Modified Rodrigues Parameters Research Articles

Adaptive Formation Tracking Control of Multiple Vertical Takeoff and Landing UAVs With Bearing-Only Measurements.

This article studies the leader-following formation tracking control problem of multiple vertical takeoff and landing (VTOL) unmanned aerial vehicles (UAVs) subject to uncertain parameters, in which the target formation configuration is defined by using the interneighbors' bearing vectors. An adaptive formation control algorithm with bearing-only measurements is proposed under a hierarchical control framework. More specifically, a saturated adaptive distributed control force is developed in the position loop with bearing-only measurements. Since the bearing vectors with respect to the neighbors can be directly measured by low-cost onboard cameras, the proposed formation algorithm can be implemented without interagent communication. Subsequently, in the attitude loop, an adaptive hybrid control scheme via two modified Rodrigues parameters (MRPs) sets is proposed, which achieves the command attitude tracking globally and also avoids the unwinding problem of MRPs. Based on the Lyapunov stability analysis and hybrid theory, we prove that the overall closed-loop error system is globally asymptotically stable. Finally, we provide a numerical example to demonstrate the effectiveness of the proposed control algorithm.

Satellite Attitude Control Using Faulty Double Gimbal Variable Speed Control Moment Gyroscope

A rigid satellite attitude control requires stable finite-time attitude convergence in the presence of rotation unwinding, external disturbances, and actuator faults. To achieve this, a new nonsingular fast fixed-time sliding mode controller is proposed to ensure finite-time convergence independent of the initial conditions of the system. Rotation unwinding is addressed using the body and shadow pair of the Modified Rodrigues Parameters (MRPs). Convergence behavior of the states errors in the neighborhood of the MRPs switching surface is also addressed. Attitude control is achieved using a double-gimbal variable-speed control moment gyro (DGVSCMG). For the first time, a DGVSCMG having all possible electrical and mechanical faults is considered for control design. A novel nonsingular fast fixed-time anti-unwinding fault-tolerant sliding control using a new sliding surface is proposed. This eliminates switching of the sliding surfaces and minimizes control chattering. Four new propositions are proposed and proved to establish fast fixed-time global stability of the system for tracking maneuver in fault and fault-free conditions with anti-unwinding. Simulation results are presented and compared with existing fixed-time sliding control approaches. The results show superiority of the proposed anti-unwinding fault-tolerant fast fixed-time control for the tracking maneuver in terms of convergence time, energy efficiency, and fault tolerance.

Negative-imaginary-based spacecraft attitude tracking control using modified Rodrigues parameters

This paper investigates a counter-clock-wise input-output map about spacecraft attitude dynamics and kinematics using Modified Rodrigues Parameters. A gyro-free attitude control scheme is established by constructing a negative imaginary system for spacecraft attitude stabilization. Attitude tracking control scheme is also explored where the negative-imaginary-based controller has no need on the angular velocity measurements. Numerical simulations are presented to show the effectiveness of the proposed attitude control scheme.

Connecting the Equinoctial Elements and Rodrigues Parameters: A New Set of Elements

A geometric interpretation of the equinoctial elements is given with a connection to orthogonal rotations and attitude dynamics in Euclidean 3-space. An identification is made between the equinoctial elements and classic Rodrigues parameters. A new set of equinoctial elements is developed using the modified Rodrigues parameters, thereby removing the coordinate singularity for retrograde equatorial orbits present in previous versions of these elements. A low-thrust trajectory optimization problem is set up using the new elements to numerically verify convergence for the two-point boundary problem, as compared to their predecessors.

