## The SDTlib (Satellite Dynamics Toolbox – Library)

### Modeling software for flexible spacecrafts

The SDTlib (Satellite Dynamics Toolbox – Library) is the result of 15 years of academic research at ISAE-SUPAERO in partnership with ESA, CNES, ONERA, and industrial partners.

Implemented in Matlab-Simulink, the SDTlib allows modeling complex systems in a multibody framework, with a unique expertise in:

- Flexible dynamics,
- Parametric uncertainties,
- Mechanisms and vibrations,

and has been extensively validated with many pilot cases during various projects with ESA and industrial partners.

A tutorial/demo version presents two simple tutorials to show how to use the toolbox to model a satellite with uncertain flexible appendages, and how to use the resulting model for robust control and analysis.

#### A multibody framework

In the following example, the SDTlib is used to easily build the model of a satellite composed of:

- A rigid body $\mathcal B$ with 6 degrees of freedom (rotation and translation),
- A flexible solar panel $\mathcal S$ modeled with Nastran, connected through a Solar Array Driving Mechanism (SADM) $\mathcal M$,
- Reaction wheels,
- Two booms $\mathcal B_1$ and $\mathcal B_2$.

Each element is represented by an independent block, which are assembled in Matlab-Simulink to obtain the model of the structure. Each individual model is parameterized with the Linear Fractional Transformation (LFT) representation. The resulting model is an LFT with minimal number of repetitions of the parameters.

#### The Matlab-Simulink library

The blocks are parameterized, compliant with uncertain parameters, and fully documented. The resulting model can be used with the standard tools of the robust control toolbox of Matlab (systune, wcgain…).

#### The TITOP approach

The modeling of uncertain flexible bodies in the SDTlib is enabled by the Two-Input Two-Output Port (TITOP) approach. The TITOP model allows describing the dynamics of a flexible body independently from the rest of the multibody structure. Here is represented a structure noted $\mathcal L_i$, connected to a parent structure $\mathcal L_{i-1}$ at point $P$ and to a child structure $\mathcal L_{i+1}$ at point $C$:

The TITOP model is the {12×12} linear dynamic model between the inputs (in red):

- $\mathbf{W}_{\mathcal L_{i+1}/\mathcal L_{i},\, C} $
*the 6×1 wrench (forces and torques) applied to $\mathcal L_{i}$ by $\mathcal L_{i+1}$ at point C*; *$\mathbf{\ddot{u}}_P$: the 6×1 inertial acceleration (linear and angular) imposed to $\mathcal L_{i}$ at point $P$;*

And the outputs (in blue):

*$\mathbf{\ddot{u}}_C$: the**6×1*components of the inertial acceleration at point $C$;- $\mathbf{W}_{\mathcal L_{i}/\mathcal L_{i-1},\, P} $ : the
*6×1*wrench applied by $\mathcal L_{i}$ to $\mathcal L_{i-1}$ in reaction at point P.

The TITOP model is constructed from the mechanical properties of the body: mass, natural frequencies, modal participation factors… which can be retrieved either from analytical models (for simple shapes: plate, beam) or from finite-elements models thanks to a interface with Nastran. Of course, the TITOP approach is extended to any number of connections with other bodies.

Then, the TITOP model can be augmented with the Linear Fractional Transformation (LFT) representation, to take into account parametric variations on any of these mechanical properties. Typically, it allows considering uncertain properties such as the natural frequencies ; but it can also be used to consider a mechanical parameter as a decision variable to be optimized simultaneously with the controller, in a so-called control/structure co-design approach.