This document summarizes an attitude determination method for multirotor unmanned aerial vehicles (UAVs) using vector measurements from a downward-facing camera. The method uses an extended Kalman filter with two different attitude representations - the quaternion extended Kalman filter (QEKF) and the multiplicative extended Kalman filter (MEKF). Computational simulations are used to evaluate the proposed multirotor attitude determination scheme using vector measurements from the camera and rate gyro data.
A Study of Non-Gaussian Error Volumes and Nonlinear Uncertainty Propagation f...Justin Spurbeck
The ever-growing resident space object population poses a continual threat in that a hyper velocity impact is likely to be catastrophic to an active satellite. To avoid these scenarios, space operators compute a probability of collision metric for each potential conjunction. Uncertainty trends are studied in the conjunction plane and operational decisions to mitigate any high-risk situations are made based off this information. There are many methods of uncertainty propagation and probability of collision formulations and knowledge of their realism is required to maintain a sustainable space environment. Thus, this research studies the effect of Chan, Alfano, Foster, Gaussian mixture, and Monte Carlo probability of collision calculations and their correlation to uncertainty realism metrics. The linear, unscented transform, entropy-based, and Monte Carlo propagation techniques are utilized alongside the collision calculations and it is shown that there are important correlations any space operator should be aware of to support maintenance of a healthy spacecraft.
This document summarizes a study that used sigmoidal parameterization and Metropolis-Hasting (MH) inversion to estimate seismic velocity models from traveltime data. The key points are:
1) Sigmoidal functions were used to parameterize discontinuous velocity fields, allowing for sharp variations while maintaining continuity.
2) Ray tracing and the MH algorithm were used to invert traveltime data and estimate model parameters.
3) Tests on synthetic models showed the MH method produced higher resolution velocity models that better fit the observed traveltime data, compared to other global optimization methods like very fast simulated annealing.
Three-dimensional structure from motion recovery of a moving object with nois...IJECEIAES
In this paper, a Nonlinear Unknown Input Observer (NLUIO) based approach is proposed for three-dimensional (3-D) structure from motion identification. Unlike the previous studies that require prior knowledge of either the motion parameters or scene geometry, the proposed approach assumes that the object motion is imperfectly known and considered as an unknown input to the perspective dynamical system. The reconstruction of the 3-D structure of the moving objects can be achieved using just twodimensional (2-D) images of a monocular vision system. The proposed scheme is illustrated with a numerical example in the presence of measurement noise for both static and dynamic scenes. Those results are used to clearly demonstrate the advantages of the proposed NLUIO.
2006 Green Incorporating Pulse to Pulse Motion Effects anto Side Looking Arra...Justin Green
This document presents a technique for incorporating pulse-to-pulse motion effects into side-looking array radar data models. The technique accounts for changes in platform attitude between pulses, which affects the geometry between the radar and targets. It uses coordinate transformations to describe how the platform orientation changes with each pulse. This allows calculation of pulse-dependent parameters like spatial frequency and Doppler, extending the existing radar data model to be sensitive to platform motion. The new motion-sensitive space-time snapshot can then be used to characterize how motion impacts the performance of space-time adaptive processing algorithms.
GPS cycle slips detection and repair through various signal combinationsIJMER
Abstract: GPS Cycle slips affect the measured spatial distance between the satellite and the receiver, thus affecting the accuracy of the derived 3D coordinates of any ground station. Therefore, cycle slips must be detected and repaired before performing any data processing. The objectives of this research are to detect the Cycle slips by using various types of GPS signal combinations with graphical and statistical tests techniques, and to repair cycle slips by using average and time difference geometry techniques. Results of detection process show that the graphical detection can be used as a primary detection
technique whereas the statistical approaches of detection are proved to be superior. On the other hand, results of repairing process show that any trial can be used for such process except for the 1st and 2nd time differences averaging all data as they give very low accuracy of the cycle slip fixation.
SLAM of Multi-Robot System Considering Its Network Topologytoukaigi
This document proposes a new solution to the multi-robot simultaneous localization and mapping (SLAM) problem that takes into account the network topology between robots. Previous multi-robot SLAM research has expanded one-robot SLAM algorithms without considering how the relationship between robots changes over time. The proposed approach models the network structure and derives the mathematical formulation for estimating the multi-robot SLAM. It presents motion and observation update equations in an information filter framework that can be implemented in a decentralized way on individual robots. Future work will focus on specific challenges in multi-robot SLAM like map merging.
This document summarizes the concept and uses of response spectra for structural engineers. Response spectra provide a way to quantify the demands of earthquake ground motion on structures of varying natural periods of vibration. They have been incorporated into building codes since the 1950s and help establish seismic design forces. Actual recorded response spectra are jagged, but design response spectra are smoothed curves. Response spectra can be used for rapid evaluation of building inventories, performance-based design, evaluation of seismic vulnerability, and post-earthquake damage estimates. They provide a useful tool for earthquake-resistant design.
A Study of Non-Gaussian Error Volumes and Nonlinear Uncertainty Propagation f...Justin Spurbeck
The ever-growing resident space object population poses a continual threat in that a hyper velocity impact is likely to be catastrophic to an active satellite. To avoid these scenarios, space operators compute a probability of collision metric for each potential conjunction. Uncertainty trends are studied in the conjunction plane and operational decisions to mitigate any high-risk situations are made based off this information. There are many methods of uncertainty propagation and probability of collision formulations and knowledge of their realism is required to maintain a sustainable space environment. Thus, this research studies the effect of Chan, Alfano, Foster, Gaussian mixture, and Monte Carlo probability of collision calculations and their correlation to uncertainty realism metrics. The linear, unscented transform, entropy-based, and Monte Carlo propagation techniques are utilized alongside the collision calculations and it is shown that there are important correlations any space operator should be aware of to support maintenance of a healthy spacecraft.
This document summarizes a study that used sigmoidal parameterization and Metropolis-Hasting (MH) inversion to estimate seismic velocity models from traveltime data. The key points are:
1) Sigmoidal functions were used to parameterize discontinuous velocity fields, allowing for sharp variations while maintaining continuity.
2) Ray tracing and the MH algorithm were used to invert traveltime data and estimate model parameters.
3) Tests on synthetic models showed the MH method produced higher resolution velocity models that better fit the observed traveltime data, compared to other global optimization methods like very fast simulated annealing.
Three-dimensional structure from motion recovery of a moving object with nois...IJECEIAES
In this paper, a Nonlinear Unknown Input Observer (NLUIO) based approach is proposed for three-dimensional (3-D) structure from motion identification. Unlike the previous studies that require prior knowledge of either the motion parameters or scene geometry, the proposed approach assumes that the object motion is imperfectly known and considered as an unknown input to the perspective dynamical system. The reconstruction of the 3-D structure of the moving objects can be achieved using just twodimensional (2-D) images of a monocular vision system. The proposed scheme is illustrated with a numerical example in the presence of measurement noise for both static and dynamic scenes. Those results are used to clearly demonstrate the advantages of the proposed NLUIO.
