A Finozzi

F Sanfedino

D Alazard
email: daniel.alazard@isae.fr

Parametric sub-structuring models of large space truss structures for structure/control co-design

Keywords:

Modern and future high precision pointing space missions face increasingly high challenges related to the widespread use of large flexible structures. The development of new modeling tools which are able to account for the multidisciplinary nature of this problem becomes extremely relevant in order to meet both structure and control performance criteria. This paper proposes a novel methodology to analytically model large truss structures in a sub-structuring framework. A three dimensional unit cube element has been designed and validated with a Finite Element commercial software. This model is composed by multiple two-dimensional sub-mechanisms assembled using block-diagram models. This constitutes the building block for constructing complex truss structures by repetitions of the element. The accurate vibration description of the system and its minimal representation, as well as the possibility of accounting for parametric uncertainties in its mechanical parameters, make it an appropriate tool to perform robust Structure/Control co-design. In order to demonstrate the strengths of the proposed approach, a structure/control co-design study case is proposed and solved using structured robust 𝐻 ∞ -synthesis. The objective is to optimize the pointing performances of an antenna, minimizing the perturbations coming from the Solar Array Driving Mechanisms (SADM) of two solar panels, performing active control by means of multiple Proof Mass Actuators (PMA), and simultaneously reduce the mass of the truss-structure which connects the antenna to the main spacecraft body.

Introduction

In order to systematically face the challenges associated with the next generation of satellites, the European Space Agency (ESA) and NASA have combined their past experiences to cope with the fine pointing requirements of high accuracy observation and Science missions [START_REF] Dennehy | Spacecraft Micro-Vibration: A Survey of Problems, Experiences, Potential Solutions, and Some Lessons Learned[END_REF]. This represent a domain which is extremely multi-disciplinary: structural, control and system engineering considerations must coalesce to limit the propagation and amplification of internally generated disturbances through the satellite's flexible structures. For these reasons, the development of rigorous methodologies and design tools that can handle all these domains is crucial at early stages of design. The works of Preda et al. [START_REF] Preda | Robust microvibration mitigation and pointing performance analysis for high stability spacecraft[END_REF] and Sanfedino et al. [START_REF] Sanfedino | Robust active mirror control based on hybrid sensing for spacecraft line-of-sight stabilization[END_REF][START_REF] Sanfedino | Integrated modeling of microvibrations induced by Solar Array Drive Mechanism for worst-case end-to-end analysis and robust disturbance estimation[END_REF] are example or this approach.

In the past decades, structural and control co-design has attracted a lot of attention due to its ability of merging these multiple multidisciplinary requirements into a single design flow. Moreover, the increasing use of large structures and appendages for Space applications has rendered flexible modal analysis mandatory for the design of proper spacecraft control laws. nature of the two sub-problems: mass reduction can increase significantly the flexibility of the system, whose low frequency modes may interfere with the satellite's Attitude Control System (ACS), as seen in [START_REF] Falcoz | Integrated control and structure design framework for spacecraft applied to biomass satellite[END_REF].

When an analytical model of the structure is available, the integration of the design variables as uncertain parameters opens the possibility of achieving control/structure co-design in a unique iteration, using the non-smooth techniques available in the robust structured 𝐻 ∞ control framework [START_REF] Gahinet | Structured 𝐻 ∞ synthesis in MATLAB[END_REF], as shown in several aerospace applications [START_REF] Perez | Linear dynamic modeling of spacecraft with open-chain assembly of flexible bodies for ACS/structure co-design[END_REF][START_REF] Alazard | Avionics/control co-design for large flexible space structures[END_REF][START_REF] Denieul | Multicontrol surface optimization for blended wing?body under handling quality constraints[END_REF]. A drawback of such a direct co-design is that it could provide a non-global optimal design, by risking to fall within one of the many local minima. On the other hand, this approach has the advantage to be very easy to implement and very efficient from the computational time point of view when the number of sizing parameters is not too high. This work uses this approach for the structure/control co-design problem presented in the case study.

The main contributions of this paper are:

• development of a sub-structured model fully parametrized by both mechanical sizing parameters and uncertain parameters, allowing the complex truss structure to be assembled by block-diagram interconnection, • validation of a 3D-cube element by a detailed comparison with a NASTRAN model, • co-design of a complex space truss structure holding an optical payload and of the LOS (Line Of Sight) stabilization controller using multiple PMAs.

This work must also be positioned in the field of the Control Structure Interactions (CSI) which motivates lots of contributions since the 90's with a particular emphasize on model uncertainties. In [START_REF] Balas | Robust Control of Flexible Structures: Theory and Experiments[END_REF], G. Balas already addressed the control of an experimental flexible truss structure (named Caltech Flexible Structure) using PMAs with a deep insight on the quantization of the uncertainties between the mathematical (or knowledge-based) model and the physical model. Uncertainties is also largely addressed in [START_REF] Campbell | Uncertainty modeling for structural control analysis and synthesis[END_REF] and discussed on the MACE (Middeck Active Control Experiment) considering the impact of the suspension system required under 1-g environment on the system dynamical behavior under 0-g environment. A first toolbox for Uncertainty Bound IDentification (UBID) was proposed in [START_REF] Lim | Structured Uncertainty Bound Determination From Data for Control and Performance Validation[END_REF]. More generally, such uncertainty quantization was required to feed robust control and analysis methods. The robust control design methods available till 2010 addressed only the full order design (i.e. the optimal controller have the same order than the design model). Thus, one can also find many contributions on reduction and associated reduction error bounds to reduce the order of the design model (and therefore of the controller). Today, effective methods to design robust structured controllers [START_REF] Gahinet | Structured 𝐻 ∞ synthesis in MATLAB[END_REF] where the order and/or the structure of the controller can be fixed rekindle the debate. In the present work, the general guideline is to push the knowledge-based model as far as possible and to consider it as the design model even if this model is a high-order model with a complex dependence on uncertain (real) parameters. Such an approach can be justified in the context of control/structure co-design during early design phases [START_REF] Bourgault | Model Uncertainty and Performance Analysis for Precision Controlled Space Structures[END_REF] when ground-based experimental mock-ups are not available. In addition, in the field of space engineering, the experiments for controls/structure interactions are very sensitive to the 1 g environment or to the gravity compensation mechanisms.

After a brief introduction on the TITOP approach used to model a flexible appendage in Section 2, the modeling of all 2D mechanism is detailed in Section 2.1. The introduction and validation of the cubic structural element is performed in Section 2.2. The in-depth description of the co-design study case is performed in Section 3. Sections 3.1 to 3.3 presents the model of the whole system from the various sub-structures. Section 3.4 details the closed-loop control strategy. Sections 3.5 to 3.7 presents the control/structure co-design formulation, results and validation.

TITOP approach

Let us consider a flexible body  𝑖 as seen in Fig. 1 (left) connected to a parent structure  𝑖-1 at the point 𝑃 and to a child structure  𝑖+1 at the point 𝐶. The resulting TITOP model D  𝑖 𝑃 𝐶 (s), schematized in Fig. 1 (right), is a {12 × 12} linear dynamic model whose inputs are:

• W  𝑖+1 ∕ 𝑖 , 𝐶 : the {6 × 1} wrench (forces and torques) applied by the body  𝑖+1 to  𝑖 at point C;

• ü𝑃 : the {6 × 1} inertial acceleration (linear and angular) imposed by the parent body  𝑖-1 at point 𝑃 to  𝑖 ; and the conjugated outputs are:

• ü𝐶 : the {6 × 1} components of the inertial acceleration of point 𝐶;

• W  𝑖 ∕ 𝑖-1 , 𝑃 : the {6 × 1} wrench applied by  𝑖 to the parent structure  𝑖-1 at point P.

All these input/output variables are projected in the body frame. Thus, to lighten the notations, the projection frame is not mentioned in the various block-diagrams presented hereafter.

As described in [START_REF] Chebbi | Linear dynamics of flexible multibody systems: A system-based approach[END_REF], the state-space representation of D  𝑖 𝑃 𝐶 (s) can be directly built from the data (flexible mode frequencies, modal participation factors and modal shapes) of the clamped at 𝑃 -free at 𝐶 model of the body  𝑖 . Then channel inversion operations can be used to derive the model under different boundary conditions.