Antiunwinding Sliding Mode Control for Rigid Spacecraft Based on Modified Rodrigues Parameters

In this paper, attitude maneuver control without unwinding phenomenon is investigated for rigid spacecraft via modified Rodrigues parameters. A novel switching function is constructed by a hyperbolic sine function to guarantee that the sliding surface contains two stable equilibria. A sliding mode controller is developed to ensure that the sliding surface is reachable. Moreover, a dynamic parameter is designed to guarantee unwinding-free performance of the closed-loop system when the system states outside the sliding surface. In addition, a boundary layer is introduced to deal with the chattering problem. Further, the unwinding-free performance within the boundary layer is guaranteed by imposing some conditions on the parameters of the switching function. Finally, comparison simulations are performed to verify the effectiveness and superiority of the presented control law.

Anti-unwinding adaptive super-twisting attitude control for spacecraft

This article addresses the attitude maneuver control without unwinding based on variable gain(i.e. adaptive) super-twisting algorithm and the modified Rodrigues parameters(MRPs). First, a selection logic for the MRP set and its shadow set is designed in order to construct an attitude control system model for rigid spacecraft. Second, a nonlinear sliding function is presented, and on the sliding phase, finite-time convergence and anti-unwinding performance are guaranteed. As a result, a first order sliding mode controller is created to ensure finite-time convergence and anti-unwinding performance on the reaching phase. To alleviate the chattering and improve convergence speed, an adaptive super-twisting algorithm is presented, and the finite-time convergence and anti-unwinding performance are guaranteed on the reaching phase. Furthermore, the adaptive gains are designed based on dual-layer adaptive law, which allows the gains to be increased and decreased as appropriate. It is further demonstrated that when the controller gains are time-varying according to adaptive law, the convergence property and anti-unwinding performance are guaranteed. Numerical simulations are carried out to illustrate the closed-loop system's performance under the proposed control scheme.

Global stabilization control for multiple input multiple output nonlinear systems with unknown function vector

AbstractIn this article, a control algorithm is proposed to solve the global stabilization control problem of multiple input multiple output (MIMO) nonlinear systems with unknown function vectors (UFVs). Firstly, a Lemma dealing with UFVs is proposed. Then, combined with the backstepping method, the controller is designed such that all signals of the closed‐loop system are globally stable. Compared with the approximation method, the algorithm in this article solves the global stabilization control problem. Compared with the assumptions of UFVs, the algorithm in this article reduces the conservatism problem. At the same time, the algorithm in this article also solves the “explosion of terms” problem of backstepping. Compared with the methods to solve this problem: dynamic surface control (DSC) and direct fuzzy control, the algorithm in this article can make all signals of the closed‐loop system converge to the origin. Finally, the algorithm is applied to the model of spacecraft with modified Rodrigues parameters, and the simulation results show the effectiveness of this algorithm.

An output feedback back-stepping attitude control for rigid satellite

This paper presents the attitude control of a rigid satellite in the presence of the total uncertainty of inertia and external disturbance while only outputs of the system are measurable. Noticing the conditions, a new robust back-stepping control for the modified Rodrigues parameters model of the rigid satellite is proposed, at the same time the upper bound of the norm of the total uncertainty vector is estimated adaptively. Due to the inaccessibility of the angular velocity of the system, a finite-time extended observer is used to estimate the angular velocity and total disturbance. The robust stability of the control system is proved by the Lyapunov theory. Finally, a numerical simulation is presented to demonstrate the effectiveness of the proposed control method.

Design of Sliding Mode Controller based on Radial Basis Function Neural Network for Spacecraft Autonomous Proximity

Since the dynamic model of spacecraft has the characteristics of non-linear, kinematic couplings, uncertainties and nonstationary disturbance, it has become a challenging problem to accurately control the relative position and attitude of the spacecraft. A radial basis function neural network (RBFNN)-based sliding mode controller (SMC) is proposed for trajectory tracking of spacecraft autonomous proximity in this paper. Firstly, a six degree-of-freedom (DOF) relative motion dynamics model is developed for close proximity operations. The modified Rodrigues parameters are applied to solve the problem of singularity. Then, a SMC that does not require accurate model information is designed. RBFNN is used to adaptively eliminated the model uncertainty impacts on the system. Finally, the stability of the relative motion dynamics is proved via Lyapunov stability theory. Simulation results illustrate that the method can attenuate the attitude and position errors, reduce the chattering of the input and decrease the overshoot of the control torque effectively.

