TRIAD Algorithm Research Articles

Implementing the TRIAD Algorithm for Nanosatellite Attitude using NUCLEO-F303RE Microcontroller: Contributing to Mexico’s Space Initiatives

Abstract Space exploration continues to inspire humanity, with nanosatellites standing at the forefront of critical sectors like remote sensing, telecommunications, and meteorology. In this panorama, accurately determining their attitude is a considerable challenge. Addressing this, our research delves into the application of the well-known TRIAD algorithm. Utilizing four distinct vectors - two reference and two observed - sourced from solar position vectors and the magnetic field, we implemented the algorithm in the NUCLEO-F303RE microcontroller, equipped with the ARM® Cortex® M4 architecture. By carefully corrupting the observed vectors using data from prospective sensors, realized through the Systems Tool Kit software, we aspired to simulate real-world conditions. While tools like STK® for simulations are not new globally, they remain relatively unknown to many Mexican students. By introducing these resources, we aim to pave the way for future researchers at the Mexican Space Agency (AEM), bolstering Mexico’s ongoing journey in space exploration.

TRIAD-Aided EKF for Satellite Attitude Estimation and Sensor Calibration

Increasing demand for the space operations, space industry turns its face to cost effective solutions. Small satellites, due to their size and cost, are receiving interest from many organizations. Two of most common sensors that are being used in nanosatellites are magnetometers and sun sensors. In this work, magnetometer and sun sensor measurements are fused together with the TRIAD algorithm to produce body angle estimation. Combining with gyroscopes measurements, an Extended Kalman Filter is used to estimate body angles, angular velocities, gyroscope and magnetometer biases.

Algorithm for determining satellite orientation based on use of earth sensor and magnetometer

An algorithm for determining the orientation of a satellite based on information from an Earth sensor and a magnetometer is considered. The effect of instrumental errors of the magnetometer on the accuracy of orientation determination is analyzed. The TRIAD algorithm is compared with an algorithm in which only the heading angle is calculated.

A combined configuration (αβ filter- TRIAD algorithm) for spacecraft attitude estimation based on in-Orbit Flight Data

A combined configuration (αβ filter- TRIAD algorithm) for spacecraft attitude estimation based on in-Orbit Flight Data

Implementation of the risk triad algorithm to determine the probability of failure in pipelines: case of external corrosion

En este documento se describe y se trata un caso de aplicación en la industria de transporte de hidrocarburos por ductos del denominado algoritmo probabilístico de la tríada de riesgos discutido por Kent Muhlbauer (o algoritmo del Modelo de Referencia de Riesgo Cuantitativo), para establecer la probabilidad de falla (PoF) de un evento, en particular de la amenaza corrosión externa. El modelo utiliza matemática probabilística para combinar las variables y capturar tanto los impactos individuales como la acumulación de efectos menores, a través de las compuertas lógicas OR y AND. El uso de estas compuertas lógicas en modelos de riesgos representa una mejora evidente sobre la mayoría de los métodos más antiguos, ya que permite obtener una mejor representación de la forma en que se comportan los parámetros que materializan una amenaza. En este modelo, las variables que afectan la PoF se agrupan en tres categorías o elementos: exposición, mitigación y resistencia, que se representan en función de la amenaza o mecanismo de daño potencial al cual está sujeto el activo en estudio. Luego de obtener el valor de PoF, este se contrasta con los valores numéricos asociados con la probabilidad de falla en la Tabla 4.2 de API RP 581, se cruza con los resultados de CoF y de este modo se obtiene un ranking de riesgo que permite optimizar los recursos al aplicarlos a los activos de mayor riesgo. Se presentan los resultados luego de aplicarlos a la amenaza corrosión externa.

