Modeling and control of multibody system with flexible appendages

Modeling and Control of Multibody System with Flexible Appendages

Álvaro Manoel de Souza Soares

Departamento de Informática

Universidade de Taubaté

12080-000 Taubaté, SP Brazil

[e-mail]

Luiz Carlos Sandoval Góes

Divisão de Engenharia Mecânica-Aeronáutica

Instituto Tecnológico de Aeronáutica

12228-900 São José dos Campos, SP Brazil

[e-mail]

Luiz Carlos Gadelha de Souza

Instituto Nacional de Pesquisas Espaciais

12201-970 S.J.Campos, SP Brazil

[e-mail]

Abstract

The objective of this work is to describe the design and the implementation of an experiment to study the dynamics and the active control of a slewing multi-link flexible structure. The experimental apparatus was designed to be representative of a flexible space structure such as a satellite with multiple flexible appendages. In this study we describe the design procedures, the analog and digital instrumentation, the analytical modeling together with model validation studies carried out through experimental modal testing and parametric system identification studies in the frequency domain. Preliminary results of a simple positional control where the sensor and the actuator are positioned physically at the same point is also described.

Keywords : Flexible Structures, Modal Analysis, Identification and Control of Structures, Identification.

Introduction

This paper presents the design procedures, the analog and digital instrumentation, the analytical modeling together with model validation studies carried out trough experimental modal tests and parametric system identification studies in the frequency domain with a multibody flexible structure prototype. The experimental setup (see the figure below) was assembled at ITA-IEMP’s Dynamics Laboratory with the aim to investigate the dynamics and control of flexible structures representative of aerospace structures like a satellite with flexible appendages. The experimental setup is composed of two flexible aluminum beams coupled to a central rigid hub. The hub is mounted on a steel disc supported on a gas bearing, in an attempt to minimize the static friction and to simulate the structure’s slew motion in space conditions. The steel disc is linked to a brushless DC motor which gives the necessary excitation to the structure. The direct-drive torque actuation avoids the introduction of spurious non-linear effects such as dry friction and backlash in the gear transmission system. The instrumentation and measurement subsystems consist of collocated and non-collocated sensors and their respective signal conditioning systems. A piezoelectric accelerometer, conditioned by a charge amplifier and an integrating preamplifier, is used to monitor the vibracional displacement of the beam tip. Two full strain-gages bridges are used to measure the elastic deformation at two known positions along the arms. The collocated sensors comprehend an angular velocity sensor (provided by the DC motor driver) and a potentiometer fixed to the motor axis. A Dynamic Signal Analyzer is used to estimate the experimental frequency response function between the actuator input signal and each one of the sensors output. The experimentally determined modal response functions together with their coherence signals are used to carry out parametric system identification studies in the frequency domain. MATLAB subroutines are used to validate the analytical model described below, by comparing the theoretical transfer functions with the experimentally determined ones. A schematic view of th experimental set up is shown in Fig. 1 below.

The Analytical Model

The generalized Lagrangean approach and the Assumed Mode Method are used to derive the analytical model of the multi-link flexible structure. In this study we assume that the elastic deformation on the beams are symmetric with respect to the hub, consequently it is necessary to model only the elastic displacement of one of the arms (Junkins and Kim, 1993). The position of a generic point on the beam is written on a local body fixed coordinate system, as shown in the Fig. 2 below :

(1)

where C_x, C_y and C_z are unit vectors of a reference system fixed on the body. The velocity of a point on the deformed beam is written as :

(2)

where w is the angular velocity () of the hub. The kinetic energy of the system is the sum of the kinetic energy of the hub, the arms and tip mass (due to the accelerometer mass loading) :

T = T_hub + T_beam + T_tip(3)

where

(4)

(5)

and

(6)

where I_hub is the hub inertia, I_t is the total moment of inertia (hub + appendages), r is the linear mass density of the beams, L₁ the appendages length and m_t is the mass of the accelerometer located at the tip of the beams. Thus we have the following expression for the kinetic energy :

(7)

The potential energy of the distributed parameter system do not take into account the shear deformation and the rotary inertia of the beam and is given by the following expression :

(8)

The discrete model of the system is obtained by Ritz’s Assumed Modes Method. In this method the elastic displacement of the beam can be described as :

(9)

where fi(x) are comparison functions and I_t are time varying coefficients to be determined. The comparison function, r, used in the numerical calculations, is chosen as the solution of an uniform cantilever Euler-Bernoulli beam (Inman, 1989) :