Dual Attitude Representations and Kinematics for Six-Degree-of-Freedom Spacecraft Dynamics

This paper focuses on the development of three newly formed representations of dual attitude and their respective kinematics equations - Dual Classical Rodrigues Parameters, Dual Modified Rodrigues Parameters, Dual Principal Axis and Principal Angle (Dual Principal Rotation Vector). These three formulations are developed by building off of the theories and concepts used with Dual Quaternions and utilizing numerous attitude identities for quaternions to create new identities using dual quaternions and dual attitudes while detailing the process used throughout the derivation. The accompanying dual kinematic equations using these dual attitude parameters are also generated using their respective dual attitude coordinate systems. Furthermore, the new dual attitude representations with their respective, distinct kinematic equations and identities are validated against a simulated 6-DOF rigid body spacecraft control problem to verify equivalency across equations using the dual quaternion as a baseline.

Common Frame Dynamics for Conically-Constrained Spacecraft Attitude Control.

Attitude control subjected to pointing constraints is a requirement for most spacecraft missions carrying sensitive on-board equipment. Pointing constraints can be divided into two categories: exclusion zones that are defined for sensitive equipment such as telescopes or cameras that can be damaged from celestial objects, and inclusion zones that are defined for communication hardware and solar arrays. This work derives common frame dynamics that are fully derived for Modified Rodrigues Parameters and introduced to an existing novel technique for constrained spacecraft attitude control, which uses a kinematic steering law and servo sub-system. Lyapunov methods are used to redevelop the steering law and servo sub-system in the common frame for the tracking problem for both static and dynamic conic constraints. A numerical example and comparison between the original frame and the common frame for the static constrained tracking problem are presented under both unbounded and limited torque capabilities. Monte Carlo simulations are performed to validate the convergence of the constrained tracking problem for static conic constraints under small perturbations of the initial conditions. The performance of dynamic conic constraints in the tracking problem is addressed and a numerical example is presented. The result of using common frame dynamics in the constrained problem shows decreased control effort required to rotate the spacecraft.

The error model based on the special Euclidean group SE(3) of the INS: Comparison and extension

In this paper, the error model based on the Lie group is investigated with main focus on the Special Euclidean group SE(3) based error model construction and filtering comparison for the inertial navigation system (INS). The conventional theory of the SE(3) based error model is only focused on the state error definition. Moreover, the attitude representation is usually used by Euler angle. In order to enrich the family of the SE(3) based error model of the INS, this paper is to establish a theoretical bridge that can connect the attitude estimation to the error model construction for the Lie group. For this theoretical bridge, the SE(3) based error model based on Rodrigues parameter and modified Rodrigues parameter is proposed to fill the gap of the family of the SE(3) based error model. Further, based on the integrity of the SE(3) based error model, a uniform error model is also proposed to unify the family of the SE(3) based error model using the attitude transformation and direction cosine matrix. The field tests are carried out to assess the comprehensive performance of the existing SE(3) based error model under different conditions for the INS.

Unwinding-Free Fast Finite-Time Sliding Mode Satellite Attitude Tracking Control

A double-gimbal variable-speed control moment gyro (DGVSCMG) is considered for attitude control of a rigid satellite for track-to-track maneuver. Satellite attitude tracking control involves precision and time limit. In addition, satellite attitude control also faces rotation kinematic singularity, rotation unwinding, and control singularity associated with DGVSCMG. The rotation singularity and unwinding problems are solved using the body and shadow sets of the modified Rodrigues parameters. A nonlinear unwinding and kinematic singularity-free fast finite-time sliding mode tracking attitude control is proposed and designed for the satellite attitude control using a single unit of DGVSCMG. The issue of control singularity is addressed using constrained optimization to avoid gimbal lock. It is proved through five new propositions that the system states errors converge first to the sliding surface and then to the origin in fast finite-time with unwinding-free dynamics showing global stability of the system for the track-to-track maneuver. As a special case, it is shown that the proposed control is also applicable to rest-to-rest maneuver. Simulation results are presented and compared with a finite-time sliding control showing superiority of the proposed fast finite-time control in terms of time of convergence and unwinding and singularity-free angular motion.

