Dynamics and attitude control of space-based synthetic aperture radar

Serhii Khoroshylov; Serhii Martyniuk; Oleksandr Sushko; Volodymyr Vasyliev; Elguja Medzmariashvili; William Woods

doi:10.1515/nleng-2022-0277

Article Open Access

Dynamics and attitude control of space-based synthetic aperture radar

Serhii Khoroshylov , Serhii Martyniuk , Oleksandr Sushko , Volodymyr Vasyliev , Elguja Medzmariashvili and William Woods

Published/Copyright: February 16, 2023

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From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

This work tackles the problem of attitude control of a space-based synthetic aperture radar with a deployable reflector antenna, representing a lightly damped uncertain vibratory system with highly nonlinear dynamics. A control strategy based on two identifiable in-orbit vector parameters is proposed to make the robust controller less conservative. The first parameter is used in the feedforward loop to achieve a trade-off between the energy efficiency of maneuvers and the amplitudes of the oscillatory response. The feedback loop utilizes the second parameter to accurately handle the controller-structure interactions by adaptive notch filters. The notch filters are included in the augmented plant at the design stage to guarantee closed-loop robustness against disturbances, unmodeled dynamics, and parametric uncertainty. The system’s robustness and specified requirements are confirmed by formal criteria and numerical simulations using a realistic model of the flexible spacecraft.

Keywords: flexible reflector; nonlinear transformation; in-orbit identification; robust control; slewing maneuvers

1 Introduction

Space-based synthetic aperture radar (SSAR) has unique capabilities for observing dynamic processes on the Earth’s surface. The SSAR makes it possible to obtain high-quality images regardless of weather conditions and the presence of sunlight and clouds [1]. Even though mainly large spacecraft (SC) with a mass of 1–2 tons are currently operational today, there has recently been a growing interest in small satellites for such purposes [2]. Low-cost small satellite systems can potentially expand the range of SSAR applications and make them more commercially attractive.

To design a SSAR with a mass of about 200 kg, it is necessary to use approaches and solutions that differ from those traditionally used for large SC. One such promising approach is to use a deployable reflector antenna (DRA) [3] instead of a conventional phased array antenna (PAA). Compared to the PAA, DRAs are less complex, polarization invariant, lighter, provide a higher gain, etc.

Despite these advantages, the design of a remote sensing SC with DRA leads to some issues in the field of attitude dynamics and control. For example, additional structure elements such as booms are needed to establish the position of the DRA relative to the feedhorns of the radar. Such a structure may be essentially asymmetric, resulting in inertia matrix elements with relatively large values. In this case, the task of fast slew maneuvers becomes significantly more complex because the dynamics of the plant are highly nonlinear, and the control channels of the SSAR are strongly coupled. Furthermore, there are constraints on mass, cost, and power consumption of the actuators, and the DRA and the deployable boom are flexible structures and may significantly vibrate during and after the SSAR maneuvers. All these features should be carefully considered when designing control algorithms, since strict requirements are imposed not only on the accuracy of the platform attitude control but also on the accuracy of the reflecting surface of the DRA and its position relative to the feedhorns. In addition, the control algorithms should be simple enough, given the limited computing power available for the mini-satellite platform.

Since the slew maneuvering flexible SC is a vibratory system with highly nonlinear dynamics, it is not straightforward to design optimal control laws which are simple enough for implementation on a small satellite [4]. For example, in Gasbarri et al. [5], a nonlinear controller is designed for the rigid plant by solving the state-dependent Riccati equation in conjunction with the command shaping to reduce vibrations of the flexible elements. Even though this approach can be employed to control a nonlinear flexible plant, it implies that the solution of the Riccati equation has to be found at each sample time during the control process. In the case of a short sample period of the control system, the SC must have significant computational capabilities to implement such algorithms. Moreover, the parameters of a flexible plant, such as natural frequencies, shapes, and damping ratios, may not be precisely known. Robust [6] or adaptive [7] control methods can be used to handle this issue. However, these methods also complicate control algorithms in the nonlinear formulation of the problem.

To facilitate control design, this attitude control task is often split into several subtasks, such as feedforward control to slew the SC along some optimal trajectory [8] and feedback control to correct deviations of the real SC trajectory from the optimal one [9,10]. Vibrations of flexible structures can be handled in the feedforward and feedback control loops or an additional vibration control loop [11,12].

Yefymenko and Kudermetov [13] demonstrated that the exact nonlinear feedforward control for fast slew maneuvers can be designed using linear methods by a preliminary nonlinear transformation of coordinates and subsequent nonlinear feedback. These methods allow designers to find analytical solutions to optimal control problems. Such control algorithms are simple enough for real implementation but were not considered for flexible plants. Meng and Zhao [14] transformed the flexible SC dynamics into an exact feedback linearization standard form, but they analyzed the closed-loop performance of flexible SC with only passive control. Zakrzhevskii [15] presented control algorithms for slewing of a flexible SC with a minimum relative acceleration. However, such an approach is justified only when the energy cost or durations of the maneuvers are not very important, but this is not the case for the considered SSAR. An efficient method of designing smooth time-optimal attitude maneuver trajectories is proposed for flexible SC in previous studies [16,17]. Although the degree of the smoothness of the trajectory can be adjusted, these results cannot be applied to cases with other optimality objectives. Closed-form solutions for slew maneuvers under a finite jerk constraint are presented in the study of Lee and Song [18]. However, the applications of the solutions are limited because only single-axis slew maneuvers were considered.

Feedforward techniques that seek to reduce residual vibrations of flexible systems by input shaping are studied [19,20], but the accuracy of this approach depends on the exact parameters of the system, including natural frequencies and damping ratios.

A class of controllers that solves the tracking problem for a flexible SC is designed in the study of Di Gennaro [21]. These controllers do not need the measure of the modal and attitude variables, but they rely on the perfect knowledge of the system parameters, such as natural frequencies and damping ratios. This is an obvious limitation since they are not usually known accurately.

