domain: spi math phys

text:
Sofiane Kraïem
email: sofiane.kraiem@onera.fr

Mathieu Rognant
email: mathieu.rognant@onera.fr

Jean-Marc Biannic
email: jean-marc.biannic@onera.fr

Yves Brière
email: yves.briere@isae-supaero.fr

Dynamics and Robust control of a space manipulator with flexible appendages for On-Orbit Servicing

Keywords: Space manipulator, Extended state observer, Nonlinear Dynamic Inversion, Disturbance compensation, Linear Matrix Inequality

Space manipulators allow to respond to a variety of problems in future space exploitation and exploration such as on-orbit deployment, active debris removal or servicing operations. However, a difficulty to autonomously control space manipulator systems arise with large and light structures presenting flexible behavior. Flexible dynamics remain a challenging study focus as its modeling may present a first difficulty while the different coupling with the manipulator may deteriorate the control quality. This paper addresses design and control problems related to autonomous space manipulator equipped with kinetic moment exchange devices for spacecraft rotation control when dealing with system internal disturbances, model uncertainties and measurement errors. One advantage of modeling the rigid-flexible dynamics of a multi-body system is the possibility of including the non-measurable states in the system decoupling and linearization. In this work, in addition to the development of an Extended State Observer (ESO) that estimate the flexible dynamics, a Nonlinear Disturbance Observer (NDO) is also introduced and included in a Nonlinear Dynamic Inversion (NDI) framework where both modeling uncertainties and measurement errors are considered. Inter-dependencies between observers and control dynamics motivate a simultaneous computation of their gains to improve system stability and control performances. This is achieved by the resolution of Linear Matrix Inequalities (LMI). In order to highlight the interest of the proposed scheme and validate our approach in a realistic environment, extensive tests of an on-orbit space telescope assembly use-case are performed on a high-fidelity simulator.

Introduction

The trend of dehumanizing Low Earth Orbit missions to improve safety in extra-vehicular operations and the space exploration ambitions have placed robotic systems as serious solutions to tackle current and future challenges that space exploitation is facing. Among the multitude of On-Orbit Servicing (OOS) and Active Debris Removal (ADR) missions, space manipulator systems (SMS) provide the required assets to perform different capture, maintenance tasks or refueling operations. This reinforces the interest in developing new technologies to become viable solutions [START_REF] Flores-Abad | A review of space robotics technologies for on-orbit servicing[END_REF]. After the successful demonstration tests of the ETS-VII mission [START_REF] Oda | Ets-vii, space robot in-orbit experiment satellite[END_REF], the use of SMS to proceed to ADR operations has since remained a challenging field for technologies improvement [START_REF] Priyant | Review of active space debris removal methods[END_REF]. The tumbling or non-cooperative properties of space debris and potential targets make the approach and grasping phases tricky as the relative dynamics between the manipulator and the target requires specific measurement or estimation means in order to reduce collision risks [START_REF] Ellery | Tutorial review on space manipulators for space debris mitigation[END_REF]. Accurate identification of target properties is also necessary for the detumbling phase, and for this purpose cooperative targets are equipped with visual features [START_REF] Flores-Abad | A review of space robotics technologies for on-orbit servicing[END_REF]. However, observation and planning phases are still open research fields to safely proceed to capture with SMS [START_REF] Papadopoulos | Robotic manipulation and capture in space: A survey[END_REF]. As illustrated by current OOS operations, either for on-orbit deployment [START_REF] Song | Review on on-orbit assembly of large space telescopes[END_REF] or maintenance scenarios [START_REF] Beyer | Caesar: Space robotics technology for assembly, maintenance, and repair[END_REF], SMS are mainly tele-operated from the earth or the International Space Station. This becomes an issue in the presence of communication delays and breakdowns, limited operator visibility or structures vibrations that limit the use of SMS [START_REF] Papadopoulos | Robotic manipulation and capture in space: A survey[END_REF]. The viability of using SMS will then reside in its capability of performing operations autonomously which remains challenging [START_REF] Li | On-orbit service (oos) of spacecraft: A review of engineering developments[END_REF].

Mission lifespan is an other key aspect of future space operations. For this purpose electrical actuators are to be preferred to thrusters to perform spacecraft attitude control. As categorized by Wilde [START_REF] Wilde | Equations of motion of free-floating spacecraft-manipulator systems: An engineer's tutorial[END_REF], space manipulator are differently considered according to either the base rotation and linear displacement are actively controlled (rotation floating SMS) or not (floating SMS). More precisely, during rotation-floating maneuvers, with the help of kinetic moment exchange devices, such as reaction wheels or control moment gyroscopes, the spacecraft attitude is actively controlled. Conversely, during floating maneuvers, only the manipulator joints are actively controlled. The use of such base actuators has recently attracted much interest in the space community for the manipulation of heavy objects whose inertia is not negligible compared to that of the spacecraft [START_REF] Li | Motion planning and coordination control of space robot using methods of calculated momentum[END_REF]. In such a case, a simultaneous control of the base rotations and the manipulator joints enables to improve manipulability [START_REF] Mathieu Rognant | Kinematic indices of rotation-floating space robots for on-orbit servicing[END_REF]. More specifically, in on-orbit deployment applications, the attitude of the base must remain as motionless as possible during the movements of the manipulator to ensure the quality of the power supply (orientation of solar panels) and communication with the ground (orientation of the antenna).This has motivated the study of combining control moment gyroscopes that provide large control torques and reaction wheels to control the platform [START_REF] Wu | Attitude control for on-orbit servicing spacecraft using hybrid actuator[END_REF].

As the spacecraft operates in free-fall environment, the different couplings between the SMS components represent a real challenge for the development of SMS control systems. Because of the invariance of the global kinetic momentum, the manipulator motions clearly induce base motions and reciprocally [START_REF] Mathieu Rognant | Kinematic indices of rotation-floating space robots for on-orbit servicing[END_REF]. Such couplings have long been established through factors allowing to evaluate the participation of each SMS element during a given motion. To enable accurate manipulator control, the influence of the actuators on each other have been described with a coupling factor [START_REF] Xu | The measure of dynamic coupling of space robot systems[END_REF]. More recently, for rotation free-floating manipulators, kinematic indices based on the system momentum conservation have been introduced for sizing a common manipulator and base control system [START_REF] Mathieu Rognant | Kinematic indices of rotation-floating space robots for on-orbit servicing[END_REF].

