Monitoring of a controlled space flexible multibody by means of embedded piezoelectric sensors and cameras synergy

Abstract

Interaction between elastic dynamics and attitude control is a serious problem in space operations, which often involve satellites with highly flexible appendages. Monitoring and eventually control of the vibrations are a major concern to avoid a decrease in the expected performance. In particular, the classic case of a central bus with two lateral appendages (solar panels) is considered. The design of a system for structural vibration monitoring is proposed both from a numerical and an experimental point of view. Piezoelectric devices are a usual solution for measuring the deformation of the structures. In the proposed work, optical sensors are also implemented: the combined use of the two sets allows for the monitoring of the elastic displacement of the solar panels and for the reconstruction of the modal shapes of the entire flexible multibody system.

Keywords

Large flexible space structures vibration control PZT sensors visual sensors experimental vibration sensing

Introduction

Space structures are particularly sensitive to the problem of residual vibrations, which decrease the performance of attitude maneuvers. In fact, lightweight requirements together with large dimensions of solar panels, antennas, and booms lead to complex flexible multibody systems that are characterized by low modal frequencies that can easily interact with the rigid controlled attitude motion. As a result, a degradation of the performance of the control (and hence of the relevant mission payload) could be suffered. In the worst cases, instability of the systems controlled could be reached (Sabatini et al., 2015).

Several studies have proposed in the past to use piezoceramic materials (lead zirconate titanate [PZT]) or other smart sensors/actuators to sense and actively damp the vibrations (Heganna and Joglekar, 2016; Lampani and Gaudenzi, 2010). As an example, Zhong et al. (2016) designed a keep tracking controller to suppress the vibration actively by means of a model-free algorithm. Pai et al. (2000) investigated the suppression of steady-state vibrations of a cantilevered skew aluminum plate using nonlinear saturation phenomena and PZT patches. Garcia-Perez et al. (2016) considered a space-frame flexible structure mounted on a rigid revolute servomechanism, as a flexible-like robotic system. For active vibration control, a combined control scheme was proposed, using a proportional derivative (PD) with direct strain feedback control, and a multiple positive position feedback control by means of a PZT stack actuator mounted into the structure. Therefore, the endogenous and exogenous vibrations on the overall structure are simultaneously attenuated by the combined controller. The use of piezoelectric transducers has been also suggested for different goals; in fact, they can be embedded inside the composites or bonded onto the surface are a possible solution for online structural health monitoring and non-destructive evaluation, as suggested by Albakri and Tarazaga (2016), Park et al. (2007), Tang et al. (2011), and Yang et al. (2008). Also in Shanker et al. (2011), the PZT patches are used to determine the natural frequencies and the strain mode shapes of the structure as well as to acquire the electro-mechanical admittance signature to facilitate an improved damage assessment. As an example of additional application of PZT patches, also the concept of energy harvesting, which is a process of capturing ambient waste energy and converting it into useable electricity, has been investigated by the use of PZT devices (Shen et al., 2008).

Even though PZT materials have this large variety of applications, active PZT control is not the only mean for vibration suppression. Hu et al. (2014) studied a new and effective approach for vibration suppression of large space structures. Collocated pairs of control moment gyroscope (CMG) and angular rate sensors were adopted as actuators/sensors. This method, though interesting, must be considered still at concept level.

A different approach consists in facing the vibration problem from the control algorithm point of view; in particular, the authors have proposed a time-delay control for space multibodies for increasing the phase margins of the controller (Sabatini et al., 2015; Zhong et al., 2016).

In this case, more attention is paid on vibration sensing rather than on active suppression. Sabatini et al. (2015) used a visual-based approach to elastic displacement detection and measurements; although acceleration sensors such as PZT patches are maybe the most accurate method, they cannot be used remotely because these are contact-type sensors (Son et al., 2015). The optical method has the advantage of needing no inclusion of ad hoc devices and relevant harness; a single camera is sufficient; recent studies also face the problem of extracting full-field structural dynamics in the presence of arbitrary rigid body motion (Dasari et al., 2018). However, small elastic displacement requires high-resolution cameras, but the image processing algorithms have a computational cost that increases with resolution. As a result, only low-frequency modes can be tracked by cameras in real time.