In previous studies [22,23], robust control methods are used to solve the problem of uncertainty of a linear flexible plant. However, the success of the robust approach largely depends on a non-conservative description of uncertainty. A technique for deriving a low-order model of a large, deformable SC is proposed in the study of Nicassio et al. [24]. This modeling tool allows for a quick search of gains, which minimize structural excitations, but in this study, modal characteristics are estimated using a finite element model and can differ from the real ones. Miao et al. [25] addressed a flexible SC attitude tracking problem by applying an adaptive strategy to estimate the boundary of unknown external disturbances. According to the reported simulation results, a fast convergence speed can be achieved while simultaneously guaranteeing the finite-time stability of the controlled system. However, the controller requires unnoisy measurements.

In previous studies [26,27], the problem of the uncertainty of the natural frequencies of a flexible structure is addressed using adaptive notch filters. Here the controller is synthesized without considering parameter variations in such filters. Thus closed-loop stability and performance cannot be guaranteed.

Angeletti et al. [28,29] used a network of piezoelectric actuators to reduce the vibrations of a large space antenna. This approach can significantly improve the damping characteristics of the antenna, but it results in a more complex design and architecture of the control system. Moreover, in the study of Angeletti et al. [29], the attitude controller is obtained via micro-synthesis and thus may have an unreasonably high order. A vibration suppression method using a distributed set of angular momentum exchange actuators is proposed in the study of Hu et al. [30]. Although the effectiveness and feasibility of the method are demonstrated, it requires additional sets of reaction wheels (RWs), which increase the mass and cost of the SC.

Even though significant progress has been achieved in the field of modeling of flexible SC [31,32,33], the accuracy of such models still relies on some parameters, which are difficult to verify without physical experiments.

The main problems of the previous methods for small SSAR applications can be summarized as follows:

The success of the controller design largely depends on the accuracy of the SSAR model. Specialized facilities, equipment, and algorithms are needed to accurately identify the parameters of these models, making the mini-satellite more expensive and reducing its commercial benefits.
The controllers are often too complex for implementation on a small satellite. For example, they may require significant computational resources or even additional sensors and actuators for vibration suppression.

These problems are addressed in this article. The main objective of this study is to make the control synthesis less dependent on knowledge of the SSAR model while keeping the controller design robust against uncertainties and pretty simple for small satellite applications. To achieve this objective and to reduce the conservativeness of the robust controller, a control strategy based on two identifiable in-orbit vector parameters is proposed as follows:
The first parameter is used in the feedforward loop to achieve a trade-off between maneuvers’ energy efficiency and oscillatory response amplitudes. The second parameter is utilized in the feedback loop to accurately handle the controller-structure interactions by adaptive notch filters.
The notch filters are included in the augmented plant at the design stage to guarantee controller robustness against the unmodeled dynamics and parametric uncertainty and to avoid making it too complex for the implementation.

The system robustness and specified requirements are confirmed by formal criteria and numerical simulations using a realistic model of the flexible SSAR considering parametric uncertainties, external perturbations, and sensor noise.

The remainder of the article is organized as follows: Section 2 describes the design of the SSAR and DRA; the model of the SSAR dynamics and the finite element model of the DRA are presented in Section 3; the control algorithms are designed in Section 4; the actuator specifics are described in Section 5; Section 6 contains the results of the computer simulations; the main conclusions are summarized in Section 7.

2 Satellite design, system data, and control goals

The SSAR operates in a circular orbit with an altitude of 500 km and an inclination of 98°. Figure 1 illustrates a model of the SSAR. The main design feature of the SSAR is a DRA [34], which is attached to the platform through a deployable boom.

Figure 1

Model of the SSAR.

The DRA is designed using the authors’ many years of experience in deployable structures for space applications [35] but taking into account the specifics of small satellites. Such specifics include high requirements on the structural stiffness and the reflecting surface accuracy but under strict constraints on the mass and stowed size of the DRA.

Two symmetrical network systems (2) of the reflector are attached to the deployable ring (1), which has the shape of an elliptical cylinder (Figure 2). The network systems, tensioned using ties, together with the reflective mesh, form the reflective surface of the required shape. The DRA is attached to the boom in two points.

Figure 2

Fragment of the SSAR (1 – deployable ring, 2 – network systems, A – V-folding bar units).

The elliptical cylinder of the reflector is approximated by 18 planes. In each plane, two beams are connected by a cylindrical hinge in the middle (the so-called pantograph). The 18 pantographs are interconnected at the upper and lower points using the hinge units and form one prismatic ring (Figure 2). The upper and lower hinge units (Figure 3) are interconnected by the horizontal V-folding bars (Figure 4), forming two support rings in the form of the polygons, which can be inscribed in the ellipse. The large diameter of the support ring is 3.538 m, and the small diameter is 3.302 m. The distance between the support rings (height of the deployed reflector) is 0.630 m.

Figure 3

Hinge unit with the deployment synchronizer.

Figure 4

Fixation unit of the V-folding bar.

The reflector is deployed using a cable system driven by two motors installed on the boom. The hinge units of the V-folding bars have mechanisms for fixing the deployed position of the reflector.

The boom consists of four carbon fiber tubes connected by means of hinge units with deployment drivers. The length of the deployed boom is 3.698 m.

The SSAR has the following characteristics: mass of the SSAR platform (including payload) is 201 kg; the mass of the DRA is 7.8 kg; the mass of the boom is 4.56 kg; and the area of the solar array is 2.8 m².

A pyramid configuration [36] of four RWs is used for the SSAR attitude control. Each RW has the following characteristics: the maximum angular momentum is 4 N m s; the maximum reaction torque is 0.3 N m; the maximum wheel angular velocity is 1,000 rad/s; the moment of inertia is 0.004 kg m².

The attitude control system shall ensure the slew of the SSAR from its initial orientation to a final one in a given time. The slewing maneuvers shall be carried out in an energy-optimal manner. At the end of the slews, the orientation errors should be less than 0.05° and 0.005°/s for attitude angles and angular velocities, respectively. The displacements of the DRA shall be less than 1 mm during and after the maneuvers.

3 SSAR dynamics

The following right-handed reference frames are used to describe the SSAR dynamics:

O I x I y I z I is the inertial reference frame (IRF). The origin O I refers to the center of the Earth, the y I -axis points to the North Pole, the z I -axis points to the vernal equinox;

O B x O y O z O is the orbital reference frame (ORF) with the origin O B at the center of mass of the SSAR, z O -axis points along the position vector of the SSAR, with respect to the Earth, y O -axis is taken along the direction normal to the plane defined by the orbital position and velocity vectors, and pointing towards the positive values of the orbital angular momentum.

