FINAL YEAR PROJECT 2003 ADAPTIVE PASSIVE CONTROL OF A TENSEGRITY STRUCTURE

Transcription

1 FACULTY OF ENGINEERING SCHOOL OF MECHANICAL ENGINEERING FINAL YEAR PROJECT 2003 ADAPTIVE PASSIVE CONTROL OF A TENSEGRITY STRUCTURE Supervisors: Dr. Ben Cazzolato Dr. Dunant Halim Authors: Diana Tembak (a ) Hafiz Md. Rashid (a ) Wilson Handoko (a )

2 Executive Summary This project is concerned with designing and building a tensegrity structure and controlling the structure using adaptive passive control such that vibration induced into the structure is minimised. Tensegrity structures consist of discontinuous bars held in space together by strings. The controller works by adaptively changing the lengths of the strings in the structure such that the stiffness of the structure increases to minimise vibration. Since the concept of tensegrity structures is relatively new in the University of Adelaide, it was necessary to design and build several prototypes and decide on a tensegrity structure that would be able to successfully suppress vibration. The final structure for control was chosen to be a 3-bar 2-stage tensegrity structure. Experiments were performed on the final structure to investigate the properties and characteristics of the structure. The symmetrical reconfiguration method was chosen for motion control of the structure. This method determines the appropriate string lengths needed to rotate the bars in the structure. The controller utilised for reduction of vibration is an adaptive passive controller which implements the Least-Mean-Squared (LMS) algorithm. The experimental results showed that vibration induced in the structure in the horizontal direction at frequencies ranging between 8-9 Hz was successfully reduced by approximately 50%. i

3 Acknowledgments Acknowledgement is due to several people whose involvement in this project has made it possible. Firstly, many thanks to our project supervisors, Dr. Ben Cazzolato and Dr. Dunant Halim, who have given us much support and guidance throughout our project. Secondly, to Mr. George Osborne whose expertise has been a great asset to us for without whom our structure could not have been built so professionally. Thanks also goes to the rest of the technical staff in the Acoustics Laboratory for their guidance and advice. Also, many thanks to Mr. Jean Paul Pinaud from the Department of Mechanical and Aerospace Engineering of the University of California for giving us information on their current research on controllable tensegrity structures. ii

4 Table Of Contents Executive Summary Acknowledgments List of Figures List of Tables Notation i ii vi xi xiii 1 Introduction Project Aim Literature Review Tensegrity Structures Control Background Theory Tensegrity Structures Structure Used Properties of Tensegrity Structures Symmetrical Reconfiguration Stiffness-String Control Control Passive Control Adaptive Control iii

5 Table of Contents 3.3 Motor R/C Servo Motor DC Stepper Motor Design Of Structure Selection of Strings Selection of String Configuration Prototype Discussion Selection of Bars Selection of Connection Ends First Stage Connection Ends Second Stage Connection Ends Selection and Setup of Motors Costs Experiments Experiments to Investigate Geometry Change Setting String Lengths String Pretension Vibration Reduction Frequency Response Initial Testing After Symmetrical Reconfiguration Symmetrical Reconfiguration Analysis Linearisation Results Controller Servomotor Controller Motion Controller Vibration Controller Controller Implementation iv

6 Table of Contents 8 Results and Discussion Manual Control Adaptive Passive Controller Conclusion Recommendations 77 References 79 Appendix A Experiments 81 Appendix B Prototype Designs 103 Appendix C Data for Symmetrical Reconfiguration 109 Appendix D Controller 117 Appendix E Pictures of Prototype 121 Appendix F Specification Lists 127 Appendix G Pictures of Final Structure 130 Appendix H CAD Drawings 133 v

7 List of Figures 1.1 Needle Tower in Hirshorn Museum The Top View and Elevation of a 3-Bar 2-Stage Tensegrity Top view Showing Existence conditions for a 3-Bar 2 Stage Tensegrity Axial Stiffness of a 3-bar 2 Stage Tensegrity Bending Stiffness of a 3-Bar 2-Stage Tensegrity Typical Error Surface ( Bowl ) Implementation of Controller to Structure Construction of the R/C Servomotors Prototype Bottom End of Bars in First Stage Top End of Bars in First Stage Top End of Bars in Second Stage : Bottom End of Bars in Second Stage...32 vi

8 List of Figures 4.6 Servomotor with Pulleys Final Tensegrity Structure Vibration Amplitude of the Structure for Option Frequency Response in Vertical Direction Setup of Frequency Response Experiment Position of Imbalance Motor and Accelerometer Frequency Response in Horizontal Direction Frequency Response in Vertical Direction Time Histories from = 55 to = Time Histories from = 39 to = Saddle, Vertical and Diagonal String Time Histories from = 55 to = Strings Tension Time Histories from = 55 to = Comparison between Symmetrical Reconfiguration and Linearisation Data Movement from = 35 to = Spring Attached to Base of Structure Pulse Width Modulation Servomotor SIMULINK Model Look-up Table SIMULINK Model Vibration Control SIMULINK Model Vibration Response at 8 Hz using Manual Mode Adaptive Filter Block Function Subsystem of LMS Block Function...62 vii

9 List of Figures 7.8 Subsystem of Update Block Function Subsystem of LMS Coefficient Update Complete SIMULINK Model for Controller Control Desk Layout Vibration Response in Relation to the Change in Saddle Length (Manual) Vibration Reduction at 25 Hz (Manual) Vibration Reduction at 11 Hz (Manual) Vibration Reduction at 9 Hz (Manual) Vibration Reduction at 8 Hz (Manual) Vibration Reduction at 25 Hz (Controller) Vibration Reduction at 11 Hz (Controller) Vibration Reduction at 9 Hz (Controller) Vibration Reduction at 8 Hz (Controller)...74 A.1 Frequency Response of the Structure in Horizontal Direction at the Centre (Hour-Glass)...86 A.2 Frequency Response of the Structure in Horizontal Direction at the Corner (Hour-Glass)...86 A.3 Frequency Response of the Structure in Vertical Direction at the Side (Hour-Glass)...88 A.4 Frequency Response of the Structure in Vertical Direction at the Corner (Hour-Glass)...88 A.5 Frequency Response of the Structure in Horizontal Direction at the Side (Beer-Barrel)...89 A.6 Frequency Response of the Structure in Horizontal Direction at the Corner (Beer-Barrel)...90 viii

10 List of Figures A.7 Frequency Response of the Structure in Vertical Direction at the Side (Beer-Barrel)...91 A.8 Frequency Response of the Structure in Vertical Direction at the Corner (Beer-Barrel)...91 A.9 Calibration of Spring Gauges...93 A.10 Option A.11 Option A.12 Option B.1 1-Stage Tensegrity with 2 controlled Strings B.2 1-Stage Tensegrity with 3 Strings B.3 Prototype B.4 Tubing System B.5 Pulley System E.1 Prototype E.2 Top Half of Prototype E.3 Bottom Half of Prototype E.4 View from Top of Prototype E.5 Prototype E.6 Connection at 1,2,3 for Prototype E.7 Connection at 5,7,9 for Prototype E.8 Connection at 4,6,8 for Prototype E.9 Connection at 10,11,12 for Prototype G.1 Full Structure ix

11 List of Figures G.2 View from Base G.3 View under the Structure G.4 View from top x

12 List of Tables 4.1 String Path for Prototype String Lengths for Option Different Set of Configurations Investigated Chosen Configurations...49 A.1 Corresponding Lengths of Strings for Hour-Glass Structure...83 A.2 Corresponding Lengths of Strings for Beer-Barrel Structure...84 A.3 Change in Lengths of Strings between Beer-Barrel and Hour-Glass Structure...84 A.4 Results for Frequency Response in Horizontal Direction for Hour-Glass Structure...85 A.5 Results for Frequency Response in Vertical Direction for Hour-Glass Structure...87 A.6 Results for Frequency Response in Horizontal Direction for Beer-Barrel Structure..89 A.7 Results for Frequency Response in Vertical Direction for Beer-Barrel Structure...90 A.8 Change in Length after Applied Tension 12 Newtons...94 A.9 Change in Length after Applied Tension 29 Newtons...95 xi

13 List of Tables A.10 Initial String Lengths...96 A.11 Amplitude Vibration from Initial Increments/Decrements...96 A.12 Amplitude of Vibration with Different Lengths of Strings...97 A.13 Amplitude of Vibration using Option A.14 Amplitude of Vibration using Option A.15 Amplitude of Vibration using Option B.1 Comparison between Motors xii

14 Notation Structure Notation b D F 3 h K k ST k S k D k V l m m b S T D T S T V T r t T Length of side base of the triangle Diagonal string Externally applied load to the top plate in the vertical direction Overlap of the bars of one stage to the adjacent stage Ratio of string stiffness to bar stiffness Stiffness of the structure Stiffness of saddle string Stiffness of diagonal string Stiffness of vertical string Length of the bar Mass of the structure Mass of the bar Saddle string Tension in the diagonal string Tension in the saddle string Tension in the vertical string A 1 x 3 tension matrix Time Finite time xiii

15 Notation V Z Ii f I f 0 dr I Vertical string Structure height Angle between the side of the base triangle and the projection of the bar Initial Final Angle between the vertical string and the bar Initial Final Prestrain in the strings Natural frequency of the structure Driving frequency Control Notation e FIR k LMS PWM RMS t T u w y Error signal Finite Impulse Response Current algorithm iteration Least-Mean-Squared Pulse-Width-Modulation Root-Mean-Squared Time Finite time Reference input signal Taps in the controller Acceleration signal Convergence coefficient xiv