Also smart materials such as PZT could be used for sensing; their field of application can be considered complementary to the one of the cameras, since they are best fit for high-frequency applications

A combined use of camera and PZT patches can be found in the work of Qiu et al. (2016), where, however, the PZT patches are used as actuators. In fact, a visual sensor is used to measure the vibration of a flexible cantilever plate. After image processing, including median filter, image segmentation for object recognition, and least squares technique, the vibrations of bending and torsional modes for flexible plate can be obtained. The visual measured signals are used as feedback to suppress the excited vibrations by means of PZT actuators.

It must be noticed that materials used in space on satellite external surfaces may have to cope with strong levels of radiation dose under proton and electron irradiations (Paulmier and et al., 2013). These high radiation dose levels received after long flight duration (several years flight) may strongly affect the material structure through fragmentation and reticulation processes, production defects, or ionization.

Therefore, it is possible that the elastic characteristics of the flexible appendages change during the mission time, and it is important to have the possibility to measure them during the operational life of the satellite.

In this work, the dual use of camera and PZT sensors is proposed as a reliable setup for measuring the elastic displacements. In fact, the measurements of the local deformations from a PZT patch and a reliable mathematical model of the structure could be sufficient to extract the elastic global shapes in terms of displacements. For this purpose, all the characteristics, both of the structure and of the PZT patches, should be precisely known. When the materials are subjected to partial failures or environmental aging, the characteristics of the structure change and such a method become imprecise.

In this article, the use of the camera is proposed in a different framework and with a different goal with respect to the referenced work. In fact, here the main role of the camera is an in-orbit calibration of the PZT sensors so that they can be used for measuring all the needed elastic characteristics. With the combined use of PZT and camera sensors, the positive features of PZT sensors (e.g. high acquisition frequency with respect to a camera) can be obtained without the drawback of the possible inaccurate knowledge of the PZT electro-mechanical coupling parameters, which would otherwise impede a correct measurement of the structural status.

At the scope of performing the ground tests for a space-like system, two composite material plates have been designed and manufactured to be mounted on a central free-floating bus, thus representing a typical space system with a central bus and solar panels. The PZT patches are embedded in the lamination sequence of one of the two panels.

This article presents at first the designs and the manufacturing process of the plates (section “Design, analysis, and manufacturing of the laminated composite plates”). The required charge amplification circuit for acquiring the measurements from the PZT sensors is explained in details in section “Charge amplifier circuit.” Section “Modal characterization performed by means of image analysis” is dedicated to image analysis as a tool for extracting information on the elastic dynamics of the structure. The calibration process implemented using PZT patches and camera jointly is explained in section “Beam analytical model for PZT measurement calibration,” together with the main results. The free-floating platform testbed and the relevant mathematical model are described in section “Free-floating platform: setup and mathematical model,” and the results to the first preliminary tests are reported in the following section “Experimental results with free-floating platform.” Final remarks conclude this article in section “Conclusion.”

Design, analysis, and manufacturing of the laminated composite plates

At the scope of simulating the behavior of real space structures like solar arrays or antennas, the first task is to design and produce a flexible test structure with the appropriate elastic characteristics. In fact, these kinds of space structures have to be as light as possible and usually have remarkable dimensions with respect to the bus (i.e. the free-floating central platform). Lightness for aerospace structures is synonymous of flexibility, and consequently, these systems are characterized by low modal frequencies. Starting from these considerations, two laboratory-scale experimental plates have been designed with the dimensions 540 mm (length) × 150 mm (width) × 1.6 mm (thickness).

The material used is a Hexcel prepreg with M34 epoxy resins system at 41% in volume and 300 g/m² woven fabric fiber glass with H8 waving. Mechanical properties of this material are listed in Table 1.

Table 1.

Mechanical properties of Hexcel M34/41%/300H8/G.

Density (kg/m³)	1880
E_L = E_T (GPa)	21
G_LT (GPa)	4
Poisson’s ratio, ν_LT	0.2

The shape of the panels and the lamination sequence have been designed by means of parametric simulations performed on numerical models with a finite element approach; the software used is MSC Patran/MSC Nastran. The mathematical model consists of 2796 two-dimensional (2D) QUAD elements and one multipoint constraint. The composite material is modeled as five layers (thickness: 0.267 mm) of orthotropic material (elastic modulus: 21 GPa; Poisson’s ratio 0.2; shear modulus: 4 GPa; density: 1880 kg/m³) with a $\pm 45 \circ$ orientation. Since only the flexural deformations of the plates are able to substantially perturb the dynamics of our planar experimental testbed, axial and torsional modes are to be avoided for low frequencies. The main driver of the design is that at least the first two modal shapes must be flexural. The shape of the plates together with the associated reference frame is shown in Figure 1.