2006 Green Incorporating Pulse to Pulse Motion Effects anto Side Looking Arra...Justin Green
This document presents a technique for incorporating pulse-to-pulse motion effects into side-looking array radar data models. The technique accounts for changes in platform attitude between pulses, which affects the geometry between the radar and targets. It uses coordinate transformations to describe how the platform orientation changes with each pulse. This allows calculation of pulse-dependent parameters like spatial frequency and Doppler, extending the existing radar data model to be sensitive to platform motion. The new motion-sensitive space-time snapshot can then be used to characterize how motion impacts the performance of space-time adaptive processing algorithms.
GPS cycle slips detection and repair through various signal combinationsIJMER
Abstract: GPS Cycle slips affect the measured spatial distance between the satellite and the receiver, thus affecting the accuracy of the derived 3D coordinates of any ground station. Therefore, cycle slips must be detected and repaired before performing any data processing. The objectives of this research are to detect the Cycle slips by using various types of GPS signal combinations with graphical and statistical tests techniques, and to repair cycle slips by using average and time difference geometry techniques. Results of detection process show that the graphical detection can be used as a primary detection
technique whereas the statistical approaches of detection are proved to be superior. On the other hand, results of repairing process show that any trial can be used for such process except for the 1st and 2nd time differences averaging all data as they give very low accuracy of the cycle slip fixation.
SLAM of Multi-Robot System Considering Its Network Topologytoukaigi
This document proposes a new solution to the multi-robot simultaneous localization and mapping (SLAM) problem that takes into account the network topology between robots. Previous multi-robot SLAM research has expanded one-robot SLAM algorithms without considering how the relationship between robots changes over time. The proposed approach models the network structure and derives the mathematical formulation for estimating the multi-robot SLAM. It presents motion and observation update equations in an information filter framework that can be implemented in a decentralized way on individual robots. Future work will focus on specific challenges in multi-robot SLAM like map merging.
This document summarizes the concept and uses of response spectra for structural engineers. Response spectra provide a way to quantify the demands of earthquake ground motion on structures of varying natural periods of vibration. They have been incorporated into building codes since the 1950s and help establish seismic design forces. Actual recorded response spectra are jagged, but design response spectra are smoothed curves. Response spectra can be used for rapid evaluation of building inventories, performance-based design, evaluation of seismic vulnerability, and post-earthquake damage estimates. They provide a useful tool for earthquake-resistant design.
Action Trajectory Reconstruction for Controlling of Vehicle Using SensorsIOSR Journals
Abstract: Inertial sensors, such as accelerometers and gyro-scopes, are rarely used by themselves to compute
velocity and position as each requires the integration of very noisy data. The variance and bias in the resulting
position and velocity estimates grow un-bounded in time. This paper proposes a solution to provide a de-biased
and de-noised estimation of position and velocity of moving vehicle actions from accelerometer measurements.
The method uses a continuous wavelet transform applied to the measurements recursively to provide reliable
action trajectory reconstruction. The results are presented from experiments performed with a MEMS accelerometer
and gyroscope.
Keywords: Action trajectory, continuous wavelet transform, inertial measurement unit.
Photogrammetry - Space Resection by Collinearity EquationsAhmed Nassar
Space resection is commonly used to determine the exterior orientation parameters (which refers to position and orientation related to an exterior coordinate system) associated with one or more photos based on measurements of ground control points (GCPs). space resection is a nonlinear problem, existing methods involve linearization of the collinearity condition and the use of an iterative process to determine the final solution using the least-squares method. The process also requires initial approximate values of the unknown parameters, some of which must be estimated by another least-squares solution.
Gaussian Orbital Determination of 1943 AnterosMatthew Li
Paper detailing the theory, methods, calculations, and results regarding the investigation of the orbit of asteroid 1943 Anteros through approximately six weeks of celestial observation and data collection.
This document describes a method for correcting smear in images captured by frame transfer CCD cameras when there are significant changes in illumination between frames. Existing smear correction algorithms assume constant illumination, which is not always valid. The proposed method models smear using a matrix equation that accounts for variable illumination levels transitioning between frames. It was developed for fast polarimetric imaging but could benefit other applications involving highly variable scenes synchronized with detector readout.
Verification of gravimetric geoidal models by a combination of gps and orthom...LENIN Quintero
This document summarizes a thesis that investigated verifying gravimetric geoid models using a combination of GPS heights and orthometric heights. It discusses height systems, the Canadian vertical datum, determination of the Canadian geoid using Stokes's formula and global geopotential models, biases and errors in GPS height differences, analysis of sample data, and analysis of field data collected in the Northwest Territories, Ontario, and Manitoba. The analyses aimed to detect any systematic dependencies between misclosures in the geoid-GPS/leveling comparison.
Mapping spiral structure on the far side of the Milky WaySérgio Sacani
Little is known about the portion of the Milky Way lying beyond the Galactic center at distances
of more than 9 kiloparsec from the Sun. These regions are opaque at optical wavelengths
because of absorption by interstellar dust, and distances are very large and hard to measure.
We report a direct trigonometric parallax distance of 20:4þ2:8
2:2 kiloparsec obtained with the Very
Long Baseline Array to a water maser source in a region of active star formation. These
measurements allow us to shed light on Galactic spiral structure by locating the ScutumCentaurus
spiral arm as it passes through the far side of the Milky Way and to validate a
kinematic method for determining distances in this region on the basis of transverse motions.
COMPLEMENTARY VISION BASED DATA FUSION FOR ROBUST POSITIONING AND DIRECTED FL...ijaia
The present paper describes an improved 4 DOF (x/y/z/yaw) vision based positioning solution for fully 6
DOF autonomous UAVs, optimised in terms of computation and development costs as well as robustness
and performance. The positioning system combines Fourier transform-based image registration (Fourier
Tracking) and differential optical flow computation to overcome the drawbacks of a single approach. The
first method is capable of recognizing movement in four degree of freedom under variable lighting conditions, but suffers from low sample rate and high computational costs. Differential optical flow computation, on the other hand, enables a very high sample rate to gain control robustness. This method, however, is limited to translational movement only and performs poor in bad lighting conditions. A reliable positioning system for autonomous flights with free heading is obtained by fusing both techniques. Although the vision system can measure the variable altitude during flight, infrared and ultrasonic sensors are used for robustness. This work is part of the AQopterI8 project, which aims to develop an autonomous
flying quadrocopter for indoor application and makes autonomous directed flight possible.
Precise Attitude Determination Using a Hexagonal GPS PlatformCSCJournals
In this paper, a method of precise attitude determination using GPS is proposed. We use a hexagonal antenna platform of 1 m diameter (called the wheel) and post-processing algorithms to calculate attitude, where we focus on yaw to prove the concept. The first part of the algorithm determines an initial absolute position using single point positioning. The second part involves double differencing (DD) the carrier phase measurements for the received GPS signals to determine relative positioning of the antennas on the wheel. The third part consists of Direct Computation Method (DCM) or Implicit Least Squares (ILS) algorithms which, given sufficiently accurate knowledge of the fixed body frame coordinates of the wheel, takes in relative positions of all the receivers and produces the attitude. Field testing results presented in this paper will show that an accuracy of 0.05 degrees in yaw can be achieved. The results will be compared with a theoretical error, which is shown by Monte Carlo simulation to be < 0.001 degrees. The improvement to the current state-of-the-art is that current methods require either very large baselines of several meters to achieve such accuracy or provide errors in yaw that are orders of magnitude greater.
