Systems, Cairo, 2009, pp. 127-132. A. J. Grunwald, E. Wahnon, and S. J. Merhav, “Estimation of Vehicle Attitude and Flight-path Angles from Low Cost Rate Gyro Measurements”, in The 25th Israel Annual Conference on Aviation Astronautics, Israel, 1983, pp. 121–127. M. D. Shuster and S. D. Oh, "Three-axis Attitude Determination from Vector Observations," AIAA Journal of Guidance, Control, and Dynamics, vol. 4, no. 1, 1981, pp. 70-77. J. M. Roberts, P. I. Corke, and G. Buskey, “Low-cost Flight Control System for Small Autonomous Helicopter”, in The Australian Conference on Robotics and Automation, Auckland, 2002, pp. 71–76. F. Markely, “Attitude Determination Using Two Vector Measurements”. Internet: ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19990052720.pdf, NASA's Goddard Space Flight Center, Greenbelt, 1998. M. D. Shuster, “A Survey of Attitude Representations,” Journal of Astronautical Sciences, vol. 41, no. 4, 1993, pp 439-517. J. Crassidis, F. Markley, and Y. Cheng, "Survey of Nonlinear Attitude Estimation Methods," Journal of Guidance, Control, and Dynamics, vol. 30, no. 1, 2007, pp. 12-28. N. K. Filyashkin and V. S. Yatskivsky, “Prediction of Inertial Navigation System Error Dynamics in INS/GPS System", in The IEEE 2nd International Conference on Actual Problems of Unmanned Air Vehicles Developments, 2013, pp.206-209. D. Choukroun, I.Y. Bar-Itzhack, and Y. Oshman, “Novel Quaternion Kalman Filter,” IEEE Transactions on Aerospace and Electronic Systems, 2006, pp. 174-190. R. Mahony, T. Hamel, and J. Pflimlin, “Non-linear Complementary Filters on the Special Orthogonal Group,” IEEE Transactions on Automatic Control, 2007, pp. 1477-1484. K. Narayan, "Performance Analysis of Attitude Determination Algorithms for Low Cost Attitude Heading Reference Systems." PhD Thesis, Auburn University, USA, 2010. J. Thienel and R. M. Sanner, “A Coupled Nonlinear Spacecraft Attitude Controller and Observer with an Unknown Constant Gyro Bias and Gyro Noise,” IEEE Transactions on Automatic Control, 2003, pp. 20112015. S. Bonnabel, P. Martin, and P. Rouchon, “A Non-Linear Symmetry Preserving Observer for Velocity-Aided Inertial Navigation”, in The American Control Conference, 2006, pp. 2910–2914. M. Euston, P. Coote, R. Mahony, J. Kim, and T. Hamel, “A Complementary Filter for Attitude Estimation of a Fixed-Wing UAV”, in The IEEE/RSJ International Conference on Intelligent Robots and Systems, Nice, France, 2008, pp. 340-345. H. Rehbindera and X. Hub, “Drift-free Attitude Estimation for Accelerated Rigid Bodies,” Automatica, vol. 40, 2004, pp. 653–659. H. Yamato, T. Furuta, and K. Tomiyama, "Attitude Determination by Globally and Asymptotically Stable Estimation of Gyroscope Bias Error with Disturbance Attenuation and Rejection," Journal of Robotics and Mechatronics, vol. 24, no. 2, 2012, pp. 389-398. U. I. Bhatti, "Improved Integrity Algorithms for Integrated GPS/INS Systems in the Presence of Slowly Growing Errors." PhD Thesis, Imperial College London, UK, 2007. IVAO HQ. Training Department, "Air Speed Definitions." Internet: https://www.ivao.aero/training/documentation/books/PP_ADC_airspeed. pdf, Sept. 29, 2016. D. H. Titerton and J. L. Weston, “Strapdown Inertial Navigation Technology.” AIAA (2nd Edition), 2004.