In order to increase the stiffness of the torsion and the compliance of the bending, that is equivalent to move up the frequencies of torsional modes and down the flexural ones, the panels are composed by six layers of composite material whose fibers form an angle of 45° with respect to the x-axis of the plate. For the same reason, triangular holes are cut off the plates. Moreover, a mass saving (lighter structure) is obtained.

The values of the first three natural frequencies of vibration for the selected configuration of the plate are reported in Table 2 for a cantilever configuration, since this is the closest condition compared to the experimental one. Even if these values are not high, space structures could be characterized by frequencies that are even lower. Therefore, dummy masses are added to reduce their value. In Table 2, last three rows, the frequency values are reported.

Table 2.

Numerical eigenfrequencies of the plate.

Mode number	Frequency (Hz)	Type
1 (no tip mass)	2.4	Flexural predominant
2 (no tip mass)	15	Flexural predominant
3 (no tip mass)	20.99	Torsional predominant
1 (250-g tip mass)	1.32	Flexural predominant
2 (250-g tip mass)	6.34	Flexural predominant
3 (250-g tip mass)	9.88	Torsional predominant

In Figures 2 and 3, the relevant mode shapes are shown.

Manufacturing process

Two plates have been manufactured, one of them with three PZT patches embedded conforming to grade PIC255 (Table 3) by PI Ceramic, Lederhose, Thuringia, Germany (Gaudenzi and Lampani, 2016; Wierach, 2013). PIC255 is a modified PZT that is especially suited to bipolar operations, in shear actuators, for dynamic operating conditions and under high ambient temperatures. The piezoelectrics have square shape, with equal length and width of 10 mm and thickness of 0.2 mm. The electrodes have been cabled with copper enamel wires of 0.2 mm diameter, to make them insulators, and they were microwelded using tin. Three squared portions of 10 mm × 10 mm were cut off in the second of six composite plies in order to locate the sensors. The thicknesses of the PZT patches are compatible with the composite ply ones (Figure 4). Laminates were cured in autoclave and, in order to maintain the integrity of the piezoelectric devices, the curing cycle has consisted of a heating phase from room temperature to 75°C with 3°C/min and a stasis of 8 h. A vacuum of 70 mbar in average has been applied during all the cycle. The piezoelectric capacity was measured both before and after the autoclave cycle. As expected from experience, after the embedment process, a significant decrease in the value of the electrical capacity of the PZT patches has been measured. The measurement of the capacity is also a check for the integrity of the patches and wired connections after curing cycle.

Table 3.

Physical and electro-mechanical properties of PIC255.

Density, $ρ$ (kg/m³)	7800
Curie temperature, T_C (°C)	350
Relative permittivity, $ε_{33} / ε_{0}$	1750
Piezoelectric charge coefficient, d₃₁ (10⁻¹² C/N)	−180
Piezoelectric voltage coefficient, g₃₁ (10⁻³ Vm/N)	−11.3
E_L = E_T (GPa)	62.1
G_LT (GPa)	21
Poisson’s ratio, ν_LT	0.32

Charge amplifier circuit

A properly designed interface circuit plays a key role in the optimal exploitation of piezoelectric sensors (Gaudenzi, 2009; Karki, 2000). In many cases, the sensors can be directly connected to the a microcontroller without any kind of interface; the signal is however noisy, sensitive to harness capacitance, and, since it ranges from a negative to a positive value of voltage, it is often not suitable for common 0–3.3 V or 0–5 V input range.

For these reasons, there is the need to design an interface circuit developed around a classical configuration of a charge amplifier circuit (see Figure 5), typically used when long wirings are present between the PZT sensors and the circuit (Sirohi and Chopra, 2000). In this manner, the output voltage is directly related to the feedback capacitance of the operational amplifier (op-amp), and it does not depend on the piezocapacitance. Also, the wire capacitance does not influence the output signal of the electronic circuit.