O B x B y B z B is the SSAR body reference frame (BRF) with the origin O B at the SSAR center of mass. The directions of the BRF axes are depicted in Figure 1. The BRF coincides with the ORF when all attitude angles are zero.

The matrix of rotation from the IRF to ORF can be obtained as follows (Figure 5):

Figure 5

Axes of IRF and ORF.

(1) T IO = T u T i T Ω T i = ci si 0 − si ci 0 0 0 1 , T Ω = c Ω 0 − s Ω 0 1 0 s Ω 0 c Ω , T u = cu 0 − su 0 1 0 su 0 cu ,

where Ω is the longitude of the ascending node; i is the inclination of the orbit; u is the argument of latitude. Hereinafter, the symbols c and s in the rotation matrices denote cos and sin functions, respectively.

The orientation of the BRF with respect to the ORF is defined by Tait–Bryan angles ϕ , ϑ , ψ (pitch, roll, yaw). The matrix of rotation from the ORF to BRF (Figure 6) is given as:

(2) T OB = T ψ T ϕ T ϑ ,

where

T ψ = 1 0 0 0 c ψ s ψ 0 − s ψ c ψ , T ϕ = c ϕ 0 − s ϕ 0 1 0 s ϕ 0 c ϕ , T ϑ = c ϑ s ϑ 0 − s ϑ c ϑ 0 0 0 1 .

Figure 6

Orientation of BRF with respect to ORF.

Tait–Bryan angles are convenient for understanding attitude maneuvers of the SSAR, but it is known that the kinematic equations for these parameters have singular states. To overcome this issue during control design and simulations, the following vector composed of the quaternion components is used Q = [ q 0 q 1 q 2 q 3 ] T = [ q 0 q ] T .

The quaternion components are expressed in terms of Tait–Bryan angles as follows:

(3) q 0 = c ( ϑ / 2 ) c ( ϕ / 2 ) c ( ψ / 2 ) + s ( ϑ / 2 ) s ( ϕ / 2 ) s ( ψ / 2 ) , q 1 = c ( ϑ / 2 ) c ( ϕ / 2 ) s ( ψ / 2 ) − s ( ϑ / 2 ) s ( ϕ / 2 ) c ( ψ / 2 ) , q 2 = c ( ϑ / 2 ) s ( ϕ / 2 ) c ( ψ / 2 ) + s ( ϑ / 2 ) c ( ϕ / 2 ) s ( ψ / 2 ) , q 4 = s ( ϑ / 2 ) c ( ϕ / 2 ) c ( ψ / 2 ) − c ( ϑ / 2 ) s ( ϕ / 2 ) s ( ψ / 2 ) .

The angular velocity of the SSAR and the quaternion derivative are connected as follows:

(4) Q ′ = 0 . 5 0 − ω BO T ω BO − ω BO × Q ,

where ω BO is the angular velocity of the BRF with respect to the ORF, given in the BRF.

In Eq. (4) and further in this work, the notation x × means the following representation of the vector x = [ x 1 x 2 x 3 ] T

(5) x × = 0 − x 3 x 2 x 3 0 − x 1 − x 2 x 1 0 .

The angular velocity of the SSAR with respect to the ORF is given as follows:

(6) ω BO = ω − T OB ω OI ,

where ω is the angular velocity of the BRF with respect to the IRF given in the BRF; ω OI is the angular velocity of the ORF with respect to the IRF given in the ORF.

For dynamic modeling, the SSAR is represented as an interconnection of the absolutely rigid platform, the flexible boom, and the flexible DRA. The displacements of the i-th node of the structure due to elastic deformations are given as follows:

(7) ρ i = Ψ i Y ,

where Ψ i is the modal shape matrix for the i-th node of the structure; Y is the vector of the modal coordinates.

The position of the i-th node of the structure in the IRF is defined as follows:

(8) R i = X 0 + T BI ( ξ i + ρ i ) ,

where X 0 is the position vector of the center of mass of the SSAR in the IRF; ξ i is the position vector of the i-th node of the structure in the BRF; T BI = T IO T T OB T is the matrix of rotation from the BRF to IRF.

In the case when the modal shapes of the unconstrained structure are chosen for the matrix Ψ i , the orbital motion of the SSAR is not coupled with its attitude motion and elastic vibrations of the structure, and the dynamics can be given in the following general form [5]:

(9) m X 0 ″ = F ( t ) ,

(10) M ( X 1 , t ) X 2 ′ + D ( X 1 , X 2 , t ) + C ( X 1 , t ) = V ( t ) ,

where m is the mass of the SSAR; F is the vector of forces acting on the SSAR; X 1 = [ Q Y ] T , X 2 = [ ω Y ′ ] T are the elements of the state vector characterizing the attitude motion of the SSAR and vibrations of the structure; M ( X , t ) is the matrix of system mass; D ( X , t ) is the vector containing gyroscopic and damping terms; C ( X , t ) is the vector representing the stiffness of the system; V ( t ) is the vector of non-conservative generalized forces caused by the action of the actuators and external perturbations.

The damping term is considered to be stiffness proportional in this study. In Eq. (10), external perturbations include gravitational, aerodynamic drag, and solar radiation pressure torques, which are calculated using equations from the study of Markley and Crassidis [36]. The lengthy procedure of deriving the governing equations of the SSAR is not reported here for brevity, but it is known and can be found, for example, in the study of Gasbarri et al. [5].

The model of the SSAR dynamics (10) incorporates modal frequencies and shapes of the flexible elements of the structure. These characteristics are calculated using the finite element modeling tool Autodesk Inventor^® software. The finite element (FE) model of the DRA consists of hinges, beams, and point masses.

Figure 7 shows a simplified FE model of the top and bottom hinge units using beam elements. Although the real geometry of these units has a different shape (Figure 3), such a representation is justified because these units have a significantly greater stiffness than other structural elements, and their stress–strain state is not under study. The main simulated properties of these hinge units are stiffness, location, and direction of the rotation axes of the hinges. Flexible beam systems model the central part of the DRA. Flexible ties connect two symmetrical systems. The tension of the nets is simulated by the temperature loads applied to the ties.

Figure 7

FE representation of the hinge unit.