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 [20] M. Butt, K. Munawar, U. Bhatti, S. Iqbal, U. Al-Saggaf, and W. Ochieng, “4D Trajectory Generation for Guidance Module of a UAV for a Gate to Gate Flight in Presence of Turbulence,” International Journal of Advanced Robotics Systems, 2016, pp. 1-14. [21] F. Hoblit, “Gust Loads on Aircraft: Concepts and Applications.” AIAA Education Series, 1988. [22] R. C. Dorf and R. H. Bishop, “Modern Control Systems.” Prentice Hall (12th Edition), 2010. [23] Kumar, N. Ravi, and E. Nagabhooshanam, "An Extended Kalman Filter for Low-Cost Positioning System in Agricultural Vehicles", in The IEEE International Conference on Wireless Communications, Signal Processing and Networking (WiSPNET), 2016. [24] Lee, J. Keun, "A Two-step Kalman/Complementary Filter for Estimation of Vertical Position using an IMU-Barometer System," Journal of Sensor Science and Technology, 2016, pp. 202-207. [25] S. Leutenegger, A. Melzer, K. Alexis, and R. Siegwart, "Robust State Estimation for Small Unmanned Airplanes", in The IEEE Conference on Control Applications (CCA), 2014, pp. 1085-1992. [26] X. Dong, Y. Zhou, Z. Ren, and Y. Zhong, "Time-varying Formation Control for Unmanned Aerial Vehicles with Switching Interaction Topologies," Control Engineering Practice, 2016, pp. 26-36. [27] Mahony, Robert, R. Beard, and V. Kumar, "Modeling and Control of Aerial Robots." Springer International Publishing, 2016, pp.1307-1334.
[28] Valenti, G. Roberto, I. Dryanovski, and J. Xiao, "Keeping a Good Attitude: A Quaternion-based Orientation Filter for IMUs and MARGs," Sensors vol. 15, no. 8, 2015, pp. 19302-19330. [29] S. Madgwick, A. Harrison, P. Sharkey, R. Vaidyanathan, and W. Harwin, "Measuring Motion with Kinematically Redundant Accelerometer Arrays: Theory, Simulation and Implementation," Elsevier-Mechatronics, vol. 23, issue, 5, 2013, pp. 518-529. [30] CMIGITS III User Manual, Systron Donner, California, USA. [31] VectorNav VN-200 User Manual, VectorNav Technologies, Texas, USA, 2008. [32] Shimmer IMU User Guide Rev. 1.4, Shimmer, Dublin, Ireland, 2006.
Highlights
Quaternion based attitude estimation by using low cost inertial sensors
True Air Speed (TAS) model development via body frame mechanization
Complementary filter for Gyroscope time growing error compensation using accelerometer and TAS measurements
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S0263-2241(17)30705-4 https://doi.org/10.1016/j.measurement.2017.11.003 MEASUR 5072
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
24 November 2016 26 September 2017 2 November 2017
Please cite this article as: S.M. Dildar Ali, U. Iqbal Bhatti, K. Munawwar, U. Al-Saggaf, S. Mansoor, J. Ali, Gyroscopic Drift Compensation by Using Low Cost Sensors for Improved Attitude Determination, Measurement (2017), doi: https://doi.org/10.1016/j.measurement.2017.11.003
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Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2
1
Gyroscopic Drift Compensation by Using Low Cost Sensors for Improved Attitude Determination S. M. Dildar Ali, U. Iqbal Bhatti, K. Munawwar, U. Al-Saggaf, Shoaib Mansoor and Jamshaid Ali
Transformation matrix from e to n
Abstract— Desire of inexpensive Electro-Optic and MicroElectro-Mechanical System (MEMS) inertial sensors has drastically been increased in the recent times for both military and commercial applications. Beside the traditional applications, reduced cost of such sensors has opened new domains in personal navigation. This paper provides a framework for attitude estimation using miniaturized and cost-effective Inertial Measurement Units (IMU). Sensor fusion of gyroscope, accelerometer and True Air Speed (TAS) sensor helps in minimizing the characteristic time growing error present in gyroscopic integrated data. A novel approach of TAS model development is implemented to generate true air speed data in the absence of TAS sensor. The presence of linear acceleration is estimated and eliminated by means of gyroscope and TAS model. Due to the slight difference in two direction vectors, an error function is estimated and constantly compensated by using a Proportional-Integral (PI) block. Coarse tuning of PI gains is applied and simulated results are produced to assess the filter performance by using real vehicle data. It has been presented that the proposed filter can be used to compute a reasonably accurate attitude solution by using low cost inertial sensors when external aiding is unavailable or not useful. Keywords — Attitude, Complementary Filter, True Air Speed Model, Air Turbulence Model, Inertial Sensors, Body Frame.
