Symmetric systems with semisimple structure algebra: The quaternionic case
Linear systems over the quaternions is discussed. The theory of real, semisimple symmetric systems is outlined.
Hazewinkel, M.↗
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Simultaneous quaternion estimation (QUEST) and bias determination
Tests of a new method for the simultaneous estimation of spacecraft attitude and sensor biases, based on a quaternion estimation algorithm minimizing Wahba's loss function are presented. The new method is compared with a conventional batch least-squares differential correction algorithm. The estimates are based on data from strapdown gyros and star trackers, simulated with varying levels of Gaussian noise for both inertially-fixed and Earth-pointing reference attitudes. Both algorithms solve for the spacecraft attitude and the gyro drift rate biases. They converge to the same estimates at the same rate for inertially-fixed attitude, but the new algorithm converges more slowly than the differential correction for Earth-pointing attitude. The slower convergence of the new method for non-zero attitude rates is believed to be due to the use of an inadequate approximation for a partial derivative matrix. The new method requires about twice the computational effort of the differential correction. Improving the approximation for the partial derivative matrix in the new method is expected to improve its convergence at the cost of increased computational effort.
Markley, F. Landis↗
Quaternion normalization in additive EKF for spacecraft attitude determination
This work introduces, examines and compares several quaternion normalization algorithms, which are shown to be an effective stage in the application of the additive extended Kalman filter to spacecraft attitude determination, which is based on vector measurements. Three new normalization schemes are introduced. They are compared with one another and with the known brute force normalization scheme, and their efficiency is examined. Simulated satellite data are used to demonstate the performance of all four schemes.
Bar-Itzhack, I. Y.↗
Quaternion normalization in spacecraft attitude determination
Methods are presented to normalize the attitude quaternion in two extended Kalman filters (EKF), namely, the multiplicative EKF (MEKF) and the additive EKF (AEKF). It is concluded that all the normalization methods work well and yield comparable results. In the AEKF, normalization is not essential, since the data chosen for the test do not have a rapidly varying attitude. In the MEKF, normalization is necessary to avoid divergence of the attitude estimate. All of the methods of the methods behave similarly when the spacecraft experiences low angular rates.
Deutschmann, Julie↗
Robust control of nonlinear flexible multibody systems using quaternion feedback and dissipative compensation
Global asymptotic stability of a class of nonlinear multibody flexible space structures under dissipative compensation is established. Two cases are considered. The first case allows unlimited nonlinear motions of the entire system and uses quaternion feedback. The second case assumes that the central body motion is in the linear range although the other bodies can undergo unrestricted nonlinear motion. The stability is proved to be robust to the inherent modeling nonlinearities and uncertainties. Furthermore, for the second case, the stability is also shown to be robust to certain actuator and sensor nonlinearities. The stability proofs use the Lyapunov approach and exploit the inherent passivity of such systems. The results are applicable to a wide class of systems, including flexible space structures with articulated flexible appendages.
Kelkar, Atul G.↗
Three-axis stabilization of spacecraft using parameter-independent nonlinear quaternion feedback
This paper considers the problem of rigid spacecraft. A nonlinear control law which uses the feedback of the unit quaternion and the measured angular velocities is proposed and is shown to provide global asymptotic stability. The control law does not require the knowledge of the system parameters, and is therefore robust to modeling errors. The significance of the control law is that it can be used for large-angle maneuvers with guaranteed stability.
Joshi, Suresh M.↗
Robust control of nonlinear flexible multibody systems using quaternion feedback and dissipative compensation
Global asymptotic stability of a class of nonlinear multibody flexible space-stnuctures under dissipative compensation is established. Two cases are considered. The first case allows unlimited nonlinear motions of the entire system and uses quaternion feedback. The second case assumes that the central body motion is in the linear range although the other bodies can undergo unrestricted nonlinear motion. The stability is proved to be robust to the inherent modeling nonlinearities and uncertainties. Furthermore for the second case the stability is also shown to be robust to certain actuator and sensor nonlinearities. The stability proofs use the Lyapunov approach and exploit the inherent passivity of such systems. The results are applicable to a wide class of systems including flexible space-structures with articulated flexible appendages.