(10)

where the l _i are the roots of the frequency equation, written as :

1+ cos(l_iL)cosh(l_iL) = 0 (11)

Using the Lagrange equation, we can derive the equations of motion, as follow :

i= 1, 2, ...n (12)

where F_i are the generalized forces and q is the generalized coordinate vector, given by :

(13)

Substituting eqs. (9) into (7) and (8), applying eq. (12), and neglecting the coupled and second order terms, we can after some manipulation, derive the system equations in the mass (M) and stiffness (K) matrix form:

(14)

where M and K have the following form :

(15)

(16)

and,

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

Now it’s simple to get the state-space representation of the system in the form :

(25)

where the A e B matrix are :

(26)

(27)

In order to obtain the analytical transfer functions, we need to define the observation matrix, C, that describe the measured signals in terms of the state variables. This matrix is obtained from the model of the available sensors. As described in the section before, the system instrumentation is composed by four types of sensors: a piezoelectric accelerometer, a potentiometer, two strain-gage bridges and a tachometer. The accelerometer is located at the free tip of the beam and, its signal is conditioned by a charge pre-amplifier and an integrator filter with a global coefficient of sensitivity given by, G_a, in V/cm units. Thus, we can write :

(28)

Substituting (9) in (28), we can rewrite the integrated accelerometer equation as :

(29)

The potentiometer provides a voltage proportional to the angular position of the hub, e_n = Gaq(t). The full strain-gage bridge gives a signal proportional to the axial strain of the beam (e_s), which can be related with the elastic deformation, y(x, t), at the point were it is located.

(30)

where, e, is the thickness of the beam. Substituting Eq. 9 in Eq. 30, we obtain :

(31)

where x₁ is the position where the strain-gages are located on the beam.

The tachometer gives a signal proportional to the angular velocity of the hub,, which combined with the other sensor equations, gives the observation vector Y = C . X, where

(32)

and,

(33)

The Analytical Transfer Functions

To obtain the analytical transfer functions, we used the physical parameters, listed in table 1 below, for the multi-link flexible system.

Applying the Laplace transform in Eq. (26) with zero initial conditions, and using the model parameters listed in table 1, we can obtain the analytical transfer functions for each sensor. The Bode plots of the open loop system are obtained by substituting (s=jw) in the Laplace transfer functions shown below:

(34)

The following figures, illustrate the analytical frequency response function between the actuator torque and the output of the four sensors :

The Experimental Frequency Response Functions

In this work, the experimental transfer functions were obtained by non-parametric system identification in the frequency domain. The flexible system was excited with band limited white noise derived from a Dynamic Signal Analyzer. In our case the excitation was applied trough the DC brushless motor driver input and the transfer functions between the armature voltage and the sensors outputs were estimated by calculating the cross- and auto-power spectral density of the discretized sensor signals. The system was analyzed by a standard FFT routines available in the Fourier Analyzer. The analyzer estimates the frequency response functions () between the measured outputs and the excitation signal, as well as the coherence functions between system input and the multiple outputs. These transfer functions are estimated through the cross spectral density function (Bendat and Piersol, 1986) as follow :

(35)

where ^ means spectral estimator; Gxy is the cross spectral density function; Gxx e Gyy are the autospectral density functions and g_xy² is the coherence function.

The experimental frequency response functions estimated by the Dynamic Signal Analyzer, as well as its respective coherence function are shown below.

Model Validation and Parametric Identification

Comparing the analytical and experimental frequency response functions we can observe some discrepancies between them. These errors are due mainly to sensor noise and unmodelled sensors and actuator dynamics. Nonetheless, the experimental transfer functions clearly show some sharpen peaks in the spectrum, which can be associated with the vibration modes of the flexible system. In order to check this assumption a more accurate comparison can be done between the predicted and the experimentally determined modal frequencies. Table 2 below shows a comparison between the analytical and the experimental results, which suggests a reasonable agreement between these analysis.

To have a parametric identification of the experimentally determined model, we used a M-file from the MATLAB signal processing toolbox, called INVFREQS.M. This routine provides a fit to the experimental FRF based in the frequency points and the imaginary and real parts of the FRF experimental vector. As the system has fourteen state variables, the fit was done using two polynomial of fourteenth order. As an example, the Eqs. (36) and (37) show the resulting identified function, obtained with the accelerometer and strain-gage FRF’s respectively :

(36)

(37)

In the figures below we show a comparison between the fitted and the real curves :

As we can see from the figures above, the fit was successful. The curves are very closed and the modes are represented very well. The configuration of poles and zeros, in the s-domain, of the fitted functions above are show in the Fig. 17 and 18, below.