NOMENCLATURE b frame e frame i frame n frame
Body frame Earth frame Inertial frame Navigation frame (NED frame) Transformation matrix from b to e
frame Transformation matrix from b to i frame Transformation matrix from b to n frame Transformation matrix from e to i frame Syed M. Dildar Ali ([email protected]) and Dr. Umar Iqbal Bhatti ([email protected]) are working with the Aeronautics and Astronautics Engineering Department at Institute of Space Technology (IST), Islamabad 48000, Pakistan. Prof. Dr. Khalid Munawar and Prof. Dr. Ubaid Al-Saggaf are working at Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdul Aziz University, Jeddah, KSA. Shoaib Mansoor is a PhD scholar at CUST Islamabad, Pakistan and is a member of the research team. Dr. Jamshaid Ali is working as a consultant in an Aerospace Industry at Islamabad, Pakistan.
frame
L l
Specific force in b frame Local gravity vector Angular rate of the Earth Gyroscope provided angular rates Angular rate of body w.r.t to e frame expressed in b frame Angular rate of Earth w.r.t to i frame expressed in e frame Roll angle (p) Pitch angle (q) Yaw angle (r) Latitude Longitude I. INTRODUCTION
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recise attitude measurement remains the foremost requirement in all aerial applications. It plays a vital role in maneuvering of an aircraft/Unmanned Aerial Vehicle (UAV) or a guided missile to follow a trajectory to achieve the set targets. Moreover, with the rapid growth of industrial automation, attitude determination has become very critical in the field of robotics and humanoids [1]. Due to recent increased demand of cheaper and reliable systems, it has become the prime concern to utilize cheaper sensors i.e., low cost IFOG (Interferometic Fiber Optic Gyroscope) and MEMS (Micro Electro-Mechanical System) based inertial sensors, with more sophisticated state estimation algorithms [2]. However, the complexity of such algorithm increases as the quality of the measuring sensor decreases. Without relying on any external reference, data fusion of available sensors can minimize the attitude errors with an intelligent and robust attitude algorithm. Over the past decades, major research has been carried out to minimize the time growing attitude error with minimum computational requirements. Earlier, Shuster and Oh [3] have used TRIAD algorithm for the deterministic attitude while the QUEST (Quaternion-Estimator) algorithm has been implemented to measure the optimal attitude of spacecraft in weighted configuration [4]. Markely [5] has also presented a comparison of different attitude representations using two vector measurement approaches. A similar work has also been
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 presented by Shuster [6], which provides different relationships between twelve attitude representations. The Kalman Filter (KF) and its variants such as the Extended Kalman Filter (EKF) are well studied techniques for error estimation and gyroscope bias elimination [7]. However, such filters cannot always guarantee the stability and optimality of the solution in case of nonlinearity or drastic change in noise model [8]. Likewise, EKF can also fail, when vehicle dynamics is highly nonlinear or having poor a-priori state estimates [7]. Furthermore, the KF and EKF are difficult to apply robustly where computational resources are limited making them unsuitable for limited scale applications. [23, 9, 4]. Other than KF variants, the most commonly class of filtration technique is the complimentary filters. Such filters have good frequency filtering properties for linear systems [24, 10]. Complementary filter combines measurements from two or more available sources with different characteristics and having minimal distortion in the signal output [11]. Recently, a lot of development has taken place on nonlinear analogous of Single Input Single Output (SISO) filters for attitude determination [12, 13]. Implementation of such schemes on UAVs uses the accelerometer provided information to estimate the gravitational direction. Such filters, however, fail when the vehicle dynamics are large and accelerometer measurements can no longer provide suitable estimates of gravitational direction [14]. Rehbindera and Hub have also proposed a solution by fusing IMU based sensor information to offer stable attitude estimates for robotic applications [15]. Euston et al. [14] work allows full estimation of vehicle attitude as well as gyroscope biases based just on the accelerometer and gyroscope outputs. The non-linear complementary filter provides an excellent attitude estimate suitable for small UAVs [25]. Furthermore, a more recent work on sensor fusion by Yamato et al. [16] provides a new framework for robotic applications, by using angular velocity vectors and direction vectors. The stated approach contains two-step measurement update methodology, namely, attitude estimation by simple rate integration and attitude determination via vector matching by using Wahba problem technique [12]. The first function is for regulation of error accumulation while the second functional block estimates the reference attitude from the bias removing feedback. Such filtering techniques are used for attitude estimation for swarm UAVs [26] and aerial robots [27]. A MARG (Magnetic, Angular Rate and Gravity) sensor is employed with a