Kelkar, Atul G.↗
Fast Quaternion Attitude Estimation from Two Vector Measurements
Many spacecraft attitude determination methods use exactly two vector measurements. The two vectors are typically the unit vector to the Sun and the Earth's magnetic field vector for coarse "sun-mag" attitude determination or unit vectors to two stars tracked by two star trackers for fine attitude determination. Existing closed-form attitude estimates based on Wahba's optimality criterion for two arbitrarily weighted observations are somewhat slow to evaluate. This paper presents two new fast quaternion attitude estimation algorithms using two vector observations, one optimal and one suboptimal. The suboptimal method gives the same estimate as the TRIAD algorithm, at reduced computational cost. Simulations show that the TRIAD estimate is almost as accurate as the optimal estimate in representative test scenarios.
Markley, F. Landis↗
Waypoint Following Dynamics of a Quaternion Error Feedback Attitude Control System
Closed-loop attitude steering can be used to implement alternative attitude maneuvers by using a conventional attitude control system to track a non-standard attitude profile sampled as a series of sequential commands. The idea has been employed in practice to perform zero-propellant maneuvers on the International Space Station and minimum time maneuvers on NASA’s TRACE space telescope. A challenge for operational implementation of the idea is the finite capacity of a space vehicle’s command storage buffer. One approach to mitigate the problem is to downsample-and-hold the attitude commands so that the attitude control system to follows a set of waypoints. This paper explores the waypoint following dynamics of a quaternion error feedback control law for downsample-and-hold. It is shown that downsample-and-hold induces a ripple between downsamples that causes the satellite angular rate to significantly overshoot the desired limit. Analysis in the z-domain is carried out in order to understand the phenomenon. An interpolating Chebyshev-type filter is proposed that allows attitude commands to be encoded in terms of a small set of filter coefficients. Using the interpolating filter, commands can be issued at the ACS rate but with significantly reduced memory requirements. The attitude control system of NASA’s Lunar Reconnaissance Orbiter is used as an example to illustrate the behavior of a practical attitude control system.
Mark Karpenko↗
Application of A Dual-Quaternion Six Degree-of-Freedom Guidance to Human-Scale Mars Entry, Descent, and Landing
Landing humans on Mars comes with many challenges, including the execution of a safe and precise entry, descent, and landing (EDL) sequence. Various NASA studies have shown that there are a variety of EDL guidance methods that potentially offer solutions to the human-scale EDL precision landing problem and work is ongoing to assess new and novel methods. As part of these ongoing studies, a dual-quaternion-based six degree-of-freedom guidance algorithm was implemented in the Program to Optimize Simulated Trajectories II (POST2), a NASA-and industry-standard spacecraft trajectory and vehicle design tool. This algorithm considers both translational and rotational dynamics and casts the trajectory optimization problem as a quadratically constrained quadratic program (QCQP) with various constraints at discrete nodes throughout the trajectory. The QCQP problem is solved via an alternating direction method of multipliers (ADMM) approach, and the output is a discretized optimal trajectory. The guidance is applied to a NASA reference human-scale Mars EDL system, and results are compared and discussed.
Optimization↗
Application of A Dual-Quaternion Six Degree-of-Freedom Guidance to Human-Scale Mars Entry, Descent, and Landing
Landing humans on Mars comes with many challenges, including the execution of a safe and precise entry, descent, and landing (EDL) sequence. Various NASA studies have shown that there are a variety of EDL guidance methods that potentially offer solutions to the human-scale EDL precision landing problem and work is ongoing to assess new and novel methods. As part of these ongoing studies, a dual-quaternion-based six degree-of-freedom guidance algorithm was implemented in the Program to Optimize Simulated Trajectories II (POST2), a NASA-and industry-standard spacecraft trajectory and vehicle design tool. This algorithm considers both translational and rotational dynamics and casts the trajectory optimization problem as a quadratically constrained quadratic program (QCQP) with various constraints at discrete nodes throughout the trajectory. The QCQP problem is solved via an alternating direction method of multipliers (ADMM) approach, and the output is a discretized optimal trajectory. The guidance is applied to a NASA reference human-scale Mars EDL system, and results are compared and discussed.