Let us denote [M(s)] -1 I the model M(s) where the channel numbered in the vector of indexes I are inverted following the procedure described in [START_REF] Chebbi | Linear dynamics of flexible multibody systems: A system-based approach[END_REF] (Appendix 1), then for instance: 3 4 6 11] models the body  𝑖 under the pinned at 𝑃 and 𝐶 boundary conditions. Both pinned axes are the y-axis of the body frame used to project the model (Fig. 1 (left)). A multibody system composed of several flexible bodies and several revolute or clamped joints can then be built by the interconnection of the TITOP models of each body and the twice 6 × 6 DCM (Direction Cosine Matrix) between the various body frames. This approach was embedded in the Satellite Dynamics Toolbox (SDT) with various features:

• [D  𝑖 𝑃 𝐶 (s)] -1 [1:6] models the body  𝑖 under the clamped at 𝑃 -clamped at 𝐶 boundary conditions, • [D  𝑖 𝑃 𝐶 (s)] -1 [1 2
• a NASTRAN/SDT interface to build the TITOP model directly from the output files (.f06 and .bdf) of the NASTRAN/PATRAN model, • an analytical TITOP model of the Euler-Bernoulli beam in the 6 d.o.fs (degree-of-freedom) case fully parametrized according to its length 𝑙 (along the x-axis of the beam reference frame), its section area 𝑆, its second moments of area 𝐼 𝑧 and 𝐼 𝑦 , its Young modulus 𝐸 and its mass density 𝜌. These parameters can be declared as varying parameters to obtain an LPV (Linear Parameter Varying) model fully compliant with the MATLAB ® Robust Control Toolbox to perform robust design and performance analysis.

This beam model, fully detailed in [START_REF] Chebbi | Linear dynamics of flexible multibody systems: A system-based approach[END_REF], is used as the basic element for the truss structure models presented in the sequel. The reader is advised to read the SDTlib Users' Manual to have a deep insight in the SDT [START_REF] Alazard | Satellite Dynamics Toolbox library SDTlib -User's Guide[END_REF].

2D mechanisms

In this section, a series of two-dimensional mechanisms is presented. The main goal of these kinematics is to act as intermediate step towards building complex three-dimensional structures. By exploiting the assembly of multiple elementary TITOP beam blocks, several multi-body mechanical systems have been implemented: the so called L-Chain mechanism, the Triangle mechanism and the Square mechanism. This section will detail the geometrical characteristics of these kinematics and their modeling using a block-diagram approach.

L-Chain mechanism

The L-Chain mechanism  is composed by two beams 𝐴𝐵 (body  1 ) and 𝐶𝐵 (body  2 ) connected at point 𝐵 in a given angular configuration 𝛼, as seen in Fig. 2(a) (in the case 𝛼 = 𝜋∕2 (𝑟𝑎𝑑)). This mechanism is linked:

• to two parent bodies  𝐴 and  𝐶 at the point 𝐴 and 𝐶, imposing accelerations ü𝐴 and ü𝐶 , • to a child body  𝐵 at point 𝐵 applying a wrench W  𝐵 ∕, 𝐵 . The 3 input -3 output ports model of the mechanism  is then described by the block-diagram depicted in Fig. 2(b). This model involves the clamped at 𝐴 -free at 𝐵 model D

 1 𝐴, 𝐵 (s) of the beam  1 and the clamped at 𝐶 -clamped at 𝐵 model [D  2 𝐴, 𝐵 (s)] -1 [1∶6] of the beam  2 .
The upper ports of these to sub-model are connected in a feedback loop to take into account:

• the loop closure constraint: the point 𝐵 on the 2 bodies must have the same accelerations ü𝐵 , • the wrench balance at the point 𝐵 of beam  1 :

W ( 2 + 𝐵 )∕ 1 ,𝐵 = W  𝐵 ∕,𝐵 -𝑅 2,1 W  1 ∕ 2 ,𝐵 .
Note that in Fig. 2 the blocks 𝑅 𝑖,𝑗 correspond to the twice 6 × 6 DCM from the body frame of beam  𝑖 to the body frame of beam  𝑗 .

This model is named clamped-free-clamped (CFC) L-Chain model and is denoted  𝐶𝐹 𝐶 (s). Indeed when the three inputs ü𝐴 , W  𝐵 ∕ , 𝐵 and ü𝐶 (in this order) are null, the  mechanism is clamped at point 𝐴, free at point 𝐵 and clamped at point 𝐶. This model described the dynamic behavior between three conjugated input-output pairs associated to the three ports (connection points) of the mechanism. Thus, using the channel inversion operation, one can also define:

 𝐶𝐶𝐶 (s) = [ 𝐶𝐹 𝐶 (s)] -1 [7∶12] ,  𝐹 𝐹 𝐶 (s) = [ 𝐶𝐹 𝐶 (s)] -1 [1∶6] , …
All the inputs and outputs of the model  𝐶𝐹 𝐶 (s) are projected in the body frame (x  , y  , z  ) of the mechanism , chosen aligned with the frame of body  1 (see Fig. 2(a)).

Triangle mechanism

The modeling of a basic triangular closed-loop mechanism is hereby considered. This mechanism is first modeled in the clamped at 𝐴, free at 𝐵, free at 𝐶 boundary conditions considering the acceleration ü𝐴 , the wrenches W  𝐵 ∕ , 𝐵 and W  𝐶 ∕ , 𝐶 as inputs applied by the external bodies  𝐴 ,  𝐵 and  𝐶 at points 𝐴, 𝐵 and 𝐶, respectively (see Fig. 3(a)).

This model, denoted  𝐶𝐹 𝐹 (s) can then be represented by the block-diagram depicted in Fig. 3(b). Indeed, since the loop closure constraint at the point 𝐵 is already taken into account in the model  𝐶𝐹 𝐶 (s) previously presented (considering now that 𝛼 = 𝜋∕4 (𝑟𝑎𝑑)), the model  𝐶𝐹 𝐹 (s) can be built by adding the model D  3 𝐴, 𝐶 (s) of the third beam 𝐴𝐶 (body  3 ) connected to the model  𝐶𝐹 𝐶 (s) to satisfy the new constraints:

• the kinematic constraint: the point 𝐴 on the 2 bodies  1 and  3 must have the same accelerations ü𝐴 , • the wrench balance at the point 𝐴:

W  ∕ 𝐴 ,𝐴 = W  1 ∕ 𝐴 ,𝐴 + 𝑅 3, 1 W  3 ∕ 𝐴 ,𝐴 ,
• the wrench balance at the point 𝐶:

W ( 𝐶 + 2 )∕ 3 ,𝐶 = 𝑅 1, 3 ( W  𝐶 ∕ ,𝐶 + 𝑅 2, 1 W  2 ∕ 3 ,𝐶 ) .
All the inputs and outputs of this 3 input-3 output port model  𝐶𝐹 𝐹 (s) are projected in the body frame (x  , y  , z  ) of the mechanism  , chosen aligned with the frame of body  1 (see Fig. 3(a)).

Square mechanism

The square mechanism presented in Fig. 4 is composed of five beams: four of them form a polygonal perimeter and the last one is positioned diagonally to create two closed loop chains. This mechanism is modeled in the clamped at 𝐴, free at 𝐵, free at 𝐶, free at 𝐶 boundary conditions considering the acceleration ü𝐴 , the wrenches W  𝐵 ∕, 𝐵 , W  𝐶 ∕, 𝐶 and W  𝐷 ∕, 𝐷 as inputs applied by the external bodies  𝐴 ,  𝐵 ,  𝐶 and  𝐷 at points 𝐴, 𝐵, 𝐶 and 𝐷, respectively (see Fig. 4(a)).

This model, denoted  𝐶𝐹 𝐹 𝐹 (s) can then be represented by the block-diagram depicted in Fig. 4(b). It involves directly the model  𝐶𝐹 𝐹 (s) of the triangular mechanism and the model  𝐶𝐹 𝐶 (s) of the L-chain mechanism with two feedback loops between their ports relative to the points 𝐵 and 𝐶, allowing to take into kinematics constraints and wrench balances at these two connection points. This allows for a drastic reduction in the assembly complexity: the model is composed only by two blocks while representing five flexible bodies in total. This showcases the power of this modular approach in structural design, whose advantages will be fully displayed in the three-dimensional approach introduced in Section 2.2.

All the inputs and outputs of this 4 input -4 output port model  𝐶𝐹 𝐹 𝐹 (s) are projected in the body frame (x  , y  , z  ) of the mechanism , chosen aligned with the frame of the  mechanism (see Fig. 4(a)). Using the channel inversion operation, one can also defined:

 𝐶𝐶𝐶𝐶 (s) = [ 𝐶𝐹 𝐹 𝐹 (s)] -1 [7∶24] .

3D Mechanisms

The two-dimensional elements introduced in Section 2.1 ans the channel inversion operation are powerful tools that can be used for the creation and assembly of complex three-dimensional structures. In the context of the multi-body approach followed in this paper, the mechanical conception focused on the definition of a unit-cube 3D module which could represent the basic building block for large space truss structures.

