Relative Dynamics Equations Research Articles

A new approach to derive a formation flying model in the presence of a perturbing body in inclined elliptical orbit: relative hovering analysis

In this paper, the modeling of satellites’ relative motion has been investigated in the presence of a third body and with consideration of main-body’s obliquity. The proposed model is based on non-simplified perturbed relative dynamic equations of satellites and combined with perturbed target orbit. From a new point of view, in the present work, the relative dynamic equations of a disturbing body in an elliptic inclined three-dimensional orbit are derived using Lagrangian mechanics. The Main-body Center based Relative Motion (MCRM) model has been introduced for validating the presented model. Furthermore, for relative hovering analysis, the parameters such as the fuel consumption, proper positioning of follower, minimum and maximum values of required thrust, and the proper timing for performing the hovering operation based on the target satellite’s position have been addressed in this study. Also, the effects of eccentricity and inclination of the third body on the abovementioned parameters have been studied. The findings of this research show the effect of obliquity on the relative hovering. In this study, several useful practical examples related to the Earth–Moon system have been provided for analyzing the abovementioned factors. Furthermore, the hybrid IWO/PSO algorithm has been applied to find the desired hovering location, and also the minimum and maximum amounts of thrust.

LEO cooperative multi-spacecraft refueling mission optimization considering J2 perturbation and target’s surplus propellant constraint

The optimization of an LEO cooperative multi-spacecraft refueling mission considering the J2 perturbation and target’s surplus propellant constraint is studied in the paper. First, a mission scenario is introduced. One service spacecraft and several target spacecraft run on an LEO near-circular orbit, the service spacecraft rendezvouses with some service positions one by one, and target spacecraft transfer to corresponding service positions respectively. Each target spacecraft returns to its original position after obtaining required propellant and the service spacecraft returns to its original position after refueling all target spacecraft. Next, an optimization model of this mission is built. The service sequence, orbital transfer time, and service position are used as deign variables, whereas the propellant cost is used as the design objective. The J2 perturbation, time constraint and the target spacecraft’s surplus propellant capability constraint are taken into account. Then, a hybrid two-level optimization approach is presented to solve the formulated mixed integer nonlinear programming (MINLP) problem. A hybrid-encoding genetic algorithm is adopted to seek the near optimal solution in the up-level optimization, while a linear relative dynamic equation considering the J2 perturbation is used to obtain the impulses of orbital transfer in the low-level optimization. Finally, the effectiveness of the proposed model and method is validated by numerical examples.

Shared Control of Teleoperation Rendezvous and Docking in Lunar Orbit

AbstractTeleoperation rendezvous and docking (RVD) in the lunar orbit may not reach the required RVD accuracy because of the inherent time delay in the communication link and the measurement error. To solve this problem, the shared control strategy for teleoperation RVD based on the potential field function was studied in this work. On the basis of the relative dynamic equations, a predictive display model to depict the predictive ability of the operator was established to overcome the time delay effect. The automatic mode was designed based on the potential field function, which regarded the relative distance and speed with respect to the safe corridor as reference quantities. In addition, this article outlines the concept of shared control weighting coefficient and its derivation on the basis of the influencing factors. Finally, a validation experiment was conducted on a nine-degree teleoperation RVD simulator. Results showed that the shared control method can overcome the influence of time delay in the...

Integrated Guidance and Control Method for the Interception of Maneuvering Hypersonic Vehicle Based on High Order Sliding Mode Approach

This paper focuses on the integrated guidance and control (IGC) method applied in the interception of maneuvering near space hypersonic vehicles using the homogeneous high order sliding mode (HOSM) approach. The IGC model is derived by combining the target-missile relative motion and dynamic equations. Then, a fourth-order sliding mode controller is implemented in the augmented IGC model. To estimate the high order derivatives of the sliding manifold which is required in the HOSM method, an Arbitrary Order Robust Exact Differentiator is presented. At last, the idea of virtual control is introduced to alleviate the chattering of the control input without using any saturation functions which may lead to a loss of the robustness. And the stability of the closed-loop system with presented fourth-order homogeneous HOSM controller is also proved theoretically. Finally, simulation results are provided and analyzed to demonstrate the effectiveness of the proposed method in three typical engagement scenarios.