Collinearity Equations
Kinds of product that can be derived by the collinearity equation
- Space Resection By Collinearity
- Space Intersection By Collinearity
- Interior Orientation
- Relative Orientation
- Absolute Orientation
- Self-Calibration
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document provides an overview of geostatistics and variogram analysis. It discusses how the variogram describes the spatial correlation of a phenomenon through parameters like the nugget effect and range. Experimental variograms are calculated from data and theoretical models like spherical, exponential, and power models are fitted. The variogram can identify different correlation scales through nested models. Components at different scales can be extracted through kriging. As an example, fertility data from France is analyzed to filter its large-scale spatial structure.
GLOBAL TRAJECTORY OPTIMISATION OF A SPACE-BASED VERY-LONG-BASELINE INTERFEROM...Mario Javier Rincón Pérez
This document summarizes a bachelor's thesis on optimizing the trajectory of a space-based very long baseline interferometry mission. The thesis develops a solver's code to determine an optimal solution for a mission involving three spacecraft orbiting Earth that take measurements of celestial sources. It studies approaches for spacecraft maneuvers, including lunar flybys, to maximize a performance index over the mission time while minimizing propellant usage. The optimal trajectory is determined using a multi-objective evolutionary algorithm to optimize resonant orbits obtained after flybys and maximize the number of measurements taken.
This document describes the development of a "smart pebble" device using MEMS sensors to monitor sediment transport in river beds. The smart pebble is designed to be about 2-3 cm in diameter, operate underwater, and autonomously record motion data. It uses accelerometers and gyroscopes to measure motion and orientation. A mixed-signal approach is used to process the sensor data, employing analog circuits like multipliers combined with a digital processor. Experimental tests showed the mixed-signal design provided reasonably accurate motion data compared to encoder measurements, demonstrating the feasibility of the smart pebble concept for further sediment transport research.
The document discusses formulas for calculating the gravitational effects of topographic-isostatic masses on airborne and satellite gravity gradiometry measurements. It derives integral formulas in ellipsoidal approximation for computing the gravitational potential, gradients, and tensor due to various topographic-isostatic models. The formulas separate the computations into spherical and ellipsoidal components. They are applied to calculate the gravitational tensor at GOCE satellite altitude using a 5-arcminute digital elevation model. The approach uses mass-lines to approximate ellipsoidal volume elements for numerical evaluation.
Assessment of the Usefulness of Egnos Differential Corrections in Conducting ...IJERA Editor
The article presents the results of research in determining the position of the base station in a static mode using
EGNOS differential corrections. The experiment was conducted in the Dęblin airport area. The experiment
comprised determining the coordinates of three base stations (REF1, VirA i VirB). The analysis of the results
for the tools adopted pointed out to the fact that the accuracy of coordinate determination for each station was in
the range of less than 2 meters.
The document contains sample tasks and answers related to engineering concepts. Task 1 covers SI base units, derived units, and unit conversions. Task 2 discusses qualitative and quantitative research methods. Task 3 provides examples of qualitative and quantitative case studies. Subsequent tasks cover topics such as forces, moments, shear force diagrams, bending moment diagrams, buoyancy, material properties, heat transfer, electrical circuits, and more. The document serves as a reference for various engineering calculations and concepts.
1. The document presents a study of rotational asymmetry in galaxies for both morphological and physical diagnostic purposes. It introduces two measures of asymmetry: A1, the existing measure, and A3, an alternative measure proposed using Spearman's correlation coefficient.
2. Methods describe existing measures of concentration, asymmetry, smoothness, and other morphological parameters. It then provides the equations for measuring A1 and introduces A3.
3. Results show correlations between A3 and concentration, and histograms comparing A1 and A3 in distinguishing galaxy types using a sample from existing galaxy catalogs. A3 proved more effective than A1 at distinguishing classes.
The document describes modeling the 3D geometry of relativistic jets based on 2D projection observations. It presents a mathematical model that uses Bayesian parameter estimation via MCMC to statistically estimate the 3D structural parameters of jet bends from measured 2D projection properties. Figures and equations show how the model accounts for projection effects to determine the intrinsic 3D geometry from observed 2D jet features like bend angles and distances.
The document discusses applying probabilistic localization and SLAM algorithms to a marine robotic middleware. It presents using Monte Carlo localization to estimate a vehicle's position based on range measurements to known beacons. It also applies FastSLAM to simultaneously estimate the vehicle position and map unknown beacon locations. Results show both algorithms accurately estimate positions, apart from an initial offset in FastSLAM, demonstrating their effectiveness for marine robot localization and mapping.
This document proposes new methods for horizon line detection in marine images captured using infrared or visible light cameras. It discusses existing methods like edge detection and Hough transform (EDHT) and introduces improvements. A new histogram-based method is proposed that segments an image into sky and sea regions by comparing regional probability distribution functions (PDFs) or histograms and selecting the line that maximizes the statistical distance between the PDFs. The document also describes combining EDHT with a statistical criterion to select the optimal line among candidate lines detected in the image. It compares these methods quantitatively and visually on test images and concludes that the introduced methods can benefit applications like tracking, navigation and target recognition from marine imagery.
Action Trajectory Reconstruction for Controlling of Vehicle Using SensorsIOSR Journals
Abstract: Inertial sensors, such as accelerometers and gyro-scopes, are rarely used by themselves to compute
velocity and position as each requires the integration of very noisy data. The variance and bias in the resulting
position and velocity estimates grow un-bounded in time. This paper proposes a solution to provide a de-biased
and de-noised estimation of position and velocity of moving vehicle actions from accelerometer measurements.
The method uses a continuous wavelet transform applied to the measurements recursively to provide reliable
action trajectory reconstruction. The results are presented from experiments performed with a MEMS accelerometer
and gyroscope.
Keywords: Action trajectory, continuous wavelet transform, inertial measurement unit.
Photogrammetry - Space Resection by Collinearity EquationsAhmed Nassar
Space resection is commonly used to determine the exterior orientation parameters (which refers to position and orientation related to an exterior coordinate system) associated with one or more photos based on measurements of ground control points (GCPs). space resection is a nonlinear problem, existing methods involve linearization of the collinearity condition and the use of an iterative process to determine the final solution using the least-squares method. The process also requires initial approximate values of the unknown parameters, some of which must be estimated by another least-squares solution.
Gaussian Orbital Determination of 1943 AnterosMatthew Li
Paper detailing the theory, methods, calculations, and results regarding the investigation of the orbit of asteroid 1943 Anteros through approximately six weeks of celestial observation and data collection.
This document describes a method for correcting smear in images captured by frame transfer CCD cameras when there are significant changes in illumination between frames. Existing smear correction algorithms assume constant illumination, which is not always valid. The proposed method models smear using a matrix equation that accounts for variable illumination levels transitioning between frames. It was developed for fast polarimetric imaging but could benefit other applications involving highly variable scenes synchronized with detector readout.
