Relative Dynamics Equations Research Articles

Gauss Principle of Least Compulsion for Relative Motion Dynamics and Differential Equations of Motion

This paper focuses on Gauss principle of least compulsion for relative motion dynamics and derives differential equations of motion from it. Firstly, starting from the dynamic equation of the relative motion of particles, we give the Gauss principle of relative motion dynamics. By constructing a compulsion function of relative motion, we prove that at any instant, its real motion minimizes the compulsion function under Gaussian variation, compared with the possible motions with the same configuration and velocity but different accelerations. Secondly, the formula of acceleration energy and the formula of compulsion function for relative motion are derived because the carried body is rigid and moving in a plane. Thirdly, the Gauss principle we obtained is expressed as Appell, Lagrange, and Nielsen forms in generalized coordinates. Utilizing Gauss principle, the dynamical equations of relative motion are established. Finally, two relative motion examples also verify the results' correctness.

Extended Kalman filters for close-range navigation to noncooperative targets

This work presents a set of dynamic filters for estimating the relative roto-translational state and the main parameters of a noncooperative target from an observing chaser satellite during close proximity operations. The proposed different options address a wide range of design possibilities for the architecture of the relative navigation system. All filters are derived from a common, general, core shaped as a dynamic multiplicative extended Kalman filter using dual quaternions. This allows exploiting the advantages of handling the pose (i.e., attitude and position) in a multiplicative fashion, while improving the accuracy in the estimation of the angular and linear relative velocities, as well as enabling the estimation of some meaningful parameters of the target spacecraft (e.g., the ratios of the moments of inertia, position and orientation of the principal axes frame). Moreover, by adopting relative kinematics and dynamics equations in dual quaternions, the inherent coupling of the six degrees-of-freedom motion is addressed with no approximations. All filters take as observations only the noisy pose measurements from an electro-optical device. For each proposed formulation, numerical simulations are carried out to show the behavior of the filter within a scenario representative of close-range target inspection at conclusion of the mid-range rendezvous.

Integrated Dynamics of Space Rigid-Flex Combination System with Time-Varying Configuration

Dual numbers were applied to the dynamics of a rigid-flexible combination system (RFCS) with time-varying configuration in this paper. The six-dimensional spinor form of the motion of flexible modules, including the dual vector, dual momentum, dual inertia operator, dual coupling coefficient operator, and dual-modal coordinates, was derived using the dual numbers that could represent spiral motion in a compact form. On this basis, the integrated dynamic model of a rigid-flexible combination system with a time-varying configuration was proposed. And then, the relative dynamics equations between two rigid-flexible combination systems which both have time-varying configuration were provided. An on-orbit assembly mission of flexible modules transported and operated by free-flying space robots (FFSRs) is presented as an exemplary application of relative dynamics. Simulation results illustrate the complex coupling effects on the relative motion between two rigid-flex combination systems with time-varying configuration.

Mechanical Analysis and Optimal Design of Wave Energy Device

At present, the global energy problem is serious. Wave energy is the kinetic energy and potential energy of waves on the ocean surface. Because of its high energy density, wide distribution, and long continuous operation of the device, it has become the focus of academic research. The key problem to be solved urgently for the large-scale application of wave energy is how to optimize the energy conversion efficiency of wave energy devices. This paper starts from the Newtonian dynamic equation, analyzes the motion state of the wave energy device in different wave environments, and studies the optimization and control of its power output. This paper first adopts the analysis method of first whole and then isolation, and establishes a one-dimensional coordinate system with the zero point of sea level. Then, the force analysis of the system is carried out, the dynamic differential equation is established, and the initial draft of the system is solved through the initial balance equation. Since the dynamic equation is a second-order linear ordinary differential inhomogeneous linear equation, the fourth-order Runge is used. The Kutta method is used to solve the problem, and the numerical solution for the position and velocity of the float is obtained. Secondly, the movable coordinate system is established with the float as the reference system, and the force analysis of the oscillator is carried out to obtain its relative dynamic equation. Then, the initial relative position of the oscillator is solved through the initial balance equation, and the relative dynamic differential equation is obtained by using the fourth-order Runge-Kutta method to obtain the numerical solution of the relative position and relative velocity of the oscillator with respect to time. Finally, the numerical solution of the absolute position and absolute velocity of the oscillator with respect to time is obtained through coordinate transformation.