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complementary filter to integrate with an IMU to provide the attitude solution for MAV (Micro-Air vehicles) [28]. Similar configuration has been tested with a sensor array for improved attitude estimation [29]. The current study proposes a continuation of above work by developing an attitude estimation algorithm for miniaturized and cost-effective systems. The gyroscope provided attitude solution typically suffers from a time growing error due to the presence of certain bias levels in gyroscope signal. The current sensor fusion configuration scheme utilizes a triad of low cost inertial sensor (MEMS/IFOG) along with a True Air Speed (TAS) sensor model to independently aid the attitude solution. A similar kind of work has already been proposed by Euston et al. but the scheme fails to address a problem when TAS sensor data is not available. The situation may also arise in the applications where TAS sensor has no use and cannot provide sufficient results e.g. robots, land rovers and platform stabilizers. A complete mathematical description for the TAS model along with certain air disturbance, is devised. For ease and available data set limitations, complete algorithm has been applied for body frame attitude computation without any known reference. The obtained results clearly exhibit a potential of the proposed scheme to estimate a more accurate attitude solution. II. PROPOSED ATTITUDE DETERMINATION TECHNIQUE The problem stated above, can be addressed either by providing an external aiding or implementing a filtration scheme to compensate and minimize growing drifts in gyroscopes. The presented work demonstrates a methodology of estimating an improved attitude solution as compared to conventional schemes through a nonlinear complementary filter, also augmented by vehicle dynamics. The attitude determination technique consists of two computational blocks. The first block computes and simulates the true air speed sensor data which later combines with the gyroscopic output to provide the estimates of direction vector. The gyroscope and TAS model fusion provide the best estimates of gravitational or direction vector. A non-linear and non-inertial acceleration model is used to compensate the accelerometer output to obtain a zero bias estimate of the gravitational direction. The model is based on a simple centripetal force model derived from the airspeed and the rate of turn of the vehicle [14].Abbreviations and Acronyms
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2
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Complementary filter performs a low pass filtering on the low frequency attitude which is obtained from accelerometers and operates as a high pass filter on biased high frequency attitude estimates, obtained from the direct integration of gyroscopes. Both the estimates are then fused together to achieve all-pass estimate of the attitude. The schematic of the proposed filter is shown in Figure1. The performance of the algorithm is examined using the body frame attitude solution in order to maintain simplicity for technique validation. Its implementation in NED frame can also be applied, however, that analysis is unnecessary. The combined system is relatively simple to implement and achieves better performance as compared to an unaided system or loosely coupled system. The algorithm is implemented and tested on a data set of a ground vehicle [17] and favorable results are obtained.
in the body frame and the answer is also expressed in the body frame. (4)
III. TRUE AIR SPEED (TAS) MODEL
The total angular rate of the body with respect to inertial frame can be expressed as the sum of two rates such that:
True Air speed sensor provides the speed of the aircraft relative to air mass in which it is flying [18]. In the absence of TAS data, the desired data set is generated using the equipped INS through a TAS model. The proposed TAS model under this scheme has brought a novelty in previous cited work in the introduction section. Although body frame mechanization is a well-known formulation for example described in Titterton and Weston [19]. However, it is typically used for control purposes and not for propagation of navigation variables. In this way it has become possible to use a conventional model for air disturbance named as Dryden Model to be incorporated in the TAS model, for example as in Maaz et al. [20]. The output of a conventional Dryden Model are typically provided in the body frame and hence are directly utilized for providing better imitation of original TAS. The TAS model requires two additional computations i.e. velocity computation in the body frame and an air turbulence model.