Verification of gravimetric geoidal models by a combination of gps and orthom...LENIN Quintero
This document summarizes a thesis that investigated verifying gravimetric geoid models using a combination of GPS heights and orthometric heights. It discusses height systems, the Canadian vertical datum, determination of the Canadian geoid using Stokes's formula and global geopotential models, biases and errors in GPS height differences, analysis of sample data, and analysis of field data collected in the Northwest Territories, Ontario, and Manitoba. The analyses aimed to detect any systematic dependencies between misclosures in the geoid-GPS/leveling comparison.
Mapping spiral structure on the far side of the Milky WaySérgio Sacani
Little is known about the portion of the Milky Way lying beyond the Galactic center at distances
of more than 9 kiloparsec from the Sun. These regions are opaque at optical wavelengths
because of absorption by interstellar dust, and distances are very large and hard to measure.
We report a direct trigonometric parallax distance of 20:4þ2:8
2:2 kiloparsec obtained with the Very
Long Baseline Array to a water maser source in a region of active star formation. These
measurements allow us to shed light on Galactic spiral structure by locating the ScutumCentaurus
spiral arm as it passes through the far side of the Milky Way and to validate a
kinematic method for determining distances in this region on the basis of transverse motions.
COMPLEMENTARY VISION BASED DATA FUSION FOR ROBUST POSITIONING AND DIRECTED FL...ijaia
The present paper describes an improved 4 DOF (x/y/z/yaw) vision based positioning solution for fully 6
DOF autonomous UAVs, optimised in terms of computation and development costs as well as robustness
and performance. The positioning system combines Fourier transform-based image registration (Fourier
Tracking) and differential optical flow computation to overcome the drawbacks of a single approach. The
first method is capable of recognizing movement in four degree of freedom under variable lighting conditions, but suffers from low sample rate and high computational costs. Differential optical flow computation, on the other hand, enables a very high sample rate to gain control robustness. This method, however, is limited to translational movement only and performs poor in bad lighting conditions. A reliable positioning system for autonomous flights with free heading is obtained by fusing both techniques. Although the vision system can measure the variable altitude during flight, infrared and ultrasonic sensors are used for robustness. This work is part of the AQopterI8 project, which aims to develop an autonomous
flying quadrocopter for indoor application and makes autonomous directed flight possible.
Precise Attitude Determination Using a Hexagonal GPS PlatformCSCJournals
In this paper, a method of precise attitude determination using GPS is proposed. We use a hexagonal antenna platform of 1 m diameter (called the wheel) and post-processing algorithms to calculate attitude, where we focus on yaw to prove the concept. The first part of the algorithm determines an initial absolute position using single point positioning. The second part involves double differencing (DD) the carrier phase measurements for the received GPS signals to determine relative positioning of the antennas on the wheel. The third part consists of Direct Computation Method (DCM) or Implicit Least Squares (ILS) algorithms which, given sufficiently accurate knowledge of the fixed body frame coordinates of the wheel, takes in relative positions of all the receivers and produces the attitude. Field testing results presented in this paper will show that an accuracy of 0.05 degrees in yaw can be achieved. The results will be compared with a theoretical error, which is shown by Monte Carlo simulation to be < 0.001 degrees. The improvement to the current state-of-the-art is that current methods require either very large baselines of several meters to achieve such accuracy or provide errors in yaw that are orders of magnitude greater.
Collinearity Equations
Kinds of product that can be derived by the collinearity equation
- Space Resection By Collinearity
- Space Intersection By Collinearity
- Interior Orientation
- Relative Orientation
- Absolute Orientation
- Self-Calibration
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document provides an overview of geostatistics and variogram analysis. It discusses how the variogram describes the spatial correlation of a phenomenon through parameters like the nugget effect and range. Experimental variograms are calculated from data and theoretical models like spherical, exponential, and power models are fitted. The variogram can identify different correlation scales through nested models. Components at different scales can be extracted through kriging. As an example, fertility data from France is analyzed to filter its large-scale spatial structure.
GLOBAL TRAJECTORY OPTIMISATION OF A SPACE-BASED VERY-LONG-BASELINE INTERFEROM...Mario Javier Rincón Pérez
This document summarizes a bachelor's thesis on optimizing the trajectory of a space-based very long baseline interferometry mission. The thesis develops a solver's code to determine an optimal solution for a mission involving three spacecraft orbiting Earth that take measurements of celestial sources. It studies approaches for spacecraft maneuvers, including lunar flybys, to maximize a performance index over the mission time while minimizing propellant usage. The optimal trajectory is determined using a multi-objective evolutionary algorithm to optimize resonant orbits obtained after flybys and maximize the number of measurements taken.
This document describes the development of a "smart pebble" device using MEMS sensors to monitor sediment transport in river beds. The smart pebble is designed to be about 2-3 cm in diameter, operate underwater, and autonomously record motion data. It uses accelerometers and gyroscopes to measure motion and orientation. A mixed-signal approach is used to process the sensor data, employing analog circuits like multipliers combined with a digital processor. Experimental tests showed the mixed-signal design provided reasonably accurate motion data compared to encoder measurements, demonstrating the feasibility of the smart pebble concept for further sediment transport research.
The document discusses formulas for calculating the gravitational effects of topographic-isostatic masses on airborne and satellite gravity gradiometry measurements. It derives integral formulas in ellipsoidal approximation for computing the gravitational potential, gradients, and tensor due to various topographic-isostatic models. The formulas separate the computations into spherical and ellipsoidal components. They are applied to calculate the gravitational tensor at GOCE satellite altitude using a 5-arcminute digital elevation model. The approach uses mass-lines to approximate ellipsoidal volume elements for numerical evaluation.
Assessment of the Usefulness of Egnos Differential Corrections in Conducting ...IJERA Editor
The article presents the results of research in determining the position of the base station in a static mode using
EGNOS differential corrections. The experiment was conducted in the Dęblin airport area. The experiment
comprised determining the coordinates of three base stations (REF1, VirA i VirB). The analysis of the results
for the tools adopted pointed out to the fact that the accuracy of coordinate determination for each station was in
the range of less than 2 meters.
The document contains sample tasks and answers related to engineering concepts. Task 1 covers SI base units, derived units, and unit conversions. Task 2 discusses qualitative and quantitative research methods. Task 3 provides examples of qualitative and quantitative case studies. Subsequent tasks cover topics such as forces, moments, shear force diagrams, bending moment diagrams, buoyancy, material properties, heat transfer, electrical circuits, and more. The document serves as a reference for various engineering calculations and concepts.
1. The document presents a study of rotational asymmetry in galaxies for both morphological and physical diagnostic purposes. It introduces two measures of asymmetry: A1, the existing measure, and A3, an alternative measure proposed using Spearman's correlation coefficient.
2. Methods describe existing measures of concentration, asymmetry, smoothness, and other morphological parameters. It then provides the equations for measuring A1 and introduces A3.
3. Results show correlations between A3 and concentration, and histograms comparing A1 and A3 in distinguishing galaxy types using a sample from existing galaxy catalogs. A3 proved more effective than A1 at distinguishing classes.