The boom FE model consists of three flexible carbon fiber tubes interconnected by rigid aluminum joints.

4 Controller design

The SSAR is a flexible system with essentially nonlinear dynamics and kinematics. It is not straightforward to synthesize optimal attitude control for such a complex system. To facilitate the controller design, we represent the control torque using two terms as follows:

T C = T C FF + T C FB .

The first term T C FF is a nonlinear feedforward control to slew along some optimal trajectory designed using a simplified plant. The second term T C FB is a linear feedback control that corrects deviations of the real SSAR trajectory from the optimal one caused by a mismatch of the real SSAR and simplified plant dynamics.

4.1 Feedforward control

The feedforward control design aims to achieve a trade-off between the energy efficiency of maneuvers and the amplitudes of the oscillatory response. To design the feedforward control laws, a simplified model of the attitude dynamics of the SSAR is extracted from Eq. (10) in the following form:

(11) J ω ′ + ω × J ω = T C FF ,

where J is the total inertia matrix of the SSAR.

In order to facilitate the control design, the SSAR dynamics model is represented in the form of second-order differential equations with respect to the quaternion components as follows [37]:

(12) Q ″ = ( I 4 − Q ⋅ Q T ) U − Q ⋅ ‖ Q ′ ‖ 2 ,

where I 4 is the identity matrix with a dimension of 4 × 4.

In Eq. (12), U ∈ R 4 is a new control vector defined as follows:

(13) U = − 0 . 5 A T J − 1 ( ω × J ω + J ( ω OI × ω + ω OI ′ ) ) ,

where A = [ − q q 0 I 3 − q × ] , I 3 is the identity matrix with a dimension of 3 × 3 .

A physically implementable control in the R 3 space can be obtained from U ∈ R 4 using the following reverse transform:

(14) T C FF = 2 JAU + ω × J ω + J ( ω OI × ω + ω OI ′ ) .

Following the results of Yefimenko [37] we introduce an unnormalized quaternion Λ defined as follows:

(15) Q = Λ / ‖ Λ ‖ .

Using this unnormalized quaternion, the equation of attitude motion can be given as follows:

(16) Λ ″ = Θ .

The required attitude motion of the SSAR can be provided by a properly designed control Θ ( t ) . Then, the control used in Eq. (14) can be obtained as follows [37]:

(17) U = ( Λ ″ − 2 δ ′ Q ′ ) / ‖ Λ ‖ ,

(18) δ ′ = Q T Λ ′ ,

(19) Q ′ = ( I 4 − Q Q T ) Λ ′ / ‖ Λ ‖ .

The plant (16) represents four uncoupled double integrators, and for such a case, known analytical solutions of the optimal control theory can be used [38]. A control law for optimal slewing of the SSAR can be found by solving the terminal control problem for the energy-optimal case. Such a task can be formulated as follows: to find the control Θ that slews the SSAR from the initial orientation Λ ( t I ) , Λ ′ ( t I ) , at the time moment t I to the terminal orientation Λ ( t T ) , Λ ′ ( t T ) , at the time moment t T , and minimizes the following cost function:

(20) V = 0 . 5 ∫ t I t T Θ T Θ d t .

The boundary conditions are set in such a way that the normalized quaternion coincides with the unnormalized one at the initial and final moment of the maneuver

(21) Λ ( t I ) = Q ( t I ) , Λ ′ ( t I ) = Q ′ ( t I ) , Λ ( t T ) = Q ( t T ) , Λ ′ ( t T ) = Q ′ ( t T ) .

A drawback of the control found in this way is the large torques at the initial and final moment of the maneuver, which may induce significant structural vibrations. This drawback can be mitigated by additional constraints on the accelerations Λ ″ ( t I ) , Λ ″ ( t T ) at the initial and final moments of the maneuver. We improve the feedforward laws from [37] by introducing boundaries on the control torque, which are defined as weighted values of the corresponding optimal control as follows:

(22) T ∼ c w ( t I ) = W I T ∼ c ( t I ) , T ∼ c w ( t T ) = W T T ∼ c ( t T ) ,

where W I = diag ( w ll I ) , W T = diag ( w ll T ) , 0 ≤ w ll I ≤ 1 , 0 ≤ w ll T ≤ 1 , l = 1 , 2 , 3 .

The optimal control in the R 4 space U ∼ ( t I ) , U ∼ ( t T ) and the corresponding control torques T ∼ c ( t I ) , T ∼ c ( t T ) are found as the solutions to the optimal control problem (20), (21). The boundary values (22) can be considered as the maximum feedforward torque, provided the specified control performance is achieved despite the excited vibrations of the structure. The trade-off between the energy efficiency of maneuvers and the amplitudes of the oscillatory response can be achieved by selecting the weights W I , W T . The appropriate choice of the weight matrices allows us to set the bounds on the control torques in the range from zero to the corresponding optimal values. The initial weight values can be determined from computer simulations and then adjusted in orbit using measurements of the attitude errors.

The boundary control torque values can be transformed into the values of control in the space R 4 using Eq. (13). To find a solution to the optimal control problem with such additional constraints, Eq. (16) is differentiated as follows:

(23) Λ ‴ = Ψ ,

where Ψ = Θ ′ .

The cost function for the plant (23) can be given as follows:

(24) V ∼ = 0 . 5 ∫ t I t T Ψ T Ψ d t .

Using the extended state vector Z = Λ Λ ′ Λ ″ T , the solution to the optimization problem (24) can be found using Pontryagin’s maximum principle in the following form [38]:

(25) Z ( t ) = Γ 11 ( t , t I ) Z ( t 0 ) + Γ 12 ( t , t I ) μ 0 ,

(26) μ ( t ) = Γ 22 ( t , t I ) μ 0 ,

where Γ ( t , t I ) = Γ 11 Γ 12 Γ 21 Γ 22 is the transition matrix of the augmented system (23); and μ ( t ) is the conjugate vector.

For the given boundary conditions Z ( t I ) and Z ( t T ) , the initial value of the conjugate vector is determined as follows:

(27) μ 0 = Γ 12 − 1 ( t T , t I ) ( Z ( t T ) − Γ 11 ( t T , t I ) Z ( t I ) ) .

The elements of the matrix Γ ( t , t I ) are given in the appendix.