Comparative Analysis of Low Earth Orbit Satellite Attitude Determination using Sensor Integration and Attitude Estimation using Kalman Filter

The objective of the research is to undergo a performance comparison in terms of accuracy, convergence time, amount of memory, etc. between the satellite attitude determination and attitude estimation using non-linear filters. The fundamental approach towards it is to design an OBC (On Board Computer) that would help in achieving a controlled output for the chosen plant (MicroMAS1 satellite). The attitude determination algorithm is implemented through TRIAD algorithm, which takes sensor readings of body frame and inertial frame of reference. Then it is used to determine the rotation matrix (DCM) by converting the matrix form into vector form and again back to matrix form to determine the 3x3 matrix, which includes all the Euler angle equations to determine the pitch, roll, yaw characteristics of the system. The attitude estimation algorithms involves the use of nonlinear filters which provide an added advantage that energy can be transferred in a designed manner and extra degrees of freedom are available in filter design. The Unscented Kalman Filter (UKF) is preferred as it addresses the problem using a deterministic sampling approach. Moreover, the non-linear filters are used to remove the noise error and disturbance caused by engine. The design of satellite attitude models involves more of a mathematical approach that would be dealt with MATLAB and SIMULINK operations.

Total Least-Squares Determination of Body Segment Attitude.

To examine segment and joint attitudes when using image-based motion capture, it is necessary to determine the rigid body transformation parameters from an inertial reference frame to a reference frame fixed in a body segment. Determine the rigid body transformation parameters must account for errors in the coordinates measured in both reference frames, a total least-squares problem. This study presents a new derivation that shows that a singular value decomposition-based method provides a total least-squares estimate of rigid body transformation parameters. The total least-squares method was compared with an algebraic method for determining rigid body attitude (TRIAD method). Two cases were examined: case 1 where the positions of a marker cluster contained noise after the transformation, and case 2 where the positions of a marker cluster contained noise both before and after the transformation. The white noise added to position data had a standard deviation from zero to 0.002 m, with 101 noise levels examined. For each noise level, 10 000 criterion attitude matrices were generated. Errors in estimating rigid body attitude were quantified by computing the angle, error angle, required to align the estimated rigid body attitude with the actual rigid body attitude. For both methods and cases, as the noise level increased the error angle increased, with errors larger for case 2 compared with case 1. The singular value decomposition (SVD)-based method was superior to the TRIAD algorithm for all noise levels and both cases, and provided a total least-squares estimate of body attitude.

RETRACTED ARTICLE: On-orbit calibration of sun sensor’s central point error for triad

ABSTRACT In this paper, a novel sun sensor which has been applied on orbit since 2018 is introduced. The remote data transmitted from the satellite is analyzed to show its performance. In the lighting area, the sun sensor can give the sun vector which used for the attitude determination. From the data, the center point error is found to be the most significant factor that affects the accuracy of the sun sensor. In the “Sun pointing” work mode, the real output of the sun sensor is different from the theoretical output. The data from the star sensor is used for compensating the center point error based on ellipse fitting. The triad algorithm is affected by the center point error of the sun sensor. The simulation only considers this kind of error. And the result shows that it can decrease the Euler angle errors, so that improves the accuracy of triad algorithm.

Globally Asymptotic Stable Attitude Estimation with Application to MEMS Sensors

Aiming at the requirement of attitude information module with high precision, small size and low power consumption for the control of miniature UAV, a practical attitude estimation algorithm based on the micro-electro-mechanical sensor is proposed in this paper, which realizes the accurate estimation of the attitude of the UAV under the condition of low acceleration. A low-cost MEMS gyroscope, accelerometer, and magnetometer are used in the system. The Euler angle is obtained by the state observer method based on Direction Cosine Matrix (DCM) which can be got by fusing the sensor data. Firstly, based on the basic idea of TRIAD algorithm, a method to determine the attitude rotation matrix by accelerometer and magnetometric measurement is proposed. Compared with the traditional method, this method does not have to calculate the inverse of the matrix. Secondly, a state observer is intended to estimate the attitude of the system. The state observer doesn't have to observe the bias of the gyroscope, but still ensures the convergence of the Euler angle. Finally, the simulation based on the actual sampling data of the MEMS sensor shows that the output of the state observer designed in this paper still has high accuracy and good dynamic characteristics under the condition of gyroscope noise and bias.