16 1 Introduction Many biological systems have the capability to alter their system properties in order to adapt to changes in their environment allowing them to function efficiently. For instance, muscles vary its stiffness via contraction/relaxation to respond to external stimuli. In this project, this concept of adapting structures is explored through the usage of a tensegrity structure to minimize vibration. A typical tensegrity structure consists of discontinuous bars held together in space by strings. This structure can vary its stiffness by changing its geometry to respond to vibration. Tensegrity structures were first introduced in 1968 by an artist named Kenneth Snelson. His first work was the Needle Tower which is now preserved in Hirshhorn Museum, Washington DC. At that time, this tensegrity structure was considered more as an architectural work of art. Until recently, tensegrity structures have been found to possess high strength to mass ratio and facilitate high precision control (Skelton, 2001). The advantages of this unique structure provides a promising paradigm for integrating structure and control design (Skelton, 2001a). This project will open opportunities for more research on vibration control using lightweight structures that adapt to its environment. Some of the applications would include radio antenna platforms, telescopic arms in space applications and elevated platforms. 1

17 Introduction (a) (b) Figure 1.1: Needle Tower in Hirshorn Museum (a) Side view (b) View from below (Fletcher,2003) 1.1 Project Aim The aim of this project was to design and built a tensegrity structure and minimize vibration induced into the structure. This was done to observe the capabilities of a tensegrity structure in reducing vibration applied to it. In achieving the aim of the project, the group designed and built a tensegrity structure that utilized the symmetrical reconfiguration method for motion control and an adaptive passive controller for changing the configuration structure. Two prototypes were built and the latter was chosen as the basis for the final structure in terms of its controllability. Various experiments were conducted on the structure to understand the properties of the structure and its constraints. It was found that the structure had various options for controlling its movement. As such, further research was done on symmetrical configuration of the structure which would allow for motion control of the structure. Symmetrical reconfiguration of the structure consisted of rotating the structure by changing lengths of certain strings in the structure whilst keeping the height of the structure constant. The adaptive passive controller with a Least-Mean-Squared algorithm was used to control the tensegrity structure. 2

18 2 Literature Review It was important to familiarise with the existing works conducted on tensegrity structures, to be aware of the extent of existing researches on the structure. The control concepts also had to be further explored to understand the appropriate control needed for dealing with a structure such as that of the tensegrity structure. 2.1 Tensegrity Structures The origin of tensegrity structures can be found in art forms dated since as far as 35 years back. The first tensegrity structure was developed by an artist named Kenneth Snelson. In 1975, research on tensegrity structures in an engineering perspective was started by Buckminster Fuller. Geometrical investigations of these structures followed and not until recently, the research has become more engineering oriented to establish the theoretical framework of the analysis and design of these structures. Static analysis was done by researches by scientists (Pellegrino, 1990; Motro, 1992 and Calladine, 1978). Linear dynamic analysis of the structures have also been conducted (Motro, 1986 and Furuya, 1992). Nonlinear dynamics and control design were undertaken by researchers (Skelton, 2001 and Sultan, 1999) from 1997 till present. 3

19 Literature Review In this project, a number of tensegrity related journals were referred to. The latest research on tensegrity structures which relates to the nonlinear dynamics and control design studies were studied. Skelton et. al. (2001a) gives an introduction to the mechanics of tensegrity structures. These structures were found to be deployable and promoted the integration of structure and control disciplines. Tensegrity structures are special kind of trusses which consist of bars and strings. The bars are held in space by strings which are connected at the end of the bars. It was found that the 3-bar 2-stage structure is the simplest three dimensional tensegrity unit. Extensive research in mechanics was done for this particular structure (Skelton et. al, 2001a). The 3-bar 2-stage structure is defined and the existence conditions are laid out. It was also found that the axial and bending stiffness of this structure is a function of geometry. This information is very important as the basis of this project. The geometry of the 3-bar 2-stage tensegrity structure is a function of the string length. A unique set of string lengths in the structure relates to a unique geometry. The derivation of the string lengths can be found in a journal by Sultan et. al. (2002). A mathematical model that describes the nonlinear dynamics of tensegrity structures was derived based on the Langrange methodology. A particular structure which is the 3-bar 2-stage tensegrity was investigated for motion control. The corresponding equations of motion were derived and conditions under which these motions occur were established. The changing of the geometry of the structure through symmetrical motion was coined as symmetrical reconfiguration. The reconfiguration procedures were laid out with numerous examples and illustrations for clearer explanations. 2.2 Control Apart from the study of tensegrity, this project requires knowledge in control. The appropriate control method is essential to ensure that the structure is controlled such that the minimum vibration in the structure can be obtained 4

20 Literature Review According to Clark et al (1998), adaptive structures are structures that can be configured with actuators and sensors and can be controlled such that its dynamic response can be modified. This book is useful for those interested in the design and analysis of adaptive structures. It focuses on the realities of practical adaptive structure designs. The tensegrity structure used in the project would be classified as an adaptive structure that utilizes servomotors to change the lengths in the strings of the structure. The configuration of the tensegrity structure changes as the lengths changes, this hence alters the stiffness of the structure which will cause either an increase or decrease in vibration induced into the structure. In order to control the vibration of the structure, an adaptive passive control algorithm was utilized. For this project, it was decided that the gradient descent algorithm would be used. Several books covered aspects of the gradient descent algorithm (Snyder, 2000 and Snyder and Hansen, 2000). Snyder (2000) gives a brief overview of the gradient descent algorithm. It serves to give a general idea of gradient descent control rather than a full understanding of the algorithm. The gradient descent algorithm described in this book gives a concise explanation of the algorithm but was insufficient to give one a complete understanding of the algorithm. Snyder and Hansen (2000) covers the theoretical and practical developments of active control of sound and vibration and the fundamental principles that govern these issues. Section 6.5 of the book focuses a great deal on the deterministic gradient descent algorithms. Active Control of Noise and Vibration has a detailed understanding of the least-mean-squared (LMS) algorithm which will be used in the design of the controller. The LMS algorithm is also known as stochastic gradient descent algorithm. Snyder and Hansen (2000) describe the algorithms for the standard adaptive use of the finite impulse response (FIR) filter which includes the development and characterization of the error criterion. Derivations of the algorithm are explained and the minimization of error in the algorithm is described through an illustration of a typical error surface in the shape of a bowl. This is further explained in chapter 3 of the report. The gradient descent algorithm is essentially used to minimise the error signal. This algorithm was used as part of the adaptive passive controller for the tensegrity structure. 5

21 Literature Review Inman (1989) gives an overview of vibration control. The design and control of vibration with details of the specific differences between active vibration control and passive vibration control were also explained. It also touches on the nature of vibration and the methods to minimize the level of vibration. From the understanding of both control techniques, it was decided which method would be more suitable for the application of controlling a tensegrity structure. Several factors which influence the type of control chosen was the type of system the structure utilizes and if it will be controlled by any external devices. 6

22 3 Background Theory This chapter deals with the necessary knowledge required for this project. This includes tensegrity structures, control methods and motors. The scope of the discussion in tensegrity structures is limited to 3-bar 2-stage tensegrity structures; the control methods are limited to the usage of passive controllers with adaptive algorithm and the analysis on motors is limited to R/C servomotors and stepper motors. 3.1 Tensegrity Structures Tensegrity structures consist of bars and strings. Each member is designated to a specific type of loading. The bars are always in compression while the strings are always in tension. In a tensegrity structure, the bars cannot be attached to each other as this will impart torques (Skelton et. al., 2001a). Thus, a tensegrity structure experiences compressive stress and tensile stress only to any type and direction of loading. 7

23 Background Theory Structure Used There are many types of tensegrity structures but the structure used for this project is a 3-bar 2- stage tensegrity which is shown in Figure 3.1. A stage of tensegrity in this case consists of 3 bars which are located at an equilateral distance with each other. These bars are twisted in either clockwise or counter-clockwise direction. The bars in the second stage are twisted in the opposite direction from the first stage (Skelton et. al, 2001a). This allows the bars in the second stage to move in the same direction as the first stage. A 3-bar tensegrity was chosen as it is the simplest three-dimensional tensegrity and it was upgraded to a 2-stage structure, as this is the minimum number of stages needed to control a tensegrity structure. There are three types of strings that connect the bars in a 3-bar 2-stage tensegrity structure as shown in Figure 3.1. The saddle string (S) connects the top of the bar of one stage to the top of the bar of the adjacent stage. The vertical string (V) connects the top of the bar of one stage to the bottom of the bar of the same stage. The diagonal string (D) connects the top of the bar of one stage to the bottom bar of the adjacent stage (Skelton et. al, 2001a). Figure 3.1: The Top View and Elevation of a 3-bar 2-Stage Tensegrity (Skelton et. al, 2001a) 8

24 Background Theory The geometry of the structure is parameterised as shown in Figure 3.1. The structure height, Z is measured from the bottom of the first stage to the top of the second stage. The overlap of the bars of one stage to the adjacent stage is denoted as h. The angle between the vertical string and the bar is known as (Skelton et. al, 2001a). The angle between the side of the base triangle and the projection of the bar is denoted as. + is the projection of the bars inside the base, i.e. hourglass type structure while - is the projection of the bars outside the base, i.e. beer-barrel type structure. Figure 3.2 shows a sketch of the top view of the structure, showing the orientation of the bars for an hour-glass type structure and a beer-barrel type structure. A particular configuration is defined by a specific and. These notations and symbols, as well as the string definitions mentioned above will be used throughout the report. - (a) Figure 3.2: Top view Showing for (a) Beer-Barrel Type Structure (b) Hour-Glass Type Structure (b) Properties of Tensegrity Structures The following section discusses the properties of a 3-bar 2-stage tensegrity structure. Existence Conditions The existence of a tensegrity structure requires that all bars are in compression and all the strings are in tension without any external load (Skelton et. al, 2001a). The values of and that satisfy the existence conditions yield an equilibrium tensegrity structure. Figure 3.3 shows the existence conditions for a 3-bar 2-stage tensegrity structure in relation to and. Every point in the graph corresponds to an equilibrium structure. For example, from Figure 3.3, a structure at =10 with 9