Optimization↗
Hardware in the Loop Performance of Terrestrial Powered Descent Dual Quaternion Guidance with a Custom First-Order Solver
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Convex Optimization↗
Spacecraft Robot Kinematics Using Dual Quaternions
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Alfredo Valverde↗
Angular-Rate Estimation Using Quaternion Measurements
In most spacecraft (SC) there is a need to know the SC angular rate. Precise angular rate is required for attitude determination, and a coarse rate is needed for attitude control damping. Classically, angular rate information is obtained from gyro measurements. These days, there is a tendency to build smaller, lighter and cheaper SC, therefore the inclination now is to do away with gyros and use other means and methods to determine the angular rate. The latter is also needed even in gyro equipped satellites when performing high rate maneuvers whose angular-rate is out of range of the on board gyros or in case of gyro failure. There are several ways to obtain the angular rate in a gyro-less SC. When the attitude is known, one can differentiate the attitude in whatever parameters it is given and use the kinematics equation that connects the derivative of the attitude with the satellite angular-rate and compute the latter. Since SC usually utilize vector measurements for attitude determination, the differentiation of the attitude introduces a considerable noise component in the computed angular-rate vector.
Azor, Ruth↗
Closed-form integrator for the quaternion (euler angle) kinematics equations
The invention is embodied in a method of integrating kinematics equations for updating a set of vehicle attitude angles of a vehicle using 3-dimensional angular velocities of the vehicle, which includes computing an integrating factor matrix from quantities corresponding to the 3-dimensional angular velocities, computing a total integrated angular rate from the quantities corresponding to a 3-dimensional angular velocities, computing a state transition matrix as a sum of (a) a first complementary function of the total integrated angular rate and (b) the integrating factor matrix multiplied by a second complementary function of the total integrated angular rate, and updating the set of vehicle attitude angles using the state transition matrix. Preferably, the method further includes computing a quanternion vector from the quantities corresponding to the 3-dimensional angular velocities, in which case the updating of the set of vehicle attitude angles using the state transition matrix is carried out by (a) updating the quanternion vector by multiplying the quanternion vector by the state transition matrix to produce an updated quanternion vector and (b) computing an updated set of vehicle attitude angles from the updated quanternion vector. The first and second trigonometric functions are complementary, such as a sine and a cosine. The quantities corresponding to the 3-dimensional angular velocities include respective averages of the 3-dimensional angular velocities over plural time frames. The updating of the quanternion vector preserves the norm of the vector, whereby the updated set of vehicle attitude angles are virtually error-free.
Whitmore, Stephen A.↗
Method and apparatus for rate integration supplement for attitude referencing with quaternion differencing
A control system for providing attitude control in spacecraft. The control system comprising a primary attitude reference system, a secondary attitude reference system, and a hyper-complex number differencing system. The hyper-complex number differencing system is connectable to the primary attitude reference system and the secondary attitude reference system.
Rodden, John James↗
Attitude Representations for Kalman Filtering
The four-component quaternion has the lowest dimensionality possible for a globally nonsingular attitude representation, it represents the attitude matrix as a homogeneous quadratic function, and its dynamic propagation equation is bilinear in the quaternion and the angular velocity. The quaternion is required to obey a unit norm constraint, though, so Kalman filters often employ a quaternion for the global attitude estimate and a three-component representation for small errors about the estimate. We consider these mixed attitude representations for both a first-order Extended Kalman filter and a second-order filter, as well for quaternion-norm-preserving attitude propagation.
Markley, F. Landis↗
RASPA3
RASPA3, a molecular simulation code for computing adsorption and diffusion in nanoporous materials and thermodynamic and transport properties of fluids. It implements force field based classical Monte Carlo/molecular dynamics in various ensembles. RASPA3 is rewritten from the ground up in C++23 with speed and code readability in mind. Transition-matrix Monte Carlo is added to compute the density of states and free energies. The Monte Carlo code for rigid molecules is based on quaternions, and the atomic positions needed in the energy evaluation are recreated from the center of mass position and quaternion orientation. The expanded ensemble methodology for fractional molecules, with a scaling parameter λ between 0 and 1, now also keeps track of analytic expressions of dU/dλ, allowing independent verification of the chemical potential using thermodynamic integration. The source code is freely available under the MIT license on GitHub.
Dubbeldam, David↗