Natural Coordinate Formulation Research Articles

Deployment dynamics and experiments of a tendon-actuated flexible manipulator

The quantity of space debris on Earth orbit has escalated tremendously in recent years, presenting a significant hazard to human space operations. It is urgent to develop effective measures to capture and remove various space debris. For this purpose, this paper presents a tendon-actuated flexible deployable manipulator. The flexible manipulator consists of several deployable units connected by Cardan joints and actuated by tendons. Compared with the present technologies for capturing space debris such as rigid robotic arm or flying net, this flexible manipulator is deployable, reusable, lightweight and applicable to the capture of large space debris. In order to investigate its deployment dynamics, an accurate dynamic model of the flexible manipulator is established based on the natural coordinate formulation (NCF) and the absolute nodal coordinate formulation (ANCF). Subsequently, numerical simulations are carried out to study the effects of system parameters and the base satellite on its deployment dynamics. Finally, ground experiments for both deployment and bending of the flexible manipulator are conducted to verify its effectiveness and feasibility.

Dynamics of reconfiguration and assembly of a stacked satellite system

Separation and reconfiguration of stacked satellites in orbit is an effective method for constructing large space structures. This paper investigates the dynamics of a stacked satellite system (SSS) during the separation and reconfiguration process. Firstly, the dynamic model of the SSS is established via the natural coordinate formulation (NCF) to conveniently handle the time-varying constraints between satellites. Then, the separation and reconfiguration strategies are proposed for the SSS including the reconstruction step between two configurations. To ensure collision-free satellite assembly, a proportional differential (PD) controller combined with a potential function is introduced. Additionally, a docking mechanism is designed, considering collision force constraints during satellite docking assembly. Finally, several numerical simulations are conducted to demonstrate the effectiveness of the separation and assembly strategies.

Orbit-attitude-structure coupled modelling method in local translational coordinate frame for multibody systems

The absolute coordinate-based (ACB) method, including absolute nodal coordinate formulation and natural coordinate formulation, is a widely-used dynamic modelling method for multibody systems. However, the problem of accuracy loss would arise when multi-scale motions are described in the same global coordinate system, such as large-scale orbital motions, small-scale attitude motions and structural vibrations, and micro-scale contact/collision. To address this problem, an ACB method in local translational coordinate system is proposed such that the large-scale orbital motions can be separated from other smaller-scale motions. The dynamic equations are derived by the Hamilton's equations and the Kronecker product. The validity of the proposed method is studied using an orbit-attitude coupled model of an orbiting rigid plate. It is found that the attitude errors of the original ACB method increase as time step size decreases. The reason for this counterintuitive phenomenon is that more serious accuracy loss is encountered for smaller time step size. In addition, more iterations are required for the original ACB method to obtain the prescribed tolerable error in each time step due to accuracy loss. In comparison, the proposed method is more accurate and efficient in numerical simulations. Finally, the proposed method is applied to the orbit-attitude-structure coupled simulation of a space robotic system.

A general formulation for simulating the dynamic deployment of thick origami

The deployment of origami is usually a rapid and complicated dynamic process. However, little has been done to simulate the entire deployment process of thick origami which is a typical over-constrained mechanical system. Based on the constrained Hamilton’s equations in conjunction with the natural coordinate formulation, a general formulation capable of tracing the dynamic deployment of thick origami is established. Particular efforts are paid to dealing with the problems including geometric description, redundant constraint, large rotational angle, and possible bifurcation paths. The proposed formulation is applied to three thick origami structures: thick Miura-ori, thick diamond origami, and thick waterbomb origami. The results show that the dynamic characteristics during the deployment can be successfully obtained, and the numerical accuracy is also validated by the ADAMS simulation and prototype experiments. Furthermore, parametric analyses are performed to investigate the effects of actuating positions and geometric parameters on deployment behaviors. The proposed formulation provides a clear view of the deployment process for thick origami structures, which will benefit the optimal design in practical applications.