25 Background Theory = 60 is an equilibrium structure. However, the values of and that satisfy the existence conditions are limited such that the chosen values must correspond to: Overlap between 0% to 100% Positive string tension < 90 With reference to Figure 3.3, it is seen that the existence conditions are limited for -30< < 30. It was assumed that for conditions with 30, the trend would be a mirror image of Figure 3.3 with similar boundaries since the bars for both stages have to rotate in the opposite direction. For example, if the bars were set to rotate clockwise for < 30, the bars need to rotate counterclockwise for >30. This is because at = 30, the bars collide with each other. Figure 3.3: Existence Conditions for a 3-bar 2-stage Tensegrity. l=0.4 m, b=0.2 m (Skelton et. al, 2001a) Structure Stiffness The stiffness of the 3-bar 2-stage tensegrity structure depends on the configuration of the structure, pretension and any externally applied load (Skelton et. al, 2001a). According to Skelton et. al. (2001a), pretension is a method of increasing the load bearing capacity through the use of 10

26 Background Theory strings that are stretched to a desired tension. The stiffness of the structure is explored in the axial and bending stiffness. Figure 3.4 and 3.5 depicts the axial stiffness and bending stiffness respectively of a 3-bar 2-stage tensegrity structure. The configuration of the structure has a significant effect on its stiffness. Both the axial and the bending stiffness would increase if becomes more negative. In other words, a beer-barrel type structure performs better under loading compared to an hour-glass type structure. The axial stiffness is high with low while the bending stiffness is high with high. Figure 3.4: Axial Stiffness of a 3-bar 2-stage Tensegrity for (a) different with = -5, 0 = 0.05%, K= 1/9 (b) different with = 35, 0 = 0.05%, K=1/9 (c) different 0 with = -5, = 35, K=1/9. l = 0.4m, b = 0.2m. (Skelton et. al, 2001a) Pretension increases the axial stiffness of the structure in the region of small external loading only and is practically negligible at large external loading. Pretension does not affect the bending stiffness but has an effect of delaying the onset of slack strings. 11

27 Background Theory The external applied loading only affects the axial stiffness whereby the stiffness is increased with increasing loading up to a point where there is no slack strings. The bending stiffness remains constant for any external loading and the loading has the effect on delaying the onset of slack strings Figure 3.5: Bending Stiffness of a 3-bar 2-Stage Tensegrity for (a) different with = -5, 0 = 0.05%, K = 1/9, (b) different angle with =35, 0 = 0.05%, K=1/9 and (c) different 0 with = -5, = 35, K=1/9. l = 0.4m, b = 0.2m (Skelton et. al, 2001a) Symmetrical Reconfiguration Symmetrical reconfiguration is a process of changing from one symmetrical configuration, characterized by i, and i, (initial configuration) to another symmetrical configuration characterized by f and f (final configuration) at a constant structure height, Z (Sultan et. al., 2002). 12

28 Background Theory The scope of symmetrical reconfiguration is focused on 3-bar 2 stage tensegrity structures. The structure has to be symmetrical in order for this method to work. A symmetrical structure was assumed to have the following properties (Sultan et. al., 2002): All bars are of identical length l and mass m b ; Top and base triangles are equal equilateral triangles with sides of length b; All saddle, diagonal and vertical strings are identical of stiffness k S, k D and k V respectively; The coefficient of friction at the joints are equal; All bars have the same angle in terms of and ; The bottom plane of the 1 st stage and the 2 nd stage are parallel to each other. A configuration can be characterized by, and Z. However, from physical investigation, several restrictions had to be imposed on the selection of, and Z which were as follows (Sultan et. al., 2002): { [0, 360) 30};. For = 30, the bars intersect each other; 0 < < 90; For = 0, the bars are perpendicular to the base and top and do not twist. For = 90, the bars collide with the base or top plate; Z > l cos (). Ensure that the ends of the bars of the second stage do not collide with the base, i.e. if Z = l cos (), impact occurs. In order for symmetrical reconfiguration to exist, several conditions had to be met. This included ensuring that the components of the force and the torque acting on the rigid top plate were zero, except those along the vertical axis. The structure height, Z also has to remain constant, which implies that the only rigid bodies in motion are the bars (Sultan et. al., 2002). 13

29 Background Theory Equations The lengths of all saddle, diagonal and vertical strings are the same for a symmetrical configuration. The equations used to determine the lengths of the strings are as follows (Sultan, 2002): Saddle (S) string length; 2 1/ b 2 S Z l l lz lbsin( )cos( 30) (3.1) 3 cos Diagonal (D) string length; ( ) 4 cos( ) / 2 2 b 2 2 D l Z 2lZ cos( ) lb sin( ) sin( ) (3.2) 3 3 Vertical (V) string length. l 2 2 1/ 2 V b 2lbsin( )sin( 30) (3.3) 3 where Z is the structure height in metres, l is the bar length in metres, b is the length of the base triangle sides in metres, is the angle between the vertical string and the bar in degrees and is the angle between the side of the base/top triangle and the projection of the bar in degrees. S, V and D correspond to saddle, vertical and diagonal string length in metres respectively. The tension in saddle, diagonal and vertical strings are given by; T r = [T S T V T D ] T = A -1 r [0 0 F 3 ] T (3.4) where T r is a 1 x 3 matrix, T S, T V and T D are tensions in the saddle, vertical and diagonal string in Newtons respectively, F 3 is the externally applied load to the top plate in the vertical direction in Newtons and A r is a 3 x 3 matrix whose elements are given by: 3(2Z 3l cos( )) l sin( ) lbcos( )cos( 30) A r 11 3S lb cos( )sin( 30) A r 12 V (3.5) (3.6) 14

30 Background Theory 3lZ sin( ) lb cos( )sin( ) A r 13 (3.7) 3D lbsin( )sin( 30) A r 21 3S lbsin( )cos( 30) A r 22 V lbsin( )cos( ) A r 23 (3.10) 3D 6Z 12l cos( ) A r 31 (3.11) S A r32 0 (3.12) 6Z 6l cos( ) A r 33 (3.13) D (3.8) (3.9) Throughout the symmetrical reconfiguration, the tensions in the strings are always positive; T S > 0, T V > 0, T D > 0. These equations determine the string lengths and tensions for saddle, diagonal and vertical strings for given, and Z. In order to change from initial configuration to the final configuration, the necessary characteristics of and as functions of time need to be determined. The following steps ensures that (t) and (t) which connects i and i to f and f exist (Sultan et. al., 2002). The transition between the two symmetrical configurations take place in a prescribed finite time, T. Since the initial and final configurations are in equilibrium, and must be constant before and after the transition, which is assumed to take place for t [0, T]. Thus: i t 0 i t 0 ( t), ( t). (3.14) f t T f t T Consider only continuous time controls, yields: 15

31 Background Theory T t T t t t f t i, 0 0 ) ( (3.15) T t t T t t t T i f i t (3.16), 0 0 ) ( T t T t t t f t i (3.17) T t t T t t t T i f i t (3.18) The functions (t) and (t) must satisfy det (A r ) Stiffness-String Control Controlling the stiffness of the structure can reduce vibrations in the structure. This is related to Equation (3.19). m k ST dr (3.19) where is the natural frequency in rad/s, dr is the driving frequency in rad/s, k ST is the stiffness of the structure in N/m and m is the mass of the structure in kg. The natural frequency of the structure depends on the stiffness and the mass of the structure. If the mass is held constant, then the natural frequency of the structure varies when the stiffness varies. For example, the structure at a particular configuration vibrates when its natural frequency is the same as the exciting frequency of the vibrating source. Thus, when the stiffness of the structure changes, the resonance frequency shifts and resonance no longer occurs. Hence, vibration is reduced. The stiffness of the structure is varied according to the configuration of the 16

32 Background Theory structure; and. The value of and is governed by a unique set of string lengths. Using the symmetrical reconfiguration method, the string lengths can be calculated to achieve the desired and. The stiffness of the structure can be controlled through modification of the string length. This was be accomplished by attaching motors at the end of the bars. The motors were controlled by a suitable controller which determines the string length required for a particular configuration. 3.2 Control A control system places an input into the system and interconnects a configuration of components in order to produce a desired response. The design and implementation of a control system to a real time system requires analysis of the structure to be controlled and an understanding of the control concepts. The control concepts were explored and it was decided that a passive controller with an adaptive algorithm would be used. This controller would serve to reduce the vibration levels in the tensegrity structure. The concepts behind this will be explained in the following section that touches on the fundamental theories behind passive control and adaptive control Passive Control A passive control system is a process of redesign that changes parameters of an existing structure to produce a more desirable response (Inman, 1989). On the other hand, an active control system uses an external adjustable or active device like an actuator to shape or control the response (Inman, 1989). In the case of the tensegrity structure, implementation of active control was not considered as the changing of string length for this structure was done slowly, therefore the speed of this control would not be able to match the driving frequency at which the structure was vibrated. Passive control works by changing properties in a structure such as mass, stiffness and damping. The effects of varying these properties are: 17