Oscillation and nonoscillation for second order neutral dynamic equations with positive and negative coefficients on time scales

Abstract We investigate oscillation and nonoscillation of certain second order neutral dynamic equations with positive and negative coefficients. We apply the results from the theory of lower and upper solutions for related dynamic equations along with some additional estimates on positive solutions and use different techniques to obtain some oscillatory theorems. Also, we apply Kranoselskii’s fixed point theorem to obtain nonoscillatory results and then give two sufficient and necessary conditions for the equations to be oscillatory. Some interesting examples are given to illustrate the versatility of our results.

공중재급유를 위한 상대운동방정식 유도 및 검증

This paper addresses the derivation of 6-DOF equation of Tanker and Receiver's aircraft for aerial refueling. The new set of nonlinear equations are derived in terms of the relative translational and rotational motion of receiver aircraft respect to the tanker aircraft body frame. Further the wind effect terms due to the tanker's turbulence are included. The derivation of absolute dynamic equation for tanker aircraft written in the inertial frame is calculated from the relative dynamics equations of receiver. The derived relative and absolute equations are implemented the simulation in the same flight conditions to verify the relative motion and compare the trim results by using the MATLAB/SIMULINK program.

Ideal proportional navigation for exoatmospheric interception

Ideal proportional navigation (IPN) is a natural choice for exoatmospheric interception for its mighty capture capability and ease of implementation. The closed-form solution of two-dimensional ideal proportional navigation was conducted in previous public literature, whereas the practical interception happens in the three-dimensional space. A novel set of relative dynamic equations is adopted in this paper, which is with the advantage of decoupling relative motion in the instantaneous rotation plane of the line of sight from the rotation of this plane. The dimension-reduced IPN is constructed in this instantaneous plane, which functions as a three-dimensional guidance law. The trajectory features of dimension-reduced IPN are explored, and the capture regions of dimension-reduced IPN with limited acceleration against nonmaneuvering and maneuvering targets are analyzed by using phase plane method. It is proved that the capture capability of IPN is much stronger than true proportional navigation (TPN), no matter the target maneuvers or not. Finally, simulation results indicate that IPN is more effective than TPN in exoatmospheric interception scenarios.

Smith fuzzy control of teleoperation rendezvous with variable time delay

The control methods of ground-based teleoperation rendezvous with variable time delay are surveyed. With the help of root locus method, the influences of time delay on the stability and performance of the control system are analyzed. Then, a time-delay compensator is designed to transfer the variable delay into a constant. A predictive model is established based on the relative dynamic equations. After that, a multivariable Smith predictor is designed based on the principle of Smith predictor, which makes the control input state variables independent of variable time delay. At the same time, the fuzzy control method is introduced to alleviate the uncertainties in the teleoperation rendezvous system. Finally, semi-physical simulations are carried out to verify the methods presented in this paper. Simulation results show that time-delay compensation and Smith-Fuzzy control are effective in alleviating the variable time delay and model uncertainties in the process of teleoperation rendezvous, and the success probability and control accuracy can be improved.

Studies on the Interaction Mechanism between Lithium Chloride and Amorphous Al(OH)<sub>3</sub>

Experiments on the interaction mechanism between lithium chloride and amorphous Al(OH)3 were designed and conducted applying simulated brines and aqueous solutions, respectively. The concentrations of lithium ion in the different solutions were determined and XRD patterns of solids formed between lithium chloride and amorphous Al(OH)3 were obtained. The data were fitted according to related dynamic equations. The results show that the variation of the concentration of lithium ion with time obeys the first order reaction dynamic equation, and the mechanism of the interaction should be chemical reaction between lithium chloride and amorphous Al(OH)3. Li/Al ratio should be 1:2 in the formula of the solid products.

