Rodrigues Parameters Research Articles

A new approach to representations of homothetic motions in Lorentz space

Abstract In this study, Rodrigues parameters have been first calculated for a homothetic rotation around spacelike and timelike axis using one parameter homothetic motions in Lorentz 3-space E 1 3 ${\mathbb{E}}_{1}^{3}$ . The behavior of the vectors during the homothetic rotation, which is the increase or decrease (about size of one of the objects in motion) notion, has been investigated as a three dimensional shape with the help of Cinema 4D program. The behavioral differences have been observed on figures. Then, Lorentz motions that corresponding to the homothetic rotation matrices in terms of spacelike and timelike axes have been examined and expressed in E 1 3 ${\mathbb{E}}_{1}^{3}$ . Some definitions, theorems, corollaries and three dimensional figures have been given in Lorentz 3-space.

Signum-Function-Based Multi-Input Multi-Output Adaptive Homogeneous Finite-Time Control for a Rigid-Body Attitude Tracking

This study is devoted to investigating the robust adaptive finite-time attitude tracking control problem for a rigid body subject to unknown uncertainties. Considering about the nonlinear attitude dynamics for a rigid body, a homogeneous sliding variable is designed by employing the signum-function technique. For the presented homogeneous sliding variable, a real sliding manifold, on which the singularity problem is avoided, could be achieved. Subsequently, a novel signum-function-based MIMO adaptive homogeneous finite-time control (SMAHFTC) algorithm is proposed. Finite-time convergence of the tracking errors to a region around the origin is rigorously proved through the Lyapunov approach. The control gains will not be overestimated, which implies that minimal control gains will be obtained by the adaptation law. Moreover, the controls’ signals are continuous for the SMAHFTC law. Then, by means of the proposed design method, an attitude tracking controller is designed for a rigid body described by the modified Rodrigues parameters’ (MRPs) representation. A comparison simulation is carried out to show the effectiveness and superiority of the proposed method.

Constrained Attitude Maneuvering via Modified-Rodrigues-Parameter-Based Motion Planning Algorithms

The attitude dynamics and control of a spacecraft becomes complex in the presence of constraints in its orientation. Such constraints consist in not pointing sensitive payload toward bright objects in space or, vice versa, keeping the same bright objects within some instrument’s field of view. This paper investigates the applicability of nonsingular attitude motion planning algorithms to navigate a three-dimensional grid in the Modified Rodrigues Parameter configuration space. The aim is to compute a constraint-compliant reference trajectory in Modified Rodrigues Parameter space that can be tracked to reorient a spacecraft from an initial attitude to a final attitude while also attempting at minimizing the required control effort. Nonzero initial and final angular rates are considered to yield a general solution. The path is constructed with B-spline curves that pass through each attitude waypoint. Further, a constant angular maneuver rate is imposed, which influences the speed at which the reference attitude traverses the B-spline solution. Simulations illustrate that constraint-compliant, control-torque-minimizing paths are found if the grid spacing is chosen to be fine enough.

Three‐dimensional weakly compressible moving particle simulation coupled with geometrically nonlinear shell for hydro‐elastic free‐surface flows

AbstractA three‐dimensional fluid–structure interaction solver based on an improved weakly compressible moving particle simulation (WC‐MPS) method and a geometrically nonlinear shell structural model is developed and applied to hydro‐elastic free‐surface flows. The fluid–structure coupling is performed by a polygon wall boundary model that can handle particles and finite elements of distinct sizes. In WC‐MPS, a tuning‐free diffusive term is introduced to the continuity equation to mitigate nonphysical pressure oscillations. Discrete divergence operators are derived and applied to the polygon wall boundary, of which the numerical stability is enhanced by a repulsive Lennard–Jones force. Additionally, an efficient technique to deal with the interaction between fluid particles placed at opposite sides of zero‐thickness walls is proposed. The geometrically nonlinear shell is modeled by an unstructured mesh of six‐node triangular elements. Finite rotations are considered with Rodrigues parameters and a hyperelastic constitutive model is adopted. Benchmark examples involving free‐surface flows and thin‐walled structures demonstrate that the proposed model is robust, numerically stable and offers more efficient computation by allowing mesh size larger than that of fluid particles.