where;
and
The update equation, from e-frame to b-frame is (5)
(6) At time t = 0;
and
A. Body Frame Mechanization Body frame velocities are computed by using the INS data. The following mathematical derivation provides a clear understanding of body frame mechanization along with necessary initial conditions. The Coriolis law for the body frame mechanization (subscript and superscript b) can be expressed as (1) Substituting the
from inertial frame mechanization as
formulated in appendix Eq. (A.9). (2) (3) In body frame,
Dot representation is used when the derivative is calculated
Fig. 1. Basic Schematic of Proposed Algorithm with Complementary Filter (BF denotes Body Frame) Fig. 2. RSS Comparison of Velocities in Body Frame and NED Frame
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2
(8)
The above derivation provides encouraging results for body frame velocity computation. The root sum square (RSS) of the velocities in the body frame is identical to NED frame velocities. Fig. 2 demonstrates the validation of body frame velocity computation by comparing RSS of velocities in different frames for a ground vehicle. This figure is of tutorial value and acts as a cross checking measure for validity of the body frame based TAS model. Furthermore this can be used as a benchmark for the case when disturbances are added for simulation of a more realistic TAS model. It is acknowledged, that the work done for body frame mechanization is not completely novel however its use in the particular context of navigation and for ease of disturbance model shows its worthiness. B. Effect of Air Turbulence Model The velocities computed through INS data in the body frame are the ideal velocities. However, in actual, velocities may differ due to the presence of certain disturbances in environment i.e., wind turbulences (side slip, wind gusts etc.). To further enhance the validity of the INS computed velocities, velocities are passed from another model i.e. Simulink programmed Dryden Wind Turbulence Model to measure and introduce turbulence intensities as a function of altitude [21]. The Dryden model adds turbulence to the aerospace model by passing band-limited white noise through appropriate forming filters and provides output as stochastic processes with the Dryden gusts power spectral densities. Wind turbulences varies with altitude and vehicle speed. As the altitude increases, the magnitude of air gusts decreases and
vehicle experience much less disturbances. The example is of aircraft cruise flight, which typically experiences much smaller air turbulences in a level flight while it may suffers from heavy turbulence during landing and takeoff. The model can be validated by varying the input parameters and analyzing the resultant turbulences in each axis. The vehicle in this case, is travelling with 40 to 50 km/hr at an altitude of 35 meters, the turbulence disturbance comparison is shown in two parts of Figure 3. The results clearly shows that
Fig. 4. Effects of Turbulence Velocities on INS Computed XY Velocities
difference in turbulence velocity is quite small for these two speeds. Hence for the complete simulation work, the speed of the vehicle has been finalized as 50 km/hr. C. TAS Simulation Results As described earlier, due to the absence of air speed sensor data, TAS model is used to provide input to the proposed filter. Once the disturbances are known, they can be added in the computed velocities to get the final velocities in the body frame. TAS model results are presented in Figure 4 which provides the simulated true air speed of the vehicle in body frame. IV.
Fig. 3. Turbulence Velocity as generated by the used model at an input speed of 40 km/hr (upper figure) and 50 km/hr (lower figure) respectively
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SIMULATION RESULTS
The IMU used for the implementation of proposed attitude estimation algorithm is the Honeywell Commercial Inertial Measurement Unit (CIMU), which contains optical gyroscopes and quartz based accelerometers. CIMU is a navigation grade IMU with gyros of accuracy range of 0.01 degree per hour. By implementing the proposed TAS model, air speed data limitation has been resolved and the availability of required input parameters for the proposed scheme is ensured. The gravitational direction vectors can now be computed from the system and quaternion gravitational estimates. The error, hence arisen due to the difference between two gravitational vectors, is compensated by using PI filter. The PI filter is conventional Proportional and Integration filter traditionally used in the realm of feedback control systems [22]. In this type of filtering algorithm, the error signal is fed to the filter and the whole loop is operated
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 in a feedback configuration. By tuning the gains of the filter which are known as Proportional and Integration gains, the closed loop operate to minimize the error. The proportional gain is simply a multiplying factor and the Integration gain is a discrete (or continuous) integration routine operated upon the input error. This small routine involves current error value and error history. The output of both of these operations are added together. It has been prove in control systems theory that this sort of filter is very successful in minimizing the input error even if it contains a bias [22]. The tuning of PI block is critical and needs to be addressed during the calibration process of IMU/INS. The gains of each block affect the final attitude solution and the effect of each block is discussed separately.