Each electrode of the PZT sensors, which is modeled as a charge generator, is directly connected to the inverting node of an op-amp, while the non-inverting node is connected to a polarization block, which is tuned to have a constant voltage of 2.5 V. Calling C_f the feedback capacitance, the voltage of the op-amp is equal to:

V_{o p - a m p} = - \frac{Q}{C_{f}}

(1)

where Q is the charge on the PZT electrode. In order to prevent the amplifier from drifting into saturation, the resistor R has the task of removing the charge from the capacitor at low rate. In addition, this circuital component provides a direct current bias path for the negative feedback. The value of R and C_f set the time constant of the amplifier $τ$ , which is defined as

τ = R C_{f}

(2)

Therefore, we can define the low cut-off frequency of the amplifier as

f_{c} = \frac{1}{2 π τ}

(3)

where f_c represents the frequency above which the reduction of the signal is lower than −3 dB. For a low-frequency measurement, an input resistance needs to be high enough so that the cut-off frequency is well below the desired operating frequency. Starting from this classical configuration, the interface circuit in Figure 6 has been developed.

In this configuration, the task of the first two op-amps is to amplify the signal coming from the sensor. In this way, also the noise, which is a common mode signal, is amplified. Therefore, the third op-amp must take the difference between the two incoming signals and cancel the common mode signal.

In the proposed configuration, the values C_f = C₁ = C₂ = 100 nF and R = R₃ = R₄ = 22 MΩ have been selected, so that the cut-off frequency is about 0.07 Hz, that is, much lower than the interesting modal frequencies of the plate. The last amplifier is configured as a differential amplifier. Its gain G₃ is defined as

G_{3} = \frac{R_{9}}{R_{8}} = \frac{R_{6} \ \ R_{5}}{R_{7}} = 1

(4)

It is important to note that the differential amplifier configuration correctly works only if R₇ = R₈ and R₉ = R₆\\R₅. Powering the interface circuit at 5 V, the output voltage of this op-amp ranges from 0.6 to 3.6 V: in this case, the bias is set to be at 2.1 V that corresponds to the mean value of the signal.

From equations (1) and (4), the relation between the output voltage of the final interface circuit and the charge on the electrodes is obtained and is equal to

V_{out} = 2 \frac{Q}{C_{1}}

(5)

Figure 7 shows the circuit integrated on a printed circuit board (PCB).

Modal characterization performed by means of image analysis

Three different image processing algorithms have been used to extract the elastic characteristics of the structure. In general, the goal is to find specific points in an image (features) and then to track their movement during the experiments. This is possible by following different approaches: one of them consists in finding simple geometries inside an image, for example, the corners or the sides of the plate (Sabatini et al., 2013). The MATLAB “Corner” (Harris and Stephens, 1988) function (in which one of the plate end points is detected and tracked) and “Hough Transform” (Duda and Hart, 1971) (in which straight lines are identified, and the end point is computed at the intercept of the lines) are the first two algorithms used.

A third approach consists in identifying areas characterized by homogeneous color, a task that has been facilitated by the use of red markers on the top and on the middle of the plate (see Figure 8). The algorithm used is the MATLAB “BLOB Analysis” that is able to find regions (in the digital image) that differ in properties, such as brightness or color, compared to surrounding regions (Moeslund, 2012). All these three methods have been used in order to extract the modal frequencies of the structure, and the results are used to tune the finite element model (FEM) in terms of elastic characteristics of the composite plate.

A value of 25 Hz is the maximum stable acquisition rate for our camera (Microsoft LifeCam Cinema, Redmond, WA, USA) we tested. As mentioned in Table 2, the second bending mode for the case with tip masses has a frequency of about 6.34 Hz. In this case, our optical sensor is able to recognize the deformations concerning the first two modal shapes, since they are both lower than the Nyquist frequency. In this tuning experiment, the base of the beam is fixed on a rigid support, as depicted in Figure 8.

Tests

The three methods previously explained are applied to the case of a plate under an impulsive excitation (Ewins, 2000; McConnell and Varoto, 2008) Redmond, WA 98052-6193, USA. The acquisition time of each experiment is 24 s that correspond to 600 recorded frames. In Figure 9, the displacement time history of the panel’s tip obtained by means of the Corner and Hough Transform function is depicted.

The measurements obtained with these two methods are very similar, and this could be considered as a cross-validation of the methods itself. In Figure 10, the fast Fourier transform (FFT) of the previous plots, which represents the frequency content of the signal, is shown. The comparison of the FFTs for the two algorithms give a good agreement; nevertheless, the Corner function algorithm is more sensitive to the noise of the acquired image, easily leading to a wrongly identified corner; for this reason, it will not be used in the following part of this work.