Multiplicative modified Rodrigues-parameters-based strong tracking unscented Kalman filter for satellite attitude estimation

In this paper, an improved strong tracking unscented Kalman filter based on multiplicative modified Rodrigues parameters (MRPs) is proposed for satellite attitude estimation. The multiplicative MRPs are superior to additive MRPs in terms of attitude representation, especially when attitude angles are large. By minimizing the loss function in Wahba’s problem, a novel method of weighted average for MRPs is derived to replace the simple procedure. The generation of Sigma points, update of state variables, and calculation of covariance matrices are all different from the existing literature to maintain the multiplicative property of MRPs. Simulation results by raw telemetry data from the on-orbit CubeSat Enlai-1 demonstrate the excellent performance of the proposed filter under large attitude angles.

Dynamics of an autonomous spacecraft control system at initial transition to a tracking mode

The problems of autonomous digital control of the information satellites and space robots during their initial transition to a tracking mode, namely in the initial orientation modes, are considered. Autonomous angular guidance and modularly limited vector digital control using a vector of the modified Rodrigues parameters are applying to bring the spacecraft’s orientation from completely arbitrary to the required one. The developed methods, algorithms and simulation results for a mini-satellite in a sun-synchronous orbit are presented.

SINS attitude algorithm based on moving-window overdetermined polynomial fitting of gyro outputs

The attitude algorithm is the most important part of the whole strapdown inertial navigation processing. It calculates the attitude of certain parameterization by integrating the gyro outputs or measurements in a specifically tailored way according to the attitude kinematic differential equation. The measurements or some angular velocity models obtained by fitting these measurements are often assumed free of errors in order to assess the numerical errors only. However, the gyro outputs and hence the models from them are by no means free of measurement errors. It is more often than not that the measurement errors dominate the numerical ones in practice. In this study, with coping with the measurement errors as the focus, we aim to improve the angular velocity model which is used as input in an attitude integration algorithm. This is achieved by exploiting the potential of overdetermined least-squares polynomial fitting. In order to avoid reducing the update rate by incorporating more measurements, the moving window trick is employed to re-use measurements in the previous update interval. The conventional attitude algorithm with second-order approximation in solving the differential equation of the equivalent rotation vector is employed as an example; however, the proposed method can be readily applied to other parameterizations such as direction cosine matrix, quaternion or Rodrigues parameters, and other high order approximations in solving the differential equation widely studied recently.

Predefined-time control for free-floating space robots in task space

This work mainly studies the position and attitude tracking control of a free-floating space robot. With the attitude represented in modified Rodrigues parameters (MRPs), a task-space controller with predefined-time stability is developed considering the external disturbance. The tuning parameters of a predefined-time controller can be formulated as functions of the prescribed upper bound of the stabilization time. Based on the backstepping technique and a novel predefined-time stabilizing function, a predefined-time control scheme is designed for the space robot system. Moreover, to avoid ’explosion of terms’, an auxiliary variable is introduced such that the controller is independent of the derivative of the virtual control law. Numerical simulations are presented to demonstrate the effectiveness of the proposed method.

Dual Quaternion Particle Filtering for Pose Estimation

This article presents a particle filter for pose estimation using unit dual quaternion kinematics. The eight-parameter unit dual quaternion is used for global representation of the pose, whereas the six parameters of the dual modified Rodrigues parameters (MRPs) are used for local pose representation in the state-space model. The dual MRPs enable estimates of the mean and the covariance to be calculated from the particles without violating the algebraic constraint of the unit dual quaternion. For verification of the filter and comparison with state of the art, we consider pose measurements available in the form of unit dual quaternions. Angular velocity and specific force measurements from a body-mounted inertial measurement unit are also considered in the filtering. We show through simulations that the suggested particle filter has comparable accuracy with a previously proposed unscented Kalman filter based on unit dual quaternions. We also consider a practical application where the pose of an arbitrary rigid object is estimated using a sequence of point clouds from a 3-D camera. A model point cloud of the object is displaced with the unit dual quaternion associated with each particle, and a fitting score is calculated between the displaced model point cloud and the measured point cloud from the 3-D camera. The likelihoods of the fitting scores are calculated from an exponential distribution and are used to update the weights of the particles. The system was verified in an experiment where the motion of a swinging payload hanging from a crane was estimated using a 3-D camera.