An Attitude Filtering and Magnetometer Calibration Approach for Nanosatellites

We propose an attitude filtering and magnetometer calibration approach for nanosatellites. Measurements from magnetometers, Sun sensor and gyros are used in the filtering algorithm to estimate the attitude of the satellite together with the bias terms for the gyros and magnetometers. In the traditional approach for the attitude filtering, the attitude sensor measurements are used in the filter with a nonlinear vector measurement model. In the proposed algorithm, the TRIAD algorithm is used in conjunction with the unscented Kalman filter (UKF) to form the nontraditional attitude filter. First the vector measurements from the magnetometer and Sun sensor are processed with the TRIAD algorithm to obtain a coarse attitude estimate for the spacecraft. In the second phase the estimated coarse attitude is used as quaternion measurements for the UKF. The UKF estimates the fine attitude, and the gyro and magnetometer biases. We evaluate the algorithm for a hypothetical nanosatellite by numerical simulations. The results show that the attitude of the satellite can be estimated with an accuracy better than 0.5 $${^{\circ }}$$ and the computational load decreases more than 25% compared to a traditional UKF algorithm. We discuss the algorithm’s performance in case of a time-variance in the magnetometer errors.

A two-step self-calibration method for three-axis field sensors

A self-calibration method was proposed to compensate the drawback generated by classic ellipsoid fitting algorithm. In this paper magnetometers and accelerometers were chosen to obtain geomagnetic field and gravitational field for TRIAD algorithm. The first step is using the classic ellipsoid fitting to calibrate the magnetic vector, which is taken as an internal reference in the second step. Due to the direction the geomagnetic vector and gravity vector are relatively invariant; therefore the projection of geomagnetic vector on gravitational vector is a constant value. The second step is using projection method to calibrate accelerometer. At last, both simulation and experiment were taken to prove the accuracy of the calibration method.

An Optimized Triad Algorithm for Attitude Determination

Abstract The classic TRIAD was used to obtain the attitude of air vehicles. However, the accuracy was dominated by the sensor noise and the calculation order. To improve that in this paper, a new method based on weighting the vectors summation and difference was proposed. Then both simulation and experiment verified the advantages of the optimized algorithm.

3축 자기장 센서 및 관성센서를 이용한 차량 방위각 추정 방법

The vehicle attitude and sideslip is critical information to control the vehicle to prevent from unintended motion. Many of estimation strategy use bicycle model or IMU integration, but both of them have limits on application. The main purpose of this paper is development of vehicle orientation estimator which is robust to various vehicle state and road shape. The suggested estimator use 3-axis magnetometer, yaw rate sensor and lateral acceleration sensor to estimate three Euler angles of vehicle. The estimator is composed of two individual observers: First, comparing the known magnetic field and gravity with measured value, the TRIAD algorithm calculates optimal rotational matrix when vehicle is in static or quasi-static condition. Next, merging 3-axis magnetometer with inertial sensors, the extended Kalman filter is used to estimate vehicle orientation under dynamic condition. A validation through simulation tools, Carsim and Simulink, is performed and the results show the feasibility of the suggested estimation method.

A New Analytic Alignment Method for a SINS.

Analytic alignment is a type of self-alignment for a Strapdown inertial navigation system (SINS) that is based solely on two non-collinear vectors, which are the gravity and rotational velocity vectors of the Earth at a stationary base on the ground. The attitude of the SINS with respect to the Earth can be obtained directly using the TRIAD algorithm given two vector measurements. For a traditional analytic coarse alignment, all six outputs from the inertial measurement unit (IMU) are used to compute the attitude. In this study, a novel analytic alignment method called selective alignment is presented. This method uses only three outputs of the IMU and a few properties from the remaining outputs such as the sign and the approximate value to calculate the attitude. Simulations and experimental results demonstrate the validity of this method, and the precision of yaw is improved using the selective alignment method compared to the traditional analytic coarse alignment method in the vehicle experiment. The selective alignment principle provides an accurate relationship between the outputs and the attitude of the SINS relative to the Earth for a stationary base, and it is an extension of the TRIAD algorithm. The selective alignment approach has potential uses in applications such as self-alignment, fault detection, and self-calibration.

Design and simulation of an attitude determination system based on the Extended Kalman Filter for Cube-Sat Colombia I

The purpose of this document is to describe the design and simulation of an attitude determination system (ADS) for the CubeSat Colombia I picosatellite, based on an extended Kalman filter EKF [1,2]. In this development the propagation of the State of the system has been implemented through a Runge-Kutta procedure instead of traditional State Transition matrix method, improving predictions from the EKF filter without making direct measurements of angular velocity, it was obtained a reduced prediction time by incorporating a deterministic algorithm (the TRIAD algorithm) within the EKF. The results validated through simulation are those shown in this article.