The same tests have been performed by means of the BLOB analysis. The displacement of the red “blob” targets is illustrated in Figure 11. Also in this case, the deformation of the panels is well acquired by the camera. The FFT of the measurements of the two targets is compared in Figure 11. It is possible to observe both the optical targets return very good behavior and are able to capture the elastic displacements which concern the first two modal shapes.

In summary, the three methods show similar performance, both in terms of modal frequency identification and in terms of computational time; Corner detection has proved to be slightly less robust to noise; the BLOB analysis determines the point of interest as the center of mass of the BLOB, and it is therefore a “filtered” measurement: smoother results are obtained with respect the point-wise tracking methods. However, this also means that small displacements could be not correctly recorded. As a consequence, Hough transform algorithm has been selected in the following experiments due to its robustness and reliability.

Beam analytical model for PZT measurement calibration

The main application of the camera sensors in this work is dedicated to the calibration of the PZT devices. In fact, the PZT sensors change their electro-mechanical characteristics after the autoclave process. Establishing a correlation between the optical measurements and the PZT measurements will allow to re-calibrate the PZT ones. At the scope, it is necessary to define a mathematical model that links the electric potential between the PZT’s electrodes to the plate’s tip displacement.

The amplitude of the PZT’s signal is related to the charges accumulated on the electrodes through the gain of the electronic circuit equation (5), it is then possible to find out the average deformation and stress inside the sensor by means of constitutive relations of the PZT sensors for the direct effect. Limited to the bending behavior, the following assumptions are adopted:

The plate is modeled as a beam;

The presence of the triangular cuts is taken into account as a uniform distribution of compliance and lack in mass along the beam;

The PZT patches cannot influence the dynamic of the system;

The PZT patches have no influence on the laminated composite thickness.

Each sensor is located with an offset of 0.4 mm from the middle plane of flexion. The piezoelectric is perfectly embedded into the composite, so the strain of the plate at 0.4 mm is assumed to be equal to the PZT patch strain

ε_{1}^{piezo} (z) = ε_{1}^{plate} (z)

(6)

where, according to Figure 1, z is directed through-the-thickness of the plate. We assume also the charges on the electrodes only depend on the stress $σ_{1}$ in x direction. In the relevant analysis, there is no electric field applied between the electrodes; therefore, from the constitutive relation of PZT material, we have

D_{3} = d_{31} σ_{1}

(7)

Integrating the aforementioned equation on the area S of the sensor, we obtain the value of the charge stored on the electrodes, that is

Q = \int_{S} D_{3} dS = \int_{S} d_{31} σ_{1} dS = d_{31} σ_{1} S

(8)

Substituting Q with the its expression shown in equation (5), we have

V_{out} \frac{C}{2} = d_{31} σ_{1} S

(9)

and then solving $σ$ , we obtain

σ_{1} = V_{out} \frac{C}{2} \frac{1}{d_{31} S}

(10)

that relates the values of $V_{out}$ to the stress inside the sensor. Because the assumption of perfect embedment of the piezoelectric in the structure, it is possible to estimate the stress inside the plate at the same position of the PZT sensor as

σ_{plate} = σ_{piezo} \frac{E_{plate}}{E_{piezo}}

(11)

In equation (11), Young’s modulus for the plate is computed by the means of the following relationship

E_{plate} = E_{L} \cos^{4} θ + E_{L} \sin^{4} θ

(12)

where $θ$ is the angle (45°) between the x-axis of the plate and the fibers direction. In this case

E_{plate} = \frac{1}{2} E_{L}

(13)

From equations (10), (11), and (13), we obtain the value of the stress $σ_{plate}$ in the same position of the sensor that is equal to

σ_{plate} = \frac{1}{2 S d_{31}} \frac{E_{L}}{E_{piezo}} \frac{C}{2} V_{out}

(14)

According to beam’s theory, the bending stress is assumed to be linear with z

σ_{plate} (z) = \frac{M}{\bar{I}} z

(15)

where M is the bending moment and $\bar{I}$ is a constant mean value of the beam section moment of inertia (purposely selected to consider the triangular holes of the plate). The bending moment is related to the beam curvature $χ_{plate}$ by the well-known relation

M = η E_{plate} \bar{I} χ_{plate}

(16)

where $η$ is a coefficient here introduced, so that the structural model uncertainties can be taken into account.