Cube mechanism

The Cube Mechanism  is a multi-body structure composed by 13 flexible appendages, assembled at 8 nodes to form a cubic outline with diagonal elements along the faces. A representation of its complex kinematics is given in Fig. 5(a). Each node 𝑖, 𝑖 = 1, … , 8 of the cube is connected to an external body  𝑖 imposing an acceleration ü𝑁 𝑖 (case of a parent body) or applying a wrench W  𝑖 ∕,𝑁 𝑖 (case of child body).

At a first glance, it can be noticed that no flexible appendix can be found to form the side of the bottom face of the Cube, along the (𝑥, 𝑦) plane. This has been done to facilitate the construction of complex mechanical systems: this structure is conceived as a unit-cube-module which can be stacked on top of other elements, serially connecting them to create an elongated system. In the same fashion, multiple cubes can then be added on the sides as well. An example of this sub-structuring modeling technique will be outlined in the case study of Section 3.

In order to facilitate this serial connection of cubes along the 𝑧-axis, the design of this mechanical system has been carried out to present four lower nodes (from 𝑁 1 to 𝑁 4 ) with accelerations imposed on the structure by the external parent bodies  𝑖 (𝑖 = 1, … , 4), while the upper nodes of the structure (from 𝑁 5 to 𝑁 8 ) are subjected to the wrenches transmitted by the external child bodies  𝑖 (𝑖 = 5, … , 8). These excitation acting on the system, highlighted in red in Fig. 5(a), will represent the inputs of the 8 input -8 output port model of the cube, denoted  𝐶𝐶𝐶𝐶-𝐹 𝐹 𝐹 𝐹 (s). Following the general NINOP model formalism, the outputs are the conjugate of the inputs.

As it can be seen in Fig. 5(b), the model  𝐶𝐶𝐶𝐶-𝐹 𝐹 𝐹 𝐹 (s) of this complex three-dimensional structure can be easily modeled by means of only two different 2D mechanisms: Clamped-Free-Clamped (CFC) L-Chains  𝐶𝐹 𝐶 𝑗 (s), repeated four times, and one single Clamped-Clamped-Clamped-Clamped (CCCC) Square Mechanism  𝐶𝐶𝐶𝐶 (s). The imposed accelerations at nodes 𝑁 1 to 𝑁 4 have been inputted directly to the two clamped ends of the L-Chains, while their free vertex receives the combined effort of the external forces applied by 𝐵 𝑖 (𝑖 = 5, … , 8), and  𝐶𝐶𝐶𝐶 (s). Exactly like in the previous cases, the feedback loops between the sub-blocks allows to satisfy the kinematics constraints and the wrench balances.

Finally, in Fig. 5(b) it can be noted that this model does not present DCM between the blocks. This is made possible by the fact that the changes of reference frames are handled internally within the elementary blocks, allowing the possibility to express all vectors in any generic common frame (for instance, the frame (𝑁 1 , x, y, z) for the cube).

Table 1

Parameters of the cube sides and TITOP Beams used for the validation of Cube model.

𝑙 𝑥 [m] 𝑙 𝑦 [m] 𝑙 𝑧 [m] 𝑆 [m 2 ] 𝜌 [kg∕m 2 ] 𝐸 [GPa] 𝜈 𝐼 𝑦 [m -4 ] 𝐼 𝑧 [m -4 ] 𝜉 1.0 1.

System validation

The sub-structuring technique presented in the previous sections has been implemented in MATLAB ® SIMULINK ® in the form of N-Input-N-Output Port block models, which depict the dynamical behavior of each mechanism. These blocks integrate the analytical beam model of the SDT library [START_REF] Alazard | Satellite Dynamics Toolbox library SDTlib -User's Guide[END_REF].

These new models of elementary truss structures are now validated by comparison with the models obtained from a widespread finite-element-model commercial software: MSC Patran/Nastran. This validation is performed on the Cube structure. Since it embeds the three elementary (L-chain, Triangle and Square) sub-structures, the validation of the Cube model acts as a general validation of all its sub-structural components.

Geometry definition and Patran/Nastran modeling. A cube mechanism, as described in Section 2.2.1 is hereby considered for verification purposes. Its geometry is fixed by means of the length of its sides, 𝑙 𝑥 , 𝑙 𝑦 and 𝑙 𝑧 , along the axis of the (𝑁 1 , 𝑥, 𝑦, 𝑧) reference frame and by the mechanical characteristics of each appendage composing the kinematics. The same homogeneous beam has been repeated for each flexible body. The full mechanical characterization of the validation model is described by Table 1, where 𝜈 is the Poisson's coefficient and 𝜉 is the damping factor.

For the comparison with the Nastran model, the cube is only clamped at node 𝑁 1 whose model (labeled SDT ) is obtained using the channel inversion operation:

 𝐶𝐹 𝐹 𝐹 -𝐹 𝐹 𝐹 𝐹 (s) = [ 𝐶𝐶𝐶𝐶-𝐹 𝐹 𝐹 𝐹 (s)] -1 [7∶24] .
That allows to compare not only the flexible mode frequencies but also their modal participation factors at this node. The same mechanical system has been implemented in MSC/Patran, using the same mechanical characteristics of the SDT model and modeling each beam using the CBEAM element property, in order to take torsional behavior into account. The 3D model (labeled Patran) created in MSC/Patran is displayed in Fig. 6, where the 5 elements used for each beam can be distinguished. The structure can be seen clamped at the origin of the (𝑥, 𝑦, 𝑧) axis, which corresponds to node 𝑁 1 .

Table 2 describes the first ten modes of the two models (SDT and Patran) by means of their natural frequencies 𝜔 𝑘 and their modal participation factors. The comparison shows a good match in both physical properties.

A more detailed validation is possible thanks to the interface between MSC/Nastran and SDT library which allows to import the MSC/Patran model directly in MATLAB ® /SIMULINK ® . From the Nastran .f06 analysis file, this interface provides the 6 × 6 transfer from the acceleration üN 1 at the node 𝑁 1 to the reaction wrench W ∕ 1 ,N 1 at this node. The frequency-domain response for the 3 translation degrees of freedom of this transfer is depicted in Fig. 7 (labeled P/N ) and compared with the ones from the proposed model (labeled SDT ). The analysis of these plots confirms a good match between the two models. In the [0, 500] rad∕s frequency range, the relative error on each transfer is below 5% except of course on the resonances which are not exactly at the same frequencies (see Table 2).

Application on space structural and control co-design study case

This section introduces a study case on a space application to demonstrate the power of the proposed multi-body sub-structuring approach for modeling and control of flexible structures. Let us consider a Telecom satellite having one High-Precision-Pointing (HPP) antenna connected to the main spacecraft (S/C) body by means of a large truss structure as in Fig. 8. Two Solar Panel Arrays are connected to the sides of the satellite and are able to rotate thanks to Solar Array Driving Mechanisms (SADM), which on the other hand introduce perturbations in the attitude of the spacecraft and in the pointing of the antenna.

In this context, the blocks introduced in Sections 2.1 and 2.2 have been used to construct the large T-Shaped truss structure 𝑇 𝑠𝑡𝑟 , which support the high-precision antenna. Given that the multi-body design is based on the TITOP beam model, this complex flexible system can be fully parametrized for any physical property characterizing the beams. A structural and robust control co-design has been implemented for the system, to showcase the strength of a parametric structural model by achieving the two following concurrent goals:

1. Reject the perturbations introduced by the SADMs acting on the Line-of-Sight (LOS) of the Antenna by performing an active control using 4 PMAs, distributed on the structure. 2. Minimize the mass of the T-Truss structure 𝑇 𝑠𝑡𝑟 by reducing the section of the beams composing the system, all while complying with the pointing and vibration rejection requirements.

The following sections will detail the design and assemblage of the T-Truss structure and the procedure chosen to implement the co-design.

Flexible T-Truss structure

The T-shaped Truss antenna support has been modeled by means of five Cube mechanism elements, connected to each other to form the structure seen in Fig. 9, where the diagonal beams on the faces have been hidden to facilitate reader's understanding of the The section area 𝑆 and bending inertia 𝐼 𝑦 , 𝐼 𝑧 of each beam is expressed in the generic beam local reference of frame  𝑙 (𝑁 𝑙 , 𝑥 𝑙 , 𝑦 𝑙 , 𝑧 𝑙 ) as function of the side parameter ℎ.

Table 3

Parameters of TITOP Beams used for the T-Truss Structure.

