Shape control of flexible structures

This research was performed by Saki Hakim, a Ph.D. student at the School of Mechanical Engineering at Tel Aviv University under the supervision of Moshe Fuchs. The general purpose of the research was to find optimal locations for sensors and actuators in flexible truss structures in order to maintain its original shape at the controlled degrees of freedom. It is assumed that the structure is subjected to a parametric family of disturbances (perturbations), such as temperature fields for space structures, traveling loads and so on. The novel idea was to design the best sensor and actuator locations without specifying a priori the value of the disturbance parameters. Although the perturbations cannot be characterized, the distortions caused by the perturbations have definite characteristics which can be studied and which form the basis for placing actuators and sensors.

Publications include:

• Hakim S. and Fuchs M.B. (1994), A New Criterion for Actuator Placement in Quasi-Static Control of Flexible Structures Under Arbitrary Distortions, 45th International Astronautical Conference, Jerusalem 9-14 Oct.
  
  This paper attempts to quantify the performance of a given configuration of actuators in an environment of a family of perturbations (of arbitrary magnitudes). Every instance of perturbations causes a distortion which is then reduced by the actuators, leaving a residual distortion. After scanning all possible perturbations one of these residual distortions will be maximum. We propose to use this worst case distortion, J, as a measure of the quality of the given configuration of actuators. It turns out that J is equal to the largest singular value of a rectangular matrix which depends on the disturbance and the control matrices.

• Hakim S. and Fuchs M.B. (1995), Optimal Actuator Placement with Minimum Worst Case Distortion Criterion, AIAA/ ASME/ ACSE/ AHS/ ASC 36th Structures, Structural Dynamics, and Materials Conference , New Orleans, April.
  
  In this paper the worst residual distortion is employed as an objective function for finding optimal configurations of actuators. The structures are a 2D truss beam and a 3D antenna truss, both representing light-weight structures floating in space. Extensional actuators are assumed to be embedded in some bars of the truss. Finding best locations for N actuators out of a total of M bars is a formidable problem. Exhaustive search must be ruled out and the paper describes some heuristic search techniques. The paper also introduces a concept of ideal actuator, which is a theoretical actuator capable of deforming the structure in any desired manner. Ideal actuators are obviously better than any other physical one. Employing ideal actuators one can predict the lowest possible residual distortions without bothering with positioning the actuator. It turns out that the residual deformation of real actuators is not very different from the theoretical lower bound. ( Postscript file, 524K). See also, Hakim S. and Fuchs M.B. (1996), Quasi-Static Optimal Actuator Placement with Minimum Worst Case Criterion, AIAA J., 34(7).

• Hakim S. and Fuchs M.B. (1995), Simulated Annealing Techniques for the Optimal Control of Space Structures, First World Congress of Structural and Multidisciplinary Optimization, Goslar, May 28 - June 2.
  
  In this paper the worst residual distortion is employed as an objective function for finding optimal configurations of actuators. The structures are a 2D truss beam and a 3D antenna truss, both representing light-weight structures floating in space. Extensional actuators are assumed to be embedded in some bars of the truss. Finding best locations for N actuators out of a total of M bars is a formidable problem. Exhaustive search must be ruled out and Simulated Annealing was used to approach the optimal solution. The paper discusses various numerical aspects of the method and assesses its merits. The solutions are compared with lower bounds based on ideal actuators. It is shown that when replacing the type of actuators, dramatic improvements can sometimes be expected. ( Postscript file, 158K).

• Hakim S. and Fuchs M.B. (1996), Shape Estimation of Distorted Flexible Structures, 6th AIAA/ NASA/ USAF Multidisciplinary Analysis & Optimization Symposium, Bellevue, Sep 4 - 6.
  
  The present work discusses the optimal placement of sensors in truss structures in order to obtain best possible information regarding the distortions of the structure. The estimation goal is to reconstruct the deformed shape of the structure, at the controlled degrees of freedom, from the sensor readings. A basic assumption is that the structure is subjected to a parametric disturbance field. We distinguish between disturbances which cause uniform or arbitrary distortions of the structure, and disturbances which cause structured distortions. The estimator is based on the least squares method, hence the estimated shape is the one with least RMS displacement for the given sensor readings. To evaluate the performance of each set of sensors the measure is the largest possible error between the estimated and the actual displacements, at the CDOF. Results show that for reasonable shape estimation a relatively large number of sensors is needed. It is also shown that when using sensors which measure mainly the distortions of the controlled degrees of freedom, significant improvements in the shape estimation can be obtained. ( Postscript file, 158K).

• Hakim S. and Fuchs M.B. (1997), Optimal Geometries of Shape Controlled Structures, Submitted
  
  This paper deals with the geometry design of trusses which must maintain a set of nodes, the controlled degrees of freedom, as undeformed as possible. In contrast with previous research the structure is subjected to a family of disturbances whose total magnitude is bounded in an overall sense but which is only loosely defined at any given point in time. Moreover, embedded in the truss are N_c<\I> ideal controllers which will control the structure in order to reduce the distortions. The actuators are assumed to apply optimal control corrections. It is shown that a measure of the distortions is the (N_c+1)<\I>th singular values of the disturbance influence matrix. The purpose is therefore to modify the geometry of the structure in order to minimize that singular value. One of the difficulties encountered during the optimization is that of repeated singular values. Their derivative is different from that of a single singular value and requires more attention. It is shown that the design tends to generate structures composed of stiff segments joined at flexible interfaces where the actuators are located. These rather peculiar designs may harbor interesting guide lines in future implementations of smart structures. (shape.ps)