Immersion and invariance based adaptive attitude control of asteroid-orbiting spacecraft using modified Rodrigues parameters

The attitude control of an asteroid-orbiting spacecraft based on immersion and invariance (I&I) theory is the subject of this paper. It is assumed that the moment of inertia matrix and the gravitational parameters are not known. The objective is to attain nadir pointing attitude on an elliptic orbit. First, based on the I&I principle, an adaptive attitude control system using the Modified Rodrigues Parameters (MRPs) is derived. Through the Lyapunov stability analysis, the asymptotic convergence of the MRP trajectories to the origin is established. Interestingly, in contrast to traditional adaptive systems, the trajectories of the closed-loop system converge to an attractive manifold in an extended state space. Then, for the purpose of comparison, this MRPs-based control law and a quaternions-based control system (also designed using the I&I principle) are simulated for the attitude control of the spacecraft in eccentric orbits around asteroid 433 Eros. It is observed that while each I&I-based control law can accomplish precise attitude control, for identical design parameters in these two control systems, the MRPs-based control law requires smaller control magnitude and accomplishes smoother convergence of trajectories to the attractive manifold, but requires larger settling time for the attitude trajectories, compared with the quaternions-based adaptive control law.

Математические модели углового движения космических аппаратов и их использование в задачах управления ориентацией

Предложена общая структура кинематических уравнений эволюции ориентации космического аппарата (КА) (системы координат, связанной с КА (ЗСК)) относительно опорной системы координат (ОСК). Предполагается, что начала систем координат совпадают и расположены в произвольной точке КА. Каждая из систем координат оборачивается произвольной абсолютной угловой скоростью (скоростью относительно инерциального пространства), заданной проекциями на их оси. В качестве параметров ориентации могут использоваться углы Эйлера-Крылова, параметры Родрига-Гамильтона, модифицированные параметры Родри-га. Показано, что известные представления уравнений эволюции ориентации ССК относительно ОСК в параметрах Родрига-Гамильтона (компонентах нормируемых кватернионов) могут быть получены из решения задачи Н.П. Ерюгина — отыскание всего множества дифференциальных уравнений, имеющих заданный интеграл движения. Отмечены преимущества и недостатки использования каждого из указанных параметров ориентации. Предложен общий для всех этих уравнений метод синтеза управления ориентацией, основанный на декомпозиции исходной задачи на кинематическую и динамическую задачи и использовании известных обобщений прямого метода Ляпунова для их решения. С помощью компьютерного моделирования проиллюстрировано свойство структурной грубости в смысле А.А. Андронова-Л.С. Понтрягина полученного алгоритма. А именно, на конкретном примере проиллюстрирована свойство даже намеренно структурно упрощенного алгоритма стабилизации заданной постоянной ориентации КА с достаточной точностью отслеживать программу ее изменения. Задача наблюдения типична в управлении стыковкой КА, спуском КА с орбиты, выполнении маршрутных съемок поверхности Земли.