𝑆 [m 2 ] 𝜌 [kg∕m 2 ] 𝐸 [GPa] 𝜈 𝐼 𝑦 [m -4 ] 𝐼 𝑧 [m -4 ] 𝜉 ℎ 2 2700 70 0.35 ℎ 4 ∕12 ℎ 4 ∕12 0.001
architecture. Overall, this complex mechanical system is composed by 65 flexible beams. Among them, it is possible to identify a core assembly of three cubes ( 1 ,  2 and  3 ) which are simply serially stacked on top of each other, along the z 𝑔𝑙 axis on the global frame of reference (𝑁 1 , x 𝑔𝑙 , y 𝑔𝑙 , z 𝑔𝑙 ). The remaining two cubes are then placed on the sides of the previous assembly, developing along the y 𝑔𝑙 axis direction. This is achieved by the use of two direction cosine matrices, which perform a rotation of the cubes around the x 𝑔𝑙 axis. This results in cube  4 facing towards the negative direction of y 𝑔𝑙 while  5 towards the positive one.

The overall structure is composed by 24 nodes in total. Four of them -nodes from 𝑁 1 to 𝑁 4 -are connected to the S/C main body and therefore are designed to have an acceleration imposed onto them. The rest of the points on the other hand can receive an external input in the form of an external wrench. This is the case for nodes 𝑁 17 and 𝑁 20 , where the PMAs and the antenna is connected, while no external forcing term are applied to other points of the structure. In terms of physical and geometrical characterization of the system, it has a total envelope of 1 m×3 m×3 m, with single cube having a volume of 1 m×1 m×1 m. The same properties have been repeated for all the 65 beams composing the an aluminum beam with a square section, a parametric model has been implemented by means of the section length ℎ. This variable drives both the section area and the second moments of area of the beam, as displayed in Fig. 9. The parameter will then be at the center of the structural-control co-design of Section 3, as it is directly related to both mass and stiffness properties of the system. Finally, the mechanical characterization of all the beams is given in Table 3.

Proof Mass Actuators (PMAs) model

The PMA mechanical system has been modeled in Fig. 10 as the body , composed by a rigid casing and a one-dimensional spring-mass-damper system. Under these assumptions, the mechanical actuator is fully defined by the following set of parameters:

•  𝑎 : local reference frame attached to the PMA at the reference point 𝑂;

• v: the unit vector along the PMA axis, expressed in  𝑎 ;

• 𝐺 and 𝑃 : the center of mass of the PMA (at rest) and the connection point to the parent body, respectively;

• 𝑀  and I  𝐺 : respectively, the mass and the inertia matrix at 𝐺 of the PMA casing; • 𝑚 𝑝 , 𝑘 𝑝 , 𝑑 𝑝 .: the mass, stiffness and damper of the spring-mass-damper system describing the dynamics of the PMA along the axis v; • 𝑢: control effort applied on the proof mass along the axis v; • 𝛿 𝑥 . Relative displacement of the proof mass with respect to the casing; • ü𝑃 : {6 × 1} acceleration twist of the PMA at the connection point expressed in the  𝑎 frame; • W ∕.,𝑃 : {6 × 1} wrench applied by the PMA at point 𝑃 , expressed in the  𝑎 frame;

The PMA mechanical system  has then been modeled as D  𝑃 (s), a {7 × 7} linear dynamic model in the TITOP approach. The system is obtained by means of the equations of the mechanical dynamics, defined as follows:

𝑚([v 𝑇 0 1×3 ] ü𝐺 + δ𝑥 ) = -𝑘𝛿 𝑥 -𝑑 δ𝑥 + 𝑢 (1) ü𝐺 = 𝝉 GP ü𝑃 (2) W ∕.,𝑃 = -𝝉 𝑇 GP ([ 𝑀I 3 0 3 0 3 I  𝐺 ] 𝝉 GP ü𝑃 + 𝑚 [ v 0 ] δ𝑥 ) (3)
𝝉 GP = [ I 3 [r GP ] × 0 3×3 I 3 ] (4)
where 𝝉 𝐺𝑃 is the Kinematic Model of the rigid link between point 𝐺 and point 𝑃 , defined thanks to the skew matrix [r 𝐺𝑃 ] × associated to the vector r 𝐺𝑃 from node 𝐺 to node 𝑃 . The PMA 7 × 7 block-diagram model D  𝑃 (s) is showcased in Fig. 10. The first 6 × 6 port of the model defines the transfer between the acceleration ü𝑃 and the wrench W ∕.,𝑃 , while the last describes the one between the control effort 𝑢 and the proof mass relative displacement 𝛿 𝑥 . The mechanical parameters which describe the PMAs used in the current case study are given in appendix, Table 6.

System modelization

Full order Linear Parameter Varying (LPV) model

The complete system in Fig. 8 is modeled by means of a block-diagram approach using elements derived from the SDT library introduced in [START_REF] Alazard | Satellite Dynamics Toolbox library SDTlib -User's Guide[END_REF]. The block-diagram representation of the whole spacecraft is then depicted in Fig. 11 and is detailed by:

• Central body (𝑆∕𝐶). This block is a static 42 × 42 multi-port rigid body model. It models the dynamics of a rigid body subjected to multiple wrenches applied at connection points 𝑃 1 , the center of mass where is located the Attitude Control System (ACS), 𝑃 2 , 𝑃 3 , 𝑃 4 , 𝑃 5 the 4 connection points with the T-truss structure, and 𝑃 6 , 𝑃 7 the connection points with the 2 solar panels. The geometry of these points is detailed in Appendix, Table 7. • Solar Panels (𝑆𝑃 𝑖 , 𝑖 = 1, 2). These 2 blocks are 6-th order single port flexible body models. Each solar panel model takes into account 3 flexible modes and is connected to the central body by a revolute joint driven along the 𝑥-axis by the SADM. The two solar panels are identical. The joint angular configuration 𝜃 is taken into account in the DCM 𝑅 𝑆𝐶∕𝑆𝑃 𝑖 . The SADM is the main source of disturbance and is modeled by a local stiffness 𝑘 𝑆𝑃 𝑖 , a viscous friction 𝑓 𝑆𝑃 𝑖 and an internal disturbing torque 𝑝 𝑆𝑃 𝑖 . The overall torque acting on the mechanisms is: • 4 PMAs subsystem (𝑃 𝑀𝐴𝑠). This 16 × 16, 8-th order block embeds the 4 PMAs as described in 3.2. The positioning of these PMAs at node 𝑁 20 and 𝑁 17 has been selected to maximize the control effect over the orientation of the LOS. Thanks to the use of two couples of PMAs acting in the same direction, the control system is able to control the elevation and the azimuth of the LOS and the 2 translations in the plane orthogonal to the LOS.

𝑢 𝑆𝑃 𝑖 =
The data of all spacecraft sub-components can be found in Appendix, Table 6.

The full order model of the whole system is thus a 1324-th order model. Its 20 outputs y, seen in blue in Fig. 11 are:

• LOS: 2 × 1 vector with LOS elevation and azimuth angular accelerations. This is one of the parameters taken into account in the performance index of the co-design, • x PMAs u : 4 × 1 vector of the linear internal displacements of the 4 PMAs.

• üP 1 : 6 × 1 acceleration dual vector of the center of mass of the central body,

• θPMA s : 8 × 1 vector of the linear and angular accelerations at nodes 𝑁 17 and 𝑁 20 along and around the x and y axes.

The 12 inputs u of the model, highlighted in red in Fig. 11,are: • 𝑝 𝑆𝑃 1 , 𝑝 𝑆𝑃 2 : the two internal perturbation torques acting at the revolute joints of the each SADM, • W P 1 : 6 × 1 wrench vector at the center of mass of the spacecraft 𝑃 1 ,

• W PMAs u : 4 × 1 vector of linear forces applied to the system at nodes 𝑁 17 and 𝑁 20 by the four PMAs.

One of the main interest of the SDT lies in the possibility of declaring the various mechanical and geometrical parameters as varying parameters or uncertain parameters. Thus, the LPV model directly computed from the block-diagram description is fully compatible with the MATLAB ® robust control toolbox to perform robust performance analyzes and robust control designs. That is particularly relevant during preliminary design of Space systems to face fabrication unpredictable inaccuracies and also uncertainties or misknowledges on some sub-systems whose the design is not finalized. It is all the more relevant for preliminary co-design where some of these parameters are tunable parameters. In this case study, two kinds of parameters are considered:

• the uncertain parameters: 𝜹 = [𝛿 𝑀 𝑆𝐶 , 𝛿 𝐼 𝑦𝑦 𝑆𝐶 , 𝛿 𝜔 1 𝑆𝑃 , 𝜏] 𝑇 composed of normalized (∈ [-1, 1]
) uncertainties on the mass of the main spacecraft hub 𝑀 𝑆𝐶 , the inertia along the 𝑦-axis of the same body 𝐼 𝑦𝑦 𝑆𝐶 , the first natural frequency of both solar panels 𝜔 1 𝑆𝑃 and the angular configuration 𝜃 ∈ [-𝜋, ±𝜋] of the solar panels parametrized by 𝜏 = tan(𝜃∕4) ∈ [-1, 1] as introduced in [START_REF] Guy | Dynamic modeling and analysis of spacecraft with variable tilt of flexible appendages[END_REF][START_REF] Dubanchet | Modeling and Control of a Flexible Space Robot to Capture a Tumbling Debris[END_REF]. The uncertainty values are defined in Appendix, Table 6, • the tunable parameter: 𝜣 = ℎ = ℎ 0 (1 + 0.95 𝛿 ℎ ), the side of the T-truss beam section with: Remark. the variation of the beam section side is quite wide. Note that the lower bound (1 mm) is certainly not representative. Constraints on ℎ required for the truss structure (in a stowed configuration) to support launch loads are not considered in this study. The main objective of this paper is to illustrate the interest of the parametric model approach for optimization and co-design purposes.