Combining equations (14) to (16), a relationship that links a structural quantity, that is, the curvature $χ_{plate}$ , to an electrical measurement is derived

χ_{plate} = \frac{1}{E_{plate} z' η} σ_{plate} (z') = \frac{1}{E_{piezo} z'} \frac{1}{S d_{31} η} \frac{C}{2} V_{out}

(17)

where $z'$ now represents the distance of the piezo-patch from the middle plane of the hosting plate.

The camera and PZT sensors provide different kinds of information. The camera is able to recognize the displacement of one point—in this case, the corner of the plate—whereas PZT patches give us information about the curvature of the structure. In addition, due to the different nature of these sensors, also the position of the observed points is different. Therefore, the data matching will be performed on the basis of a global variable (i.e. the modal amplitude); at the scope, a modal decomposition is needed. The equation of free dynamics of a beam can be written as

E_{plate} \bar{I} \frac{\partial^{4} w (x, t)}{\partial x^{4}} = - \bar{μ} \frac{\partial^{2} w (x, t)}{\partial t^{2}}

(18)

where a constant—mean value—of the linear density, $\bar{μ}$ , is used. This assumption is supported by the fact that by comparing the values of the natural frequencies of the equivalent beam model (equation (18)) and the ones obtained from the FEM plate, there are negligible differences. Also the modal shapes of the first modes are practically equivalent.

The classical modal approach can be used to transform the partial differential equation (18) in space and time in a time-dependent ordinary differential system. As a matter of fact, the flexural displacement $w (x, t)$ can be written as a combination of modal shapes functions

w (x, t) = \sum_{n = 1}^{N} K_{n} (t) φ_{n} (x)

(19)

where the n-term $K_{n} (t)$ is the modal amplitude of the nth mode $φ_{n} (x)$ (which can be computed analytically using equation (18) and the relevant the boundary conditions).

Image analysis returns the displacement of the tip of the beam, $w (L, t)$ . From equation (19), limiting the study to the first mode of vibration (which is a strong approximation, but that is satisfactory for the proposed experiments), it is possible to find the relevant modal amplitude, $K (t)$

K (t) = \frac{w (L, t)}{φ (L)}

(20)

However, the same (global) variable can be computed from the PZT measurements. In fact, from equation (17), the curvature can be evaluated for each PZT sensor and hence the modal amplitude

K_{i} (t) = \frac{χ (x_{i_{piezo}}, t)}{φ ″ (x_{i_{piezo}})}, i = 1, 2, 3

(21)

where $x_{i_{piezo}}$ represents the position of the ith piezoelectric inside the beam. The value of $K (t)$ , which is obtained by means of the image analysis (equation (20)), has to be equal to the value $K_{i} (t)$ for each sensor (i = 1, 2, 3). If it does not happen, it means that the value of $χ (x_{i_{piezo}}, t)$ is computed relying on a wrong value of the electro-mechanical coupling coefficient $d_{31}$ and on the structural model uncertainties $η$ (see equation (17)).

Comparing the values of K(t) measured from the camera and the ones $K_{i} (t)$ obtained from the PZT patches, and performing a best fitting procedure, the value of the calibrated product $d_{31} η$ is calculated for each PZT patch.

Calibration results

For the calibration, a cantilever configuration is applied to the plate, as in Figure 8. In Figure 12, modal amplitude computed by means of the camera and of the three PZT sensors are reported.

It is evident that camera and PZT patches do not return the same results and also that the two sensors placed near the fixed joint do not return the same value of modal amplitude. This means that the same deformations of the two sensors generate a different charge density on the electrodes. This behavior, which is different from the nominal one, is due to the change in electro-mechanical coupling factor, $d_{31}$ , after the autoclave process and the presence of the triangular cuts in the plate.

The results of the calibration are reported in Table 4; it is evident that each of the sensors was subjected to a reduction of its coupling coefficient. With the newly found $d_{31} η$ parameters, we are now able to perform an elastic displacement reconstruction using the information coming from the piezoelectric sensors as a standalone measurement (i.e. from now on independent from the camera).

Table 4.

Values of the product $d_{31} η$ of the three piezoelectric sensors after the autoclave process.