The document describes modeling the 3D geometry of relativistic jets based on 2D projection observations. It presents a mathematical model that uses Bayesian parameter estimation via MCMC to statistically estimate the 3D structural parameters of jet bends from measured 2D projection properties. Figures and equations show how the model accounts for projection effects to determine the intrinsic 3D geometry from observed 2D jet features like bend angles and distances.
The document discusses applying probabilistic localization and SLAM algorithms to a marine robotic middleware. It presents using Monte Carlo localization to estimate a vehicle's position based on range measurements to known beacons. It also applies FastSLAM to simultaneously estimate the vehicle position and map unknown beacon locations. Results show both algorithms accurately estimate positions, apart from an initial offset in FastSLAM, demonstrating their effectiveness for marine robot localization and mapping.
This document proposes new methods for horizon line detection in marine images captured using infrared or visible light cameras. It discusses existing methods like edge detection and Hough transform (EDHT) and introduces improvements. A new histogram-based method is proposed that segments an image into sky and sea regions by comparing regional probability distribution functions (PDFs) or histograms and selecting the line that maximizes the statistical distance between the PDFs. The document also describes combining EDHT with a statistical criterion to select the optimal line among candidate lines detected in the image. It compares these methods quantitatively and visually on test images and concludes that the introduced methods can benefit applications like tracking, navigation and target recognition from marine imagery.
Flight Dynamics Software Presentation Part I Version 5Antonios Arkas
This document describes an orbit determination simulator and its key features:
- It uses a weighted least-squares estimator to process range and angular tracking measurements from multiple Earth stations to determine orbital state. It can estimate parameters like reflectivity coefficient, ballistic coefficient, and antenna biases.
- It provides outputs like the determined orbit, validity metrics, covariance analysis, residuals graphs, and confidence ellipsoids. It can also propagate determined state covariance over time.
- The simulator was validated against another flight dynamics software by comparing results from processing real tracking data. Determined states and other parameters showed close agreement.
- Consider covariance analysis is performed to assess impact of neglected parameters like antenna biases. This is done through formal
The Geometric Characteristics of the Linear Features in Close Range Photogram...IJERD Editor
The accuracy of photogrammetry can be increased with better instruments, careful geometric
characteristics of the system, more observations and rigorous adjustment. The main objective of this research is
to develop a new mathematical model of two types of linear features (straight line, spline curve) in addition to
relating linear features in object space to the image space using the Direct Linear Transformation (DLT). The
second main objective of the present paper is to study of some geometric characteristics of the system, when the
linear features are used in close range photogrammetric reduction processes. In this research, the accuracy
improvement has been evaluated by adopting certain assessment criteria, this will be performed by computing
the positional discrepancies between the photogrammetrically calculated object space coordinates of some check
object points, with the original check points of the test field, in terms of their respective RMS errors values. In
addition, the resulting least squares estimated covariance matrices of the check object point's space coordinates.
To perform the above purposes, some experiments are performed with synthetic images. The obtained results
showed significant improvements in the positional accuracy of close range photogrammetry, when starting node,
end nodes, and interior node on straight line and spline curve are increased with certain specifications regarding
the location and magnitude of each type of them.
POSITION ESTIMATION OF AUTONOMOUS UNDERWATER SENSORS USING THE VIRTUAL LONG B...ijwmn
This document summarizes a study on using the virtual long baseline (VLBL) method for positioning autonomous underwater sensors. The study describes a mathematical model for the VLBL positioning system and experimental setup. In field experiments, an autonomous underwater vehicle measured distances from deployed responder beacons at different locations to estimate their positions. Results showed the VLBL method could accurately position the beacons within 1 meter using a preliminary one-dimensional optimization approach to avoid false minima in position estimation.
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Attitude determination of multirotors using camera
1. 22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
Copyright c 2013 by ABCM
ATTITUDE DETERMINATION OF MULTIROTORS USING CAMERA
VECTOR MEASUREMENTS
Pedro Filizola Sousa Maia Gonçalves
Roberto Brusnicki
Davi Antônio dos Santos
Instituto Tecnológico de Aeronáutica, Praça Marechal Eduardo Gomes, 50, Vila das Acácias, 12228-900, São José dos Campos, SP,
Brazil
pedrofsm@ita.br, rbrusnicki@gmail.com, davists@ita.br
Abstract. The employment of camera in low-cost navigation and guidance of multirotor unmanned aerial vehicles (UAV)
has recently been the focus of many investigations. Nevertheless, in the previous works, camera measurements was
adopted either to aid in the position/velocity estimation or to directly provide feedback for guidance, but not specifically
for assisting in the attitude determination process. This work is concerned with the attitude determination of multirotor
UAVs using vector measurements taken from a camera. The vehicle is assumed to be equipped with an altimeter, a triad of
rate-gyros, and a downward-facing strapdown camera. It is assumed to fly in an indoor environment containing various
landmarks placed in known positions on the floor. The quantity and positions of the landmarks are chosen in such a way
that at least two of them always remain in the camera field of view. Therefore, at each time instant, two noncollinear unit
vectors directed from the camera to the center of area of the landmarks can be computed. In order to carry out attitude
determination, two quaternion estimation methods are adopted: the multiplicative extended Kalman filter (MEKF) and
the quaternion extended Kalman filter (QEKF). The proposed multirotor attitude determination scheme is evaluated by
computational simulations.
Keywords: aerial robotics, attitude determination, Kalman filtering, computer vision
1. INTRODUCTION
The attitude determination (AD) is a fundamental part of any control system for unmanned aerial vehicles (UAV). In
general, it is concerned with the estimation of the vehicle’s attitude and angular velocity with respect to a given reference
coordinate system. The estimates computed by the AD function is then used to provide the attitude control laws with
feedback information.
The literature on AD is very extensive and has mainly been developed in the aerospace field (Wertz, 1978), (Yang,
2012). The AD methods stems from the Wabba Problem (Markley, 1988), which defined a framework to estimate attitude
from vector measurements. (Cheng et al., 2008) uses the extended Kalman filter (EKF) to estimate pitch and roll angles
of a Micro Aerial Vehicle (MAV). The third column of the Direction Cosine Matrix (DCM) and the rate gyro bias are used
as state variables. Gravity is used as the observation vector in the measurement model. The yaw angle is obtained from
geomagnetic field vector. Gebre-Egziabher and Elkaim (2008) use both gravity and geomagnetic field as observation
vectors in two different approaches to estimate the attitude quaternion. The first approach is an iterated least-square
estimator (LSE) and the second is an EKF. The LSE executes a global search of the attitude at each time step. On the
contrary, the EKF algorithm accounts for a priori information, resulting in a better performance. The above two methods
were designed to be gyro-free and GPS assisted. (Bar-Itzhack and Oshman, 1985) proposes a quaternion extended Kalman
filter (QEKF), which, to ensure estimates with unit norm, realizes an Euclidian normalization step after each measurement
update. (Idan, 1996) proposes a minimum-variance filter to estimate attitude parameterized by Rodrigues parameters. Due
to simpler algebraic expressions, this approach has a relative computational advantage over the quaternion estimators.
(Markley and Crassidis, 1996) presents a multiplicative extended Kalman filter (MEKF) that estimates an attitude error in
MRP and updates the total attitude represented by quaternion by means of quaternion multiplication.