33 Background Theory Mass: Increasing the mass lowers the natural frequency; Stiffness: Increasing stiffness increases the natural frequency; Damping: Adding damping reduces the resonant response of the system. In the case of the tensegrity structure, passive control was utilised by changing the lengths of the structure which therefore changes the stiffness of the structure. Passive control is generally an efficient way to control the response of a system and would be easier to implement in comparison to utilising active control methods Adaptive Control Adaptive control techniques can cope with increasingly complex systems that require extreme changes in system parameters and input signals (Chalam, 1978). Thus, adaptive control is a form of control that can modify its behaviour in response to changes in the dynamics of the process and the character of the disturbances. Adaptive control is a suitable method for the tensegrity structure as the geometry and hence the properties of the structure can be altered from changing the string lengths. Adaptive Structures A structure that can be configured with distributed actuators and sensors and directed by a controller capable of modifying the dynamic response of the structure in the presence of timevarying environmental and operational conditions is known as an adaptive structure (Clark, 1998). There are two fundamental forms of adaptive structures, which are constrained adaptive and purely adaptive. Constrained adaptive structures are operated using a fixed-gain control law and is limited to certain performance limits that are in the range of space of the controlled system (Chalam, 1978). However, a purely adaptive structure operates in the presence of a control law whose parameters are modified as a function of system output measurements (Chalam, 1998). The tensegrity structure would be classed as a constrained adaptive structure as the lengths of the strings are dependent on each other. 18

34 Background Theory Adaptive Algorithm An adaptive algorithm was required to generate the control input by controlling the values of its weights to minimise the level of vibration in the structure. This type of algorithm is used for modifying the weights of the digital filter so that attenuation of the unwanted disturbance of vibration of the structure is minimised (Snyder, 2000). Gradient Descent Algorithm The gradient descent algorithm works by adding small percentage of taps of the negative gradient of the error surface to the current value of the filter (Snyder and Hansen, 2000). This form of adaptive algorithm modifies the filter weight values to arrive at the optimum set to minimize the error criterion, which will be discussed later. The generic gradient descent algorithm (Snyder, 2000) can be expressed as: w( k 1) w( k) e( k) (3.20) where k is the current algorithm iteration, w is taps for the controller, e(k) is the error between the desired and actual output and in is the convergence coefficient In Equation (3.20), the convergence coefficient defines the part of the negative gradient to be added and also controls the stability of the system. The selection of is important as it affects the speed of the adaptation and the stability of the algorithm (Snyder, 2000). A that is too small will cause the weights to adapt slowly and possibly stop adapting before optimum values are reached. Also, if the is too large, the weights will fail to stay close to the optimum values and either change constantly or diverge completely. Error Criterion The error signal labelled as e(k) is defined as the difference between the desired output, d(k) and the actual output, y(k) (Snyder, 2000). However, in the case for the control of the tensegrity structure, the error signal was set as the difference between the current and previous acceleration of the structure. 19

35 Background Theory The plot for a typical error surface for the case of two weight coefficients in three-dimensional space is shown in Figure 3.6. The error surface shown can apply for the mean square error or the root mean square (RMS) error. Figure 3.6: Typical Error Surface ( Bowl ) (Snyder and Hansen, 2000) From Figure 3.6, it can be seen that the error criterion has only one extremum at the minimum. However, this is not always the case as there can be more than one local minimum in an error surface. The aim of this adaptive filtering process through the least-mean-squared (LMS) algorithm is to derive an optimum set of coefficients such that the value of the taps reaches the global minimum (Snyder and Hansen, 2000). LMS Algorithm The LMS algorithm uses the gradient descent algorithm described previously but includes an approximation to avoid the limitations when calculating quantities over time (Snyder and Hansen, 2000). The LMS algorithm can be described as: w( k 1) w( k) 2e( k) u( k) (3.21) where e(k) is the error signal, is the convergence coefficient and w is the taps for the controller and u(k) is the reference signal. For the case of the tensegrity structure, the minimum vibration desired would be zero, but this would cause the second term in the equation to equal to zero. As such, in this case, u(k) was set to a value, which depended on the least amount of reduction of vibration in the structure. It is also known as the stochastic gradient descent algorithm as it is the stochastic approximation of the deterministic gradient descent algorithm (Snyder and Hansen, 2000). 20

36 Background Theory Implementation of Adaptive Algorithm to the Structure Error Signal e(k) LMS Algorithm w 0 w 1 w 2 w 3 w 4 w 5 Σ Previous acceleration y(k-1) y(k) PWM PWM PWM PWM PWM PWM Conversion Servo motor 1 Servo motor 2 Servo motor 3 Servo motor 4 Servo motor 5 Servo motor 6 Band Pass Filter Disturbance Input Controllable Tensegrity Structure Current acceleration y(t) Figure 3.7: Implementation of Controller to Structure Figure 3.7 shows a basic flow diagram for the implementation of the controller with the adaptive algorithm which will be used to control the tensegrity structure. As explained in previous sections, the purpose of the LMS algorithm is to determine at which configuration in the structure, the vibration is minimised the most. In Figure 3.7, it can be seen that the error signal (in RMS terms) is an input into the LMS algorithm. 21

37 Background Theory The signal sent to the servomotors was calibrated according to the calculation done in the LMS algorithm. If the error starts minimizing, then the taps will move in the same direction so that the error will minimize even further. However, if the error starts to increase, then the taps will move in the opposite direction. In order to control the servomotors, the signal must be sent using pulse width modulation (PWM). The six servomotors will be controlled to increment or decrement the lengths of the strings in the tensegrity structure. In the case of the tensegrity structure, the current acceleration signal has to be converted to a mean square value or a root mean square (RMS) value as the square value of this vibration associate with the disturbance input s potential energy. This conversion is done before the signal is filtered through the band pass filter (as seen in Figure 3.7). The mean square and the RMS value are shown in Equation (3.22) and Equation (3.23) respectively. y 2 T 1 2 ( t) y( t) ; t = 1,2,., T (3.22) T i1 T 1 2 y( t) y( t) ; t = 1,2,., T (3.23) T i1 where y(t) is the vibration signal at particular time and T is the finite time of averaging. This conversion method using either mean square or RMS was determined once vibration reduction experiments were conducted on the structure. This process will iterate until the error signal is minimized which is until the vibration levels in the tensegrity structure is reduced to the vibration reference signal described in Equation (3.20). The controller for this system is discussed in Chapter 7 of this report. 3.3 Motor There were two types of motor suitable for application of vibration control which were the R/C servomotor and DC stepping motor. They are discussed individually below and based from the 22

38 Background Theory characteristic comparisons, the suitable motor will be chosen. The selection process and the setup of motors in the structure is discussed later in Section R/C Servo Motor R/C servomotors are frequently used in radio-controlled models. The R/C servomotor has a limited rotation of about 180 degrees or less. Generally, R/C servomotors are constructed from four basic components, namely a DC motor, gears, a feedback device which is typically a potentiometer (variable resistor) and a control board (Funakubo, 1991). The construction of R/C servomotor is shown in Figure 3.9. Figure 3.8: Construction of the R/C Servomotors (Funakubo, 1991) The R/C servomotors have low rotor inertia in proportion to the generated torque. It also has fast built-up response during starting which is in a range of a few milliseconds to a few tens of milliseconds. A potentiometer is fed into the servo control circuit for rotational angle and rotation direction discretions. In addition, the R/C servomotors are typically used for applications involving repetitive start and stop sequences. They operate equally in forward and reverse directions, rotate smoothly at all speeds and show resistivity to environmental conditions such as temperature, humidity and vibration (Funakubo, 1991) DC Stepper Motor The DC stepper motor rotates in a stepwise manner from one equilibrium position to the next; therefore the stepper motor s speed is a function of the frequency at which the windings are energized. Stepper motors produce a high output torque at low angular velocities. A holding 23

39 Background Theory torque can be applied to the load solely with DC excitation of the stepper motors winding. They are robust due to the absence of brushes and commutator and this leads to high mechanical reliability. Stepper motors are able to operate with a basic accuracy of ± 1 step, in an open loop system. The step angular error does not accumulate even with multiple steps. This accuracy removes the need to use additional positional or speed transducer (Crowder, 1995). 24

40 4 Design Of Structure This chapter deals with the design of the final tensegrity structure. These design considerations were based on the background theory discussed in Chapter Selection of Strings The strings for a tensegrity structure must exhibit high strength and low stretch properties. These characteristics were required to hold the equilateral triangle aluminium base in place and to withstand a reasonably high tension that was applied to the structure. The low stretch characteristic was important to ensure accurate pretensioning to the structure. Kevlar string with 80 kg breaking strength was used for the tension members (i.e. the strings) since it fulfils the characteristics as required above. Kevlar string is extremely stiff, tough and resistant to stretching. 4.2 Selection of String Configuration The string configuration of the structure was important to ensure smooth control of the movement of the bars. It was found that the least number of strings used (while adhere to the standard string configurations, namely saddle, vertical and diagonal string) promotes a better controllable 25