ℎ 0 = 2 𝑐𝑚 𝛿 ℎ ∈ [-1, 1] . ( 6
)
From the block-diagram model depicted in Fig. 8, the LPV full order model (𝜣, 𝜹) is directly computed thanks to the MATLAB ® function ulinearize and fully characterized by the arguments of the following Linear Fractional Transformation (LFT):

(𝜣, 𝜹) =  𝑢 ((Θ), 𝜟) and (Θ) =  𝑢 (G(s), ∆ Θ ) (7)
with 𝜟 the uncertain parameter block and ∆ Θ the tunable parameter block defined by:

𝜟 = diag ( 𝜟 𝜏 , ∆ 𝑝 ) , 𝜟 𝜏 = 𝜏 I 16 , ∆ 𝑝 = diag ( 𝛿 𝑀 𝑆𝐶 I 3 , 𝛿 𝐼 𝑦𝑦 , 𝛿 𝜔 1 𝑆𝑃 I 4 ) , ∆ Θ = 𝛿 ℎ I 877 (8)
where I 𝑛 is the identity matrix of size 𝑛. This size indicates the number of occurrences of each parameters. Thus the system presents an extremely high number of occurrences for the tunable parameter ℎ, amounting to 877 occurrences. These upper LFTs  𝑢 (., .) can be interpreted by the block-diagram interconnections as depicted in Fig. 12.

System reduction

As mentioned in the previous section, the order of G(s) is very high (1324). A reduction in the modal state-space representation of G(s) is thus performed to remove the very slow poles and the high frequency flexible modes based on the following considerations:

• in the modeling of the T-truss structure, the numerous loop closure constraint are solved thanks to the channel inversion operation. Each closed-loop kinematic chain concerns the 6 d.o..fs and leads to 12 poles at 0. But numerical errors during the inversion make that these poles ar not exactly at 0. These very slow poles do not contribute in the input-output transfer of G(s) and they can be removed by a reduction in the modal realization of G(s) of the stable poles with a magnitude lower than 0.01 rad∕s. This reduction removes 540 (= 45 × 12 considering the 45 closed-loop kinematic chains inside the T-truss structure) poles. This reduced order model is denoted G r𝑙𝑜𝑤 (s), • very high frequency flexible mode, with a frequency greater than 700 rad∕s, are out of the frequency range considered for the active control with the PMAs. They are also reduced in the modal realization of G(s). This final reduced order model is denoted G r (s) and its order is 296.

The goodness of the reduction can be appreciated by comparing the frequency-domain responses of the input-output transfer (u → y) between the full order model G(s) and the 2 reduced order models G r𝑙𝑜𝑤 (𝑠) and G r (s), as done in Fig. 13. The magnitude of G(s) -G r𝑙𝑜𝑤 (s) is totally neglectable, showing the irrelevance of the loop-closure poles, while the magnitude G(s) -G r (s) shows an acceptable error. The final model G r (𝑠) correctly represents the dynamics of the open-loop system in the frequency domain of interest and will be used to perform all further analysis and controller synthesis using the upper linear fractional transformations  𝑟 (Θ) =  𝑢 (G r (s), Θ) for the LPV system and  𝑢 ( 𝑟 (Θ), 𝜟) for the overall uncertain open-loop system. Nevertheless, the performance robustness analysis proposed in Section 3.7 will consider the model G r𝑙𝑜𝑤 (s) with all the flexible modes.

Closed-loop system

As previously stated, the main objective of this case study is to demonstrate the capability of performing structural and control codesign using parametric models for the flexible appendages, suited for robust analysis. In this context, the open-loop linear model described in the previous section can be augmented with the controllers acting on the system and the pointing performance indexes as shown in Fig. 14.

The closed loop system is composed by two controllers: • The Attitude Control System 𝐴𝐶𝑆. The ACS implemented for this mission is a basic proportional-derivative (PD) controller aimed at fixing the attitude of the satellite in space.

K 𝐴𝐶𝑆 (s) = -[0 3×3 I 3 ] 𝑇 k 𝑝 + k 𝑣 s s 2 [0 3×3 I 3 ] (9)
K 𝐴𝐶𝑆 (s) includes the rigid motion of the whole spacecraft. The decentralized (diagonal) controller gains have been chosen in order to fix the poles of the ACS at a frequency of 𝜔 𝐴𝐶𝑆 = 0.1 rad/s with a damping of 𝜉 𝐴𝐶𝑆 = 0.7 assuming the whole spacecraft is rigid:

k 𝑝 = 𝜔 2 𝐴𝐶𝑆 diag(J 𝑡𝑜𝑡 ), k 𝑣 = 2𝜉 𝐴𝐶𝑆 𝜔 𝐴𝐶𝑆 diag(J 𝑡𝑜𝑡 ) (10)
where J 𝑡𝑜𝑡 is the total inertia matrix of the satellite computed at point 𝑃 1 . • PMAs Controller. The controller for the actuator was implemented as by means of the filter F PMAs (s) and a static gain given by matrix K PMA . F PMAs (s) is a diagonal 4 × 4 filter composed of an integrator, required to estimate the velocity from the measured acceleration, a wash-out filter with a cut-off frequency 𝜔 𝑃 𝑀𝐴 = 5𝜔 𝐴𝐶𝑆 = 0.5 rad/s and a low-pass filter to prevent spillover on neglected flexible modes with a cut-off frequency 𝜔 𝐿𝑃 = 400 rad/s:

F 𝑃 𝑀𝐴 (s) = s s 2 + 1.4𝜔 𝑃 𝑀𝐴 s + 𝜔 2 𝑃 𝑀𝐴 𝜔 2 𝐿𝑃 s 2 + 1.4𝜔 𝐿𝑃 s + 𝜔 2 𝐿𝑃 I 4 . ( 11
)
The fine pointing performance is expressed in form of Relative Performance Error (RPE) as defined in [START_REF] Ott | ESA pointing error engineering handbook[END_REF]. The RPE index represents the statistics of the instantaneous angular difference between the pointing vector LOS and its short-time average in a time interval 𝑡 𝛥 . In the frequency domain this filter corresponds to an high-pass weight [START_REF] Ott | ESA pointing error engineering handbook[END_REF][START_REF] Pittelkau | Pointing error definitions, metrics, and algorithms[END_REF][START_REF] Ott | Precision pointing 𝐻 ∞ control design for absolute, window-, and stability-time errors[END_REF] applied to the system:

F 𝑅𝑃 𝐸 (s) = 𝜖 -1 𝑚𝑎𝑥 𝑡 𝛥 s (𝑡 𝛥 s + √ 12) (𝑡 𝛥 s) 2 + 6(𝑡 𝛥 s) + 12 I 2 (12)
where 𝜖 𝑚𝑎𝑥 is the maximum RPE deviation allowed, which bounds the transfer of the filter. For the present study case the following values were selected: 𝜖 𝑚𝑎𝑥 = 50 μrad, 𝑡 𝛥 = 3 ms. Moreover, the SADM perturbation inputs 𝑝 𝑆𝑃 1 and 𝑝 𝑆𝑃 2 on the system have been normalized by an estimated maximum torque acting on the system, indicated by the upper bound 𝑝 𝑚𝑎𝑥 : 𝑇 𝑝,𝑛𝑜𝑟𝑚 = 𝑝 𝑚𝑎𝑥 = 0.3820 Nm. The output x PMA u is used to monitor and limit the internal displacements of the 4 PMAs.

Finally the transfer from p PMA to W PMA u is the input sensitivity function of the PMA control loop and can be used to impose stability margins (disc margin).