Influences of space perturbations on robotic assembly process of ultra-large structures

The space assembly of two flexible beams by a dual-arm space robot is a typical assembly scenario to construct ultra-large space structure. Yet, previous studies mainly focused on the assembly of small structures, neglecting the influences of space perturbations. Two models are developed in this research to investigate the effects of space perturbations on the space assembly process of ultra-large space structures. Firstly, a theoretical modelling method is proposed based on quasi-static hypothesis and linear structural mechanics. The theoretical model can be utilized for analytically estimating the transverse and axial distributed forces of the flexible beams, structural vibrations, and the control moments of the space robot. An orbit–attitude–structure coupled simulation model is then established to validate the theoretical model and study the dynamic behaviours more accurately, using absolute nodal coordinate formulation and natural coordinate formulation. Finally, the effects of the attitude angle, orbital radius, and lengths of beams on the dynamic responses during assembly are investigated. Theoretical and simulation results show that the control moments and structural vibration amplitude increase dramatically with the length of the beams. The effects of Coriolis force and gravity gradient must be considered for ultra-large space structures during assembly, otherwise the control moments and structural vibrations would be substantially underestimated. The results are instructive to the assembly strategy design as well as modular component design of ultra-large space structures.

Dynamic analysis of a rigid-flexible inflatable space structure coupled with control moment gyroscopes

The vibration generated by the inflatable structure after deployment has a great impact on the performance of the payloads. In this paper, the influence of the control moment gyroscopes (CMGs) on the dynamic responses and characteristics of an inflatable space structure is studied, based on the flexible multibody dynamics in a combination of the absolute nodal coordinate formulation (ANCF) and the natural coordinate formulation (NCF). Firstly, the ANCF and NCF are used to accurately describe the large deformations and large overall motions of flexible inflatable tubes and rigid satellites, respectively. Then, instead of modeling gyroscopic flexible bodies, this paper pioneers a rigid body dynamic model of the CMG in detail by using the NCF modeling scheme, which can be attached to and coupled with any flexible bodies without any assumptions. Then, the orbital dynamic equations of the inflatable space structure coupled with distributed CMGs are obtained by considering the effects of Coriolis force, centrifugal force, and gravity gradient through coordinate transformation. The dynamic characteristics of the inflatable space structure are also analyzed by deriving the eigenvalue problem of a flexible multibody system. Finally, the accuracy of the CMG dynamic model is verified via a classic heavy top example. Several numerical examples are presented to study the influence of the magnitudes and directions of the rotor angular momentum of the CMGs on the dynamic responses and characteristics of the inflatable space structure.

Dynamics modeling and experiment of a large space umbrella truss structure

This paper studies the dynamics of a large space umbrella truss structure (SUTS) by using a flexible multibody dynamic method. The SUTS is treated as a rigid-flexible multibody system that consists of a number of horizontal and vertical elastic rods, and connecting rigid joints. A SUTS model is established by the absolute-coordinate-based (ACB) modelling strategy, which is composed of the methods of natural coordinate formulation (NCF) and absolute nodal coordinate formulation (ANCF). The NCF method is adopted to describe the global motions of the rigid joints, while the ANCF method accounting for the large displacement and large deformation is utilized to establish the dynamic model of the elastic rods. The method of Lagrangian multipliers is used to obtain the dynamic equations of the system, which are solved based on the generalized-alpha method. The accuracy of the established SUTS model is verified via the ground air-bearing vibration experiment. Finally, numerical case studies are presented for investigating the dynamics of the SUTS model by comparisons.

Autonomous separation deployment dynamics of a space multi-rigid-body system with uncertain parameters

In this study, we investigate nonlinear dynamics of the autonomous separation deployment maneuver for a space multi-rigid-body system with uncertain-but-bounded parameters by using a high-efficiency interval analysis method. Based on the virtual work principle, the natural coordinate formulation method is used to describe the variations in displacements and attitude motions for rigid bodies. In addition, the orbital dynamics of the system and varying constraints between rigid bodies are incorporated into the governing equations of motion to improve the dynamic model accuracy. A multistage deployment strategy is utilized by regulating the separation sequences and directions to achieve a collision-free separation maneuver. Considering the uncertainties in the system parameters, a bounds estimation problem is formulated for the separation deployment maneuver. Furthermore, an interval analysis method utilizing the Chebyshev series is adopted to solve this problem using the lower and upper bounds of uncertain parameters. Finally, numerical simulations are included to validate the effectiveness of the interval analysis method and evaluate the influence of uncertain parameters on the system dynamics.