Attitude cooperative control of spacecraft formation via output‐feedback

PurposeThe purpose of this paper is to propose a decentralized output feedback controller for cooperative attitude regulation of spacecraft formation in absence of angular velocity feedback.Design/methodology/approachThe nonlinear relative attitude dynamic and kinematic equations represented by relative quaternion and relative angular velocity, respectively, are considered in this paper. The lead filter is employed to synthesize virtual angular velocity signal so that the design of output feedback controller is achieved. Lyapunov method is adopted to prove the stability of closed‐loop system. Considering the external disturbance, the theory of L2‐gain disturbance attenuation is employed to improve the designed controller. Numerical simulations are carried out to verify the controllers proposed.FindingsIt is found that the closed‐loop system can be guaranteed asymptotically stable in absence of external disturbance. When disturbance is considered, as long as the sufficient condition proposed is satisfied, the improved controller can render system uniformly ultimately bounded stable.Practical implicationsThe proposed output feedback control scheme can be considered as a fall‐back alternative for the case that the angular velocity sensors fail, or seen as another option for the system without angular velocity sensors at all.Originality/valueUnlike most classical works in the field of output feedback which focus on centralized scheme and neglect the disturbance, the controller proposed in this paper is able to handle the output feedback control problem of multi‐agent formation in a decentralized fashion, so as to avoid the single failure point of a centralized scheme. Meanwhile, the capability of L2‐gain disturbance attenuation is also achieved simultaneously.

Optimal multi-objective trajectory design based on close-looped control for autonomous rendezvous

This paper considers the problem of optimal multi-objective trajectory design for autonomous rendezvous. Total velocity cost and relative state robustness of close-looped control are selected as the objective functions. Based on relative dynamics equations, the state equations and measurement equations for angles-only relative navigation between spacecrafts are set forth. According to the method of linear covariance analysis, the close-looped control covariance of the true relative state from the reference relative state is analyzed, and the objective functions of relative state robustness are formulated. Considering the total velocity cost and the relative state robustness, the multi-objective optimization algorithm of NSGA-II is employed to solve this multi-impulsive rendezvous problem. Lastly, the validity of the objective functions and the covariance results are demonstrated through 100 times Monte Carlo simulation.

Spacecraft Attitude and Orbit Coupled Nonlinear Adaptive Synchronization Control

Precise dynamic model of spacecraft is essential for the space missions, to be completed successfully. Nevertheless, the independent orbit or attitude dynamic models can not meet high precision tasks. This paper developed a 6-DOF relative coupling dynamic model based upon the nonlinear relative motion dynamics equations and attitude kinematics equations described by MRP. Nonlinear synchronization control law was designed for the coupled nonlinear dynamic model, whose close-loop system was proved to be global asymptotic stable by Lyapunov direct method. Finallly, the simulation results illustrate that the nonlinear adaptive synchronization control algorithm can robustly drive the orbit and attitude errors to converge to zero.

Differential geometric modeling of guidance problem for interceptors

It is a comparatively convenient technique to investigate the motion of a particle with the help of the differential geometry theory, rather than directly decomposing the motion in the Cartesian coordinates. The new model of three-dimensional (3D) guidance problem for interceptors is presented in this paper, based on the classical differential geometry curve theory. Firstly, the kinematical equations of the line of sight (LOS) are gained by carefully investigating the rotation principle of LOS, the kinematic equations of LOS are established, and the concepts of curvature and torsion of LOS are proposed. Simultaneously, the new relative dynamic equations between interceptor and target are constructed. Secondly, it is found that there is an instantaneous rotation plane of LOS (IRPL) in the space, in which two-dimensional (2D) guidance laws could be constructed to solve 3D interception guidance problems. The spatial 3D true proportional navigation (TPN) guidance law could be directly introduced in IRPL without approximation and linearization for dimension-reduced 2D TPN. In addition, the new series of augmented TPN (APN) and LOS angular acceleration guidance laws (AAG) could also be gained in IRPL. After that, the differential geometric guidance commands (DGGC) of guidance laws in IRPL are advanced, and we prove that the guidance commands in arc-length system proposed by Chiou and Kuo are just a special case of DGGC. Moreover, the performance of the original guidance laws will be reduced after the differential geometric transformation. At last, an exoatmospheric interception is taken for simulation to demonstrate the differential geometric modeling proposed in this paper.

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.