By assembling all these blocks, the closed-loop LPV system M r (Θ) displayed in Fig. 14 is obtained. The uncertainties are added to the model by means of the LFT form  𝑢 (M r (Θ), 𝜟). The model can then be augmented with the PMAs controller gains contained in the {4 × 8} scalar matrix K PMA by means of a lower Linear Fractional Transformation

 𝑙 (  𝑢 ( M r (Θ), 𝜟 ) , K PMA )
. The selection of the proper gains to drive the actuation of the PMAs will be the main focus of the structural and control co-design of Section 3.5.

Robust structure -control co-design

The objective of this study case is to implement a robust co-design of both the 𝑇 𝑠𝑡𝑟 structure and control laws of the PMAs to maximize pointing performances and minimize the mass 𝑚(Θ) of the T-truss structure at the same time. For our study, this means minimizing the reducing the side ℎ of the T-truss beam while constraining the transfer between the normalized SADM perturbation torques p = [p 𝑆𝑃 1 , p𝑆𝑃 1 ] and the normalized 𝑅𝑃 𝐸 index below 1 for any values of the uncertain parameters 𝜹. In addition, two hard constraints are taken into account:

• the internal displacement of each PMA in response to the SDM disturbance torques must be lower that the maximal value 𝑥 PMA = 3 mm, • the PMA control loop must satisfied stability margins expressed as a disc margin on the input sensitivity function.

Let us consider the LFT closed-loop system  𝑙 (  𝑢 ( M r (Θ), 𝜟 ) , K 𝑃 𝑀𝐴 ) depicted in Fig. 14, the robust co-design problem reads:

{ Θ, K𝑃 𝑀𝐴 } = arg min Θ,K 𝑃 𝑀𝐴 𝑚(Θ) = arg min ℎ,K 𝑃 𝑀𝐴 ℎ such that: ( 13
)
𝛾 1 = max 𝜟 ‖ ‖ ‖  𝑙 (  𝑢 ( M r (Θ), 𝜟 ) , K 𝑃 𝑀𝐴 ) p→RPE ‖ ‖ ‖ ∞
≤ 1 (worst-case pointing performance requirement) ( 14)

𝛾 2 = max 𝜟 ‖ ‖ ‖  𝑙 (  𝑢 ( M r (Θ), 𝜟 ) , K 𝑃 𝑀𝐴 ) p→x PMA u ‖ ‖ ‖ ∞ ≤ 𝑥 PMA (worst-case PMA displacement requirement) ( 15
)
𝛾 3 = max 𝜟 ‖ ‖ ‖  𝑙 (  𝑢 ( M r (Θ), 𝜟 ) , K 𝑃 𝑀𝐴 ) p PMA →W PMA u ‖ ‖ ‖ ∞ ≤ 𝛾 𝐷𝑀 (worst-case disc margin requirement) (16)
where ⋆ i→o denotes the subsystem from input i to output o in the system ⋆.

In the following application 𝛾 𝐷𝑀 = 1.5. This value ensures:

• a modulus margin > 1∕𝛾 𝐷𝑀 = 0.666,

• a gain margin > 𝛾 𝐷𝑀 𝛾 𝐷𝑀 -1 = 3 (6 dB), • a phase margin > 2 arcsin 1 2𝛾 𝐷𝑀 = 38.9 (deg).

The Fig. 15 displays the frequency-domain response relative the pointing performance requirement for the nominal value of the beam side Θ 0 = ℎ 0 = 2 cm without PMA control. One can check that some flexible modes do not meet the constraint.

Co-design optimization results

The non-smooth robust structured 𝐻 ∞ problem defined by the performance index [START_REF] Holm-Jørgensen | A component mode synthesis algorithm for multibody dynamics of wind turbines[END_REF] and the 3 hard constraints ( 14), [START_REF] Yu | Element-by-element model updating of large-scale structures based on component mode synthesis method[END_REF] and ( 16) is solved using MATLAB ® 's systune routine [START_REF] Gahinet | Structured 𝐻 ∞ synthesis in MATLAB[END_REF]. In terms of computational efficiency, the optimization takes around 120 min on a standard laptop computer to converge to the solution summarized in Table 4. This yields to a mass reduction of 76% of the T-Truss structure original mass, saving a total of 62.15 kg with respect to the initial design having 𝑚 0 = 81.38 kg. The impact

Table 4

Co-design results in terms of ĥ, structural mass reduction of 𝑇 𝑠𝑡𝑟 and normalized constraints. of the saved T-truss mass/inertia on the whole spacecraft inertia J 𝑡𝑜𝑡 involved in the tuning of the ACS gains (Eq. ( 10)) is very small so that the ACS gains do not need to be updated to the new optimal mass/inertia. The 3 hard constraints are saturated since the weighted 𝛾 𝑖 , 𝑖 = 1, 2, 3 values are very close to 1. The frequency domain interpretation of the 3 hard constraints presented in Fig. 16 for several models randomly sampled in the uncertain parametric space highlights the frequencies of interest limiting the performances. The PMA control allows to damp pointing performance critical flexible modes (around 100 rad∕s as showcased in Fig. 15 and around 20 rad∕s in Fig. 16). thanks to the positivity between the 4 PMAs and the integration of the 4 linear accelerometers but this positivity is limited by the low-pass filter introduced to prevent the spillover on the neglected modes. In addition, this positivity is no more guaranteed regarding the feedback of the 4 angular accelerator (integrated) to the 4 PMAs. That is why the disc margin requirement is very determining for the control design. A more accurate robust performance analysis on the full order model, is presented in the next section.

Θ [cm] 𝑚( Θ) [kg] 𝑚 0 [kg] ( 𝑚( Θ) -𝑚 0 ) ∕𝑚 0 𝛾 1 𝛾 2 ∕

Worst-case pointing analysis

Let us consider the closed-loop model described in Section 3.4. Previously, the model M r (Θ) displayed in Fig. 14 used for the codesign has been constructed based on the reduced order model G r (s). Let us now consider the full order model M r𝑙𝑜𝑤 (Θ) obtained just simply changing G r (s) by G r𝑙𝑜𝑤 (s) in the block diagram interconnection shown in Fig. 14.

In order to fully validate the performance of the optimal solution { Θ, K𝑃 𝑀𝐴 }, a worst-case analysis has been implemented on the optimal closed-loop system. The objective is to find the worst-case values of the uncertain parametric vector 𝜹 = [𝛿 𝑀 𝑆𝐶 , 𝛿 𝐼 𝑦𝑦 𝑆𝐶 , 𝛿 𝜔 1 𝑆𝑃 , 𝜏] 𝑇 for the 3 hard constraints ( 14), ( 15) and ( 16) evaluated on the full-order optimal closed-loop system. The solar panel geometrical configuration parameter 𝜏 = tan 𝜃∕4 is repeated 16 times in the uncertain block ∆ (Eq. ( 8)) leading to unacceptable computational time using the standard worst-case analysis tools. This problem is circumvented by computing the whole closed-loop model for a given value of 𝜏 and by sampling 𝜏 on a grid of 𝑁 𝜏 = 50 points regularly distributed on ) , the full uncertain closed-loop system associated with the optimal solution reads:

 𝐶𝐿 (𝜃, 𝜟 𝑝 ) =  𝑙 (  𝑢 ( (𝜃), 𝜟 𝑝 ) , K𝑃 𝑀𝐴 ) (17)
where all the remaining uncertainties on 𝑀 𝑆𝐶 , 𝐼 𝑦𝑦 𝑆𝐶 and 𝜔 1 𝑆𝑃 are contained in the matrix 𝜟 𝑝 . The objective is thus to solve the 3 𝜇-analysis problems: 14), ( 15) and ( 16) for the optimal side Θ0 and the optimal PMA control K𝑃 𝑀𝐴 . 𝜇 3 (𝜃) = max

𝜇 1 (𝜃) = max 𝜟 𝑝 ‖ ‖ ‖  𝐶𝐿 (𝜃, 𝜟 𝑝 ) p→RPE ‖ ‖ ‖∞ , ( 18
) 𝜇 2 (𝜃) = max 𝜟 𝑝 ‖ ‖ ‖ 1 𝑥 PMA  𝐶𝐿 (𝜃, 𝜟 𝑝 ) p→x PMA u ‖ ‖ ‖∞ , ( 19)
𝜟 𝑝 ‖ ‖ ‖ 1 𝛾 𝐷𝑀  𝐶𝐿 (𝜃, 𝜟 𝑝 ) p→W PMA u ‖ ‖ ‖∞ . ( 20
)
For the 𝑖 = 1, 2, 3 and for the 𝑁 𝜏 values of 𝜏 (or 𝜃), these 𝜇-analysis problems 𝜇 𝑖 (𝜃) are solved by means of the wcgain routine in MATLAB ® 's Robust Control Toolbox [START_REF] Balas | Robust Control Toolbox user's guide[END_REF]. This function provides an upper bound 𝜇 𝑖 (𝜃) and a lower bound 𝜇 𝑖 (𝜃) of 𝜇 𝑖 (𝜃). The worst-case parametric configuration is associated to this lower bound. The function also returns the critical frequency where the worst gain occurs. The worst-case performances and the associated worst-case parametric combinations are summarized in Table 5. One can check that the gaps between the upper bounds and the lower bounds are very small for the 3 transfers of interest showing that the worst-case performance is accurately evaluated. The hard constraints for the pointing performance (𝑖 = 1) and the PMAs displacements (𝑖 = 2) are thus validated on the full order model. The hard constraint on the disc margin (𝑖 = 3) is slightly above 1 but can be considered as satisfying.