In satellite AD methods, vector measurements are typically taken from solar sensors (Sun direction), magnetometers
(local geomagnetic field vector), horizon sensors (direction of nadir), star sensors (direction of stars) (Wertz, 1978). On
the other hand, the multirotor UAV literature usually relies only on two vector measurements taken, respectively, from
accelerometers (local vertical) and magnetometers.
This work presents a multirotor UAV attitude determination method using vector measurements taken from images. It
is assumed that the vehicle is equipped with three strapdown sensors: a downward-facing camera, a triad of rate-gyros and
an altimeter. The vehicle is assumed to fly indoors over a flat ground with various landmarks. Both vehicle and landmarks
have known positions with respect to the adopted reference coordinate system. The landmarks are disposed, in quantity
and positions, in such a way that at least two of them always remain in the camera field of view (FOV). Using measure-
ments taken from the camera and the altimeter, two noncollinear vector measurements pointing from the camera to the
landmarks’ centers can be computed. In order to obtain a scheme for attitude determination of multirotor UAVs, these
2. Gonçalves, Brusnicki and Santos
Attitude Determination of Multirotors Using Camera Vector Measurements
vector measurements as well as rate-gyro data are considered in two attitude estimation methods: the QEKF (Bar-Itzhack
and Oshman, 1985) and the MEKF (Markley and Crassidis, 1996). The proposed scheme is evaluated by computational
simulation. The remaining text is organized in the following manner. Section II defines the paper problem. Section III
reformulates the attitude estimation methods. Section IV presents some simulation results. Finally, Section V presents
the paper’s conclusions.
Notation. IN is the N × N identity matrix, [•×] denotes the cross product matrix and • defines the matrix transpose.
2. PROBLEM STATEMENT
Consider the multirotor helicopter and the three Cartesian coordinate systems (CCS) illustrated in Fig. 1. The body
CCS SB = {XB, YB, ZB} is attached to the vehicle at its center of mass (CM). The ground CCS SG = {XG, YG, ZG} is
fixed on the ground at point O. The reference CCS SR = {XR, YR, ZR} is parallel to SG but is centered at CM.
𝑋B
𝑌B
𝑍B
𝑂
𝑍G
𝑋G
𝑌G
𝑋R
𝑌R
𝑍R
𝑓3
𝑓2
𝑓1
𝑓4
𝐬(1)
𝐬(2)
𝑀(1) 𝑀(2)
𝑀(𝑙)
𝐶𝑀
𝑀(3)
Figure 1: The Cartesian coordinate systems and the flight environment.
Assume that the camera is positioned at the CM and the triad of rate-gyros is aligned with SB. Define the set of
landmark indexes to be I {1, 2, ..., l}. Denote the center of the i-th landmark by M(i)
. Define s
(i)
to be the unit
geometric vector pointing from CM to M(i)
. Denote the representations of s
(i)
in SB and SR by b(i)
∈ R3
and r(i)
∈ R3
,
respectively. The representations b(i)
and r(i)
are interrelated by b(i)
= Dr(i)
, where D ∈ SO(3) is the attitude matrix of
SB with respect to SR. In order to measure two noncollinear pairs (b(i)
, r(i)
), one assumes that both CM and landmarks
have known positions and, moreover, at least two landmarks are measured by the camera at each sample instant. This
yields the following two pairs of vector measurements:
Vk
ˆb
(i1)
k ,ˆr
(i1)
k , ˆb
(i2)
k ,ˆr
(i2)
k , i1 ∈ I, i2 ∈ I, i1 = i2, (1)
where k denotes the discrete-time instant and, for i = i1, i2,
ˆb
(i)
k = D(ak)r
(i)
k + δb
(i)
k , (2)
r
(i)
k = ˆr
(i)
k + δr
(i)
k , (3)
where ˆr
(i)
k is a sample of r(i)
at instant k, δb
(i)
k and δr
(i)
k are zero-mean Gaussian white sequences with covariances R
(i)
b,k
3. 22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
and R
(i)
r,k, respectively, and ak ∈ Rn
is a discrete-time attitude representation vector which parameterizes the attitude
matrix D(ak).
Let the attitude kinematics be modeled by the differential equation (Wertz, 1978)
˙a(t) = f(a(t), ω(t)), (4)
where a(t) is a continuous-time version of ak, ω(t) ∈ R3
is the true angular velocity. Since the rate-gyros are not perfect,
the true angular velocity is given by the following stochastic model:
ω(t) = ˆω(t) + δω(t), (5)
where ˆω(t) ∈ R3
is the measured angular velocity and δω(t) ∈ R3
is the rate-gyro measurement noise, which is
assumed to be a zero-mean Gaussian white sequence with covariance Q. A discrete-time version of Eq.(5) is given by
ωk = ˆωk + δωk, where δωk has the same characteristics of δω(t).
The main problem of the paper is to recursively compute the minimum-variance (MV) estimate ˆak|k of the true attitude
vector ak using the dynamic equation (4), the sequence of angular velocity measurements ˆω1:k, and sequence of vector
measurements V1:k.
3. PROBLEM SOLUTION
This section presents two estimation methods to face the afore-defined problem: The Quaternion Extended Kalman
Filter (QEKF) (Bar-Itzhack and Oshman, 1985) and the Multiplicative Extended Kalman Filter (MEKF) (Markley and
Crassidis, 1996).
3.1 Quaternion extended Kalman filter - QEKF
Bar-Itzhack and Oshman (1985) proposed a discrete-time extended Kalman filter to estimate the attitude quaternion.