Retrieval dynamics and control for approach of tethered on-orbit service satellite

Approach and docking with space tether system has the advantages of less system complexity and lower cost compared with conventional rendezvous and docking which are usually essential operations in on-orbit service of spacecraft. This paper deals with the relative dynamics and control of the tethered on-orbit service satellite for the process of approach and docking by retrieving tether. Considering the system safety, it is necessary to keep the pitch angle constant or it varying in a small range during the approach. Therefore, we propose a control law for tether retrieval. Specifically, the relative dynamics equations are established to analyze the effects of orbit eccentricity, orbit altitude, and tether length rate on the system dynamics as well as investigate the property of pitch motion. Then the retrieval control law with constant pitch angle constraint is proposed using the dynamic inverse method based on the dynamics analyses. We further discuss the feasible region of orbit eccentricity and constant pitch angle for the control law to retrieve the tether. The numerical simulation experiments show that the proposed law can realize stable retrieval while maintaining a constant pitch angle.

Intelligent Attitude Fault-Tolerant Control of Space Tumbling Target Fly-Around Based on RBF Neural Network

The flying around monitoring task of space tumbling target is one of the key links of its on-orbit service. Considering the practical constraints of system inertia uncertainty, external disturbance, actuator saturation, and fault in engineering practice, a robust composite controller based on radial basis function (RBF) neural network is proposed. First, in a new line of sight rotation (RLOS) coordinate system, the relative attitude kinematics and dynamics equations between the tracker and tumbling target based on the error quaternion are established; second, the RBF neural network is used to estimate the additive and multiplicative faults of the system, and the fast nonsingular terminal sliding mode surface (FNTSMS) is combined with the active disturbance rejection control (ADRC) technology to design a finite-time fault-tolerant control (FTC) strategy with high accuracy, strong robustness, and anti-saturation based on the RBF neural network. It is proven that the designed robust fault-tolerant controller can ensure that the system state error converges to a small region containing the origin in a limited time under the Lyapunov framework. Finally, the effectiveness and superiority of the control strategy were verified by numerical simulation.

Cross-coupling relative dynamics for on-orbit assembly in rotating frame of reference

Dual numbers are applied in this paper to relative dynamics between rigid-flex combination systems. In space cooperative missions, the relative dynamic equations with engineering applicability require description in rotating frame of reference. Taking advantages of dual numbers which can represent spiral movement in a compact form with respect to the rotating frame, the dual vector, dual momentum, dual inertia operator, dual coupling coefficient operator and dual modal coordinates of flexible modules are derived. The algebra of dual numbers for rigid-flex combination system is further developed. On this basis, relative equations of motion between two rigid-flex combination systems in rotating frame of reference are proposed. The behaviors of relative dynamics present complex cross-coupling effects between attitude-orbit coupling and rigid-flex coupling. An on-orbit assembly mission of flexible modules via free flying robots is presented as an exemplary application of the cross-coupling relative dynamics. Ground testing results illustrate the cross-coupling effects on the six degree-of-freedom motion of rigid-flex combination system. Simulation results prove the engineering significance of the cross-coupling relative dynamic model.

Close-range leader–follower flight control technology for near-circular low-orbit satellites

Abstract Based on the characteristics of near-circular orbits and close-range leader–follower flights, the relative dynamics equations of the eccentricity/inclination ( e/i ) vector method are introduced herein. Additionally, the constraint terms in the design of the leader–follower flight formation are found to satisfy the conditions of the line-of-sight angle and inter-satellite distance. The control box algorithm is proposed under the flying task’s constraints, such as the line-of-sight angle and distance between the satellites according to the e/i vector and Gauss perturbation equations. The algorithm comprehensively takes into account the relationship between the relative motion variations in the satellite formation in near-circular orbits as well as their relationship with the velocity increment applied to the satellites. The example simulated in this study not only illustrates the existence of a coupling relationship between the flight-following distance and flight-following line-of-sight angle but also verifies the influence of the relative eccentricity of the two satellites on the leader–follower flight stability. The simulation results show that when the control box algorithm was used to maintain the leader–follower flight, this method was simple, intuitive, and may be feasibly introduced as a flight-following control strategy.