In Fig. 17 a complete representation on the worst-case analysis results is presented for the main transfer of interest, i.e. the pointing performance (𝜇 1 (𝜃)), in terms of its 𝜇-bounds as a function of 𝜃, the frequency-domain response and the worst case parameter combination. It can be seen that the 𝜇 analysis identified the two bounds close to each other and always with a value below the unity, successfully validating the robustness of the design. Moreover, it can be remarked that the bounds show a visible trend, which justifies the choice of a relative low number of 𝑁 𝜏 for this analysis. Furthermore, the study of the singular value responses reveals that the worst-case frequency response of the system remains always below requirement.

Conclusions

This paper aimed at introducing new analytical tools to model large complex truss structures for space applications in the TITOP/NINOP framework with the specific objective of developing models for structure/control co-design and robust analysis and control. A series of 2D mechanisms block has been introduced to build a unitary 3D cubic element which serves as a building block for truss structures in a sub-structuring approach. The analysis displayed the potentialities of the approach, as large structures composed by a high number of beam elements can be easily assembled by using blocks of decreasing complexity. In addition, a full validation campaign by comparison with Nastran validated the representativeness of these models. A case study was then introduced to represent the strengths of the TITOP/NINOP approach in performing robust structure/control co-design in presence of parametric uncertainties. A complex 3D truss structure was built using the previously introduced cube elements and attached to a spacecraft to act as support for an high precision antenna. The objective of the co-design was to reduce the structural mass of the system while satisfying a fine pointing requirement. This study case highlighted the potential of these analytical blocks in performing complex multi disciplinary optimization problems. The direct co-design using structured 𝐻 ∞ control synthesis allowed for computational cost reduction and brought to a mass saving of almost 76% of the original structural mass, while coping with stringent pointing performances and a large set of uncertainties in the mechanical design parameters. The main conclusion of this work is that it is possible to develop knowledge-based models for complex truss structures with an analytical dependence on the sizing, variable or uncertain parameters. These models can be directly used for robust design, robustness analysis or parametric optimization, thus simplifying model uncertainty quantification and model reduction in the overall control/structure integrated design process.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix

See Tables 6 and7.

Fig. 1 .

 1 Fig. 1. TITOP scheme and nomenclature for a generic flexible appendage  𝑖 .

Fig. 2 .

 2 Fig. 2. Representation of the Clamped-Free-Clamped (CFC) L-Chain Mechanism  𝐶𝐹 𝐶 (𝑠) (Fig. 3(a)) and its block-diagram model (Fig. 3(b)).

Fig. 3 .

 3 Fig. 3. Representation of the Clamped-Free-Free (CFF) Triangle Mechanism (Fig. 3(a)) and corresponding block-diagram model of the system (Fig. 3(b)).

Fig. 4 .

 4 Fig. 4. Representation of the Clamped-Free-Free-Free (CFFF) Square Mechanism  𝐶𝐹 𝐹 𝐹 (s) (a) and corresponding block-diagram model of the system (b).

Fig. 5 .

 5 Fig. 5. Geometrical representation (a) and block diagram system (b) of the Cubic Element (s).

Fig. 6 .

 6 Fig. 6. Validation FEM model implemented in MSC/Patran.

Fig. 7 .

 7 Fig. 7. Verification of the Cube element's frequency response (SDT) by comparison with the MSC Patran/Nastran model (P/N).

Fig. 8 .

 8 Fig. 8. Overall view of satellite's case study. The spacecraft composed by a main central spacecraft body (𝑆∕𝐶), two Solar Panels (𝑆𝑃 1 , 𝑆𝑃 2 ) rotated around axis 𝑥 of an angle 𝜃. The HPP antenna (𝐴𝑁𝑇 ) and the 4 PMAs are both placed on top of the truss structure 𝑇 𝑠𝑡𝑟 .

Fig. 9 .

 9 Fig. 9. Simplified representation of the T-Shaped Truss Structure, without any diagonal beam in the cubes' faces, and the beams square sections. For clarity sake, the Cube elements  𝑖 are represented with their local reference frames  𝑖 to indicate their orientation in the global reference frame  𝑔𝑙 (𝑁 1 , x 𝑔𝑙 , y 𝑔𝑙 , z 𝑔𝑙 ).The section area 𝑆 and bending inertia 𝐼 𝑦 , 𝐼 𝑧 of each beam is expressed in the generic beam local reference of frame  𝑙 (𝑁 𝑙 , 𝑥 𝑙 , 𝑦 𝑙 , 𝑧 𝑙 ) as function of the side parameter ℎ.

Fig. 10 .

 10 Fig. 10. PMA mechanical system and its associated block diagram system D  𝑃 (s) in the TITOP approach.

Fig. 11 .

 11 Fig. 11. Open Loop System of the spacecraft in Fig. 8.

Fig. 12 .

 12 Fig. 12. LFT representation of the LPV full order model (𝜣, 𝜹).

Fig. 13 .

 13 Fig. 13. Comparison on the input-output transfer singular value responses between the full order model G 𝑟 (𝑠), and the reduced order models G r𝑙𝑜𝑤 (𝑠), truncated at low frequencies, and G 𝑟 (𝑠), truncated both at low and high frequencies.

Fig. 14 .

 14 Fig. 14. Closed Loop System for Synthesis.

Fig. 15 .

 15 Fig. 15. Frequency-domain illustration of the hard constraint (14) for the nominal side ℎ 0 without PMA control K 𝑃 𝑀𝐴 = 0 4×8 .

  [0, 1] (or 50 values of 𝜃 in [0, 180] • ). This subset has been chosen to account for the symmetric configuration of the model in the 𝜃 ∈ [0, 180] • and 𝜃 ∈ [-180, 0] • intervals. By defining (𝜃) =  𝑢 ( M r𝑙𝑜𝑤 ( Θ), tan(𝜃∕4)I 16

Fig. 16 .

 16 Fig.16. Frequency-domain illustrations of the hard constraints (14), (15) and (16) for the optimal side Θ0 and the optimal PMA control K𝑃 𝑀𝐴 .

Fig. 17 .

 17 Fig. 17. Worst-case analysis for the pointing performance transfer  𝐶𝐿 (𝜃, 𝜟 𝑝 ) p→RPE as a function of 𝜃. On the top part of the figure, the 𝜇-bounds are displayed together with the worst-case parameter combinations. Down below, the singular values of the transfer and its worst-case are showcased.

Table 2

 2 Comparison

	Mode	𝝎 k [rad/s]		T1		T2		T3		R1		R2		R3
		Patran	SDT	Patran	SDT	Patran	SDT	Patran	SDT	Patran	SDT	Patran	SDT	Patran	SDT
	1	16.77	17.43	-3.830	-3.912	2.040	2.207	1.714	1.474	0.113	-0.292	-3.746	-3.704	5.127	5.236
	2	17.51	17.75	-0.716	-0.432	2.211	2.091	-3.890	-4.021	-5.495	-5.475	1.684	1.964	2.167	1.815
	3	34.37	34.90	2.279	-2.271	1.111	-1.256	-0.993	0.921	0.069	-0.018	4.423	-4.315	0.676	-0.765
	4	43.66	44.34	2.066	-2.004	4.546	-4.534	0.749	-0.824	-2.424	2.421	0.031	0.074	0.503	-0.439
	5	70.36	72.18	1.711	-1.746	1.737	1.616	1.288	1.239	-0.897	-0.844	-0.677	-0.651	-0.128	-0.149
	6	97.26	99.54	0.482	-0,428	0.402	-0.389	0.398	-0.392	-0.192	0.187	-0.079	0.086	0.007	-0.002
	7	112.83	115.31	0.667	-0.656	0.401	-0.398	0.561	-0.567	-0.144	0.140	0.173	-0.168	0.129	-0.132
	8	117.91	120.84	-0.554	-0.552	0.067	0.059	-0.074	-0.095	-0.026	-0.022	-0.005	-0.005	0.043	0.042
	9	348.92	354.30	-0.184	-0.139	0.126	0.118	0.386	0.472	-0.034	-0.031	-0.071	-0.077	0.031	0.032
	10	361.72	365.24	-0.727	-0.761	-0.108	-0.123	-1.101	-1.078	0.023	0.027	0.035	0.026	-0.042	-0.047

between the Patran/Nastran model and the SDT cube: modal frequency 𝜔 𝑘 and participation factors (𝑇 1 , 𝑇 2 , 𝑇 3 , 𝑅 1 , 𝑅 2 , 𝑅 3 ).