Optimal Factorization of the State-Dependent Riccati Equation Technique in a Satellite Attitude and Orbit Control System

The satellite attitude and orbit control system (AOCS) can be designed with success by linear control theory if the satellite has slow angular motions and small attitude maneuver. However, for large and fast maneuvers, the linearized models are not able to represent all the perturbations due to the effects of the nonlinear terms present in the dynamics and in the actuators (e.g., saturation). Therefore, in such cases, it is expected that nonlinear control techniques yield better performance than the linear control techniques. One candidate technique for the design of AOCS control law under a large maneuver is the State-Dependent Riccati Equation (SDRE). SDRE entails factorization (that is, parameterization) of the nonlinear dynamics into the state vector and the product of a matrix-valued function that depends on the state itself. In doing so, SDRE brings the nonlinear system to a (nonunique) linear structure having state-dependent coefficient (SDC) matrices and then it minimizes a nonlinear performance index having a quadratic-like structure. The nonuniqueness of the SDC matrices creates extra degrees of freedom, which can be used to enhance controller performance, however, it poses challenges since not all SDC matrices fulfill the SDRE requirements. Moreover, regarding the satellite's kinematics, there is a plethora of options, e.g., Euler angles, Gibbs vector, modified Rodrigues parameters (MRPs), quaternions, etc. Once again, some kinematics formulation of the AOCS do not fulfill the SDRE requirements. In this paper, we evaluate the factorization options (SDC matrices) for the AOCS exploring the requirements of the SDRE technique. Considering a Brazilian National Institute for Space Research (INPE) typical mission, in which the AOCS must stabilize a satellite in three-axis, the application of the SDRE technique equipped with the optimal SDC matrices can yield gains in the missions. The initial results show that MRPs for kinematics provides an optimal SDC matrix.

Anti-unwinding sliding mode attitude control via two modified Rodrigues parameter sets for spacecraft

In this paper, the attitude maneuver control without unwinding for rigid spacecraft is addressed based on the modified Rodrigues parameters (MRPs). First of all, an attitude control system model for rigid spacecraft is constructed by designing a selection logic for the MRP set and its shadow set. To guarantee the asymptotic stability in the large and unwinding-free performance of the closed-loop system on the sliding phase, a sliding function is presented. Then, a sliding mode control law is developed to ensure that the system states are driven to the sliding surface. Moreover, a dynamic parameter is designed in the proposed control scheme, such that the unwinding phenomenon is avoided when the system states are outside the sliding surface. Further, a boundary layer is introduced for the constructed controller to avoid the chattering phenomenon. It is also proven that the convergence property and anti-unwinding performance are ensured when the system states are within the boundary layer. Numerical simulations are performed to demonstrate the unwinding-free performance of the closed-loop system under the proposed control scheme.

Observer-based fixed-time tracking control for space robots in task space

The problem of task-space tracking control of a free-floating space robot is addressed in this paper. Based on a fixed-time extended state observer (FXESO), a fixed-time position and attitude control (FXPAC) method is developed. The attitude of the end-effector is represented in modified Rodrigues parameters (MRPs). The FXESO provides the estimation of the joint velocity and the lumped disturbance for the control system. Based on the backstepping technique and a power integrator, the FXPAC method is designed. This control scheme can guarantee tracking errors converging to a neighborhood of the origin within a fixed time in the presence of the external disturbance. Moreover, since the inverse of the generalized Jacobian matrix is involved in the controller, it is necessary to consider dynamic singularities. To handle these singularities, a method of combining the singularity separation and damped reciprocal (SSDR) is applied to the space robot. Numerical simulations are presented to demonstrate the effectiveness of the proposed method.

Trajectory Design via Convex Optimization for Six-Degree-of-Freedom Asteroid Powered Landing

This paper presents a convex programming algorithm based on modified Rodrigues parameters for a six-degree-of-freedom asteroid powered landing trajectory design problem. The trajectory design problem is formulated as a nonconvex fuel-optimal control problem with nonlinear coupled translational and rotational dynamics, nonconvex state and control constraints. In order to tackle the fuel-optimal control problem in the convexification framework, the original nonconvex continuous-time optimization problem is converted into a convex discrete-time optimization subproblem by linearizing and discretizing the dynamics and constraints. The effect of the coupling of translational and rotational dynamics on the convexification of nonconvex control constraints is discussed. The successive convexification method of solving a sequence of constrained convex subproblem is used to generate the optimal trajectory. The validity of the proposed algorithm for generating fuel-optimal trajectories and the effect of asteroid gravity on generated trajectories are examined through simulations of landing on different irregular asteroids. The Monte Carlo simulation is performed to examine the calculation performance of the proposed algorithm comparing to the Radau pseudospectral method algorithm and the proposed algorithm with the rotational motion parameterized by quaternions.

