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Attitude Estimation for Quadrotor UAV by Unscented Kalman Filter
Abstract
The paper deals with the attitude estimation of a quadrotor UAV. The attitude is represented using quaternions and estimated by the unscented Kalman filter (UKF). Two UKF implementations respecting the quaternion normalization are compared. The first one explicitly normalizes the quaternion estimate while the second one uses an error vector as the model state. Comparison of both implementations is based on the flight experiment data, where the ground truth is provided by the motion capture system VICON. Index Terms-inertial measurement unit, quaternion, Un-manned Aerial Vehicle, unscented Kalman filter
This work deals with the trajectory planning problem for the Unmanned Aerial Vehicle (UAV). It is critical for every application that employs the fully autonomous UAV because the quality of trajectory can significantly affect the UAV mission. The work contains a description of UAV behavior and additional aerodynamic effects using mathematical modeling and presentation of algorithms for the state estimation. Further, two classes of methods employed in trajectory planning are described. The first class is composed of path planning methods, which contain graph-based methods such as A*, Theta*, and Rapidly-exploring Random Tree* (RRT*) and the Artificial Potential Field Method. The second class is based on the description of a problem in a manner of Optimal Control Problem (OCP). Due to the difficulty in finding the analytic solution, the numerical methods are presented. Thanks to its good convergence and precise solution, the main interest is given to Pseudospectral (PSM). Subsequently, the trajectory tracking problem is discussed, the state-of-the-art tracking method Model Predictive Control (MPC) is presented. The examples of MPC tracking application are presented and the novel alternative method called Interpolating Control (IC) is described and compared to MPC. At the end of the work, the goals of the dissertation thesis and the direction of future work are stated.
This paper presents three iterative methods for orientation estimation. The first two are based on iterated Extended Kalman filter (IEKF) formulations with different state representations. The first is using the well-known unit quaternion as state (q-IEKF) while the other is using orientation deviation which we call IMEKF. The third method is based on nonlinear least squares (NLS) estimation of the angular velocity which is used to parametrise the orientation. The results are obtained using Monte Carlo simulations and the comparison is done with the non-iterative EKF and multiplicative EKF (MEKF) as baseline. The result clearly shows that the IMEKF and the NLS-based method are superior to q-IEKF and all three outperform the non-iterative methods.
The Kalman filter is considered as an optimal filter with the hypothesis of gaussian noise and linear model. For nonlinear model several approaches have been proposed and Unscented Kalman Filter (UKF) seems to be one of the most accurate. In this study, we wonder if an appropriate constraint can enhance the efficiency of UKF. We propose a new algorithm called Constrained Sigma Points (CSP) that constrained the sigma points with a nonlinear observer constraint. Here, our research is based on attitude estimation and the constraint is related to attitude. We evaluate its performance compared to the state of the art of non-linear fusion filters, i.e. Multiplicative Extended Kalman Filter (MEKF), UnScented QUaternion Esti-mator, Quaternion estimate (QUEST) and a nonlinear observer (CGO). Our results show that our algorithm leads to better results in term of accuracy with an effective duration of computation. In future works, we will determine how this new constraint can be generalised to different kind of nonlinear models.
Measurements from magnetometers and inertial sensors (accelerometers and gyroscopes) can be combined to give 3D orientation estimates. In order to obtain accurate orientation estimates it is imperative that the magnetometer and inertial sensor axes are aligned and that the magnetometer is properly calibrated for both sensor errors as well as presence of magnetic distortions. In this work we derive an easy-to-use calibration algorithm that can be used to calibrate a combination of a magnetometer and inertial sensors. The algorithm compensates for any static magnetic distortions created by the sensor platform, magnetometer sensor errors and determines the alignment between the magnetometer and the inertial sensor axes. The resulting calibration procedure does not require any additional hardware. We make use of probabilistic models and obtain the calibration algorithm as the solution to a maximum likelihood problem. The efficacy of the proposed algorithm is illustrated using experimental data collected from a sensor unit placed in a magnetically disturbed environment onboard a jet aircraft.