A second obstacle to autonomous SMS is brought with the larger spacecraft sizes and lighter structures synonym of flexible behaviors. In addition to a first difficulty to establishing the flexible dynamics, it also prevents the use of the conservation assumption on the angular momentum of the system. Nevertheless, based on this assumption, when applicable, many approaches have been developed to reduce interactions between the manipulator and the while guaranteeing system stability. Manipulator workspace adjustments [START_REF] Giordano | Workspace fixation for free-floating space robot operations[END_REF], simultaneous control of the global Center-of-Mass (CoM) and base rotations [START_REF] Massimo | Coordinated control of spacecraft's attitude and end-effector for space robots[END_REF], reaction null-space control techniques [START_REF] Pisculli | A minimum state multibody/fem approach for modeling flexible orbiting space systems[END_REF] are few examples of such a strategy. On the contrary, when the momentum conservation is no longer verified because of flexible vibrations or external torques or forces, motion planning methods have been proposed [START_REF] Fatina | Space robot motion planning in the presence of nonconserved linear and angular momenta[END_REF]. Manipulator trajectory planning is often a first approach to reduce the flexible vibrations as the manipulator motions are indirectly responsible of those undesired internal disturbances [START_REF] Meng | Space robots with flexible appendages: dynamic modeling, coupling measurement, and vibration suppression[END_REF]. Solutions minimizing flexible appendages coupling with the SMS through optimized trajectories [START_REF] Li | Assembly dynamics of a large space modular satellite antenna[END_REF] or using coupling factors in the control strategy [START_REF] Meng | Vibration suppression control of free-floating space robots with flexible appendages for autonomous target capturing[END_REF], nonlinear Model Predictive Control minimizing disturbances form manipulator motions [START_REF] Rybus | Control system for free-floating space manipulator based on nonlinear model predictive control (nmpc)[END_REF] have been adopted. Active control of vibrations disturbances, such as the use of piezoelectric actuators placed on the flexible elements [START_REF] Zarafshan | Adaptive hybrid suppression control of space free-flying robots with flexible appendages[END_REF] or joint variable stiffness control actuators [START_REF] Hu | Semi-active vibration control of two flexible plates using an innovative joint mechanism[END_REF], has shown limitations for actual missions. This can be illustrated with capture scenarios when target presents flexible appendages for instance.

Trajectory planning in Cartesian space minimizing base disturbances [START_REF] Lu | Trajectory planning of satellite base attitude disturbance optimization for space robot[END_REF] or robust control to unknown disturbances [START_REF] Qiao | High-precision attitude tracking control of space manipulator system under multiple disturbances[END_REF] or both internal and external perturbations [START_REF] Seddaoui | Combined nonlinear ℎ ∞ controller for a controlled-floating space robot[END_REF] are solutions more easily envisaged than active control.

Despite the challenges to derive the dynamics of a rigid-multi-body SMS with flexible appendages, this effort of system modeling allows to develop control strategies with efficient disturbances rejection. For instance, a manipulator's end-effector trajectory optimization can include vibration suppression objectives while capturing a defined target [START_REF] Liu | Trajectory planning and coordination control of a space robot for detumbling a flexible tumbling target in post-capture phase[END_REF] or an input disturbance torque can be included to improve control performances for an OOS operation [START_REF] Kraïem | Control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF]. The Newton-Euler approaches are mostly used to obtain the dynamics for a combination of rigid and flexible bodies system [START_REF] Alazard | Linear dynamic modeling of spacecraft with various flexible appendages and on-board angular momentums[END_REF][START_REF] Dubanchet | Modeling and control of a flexible space robot to capture a tumbling debris[END_REF] but Lagrangian methods which are well suited to control design can also be adopted [START_REF] Ebrahimi | Dynamics of space free-flying robots with flexible appendages[END_REF].

Focusing on the above flexible aspects, systems uncertainties and large variations during OOS operations, this paper builds upon preliminary works [START_REF] Mathieu Rognant | Kinematic indices of rotation-floating space robots for on-orbit servicing[END_REF][START_REF] Kraïem | Control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF][START_REF] Kraïem | Robust control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF] where the benefits of developing a common base and manipulator control strategy using an extended state observer to reject the internal perturbations despite systems variations were clearly demonstrated.

The paper is organized as follow. The rigid and flexible dynamics of a multi-body system are first detailed in Section 2 using a Lagrangian approach. Next, the control strategy is presented in Section 3. The system is first linearized and decoupled with an NDI-based controller including an estimation of non measurable states and disturbance torques that gather modeling uncertainties and measurement errors. Then, a simultaneous gains synthesis is developed to ensure accurate control performances for a multi-task on-orbit servicing scenario. Finally the control method is illustrated in Section 4 through realistic simulations on an orbital deployment of a space telescope (https://www.h2020-pulsar.eu). Some concluding comments and perspectives end the paper in Section 5.

System open-loop dynamics

From a general point of view, a free-floating space manipulator can be decomposed as a satellite base, a potential payload, and a degree-of-freedoms (DOFs) robotic manipulator as illustrated in Figure 1. By adding reaction-wheels to the satellite base to actively control its rotations the SMS becomes rotation-free-floating [START_REF] Wilde | Equations of motion of free-floating spacecraft-manipulator systems: An engineer's tutorial[END_REF]. The total number of actuators between the manipulator and the reaction-wheels gives = + DoFs. Moreover, only the appendages attached to the base may exhibit flexible modes while the manipulator is assumed to be rigid. The study is restricted to OOS applications where no external forces or torques is applied to the SMS in order to focus on the system internal disturbances induced by the flexibility of the appendages.

Using a standard Lagrangian approach, the derivation of the SMS rigid dynamics is first detailed. Then the Lagrangian approach is adapted to introduce the flexible dynamics of the appendages. This allows to describe the open-loop behavior of a rotation-free-floating space manipulator in the presence of flexible appendages.

Dynamics of a rigid rotation-free-floating space manipulator

To detail a rigid multi-body rotation-free-floating system, a Lagrangian approach is well suited to derive the system's dynamics based upon the energies present as well as providing a systematic computation method [START_REF] Wilde | Equations of motion of free-floating spacecraft-manipulator systems: An engineer's tutorial[END_REF]. Using the Denavit-Hartenberg (DH) convention, one can associate a twist vector to each link of the manipulator and reaction-wheels, t = r , with ∈ R 3×1 and r ∈ R 3×1 respectively the angular and linear velocities of center of mass of the ℎ element and expressed in the inertial frame. Denoting t 0 = 0 r 0 the twist of the satellite base expressed in the inertial frame R , q ∈ R ×1 and q ∈ R ×1 respectively the reaction-wheels and manipulator joints positions/rotations, one can express each element of the twist as [START_REF] Wilde | Equations of motion of free-floating spacecraft-manipulator systems: An engineer's tutorial[END_REF]:

t = J 0 t 0 + J q + J q (1)
The manipulator and the reaction-wheels are in separated kinematic chains which leads to J = 0 6× for the twist of a reaction wheel and J = 0 6× for the twist of a manipulator link. With the DH convention, Jacobians J 0 and J are rotational matrices to express the twists from the referential linked to the center of mass of the ℎ element to the inertial frame, J also includes the linear transformation from the center of mass to the actuator position.