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B. Effect of Integral Term (KI) Upon the optimization of KP gain, the effect of integral term on filter performance needs to be investigated further. The integral gain (KI) is applied, once the solution is in an acceptable range after KP tuning i.e., KP = 0.001. KI is time
A. Effect of Proportional Term (KP) Initially, the performance of the filter is restricted only to the proportional term (KP), by nulling the integral gain i.e., KI=0. This helps in understanding the direct effect of KP on attitude solution. KP is kept small primarily, which may later be tuned with appropriate gain value for desirable performance. The Fig. 6. Roll Angle at Final PI Tuning
Fig. 5. Attitude Solution with Only Proportional Block (KP= 0.001)
results are compared with the unaided attitude solution i.e., attitude solution from raw gyroscope rates. Starting with a unity magnitude for proportional gain (KP=1) and further decreasing the KP gain up to 0.001, a reasonable roll, pitch and yaw angles are achieved, as shown in Figure 5.The attitude computed directly from the raw gyroscope rates is termed as ‘Raw Gyro Rate’ with red legend and will be treated as the reference for current study. Whereas, the proposed filter provided attitude solution after the error/bias drift compensation, has been termed as ‘Drift Compensated Rate’, with a green legend. From the series of simulations, it is observed that filter solution becomes divergent at higher values of KP and hence the value for KP should be small. The filter provided attitude solution follows a similar trend of rotations as experienced by the body mounted IMU. The obtained result are much similar to that of the raw gyro solution but maintain a slight difference as we expect to observe since the raw gyro attitude solution is prone to suffer from time integrated bias error.
Fig. 7. Pitch Angle at Final PI Tuning
Fig. 8. Yaw Angle at Final PI Tuning
integrated gain, therefore, magnitude of integral gain should be much smaller than KP (about 10 to 100 times). Initially, with only the proportional gain, solution followed a similar trend to raw gyro attitude solution. But with the
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 addition of larger integral gain the solution deviates with time. Yaw angle is also disturbed at higher value of KI. Keeping the KP gain constant and lowering the integral gain, the resultant solution becomes less divergent. The results obtained with the addition of integral term in previous solution are shown in Figures 6, 7 and 8. These results provide a clear picture for gain range of the integral term. KI is much smaller than KP (KI=0.00001, 100 times smaller in this case). The obtained results are better as compared to previously simulated results. The estimated roll, pitch and yaw angles are largely improved and have become similar to raw gyro rate solution. A slight difference in roll angle may further be minimized with fine tuning. However, no significant improvement in yaw angle is observed. This also validates the proposed attitude estimation algorithm as gravitational direction vectors cannot compensate the azimuth error. A further tuning of the filter is necessary to fully optimize the attitude solution which may be achieved by change in PI gains. Furthermore, the integral term in proposed mechanism allow to eliminate the gyro bias and helps in providing a more precise solution.
Fig. 9. Drift Compensation in Roll, Pitch and Yaw Channels for KP=0.001 and KI=0.00001
V.
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SOLUTION COMPARISON
The technique is also applied to data set obtained from commercially available system known as CMIGITS. This system is developed by Systron Donner and lies in the
Fig. 11. Attitude Difference between Compensated and Raw Output
medium accuracy range of 3.0 deg/hr [30]. It must also be mentioned here that the technique is also tested for very lower grade MEMS based systems without success due to relatively large amount of noise. The systems tried are developed by VectorNav (VN-200) and Shimmer (Shimmer 3) [31, 32]. The results for drift compensation obtained are presented as below in Figure 9 and Figure 10. It can be seen that compensation has indeed been achieved in the roll and pitch channels as expected. Based on this compensation, it is further studied how much is the effect of these corrections on the performance of the system. The attitude difference between the corrected and the raw system outputs is plotted in Figure 11. It is claimed that this difference in accuracy is achieved by use of the proposed technique. Since the data is limited to 200 seconds, typical performance indicator of deg per hour cannot be compensated. However, by an approximate extrapolation of results reveal that the technique is effective in curtailing the error growth around one deg per hour in a medium accuracy system. It is pertinent to mention here that further verification of the algorithm requires use of much higher quality system to act as reference purpose which is quite expensive and not accessible. VI. CONCLUSION
Fig. 10. Drift Compensation in Roll, Pitch and Yaw Channels for KP=0.0001 and KI=0.000001
The main purpose of this study is to develop a better attitude solution algorithm for UAVs, robots and other aerial applications by utilizing cheaper and low grade sensors. A filtering technique is implemented via data fusion of inertial sensors and true air speed model. TAS model encompasses air turbulence effects which adds effectiveness in proposed work for real world applications. The proposed algorithm is implemented in MATLAB and results are produced using ground vehicle data. Simulation results clearly illustrates that the proposed technique solution is adaptable and flexible which can be tuned with an available
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 reference attitude profile. The resultant solution exhibits relatively small effect of gyroscopic drift i.e., the drift in gyroscope integrated data is minimized. The error growth is continuously monitored and measured by taking a feedback input from gravitational direction vectors. Further, the instantaneous error is compensated by adjusting appropriate gains of the PI block. Coarse tuning PI gains may bring solution in an acceptable range and consequently, fine tuning may further optimize the final solution. The presented filtering methodology is simple and easy to implement. Furthermore, it requires lesser computational power as compared to other filters (particle filter, Kalman filter etc.) making it more suitable for medium range applicants.
