Structural and control optimization of a space structure subjected to the gravity-gradient torque

doi:10.1016/S0094-5765(02)00024-3

Acta Astronautica

Volume 51, Issue 10, November 2002, Pages 673-681

https://doi.org/10.1016/S0094-5765(02)00024-3 Get rights and content

Abstract

This paper focuses on the integrated structural/control optimization of a large space structure (LSS) with a robot arm and tip payloads subjected to the gravity-gradient torque. Constraints from the structure and control disciplines are imposed on the integrated optimization process with the aim of obtaining the structure's minimum weight and the optimum control performance. The result demonstrates the feasibility to solve the problem by taking into account combined constraints from the separate disciplines, structures and control.

Further, the resulting minimum structural weight represents a saving of about 20% with respect to the weight of the initial given structure while the control design accomplishes the objectives of suppressing and controlling the structural vibration and attitude without violation of the control damping factor imposed on the problem.

Introduction

Optimization has been playing a very important role in the orbit and attitude control, and structural design of spacecraft since the advent of the space era. It is well known that weight, size, energy, and time are among the most important constraints that guide the space mission planning. Optimization has a wide application concerning this subject. The structural and the control optimization are separate disciplines that have had application in space missions since the early days of the space conquest. However, the integrated problem is recent. The word “Integrated” here stands for an optimization process that takes into account simultaneously aspects of both areas: structures and control [1], [2], [3], [4]. In this sense this approach creates a bridge between structural and control groups for they have been working separately and facing integration problems during the whole history of the space conquest, mainly when the spacecraft is large and has a complex structural configuration.

In the case treated here the optimization aims to obtain the minimum weight of a structure while satisfying constraints involving frequencies, control damping and weight of structural appendages. The control is designed together with the structure to damp the structural vibration and the pitch motion (attitude control). The gravity-gradient is considered as a source of external torque, characterizing the space environment. One of the strongest challenges to solve the integrated structural/control optimization problem is that of software integration. If computer codes are to be developed they must have structural and control optimization capabilities. The software development to implement the integrated structural and control optimization may be by itself a separate problem. The idea that has guided this work is the use of existing software. The Optimal Regulator Algorithms for the Control of Linear Systems (ORACLS) software was chosen from the control side. From the structural area the New Sequential Unconstrained Minimization Technique (NEWSUMT-A) and OPT (OPTmization) computer programs were chosen. MATLAB^® (MATrix LABoratory) software has also been used for the control transient phase. The linear quadratic regulator (LQR) technique was used to solve the vibrational/rotational and control problem while the sequential unconstrained minimization techniques (SUMT) were used to attack the structural part of the problem. This means that the SUMT and the LQR were used simultaneously through two integrated computer programs to solve the structural and control problem.

The improvements in the weight and control efforts are significant as compared with the original (non-optimal) design. However, the computer cost to solve problems with large numbers of design variables and constraints must be balanced; otherwise the problem solution may become prohibitive under cost aspects.

Section snippets

Problem formulation

The Lagrangian formulation is used in conjunction with the finite element technique for modeling purposes. By the Lagrangian formulation [5] the basic steps to obtain the equations of motion are the derivation of the kinetic and potential energies. The potential energy here includes the expression of the gravitational potential in order to yield the expression for the gravity-gradient torque. By combining finite element techniques with the Lagrangian formulation we can derive the kinetic and

The control optimization

In this section, we formulate the control optimization problem to be solved integrated with the structural optimization. We consider the optimal time invariant linear regulator for actively controlling the attitude (pitch motion) and vibration. To formulate the optimization problem we first re-write the equation of motion given by eqn (25) as $M q ̈ + Sq = Df,$ where the Q_q was replaced by Df. D is the distribution matrix that relates the control input vector f to the coordinate system.

The state space

The structural optimization

The optimization problem for this structure is stated as $Minimize the weight W(X_{i})= ∑ i=1 n_{e} ρX_{i} l_{i} subject to λ≥λ_{0} (first natural frequency), ξ_{1} ≥ξ_{0} (damping from control), ∑ i=1 2 ρ_{a} X_{a} (i)l_{a} =W_{a}^{0} (arm weight), X_{i} ′≤X_{i} ≤X_{i}^{u} (side constraints, where X is the cross-sectional area).$ Now let us define the inequality constraints G and the equality constraint H as $G(1)=λ−λ_{0} ≥0, G(2)=ξ−ξ_{0} ≥0,H= ∑ i=1 2 ρl_{a} X_{a} (i).$

Computer simulations and results

The NESUMT-A and the ORACLS software package have been integrated to solve the structural/control optimization problem simultaneously. The transient phase was solved by using the MATLAB package. The input data to solve the problem are given in Table 1.

The initial design vector is $X={9.72 9.72 9.72 9.72 4.8 4.8}^{T} .10^{−3} m^{2} .$ We have considered four actuators:

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One pitch torque actuator at the middle of the main bus.
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One rotational actuator for the robot arm.
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Two force actuators, one at each tip of the main bus.

Conclusions

The integrated structural and control optimization of a LSS with a robot arm and tip payloads subjected to the gravity-gradient torque has been solved by using the NEWSUMT-A and the ORACLS software packages. The results of simulations show important weight saving for the optimized structure and better performance of the attitude and vibrational control for the optimal integrated system.

For the model and the approach solution considered in this paper it is possible for the control designer to

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^☆: Based on paper IAF-98-I.2.03 presented at the 49th International Astronautical Congress, Sept. 28–Oct. 2, 1998, Melbourne, Australia. Research supported by CNPq/INPE (Brazil) and Howard University (USA).

¹: Distinguished Professor of Aerospace Engineering, Dept. of Mechanical Engineering, Howard University, Washington D.C.; Fellow AIAA, Fellow AAS, Member IAA.

²: Professor and Control Engineer, Space Mechanics and Control Division, Brazilian Institute for Space Research, INPE, S.J. Campos, SP, Brazil.

³: Professor, UNIVAP, S.J. Campos, SP, Brazil.

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Structural and control optimization of a space structure subjected to the gravity-gradient torque☆