The optimal control and trajectory of the SSAR motion for the state vector Λ and its derivatives are defined as follows:

For task (20): Θ ∼ = − μ 2 , Λ ∼ = z 1 , Λ ∼ ′ = z 2 ,
For task (24): Ψ ∼ = − μ 3 , Λ ∼ = z 1 , Λ ∼ ′ = z 2 , Λ ∼ ″ = z 3 .

The feedforward term of control U ∼ and the optimal trajectory for the normalized quaternion Q ∼ can be found by substituting these values in expressions (17)–(19).

4.2 Feedback control

Assuming small deviations in the real trajectories of the SSAR motion from the optimal trajectories obtained using the simplified model (11), the nonlinear model (10) can be linearized and represented in the state space form as follows:

(28) ∆ X ′ = A ∆ X + B C T C FB + B D T D ,

where ∆ X is the deviation of the real state vector from the optimal one (error); A is the system matrix; B C and B D are the control and external perturbation matrices, respectively; T D is the torque of the external perturbations.

The plant (28) is a high-order equation and includes many elastic and dissipative parameters of the structure, such as natural frequencies, shapes, and damping ratios. Being motivated to synthesize control with minimal knowledge of these parameters, Eq. (28) is reduced by eliminating the flexible dynamics as follows:

(29) ∆ X ˆ ′ = A ˆ ∆ X ˆ + B ˆ C T C FB + B ˆ D T D ,

where ˆ denotes the corresponding reduced elements of Eq. (28).

Then, we represent the discarded flexible dynamics as a multiplicative uncertainty Δ ( s ) . Using this representation, the transfer function of the plant (28) can be given as follows:

(30) P ( s ) = ( 1 + Δ ( s ) ) P 0 ( s ) ,

where P 0 ( s ) is the transfer function of the reduced plant.

The closed-loop robust stability can be verified using the following condition [39]:

(31) σ ¯ ( T ( j ω ) ) < σ ¯ ( 1 / Δ ( j ω ) ) ,

where σ ¯ and σ ¯ are the minimal and maximal singular values, respectively; T ( j ω ) = ( 1 + K P 0 ) − 1 K P 0 is the complementary sensitivity function.

To avoid the controller interaction with the unmodeled flexible dynamics of the SSAR, we utilized the following notch filters (NFs):

N ∼ m ( s ) = s 2 + 2 ( d ˆ m / c ˆ m ) Ω ˆ m s + Ω ˆ m 2 s 2 + 2 ( 1 / c ˆ m ) Ω ˆ m s + Ω ˆ m 2 ,

where Ω ˆ m is the frequency of the rejected natural mode, c ˆ m is the width of the rejected band, d ˆ m is the gain at notch frequency, and m is the mode number.

The frequencies of the rejected natural modes are considered as uncertain parameters and will be adjusted in orbit to mitigate the conservatism of the robust controller. It is known that filtering may add a phase lag in a control loop. Therefore, the NFs are included as a transfer matrix N ( s ) in the augmented plant (Figure 8) at the controller design stage to guarantee closed-loop stability and performance against uncertainties of the filters. This approach differs from other works, for example [26,27], where the main feedback controller and the NFs are designed separately.

Figure 8

Augmented plant.

H ∞ framework [39] provides convenient tools to design robust controllers. Such an approach minimizes the closed-loop impact of perturbations for multivariate systems with cross-coupling between channels. This is a worst-case design method and suits well for the uncertainties presented in the system.

Figure 8 shows the block diagram of the augmented plant P 0 ′ ( s ) , which is used for the controller design. This plant has the following exogenous input: d 1 is the external perturbations, d 2 is the reference signal, and d 3 is the sensor noise. Gains G i are used to scale input and output signals. Since the design goal is to minimize the attitude errors limiting the controller’s effort, the minimized vector includes the control error p 1 and the output signal of the controller p 2 . The functions W 1 ( s ) and W 2 ( s ) were employed for weighting the p 1 and p 2 outputs, respectively, to impose performance conditions on the system and constrain control effort.

The weights W 1 ( s ) and W 2 ( s ) are square diagonal matrices with the following diagonal elements:

(32) W ∼ 1 ( s ) = s / n 1 + Ω 1 s + a 1 Ω 1 , W ∼ 2 ( s ) = s / n 2 + Ω 2 s + a 2 Ω 2 .

The controller closed-loop bandwidth, steady-state errors, and overshoot can be tuned by selecting the crossover frequency Ω 1 , low-frequency gain a 1 , and high-frequency gain n 1 , respectively. The actuator’s bandwidth can be taken into account by selecting the parameters of the weight W 2 ( s ) in such a way as to minimize the output signal of the controller at high frequencies.

Given a pre-specified attenuation level γ min , an H ∞ suboptimal control problem is to design a controller K that internally stabilizes the closed-loop system and ensures the following”

(33) ‖ F L ( P , K ) ‖ ∞ ≤ γ min .

An optimization technique [40] using linear matrix inequalities is applied to find a suboptimal controller in the following state space form:

X ̇ K = A K X K + B K v ,

T C FB = C K X K + D K v ,

where v is the measurement of the vector ∆ X .

Here we do not follow the conventional mix-sensitivity approach and do not include in the augmented plant (Figure 8) the uncertainty weight W 3 ( s ) , but we apply robust stability and performance tests a posteriori.

The linear fractional transformation [39] is used to obtain an accurate and non-conservative representation of the uncertain NFs. To apply this technique, we represent the uncertain parameters Ω ˆ m as follows:

(34) Ω ˆ i = Ω ˆ m n + δ Ω ˆ m Δ m ,

where Δ m ∈ [ − 1 , 1 ] ; the superscript n stands for the nominal value of the parameters, and the symbol δ denotes the maximum deviation of the parameters from its nominal value.

The block diagram of the control system can be represented using the description of the parameters (34) as a system consisting of the main block N (the nominal plant and the controller) and a perturbation block Δ ∼ [39]. The overall perturbation Δ ∼ in Figure 9 is structured and has the form of a square diagonal matrix with the diagonal elements Δ m .

Figure 9

Representation of an uncertain plant.