Numerical solution for elliptical orbit pursuit-evasion game via deep neural networks and pseudospectral method

This paper presents an efficient and stable DNNs-based Radau pseudospectral method for the free-time elliptical orbit pursuit-evasion game based on the equivalent reconstruction of the game model. Firstly, the relative dynamics equations are established by adding the nonlinear terms caused by the eccentricity to the Hill–Clohessy–Wilshire equations. Then the original pursuit-evasion problem is deduced to a 4-dimensional one-sided optimal control problem (OCP) based on the equivalent reconstruction. Secondly, in order to apply the deep neural networks (DNNs) to map the relationship between the OCP and the solution, the normalization of costates is introduced to eliminate the non-uniqueness of solutions when generating samples for training DNNs. Thirdly, the DNNs-based Radau pseudospectral method is proposed where the DNNs output the guesses of solutions to the derived OCP and the Radau pseudospectral method optimizes the histories of controls obtained by the guesses to the convergence. The simulation results demonstrate that the proposed method converges more stably and decreases the calculation time greatly compared with the traditional indirect method.

Effects of new micro-pocketed bore surface topographies on the performance behaviours of aerodynamic journal bearing

This paper presents the exploration for improving the static and dynamic performance behaviours of a self-acting rigid gas journal bearing employing some new conceived micro-depth pocketed surface topographies at the bore. The conceived micro-pockets comprise of relatively large size rectangular pocket having micro-depth at bearing bore in the converging zone followed by placing of different designs of sub-pockets at the trailing edge of the previous one in the direction of journal rotation. The form of equation achieved from Patir and Cheng’s model for the case of hydrodynamic lubrication regime and the related dynamic pressure equations have been solved using the finite volume method discretisation scheme followed by the solution of the algebraic equations using the Gauss-Seidel iterative method. The minimum film thickness, frictional force, side leakage, bearing dynamic coefficients, effective stiffness, effective damping, and critical mass parameters have been investigated with each new bore surface topography and compared with the performances of conventional aerodynamic journal bearing. Substantial improvements in both static and dynamic performances have been found with the new micro-pocketed bore surface topographies as compared to conventional one. Moreover, the established best bore design has produced significant increase (21%) in minimum film thickness, substantial reduction (12%) in coefficient of friction, and excellent improvement (170%) in the stability parameter (critical mass) as compared to the conventional case.

Finite-time integrated guidance and control system for hypersonic vehicles

A finite-time integrated guidance and control (IGC) method is proposed in this study for hypersonic vehicles. The IGC dynamic model is initially built by combining the 3D relative kinematics and dynamics equations. Then, by introducing the adaptive control technology and the backstepping approach, an IGC scheme with adaptive parameters is presented to guarantee the finite-time stability of a closed-loop control system on the basis of Lyapunov stability theory. Nonlinear simulation results demonstrate the effectiveness and robustness of the proposed IGC method for hypersonic vehicles compared with other robust IGC methods.

Motion and inertial parameter estimation of non-cooperative target on orbit using stereo vision