Contact-free release dynamics of tens of stacked satellites with multiaxial rotations

Stacked satellites have a promising application in aerospace engineering for the merits of high launch efficiency and networking capability. As one of the key technical points, releasing tens or even hundreds of satellites from the stacked state in a simple and contact-free manner is of importance. In this paper, the contact-free release dynamics of tens of stacked satellites is studied with only the initial multiaxial rotations of the system. First of all, a rigid multibody dynamic model of the stacked satellite system is established via the natural coordinate formulation (NCF). The NCF modeling scheme is able to describe the large overall motions of the satellites without any singularity and makes it possible to simplify the varying constraints between the satellites. Then, the orbital dynamics of the stacked satellite system is derived via coordinate transformation by taking the Coriolis forces, centrifugal forces, and gravity gradient into consideration. In order to rapidly and accurately detect the possible contact between satellites, a convex optimization model for minimum distance computation is proposed by using a hyperelliptic approximation for a cubic satellite. Finally, a benchmark example is given to validate the contact detection algorithm and three release dynamic cases for the stacked satellites are presented to demonstrate the effectiveness of the contact-free releasing approach with multiaxial rotations.

Deployment Strategy and Dynamic Analysis of Large Ring Truss Antenna

The deployment strategy and dynamic analysis of the AM-2 AstroMesh ring truss antenna reflector are studied in this paper. The rigid multibody dynamic models of the truss structures with alterable and fixed diagonal length are established, respectively, by using the natural coordinate formulation (NCF). The driving scheme and the synchronous constraint scheme are proposed for two types of truss structure, respectively. The degree of freedom (DOF) analysis of the truss structure is carried out according to the redundant constraint processing method. The deployment strategies of the two different types of truss structures are discussed. The dynamic simulation of the deployment of a 30-side AM-2 AstroMesh reflector truss structure with fixed diagonal length without gravity is carried out. The deployment characteristics of the truss structure are obtained. The driving forces are predicted according to the dynamic simulation.

Spin dynamics of a long tethered sub-satellite system in geostationary orbit

A long tethered sub-satellite system (LTSS) has a promising application for earth observation, which requires that the LTSS spins towards the earth. Due to the long flexible tethers and the spinning of the LTSS, the system exhibits dynamics of high nonlinearity and high complexity. While most previous studies did not consider more than three satellites in a tethered satellite system, the computational method developed in this work allows considering an arbitrary number of satellites as demonstrated by the numerical examples presented. First of all, a nonlinear flexible multibody dynamic model of the spinning LTSS in the orbital coordinate frame is established and the corresponding dynamic analysis is studied. Here, the absolute nodal coordinate formulation (ANCF) and the natural coordinate formulation (NCF) are used to accurately describe the large deformations and large overall motions of the flexible tethers and rigid satellites, respectively. Afterwards, through coordinate transformation, the orbital dynamics of the LTSS is obtained and the effects of Coriolis force, centrifugal force, and gravity gradient are analytically derived. Finally, several numerical examples are presented, and the vibrations and stabilities of the LTSS are analyzed via parameter study.

Rigid-flexible coupling effect on attitude dynamics of electric solar wind sail

This paper investigates the modelling of rigid-flexible coupling effect on the attitude dynamics and spin control of an electric solar wind sail (E-sail) by developing a rigid-flexible coupling dynamic model. The model considers the attitude dynamics of the central spacecraft, the elastic deformation of the tethers and the rigid-flexible coupling between the spacecraft and the tether. The attitude and translation dynamics of the central spacecraft is described by the natural coordinate formulation, while the tether deformation is described by the high-fidelity nodal position finite element method. The latter enables a natural coupling between the motion of the flexible tethers and the rigid-body dynamics of the central spacecraft at the anchor points where the tethers connected to the spacecraft by Lagrange multipliers. Based on the model, the influence of the rigid-flexible coupling, E-sail orientation and geometrical configuration on the dynamic characteristics of the E-sail is investigated by a parametric analysis. It is found that the deformation motion of flexible tethers will cause the offset of centres of mass and thrust of E-sail, which generates disturbance torques on the central spacecraft. Through the nonlinear rigid-flexible coupling, the disturbance causes the tension fluctuations and the undesired fluctuations of the E-sail's attitude and spin rate. The parametric analysis indicates that the E-sail is more stable if the spin plane passes the centre of mass of the central spacecraft. Finally, the controllability of E-sail spin rate is investigated by applying simple feedback torque controls at the central spacecraft or at the central spacecraft and the remote units simultaneously. The analysis demonstrates the spin rate cannot be controlled by the central spacecraft along due to the rigid-flexible coupling and must be controlled at the remote units with finite control input.

Deployment Dynamics for a Flexible Solar Array Composed of Composite-Laminated Plates

Modern spacecraft often deploy large-scale and lightweight solar arrays consisting of composite-laminated plates because of their high reliability, superior mechanical properties, and low manufacturing cost. This paper presents the deployment dynamics of a flexible solar array composed of composite-laminated plates undergoing large rotation and large deformation motions. The subject is a constrained rigid–flexible coupling spacecraft consisting of a rigid main body and a flexible solar array composed of composite-laminated plates. The rigid main body and the flexible solar array are described by the natural coordinate formulation and the absolute nodal coordinate formulation, respectively. A constitutive model of a laminated shell formed of fiber-reinforced composite material is established, and the corresponding generalized elastic force is derived. Thus, the spacecraft’s equations of motion are derived as a set of differential algebraic equations and a computational scheme based on the Hilber-Hughes-Taylor (HHT)-I3 method is illustrated. A series of numerical calculations and simulations are conducted to investigate the solar array deployment dynamics. The results show that the solar panel flexibility and fiber orientation of fiber-reinforced composite materials have obvious effects on spacecraft dynamic responses during solar array deployment.

RESEARCH ON SEPARATION STRATEGY AND DEPLOYMENT DYNAMICS OF A SPACE MULTI-RIGID-BODY SYSTEM 1)

The paper focuses on the separation and deployment dynamics of an on-orbit compactly connected multi-rigid-body (MRB) system, which could separate autonomously from a carrier spacecraft. Based on the focused MRB system, it is not necessary to repeatedly use the launcher of the carrier spacecraft or install multiple launchers in the spacecraft to separate the MRB system. This is advantageous because it can effectively improve the space utilization rate of the spacecraft, simplify the separation deployment operations and reduce the risk of collision between rigid bodies. To realize the separation of such a MRB system, the paper presents an investigation on its on-orbit dynamics and the design of collision-free separation deployment schemes. Firstly, a dynamic model of a single rigid body is established based on the principle of virtual work and the Natural Coordinate Formulation (NCF) method accounting for the relative motion between rigid bodies and attitude changes of each rigid body. Considering the orbital motion, the variations of connecting constraints of the MRB system and the interactions between rigid bodies during the separation, the governing nonlinear dynamic equations including constraints of the system are obtained with a method of Lagrange multipliers. With practical engineering applications taken into consideration, the separation deployment of MRB system is realized through ejection mechanisms mounted on the four corners of each contact surface between rigid bodies. Secondly, the timing sequences of separation maneuvers are specially programmed and two separation schemes are developed by adjusting different ejection directions and ejection sequences to guarantee the non-collision between rigid bodies in the separation deployment. Finally, numerical case studies are presented for investigating the nonlinear dynamic behaviors of rigid bodies and demonstrating the effectiveness of separation schemes.

Dynamics of a tethered satellite formation for space exploration modeled via ANCF

This paper studies the dynamics of a spinning three-body tethered satellite formation by a flexible multibody dynamics method. The tethered system is specially designed to constitute a Space Infrared Interferometric Telescope for a space exploration mission. Two sub-satellites are deployed from the main-satellite by using long flexible tethers and moving along trajectories of Archimedes spirals. Both of the large displacement and large deformation of the variable-length tethers are described by utilizing the Absolute Nodal Coordinate Formulation method in a framework of Arbitrary Lagrange-Euler (ANCF-ALE) description. The Natural Coordinate Formulation (NCF) method is used for describing the overall rigid motions of the three satellites. Considering the mass flow of tether element, the dynamic equations of tether element are derived by using the D'Alembert's principle. The governing dynamic equations of the system are obtained with a method of Lagrange multipliers. The tether deployment is achieved by regulating deployed velocity and applied jet forces. Numerical case studies are presented for investigating the parametric influences on the deployment dynamics of the tethered system by comparisons.

Assembly dynamics of a large space modular satellite antenna

The autonomous assembly technology can be used to construct large space structures such as space telescopes, satellite antennas and space stations. A novel computation methodology to study the assembly dynamics of a large space modular satellite antenna is proposed. The system rigid body and flexible body are respectively described by the natural coordinate formulation (NCF) and the absolute nodal coordinate formulation (ANCF). A parameterized inverse kinematics resolution method is used to evaluate the arm joint rotation angles of a rigid assembly manipulator with 7 degree-of-freedom (DOF). To minimize the disturbance of the floating satellite base attitude, an efficient particle swarm optimization (PSO) algorithm with Latin hypercube sampling skill (LHS-PSO) is used to find the optimized arm joint trajectories of the assembly manipulator arms. In the optimization process the oriented bounding box (OBB) contact detection method is also embedded to eliminate the trajectories which may lead to the arm-satellite or arm-arm impact problems. Finally, two numerical examples are provided to validate the proposed computation methodology. The results show that the satellite base attitude almost return back to its initial state after completing the assembly process by using the optimized joint trajectories.

A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems

The goal of this investigation is to perform a comparative analysis of the principal methodologies employed for the analytical formulation and the numerical solution of the equations of motion of rigid multibody mechanical systems. In particular, three formulation approaches are considered in this work for the analytical formulation of the equations of motion. The multibody formulation strategies discussed in this paper are the Reference Point Coordinate Formulation with Euler Angles (RPCF-EA), the Reference Point Coordinate Formulation with Euler Parameters (RPCF-EP), and the Natural Absolute Coordinate Formulation (NACF). Moreover, five computational algorithms are considered in this investigation for the development of effective and efficient solution procedures suitable for the numerical solution of the equations of motion. The multibody computational algorithms discussed in this paper are the Augmented Formulation (AF), the Embedding Technique (ET), the Amalgamated Formulation (AMF), the Projection Method (PM), and the Udwadia-Kalaba Equations (UKE). The multibody formulation approaches and solution procedures analyzed in this work are compared in terms of generality, versatility, ease of implementation, accuracy, effectiveness, and efficiency. In order to perform a general comparative study, four benchmark multibody systems are considered as numerical examples. The comparative study carried out in this paper demonstrates that all the methodologies considered can handle general multibody problems, are computationally effective and efficient, and lead to consistent numerical solutions.

On the Computational Methods for Solving the Differential-Algebraic Equations of Motion of Multibody Systems

In this investigation, different computational methods for the analytical development and the computer implementation of the differential-algebraic dynamic equations of rigid multibody systems are examined. The analytical formulations considered in this paper are the Reference Point Coordinate Formulation based on Euler Parameters (RPCF-EP) and the Natural Absolute Coordinate Formulation (NACF). Moreover, the solution approaches of interest for this study are the Augmented Formulation (AF) and the Udwadia–Kalaba Equations (UKE). As shown in this paper, the combination of all the methodologies analyzed in this work leads to general, effective, and efficient multibody algorithms that can be readily implemented in a general-purpose computer code for analyzing the time evolution of mechanical systems constrained by kinematic joints. This study demonstrates that multibody algorithm based on the combination of the NACF with the UKE turned out to be the most effective and efficient computational method. The conclusions drawn in this paper are based on the numerical results obtained for a benchmark multibody system analyzed by means of dynamical simulations.

A simple orbit-attitude coupled modelling method for large solar power satellites

A simple modelling method is proposed to study the orbit-attitude coupled dynamics of large solar power satellites based on natural coordinate formulation. The generalized coordinates are composed of Cartesian coordinates of two points and Cartesian components of two unitary vectors instead of Euler angles and angular velocities, which is the reason for its simplicity. Firstly, in order to develop natural coordinate formulation to take gravitational force and gravity gradient torque of a rigid body into account, Taylor series expansion is adopted to approximate the gravitational potential energy. The equations of motion are constructed through constrained Hamilton's equations. Then, an energy- and constraint-conserving algorithm is presented to solve the differential-algebraic equations. Finally, the proposed method is applied to simulate the orbit-attitude coupled dynamics and control of a large solar power satellite considering gravity gradient torque and solar radiation pressure. This method is also applicable to dynamic modelling of other rigid multibody aerospace systems.

Simulating coupled dynamics of a rigid-flexible multibody system and compressible fluid

As a subsequent work of previous studies of authors, a new parallel computation approach is proposed to simulate the coupled dynamics of a rigid-flexible multibody system and compressible fluid. In this approach, the smoothed particle hydrodynamics (SPH) method is used to model the compressible fluid, the natural coordinate formulation (NCF) and absolute nodal coordinate formulation (ANCF) are used to model the rigid and flexible bodies, respectively. In order to model the compressible fluid properly and efficiently via SPH method, three measures are taken as follows. The first is to use the Riemann solver to cope with the fluid compressibility, the second is to define virtual particles of SPH to model the dynamic interaction between the fluid and the multibody system, and the third is to impose the boundary conditions of periodical inflow and outflow to reduce the number of SPH particles involved in the computation process. Afterwards, a parallel computation strategy is proposed based on the graphics processing unit (GPU) to detect the neighboring SPH particles and to solve the dynamic equations of SPH particles in order to improve the computation efficiency. Meanwhile, the generalized-alpha algorithm is used to solve the dynamic equations of the multibody system. Finally, four case studies are given to validate the proposed parallel computation approach.