ACKNOWLEDGMENT The authors duly acknowledge the technical and research support of Aeronautics and Astronautics Department, Institute of Space Technology (IST), Islamabad. Authors would also like to thank Dr. M. Aleem Mirza for his consistent encouragement. The authors also acknowledge the technical support of Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdul Aziz University, Jeddah, KSA. REFERENCES [1]
[2]
APPENDIX In order to obtain a relation for velocity computation in inertial frame, we start from the basic definitions of acceleration ‘a’ and accelerometer output ‘f’ i.e. the specific force: (A.1)
[3]
[4]
[5]
(A.2) [6]
(A.3)
[7]
By Coriolis Law [6] (A.4)
[8]
where [9]
Differentiating Eq. (A.4) [10]
(A.5)
[11]
Since [12]
and [13]
Therefore Eq. (A.5) becomes [14]
(A.6) [15]
Substituting Eq. (A.6) in Eq. (A.3) and rearranging
[16]
(A.7) [17]
Also, (A.8)
[18]
Comparing Eq. (A.7) and Eq. (A.8) results as:
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(A.9) [19]
E. North, J. Georgy, M. Tarbouchi, U. Iqbal, and A. Noureldin, "Enhanced Mobile Robot Outdoor Localization using INS/GPS Integration", in The IEEE International Conference on Computer Engineering and Systems, Cairo, 2009, pp. 127-132. A. J. Grunwald, E. Wahnon, and S. J. Merhav, “Estimation of Vehicle Attitude and Flight-path Angles from Low Cost Rate Gyro Measurements”, in The 25th Israel Annual Conference on Aviation Astronautics, Israel, 1983, pp. 121–127. M. D. Shuster and S. D. Oh, "Three-axis Attitude Determination from Vector Observations," AIAA Journal of Guidance, Control, and Dynamics, vol. 4, no. 1, 1981, pp. 70-77. J. M. Roberts, P. I. Corke, and G. Buskey, “Low-cost Flight Control System for Small Autonomous Helicopter”, in The Australian Conference on Robotics and Automation, Auckland, 2002, pp. 71–76. F. Markely, “Attitude Determination Using Two Vector Measurements”. Internet: ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19990052720.pdf, NASA's Goddard Space Flight Center, Greenbelt, 1998. M. D. Shuster, “A Survey of Attitude Representations,” Journal of Astronautical Sciences, vol. 41, no. 4, 1993, pp 439-517. J. Crassidis, F. Markley, and Y. Cheng, "Survey of Nonlinear Attitude Estimation Methods," Journal of Guidance, Control, and Dynamics, vol. 30, no. 1, 2007, pp. 12-28. N. K. Filyashkin and V. S. Yatskivsky, “Prediction of Inertial Navigation System Error Dynamics in INS/GPS System", in The IEEE 2nd International Conference on Actual Problems of Unmanned Air Vehicles Developments, 2013, pp.206-209. D. Choukroun, I.Y. Bar-Itzhack, and Y. Oshman, “Novel Quaternion Kalman Filter,” IEEE Transactions on Aerospace and Electronic Systems, 2006, pp. 174-190. R. Mahony, T. Hamel, and J. Pflimlin, “Non-linear Complementary Filters on the Special Orthogonal Group,” IEEE Transactions on Automatic Control, 2007, pp. 1477-1484. K. Narayan, "Performance Analysis of Attitude Determination Algorithms for Low Cost Attitude Heading Reference Systems." PhD Thesis, Auburn University, USA, 2010. J. Thienel and R. M. Sanner, “A Coupled Nonlinear Spacecraft Attitude Controller and Observer with an Unknown Constant Gyro Bias and Gyro Noise,” IEEE Transactions on Automatic Control, 2003, pp. 20112015. S. Bonnabel, P. Martin, and P. Rouchon, “A Non-Linear Symmetry Preserving Observer for Velocity-Aided Inertial Navigation”, in The American Control Conference, 2006, pp. 2910–2914. M. Euston, P. Coote, R. Mahony, J. Kim, and