Nonlinear relative position control of precise formation flying using polynomial eigenstructure assignment

A nonlinear relative position control algorithm is designed for spacecraft precise formation flying. Taking into account the effect of J 2 gravitational perturbations and atmospheric drag, the relative motion dynamic equation of the formation flying is developed in a quasi-linear parameter-varying (QLPV) form without approximation. Base on this QLPV model, polynomial eigenstructure assignment (PEA) is applied to design the controller. The resulting PEA controller is a function of system state and parameters, and produces a closed-loop system with invariant performance over a wide range of conditions. Numerical simulation results show that the performance can fulfill precise formation flying requirements.

Double line-of-sight measuring relative navigation for spacecraft autonomous rendezvous

Abstract This paper presents a double line-of-sight (LOS) measuring relative navigation technique for far or medium range (tens of kilometers ∼ several kilometers) autonomous rendezvous. In this new relative navigation technique, two chase spacecraft with inter-spacecraft crosslink form a measuring baseline relative to the target spacecraft in space and, respectively, measure LOS angles of the target and their own inertial states. The processing noise model of the relative navigation system is developed based on the spacecraft relative dynamics equations described in inertial frame. The measurement model including LOS angles of the target and the bearing and length of the measuring baseline is established. A discrete-time extended Kalman filter (EKF) is developed to estimate the relative motion of the target. The degree of observability of the navigation system is analyzed by the condition number. Numerical simulation results indicate that: (1) the relative position and velocity of the target can be estimated with sufficient accuracy using the double LOS measuring relative navigation technique, (2) compared with that of single LOS angles-only navigation, the observability of double LOS measuring relative navigation is improved, (3) the separation angle between two LOS vectors influences the estimation errors and the observability of the navigation system.

Asymptotic behavior of second-order nonlinear dynamic equations on time scales

In this paper, we consider the second-order nonlinear dynamic equation $(p(t)y^{\Delta}(t))^{\Delta}+f(t, y^{\sigma})g(p(t)y^{\Delta})=0,$ on a time scale $\mathbb{T}$. Our goal is to establish necessary and sufficient conditions for the existence of certain types of solutions of this dynamic equation. We apply results from the theory of lower and upper solutions for related dynamic equations and use several results from calculus.

Characteristics of a Reverse Guided Structure for Vibration Suppression

In this paper, based on the concept of vibration waves' guiding and suppression, we present mechanisms that can guide the vibration wave in the reverse direction and apply these mechanisms to design a reverse guided structure for vibration suppression system. The related dynamic equations are derived. Both the frequency responses and the transient responses are conducted from the dynamic model. The simulation results are good in showing the characteristics of the mechanism and the designed system of a reverse guided structure for vibration suppression. After observing the effects of system parameter variation on the performance of vibration suppression, we conclude some design guidelines for implementation in future design references.

The bifurcation of a kind of relative rotational dynamic equation with hysteresis and its approximate solution

Based on the Lagrange function, a new nonlinear relative rotational equation with Davidenkov hysteresis is established. Firstly, the bifurcation characteristics of the hysteretic relative rotational autonomous function and non-antonomous function are discussed. Secondly, the approximate solution of the nonlinear function under periodic force excitation is obtained by KBM method. At last, by numerical simulation, several bifurcation structures are obtained, and the comparisons result indicate the approximate solution of KBM method has higher accuracy and better reflects the dynamic characteristic of equation effectively.

Effects of an absorber on impact characteristics in machine cushion design with area ratio modified guiding structure

A machine cushion with guiding structure can transfer impact and vibration waves into branches of balanced forces along pre-designed paths, to reduce the effects of those waves between the machine and the ground. To improve impact load absorption and vibration wave suppression, we designed a machine cushion with an attached absorber and “area ratio modified” (ARM) guiding structure. We compared variations in vibration using impulse responses of machine cushions as a performance index, of four alternative machine cushion designs: One was an ARM with absorber design; the second was the same machine cushion but without an absorber; the third and fourth were simple passive machine cushions with and without an absorber. The related dynamic equations for these structures were derived and the related transient responses and frequency responses were evaluated. The main objectives of the study were to demonstrate the capacity of our ARM with absorber structure to absorb impact forces and vibration waves, and to measure the effect of parameter variation on the absorption of impact forces and vibration waves. As expected, simulation results showed that under test conditions, the proposed ARM with absorber machine cushion had the best impact and vibration suppression capability.