Sensor no.	Unit	$d_{31} η$
1	10⁻¹² C/N	−76.3
2	10⁻¹² C/N	−53.9
3	10⁻¹² C/N	−69.9

Free-floating platform: setup and mathematical model

The calibration makes it possible to avoid using the camera for the following experiment on a multibody structure for measuring the elastic displacements. The tests are performed using the PINOCCHIO (Platform Integrating Navigation and Orbital Control Capabilities Hosting Intelligence Onboard) system; detailed information about this platform can be found in Gasbarri et al. (2014) and Sabatini et al. (2015). This is a free-floating platform, which consists of a 10-kg central bus, where the different subsystems and the pressured air tanks are stowed, plus the flexible appendages (Figure 13). A lower air bearing allows the frictionless motion over the flat surface. Immediately above, eight thrusters’ nozzles are placed and these are connected to the tanks by rubber pipelines. In particular, the air flow of the bearing is continuous and regulated at pressure of 3 bar, while the air flow to the thrusters is controlled by on–off electrovalves, at a pressure of 5 bar. The residual dynamic friction coefficient has been measured to be of the order of $μ = 10^{- 5}$ ; the control actions can assume only two values: zero and the maximum, which is about 0.017 N m for the torque. Inside the hexagonal box, on the upper side of the bus, the central process unit (CPU) and the inertial measurement unit are stowed. The CPU is 1.6 GHz main processor, belonging to the Atom Intel class, and it manages the communication with all the sensors and actuators, and computes the reference trajectories and the required actions to be performed. The gyro that is part of the inertial measurement unit is characterized by a 0.17°/s root mean square (rms) noise. Finally, on the top of the structure, an Arduino Uno board, connected to the central processing unit, receives and digitalizes the signal from the acquisition electronic circuits connected to the PZT patches.

Multibody model

The mathematical model of the PINOCCHIO platform has been developed in MSC Adams environment. This is a commercial multibody simulation software used for mechanical system design. It is able to simulate the dynamics of different systems, for example, in aerospace, automotive, and rail applications (more information can be found in McConville, 2015). The PINOCCHIO testbed is modeled as a rigid central bus with two flexible appendages connected to it. The geometry, the structural features, and the FEM model of the panels are imported directly from MSC Nastran software. Each panel is linked to the bus by means of an ideal lock: this constraint is placed on the interface node related to the plate. To simulate the fact that in the experiment only a 2D motion is allowed on the working surface, also a planar constraint is added. As a result of the interaction of the three bodies of the system, the natural frequencies of the multibody system differ from the frequencies of the panels considered as standalone (cantilever) bodies. Performing a linear analysis in Adams, the values of the first four natural frequencies of vibration are computed and reported in Table 5; as specified, each couple of close frequencies are relevant to a similar mode of vibration of the plates, but distributed according to symmetric and anti-symmetric configurations, which make a great difference at system level. Figures 14 and 15 show the first and second modes of the system, respectively.

Table 5.

Modal frequencies of the PINOCCHIO system.

Mode number	Frequency (Hz)	Type
1	0.95	Symmetric flexural
2	1.65	Anti-symmetric flexural
3	9.36	Symmetric Torsional
4	10.24	Anti-symmetric Torsional
5	11.09	Symmetric flexural
6	11.91	Anti-symmetric flexural

PINOCCHIO: Platform Integrating Navigation and Orbital Control Capabilities Hosting Intelligence Onboard.

Experimental results with free-floating platform

The experiment in this phase of the project concerns the possibility to evaluate the natural frequencies of the system by means of the PZT patches and to determine possible undesired behavior during a controlled attitude maneuver of the pseudo-satellite testbed.

Free dynamics

The first experimental campaign concerns the free dynamics of the system with the free-floating base. An impulsive force was applied on the tip of one panel, typically the one with PZT sensors, in order to excite all the natural frequencies. These tests have been performed to verify if the modal frequencies of the system, which are obtained analyzing the output of the PZT sensors, are comparable to the data extracted from the ADAMS linear analysis. If it happens, it means that the relevant numerical model is very similar to the real one, and therefore, we are able to predict the behavior of the system also if controlled dynamics is taken in consideration. Figure 16 shows the time history plot of the PZT sensors’ output signal.

The shape of the curve, represented in the picture, is typical of a multibody system inasmuch the first two frequencies are very close. The frequency content of the previous signal is shown in Figure 17.

The natural frequencies extracted are 0.93 Hz for the first flexural symmetric mode and 1.72 Hz for the first flexural anti-symmetric mode. These values are close to the one calculated with MSC ADAMS software (see Table 5). The residual differences are due to the fact that the bus is modeled as a monolithic rigid body while the real structure is made of wooden support and panels, which are not completely rigid.