In the on-orbit servicing missions, autonomous close proximity operations require knowledge of the target’s motion and inertial parameters in order to safely interact with the target. In particular, the motion and parameter estimation of an uncooperative target is a challenging task because of the lack of some prior information about the target in unfamiliar environments. In this paper, a novel method is developed for an accurate estimation of the motion and inertial parameters of an unknown target using stereo vision measurements only. Instead of using relative position and pose as observation information, this paper chooses three non-collinear feature points on the target as estimation measurements. This treatment allows us to estimate the principal axis of inertia of the target directly, and avoid estimating the quaternion between the target measurement coordinate system and the principal axis of inertia coordinate system, which is a bilinear problem, and is difficult to obtain global convergence. Based on the relative kinematics and dynamics equations of the target, the state equation and observation equation are derived, and the Extended Kalman Filter (EKF) is designed. As a result, the developed algorithm realizes the estimation of the complete unknown information of the target, including relative position, relative velocity, attitude quaternion, angular velocity, the direction of the principal axis of inertia, inertia ratio, and the position of center of mass. Numerical simulations and experimental results verify the convergence and effectiveness of the proposed filter estimation method.

Relative Motion Dynamics Modeling and Control of DFP Vibration Isolation Spacecraft

In order to better meet the future high precision task requirements, the DFP(Disturbance-Free Payload) spacecraft composed of non-contact PM(Payload Module)and SM(Support Module)is taken as the object to study the relative motion dynamics modeling and control between the two modules and verify the system vibration isolation performance. Firstly, the force and torque expressions of the two modules are derived by simplifying the configuration and analyzing the stress. In view of that couple effect, the relative motion dynamics equations between two modules of DFP spacecraft with high model accuracy, and simple and uniform format are established with the dual quaternions. Based on this, the PD control law is designed, and the relative motion of PM and SM could meet DFP spacecraft working requirements when the measurability of control quantity and the measurement error of sensors were taken into account. Simulation results verify the advantage of vibration isolation and attitude maneuverability of DFP spacecraft.

Output Regulation Control for Satellite Formation Flying Using Differential Drag

This paper proposes a new approach of using differentials in aerodynamic drag in combination with thrusters to control satellite formation flying in low Earth orbits. Parameterized output regulation theory for formation-flying missions with combined control action is developed based on the Schweighart–Sedwick relative dynamics equations. The theory is implemented to precisely track the different trajectories of reference relative motion and eliminates the effects of the perturbations. The parametric Lyapunov algebraic equation is proposed to ensure the stability of the linear relative model subject to saturated inputs. The main goal of this study is to approve the viability of using the differentials in aerodynamic drag to precisely control different formation-flying missions. Numerical simulations using a high-fidelity relative dynamics model and a high-precision orbit propagator are implemented to validate and analyze the performance of the proposed control algorithm in comparison with the linear quadratic regulator algorithm based on actual satellite models.

Perturbed HEO Satellite Hovering Investigation in the Earth-Moon System

The relative hovering of satellites in highly elliptic orbits (HEO), as one of the most crucial space operations, is modeled and analyzed in this paper. The proposed modeling is based on the new perturbed relative dynamics equations, uses the time-varied parameters depending on the motion of the target satellite. This proposed model considered the dynamic air drag, oblateness of Earth (including all zonal harmonics coefficients) and the Lunar perturbation as an inclined third-body disturbing effect on both follower and target orbits. The non-simplified relative motion model has been obtained by employing the Lagrangian mechanics principals and completed along with the target satellite’s motion characteristic. Then the required thrust for relative hovering mission has been obtained without any simplifications to ensure the accuracy of long-duration flight analyses. To validate the presented model, another model has been built as an ECI based Relative Motion (ERM) model. Then, according to effective parameters on hovering mission design around HEOs such as the eccentricity and inclination of the target obit, the fuel consumption, optimal positioning of the follower, maximum required thrust, and the appropriate time to perform the operation, several examples are provided. Furthermore, the hybrid IWO/PSO algorithm has been used to find the location and the minimum/maximum amounts of thrust force.

Satellites Autonomous Navigation With an Extended State and Disturbance Sliding Mode Observer Method

This paper considers the satellites autonomous navigation problem with an extended sliding mode observer method. First, on consideration that the relative dynamic equation of chaser and target spacecrafts refers to a nonlinear time-varying system, a transformation is performed on the original relative dynamic equation to obtain a linear time-invariant system with a nonlinear part. Second, an augmentation strategy is made on the new system to obtain a descriptor system. A new extended sliding mode observer is then designed for the descriptor system which can generate the simultaneous estimation of state and output measurement noise. Based on the observer estimates, the equivalent output error injection approach is used to estimate the space disturbance moment. Finally, an example is presented to demonstrate the effectiveness of the developed satellites autonomous navigation observer method.