Use of least square criterion in triad algorithm

Use of least square criterion in triad algorithm

A Triad Algorithm for Analysing Individual Ante- and Post-Mortem Findings to Improve the Quality of Intensive Care

Autopsy is an important source of data for education and quality control. The aim of this study was comparison of ante- to post-mortem findings to detect weak points of intensive care unit (ICU) care. Patients who died in our 14-bed university medical ICU care and underwent an autopsy examination over 20 months (September 2007 to May 2009) were included. Modified Goldman's criteria were used to categorise discrepancies between diagnoses and post-mortem findings. A triad algorithm was constructed to analyse individual ante- to post-mortem findings. One hundred and seventy post-mortem examinations were conducted (45.6% autopsy rate). Major diagnostic discrepancies were detected in 20 patients (11.8%); four class I (2.4%) and 16 class II (9.4%). Massive pulmonary embolism with cardiac arrest was the most common class I discrepancy (75%). Triad analysis of major class I discrepancies showed that all patients had a history of chronic disease; the majority (75%) had a short ICU length of stay. In 75% adequate tests were used to detect disorders. There were interpretation problems of bedside data in complex emergency clinical conditions, especially with less experienced ICU physicians. Inappropriate or incorrectly interpreted diagnostic procedures were performed in more than half of cases with class II discrepancies (9/16, 56%). Abdominal ultrasonography was misleading in 31% (5/16) cases with class II discrepancies. In conclusion, triad algorithm analysis revealed problematic interpretation of bedside diagnostics in emergency cases by inexperienced physicians in class I major discrepancies detected at autopsy. No correct test and wrong interpretation of abdominal ultrasonography were major causes of class II discrepancies.

평균 TRIAD를 이용한 자세 결정

In general, accurate attitude information is essential to perform the mission. Two algorithms are well-known to determine the attitude through two or more vector observations. One is deterministic method such as TRIAD algorithm, the other is optimal method such as QUEST algorithm. This Paper suggests the idea to improve performance of the TRIAD algorithm and to determine the attitude by combination of different sensors. First, we change the attitude matrix to Euler angle instead of using orthogonalization method and also use covariance in place of variance, then apply an unbiased minimum variance formula for more accurate solutions. We also suggest the methodology to determine the attitude when more than two measurements are given. The performance of the Averaging TRIAD algorithm upon the combination of different sensors is analyzed by numerical simulation in terms of standard deviation and probability.

Optimal Attitude Matrix from Two Vector Measurements

A SIMPLE counting argument shows that a minimum of two unit-vector measurements, with two independent scalar pieces of information each, are required to determine the 3 degrees of freedom needed to specify a spacecraft’s attitude. The earliest published algorithm for attitude determination from two vector measurements was Black’s TRIAD algorithm [1,2], also known as the algebraic method [3]. TRIAD has been applied to both groundbased and onboard attitude determination, using either unit vectors to the sun and along the Earth’s magnetic field for a coarse “sun-mag” attitude determination or unit vectors to two stars for a precise attitude determination. TRIAD is suboptimal because it ignores one piece of information from one of the unit vectors. Modern star trackers can track 5, 6, or even 50 stars at a time, creating a need for methods that can use all of the information from two or more unit vectors, weighted in some “optimal” fashion. This need had been anticipated in 1965, when Wahba proposed her famous problem to find the proper orthogonal matrix A (i.e., with determinant 1) that minimizes the loss function [4]

Effective direction measurements for spacecraft attitude: II. Predicted directions

The predicted directions are examined as effective measurements of spacecraft attitude. It is shown that two linearly independent predicted directions are a sufficient statistic for the attitude, and that these two effective measurements can be chosen to be arbitrary directions as long as they are not parallel or antiparallel. However, the correct implementation of the predicted directions in maximum-likelihood estimation of the attitude is complicated by the form of the covariance matrix for each and their mutual correlation. Thus, the predicted directions, which differ greatly from the equivalent direction measurements presented earlier, are not useful as a practical vehicle for attitude data fusion. Unlike the equivalent directions, the predicted directions are always physically meaningful. Unfortunately, they are also almost physically meaningless, as we shall see. Nonetheless, they are of obvious theoretical interest and worthy of a careful examination. The predicted directions can be shown to be “equivalent” measurements for the TRIAD algorithm.