41 Design of Structure structure. However, too few strings in the structure do not give a symmetrical geometry movement in the structure. Four trial configurations and two prototypes were built from wood before a final structure was agreed upon. The second prototype that was built was used for the basis of the final structure. The first three trial configurations that were designed are detailed in Appendix B.1, B.2 and B.3. These string configurations were designed and compared, starting from a one stage tensegrity through to a two stage tensegrity. The following section discusses the setup of the fourth trial configuration, comparisons with other prototypes and previous string configurations Prototype 2 The second prototype using the fourth trial configuration is shown in Figure 4.1. The string configuration for the structure was advised by Pinaud, a postgraduate student working under Skelton (Pinaud, 2003). The string path configuration taken is shown in Table 4.1. The numbers shown in the column for the path of the strings are based on the numbering of the points in Figure 4.1. The specifications of the string configuration for prototype 2 were as follows: 18 strings altogether in the structure; Vertical strings were set as black strings; Diagonal strings were set as blue and white strings; Saddle strings were set as red and green strings; 12 strings were controlled which were three pairs of saddle strings and three pairs of diagonal strings; The red and green strings were paired such that one is on spooling and one is off spooling. This has the effect of changing and of the first stage; The white and blue strings were paired such that one is on spooling and one is off spooling. This has the effect of changing and of the second stage; 26

42 Design of Structure The six black strings (three strings for each stage) which are of fixed length restricts the angle and to a specific range for both stages. Figure 4.1: Prototype 2. Table 4.1: String Path for Prototype 2 Type of String Colour of String Path Corner 11 Corner 10 Corner 12 Diagonal 1 st Stage Blue Diagonal 2 nd Stage White Saddle Red Saddle Green Vertical 1 st Stage Black Vertical 2 nd Stage Black

43 Design of Structure The construction of this structure required three i-hooks for the ends of the bars that met at the centre of the structure. The configuration for the connection ends of the bars is shown in Appendix E.2. This string configuration allows six pairs of strings to be controlled. It also offers a symmetrical movement of the bars and a better controllable structure compared to the previous trials Discussion Among the four trials conducted which included the two prototypes, it was decided that the second prototype would be used as a model for the final structure. Prototype 2 allowed the best controllability in terms of the ability to easily manipulate the structure into different configurations. This would mean that if vibration were to be induced into the structure, the second prototype would be able to control the structure faster to an equilibrium state compared to the first prototype. Although the controller design for the second prototype would be more complex due to the fact that 12 strings needed to be controlled in comparison with the previous prototypes, it would achieve the aim of reducing vibration levels in the structure more efficiently. Due to cost limitations, it was decided that the strings would be controlled in pairs and thus only six motors would be needed to control the strings. The pair of strings that were initially controlled was pairs of saddle strings (red and green) and pairs of diagonal strings (blue and white strings). However, after the symmetrical reconfiguration method was analysed, it was found that all the saddle strings must have the same length at all times. The same condition applies to the diagonal strings and the vertical strings. The pairing of red strings with green strings and blue strings with white strings together had no effect in changing the saddle string length and the diagonal string length respectively. Thus, it was decided that the pairs had to be one saddle string with one diagonal string. 28

44 Design of Structure 4.3 Selection of Bars The tensegrity structure required the material of the bar to adhere with certain characteristics as follows; the bar needs to be rigid enough to endure any lateral force therefore only producing global bending moments in the structure, lightweight and strong. Several materials had been considered for the compression members of the structure. These were stainless steel, aluminium alloys, fibreglass and carbon fibre. The selection process was done by analysing each advantage of the material. From the comparison of these four materials, the carbon fibre was selected as it was rigid, lightweight, high strength and aesthetically pleasing. The cost of the carbon fibre was quite expensive in comparison to other materials but for such desired characteristics, the carbon fibre was selected. 4.4 Selection of Connection Ends Based on the second prototype, the connections of each string were studied. The connection ends consisted of the ball joints and the caps at the ends of the bars. The design of the ball joint needed to be studied for movement between a negative to a positive. The angle of the bars would not be able to move for a large range as the bars hit at = 30 on the equilateral triangle. The ball joint was designed to ensure maximum movement before the bars of the structure hit. However, the ball joint movements had to be restricted to ensure that the bars did not come out of the socket and cause the structure to collapse. As such, the ball joint was designed to rotate between a range of from -20 to 25 or 35 to 80 The rotation of the bars was clockwise in the 1 st stage for from -20 to 25 and counter-clockwise for from 35 to 80. The design of the caps had to ensure that movement of the strings were smooth in order to minimise friction force in the structure. The caps also had to be easily adjustable and removable for ease of assembly and disassembly. Three systems were studied for use as connection ends. The tubing system, pulley system and the doughnut shaped system. The designs for the tubing system and pulley system are discussed in Appendix B.4. The doughnut shape system was 29

45 Design of Structure selected because it allowed a significant amount of string movement, was a simple design and provided smooth connections. This system also allowed the strings to reorientate itself to a set position as the strings followed the curve of the cap. The caps were designed to snap fit the ends of the bars so that they could be easily rotated to the desired positions. There were different configurations of string connection at each joint. The connection ends for the bars in the first and second stage are discussed in the following sections First Stage Connection Ends One of the three bottom connection ends of the first stage are shown in Figure 4.2. There were three strings altogether inside the bar; one diagonal string that connects to the second stage and two saddle strings. The saddle strings connect straight down the hollow carbon fibre bar through the ball joint and the bottom of the plate. The diagonal string for the first stage enters the bottom end of the bar through a plastic tubing. The vertical string was fixed at the top end and connected to the bottom end of the bar. The strings were clamped down to keep them in place before they were connected to the motors. Vertical string Diagonal string 2 Saddle string 1 Diagonal string Figure 4.2: Bottom End of Bars in First Stage 30

46 Design of Structure The top ends of the three bars are identical. Figure 4.3 shows the connection ends for all four strings that were connected to the top ends of the bars. Figure 4.3: Top End of Bars in First Stage Second Stage Connection Ends The top connection ends of the bars in the second stage were almost identical to the bottom end bars in the first stage. The only difference was that the top connection ends do not have strings that flow out of the ball joints through the top plate. The ball joints were also easily adjustable and could be rotated to any angle. The bottom ends of the bars in the second stage are identical to the top ends of the bars in the first stage. These are shown in Figure

47 Design of Structure 2 saddle Strings Vertical String Diagonal String Figure 4.4: Top End of Bars in Second Stage Figure 4.5: Bottom End of Bars in Second Stage 32

48 Design of Structure 4.5 Selection and Setup of Motors In achieving the outcomes of the project, a motor with high torque with position and speed control was required. The motor had to rotate smoothly at all speeds without jerking and to start and stop repeatedly for controlling the 12 specified strings in the structure. The motor also had to be able to provide a position feedback for controlling purposes. A selection process was conducted between the DC stepper motors and R/C servomotors and shown in Appendix B.5. From the comparisons, the R/C servomotor was chosen. The torque for the motor was calculated and this is shown in Appendix C.3. The design torque was found to be 11.6 kg/cm. Each of the motor is connected to a pair of saddle and diagonal strings. For one pair of strings, one string must rotate clockwise while the other string rotates counter clockwise on the motor. This was done to reduce the amount of torque required by each motor. The motors were placed at the corner of the base triangle in such a way that the string extends out tangentially to the pulley. This ensures that the strings do not slip out from the pulley, as long as the strings are in tension. Figure 4.6 shows the pulleys attached to the servomotor. Figure 4.6: Servomotors with Pulleys 33

49 Design of Structure 4.6 Costs Costs were incurred during the period of the project for purchasing parts for the prototypes and materials for the final structure. The costs of the prototypes are included in Appendix F.2: Specification List. The cost of the final tensegrity structure is included in Appendix F.1: Specification List. The cost of the final rig including the structure and the servomotors is AUD$ Figure 4.7 shows the final structure with the servomotors attached at the base of the structure. 34

50 Design of Structure Figure 4.7: Final Tensegrity Structure 35

51

52 5 Experiments Experiments were conducted on the final structure to understand the physical properties and the limitations of the tensegrity structure. Initially, experiments were conducted on the second prototype which was chosen as the basis of the final structure to familiarize with the equipment. This chapter is divided into two sections, the first consists of experiments which were conducted to investigate the change in geometry of the structure and the second section details the experiment that was conducted to determine the characteristics of the structure for e.g. resonance frequency of the structure. In the first section, the experiments conducted were setting string lengths, applying string pretension and an experiment on vibration reduction. The second part of the experiments consisted of the frequency response experiment. These experiments were conducted on both the hour-glass tensegrity structure (positive ) and the beer-barrel tensegrity structure (negative ). 37

53 Experiments 5.1 Experiments to Investigate Geometry Change The first two experiments described below were conducted to investigate on how the structure would change as the length of the strings and the tension of the strings was to differ. The third experiment was to explore the effects of these parameters in reducing the vibration Setting String Lengths According to Skelton et. al. (2001a), a beer-barrel type structure is stiffer than an hour-glass structure. This would mean that vibration applied to the structure can be noticeably suppressed if the geometry changes occur from an hour-glass shape to a beer-barrel shape. This experiment was conducted to investigate the changing of string lengths that would effect the configuration of the structure. In this experiment, the required string lengths had to be determined to achieve the following configuration; from -5 to 10 and from 35 to 60. These values were chosen arbitrarily in order to obtain an hour-glass and beer-barrel structure. Several methods were used to set the string lengths to obtain the desired symmetrical configuration. From the different methods that were experimented with, a procedure was developed to determine the most suitable way for reconfiguration of the structure. This is shown in section Appendix A.1.1. The string lengths that were found for the hour-glass structure and the beer-barrel structure using this procedure are shown in Appendix A.1.2 and A.1.3. This experiment was found to be very time consuming as each of the 18 strings in the structure had to be adjusted one by one accordingly. The main challenge was to approximate a particular length but many problems arised as the increment/decrement of one string may restrict the length of another string. It can be concluded that there were too many possibilities of the string lengths to achieve a particular configuration. According to Pinaud (2003), the string lengths had to be calculated to achieve a better result. As such, the investigation of motion control through means of symmetrical configuration was further explored. 38