This method is described in the sequel. Let the vector a(t) assumes the form of the attitude quaternion
q(t)
q1,t
et
, (6)
subject to the unit norm constrain
q(t) = q2
1,t + et = 1, (7)
where q1,t and et are, respectively, the scalar and the complex part of the attitude quaternion. This gives rise to the
following attitude kinematic equation (Wertz, 1978):
˙q(t) = Ω(t)q(t), (8)
where
Ω(t) =
1
2
0 −ω(t)
ω(t) −[ω(t)×]
. (9)
Integrating Eq.(9) from tk to tk+1, yields
qk+1 = Φ(tk+1, tk)qk, (10)
where Φ(tk+1, tk) ∈ R4×4
is the state transition matrix. Define Ts (tk+1 − tk) as the sampling time. Assuming
constant angular velocity ω(t) during the interval Ts, the state transition matrix can be written as
Φ(tk+1, tk) = eΩkTs
, (11)
where Ωk has the same form of Eq.(9), however it is computed using ωk. Rewriting Eq.(11) by using the discrete-time
version of Eq.(5), results
4. Gonçalves, Brusnicki and Santos
Attitude Determination of Multirotors Using Camera Vector Measurements
Φ(tk+1, tk) = e
ˆΩkTs
eδΩkTs
, (12)
where ˆΩk and δΩk are given by Eq.(9), but computed using ˆωk and δωk, respectively. The second factor at the right-hand
side of Eq.(12) is expanded in power series, yielding
Φ(tk+1, tk) = e
ˆΩkTs
(I4 + δΩkTs + ...). (13)
By truncating the series in Eq.(13) after the first order term, it is possible to approximate Eq.(10) by
qk+1 ≈ e
ˆΩkTs
qk + e
ˆΩkTs
δΩkTsqk. (14)
Manipulating the second term in the right-hand side of Eq.(14), which is the state noise, one can obtain the discrete-
time state model as follows:
qk+1 = e
ˆΩkTs
qk +
Ts
2
e
ˆΩkTs
Ξkδωk, (15)
where
Ξk
−ek
[ek×] + q1,kI3
. (16)
Let Γk be defined by
Γk =
Ts
2
e
ˆΩkTs ˆΞk, (17)
where ˆΞk is given by Eq.(16), but computed using ˆqk|k. The state noise covariance is approximated as follows:
Qq
k = ΓkQΓk. (18)
The discrete-time nonlinear measurement model is now described in quaternion as follows:
ˆb
(i)
k = D(qk)r
(i)
k + δb
(i)
k , (19)
where
D(qk) = (q2
1,k − |ek|2
)I3 + 2ekek − 2q1,k[ek×]. (20)
The QEKF requires the Jacobian matrix of the nonlinear measurement model of Eq.(20), which is defined as
H
(i)
q,k+1
∂D(q)r
(i)
k+1
∂q
|q=ˆqk+1|k
= ∂D(q)
∂q1
r
(i)
k+1
∂D(q)
∂q2
r
(i)
k+1
∂D(q)
∂q3
r
(i)
k+1
∂D(q)
∂q4
r
(i)
k+1
q=ˆqk+1|k
, (21)
where the partial derivatives are given by
∂D(q)
∂q1
|q=ˆqk+1|k
= 2
ˆq1 ˆq4 −ˆq3
−ˆq4 ˆq1 ˆq2
ˆq3 −ˆq2 ˆq1
k+1|k
, (22)
∂D(q)
∂q2
|q=ˆqk+1|k
= 2
ˆq2 ˆq3 ˆq4
ˆq3 −ˆq2 ˆq1
ˆq4 −ˆq1 −ˆq2
k+1|k
, (23)
5. 22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
∂D(q)
∂q3
|q=ˆqk+1|k
= 2
−ˆq3 ˆq2 −ˆq1
ˆq2 ˆq3 ˆq4
ˆq1 ˆq4 −ˆq3
k+1|k
, (24)
∂D(q)
∂q4
|q=ˆqk+1|k
= 2
−ˆq4 ˆq1 ˆq2
−ˆq1 −ˆq4 ˆq3
ˆq2 ˆq3 ˆq4
k+1|k
. (25)
The QEKF consists of a discrete-time formulation of the extended Kalman filter (Bar-Shalom and Li, 1993) applied to
the system modeled by the state equation (15) and the measurement equation (19). In order to force an unit norm property,
an Euclidean normalization is carried out at each filter iteration, after the measurement update. For simplicity, the error
covariance of the normalized estimate is approximated by P∗
k+1|k+1 = Pk+1|k+1.
The QEKF algorithm is summarized as follows:
Initial conditions
ˆq∗
0|0 = ˆq0
P∗
0|0 = Pq
0
State propagation
ˆqk+1|k = e
ˆΩkTs ˆq∗
k|k
Pk+1|k = (e
ˆΩkTs
)P∗
k|k(e
ˆΩkTs
) + Qq
k
Measurement prediction
(ˆb
(i1)
k+1|k) (ˆb
(i2)
k+1|k) = D(ˆqk+1|k)(r
(i1)
k+1) D(ˆqk+1|k)(r
(i2)
k+1)
Pb
k+1|k = Hq,k+1Pk+1|kHq,k+1 + Rk+1
Update
Kk+1 = Pk+1|kHq,k+1(Pb
k+1|k)−1
ˆqk+1|k+1 = ˆqk+1|k + Kk+1 (b
(i1)
k+1) (b
(i2)
k+1) − (ˆb
(i1)
k+1|k) (ˆb
(i2)
k+1|k)
Pk+1|k+1 = Pk+1|k − Kk+1Pb
k+1|kKk+1
Normalization
ˆq∗
k+1|k+1 =
ˆqk+1|k+1
||ˆqk+1|k+1||
P∗
k+1|k+1 = Pk+1|k+1
Note that
Hq,k+1 = H
(i1)
q,k+1 H
(i2)
q,k+1
, (26)
and
Rk+1 =
R
(i1)
b,k+1 03×3
03×3 R
(i2)
b,k+1
. (27)
The estate transition matrix is solved by [(Wertz, 1978), pp.567]
e
ˆΩkTs
= cos( ˆωk
Ts
2
)I4 +
1
ˆωk
sin( ˆωk
Ts
2
)ˆΩk. (28)
6. Gonçalves, Brusnicki and Santos
Attitude Determination of Multirotors Using Camera Vector Measurements
3.2 Multiplicative extended Kalman filter - MEKF
Markley and Crassidis (1996) proposed a continuous/discrete-time filter which represents the true attitude quaternion
by
q(t) = δq(p(t)) ⊗ ˆq(t), (29)
where ˆq(t) is a reference quaternion, δq(p(t)) is the multiplicative error quaternion parameterized by modified Rodrigues
parameters p(t), and ⊗ denotes the quaternion product (Shuster, 1993). The reference quaternion ˆq(t) is considered
the best estimate of the true quaternion q(t) between the interval [tk, tk+1). Thus, the MRP assumes p(t) = 0 for
t ∈ [tk, tk+1), which eliminates the redundancy of two paramerizations use.
Let Eq.(4) be redefined by the MRP p(t) p1,t p2,t p3,t as follows:
˙p(t) = f(p(t), ω(t)). (30)
The nonlinear function f(p(t), ω(t)) is defined by
f(p(t), ω(t)) = G(p(t))ω(t), (31)
where (Schaub, 1998)
G(p(t)) =
1
4
(1 − ||p(t)||2
)I3 + 2[p(t)×] + 2p(t)p(t) . (32)
Applying Eq.(5) in Eq.(31), and the result in Eq.(30), yields in the state model as follows:
˙p(t) = G(p(t))ˆω(t) + G(p(t))δω(t), (33)
where the second term in the right-hand side of Eq.(33) is the state noise. Its covariance is approximated as:
Qp
(t) = Γ(t)QΓ(t) , (34)
where Γ(t) = G(ˆp(t)), ∀t ∈ [tk, tk+1).