As we only deal with structured uncertainty, we utilize a robustness measure based on the notion of structured singular values [39]. The structured singular value for a complex-valued matrix N is the inverse of the norm of the smallest perturbation (within the given class D) that makes the matrix I + N Δ ∼ singular. The structured singular value μ ( N ) is defined as:

1 μ ( N ) = inf σ ̅ ( Δ ∼ ) , Δ ∼ ∈ D , det ( I − N Δ ∼ ) = 0 .

Performance is said to be robust if the perturbed system remains stable under all perturbations and the norm of the transfer matrix ‖ N Δ ∼ ‖ ∞ of the perturbed system remains less than one under all perturbations. Necessary and sufficient for this [39] is that

(35) μ ( N ) < 1 .

Overall, the robustness and performance of the designed controller against the unmodeled flexible dynamics and uncertainties of the NFs are guaranteed if conditions (31) and (35) are both satisfied.

4.3 Identification of natural frequencies

It is necessary to know the natural frequencies of the SSAR to successfully implement the controller with the NFs. Since the real natural frequencies may differ from those calculated using the FE model, these parameters are identified in orbit using adaptive notch filters (ANFs). We use such ANFs that allow us to determine the frequencies of harmonic signals of unknown amplitudes [41]. Following this methodology, the dynamics of the ANF can be given as follows:

x ∼ m '' + 2 ζ ∼ Ω ∼ m x ∼ m ′ + Ω ∼ m 2 x ∼ m = 2 ζ ∼ Ω ∼ m 2 y ∼ ( t ) ,

Ω ∼ m ′ = − γ ∼ x ∼ m ( 2 ζ ∼ Ω ∼ m 2 y ∼ ( t ) − 2 ζ ∼ Ω ∼ m x ∼ m ′ ) ,

where x ∼ m is the state vector of the ANF; y ∼ ( t ) is the input signal; Ω ∼ m is the estimated natural frequency; ζ ∼ and γ ∼ are the damping ratio and adaptation gain, respectively.

After the corresponding NF parameters have been adjusted using identified frequencies, the main controller engages. Since the controller is designed taking into account the NF uncertainties, robust stability and performance are guaranteed when the filter frequencies vary within a specified range.

5 Actuators

For a pyramid configuration of the RWs, the orientation of the RW axes with respect to the BRF can be determined by the angles α and β , as shown in Figure 10. In this figure, h j denotes the angular momentum of the j-th RW.

Figure 10

RW configuration.

The installation angles of the RWs can be given as follows:

(36) β = arctan ( e y / e x ) , α = arctan ( e x / ( e z cos β ) ) ,

where e x , e y , and e z are the efficiency coefficients of SSAR control about the x, y, and z axes, respectively.

According to Figure 10, the rotation matrix from the RW reference frames to BRF is given by

(37) G = s α c β − s α c β − s α c β s α c β s α s β s α s β − s α s β − s α s β c α c α c α c α .

The control torque generated by an RW system is determined as follows:

(38) T c = H ′ + ω × H ,

where H is the angular momentum of the RW system in the BRF.

The reaction torque of the RW system can be obtained from Eq. (38) as

(39) T r c = T c − ω × H .

Assuming that the rate of electrical processes is much higher than the rate of mechanical processes of the RW, the dynamics of an RW can be described by the following equation:

(40) J r ω ̇ r j = T r j − T k sign ( ω r j ) − J r ω r j / K t ,

where ω r j is the angular velocity of the j-th RW; J r is the inertia moment of the RW; T r j is the motor torque of the j-th RW; T k is the Coulomb friction coefficient; and K t is the electromechanical time constant.

Using the SSAR control torque T c , obtained in the previous section, the RW control torque vector T r = [ T r 1 T r 2 T r 3 T r 4 ] T can be found from the following equation:

(41) T rc = G T r .

This is a heterogeneous linear equation system with more than one solution. In this work, the pseudo-inversion of the matrix G was used to find the solution in the following form [36]:

(42) T r = G T ( G G T ) − 1 T rc .

The angular momentum of the RW system is given as follows:

(43) H = G J r ω r ,

where ω r = [ ω r 1 ω r 2 ω r 3 ω r 4 ] T .

6 Simulation, results, and discussion

The parameters of the FE model of the DRA are summarized in Table 1. The frequencies of the natural vibrations of the DRA, which is fixed at the interface points, and the SSAR in a free-free configuration with booms having a diameter/wall thickness of 60/2 mm, are presented in Table 2. Figures 11 and 12 show the modal shapes of the SSAR with such a boom.

Table 1

Parameters of the FE model of the DRA

Parameters	DRA elements
Parameters	Beams	Hinges	Nets	Tension ties
Young’s modulus (GPa)	120	1,000	120	210
Poisson’s ratio	0.3	0.3	0.3	0.3
Density (kg/m³)	1,800	2,700	1,800	1,800
FE section type	Tube	Rectangle	Rectangle	Circle
Diameter/wall thickness (mm)	12/1	—	—	0.5
Width × height (mm)	—	10 × 10	10 × 0.5	—

Table 2

Natural frequencies

Mode	1	2	3	4	5	6
DRA natural frequencies (Hz)	1.133	8.768	20.905	32.290	34.705	40.844
SSAR natural frequencies (Hz)	1.673	2.697	4.061	8.503	10.741	31.398

Figure 11

First mode shape of the SSAR.

Figure 12

Second mode shape of the SSAR.

The validity of the SSAR model presented in Section 3 is verified by comparing the natural frequencies obtained from the characteristic equation with those obtained using the commercial FE modeling tool Autodesk Inventor^® software.

It was assumed for simulation purposes that the data in Table 2 are the real SSAR modal properties, but they are known only approximately at the controller design stage. Scaling gains are selected based on the estimates of the input magnitudes and have the following values:

G 1 = 0 . 009 , G 2 = 10 − 6 · diag ( 1 . 5 1 . 5 1 . 5 3 . 7 3 . 7 3 . 7 ) , G 3 = 10 − 4 , G 4 = 111 . 1 .

The crossover frequency Ω 1 of the low past filters W 1 ( s ) was chosen to provide a minimum closed-loop bandwidth of 0.8 rad/s. The steady-state errors were tuned by the low-frequency gain a 1 , and the high-frequency gain n 1 was selected to limit the overshoot to 20%. The parameters of the weight W 2 ( s ) were selected to account the actuator’s bandwidth by minimizing the controller’s output at high frequencies. The parameters of the weights used to design the controller are summarized in Table 3.