54 Experiments String Pretension According to Skelton et. al. (2001a), pretension is a method of increasing the load bearing capacity of a structure by stretching the string to a desired tension. Pretensioning the strings in the structure can delay the onset of slack string and increase structure stiffness only when subjected to small external loading (Skelton et.al., 1998). This aim of this experiment was to pretension the structure at a particular configuration and observe the resulting effect in reducing vibration. The configuration was set to = -5 with = 35. An aura shaker with a weight of approximately 800 g was placed on the top plate to induce vibration to the structure in a vertical direction at 11 Hz. This frequency was chosen as it was found from the frequency response experiment that the resonance frequency of the structure in the vertical direction ranged between 9-13 Hz. The aura shaker was used for inducing vibration eventhough from frequency response experiments there was more vibration in the horizontal direction. This was because of its availability at the time of the experiment. The detailed procedure of this experiment is shown in Appendix A.3 The structure was pretensioned at 11.4 N and 28.4 N using a self-made calibrated spring gauge. From the results of this experiment, it was found that the amplitude of vibration of the structure increased with pre-tensioning. However, inaccuracies occurred while setting the tension in the strings as it was difficult to do so without affecting the lengths of the strings. The change in string lengths due to the pre-tensioning were not consistent and were found to be in the range of 10 mm. This may affect the configuration although the change was not obvious. It was concluded that the stiffness of the structure was not affected by pre-tensioning of strings in the structure. The results of the experiment can be improved by modifying the spring gauge to prevent the movement of the string. This can be done by attaching a similar type of tightening mechanism which was attached to the spring at the base of the structure as shown in Figure 6.7 to tighten the string accurately. 39

55 Experiments Vibration Reduction The aim of this experiment was to investigate the efficiency of the structure in reducing vibration by varying the string lengths. The vibration was induced by a shaker placed on the top plate in a vertical direction at 11 Hz. The strings were initially set to obtain an hour-glass type structure at = 10 with = 35 : The corresponding lengths for these strings are shown in Table A.1 in Appendix A. Based from the initial testing (refer to Appendix A.4.1), the process of incrementing or decrementing the string were devised and conducted on the tensegrity structure. Three options were investigated as follows: 1) Increase lengths of blue and white strings simultaneously. 2) Increment/decrement red and green strings twice and increment/decrement blue and white strings once and continue incrementing/decrementing the lengths. 3) Increment/decrement blue and white strings twice and increment/decrement red and green strings once and continue incrementing/decrementing the lengths. The results of these options are in Appendix A.4.3.1, A and A Option 3 showed the best reduction in vibration of all three options. Three tests which correspond to the increment/decrement of the appropriate string lengths were conducted for this option. The string lengths for the tests are shown in Table 5.1. Figure 5.1 shows the vibration amplitude for option 3 showing the comparisons of the three tests. The third test showed the most vibration reduction where the change in lengths is maximum. The amplitude of the structure was successfully decreased by 63%. Table 5.1: String Lengths for Option 3 String Lengths String Type Test 1 Lengths (cm) Test 2 Lengths (cm) Test 3 Lengths (cm) Saddle String (Red) Saddle String (Green) Diagonal String (Blue) Diagonal String (White)

56 Experiments Figure 5.1: Vibration amplitude of the structure for Option 3 The main challenge of this experiment is to reconfigure the structure. Changing the lengths of the strings manually to set a certain bar angle was a tedious task as repetitive modifications of each string length had to be done. Improvements in this area have been discussed previously. The vibration reduction from Figure 5.1 implies that this structure is capable of reducing vibration when the string lengths were varied. From this experiment, it can be concluded that vibration is reduced when the structure change to a different configuration. 5.2 Frequency Response Frequency response tests were conducted on the structure to determine the resonance frequency of the structure at different configurations. As the natural frequency of the structure equalled to the driving frequency of the shaker/imbalance motor, the vibration amplitude of the structure became unbounded causing maximum vibration. These tests also helped to determine the best position of the shaker/imbalance motor and to determine the direction of vibration which has a higher amplitude. 41

57 Experiments This experiment was conducted by hitting at the side and at the corner of the top plate of the structure. The vibrations in the structure were measured in the horizontal and vertical direction. Each test was conducted three times at each position in order to obtain more accurate results. The frequency and amplitude of the structure was measured using an accelerometer. Figure 5.2 shows the position of the hammer as it hits the structure in the vertical direction. The results of the frequency response were recorded using a Hewlett Packard Signal Analyser and the signal was checked using an oscilloscope. The data from the signal was converted and transferred to the computer and converted into graphs using MATLAB. The setup for this experiment is shown in Figure 5.3. This experiment was conducted on both the hour-glass and beer-barrel structure. Accelerometer Hammer Figure 5.2: Frequency Response in Vertical Direction Tensegrity Structure Charge Amplifier Signal Analyser Oscilloscope Figure 5.3: Setup of Frequency Response Experiment Initially, testing was conducted using the configuration which was determined in the setting string length experiment for the hour-glass and beer-barrel structure. The second frequency 42

58 Experiments response experiment was conducted on the structure after it was determined that symmetrical reconfiguration would be used for motion control of the structure Initial Testing The hour-glass and beer-barrel structure lengths for the tensegrity structure used in this experiment were previously determined in the setting string length experiment and are shown in Table A.1 and A.2 respectively. Bode plots showing the amplitude of the vibration and the coherence of the signal were done using Matlab, For these plots, as long as coherence was above 0.75, the data in the Bode diagram displaying the magnitude was taken as accurate readings. The results of the testing are shown in Appendix A.2. From the results of the experiment for the hour-glass and beer-barrel structure, it can be seen that there was more vibration if the structure was applied with horizontal vibration. It was also found that range of frequencies for the structure was found to be from 9-13 Hertz depending on the type of structure, direction of vibration and position of the accelerometer. The imbalance motor used for the structure was designed to vibrate the structure horizontally and vertically with more vibration in the horizontal direction. The imbalance motor consists of a DC motor attached with an imbalance mass weighing 6 grams. The motor was screwed in place at the centre of the top plate of the structure to ensure that the resultant horizontal force on the top plate is zero. The accelerometer used was positioned at the side of the top plate and was fixed to measure vibration in the horizontal direction. Figure 5.4: Position of Imbalance Motor and Accelerometer 43

59 Experiments After Symmetrical Reconfiguration Once symmetrical reconfiguration was implemented on the structure, it was found necessary to conduct the frequency response tests on the structure again. Symmetrical reconfiguration of the structure was done from angle = 35 o to = 55 o. The frequency response of this structure was conducted to verify that the resonance frequency had not shifted significantly for these two new configurations. The results of this experiment as the structure was vibrated in the horizontal and vertical direction are shown in Figure 5.5 and Figure 5.6 respectively. Figure 5.5: Frequency Response in Horizontal Direction 44

60 Experiments Figure 5.6: Frequency Response in Vertical Direction From Figure 5.5, it can be seen that the amplitude of vibration was found to be higher at = 35 o. The resonance frequency measured in the horizontal direction was also about 7.5 Hz for the structure at = 55 o and at 9 Hz for = 35 o. For the vibration in the horizontal direction, the amplitude of vibration between the two configurations differed by a factor of 3 (about 16 db difference). From the response of vibration in the vertical direction shown in Figure 5.6, it can be seen that there were two peaks. The first peaks were at frequencies ranging between 9-10 Hz and the second peaks were at higher amplitudes that ranged between Hz. In comparison to frequency response experiments conducted before symmetrical reconfiguration, the second peak in the vertical direction did not exist. The difference between the setup of the two experiments was that in the second test, the base of the structure for mounting the motors were still connected to the structure. As such, the second peak could have possibly resulted from vibrating components in the base of the structure. However, exciting the structure at this frequency did not result in significant vibration. Thus, the resonance peak in the vertical direction was taken to be at 45

61 Experiments about 9 Hz. The amplitude of vibration in the vertical direction for both configurations only differed by about 2 db. From analysing Figure 5.5 and Figure 5.6, it was also concluded that there was negligible damping in the structure as the widths of resonance peaks in the graphs do not increase or decrease. From these results, it was decided that the structure would be vibrated at a frequency ranging between 8-9 Hz and the accelerometer would measure the amplitude of vibration in the horizontal direction. Vibration in the horizontal direction was chosen because as seen in Figure 5.5, the structure has the ability for reducing more vibration when the frequency of the imbalance motor shifts between the two configurations. From this experiment, it was also shown that the amplitude of vibration in the horizontal direction is lower at = 55 o, thus when the adaptive passive controller is implemented, the structure should tend to move to the = 55 o configuration. 46

62 6 Symmetrical Reconfiguration From the setting string lengths experiment in Section 5.1.1, it was found that the configuration of the structure could not be reconfigured properly by determining the string lengths through trial and error. In addition, the fixed vertical strings limited the movement range of the structure as it was constrained to rotate within the set lengths of the strings. Thus, the change from an hourglass shaped structure to a beer-barrel shaped structure was not possible with the fixed vertical strings. As such, calculations were done to accurately determine the appropriate string lengths needed in order to set the structure to a particular configuration. This was achieved via symmetrical reconfiguration. Symmetrical reconfiguration is a process of changing from one configuration, characterized by i and i, to another configuration characterized by f and f at a constant height, Z (Sultan, 2002). 6.1 Analysis Symmetrical reconfiguration process was analysed and the necessary calculations were written in MATLAB. Code C.1.1 in Appendix C was used to investigate the possibility of existence of a configuration for various, and Z values. Typical results are shown in Section C in Appendix C. A configuration exists when all the tension in the strings are positive; i.e. T S >0, 47

63 Symmetrical Reconfiguration T V > 0, T D > 0 and det (A r ) 0. A few configurations have been investigated and are shown in Table 6.1. Table 6.1: Different Set of Configurations Investigated Set () () 1-5 to to to to to to to to to to to to to to 70 From all the sets of values of shown in Table 6.1, it was found that there was no possible configuration for sets 1,2,3 and 4 as all the values of tension were less than zero. Configuration existed for sets 5 to 10 and the data for all these 5 sets were compared to obtain the same value for the structure height (Z) and length of the vertical string. Configuration for sets 5 to 10 have 30 o and thus the 1 st stage of the structure had to be twisted in the clockwise direction and the second stage in the counter-clockwise direction. For the experiments, the structure was initially set in the opposite direction which was for 30 o. As a larger movement was more desirable for a more significant movement of the structure, data from sets 5 to 10 were compared to determine the initial configuration and the final configuration for motion control of the structure. Comparisons of were done for an = 30 o change (maximum change available), followed by = 25 o change, = 20 o change, = 15 o change and =10 o change. The selection of the two configurations for symmetrical reconfiguration were limited to several factors: Both configurations have the same vertical string length; Both configurations have the same structure height; Ball joint can be rotated for = -20 to 25 or = 35 to 80; 48

64 Symmetrical Reconfiguration Maximum travel of servomotor was 180. From comparisons of the different sets, it was found that the maximum possible change of is 20 o. This corresponds to set 5 with set 8 or set 6 with set 9. The initial and final configurations were chosen arbitrarily between the two choices. Set 5 with set 8 was chosen and the corresponding parameters for each configuration are shown in Table 6.2. Table 6.2: Chosen Configurations Parameters = 55 = 35 (degrees) Saddle string length (cm) Diagonal string length (cm) Vertical string length (cm) Structure Height (cm) The next step was to connect the two configurations in a finite time. This process is done via symmetrical reconfiguration. The necessary equations were coded in MATLAB and can be referred to Appendix C.1.2. The results of the symmetrical reconfiguration are shown in Figure 6.1, 6.2, 6.3 and 6.4. Figure 6.1 and Figure 6.2 shows the change in the angles for and respectively when plotted against time. From these two graphs, it can be seen that within a period of 5 seconds, decreases and the increases. From Figure 6.3, it can be seen that the vertical string is relatively constant with variations up to 4 mm. The diagonal string length increased while the saddle string length decreased. The changes in the string length were non-linear at the start and at the end of the motion. Figure 6.4 depicts the tension in the saddle, diagonal and vertical string. The tension is maximum when the structure is at = 55. The tension decreased by approximately 5 N for all strings when the structure move to =35. The tension in the diagonal and vertical string is less than 1 N which may pose a problem of slack strings (i.e. slack string occurs when T < 0 N). 49

65 Symmetrical Reconfiguration Figure 6.1: Time Histories from =55 to = 35 Figure 6.2: Time Histories from = 39 to = 44 50

66 Symmetrical Reconfiguration Figure 6.3: Saddle, Vertical and Diagonal String Length Time Histories from = 55 to = 35 Figure 6.4: Strings Tension Time Histories from = 55 to = 35 51

67 Symmetrical Reconfiguration 6.2 Linearisation The non-linear change of the string length was linearised to simplify the complexity of the pulleys to circular shaped pulleys. Linearisation was done by taking the string length change and dividing this by the finite time, which was 5 seconds in this case. The rate of change for the saddle and diagonal strings were: Saddle rate of change: ds S mm / s 5 (6.1) Diagonal rate of change: dd D mm / s 5 (6.2) The calculations to determine to pulley size are shown in Appendix C.2. Figure 6.5 shows the comparisons of string lengths between symmetrical reconfiguration and the linearisation data. Figure 6.5: Comparison of String Lengths between Symmetrical Reconfiguration data and Linearisation data. 52

68 Symmetrical Reconfiguration From Figure 6.5, it can be seen that the actual saddle and diagonal string lengths differ from the data found from using symmetrical reconfiguration due to the differing pulley sizes. There was a maximum of 1 cm difference between the actual lengths from linearisation and the lengths from symmetrical reconfiguration. 6.3 Results The motion control was done on SIMULINK and real time was set up in dspace. The strings in the structure were set to the lengths found using Matlab for the = 55 configuration (shown in Table 6.2) and were connected to the six servomotors. The lengths of the saddle strings in the structure had to be decremented by 10 mm as the original length for the string could not be achieved in the structure, as there was too much tension in the strings. The structure successfully rotated from = 55 to = 35. Figure 6.6 shows symmetrical reconfiguration. In this figure, it can be seen the bar at the far end of the triangle starts to rotate outwards from = 35 to = 55. Figure 6.6: Movement from = 35 to = 55. However, during the transition of movement between the two angles, the structure did not rotate smoothly. This was due to the linearisation of the pulleys that were done and the adjustments made to the saddle string lengths. At the = 35 configuration, the structure height was higher than the initial height by about 10mm. The lengths of the vertical strings were also slightly slack in that configuration. The instability that occurred during the transition between the two configurations were overcome by attaching a spring to the base of the tensegrity structure and attaching a string from the spring to extend to the top of the structure Refer to Figure 6.7 for a picture of the spring (denoted by the red circle) attached to the bottom triangle base of the structure. The vertical strings at the = 55 53

69 Symmetrical Reconfiguration configuration were also modified using trial and error to accommodate for the slack strings that occur at that configuration. Motion control was implemented again and the structure remained stable during transition and the lengths of the strings for both configurations remained in tension. Figure 6.7: Spring Attached to Base of Structure 54

70 7 Controller The controller for the tensegrity structure was run via a dspace controller board, DS1102 that acts as a real time system interface. The input to the control system was a vibration signal induced by a DC imbalanced motor while the output was a signal to six servomotors that control the 12 strings in the structure. The controller was coded in the form of block function programming in MATLAB SIMULINK, while data acquisitions of the real time system were obtained using Control Desk. The tensegrity controller was divided into three main sections; the servomotor controller, the motion controller for the tensegrity structure and the vibration controller. These three controllers were combined to form the adaptive passive control of the tensegrity structure to minimize vibration. 7.1 Servomotor Controller The R/C servomotors were controlled by sending a pulse of variable width. The pulse parameters consist of a minimum width, a maximum width and a repetition rate. These parameters control the position of the neutral point where the servomotor has exactly the same amount of potential rotation in the counter clockwise directions as it does in the clockwise 55

71 Controller direction. These are demonstrated in Figure 7.1 The R/C servo is an active device, meaning that it will actively hold their commanded position as long as the command signal is present. The maximum amount of force that the servomotor can exert to hold this position is the torque rating of the servomotor. The maximum amount of time that servomotor can hold this position for one position pulse correspond to the command repetition rate. To hold the servo position perpetually, the position pulses must be sent repetitively. Figure 7.1 Pulse Width Modulation. (Engdahl, 2000) The six servomotors were controlled by a single pulse width modulation (PWM) that changed its value repetitively through the DS1102 controller board. The reason for controlling the six servomotors by a single command signal was to ensure that the structure moved symmetrically. As the duty cycle of the PWM was controlled, the required position in the motor was obtained. The DS1102 PWM block function accepts duty cycle values between 0 and 1, which correspond to 0% and 100% of the duty cycle respectively. However, the 50 % duty cycle has been set as the initial value and as a result only values between 0.5 and 1 was used in the DS1102 PWM. In the specification sheet, it was recommended that the servomotor be controlled by sending a PWM between 0.9 and 2.1 milliseconds. However, additional pulse width was required to rotate the pulley to a 90 and 90 position. Therefore the PWM was modified so that the pulse width 56

72 Controller lies between 0.8 to 2.2 milliseconds. This extra width in the pulse was allowed as long as the servomotor does not hit the physical stop. The physical stop would hit when the servomotor makes a jerking noise. The servomotor requires the PWM signal to be refreshed every 20 milliseconds as the specified frequency in the servomotor is 50 HZ. The 20 milliseconds was the repetition rate of the HS-700 BB servomotor. In Figure 7.2, 1.5 milliseconds was set as the neutral position, therefore as the value in the slider is placed between 0.7 and 0.7, the pulse width between 0.8 and 2.2 milliseconds is achieved via a summation block function. The value in the summation block was normalised by dividing the summation value with the servomotor repetition rate and this value was automatically multiplied by a gain of 0.1 as the requirement for the output in dspace. This resulted in a duty cycle of 0.4 and 1.1. Figure7.2 Servomotor SIMULINK Model 7.2 Motion Controller Symmetrical reconfiguration was chosen for motion control of the tensegrity structure. The symmetrical reconfiguration of the structure was implemented between = 55 º and = 35 º. The changing saddle string length data was sampled every 0.01 seconds for a period of 5 seconds. This data was the tabulated in a MATLAB m-file together with the time taken for these changes. This table is shown in Appendix D.1. The look up table block function in SIMULINK was used to link the data in the MATLAB m-file with the SIMULINK control system. The data in the MATLAB m-file was assigned as time and 57

73 Controller saddle variables where their values have been set for motion control. The SIMULINK diagram for motion control is shown in Figure 7.3. Figure 7.3: Look-up table SIMULINK Model The timing pointer in Figure 7.3 followed the time data in the look-up table while the output from this look-up table is exhibited in the display block function that was called Saddle current value. The interpolation and extrapolation method was implemented in the look-up table to give an accurate value of saddle string length. The timing pointer was replaced by a tap for the purpose of vibration control as will be discussed in the next section. The pointer was the output from the LMS block function while the current value of the saddle string length was calibrated in relation to the duty cycle of the DS1102 PWM. 7.3 Vibration Controller Vibration control integrated the control of the servomotors and the motion controller as shown in Figure 7.4. The vibration control was done through the use of an adaptive passive controller. The controller was in the form of an adaptive filter that implements an adaptation algorithm. The Least Mean Square (LMS) algorithm was chosen to track the symmetrical reconfiguration data. The LMS filter acts as a pointer that moves along the look-up table resulting in the motion control that changes the position of the motor. The motor will continue moving in the same direction if the vibration in the structure is reduced but if vibration increases, the servomotors will rotate in the opposite direction. As the motors move clockwise or counter-clockwise, the lengths of the saddle and diagonal strings change and thus the geometry of the structure is altered. The change in the geometry of the structure causes the stiffness of the structure to increase or decrease accordingly and therefore the vibration induced in the structure can be reduced. 58

74 Controller dspace 1102 Servomotor Tensegrity Structure Figure 7.4 Vibration Control The vibration in the structure was induced by a DC motor with an imbalance mass. The voltage to this motor was controlled via a Digital Analog Converter (DAC) block function in dspace while the frequency which corresponded to the servomotor was obtained via an oscilloscope. Initially, the input signal from the accelerometer was filtered using a Butterworth band pass filter. The relevant range of frequency that was allowed through was limited between frequencies ranging between 5 Hz and 20 Hz. These range of values were chosen as they correspond to the resonance frequency at different configuration of the structure. These values of resonance frequency at configurations of = 55º and = 35º were obtained from the frequency response experiments. The filtered acceleration signal was then converted to a root mean square (RMS) value. A running RMS was selected to average out the vibration signal over a period of time. The Zero Order Hold (ZOH) block function was placed before the RMS block to sample the data which was averaged every one-second. The mathematical description of the RMS value for each interval was defined in Equation (3.23). 59

75 Controller The ZOH block function accepts the RMS signal and samples it every one-second to allow enough time for the structure to move from one position to another. The vibration reference was set to equal 0.3 and this signal was held every 1 second via a ZOH block function and was then compared with the current acceleration signal. The summation block evaluates these two signals hence producing an error signal. The error signal was then passed through a unit delay and a summation block. The unit delay block delays its input by a specific sample period, which was inherited from the previous ZOH sample period. The initial condition in this unit delay was set to equal to zero. The sample time inherits the sample time of the ZOH block which was 1 second. In other words, the unit delay held the previous acceleration signal for a period of 1 second. The initial value of the vibration reference of 0.3 was obtained from analysing the RMS vibration signal in response to the change of saddle length. This was achieved by increasing the time data in the look-up table and noting the changes in vibration. The vibration response of the structure in RMS and mean absolute value at 8 Hz is shown in Figure 7.5. Vibration 8 Hz ( Root Mean Square Value) in Manual Mode Vibration 8 Hz (Mean Absolute Value) in Manual Mode Vibration Magnitude Vibration Magnitude Time (seconds) Time (seconds) (a) (b) Figure 7.5 Vibration response at 8 Hz using Manual Mode in (a) Root Mean Square value (b) Mean Absolute value The x-axis of the plot is the capture setting time, which corresponds to data time in the look up table from 0 to 5 seconds which was saddle string length data from = 55 o to = 35 o. The 60

76 Controller vibration signal in Figure 7.5 (a) was obtained using RMS while Figure 7.5 (b) uses the mean absolute value. Based on this analysis, the RMS value was used to reduce the occurrence of the local minimums as the structure was rotated between the two configurations. From these graphs, the bars in the tensegrity structure was expected to move from = 35 o to = 55 o, which corresponds to 5 and 0 seconds respectively in the look-up table. The adaptive filter controller uses the gradient descent algorithm. This algorithm provided in MATLAB is governed by Equation (7.2). u( n) ˆ ( n) ˆ( n 1) e * ( n) (7.2) H u ( n) u( n) where n is the current algorithm iteration, u(n) is the reference input signal, w(n) is the filter taps corresponds to time pointer that moves along the look-up table, e(n) is the estimation error of the current vibration with the previous vibration and is the convergence coefficient. The normalization function was switched off for this controller. The block then computes the filter-tap estimate as shown in Equation (7.3). ˆ ( n) ˆ( n 1) u( n) e * ( n) (7.3) The block function of this algorithm is shown in Figure 7.6. The subsystem of the LMS block function where the signals were checked for its validity is shown in Figure 7.7. The current and previous taps are updated in Figure 7.8 while the coefficient calculation for the taps occur in Figure 7.9. Figure 7.6: Adaptive Filter Block Function. 61

77 Controller Figure 7.7: Subsystem of LMS Block Function Figure 7.8: Subsystem of Update Block Function Figure7.9 Subsystem of LMS Coefficient Update 62

78 Controller An adapt input check box was included in the LMS block function. This checkbox, when enabled (equal to 1) allows the filter coefficients to adapt continuously. On contrary, a zero value to this adapt input would cause the block to stop adapting and hold the filter coefficient at their current values until the next nonzero value is an input. The reset check box was activated at either edge. As the reset check box was activated to either 0 or 1, the adaptive filter parameters are reset to the initial condition. The leakage factor was also included in LMS to avoid taps from getting trapped inside the local minima of vibration. The leakage factor moves in the opposite direction to the taps. The convergence coefficient, was set to 275. The taps that point to the corresponding saddle length at a particular instant in time was set as the output of the adaptive filter. The initial value of filter tap was set to 2.5 so that the structure would initially rotate between the two configurations. This was also done to verify that if the motor was set between the minimum and maximum values in the look-up table, the servomotors would still be able to tend towards the configuration which was much stiffer. The adaptive passive controller continuously adapts to a real time environment and tries to reduce the vibration in the structure until it is minimised. In other words, the controller will rotate the servomotors clockwise and counter-clockwise until it can reach the global minimum of vibration between the two configurations. However, vibration reduction of the structure was only limited to the values of saddle string data in the look-up table, which related dependently with the physical limitation of the servomotors that could only move in 180. As such, a saturation block was added to impose upper and lower boundaries on a signal such that when an input signal was outside these bounds, the signal was clipped to the upper and lower values. A switch was also incorporated in the controller in order to ease switching between manual control and the usage of the adaptive passive controller. 7.4 Controller Implementation Implementation for the servomotor controller, motion controller and the vibration controller was done using SIMULINK. The complete SIMULINK diagram utilised for implementation of the adaptive passive controller is shown in Figure

79 Controller The interaction between the dspace board, DS1102 and the control system in SIMULINK was done using Control Desk. The control desk layout used for creating virtual experiments for changing different control parameters and observing vibration reduction in the structure is shown in Figure From Figure 7.11, the blue signal corresponds to the vibration signal in RMS. The yellow line is the vibration reference. The red signal is the taps (time data in the look up table) while the green line is the error signal. The taps and the error signals run in opposite direction. This is because when the error increases the taps move along to =55 o. The pink signal corresponds to the true vibration in the structure before any filtering and conversion. The numerical displays of these signals are presented in the numerical blocks. 64

80 Controller Figure 7.10:Complete SIMULINK Model for Controller 65

81 Controller Figure 7.11: Control Desk Layout 66

82 8 Results and Discussion The results of vibration reduction of the tensegrity structure at different frequencies are shown in this chapter. The first section covers the results which were obtained using manual control and the second section of this chapter shows the results obtained from the utilisation of the adaptive passive controller to control vibration in the structure. 8.1 Manual Control The analyses of vibration response in relation to the change in the saddle string length using manual control were conducted at several frequencies. The excitation frequencies at 8 Hz, 9 Hz, 11 Hz and 25 Hz were applied to the structure. These frequencies were all random frequencies chosen except for vibration at 8 and 9 Hz which was the range for the resonance frequency of the structure at for α = 55 o and α = 35 o. These vibration response experiments were conducted three times and the same results were obtained each time. Figure 8.1 is the result of the manual mode time showing the vibration magnitude (RMS) as the bar rotates from α = 55 o to α = 35 o. Time at 0 seconds corresponded to α = 55 o while time at 500 seconds corresponded to α = 35 o. 67

83 Results and Discussion Figure 8.1: Vibration Response in Relation to the Change in Saddle Length (Manual) From Figure 8.1, it was noted that the direction of vibration magnitude was different for each excitation frequency. The analyses of potential vibration reduction in manual mode were conducted for each of the frequencies defined previously. The vibration response graphs for 25 Hz, 11 Hz, 9 Hz and 8 Hz are shown in Figures 8.2, 8.3, 8.4 and 8.5 respectively. Figure 8.2 shows the results at 25 Hz. At 25 Hz, only a small amount of vibration reduction was achieved as the bar moved along from α = 55 to α = 35. Based on the experiment, at high frequencies in the range more than 20Hz, vibration reduction could not be obtained. At 25 Hz, the vibration response was found to beat, as there were two vibration signals with a small frequency difference and equal amplitude. The higher frequency noise probably resulted from the servomotor while the lower frequency was the excitation frequency. The result was a linear addition of two components whose frequencies were close to one another. Figure 8.3 shows the results of vibration reduction of the structure at 11 Hz. Approximately 50 % reduction was achieved as the bar moved along from α = 55 to α =

84 Results and Discussion Vibration 25 Hz Vibration Magnitude (V) Time (seconds) