Khalid Hasan Tantawi, Dr. Christelle Cumer (ONERA, France), and Dr. Daniel Alazard (Lead researcher, INSAE, France)
Abstract
A method and some tools developed to build linear models of multi-body systems for space applications (typically satellites) are presented. The multi-body system is composed of a main body (hub) fitted with rigid and flexible appendages (solar panels, antennas, propellant tanks, ...etc). Each appendage can be connected to the hub by a cantilever joint or a pivot joint. More generally, our method can be applied to any open mechanical chain. In our approach, the rigid six degrees of freedom (d.o.f) (three translational and three rotational) can be treated all together, and that is very convenient to build linear models of complex multi-body systems. Each added pivot joint, adds a new degree of freedom to the six rigid d.o.fs of the system. The satellite equations of motion which relate the forces and torques applied on the main satellite hub to the angular and linear position and velocity of the hub are found. A Matlab toolbox is developed to calculate this dynamic model, hence making it a very convenient and handy tool for the design of the Attitude and Orbit Control System (AOCS) easily. This model can be interpreted using block diagram representation.
Keywords
Dynamic Model, Cantilever Hybrid Model, Flexibility, Effective Mass, Linear system, Modal Participation, Multi-body system, Pivot Joint.
Satellites and other spacecraft are typical multi-body mechanical systems that include both rigid and flexible bodies. Rigid bodies have been shown to be stabilized by applying torques on them [1], complex mechanical systems such as spacecraft are stabilized by the Attitude and Orbit Control System (AOCS) components, that may include reaction wheels and thrusters. The design of the AOCS requires a linear model taking into account all the rigid and flexible couplings between the hub (where the AOCS acts) and the various appendages.
In this work, the linear model is found for a typical satellite system that consists of a central body attached to it rigid and flexible appendages, that may be cantilevered to it or attached through a pivot joint. The dynamic model, which relates the forces and torques applied on a system, and the resulting linear and angular accelerations, is assumed to be linear. This linear assumption is quite realistic for such systems since perturbations and so motions are very small (except for very dexterous observation satellites). This linear assumption is furthermore valid in the field of future missions for deep space exploration involving formation flying of several spacecraft. For this kind of formation flying mission, it is more and more accepted that the 3 rotational degrees of freedom (d.o.f) and the 3 translational d.o.f’s must be treated all together [2].
When deformable bodies make up a part of the multi-body system, then flexibility must be taken into account, many formulations for handling flexible multi-body dynamics have been investigated in [3]. Flexibility in multi-body systems, may be treated by finite element analysis[4-5]. In this work, the Effective Mass Representation [6-7] has been used in generating the flexible model of the mechanical elements in which flexibility is present, this use was made possible by applying the Cantilever Hybrid Model [7].
Hence a 6 d.o.f. model including couplings between rotations and translations must be developed. Many multibody dynamics software are available to build such kind of models but they address the nonlinear behavior and they are too much loud to be handled at the early prototyping phase. So a tool is required to develop quickly the dynamic model and to prototype the AOCS or to analyze and to optimize the main dynamic parameters of the mechanical structure or AOCS and finally to assess the global performance of the system. Here we propose some tools developed with Matlab/SIMULINK to efficiently build the linear dynamic model of any open mechanical chain.
Figure 1.1. The study considers a linear model
More specifically, the linear multi-body model that we consider here is depicted on figure 1.1 and gives the relationship between:
• The inputs, which are composed of:
- The six external forces and torques applied on the body by the Attitude and Orbit Control System(AOSC) in a reference point O.
- The n drive torques Cm(i) applied at the ith pivot joint between an appendage and the hub.
• The output, which is composed of:
- The six linear and angular accelerations of the hub at point O.
- The angular acceleration <theta>'' (i) of the pivot point i (for i = 1,..,n)
Assumptions:
Throughout this report the following assumptions are made for this study:
· Linear approximation: All motions are considered small enough for linearity to be an acceptable assumption.
· All the external forces and torques are applied on the satellite hub (by the AOCS).
· Rigid and flexible appendages are only subject to reaction forces and torques with the hub at the connecting points, except when there is a pivot joint fitted with a drive motor. (see figure 1.2)
Figure 1.2. The principle scheme of the linear model of the satellite: All external forces and torques are applied on the hub, a torque can also be applied about the axis of a pivot if there is one.
To download the rest of the thesis:
Linear Dynamic Modeling of Satellites With Various Flexible Appendages
Publications in Linear Dynamic Modeling Research:
Khalid H.M. Tantawi, Daniel Alazard, and Christelle Cumer, Linear Dynamic Modeling of Spacecraft with Various Flexible Appendages, Journal of the International Federation of Automatic Control (IFAC), Seol, Korea, (July 2008)
Daniel Alazard, Christelle Cumer, and Khalid H.M. Tantawi, “Linear Dynamic Modeling of Spacecraft with Various Flexible Appendages and On-board Angular Momentum”, 7th International ESA Conference on Guidance, Navigation & Control Systems, 01-05 Jun 2008, Tralee, Ireland (June 2008)