To detail the Lagrangian approach, the system energy is firstly developed. For a rigid rotation-freefloating space manipulator only the kinetic energy is considered. Associating for the ℎ element, , a mass and an inertia I expressed in the inertial frame, one can express the total system kinetic energy as:

L = T = 1 2 =0 I + r r = 1 2 t 0 I 0 0 3 0 3 0 I 3 t 0 + 1 2 =1 t I 0 3 0 3 I 3 t (2)
Or expressed in function of reaction-wheels and manipulator joint actuator's velocities with (1):

L = 1 2 t 0 I 0 0 3 0 3 0 I 3 t 0 + 1 2 =1 t I 0 3 0 3 I 3 t + 1 2 =1 t I 0 3 0 3 I 3 t = 1 2 t 0 I 0 0 3 0 3 0 I 3 t 0 + t 0 =1 J 0 I 0 3 0 3 I 3 J 0 t 0 + 1 2 t 0 =1 J 0 I 0 3 0 3 I 3 J q + t 0 =1 J 0 I 0 3 0 3 I 3 J q + 1 2 q =1 J I 0 3 0 3 I 3 J 0 t 0 + q =1 J I 0 3 0 3 I 3 J 0 t 0 + 1 2 q =1 J I 0 3 0 3 I 3 J q + q =1 J I 0 3 0 3 I 3 J q (3)
Or in a compact form:

L = 1 2 t 0 q q        M 0 M 0 M 0 M 0 M 0 × M 0 0 × M               t 0 q q        = 1 2 t 0 q q M(x 0 , q , q )        t 0 q q        (4)
with x 0 the state vector composed of the base Euler angles, 0 , and position in the inertial frame such that x 0 = 0 r 0 . Matrix M is the inertia matrix of the system which is dependent on the spacecraft configurations and rotations, by construction it is a nonlinear positive definite matrix and can be detailed as:

                                                   M 0 = I 0 0 3 0 3 0 I 3 + =1 J 0 I 0 3 0 3 I 3 J 0 M 0 = =1 J 0 I 0 3 0 3 I 3 J M 0 = =1 J 0 I 0 3 0 3 I 3 J M = =1 J I 0 3 0 3 I 3 J M = =1 J I 0 3 0 3 I 3 J
System dynamics are obtained evaluating the Euler-Lagrange equation of (4) on each DoF as:

                       T t 0 - T x 0 = 0 + 0 T q - T q = + T q - T q = + (6a) (6b) (6c)
where 0 , and are external torques that respectively apply to the base, the manipulator and the reaction-wheels. Moreover, rotation-free-floating manipulators base motions are not actively controlled, meaning 0 = 0 6×1 .

For control purposes, the spacecraft angular velocity is preferably expressed in the base body frame, R . Introducing a joint-state vector q 0 = 0 r 0 , such that the base spacecraft angular velocity is expressed in R and its linear velocity is expressed in the inertial frame, the base twist is expressed as [START_REF] Virgili-Llop | Spacecraft robotics toolkit: an open-source simulator for spacecraft robotic arm dynamic modeling and control[END_REF]:

t 0 = R 0,R 0 3 0 3 I 3 q 0 = P 0 q 0 ( 7
)
where R 0,R is the Direction-Cosine-Matrix (DCM) that describe the base rotations in R from the Euler angles. With the DH conventions, the DCM is used to recursively expressed the rotations of the ℎ link in the inertial frame and give the rotation direction of the corresponding joint. Notably, this allows to express the system inertias from their body frame to R as [START_REF] Virgili-Llop | Spacecraft robotics toolkit: an open-source simulator for spacecraft robotic arm dynamic modeling and control[END_REF]:

I = R ,R I R ,R (8)
where R ,R is the DCM of the ℎ body expressed in the inertial frame.

Thus, developing the Euler-Lagrange equation ( 6) and expressing the base angular velocity in its body frame, a rigid space manipulator dynamics can be written as [START_REF] Virgili-Llop | Spacecraft robotics toolkit: an open-source simulator for spacecraft robotic arm dynamic modeling and control[END_REF];

       H 0 H 0 H 0 H 0 H 0 × H 0 0 × H        H(q 0 ,q)        q 0 q q        +        C 0 C 0 C 0 C 0 C 0 × C 0 0 × C        C(q 0 ,q, q, q 0 )        q 0 q q        =        0 6×1        (9)
C is the nonlinear convective matrix dependent on the spacecraft configurations and velocities corresponding to the Coriolis and centrifugal terms. In the rest of the paper reaction-wheels and manipulator actuators state are gathered into the vector q as q = q q and subscript denotes the association of reaction-wheels and manipulator quantities, such that H 0 = H 0 H 0 and H = H 0 0 H

. Moreover, to alleviate notations on matrices that depend on the spacecraft configuration (i.e. q 0 and q), in the following of the document it will be referred with only q as q 0 dependency corresponds to spacecraft position and orientation in the inertial frame.

Integration of the flexible dynamics

Flexible dynamics remain challenging to model and incorporate into the rigid behavior of a system. Some assumptions need to be made to simplify its derivation. The first one consist of assuming that small satellite base rotations lead to small flexible deformations. The second assumption is made when the flexible dynamics are obtained from a modal approach and a superposition of modes as detailed in the Two Input Two Output Ports (TITOP) modeling [START_REF] Sanfedino | Finite element based n-port model for preliminary design of multibody systems[END_REF]. The dynamics expression of the satellite and its flexible appendages is developed by considering the internal DoFs of the flexible appendages, and the rigid junction DoFs. The interactions between a flexible body and the rigid spacecraft take place on the common junction such that for an external force/torque applying or exciting the appendage a linear or angular displacement occurs and reciprocally any excitation lead to a force/torque response on the junction DoFs. From a finite element method, the flexible dynamic of an appendage attached at point p to the base as a second order equation that can be written with an appropriate choice of the flexible modal displacements base as [START_REF] Sanfedino | Finite element based n-port model for preliminary design of multibody systems[END_REF]: The flexible

I L p L p M t + c 0 0 0 t + k 0 0 0 0 6×1 = F F (10)
where:

• is the modal displacement vector • t is the twist vector of the fixed joint between the appendages and the satellite base • L p is the matrix of the participation factors • M is the condensed mass matrix of the rigid bodies composed of the junction and the appendage • c is the damping matrix assumed diagonal • k is the diagonal generalized stiffness matrix • F is the forces/torques imposed on the internal DoFs • F is the forces/torques imposed on the appendage by the junction With the use of the Craig-Bampton transformation [START_REF] Craig | Coupling of substructures for dynamic analyses-an overview. 41st structures, structural dynamics, and materials conference and exhibit[END_REF], the generalized internal DoFs mass matrix has been diagonalized with the normalized eigenvectors Ψ which allows to detail:

• L p = Ψ M , with M the coupling matrix between flexible and rigid DoFs • c = diag( 2), with and respectively the flexible DoFs' dampings and natural frequencies

• k = diag( 2 )
In the present study, flexible bodies are considered connected to the robot's base, however the following modeling method could be adapted to include a flexible element in the kinematic chain of the manipulator. As expressed in [START_REF] Li | Motion planning and coordination control of space robot using methods of calculated momentum[END_REF], hypothesis on low base velocities are considered. Taking into account couplings between the base and the flexible appendages, the Lagrangian formalism is adopted to include the rigid SMS equations of motions [START_REF] Wilde | Equations of motion of free-floating spacecraft-manipulator systems: An engineer's tutorial[END_REF]. Generalizing to appendages rigidly attached to the spacecraft's base, adapting the notations of the inertia matrix in [START_REF] Li | Motion planning and coordination control of space robot using methods of calculated momentum[END_REF] allows to detail energies and torques/forces present in a rotation-free-floating SMS with flexible appendages subject no external torques/forces as:

                                   T = 1 2 t 0        I L p L p =1 J 0 I 0 3 0 3 I 3 J 0        t 0 T = 1 2 t 0 I 0 0 3 0 3 0 I 3 t 0 + 1 2 =1 t I 0 3 0 3 I 3 t V = - 1 2 K F = -C (11a) (11b) (11c) (11d)
where t = J 0 t 0 , ∀ ∈ 1;

. Flexible appendages induce both a dissipative force, F , and a variation of the potential energy V , the Lagrangian expression (4) then becomes L = T +T -V . The Euler-Lagrange equation with the additional flexible DoFs is detailed as:

                       L t 0 - L x 0 = 0 6×1 L q - L q = L - L = F (12a) (12b) (12c)
Thus, developing the Euler-Lagrange ( 12) and expressing the base velocity in its body frame with [START_REF] Beyer | Caesar: Space robotics technology for assembly, maintenance, and repair[END_REF], the motion equations of a rotation-free-floating SMS with with rigid links and flexible appendages can be written as:

       H 0 H 0 H 0 H 0 H 0 × H 0 0 × H        H(q)        q 0 q        +        C 0 C 0 C 0 C 0 C 0 × C 0 0 × C        C(q, q, q 0 )        q 0 q        +        0 6 0 6× 0 6× 0 ×6 0 0 × 0 ×6 0 × K        K        q 0 q        =        0 6×1 0 ×1        (13)
The inertia matrix holds the symmetric and positive properties and still depends on the spacecraft configuration. One can note by considering both inertia and convective matrices in (13) that the manipulator and reaction-wheels have no direct impact on the flexible dynamics. These properties imply that actuators motions induce vibrations of the flexible elements if they affect the base motions. In that matter, SMS control strategies have been developed to reduce impact of actuators on a free-floating base such as based on the use of null-space projector [START_REF] Dragomir N Nenchev | Reaction null-space control of flexible structure mounted manipulator systems[END_REF] or with mapping of coupling between the manipulator and flexible appendages [START_REF] Meng | Space robots with flexible appendages: dynamic modeling, coupling measurement, and vibration suppression[END_REF].

Control strategy

Fig. 2 Block diagram of the proposed joint space control method

In this section, internal disturbances, model uncertainties and system variations are considered to develop the joint space control strategy. As illustrated with figure 2, the control law is based on a system dynamic inversion in which the unmeasured states (i.e. the flexible and linear spacecraft dynamics) are included with an extended state observer as well as an estimation of a disturbance torques induced by the modeling error. Besides allowing the decoupling of actuators and adequately linearize the system, the ESO allows to efficiently use reaction-wheels as a rejection mean of spacecraft base rotations induced by the internal disturbance torque [START_REF] Kraïem | Control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF]. The addition of a nonlinear disturbance torque (NDO) has a similar purpose for the compensation of modeling errors. The state observer is developed capitalizing on the derivation of a rotation-free-floating space manipulator with flexible appendages dynamics, detailed in section 2.1. It is designed such that actuators' velocities and control torques are used as input and the angular spacecraft's velocities as output measure. On another hand, the NDO is designed using only actuators' velocities, as accelerations are not available measures. With the proposed method, both control and observers performances are interdependent [START_REF] Kraïem | Control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF] which motivates a common gains synthesis to insure system stability. From our previous works [START_REF] Kraïem | Control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF][START_REF] Kraïem | Robust control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF], the novelty of the present study resides in the adaptation of gains synthesis method that includes discussion on significant system evolution, modeling uncertainties and measurement errors.

Before developing the open-loop dynamics, notations are introduced in order to consider errors in evaluation of inverse kinematics/dynamics due to modeling uncertainties and measurement errors. A quantity X obtained or evaluated from measurement is indicated as X such that X = X + ΔX with ΔX representing the difference between the actual value and the measured one. Thus, the system dynamics (13) obtained from the modeled spacecraft can be expressed as:

Ĥ( q)        q 0 q        + Ĉ( q, q, q0 )        q 0 q        + K        q 0 q        =        0 6×1 0 ×1        -        Δ 1 ( q, q, q0 ) Δ 2 ( q, q, q0 ) Δ 3 ( q, q, q0 )        (14)
with the following hypothesis:

• the terms ΔxΔy are neglected,

• the base linear velocity is estimated by the state observer detailed in section 3.2 such that q0 = ˆ 0 r 0 ,

• measurements of actuators position and velocities are considered good enough to pose q = q and q = q Modeling and measurement errors are then gathered in a disturbance torque defined as:

Δ ( q, q, q0 ) =        Δ 1 ( q, q, q0 ) Δ 2 ( q, q, q0 ) Δ 3 ( q, q, q0 )        = ΔH        q 0 q        + ΔC        q 0 q        + ΔK        q 0 q        -Ĥ(q)Δ        q 0 q        -Ĉ( q, q, q0 )Δ        q 0 q        -KΔ        q 0 q        (15)

Joint open-loop behavior

In order to establish a joint space control, a rewriting effort of the full system dynamic followed by the modeled manipulator ( 14) is required. This effort aims at expressing the joints' dynamics in function of either measurable states or quantities that can be estimated with the observers detailed in sections 3.2 and 3.3. First, with ( 14) actuators' dynamics are expressed as:

Ĥ q + Ĉ q = -Ĥ 0 0 q 0 -0 Ĉ 0 0        q 0        + Δ 2 + (16)
Secondly, from ( 14), the unmeasured states are isolated as:

Ĥ0 Ĥ0 Ĥ 0 Ĥ q 0 + Ĥ0 0 q + 0 Ĉ0 Ĉ0 K Ĉ 0 Ĉ        q 0        + Ĉ0 0 q = Δ 1 Δ 3 ( 17
)
One can note that the matrix Ĥ0 Ĥ0 Ĥ 0 Ĥ is an inertia matrix which by definition has an inverse given by:

Ĥ0 Ĥ0 Ĥ 0 Ĥ -1 = ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 -( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĥ0 -Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 I + Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĥ0 ( 18
)
with Ĥ = I . The inertial term ( Ĥ0 -Ĥ0 Ĥ 0 ) corresponds to a scaled inertia of the base for which the flexible inertia have been subtracted of the spacecraft base. Then by injecting [START_REF] Fatina | Space robot motion planning in the presence of nonconserved linear and angular momenta[END_REF] in [START_REF] Pisculli | A minimum state multibody/fem approach for modeling flexible orbiting space systems[END_REF] and by introducing the state vector x = q 0 , actuators' dynamics are obtained as:

M (q) q + D (q, q, q0 ) q + D (q, q, q0 )x = + J Δ (q) Δ (19)
with:

                         M = Ĥ -Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĥ0 D = Ĉ -Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĉ0 D = 0 Ĉ 0 0 -Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 -( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĥ0 0 Ĉ0 Ĉ0 K Ĉ 0 Ĉ J Δ = -Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 I Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĥ0 (20a) (20b) (20c) (20d)
Thus, the joints' dynamics depends on an un-measurable state vector x and a disturbance torque = J Δ (q) Δ . The state vector x includes spacecraft linear drift and flexible dynamics while the disturbance torque gathers modeling and measurement errors. Splitting the sources of disturbances applying on actuators presents the advantage of clearly identifying the perturbations with different dynamics properties such that a rejection strategy can be developed.