The MEKF requires the Jacobian matrix of Eq.(33), as follows:
F(ˆp(t), ˆω(t))
∂G(p(t))ω(t)
∂p
|p=ˆp(t). (35)
Assuming null MRP for [tk, tk+1), Eq.(35) results in
F(ˆp(t), ˆω(t)) =
1
2
(−[ˆω×]). (36)
Let discrete-time measurement model be defined in MRP by
ˆb
(i)
k = D(pk)r
(i)
k + δb
(i)
k , (37)
where (Shuster, 1993)
D(pk) = I3 +
8[pk×]2
− 4(1 − pk
2
)[pk×]
(1 + pk
2)2
, (38)
is the attitude matrix in MRP. In order to obtain a linear model of Eq.(37), first order Taylor expansion is applied. The
Jacobian of this operation is given by
7. 22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
H
(i)
p,k+1
∂D(p)r
(i)
k+1
∂p
|p=ˆpk+1|k
= ∂D(p)
∂p1
r
(i)
k+1
∂D(p)
∂p2
r
(i)
k+1
∂D(p)
∂p3
r
(i)
k+1
p=ˆpk+1|k
, (39)
where the partial derivatives, assuming null MRP ∀t ∈ [tk, tk+1), yield
∂D(p)
∂p1
|p=ˆpk+1|k
=
0 0 0
0 0 4
0 −4 0
, (40)
∂D(p)
∂p2
|p=ˆpk+1|k
=
0 0 −4
0 0 0
4 0 0
, (41)
∂D(p)
∂p3
|p=ˆpk+1|k
=
0 4 0
−4 0 0
0 0 0
. (42)
By means of the continuous-discrete EKF, both state model given by Eq.(33) and measurement model given by Eq.(37)
are fused in order to estimate the attitude error in MRP. The global nonsingular attitude is propagated in quaternion by
Eq.(15) in the interval [tk, tk+1). The update of the global attitude is given by the discrete-time version of Eq.(29), where
δq(pk+1|k+1) =
1−||pk+1|k+1||2
1+||pk+1|k+1||2
2p1,k+1|k+1
1+||pk+1|k+1||2
2p2,k+1|k+1
1+||pk+1|k+1||2
2p3,k+1|k+1
1+||pk+1|k+1||2
. (43)
The MEKF algorithm is summarized as follows:
Initial conditions
ˆq0|0 = ˆq0
Pp
0|0 = Pp
0
ˆm0|0 = 0
State propagation
ˆp(t) = 0, t ∈ [tk, tk+1)
˙P
p
(t) = F(ˆp(t), ˆω(t))Pp
(t) + Pp
(t)F(ˆp(t), ˆω(t)) + Qp
(t)
ˆqk+1|k = e
ˆΩkTs ˆq∗
k|k
Measurement prediction
(ˆb
(i1)
k+1|k) (ˆb
(i2)
k+1|k) = D(ˆqk+1|k)(r
(i1)
k+1) D(ˆqk+1|k)(r
(i2)
k+1)
Pb
k+1|k = Hp,k+1Pp
k+1|kHp,k+1 + Rk+1
Update
Kk+1 = Pp
k+1|kHp,k+1(Pb
k+1|k)−1
ˆpk+1|k+1 = Kk+1 (b
(i1)
k+1) (b
(i2)
k+1) − (ˆb
(i1)
k+1|k) (ˆb
(i2)
k+1|k)
Pp
k+1|k+1 = Pp
k+1|k − Kk+1Pb
k+1|kKk+1
8. Gonçalves, Brusnicki and Santos
Attitude Determination of Multirotors Using Camera Vector Measurements
ˆqk+1|k+1 = δq(pk+1|k+1) ⊗ ˆqk+1|k
Note that
Hq,k+1 = H
(i1)
q,k+1 H
(i2)
q,k+1
, (44)
and
Rk+1 =
R
(i1)
b,k+1 03×3
03×3 R
(i2)
b,k+1
. (45)
4. SIMULATION AND RESULTS
The performance of both presented estimators will be compared using simulated data. The multirotor true attitude qk
is propagated by Eq.(15) with the following angular velocity:
ωk =
0.1 sin(kTs)
0.1 cos(kTs)
−0.1 sin(kTs) cos(kTs)
, (46)
where Ts = 0.1s. The camera vector measurements were generated using Eq.(2), where
r
(1)
k =
0
5
13
−12
13
, (47)
r
(2)
k =
0
− 5
13
−12
13
. (48)
Both rate-gyro and camera noise covariances were tuned in order to not diverge the filter estimate. Table 1 shows the
assumed measurement noise covariances.
Table 1: Measurement noise covariances.
Sensor Covariance
Rate-gyro Qk = (0.005)2
I3 (rad/s)2
Camera R
(i)
b,k = (0.01)2
I3
Using the simulated measurements, both QEKF and MEKF were submitted to one hundred Monte-Carlo runs with
1000s of duration each. The integration for MEKF is given by fourth order Runge-Kutta. The initial conditions assumed
are shown in Tab.2.
Table 2: Initial conditions.
Parameter QEKF MEKF
True attitude q0 ∼ N 1 0 0 0 , P0|0 q0 ∼ N 1 0 0 0 , P0|0
State ˆq0|0 = 1 0 0 0 ˆq0|0 = 1 0 0 0 , ˆp0|0 = 0 0 0
Covariance P0|0 = 0.01I4 Pp
0|0 = 0.01I3
Accuracy, orthogonality and relative computational burden are the parameters to be examined. The accuracy is mea-
sured as follows (Wertz, 1978):
Ik = |acos
1
2
tr D(ˆqk|k) D(qk) − 1 |, (49)
9. 22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
(a) Zoom in 0-200 seconds (b) 0-1000 seconds
Figure 2: MEKF angular error in degree
(a) Zoom in 0-200 seconds (b) 0-1000 seconds
Figure 3: QEKF angular error in degree
where the index Ik corresponds to the error angle between the true and the estimate attitudes in the Euler principal angle
notation. The orthogonality is given by
Jk = tr D(ˆqk|k) D(ˆqk|k) − I3 D(ˆqk|k) D(ˆqk|k) − I3 (50)
where the index Jk describes how close the estimate attitude matrix is to the orthogonal matrix, as it gets closer to zero.
Since the CPU performs tasks parallel to the simulation, it is not possible to use the cycle time for measure an absolute
computational burden of each filter. Rather, the cycle time is used to measure how fast is one algorithm relative to the
other.
Defined the simulation conditions, the mean and the standard deviation values of both indexes Ik and Jk are calculated.
Figure (2) shows the MEKF mean accuracy index between 0.4 and 0.6 degrees, while Fig.(3) presents same index for
QEKF approximately equal to 0.2 degrees. For this simulation conditions, the QEKF shows better accuracy than the
MEKF. From Figs. (5) and (4), one can conclude that the QEKF attitude matrix is closer to the orthogonal matrix than
the MEKF one. The QEKF spent an average of 0.115ms per cycle while the MEKF spent 0.152ms, resulting in 24.34%
more time consumption for the MEKF over the QEKF.
10. Gonçalves, Brusnicki and Santos
Attitude Determination of Multirotors Using Camera Vector Measurements
Figure 4: QEKF orthogonality Index Figure 5: MEKF orthogonality Index
5. CONCLUSION
Two attitude determination methods based on camera vector measurements were presented. The quaternion extended
Kalman filter performed better than the multiplicative extended Kalman filter for the proposed simulation scheme. How-
ever, MEKF does not need a normalization step after the state update. These methods are suitable for indoor environments,
since they do not use GPS. An alternative upgrade for outdoor flight is to use gravity direction and geomagnetic field vector
along with camera vector measurements. An experimental flight test is being prepared in order to evaluate the embedded
computational burden.
6. ACKNOWLEDGEMENTS
The first author is plenty thankful to be supported by Fundação de Amparo à Pesquisa do Estado do Amazonas -
FAPEAM.
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The authors are the only responsible for the printed material included in this paper.