The transfer functions determined by Eqs. (36) and (37), characterizes a non-minimum phase transfer function system, which could be expected since the sensors in question are non-collocated with the system actuator (Miu, 1991). The table 3 below show the numerical values of the poles and zeros of the identified open-loop transfer functions.

Position Control

Position control of mechanical system with structural flexibility has been an important research topic in recent years. In this section we show preliminary experimental results of two digital position control, with sensor feedback. The first control strategy consists of a proportional controller, while the second approach uses a proportional, integral and derivative controller. These position control was developed to control the angular position of the hub in real time, with no active control of the flexural modes.

A standard computer interface board (AD/DA), with eight analog inputs and two analog outputs, was used to implement the digital controller. A scheme of the proportional controller is shown in the Fig. 19 below. The gain of the controller was adjusted to give a proper system response in terms of overshoot and settling time, as shown in the Fig. 20.

As one can see in the figures below, the positional control is able to excite a high level of residual vibration in the beam, detected by the accelerometer and strain-gage sensors, even after the positional controller has reached its final position. This is expected since the collocated sensors used by the position controller does not take into account the flexible behavior of the beam.

The second controller implemented was a proportional, integral and derivative (PID). which has its gain also adjusted in a empirical way. The Pascal algorithm used to implement this controller is similar to the one shown above, except that the error margin was used in such a way that if the error was inside this margin the signal send to the DC motor was set to zero. In this way, when the real position was close to the desired one, the error decreased and the program sent a null signal to the DC motor. The following figures shown the results with the PID controller.

The Figures 20 up to ²⁴ show that the proportional controller reached the commanded position quickly, but the its stabilization time was very high when compared with the PID controller. The PID controller reached the reference position in a time greater than with the proportional controller, but with a short stabilization time. This work is still in progress, and we intend to implement others types of control strategy including the LQG/LTR, which due to the system inaccuracies, can be shown to be more robust to unmodelled dynamics and sensor noise.

Conclusions

This paper reports preliminaries results obtained with an experimental apparatus with multiple flexible bodies. The model was derived using the Lagrangean approach and its discretization was done with the Assumed Modes Method. The model validation and identification studies were done, first matching the analytical and experimental frequency response functions of a SIMO class model, and second through a parametric identification in frequency domain of the experimentally determined system transfer functions. Comparing the analytical and experimental frequency response functions we can note some discrepancies between them. These errors are due to sensor noise and unmodelled sensor dynamics. A more accurate comparison between models can be done using the results shown in the table 2, that contain the numerical values of the modal parameters. This comparison suggest a good agreement between the analytical and the experimental models. To have some insight in the control area, we implemented two real time positional controllers. The results in control section shown that the proportional controller reach the reference position quickly, but its stabilization time is very high when compared with the PID controller. The PID controller reached the reference position in a time a little bit greater than the proportional controller but with a short stabilization time. This work is still in progress and we intend to implement others types of control strategy. Due to the system inaccuracies a robust control synthesis like LQG/LTR should be more suitable for this system (Soares, Goes and Souza, 1996).

Acknowledgments

This experiment has been partially supported by the EstraFlex project n^o 52-0182/93.6 of CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico)

Presented at DINAME 97 - 7th International Conference on Dynamic Problems in Mechanics, 3 - 7 March 1997, Angra dos Reis, RJ, Brazil. Technical Editor: Agenor de Toledo Fleury.

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• Inman, D. J., 1989 "Vibration With Control, Measurement, and Stability", Prentice-Hall International Inc., USA, pp. 351.
• Junkins, J. L. and Kim, Y., 1993 "Introduction to Dynamics and Control of Flexible Structures", AIAA Educational Series, United States, pp. 452.
• Miu, D. K., 1991, "Physical interpretation of transfer function zeros for simple control system with mechanical flexibility", Journal of Dynamic System, Measurement, and Control, Vol. 113, September, pp.419-424.
• Soares A. M. S., Souza, L. C. G. and Goes L C S. 1997, "Modal Analysis of a Multibody System with Flexible Appendages", In Nonlinear Dynamics, Chaos, Control and Their Applications to Engineering Sciences, Vol. I, J. M. Balthazar, D. T. Mook and J. M. Rosário, Editorr, 312-316.

Publication Dates

• Publication in this collection
  20 Nov 2002
• Date of issue
  Sept 1999