Moreover, one can note that matrix D gathers an equivalent stiffness and a convective term multiplied by a scaled inertia. This scaling corresponds to the impact of flexible dynamics onto the actuators ones.

In order to linearize the system a state observer and a nonlinear disturbance observer are detailed in the following sections.

State Observer model

An extended state observer is developed such that the flexible and the spacecraft's linear dynamics can be included in the system linearization detailed in section 3.4. The ESO is established using the control torque and the measure of actuators' velocities as input and the spacecraft's angular velocities as output measurements.

As the actuators' accelerations are not available, by injecting ( 16) in [START_REF] Fatina | Space robot motion planning in the presence of nonconserved linear and angular momenta[END_REF], the dynamic of the un-measured state can be written as:

H ★ 0 q 0 + C ★ 0        q 0        + C ★ q = J ★ Δ Δ + J ★ (21)
with:

                                                   H ★ 0 = Ĥ0 Ĥ0 Ĥ 0 Ĥ - Ĥ0 0 Ĥ-1 Ĥ 0 0 C ★ 0 = 0 Ĉ0 Ĉ0 K Ĉ 0 Ĉ - Ĥ0 0 Ĥ-1 0 Ĉ 0 0 C ★ = Ĉ0 0 - Ĥ0 0 Ĥ-1 Ĉ J ★ Δ = I -Ĥ0 Ĥ-1 0 0 0 I J ★ = - Ĥ0 0 Ĥ-1 (22a) (22b) (22c) (22d) (22e)
Introducing the state vector to be estimated, x = q 0 , one can re-write [START_REF] Rybus | Control system for free-floating space manipulator based on nonlinear model predictive control (nmpc)[END_REF] as:

                       x =        q 0        = 0 0 × I -H ★ -1 0 C ★ 0 x + 0 × 0 × -H ★ -1 0 C ★ H ★ -1 0 J ★ q + 0 ×(6+ + ) H ★ -1 0 J ★ Δ Δ = A (q, q, q0 )x + B (q, q, q0 )u + B Δ (q, q, q0 ) Δ y = 0 I 0 0 x = C x (23a) (23b)
The ESO dynamics include the disturbance torque which requires to be estimated in order to linearize the system as well as insuring an accurate state estimation. By introducing the linear estimation gain L and ˆ the estimation of , the state vector is estimated as x such that:

x = A (q, q, q0 )x + B (q, q, q0 )u + B Δ (q, q, q0 )J + Δ (q) ˆ + L (y -C x ) (24)
with X + denoting the pseudo-inverse of X.

Nonlinear Disturbance Observer model

Similarly to the state observer, including an estimation of the disturbance torques in the system linearization aims at improving the control performances. In particular, the present disturbance torque includes both modeling error and measurement errors which allows to compensate different uncertainties of the manipulator model and maintain high control performances.

The disturbance observer is developed capitalizing on the structure of the joint's dynamics [START_REF] Li | Assembly dynamics of a large space modular satellite antenna[END_REF] by introducing a gain L , the disturbance observer is given by [START_REF] Mohammadi | Nonlinear disturbance observer design for robotic manipulators[END_REF]:

ˆ = -L ˆ + L (M q + D q + D x -) (25)
However, actuators' accelerations are usually not available measurements. To overcome this drawback, a new variable w is introduced, such that [START_REF] Chen | A nonlinear disturbance observer for robotic manipulators[END_REF]:

w = ˆ -p(q, q) (26)
where the vector p(q, q) can be computed from the nonlinear gain L (q, q) as [START_REF] Mohammadi | Nonlinear disturbance observer design for robotic manipulators[END_REF]:

p(q, q) = L (q, q)M (q) q (27)
With ( 19), ( 25) and ( 27), one can describe the nonlinear disturbance observer with the time-derivative of (26): w = ˆp(q, q) = -L (q, q) + L (M (q) q + D (q, q, q0 ) q + D (q, q, q0 )x -) = -L (q, q) ˆ + L (q, q) D (q, q, q0 ) q + D (q, q, q0 )x -+ L (q, q)D (q, q, q0 ) (xx)

= -L (q, q)w + L (q, q) D (q, q, q0 ) q + D (q, q, q0 )x -p(q, q)

+ L (q, q)D (q, q, q0 ) (xx)

Hence the nonlinear disturbance with only measurable or estimated states is defined as:

            
w = -L (q, q)w + L (q, q) D (q, q, q0 ) q + D (q, q, q0 )x -p(q, q) ˆ = w + p(q, q) p(q, q) = L (q, q)M (q) q (29a) (29b) (29c)

Closed-loop dynamics

After establishing the different observer models and the open-loop control behavior, one can note the different inter-dependencies between each dynamics. In order to insure stability of the closed-loop system, the tracking control error signal and the observers error signals are considered in this section to highlight the notable dependencies between the dynamics. The tracking error control signal is defined as = (qq), with q the desired joint space trajectory, the state observer error signal = (xx ) and the nonlinear disturbance error signal = ( -ˆ ).

The state observer error signal dynamics can be obtained with ( 23) and ( 24) such that:

= x -x = A x + B u + B Δ J + Δ -A x + B u + B Δ J + Δ ˆ + L (y -C x ) = (A -L C ) + B Δ J + Δ (30)
In order to obtain the nonlinear disturbance observer error dynamics, one can make a first assumption on the relative convergence rate of the error signals compared with the evolution of the torque disturbance. By construction, the disturbance torque present slow dynamics as the Δ matrices have relatively low amplitudes. One can then assume that ≈ 0 [START_REF] Chen | A nonlinear disturbance observer for robotic manipulators[END_REF] and in the case of fast disturbance dynamics, one can adapt the gain synthesis detailed in section 3.5 such that the convergence rate of is exponential when is bounded [START_REF] Mohammadi | Nonlinear disturbance observer design for robotic manipulators[END_REF]. Thus, the respective error dynamics is approximated by:

= -ˆ ≈ -ˆ = -w -p (31)
Then with ( 29) and ( 31), the error signal of the NDO in function of the state observer error signal can be detailed as:

= L w -L D q + D x --p + L D -p = L (w + p) -L M q + D q + D x - + L D = -L + L D (32)
Before establishing the tracking control error dynamics, a control torque that allows to linearize and decouple the system is introduced. For a desired actuator dynamics, v, such that:

v = K q -q q -q = K ( 33
)
where K is a linear control gain, a control torque that realizes this objectives is given by:

= M K + D q + D x -ˆ (34)
By injecting [START_REF] Sanfedino | Finite element based n-port model for preliminary design of multibody systems[END_REF] in [START_REF] Li | Assembly dynamics of a large space modular satellite antenna[END_REF], the actuators closed-loop dynamics is given as:

M q + D q + D x -= M K + D q + D x -ˆ ( 35
)
matrix M is defined positive and symmetric as it an inertia matrix, thus by rewriting the close-loop dynamics, the tracking control error dynamics is obtained as:

= -K + M -1 D -M -1 (36)
Let's introduce the state vector z = , then ( 36) can be re-written as a state representation:

                 z = 0 I 0 0 + 0 -I K z + 0 M -1 D + 0 -M -1 = (A + B K) z + B + B = I 0 z = C z (37a) (37b)
Thus, by considering ( 30), ( 32) and ( 37) one can note that the system's stability depends on the convergence of coupled observers dynamics. A new hypothesis on the convergence rates can be made to reduce the coupling of the observers dynamics and enforce the hypothesis of ≈ 0. The disturbance observer error present lower amplitudes than x as it mainly corresponds to residual modeling errors. In consequence, the emphasis on an accurate estimation of x should be made over ˆ estimation. Imposing a faster convergence time on the state estimation than on the disturbances allows to simplify the coupling dependency in [START_REF] Dubanchet | Modeling and control of a flexible space robot to capture a tumbling debris[END_REF] such that:

= (A -L C ) (38)
With this last hypothesis, ( 32) and ( 37), a compact version of the closed-loop dynamics is obtained by introducing X = z :

X =        (A + B K) B B 0 (A -L C ) 0 0 L D -L        X = AX (39)

Simultaneous gains synthesis

Observing the relations between control and observer performances, the gains are synthesized simultaneously to insure both stability and disturbance rejections. The synthesis is firstly developed for a linearized system and then extended to deal with workspace considerations. Secondly, the synthesis method is adapted to deal with large system evolutions mainly due to the mass distribution variations.

Robustness to modeling and measurement errors

With the following proposition, the gains synthesis allows to insure control performances in presence of modeling uncertainties and measurement errors for a given linearized system.

Proposition:

If there exist symmetric definite matrices Q , Q and P and matrices W and W of appropriate dimensions such that for a given scalar > 0 the following LMI constraint holds:

Θ =          (A Q + B K) s B B Q Q C * (P A -W C ) s D 0 * * (-Q + D ) s 0 * * * -2 I          < 0 (40)
with X s = X + X , Then system is quadratically stabilized with K = W Q -1 , L = P -1 W and L = P -1 M -1 . Moreover, the outputs verify:

∫ ∞ 0 ( ) ( ) < 2 (41)
for any conditions z(0) = 0 and 0 , 0 ∈ { | E ≤ 1}.

Proof:

As the matrix M is positive definite, one can choose the following Lyapunov function:

V(X) = z P z + P + P M P ≥ 0 = X        P 0 0 0 P 0 0 0 (P M P )        X = X PX (42)
such that V + -2 ≤ 0 for a given > 0.

Then by integration of this constraint:

∀ T > 0, ∫ T 0 V + -2 < 0 ⇒ 2 ∫ T 0 < 2 (V 0 -V ) ⇒ 2 ∫ T 0 < 2 V 0 = 2 0 0 P 0 0 P M P 0 0 < 0 0 E 0 0 < 2 (43)
this condition is enforced by P 0 0 P M P ≤ E.

The time derivative of the proposed Lyapunov function is then given by:

V(X) = X A P + PA +        0 0 0 0 0 0 0 0 P M P        X = X        (P A + P B K) s P B P B * (P (A -L C )) s (L D ) (P M P ) * * (-P M P ) s + P M P        X (44)
where the time derivative of the equivalent inertia matrix can be defined as a function of the convective matrix as [START_REF] Mohammadi | Nonlinear disturbance observer design for robotic manipulators[END_REF]:

M = D + D (45)
As proposed by Mohammadi and al. [START_REF] Mohammadi | Nonlinear disturbance observer design for robotic manipulators[END_REF], a candidate nonlinear gain observer can be chosen as:

L = P -1 M -1 (46)
which leads with (29) to the auxiliary variable:

p = P -1 q (47)
Thus, simplification can be made to obtain the expression of the time-derivative of the Lyapunov function:

V(X) = X        (P (A + B K)) s P B P B * (P (A -L C )) s D P * * (-P ) + (P M P )        X (48)
Then the constraint V + -2 ≤ 0 is equivalent to:

       (P (A + B K)) s + -2 C C P B P B * (P (A -L C )) s D P * * (-P ) + (P M P )        < 0 (49)
By applying the Schur complement, the condition becomes:

         (P (A + B K)) s P B P B C * (P (A -L C )) s D P 0 * * (-P ) + (P M P ) 0 * * * 2          < 0 (50)
By introducing the variable changes W = KQ and W = P -1 L , and pre and post multiplying the above matrix by diag(Q , I, Q , I) = diag(P -1 , I, P -1 , I) one obtains the LMI constraint (40).

Robustness to system variations

During an on-orbit servicing operation, the distribution of mass is led to face significant changes through the different manipulations. Thus the resolution of the LMI [START_REF] Mathieu Rognant | Autonomous assembly of large structures in space: a technology review[END_REF], which is verified around a system's equilibrium state, need to be slightly changed in order to consider those system variations without considering multiple equilibrium points which would prohibitively increase the number of LMI constraints in the design process. In this section, under mild assumptions, it is highlighted that system velocities can be neglected in a variations preliminary analysis to focus on a workspace analysis. In other words, in the inequality [START_REF] Mathieu Rognant | Autonomous assembly of large structures in space: a technology review[END_REF], convective terms have much less influence than inertial terms.

A first modification of the gains synthesis can be observed by considering a previous assumption on the convergence rate of the disturbance torque estimation. Compared to both control and state observer error signals, the dynamics of the disturbance torque estimator are slow enough to be neglected. Consequently, the tracking error dynamics (37) can be approximated as:

z = (A + B K) z + B (51)
With these considerations, the LMI (40) including system variations during OOS operations is modified as follows:

Θ =          (A Q + B K) s B + ΔB 0 Q C * (P A -W C ) s D + ΔD 0 * * (-Q + D ) s + ΔD + ΔD 0 * * * -2 I          < 0 (52)
Let us now introduce bounds 1 , 2 and 3 on the maximum singular values of the interest variations:

           ¯ ΔB < 1 ¯ ΔD < 2 ¯ ΔD + ΔD < 3 (53a) (53b) (53c)
Using a Schur-complement based argument as in [START_REF] Kraïem | Robust control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF], it is readily checked that the inequalities (52) are enforced by:

Θ =          (A Q + B K) s + 1 I B 0 Q C * (P A -W C ) s + ( 1 + 2 )I D 0 * * (-Q + D ) s + ( 2 + 3 )I 0 * * * -2 I          < 0 (54)
Let us now evaluate the bounds introduced in (53). From equations ( 20) and [START_REF] Mohammadi | Nonlinear disturbance observer design for robotic manipulators[END_REF], one can identify the matrix terms that are dependent on the workspace and those that also require system velocities to be evaluated. Decomposing D with a stiffness and a convective equivalent term such that D (q, q) = K (q) D , (q, q) defined as:

               K = Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĥ0 K D , = Ĉ 0 -Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĉ + Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĥ0 Ĉ 0 -Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĉ0 + Ĥ 0 ( Ĥ0 -Ĥ0 Ĥ 0 ) -1 Ĥ0 Ĉ (55a) (55b)
one can consider matrices M (q) and K (q) on one side and velocity dependent term matrices D and D , on an other side. Firstly, the common factor term Ĥ

0 Ĥ0 -Ĥ0 Ĥ 0 -1
, which corresponds to a scaled inertia can be constrained as:

Ĥ 0 Ĥ0 -Ĥ0 Ĥ 0 -1 ≤ ¯ Ĥ 0 H 0 -H 0 H 0 = (56)
where (A) denotes the minimum singular value of matrix A. Thus, matrices M (q) and K (q) can be bounded as:

       M I ≤ M ≤ ¯ M I K ≤ ¯ Ĥ0 K (57a) (57b)
Then, the velocity dependent terms can be simplified thanks to specific assumptions in space robotic applications. In order to maintain system stability, actuators velocities remain low enough to develop slow manipulator motions such that flexible modes excitation are limited. Moreover, system variations correspond to the different mass distributions which are not affected by the reaction-wheels velocities. The study of matrices D , and D results in identifying the impact of the manipulator motions in the considered workspace.