A new spacecraft attitude estimation approach based on the Unscented Filter is derived. For nonlin-ear systems the Unscented Filter uses a carefully se-lected set of sample points to more accurately map the probability distribution than the linearization of the standard Extended Kalman Filter, leading to faster convergence from inaccurate initial conditions in at-titude estimation problems. The filter formulation is based on standard attitude-vector measurements us-ing a gyro-based model for attitude propagation. The global attitude parameterization is given by a quater-nion, while a generalized three-dimensional attitude representation is used to define the local attitude error. A multiplicative quaternion-error approach is derived from the local attitude error, which guarantees that quaternion normalization is maintained in the filter. Simulation results indicate that the Unscented Filter is more robust than the Extended Kalman Filter under realistic initial attitude-error conditions.
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This paper considers the problem of obtaining high quality attitude extraction and gyros bias estimation from typical low cost intertial measurement units for applications in control of unmanned aerial vehiccles. Two different non-linear complementary filters are proposed: Direct complementary filter and Passive non-linear complementary filter. Both filters evolve explicity on the special orthogonal group SO(3) and can be expressed in quaternion form for easy implementation. An extension to the passive ocmplementary filter is proposed to provide adaptive gyro bias estimation.
This paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units. The outputs of such systems are characterized by high noise levels and time varying additive biases. We formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group SO (3) driven by reconstructed attitude and angular velocity measurements. Lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error. The approach taken leads to an observer that we term the direct complementary filter. By exploiting the geometry of the special orthogonal group a related observer, termed the passive complementary filter, is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs. Both the direct and passive filters can be extended to estimate gyro bias online. The passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude. This leads to an observer on SO(3), termed the explicit complementary filter, that requires only accelerometer and gyro outputs; is suitable for implementation on embedded hardware; and provides good attitude estimates as well as estimating the gyro biases online. The performance of the observers are demonstrated with a set of experiments performed on a robotic test-bed and a radio controlled unmanned aerial vehicle.
Filtering and smoothing methods are used to produce an accurate estimate of the state of a time-varying system based on multiple observational inputs (data). Interest in these methods has exploded in recent years, with numerous applications emerging in fields such as navigation, aerospace engineering, telecommunications and medicine. This compact, informal introduction for graduate students and advanced undergraduates presents the current state-of-the-art filtering and smoothing methods in a unified Bayesian framework. Readers learn what non-linear Kalman filters and particle filters are, how they are related, and their relative advantages and disadvantages. They also discover how state-of-the-art Bayesian parameter estimation methods can be combined with state-of-the-art filtering and smoothing algorithms. The book's practical and algorithmic approach assumes only modest mathematical prerequisites. Examples include Matlab computations, and the numerous end-of-chapter exercises include computational assignments. Matlab code is available for download at www.cambridge.org/sarkka, promoting hands-on work with the methods.
This article provides a tutorial introduction to modeling, estimation, and control for multirotor aerial vehicles that includes the common four-rotor or quadrotor case.
This paper proposes a calibration procedure in order to minimize the process time and cost. It relies on the suggestion of optimal positions, in which the calibration procedure takes place, and on position number optimization. Furthermore, this paper describes and compares three useful calibration algorithms applicable on triaxial accelerometer to determine its mathematical error model without a need to use an expensive and precise calibration means, which is commonly required. The sensor error model (SEM) of triaxial accelerometer consists of three scale-factor errors, three nonorthogonality angles, and three offsets. For purposes of calibration, two algorithms were tested—the Levenberg–Marquardt and the Thin-Shell algorithm. Both were then related to algorithm based on Matlab fminunc function to analyze their efficiency and results. The proposed calibration procedure and applied algorithms were experimentally verified on accelerometers available on market. We performed various analyses of proposed procedure and proved its capability to estimate the parameters of SEM without a need of precise calibration means, with minimum number of iteration, both saving time, workload, and costs.
We present the three main mathematical constructs used to represent the attitude of a rigid body in three-dimensional space. These are (1) the rotation matrix, (2) a triple of Euler angles, and (3) the unit quaternion. To these we add a fourth, the rotation vector, which has many of the benefits of both Euler angles and quaternions, but neither the singularities of the former, nor the quadratic constraint of the latter. There are several other subsidiary representations, such as Cayley-Klein parameters and the axis-angle representation, whose relations to the three main representations are also described. Our exposition is catered to those who seek a thorough and unified reference on the whole subject; detailed derivations of some results are not presented.
The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.