Position Control for Closing Phase of ARD

Adaptive sliding mode control (ASMC) is used to achieve high precision position control for the closing phase of autonomous rendezvous and docking (ARD). The nonlinear relative dynamic equation is given; the gain is adjusted according to the adaptive law. If the error is small, the chattering is reduced, the precision is improved. The stability is proved. To decouple the system, time sharing strategy is used. The simulation results show ASMC has higher precision than SMD, and is faster than SMD.

Sliding mode predictive guidance for terminal rendezvous in eccentric orbits

This paper presents a robust guidance algorithm for a chaser spacecraft to rendezvous with a target spacecraft in Earth orbit. The basis of the proposed guidance method is finding an appropriate set of states as close as possible to the current states that would lead the spacecraft to the target in the desired mission time. In order to provide the prediction of states, the relative dynamics equations of motion are solved analytically for the chaser spacecraft rendezvous considering constant acceleration. Although the equations are solved for rendezvous with circular orbit target, it is shown by several simulations that the proposed guidance algorithm is applicable in perturbed elliptical orbits rendezvous as well. The sliding mode method as a robust nonlinear method is utilized as the steering law. The robust steering law tracks the desired states computed by the predictive guidance method. The Lyapunov stability method proves the asymptotic stability of the integrated guidance and steering laws. Because the proposed closed-loop guidance is simple and computationally easy, it is suitable for implementation in real-time applications. Some numerical simulations are conducted to show the performance of the proposed guidance method in different conditions. It is illustrated that compared with other steering laws, the fuel consumption is reduced utilizing the proposed guidance approach. The results reveal that the sliding mode guarantees the tracking of the required states and minimum final errors even in the presence of uncertainties and disturbances.

Shared control on lunar spacecraft teleoperation rendezvous operations with large time delay

Teleoperation could be used in space on-orbit serving missions, such as object deorbits, spacecraft approaches, and automatic rendezvous and docking back-up systems. Teleoperation rendezvous and docking in lunar orbit may encounter bottlenecks for the inherent time delay in the communication link and the limited measurement accuracy of sensors. Moreover, human intervention is unsuitable in view of the partial communication coverage problem. To solve these problems, a shared control strategy for teleoperation rendezvous and docking is detailed. The control authority in lunar orbital maneuvers that involves two spacecraft as rendezvous and docking in the final phase was discussed in this paper. The predictive display model based on the relative dynamic equations is established to overcome the influence of the large time delay in communication link. We discuss and attempt to prove via consistent, ground-based simulations the relative merits of fully autonomous control mode (i.e., onboard computer-based), fully manual control (i.e., human-driven at the ground station) and shared control mode. The simulation experiments were conducted on the nine-degrees-of-freedom teleoperation rendezvous and docking simulation platform. Simulation results indicated that the shared control methods can overcome the influence of time delay effects. In addition, the docking success probability of shared control method was enhanced compared with automatic and manual modes.

Novel Guidance Law for Interception for Maneuvering Target with High-speed

Aiming to intercept high-speed maneuvering targets, nonlinear model was established and relative dynamic equations between the interceptor and the target were analyzed. An omni-directional true proportional navigation (OTPN) guidance law was developed. In high-speed target interception, OTPN guidance law has a wide capture region and can automatically shift interception mode. Formulary analysis proved that target maneuver has no effect on the expression form of OTPN guidance law. Without any additional conditions, OTPN guidance law can meet the requirement to intercepting high-speed maneuvering targets. Numerical simulation results showed that OTPN guidance law can effectively intercept high-speed maneuvering targets and its capture region is influenced by targets’ maneuver ability and initial LOS angle. The parameters of OTPN, such as capture region, ballistic characteristics, and overload variation and control efforts, show that OTPN are superior to traditional guidance schemes.