The rewriting effort made to obtain the joint-space dynamics [START_REF] Li | Assembly dynamics of a large space modular satellite antenna[END_REF] has allowed to define D as an equivalent Coriolis matrix. This convective term can be bounded as a function of the SMS configuration and the actuators capacity such that [START_REF] Mulero-Martinez | Uniform bounds of the coriolis/centripetal matrix of serial robot manipulators[END_REF]: D (q, q) ≤ ¯ D (q, q) ≤ q 2 (58)

where the parameter is defined as a function of the studied workspace [START_REF] Mulero-Martinez | Uniform bounds of the coriolis/centripetal matrix of serial robot manipulators[END_REF]:

= 3 2 sup =1 M (q) q (59)
Furthermore, with the assumption of slow manipulator motions, the evaluation of D can be reduced to the variations of K . By definition, convective matrices correspond to the cross product terms between q, and 0 . For slow manipulator and base motions, flexible modes vibration amplitudes are limited. These general assumptions enable to neglect convective term variations in front of inertia/mass ones. Bounding D as follows

D ≤ ¯ K + ¯ D , (60)
and D , with only convective terms such that:

D , ≤ ¯ Ĉ 0 + ¯ Ĥ 0 ¯ Ĉ + ¯ Ĥ0 Ĉ 0 + ¯ Ĉ0 + ¯ Ĥ0 Ĉ (61)
four identical sun-shield's beams highlight similar flexible modes with the following natural frequencies: [14.1, 44.4, 86.9] / . Likewise, the two identical solar arrays exhibit similar flexible modes with the following natural frequencies: [1.01, 4.39, 6.79, 7.6, 19.17] / . The total spacecraft mass is 6892kg. The base weighs 1960 kg while the robotic arm is limited to 60 kg. More details on the system and the entire assembly scenario are available at https://www.h2020-pulsar.eu/listings.

Simulations results

Simulation tools and gain synthesis solver

Based on the dynamics of a rigid-flexible multi-body system, a Matlab-Simulink tool has been developed by integrating the Satellite Dynamic Toolbox [START_REF] Alazard | Linear dynamic modeling of spacecraft with various flexible appendages and on-board angular momentums[END_REF] into the SPART [START_REF] Virgili-Llop | SPART: an open-source modeling and control toolkit for mobile-base robotic multibody systems with kinematic tree topologies[END_REF] toolkit. From an XML description of the system, this integration of the tools allows for both numerical and symbolic analysis. Moreover, Simulink-based time-domain simulations can also be easily performed [START_REF] Cumer | Modelling and attitude control design for autonomous in-orbit assembly[END_REF].

The LMI resolution involves high-dimensional matrices and numerous constraints (according to the considered scenario) which may lead to significant computational times. In the present application, each constraint [START_REF] Mathieu Rognant | Autonomous assembly of large structures in space: a technology review[END_REF] corresponds to a 106 × 106 matrix inequality where the decision variables Q , Q and P are respectively 28 × 28, 14 × 14 and 50 × 50 symmetric matrices. In order to better control the resolution time, the YALMIP toolbox [START_REF] Lofberg | Yalmip: A toolbox for modeling and optimization in matlab[END_REF] is interfaced here with the MOSEK solver. To assess the computation cost, table 1 gathers, for different sizes of the set of tasks T , the resolution times which have been obtained on a standard computer equipped with an Intel i7 Processor.

As expected, the CPU time considerably increases with the size of T . However, the number of needed iterations is not much affected. It should also be emphasized that, without major degradation, the controllers gains can be restricted to diagonal matrices thus slightly reducing the number of decision variables. If the time resolution becomes an issue, as it directly depends on the number of decision variables and LMI constraints, the use of furthers relaxations can be considered at the cost of possibly increased conservatism.

Analysis of system variations

During the on-orbit assembly of the PULSAR telescope, important system variations occur due to mass distribution changes. A preliminary analysis of the impact of such variations -through the evaluation of the bounds detailed in (53) for each sub-task of the deployment scenario -is useful to reduce the size of T in the design process. Sub-tasks presenting lowest variations correspond to verification maneuvers or small adjustment of end-effector rotations. Consequently such tasks share similar physical properties with the following longer manipulation task justifying the reduction of tasks considered in the gains synthesis. Figure 4 illustrates variations of the three bounds for each task ∈ T for unused reaction-wheels (i.e. q = 0) as they do not affect the mass distribution in the workspace. For each task, the norms are evaluated for the maximal joint velocities and uniformly distributed manipulator joint configurations in the corresponding workspace . As a first observation, it can be noted that a workspace approach is a suitable method to evaluate the bounds . Observing from the upper subplot of Figure 4, the similar evolutions of ¯ M -1 D and ¯ M -1 K and then comparing ¯ D with ¯ K on the central subplot, we can validate the approximations of section 3.5.2 from which the bounds (63) are derived. Regarding the actuators' velocities and the variations of 3 from the lower subplot of Figure 4 in comparison to the evolutions of 1 and 2 the study of system variations with a workspace analysis can be justified.

A second observation can be made on the large system variations for each task by considering the evolution ¯ M -1 K , whose maximum (red full line) and minimum (blue dashed line) values are visualized on the upper subplot of figure 4. These variations motivate the use of bounds to be incorporated as a single constraint instead of considering multiple equilibrium points for each task which would generate numerous constraints and would lead to an unsolvable optimization problem.

Moreover, according to the considered scenario, some successive tasks may exhibit similar workspaces and mass distributions. Then, a reduced set of tasks can be used in the design procedure. In our context, further simplifications could be achieved by eliminating short tasks or those whose impact on the system variations remains small enough.

System performances in simulation

In this section, the interest of the method is illustrated with time-domain simulations. In order to validate the efficiency of the design procedure, simulation are performed on the following set of tasks T = {4, 124, 298}. These tasks are chosen as they last the longest and have a significant impact on the system variations ( 1 (4) = 155, 1 (124) = 136 and 1 (298) = 102). As illustrated with figure 3, 4 is a simple manipulator motion, in 124 a bundle of 5 tiles is positioned in the mirror and in 298 1 tile is moved from its stack to create a temporary bundle. Thus, different inertia properties and mass displacement are considered to highlight the robustness of the control method. Regarding modeling errors, which impact the evaluation of the inverse kinematics/dynamics, a degraded space manipulator model is considered. From the nominal model of the PULSAR telescope, the following levels of uncertainties are considered: ±3% on the inertia I , +10% on the flexible parameters and ±10% on the position of the CoM. Likewise, a noise bias is included on the measure of 0 corresponding to a ±10% of error such that state observer efficiency can be tested.