Uniform Nonlinear Error Model Based on Gibbs Parameter for the INS

In this paper, a uniform form for the nonlinear error model is proposed for the inertial navigation system (INS). A modified quaternion-based nonlinear error model is derived using the uniform coordinate frame for the INS. Based on the modified quaternion-based nonlinear error model, a uniform form of the nonlinear error model is proposed based on the Gibbs parameter. The uniform form can present the nonlinear error model based on low-order Rodrigues parameters or Rodrigues-like parameters for the INS. Field tests are carried out to assess the performance of the existing typical nonlinear error models.

Attitude dynamics of a rigid body in Keplerian motion

This paper studies the attitude dynamics of a rigid body in a Keplerian orbit. We show that the use of Classical Rodrigues Parameters for the attitude motion of the rigid body subject to gravity-gradient torques enables us to characterize the equilibria associated with the rotational motion about its mass center. A parametric study of the stability of equilibria is conducted to show that large oscillations are induced due to the energy exchange between the pitch and roll–yaw motions, specifically near the 2:1 resonant commensurability regions. A visualization tool is developed to study these pitch oscillations and gain insight into the rigid body motion near internal resonance conditions. A measure of coupling between the pitching and roll–yaw motions is developed to quantify the energy exchange utilizing information from the state transition matrix.

Autonomous Control of a Small Spacecraft in Initial Orientation Modes

Abstract The problems of autonomous digital control of the small information satellites and space robots in the initial orientation modes are considered. Autonomous angular guidance and modularly limited vector digital control using a vector of the modified Rodrigues parameters are applying to bring the spacecraft’s orientation from completely arbitrary to the required one. The developed methods, algorithms and simulation results for a mini-satellite in a sun-synchronous orbit are presented.

Robust Nonsingular Terminal Sliding Mode Attitude Control of Satellites

This paper proposes a time-varying nonsingular terminal sliding mode control (TVNTSMC) law to control the attitude of rigid spacecraft. It provides an advantage for users to stipulate convergence time beforehand, and the proposed control algorithm will ensure global robustness to uncertainties as well as to external disturbances. A minimal (three-dimensional) modified Rodrigues parameter (MRP) explanation is used to represent the dynamics of a rigid satellite; then, the state-space representation of the dynamics is developed by transforming the dynamics into Lagrangian form. MRPs are used to design a new nonlinear sliding surface. The attitude control algorithm is expressed by a piecewise continuous function with time-varying properties. The Lyapunov theory is used to calculate the convergence and settling time to reach the equilibrium point. The time-varying function parameters are derived from convergence time prespecified by the user, which will ensure the system trajectories converge to the origin within the time specified. The global finite-time stability of the overall system is presented using the Lyapunov theorem in the presence of external disturbance. The simulation results are presented for different convergence times as input by the user to show the efficacy of the proposed controller.

Performance evaluation of low-cost GPS/INS in-motion alignment model under ECEF frame

ABSTRACT Low-cost GPS/INS integration system is the ideal combination of navigation and positioning. However, the sensitivity of low-cost INS is not good enough for the initial alignment in statics base before navigation. Considering this problem, this paper presents an arbitrary misalignment angle error propagation model which does not rely on small misalignment angles assumption and two simplified versions. These models are presented in the ECEF frame approach and are suitable to implement the in-motion alignment with GPS aided. Another three error models based on quaternion, Rodrigues parameters and modified Rodrigues parameters were also proposed. Three experiments were designed to verify the accuracy and computational efficiency of these error models. The experiments’ results showed that the small misalignment angle model was applicable to the small misalignment angle for higher computational efficiency but without improvement of accuracy. And the large misalignment angle is the best error model for the initial alignment of the arbitrary misalignment angle.