T. Hamel, “A Complementary Filter for Attitude Estimation of a Fixed-Wing UAV”, in The IEEE/RSJ International Conference on Intelligent Robots and Systems, Nice, France, 2008, pp. 340-345. H. Rehbindera and X. Hub, “Drift-free Attitude Estimation for Accelerated Rigid Bodies,” Automatica, vol. 40, 2004, pp. 653–659. H. Yamato, T. Furuta, and K. Tomiyama, "Attitude Determination by Globally and Asymptotically Stable Estimation of Gyroscope Bias Error with Disturbance Attenuation and Rejection," Journal of Robotics and Mechatronics, vol. 24, no. 2, 2012, pp. 389-398. U. I. Bhatti, "Improved Integrity Algorithms for Integrated GPS/INS Systems in the Presence of Slowly Growing Errors." PhD Thesis, Imperial College London, UK, 2007. IVAO HQ. Training Department, "Air Speed Definitions." Internet: https://www.ivao.aero/training/documentation/books/PP_ADC_airspeed. pdf, Sept. 29, 2016. D. H. Titerton and J. L. Weston, “Strapdown Inertial Navigation Technology.” AIAA (2nd Edition), 2004.
Elsevier Editorial System(tm) for Measurement (Manuscript Draft): Manuscript Number: MEAS-D-16-01885R2 [20] M. Butt, K. Munawar, U. Bhatti, S. Iqbal, U. Al-Saggaf, and W. Ochieng, “4D Trajectory Generation for Guidance Module of a UAV for a Gate to Gate Flight in Presence of Turbulence,” International Journal of Advanced Robotics Systems, 2016, pp. 1-14. [21] F. Hoblit, “Gust Loads on Aircraft: Concepts and Applications.” AIAA Education Series, 1988. [22] R. C. Dorf and R. H. Bishop, “Modern Control Systems.” Prentice Hall (12th Edition), 2010. [23] Kumar, N. Ravi, and E. Nagabhooshanam, "An Extended Kalman Filter for Low-Cost Positioning System in Agricultural Vehicles", in The IEEE International Conference on Wireless Communications, Signal Processing and Networking (WiSPNET), 2016. [24] Lee, J. Keun, "A Two-step Kalman/Complementary Filter for Estimation of Vertical Position using an IMU-Barometer System," Journal of Sensor Science and Technology, 2016, pp. 202-207. [25] S. Leutenegger, A. Melzer, K. Alexis, and R. Siegwart, "Robust State Estimation for Small Unmanned Airplanes", in The IEEE Conference on Control Applications (CCA), 2014, pp. 1085-1992. [26] X. Dong, Y. Zhou, Z. Ren, and Y. Zhong, "Time-varying Formation Control for Unmanned Aerial Vehicles with Switching Interaction Topologies," Control Engineering Practice, 2016, pp. 26-36. [27] Mahony, Robert, R. Beard, and V. Kumar, "Modeling and Control of Aerial Robots." Springer International Publishing, 2016, pp.1307-1334.
[28] Valenti, G. Roberto, I. Dryanovski, and J. Xiao, "Keeping a Good Attitude: A Quaternion-based Orientation Filter for IMUs and MARGs," Sensors vol. 15, no. 8, 2015, pp. 19302-19330. [29] S. Madgwick, A. Harrison, P. Sharkey, R. Vaidyanathan, and W. Harwin, "Measuring Motion with Kinematically Redundant Accelerometer Arrays: Theory, Simulation and Implementation," Elsevier-Mechatronics, vol. 23, issue, 5, 2013, pp. 518-529. [30] CMIGITS III User Manual, Systron Donner, California, USA. [31] VectorNav VN-200 User Manual, VectorNav Technologies, Texas, USA, 2008. [32] Shimmer IMU User Guide Rev. 1.4, Shimmer, Dublin, Ireland, 2006.
Highlights
Quaternion based attitude estimation by using low cost inertial sensors
True Air Speed (TAS) model development via body frame mechanization
Complementary filter for Gyroscope time growing error compensation using accelerometer and TAS measurements
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