In the studied application, manipulator trajectories and velocities are provided with singularity free path-planning strategies also taking into account the actuators capacities (https://www.h2020pulsar.eu). Reaction-wheels desired velocities are deduced from those of the manipulator joints assuming a kinetic momentum conservation and by imposing a fixed based such that:

q = -H + 0 H 0 q + H 0 q 0 = -H + 0 H 0 q (64)
with an initial moment considered null. The choice of imposing a fixed spacecraft attitude is justified by reduction of the impact of the indirect coupling effects between the manipulator and the flexible appendages [START_REF] Kraïem | Control of rotation-floating space robots with flexible appendages for on-orbit servicing[END_REF].

Control performances are illustrated with figures 5 and 6 in which joint space velocities and tracking error signals are plotted. In both figures, the first row of sub-figures represents the evolution of the joint velocities for the three tasks in T while the second row represents the respective tracking errors, . The tracking errors highlight the precision of the control scheme. The fast dynamics required is responsible of the largest amplitudes in the transient states. Moreover, the tracking requirements remain satisfied and are not affected by the significant evolution of the mass distribution during the complete task either it was explicitly considered or not in the design process.

In order to illustrate the usefulness of disturbance and state observers to properly linearize and decouple the system, the detailed components of the control torques [START_REF] Sanfedino | Finite element based n-port model for preliminary design of multibody systems[END_REF] are visualized in figures 7 and 8. The control torques can be decomposed into three signals. The first one (M v), visualized in the upper plots of figure 7 corresponds to the feedback control torque. The second one ( D q) is a linearizing torque and is visualized on the lower plots of figure 7. The last one (D x + ˆ ) can be interpreted as a generalized disturbance torque. It appears on the upper plots of figure 8. Since the joints velocities and mass to move are significantly different during each of three tasks, it can be noted that disturbances differently impact the control performances. Thus, for the three tasks comparing amplitudes of signals composing the control torques provides information on the linearization quality. As the linearizing and disturbances torques present similar amplitudes, including the estimated states provides a more precised decoupling of actuators. Furthermore, with an usage of reaction-wheels aiming at restraining base motions during manipulator operations leads to disturbances attenuation and rejection. This can be observed when the manipulator's joints rotates at constant speeds, flexible modes are reducing velocities to rest.

Regarding the time-domain simulations, the proposed control method with its associated gains synthesis allows to provide robustness to system properties variations as well as modeling and measurement errors. Similar performances are obtained when moving various size/mass objects with for different mass arrangement highlighting the interest of relaxation terms efficiency in the gain synthesis.

Conclusion and perspectives

As a synthesis of our previous works, this paper presents a joint space control method for rotationfree-floating space manipulators in presence of flexible appendages. The proposed design strategy is based on the estimation of the disturbances induced by the flexible appendages and spacecraft's linear dynamics which are then used in a compensation scheme strongly inspired by NDI structures. The development of such a control structure strongly relies on a relevant model of the rigid/flexible multi-body system to be controlled. This non trivial modeling process which is a significant contribution of the paper was carried out via a Lagrangian approach. Residual errors induced by uncertainties on system properties (mass distribution, flexible modes,...) are depicted as an external disturbance torque. A second contribution is the proposed gains synthesis based on a LMI resolution. The method allows to simultaneously obtain control and observers gains stabilizing a coupled rigid-flexible rotation-free-floating system. Not only modeling and measurements errors are tackled, but also large system variations occurring in a multi-tasks on-obit-servicing mission performed with a given SMS. In the overall validation process, simulations on an actual on-orbit deployment scenario (https://www.h2020-pulsar.eu) highlighted the control method robustness to significant inertia changes. Future works will develop manipulator's end-effector and spacecraft base control capitalizing on the proposed robust joint-space control.

Fig. 1

 1 Fig. 1 Notations for a rotation-free-floating spacecraft with = 1 reaction-wheel and -DOFs manipulator with two flexible appendages

Fig. 3

 3 Fig. 3 Illustration of the PULSAR telescope on-orbit deployment https://www.h2020-pulsar.eu

Fig. 4

 4 Fig. 4 Evolution of relaxation terms for each tasks of the PULSAR scenario

Fig. 6 Fig. 8

 68 Fig. 6 Sub-figures in the first row represent the evolution of q measured for ∈ T ; Sub-figures in the second row illustrate the tracking error, , and highlight the precision of the proposed control law

Table 1 LMI resolution times according to the size of T

 1

	size of T	10 20	30	40	50	80 100 154
	LMI resolution time (s) 35 130 122 186 245 426 550 833
	Iterations	27 53	33	38	39	38	39	43

Sub-figures in the first row represent the evolution of q measured for ∈ T ; Sub-figures in the second row illustrate the tracking error, , and highlight the precision of the proposed control law

Acknowledgments

The PULSAR project is funded under the European Commission's Horizon 2020 Space Strategic Research Cluster Operational Grants, grant number 821858.

one finally obtains:

Thus, with (57), ( 58) and (62) the required bounds can be evaluated as:

(63c)

Proposed design methodology

The design procedure is now generalized to a multi-task on-orbit servicing scenario in which from one task to another the mass distribution may significantly differ as well as the size of the workspace. To optimally consider the system changes through the different tasks and obtain optimal constant control and observer gains, let us define tasks, denoted 1≤ ≤ ∈ T , where T denotes the entire of tasks corresponding to the total servicing scenario. The details of the design procedure are given by the following algorithm: Algorithm 1 Design procedure for a multi-task on-orbit servicing scenario Define the set of tasks T and for each task 1≤ ≤ with an associated workspace 1≤ ≤ Define the LMI variable as in the proposition in section 3.5.1 for ≤ do Define an equilibrium point to evaluate matrices in the LMI constraint (40) from ( 14) evaluate the relaxation terms 1 , 2 and 3 as in (63) Define a as in (54) end for Minimize the LMI variable > 0 such that 1≤ ≤ < 0 Return K, L , P = Q -1 Thereby, algorithm 1 provides controller and observer gains suitable for a space manipulator system to perform different tasks while preserving similar control performances for each task.

Illustration of the proposed method 4.1 Study case

In order to illustrate our proposed method, the on-orbit deployment of the PULSAR telescope presented in Rognant and al. [START_REF] Mathieu Rognant | Autonomous assembly of large structures in space: a technology review[END_REF] is used. An 8-DoFs manipulator with a first prismatic joint allows to move each of the 44 mirror tiles from stacks to a temporary bundle and then to the telescope's mirror as illustrated by figure 3. The scenario of a complete assembly can be divided into 372 sub-tasks 1≤ ≤372 . In this sequencing each task is either a motion of the manipulator to reach a tile or displacement of a tile to a bundle or the positioning of a bundle on the mirror. In that way, the system variation for a c corresponds to the changes in the system inertia distribution. In addition to the 8-DoFs manipulator, 6 reaction-wheels, positioned inside the spacecraft's base, allow to actively control the 3 rotational DoFs of the space telescope. Moreover, two solar arrays and one sun-shield, represented by four flexible beams, introduce 22 flexible modes which are integrated to the model as detailed in section 2.2. The