Controlled dynamics: PD control

Once the ADAMS model has been tested by means of the free dynamics experiments, the controlled dynamics is considered. To simulate a control strategy of the free-floating platform, the co-simulation process has been performed.

In fact, it is hard (if not impossible) to simulate in Adams the details of the real guidance, navigation, and control (GNC) system. In particular, the simulation must contain the following characteristics which are critical for the control performance:

The control is based on the gyro and PZT measurements, which are affected by errors;

The control torque of the platform is not continuous, but it is achieved thanks to the on–off firing of the thrusters; a pulse width modulation is used to pass from the continuous to the pulsed control;

The control impulses are not applied exactly at the desired time, but they suffer from a certain delay due to electronic (computation time) and mechanical causes.

All these aspects can be easily simulated with Simulink; however, the dynamics behavior of a flexible multibody system can be easily simulated in Adams. The solution is therefore to run the Adams and Simulink software interactively; the Adams model becomes the plant block in the Simulink model, which contains the navigation and control blocks for a realistic simulation of the GNC loop.

In this work, a good performance of the control is not the objective; instead, the focus is in the capability of the sensor to supply important data to investigate the system behavior. In future works, these data, together with the detailed dynamics model, will be used to predict and eventually improve the system behavior. At present, a simple PD control is designed, taking into account the only variable of interest for a space application, that is, the attitude

τ = - k_{p} (θ - θ_{ref}) - k_{d} \overset{\cdot}{θ}

(22)

In the above equation, $ϑ_{ref}$ is the desired attitude to acquire, equal to 30° both in the simulations and in the experiments. Selecting the gains ( $k_{p} = 0.1$ and $k_{d} = 0.38$ ) in such a way that in the continuous and ideal (rigid) model the desired attitude would be acquired within 20 s without overshoot, the real (elastic, time discretized, and delayed) multibody system shows an unstable behavior.

In Figure 18, the platform’s attitude obtained by means of a simulation is compared with the attitude obtained through an experimental test. As we can see in both cases, the multibody system presents an unstable behavior. The cause of this behavior is in the interaction between the flexibility of the plates and the attitude dynamics, which leads to a high energy transfer between rigid and flexible dynamics. Therefore, as shown in Figure 18, the trajectory described by the satellite presents oscillations.

In Figure 19, the displacement of the panel’s tip evaluated by means of the simulation and the same displacement computed starting from the PZT sensors are reported. It can be seen that the data returned from the software are very similar to the measured ones. The difference between the experimental and numerical results is due to an incorrect tuning of the damping ratio in the mathematical model. This problem is particularly hard to solve in flexible multibody systems, where the overall damping depends on a complex combination of the damping matrices of the different parts.

In these conditions, the PD control is not able to produce an acceptable result and so the maneuver fails. In Figure 20, time history of the control actions applied to PINOCCHIO system is reported. In this case, the control is turned ON about 16.13 s with respect to the 20 s of the simulation, which corresponds to an unacceptable propellant consumption. A movie of the video can be found at the website indicated in GN LAB Sapienza YouTube Channel (2017). This unsatisfactory maneuver will be the starting point of future studies aimed to the reduction of the elastic–attitude interaction thanks to the real-time measurements from the PZT sensors.

Conclusion

The combined use of cameras and piezoelectric devices is proposed in this research for monitoring the interaction between attitude and elastic dynamics of a space system. The cameras could be used as a complementary tool with respect to the piezoelectric device: in fact, the frequency range of the two sensors is different (low frequency for cameras and high frequency for PZT sensors), and so are the measured quantities (displacement for cameras and strain for PZT sensors). In the proposed approach, the camera is used as a calibration tool for the PZT sensors. In this way, the PZT sensors are characterized in terms of their exact electro-mechanical parameters (which are usually different from the nominal ones, due to manufacturing processes or to space environment) and can be then used as standalone sensors for the reconstruction of the elastic displacement.

The performance of the measurement system is first verified numerically and then tested with an experimental free-floating platform, both in free dynamics conditions and in an (unstable) controlled maneuver. The promising results will pave the way for future studies for limiting the risk of unstable maneuvers thanks to the knowledge of the elastic state provided by the proposed calibrated system.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Marco Sabatini

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