  𝑘 𝑆𝑃 𝑖 𝛿𝜃 𝑖 + 𝑓 𝑆𝑃 𝑖 𝛿 θ𝑖 + 𝑝 𝑆𝑃 𝑖[START_REF] Theodore | Comparison of the assumed modes and finite element models for flexible multilink manipulators[END_REF] where 𝛿𝜃 𝑖 and 𝛿 θ𝑖 are obtained integrating the relative acceleration 𝛿 θ𝑖 of 𝑆𝑃 𝑖 with respect to 𝑆∕𝐶. The two coefficients 𝑘 𝑆𝑃 𝑖 , 𝑓 𝑆𝑃 𝑖 are contained in the matrices:K 𝑆𝑃 𝑖 = [𝑓 𝑆𝑃 𝑖 , 𝑘 𝑆𝑃 𝑖 ].• Antenna (𝐴𝑁𝑇 ). The antenna is modeled as a rigid body described by its mass 𝑀 𝐴𝑁𝑇 and its inertia matrix 𝐼 𝐴𝑁𝑇 at the node 𝑁

[START_REF] Tantawi | Linear dynamic modeling of spacecraft with various flexible appendages[END_REF]

in its local frame aligned with the S/C reference frame. The LOS direction of the antenna therefore coincides with the 𝑧-axis of the global reference frame  𝑆∕𝐶 (𝑃 1 , x, y, z).

• T-truss structure (𝑇 𝑠𝑡𝑟 ). This 36 × 36 block is the 6 input -6 output port model described in Section 3.1. The 6 connection points are 𝑁 1 , 𝑁 2 , 𝑁 3 , 𝑁 4 (with the parent central body), 𝑁 17 and 𝑁 20 (with the child antenna and PMAs). Since it is composed of 65 flexible beams and since the SDT analytical model of a beam is a 20-th order model, this block is a 1300-th order model.

  xPMA 𝛾 3 ∕𝛾 𝐷𝑀

	0.972	19.23	81.38	-76.37%	0.999	0.957	0.999

Table 5

 5 Worst-case analysis results for the three transfers of interest (𝜃 𝑊 𝐶 is the worst-case solar panel geometrical configuration).𝑖max 𝜃 𝜇 𝑖 (𝜃) 𝜃 𝑊 𝐶 = arg max 𝜃 𝜇 𝑖 (𝜃) [deg] 𝜇 Critical freq. [rad/s] Worst-case 𝑀 𝑆𝐶 [kg] Worst-case 𝐼 𝑦𝑦 𝑆𝐶 [kg m 2 ] Worst-case 𝜔 1 𝑆𝑃 [rad/s]

	1	0.9767	94.5	0.9754	20.4	945	871	2.854
	2	0.974563	42.5	0.97234	3.99	720	1100	2.254
	3	1.0471	0	1.0469	473.	800	990	2.472

𝑖

(𝜃 𝑊 𝐶 )

Table 6

 6 Spacecraft mechanical characteristics. 𝑆𝑃 , 𝜔 2 𝑆𝑃 , 𝜔 3 𝑆𝑃 ] Flexible modes' frequencies [2.51 ± 20%, 3.77, 9.42] rad∕s [𝜉 1 𝑆𝑃 , 𝜉 2 𝑆𝑃 , 𝜉 3 𝑆𝑃 ]

		Parameter					Description	Value & Uncertainty
		𝑃 1						Spacecraft C.o.G.	[0, 0, 0] m
	Spacecraft 𝑆∕𝐶	𝑀 𝑆𝐶						Mass	800 kg ± 20%
		⎡ ⎢ ⎢ ⎢ 𝐼 𝑥𝑥 𝑆𝐶	𝐼 𝑥𝑦 𝑆𝐶 𝐼 𝑦𝑦 𝑆𝐶	𝐼 𝑥𝑧 𝑆𝐶 𝐼 𝑦𝑧 𝑆𝐶	⎤ ⎥ ⎥ ⎥		Inertia in  𝑆𝐶 frame	⎡ ⎢ ⎢ ⎢	1000	0 1000 ± 20%	0 0	⎤ ⎥ ⎥ ⎥	kg m 2
		⎢ ⎣		𝐼 𝑧𝑧 𝑆𝐶	⎥ ⎦			⎢ ⎣				200	⎥ ⎦
		r 𝑆𝑃 𝑂𝐺						Solar panel C.o.G. in  𝑆𝑃	[0; -2; 0.03] m
		𝑀 𝑆𝑃 1						Mass	80 kg
	Solar Panels 𝑆𝑃	⎡ ⎢ ⎢ ⎢ 𝐼 𝑥𝑥 𝑆𝑃	𝐼 𝑥𝑦 𝑆𝑃 𝐼 𝑦𝑦 𝑆𝑃	𝐼 𝑥𝑧 𝑆𝑃 𝐼 𝑦𝑧 𝑆𝑃	⎤ ⎥ ⎥ ⎥		Inertia in  𝑆𝑃	⎡ ⎢ ⎢ ⎢	80	0 20	-0.1 22	⎤ ⎥ ⎥ ⎥	kg m 2
		⎢ ⎣		𝐼 𝑧𝑧 𝑆𝑃		⎥ ⎦			⎢ ⎣			100	⎥ ⎦
		[𝜔 1 Flexible modes' damping	0.003
									⎡	-0.002	-1.5	-5	14	0.02	-0.01	⎤
		L 𝑆𝑃						Modal participation factors	⎢ ⎢	5		1	-0.1	0	2	15	⎥ ⎥
									⎢				⎥
									⎢ ⎣	0.3		0.002	0.03	-0.02	3.2	-0.2	⎥ ⎦
	HPP Antenna 𝐴𝑁𝑇	𝑀 𝐴𝑁𝑇						Mass	20 kg
		⎡ ⎢ ⎢ ⎢ 𝐼 𝑥𝑥 𝐴𝑁𝑇	𝐼 𝑥𝑦 𝐴𝑁𝑇 𝐼 𝑦𝑦 𝐴𝑁𝑇	𝐼 𝑥𝑧 𝐴𝑁𝑇 𝐼 𝑦𝑧 𝐴𝑁𝑇	⎤ ⎥ ⎥ ⎥	Inertia in   	⎡ ⎢ ⎢ ⎢	1.32	0 1.32	0 0	⎤ ⎥ ⎥ ⎥	kg m 2
		⎢ ⎣		𝐼 𝑧𝑧 𝐴𝑁𝑇	⎥ ⎦		⎢ ⎣			2.5	⎥ ⎦
		𝑀 𝐵						Casing mass	0.5 kg
		⎡ ⎢ 𝐼 𝑥𝑥 	𝐼 𝑥𝑦 	𝐼 𝑥𝑧 
		⎢ ⎢	𝐼 𝑦𝑦 	𝐼 𝑦𝑧 
	PMAs	⎢ ⎣		𝐼 𝑧𝑧 

Table 7

 7 Definition of connection points for the spacecraft sub-systems in the spacecraft reference frame  𝑆𝐶 (𝑃 1 , 𝑥, 𝑦, 𝑧).

	Description	Point	Coordinates	Unit	Description	Point	Coordinates	Unit
	𝑇 𝑠𝑡𝑟 connection to 𝑆∕𝐶 at Node 1	𝑃 2	[-0.5, -0.5, 1]	m	𝑇 𝑠𝑡𝑟 connection to 𝑆∕𝐶 at Node 4	𝑃 5	[-0.5, 0.5, 1]	m
	𝑇 𝑠𝑡𝑟 connection to 𝑆∕𝐶 at Node 2	𝑃 3	[0.5, -0.5, 1]	m	Solar Panel 1 𝑆𝑃 1 connection to 𝑆∕𝐶	𝑃 6	[1, 0, 0]	m
	𝑇 𝑠𝑡𝑟 connection to 𝑆∕𝐶 at Node 3	𝑃 4	[0.5, 0.5, 1]	m	Solar Panel 2 𝑆𝑃 2 connection to 𝑆∕𝐶	𝑃 7	[-1, 0, 0]	m