Table 3

Parameters of the weights

Parameter	Ω 1	Ω 2	Ω 3	a 1	a 2	a 3	n 1	n 2	n 3
Value	0.3	0.75	0.3	0.01	200	0.5	1.2	0.1	800

Taking into account the results of the modal analysis and robustness test (Figure 13), these filters should be tuned to the first mode frequency in the y and z channels and to the second mode frequency in the x channel. Given this consideration, the NF matrix is composed as follows:

N ( s ) = diag ( N ∼ 2 ( s ) , N ∼ 1 ( s ) , N ∼ 1 ( s ) ) .

Figure 13

Robust stability test.

The parameters of the NFs are presented in Table 4.

Table 4

Parameters of the NFs

Parameter	Ω ˆ 1	Ω ˆ 2	d ˆ 1	d ˆ 2	c ˆ 1	c ˆ 2
Value	10.5	17	0.006	0.01	2	2

For the augmented plant with NFs, a controller of 18th order with an attenuation level of γ min = 0.92 was synthesized. Taking into account the results of the modal analysis, the yaw angle signal was used to estimate the first mode frequency, and the pitch signal was used for the second mode. Before being put to the ANF input, these signals pass through low-pass filters to remove harmonics, whose frequencies are significantly lower than the estimated frequency, and then they are scaled. The low-pass filters with cutoff frequencies of 4 rad/s and 10 rad/s and scaling gains of 1.5 × 10 4 and 3 × 10 4 were selected for the yaw and pitch channels, respectively. Both ANFs use the following parameters: ζ ∼ = 0.25 and γ ∼ = 0.08.

As shown in Figure 13, only the first two vibration modes may affect the closed-loop robust stability and must be identified and adjusted in orbit. Images of the first two modes of natural vibrations (Figures 11 and 12) give a clue that to induce the first and second modes of vibrations, it is necessary to perform a yaw and roll slews, respectively.

The blue line in Figure 14 shows the ANF output during the natural frequency estimation. These results are obtained for the cases when the SSAR executes 180° yaw and 80° roll slews to identify the first and second mode natural frequencies, respectively. Feedforward control without restrictions on the control torque is used for these purposes since this control induces oscillations with significant amplitude. A simple proportional-differential controller with a bandwidth of 0.01 rad/s was used to minimize the interaction between the structure oscillations and feedback control during frequency identification. The green lines in Figure 14 show the filtered attitude signals for such a case. As can be seen from these figures, the output signal of the ANFs (blue line) converges pretty quickly to the real values of the natural frequencies.

Figure 14

Identification of natural frequencies.

The formal criteria (31) and (35) were applied to verify the closed-loop robust stability and performance against unmodeled dynamics and variations of the adaptive parameters of the NFs. In Figure 15, the green line shows the complementary sensitivity function for the controller with the NF. This figure demonstrates that this controller provides robustness against the flexible modes of the SSAR. The complementary sensitivity function for this controller but without NFs in the loop is plotted in Figure 13 by the red line. As seen in this case, the controller is not robust to the lower modes of structural vibrations.

Figure 15

Sensitivity functions.

Figure 16 depicts the upper bound on the structured singular values of the closed-loop system for δ Ω ˆ m = 0.2 Ω ˆ m n . As can be seen from this figure, the structured singular values are less than one ( μ max = 0.97 ). This result states that the designed controller provides robust stability and performance of the closed-loop system for all variations of the adaptive parameters of the NFs.

Figure 16

Upper bound on structured singular values.

In addition to the robustness tests (31) and (35), the performance of the designed controller is validated by the computer simulations using the full nonlinear model of the flexible SSAR (10), taking into account twenty modes of the structure’s natural vibrations, external perturbations, parametric uncertainties of the plant, and measurement errors. Deviations in the range of ±10% from nominal values are considered for the elements of the inertia matrix. The measurement errors are simulated by colored Gaussian noise. The control efficiency coefficients for the considered SSAR are selected as follows: e x = 1 , e y = 1 , e z = 0.5 . For these coefficients, the angles of the RW configuration have the following values: α = 70.5 ° and β = 45 ° .

This section contains the simulation results for the following slew maneuvers of the SSAR:

45° yaw slew.
24° pitch slew.
24° roll slew.

The duration of these maneuvers of 25 s corresponds to the fastest possible slews, given the proposed control algorithms and the maximum control torque of the RWs.

The feedforward control laws are designed for the following inertia matrix of the rigid plant:

J = 186.51 0 . 005 0 0 . 005 185 . 66 − 4 . 34 0 − 4 . 34 45 . 33 kg m 2 .

The elements of the matrices W I and W T have the same values at the initial and final moment of the maneuvers and for all control channels, such as

(44) w ii I = w ii T = w .

The variations in the RW torques and angular velocities are plotted in Figures 17–19 for the case when only feedforward control is applied. The figures on the left show the simulation results obtained using control without constraints on the torque at the initial and final moments of the maneuvers (w = 1); hereinafter this control is referred to as w1. The control with zero control torque at the initial and final moments of the maneuver (w = 0, hereinafter just control w0) is used for the cases in the figures on the left. As can be seen from these figures, w1-control executes the same maneuver with smaller RW angular velocities than w0-control. Thus w1-control is more energy efficient.

Figure 17

RW control torques and angular velocities for case 1.

Figure 18

RW control torques and angular velocities for case 2.

Figure 19

RW control torques and angular velocities for case 3.

Another positive feature of w1-control is that it requires large control torques of the RWs at significantly smaller angular velocities of their wheels than w0-control. This advantage makes it possible to implement faster slew maneuvers since an RW maximum control torque usually goes down significantly when its angular velocity is high. A specific choice of the parameter w in the range of 0–1 can provide the achievable pairs of “control torque – wheel angular velocity” for a selected RW.

w1-control leads to a smooth variation in the RW torques, while w0-control results in jerks of the RW torques at the initial and final moment of the maneuver, which excite vibrations of the structure. This drawback can be seen in Figures 20–22, which show the displacements of the DRA due to elastic deformations. For example, for w1-control, the amplitudes of the elastic deformations of the DRA at the end of the maneuver is approximately 20, 23, and 28 times greater than when w0-control is used for cases 1, 2, and 3, respectively. These vibrations degrade the accuracy of attitude control.