An Adaptive Robust Attitude Tracking Control of Quadrotor UAV with the Modified Rodrigues Parameter

In this paper, a novel adaptive attitude tracking control method is investigated to enhance tracking performance and robustness for a quadrotor unmanned aerial vehicle (UAV) against the modeling uncertainties and external disturbances. First, the attitude dynamics of a quadrotor UAV based on the modified Rodrigues parameter (MRP) is derived. Then, an adaptive dynamic surface technique with robust integral of the sign of the error (RISE) control approach are designed to improve the tracking performance and robustness of the quadrotor attitude control system suffering from the uncertainties and disturbances. A prescribed performance function is employed to improve the tracking performance of the quadrotor UAV by restraining the tracking errors range. The stability analysis proves the presented control scheme can guarantee all the close-loop control system signals are bounded and control system is uniformly ultimately bounded. Simulation results are carried out to demonstrate the superior and effective performance of the presented control method.

Multiplexed MPC attitude control of a moving mass satellite using dual-rate piecewise affine model

This paper proposes a Multiplexed MPC algorithm for the dual-rate piecewise affine (PWA) model to control the attitude of a low-Earth-orbit (LEO) moving mass satellite. Although still facing an inherent inertial disturbance problem, the moving mass technology is a promising way to counter the strong aerodynamic torques in LEO. To make the satellite attitude fully actuated, a magnetorquer is used to cooperate with the moving mass system, resulting in a two-input control system. Through a strategy that updates inputs asynchronously, the proposed algorithm effectively balances the inertial disturbances caused by mass motions, the system's dynamic performance, and the controller's computational complexity. Error attitude kinematic and dynamic equations are derived, to convert the tracking problem into an equivalent regulation problem. The PWA approximation is used to transform the nonlinear attitude control equations described by error modified Rodrigues parameter to a linear state-space model for MPC design. Then, a new form of Multiplexed MPC, namely dual-rate PWA-MMPC, is developed for this dual-rate PWA model, which updates two control inputs at different rates. Despite only finding sub-optimal solutions to the original problem by stability analysis, this algorithm outperforms other control strategies through its ability to coordinate multiple inputs, deal with constraints, and reduce computational complexity. Hardware-in-the-loop simulations are conducted to demonstrate the effectiveness of the proposed methodology for the attitude control.

QUEST Aided EKF for Attitude and Rate Estimation Using Modified Rodrigues Parameters

A conventional attitude estimation system for a nanosatellite involves direct input of the attitude sensor measurements to a Kalman filter. However, in case of using an extended Kalman filter (EKF) for the attitude filtering, frequent calculations of the Jacobian matrices bring an excessive computational burden which may not be practical for a nanosatellite on-board computer. In order to deal with this problem, in this study, a QUEST aided EKF attitude and attitude rate estimation system is proposed. QUEST algorithm is used to obtain an initial coarse attitude estimation and then, this estimation is filtered via an EKF. The proposed integrated system reduces the computational burden that an EKF brings since the direct input of the attitude measurements to the filter makes the measurement model linear. For the attitude representation, modified Rodrigues parameters (MRPs) are used unlike widely used quaternions due to the advantages they provide. MRP representation has a singularity at only at the multiples of 2π, therefore, any rotation can be represented by MRPs, except a complete 〖360〗^∘ rotation. This singularity can be easily avoided switching between alternate MRP sets which is also discussed in this study. The performance of the proposed system is tested with several simulations and the results are presented together with the estimation errors and variances.

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.

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.