Figure 20

Reflector displacements for case 1.

Figure 21

Reflector displacements for case 2.

Figure 22

Reflector displacements for case 3.

Although w0-control provides better attitude accuracy, both controls allow the control system to meet the specified angular error requirements (Figures 23–25). As can be seen in Figures 26–28, the requirements on the angular velocity accuracy are not met when w1-control is used. At the same, time w0-control helps to achieve these requirements with a significant margin.

Figure 23

Angle attitude errors for case 1.

Figure 24

Angle attitude errors for case 2.

Figure 25

Angle attitude errors for case 3.

Figure 26

Rate attitude errors for case 1.

Figure 27

Rate attitude errors for case 2.

Figure 28

Rate attitude errors for case 3.

A value of the parameter w in the range of 0–1 can be selected in such a way as to achieve a trade-off between control accuracy and energy efficiency. Figures 29–32 demonstrate that angular velocity requirements are met when the feedforward parameter is selected as w = 0.3.

Figure 29

Reflector displacements for case 3 (w = 0.3).

Figure 30

Angle attitude errors for case 3 (w = 0.3).

Figure 31

Rate attitude errors for case 3 (w = 0.3).

Figure 32

RW control torques and angular velocities for case 3 (w = 0.3).

Next we compare the designed controller with two other robust controllers in terms of their complexity and performance. The key features of all three controllers are summarized in Table 5. The first controller for comparison was synthesized using the mixed-sensitivity technique (MST) [42]. According to this approach, a weighted complementary sensitivity function was added to the optimized output. In this case, robustness requirements were specified by the square diagonal matrix weight W 3 ( s ) with the following diagonal elements:

W ∼ 3 ( s ) = s + Ω 3 a 3 s / n 3 + Ω 3 2 ,

and a controller of 15th order was synthesized with an attenuation level of γ min = 1.65. This approach narrows down the controller bandwidth, which can be seen in Figure 7, where the red line shows the singular values of the sensitivity function for this case.

Table 5

Features of the controllers

Controller	Order	Attenuation level	Maximum errors for case 3
			Angle (deg)		Rate (deg/s)
			w = 0.3	w = 1	w = 0.3	w = 1
Authors’ design	18	0.92	4.14 × 10 − 4	1.52 × 10 − 3	4.91 × 10 − 3	0.013
MST	15	1.65	1.26 × 10 − 3	3.82 × 10 − 3	0.011	0.028
Micro-Synthesis	68	0.98	4.23 × 10 − 4	1.58 × 10 − 3	5.05 × 10 − 3	0.014

The second controller for comparison was obtained by micro-synthesis using the D-K iterations algorithm [43]. In this case, the controller was synthesized using a plant model with two flexible modes, the frequency and damping parameters were set with 20% uncertainty. This approach led to a controller of 68th order with an attenuation level of γ min = 0.98. Comparing these three controllers, we can conclude that the controller with the NF outperforms the mixed sensitivity controller by 44% despite the fact they both have almost the same order, and the micro-controller has a substantially higher order than the controller with the NF, but they both provide nearly the same attenuation level and control accuracy.

The designed control laws are not computationally complex. For example, the main operations for calculating the feedforward and feedback control terms include only three multiplications of the matrixes of a dimension of 12 and four multiplications of the matrixes of a dimension of 18, respectively. In this work, we use a 100 ms sampling period, and we expect that these control algorithms would easily run on modern flight computers for small satellites.

Summarizing the results of this section, we can state that the designed controller is robust against uncertain flexible dynamics of the plant and allows the SSAR to meet the specified attitude control requirements.

7 Conclusion

This article presents an attitude controller of a mini-SSAR with a DRA, which is synthesized assuming that the modal properties of the structure are not precisely known at the design stage. The performance of the attitude controller depends on two parameters, which can be identified in orbit. The first parameter is used in the feedforward control loop to achieve a trade-off between the energy efficiency of the maneuvers and the amplitudes of the oscillatory response of the structure. The second vector parameter is utilized in the feedback loop to accurately handle the controller-structure interactions by adaptive notch filters. The controller is synthesized in such a way as to guarantee robustness against both the unmodeled dynamics and the variations in the rejected frequencies. The formal criteria and results of the numerical simulations of the flexible SSAR dynamics confirm the system’s robustness and specified requirements. This approach allows engineers to facilitate the controller design and reduce experimental testing of the SSAR.

Acknowledgements

The authors would like to express their sincere gratitude to EOS Data Analytics Inc. (eos.com, eossar.com) and its founder, Dr. Max Polyakov for support during this study.

Funding information: This research received no external funding.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

The transition matrix Γ ( t , t I ) has the following elements for the task (20):

Γ 11 ( t , t I ) = I 4 I 4 ( t − t I ) 0 I 4 , Γ 12 ( t , t I ) = I 4 ( t − t I ) 3 / 6 − I 4 ( t − t I ) 2 / 2 I 4 ( t − t I ) 2 / 2 I 4 ( t − t I ) , Γ 22 ( t , t I ) = I 4 0 − I 4 ( t − t I ) I 4 ;

for the task (24):

Γ 11 ( t , t I ) = I 4 I 4 ( t − t I ) I 4 ( t − t I ) 2 / 2 0 I 4 I 4 ( t − t I ) 0 0 I 4 ,

Γ 12 ( t , t I ) = − I 4 ( t − t I ) 5 / 120 I 4 ( t − t I ) 4 / 24 − I 4 ( t − t I ) 3 / 6 − I 4 ( t − t I ) 4 / 24 I 4 ( t − t I ) 3 / 6 − I 4 ( t − t I ) 2 / 2 − I 4 ( t − t I ) 3 / 6 I 4 ( t − t I ) 2 / 2 − I 4 ( t − t I ) ,

Γ 22 ( t , t I ) = I 4 0 0 I 4 ( t − t I ) I 4 0 I 4 ( t − t I ) 2 / 2 − I 4 ( t − t I ) I 4 .

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Received: 2021-08-19

Revised: 2023-01-08

Accepted: 2023-01-15

Published Online: 2023-02-16

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0277

Keywords for this article

flexible reflector; nonlinear transformation; in-orbit identification; robust control; slewing maneuvers

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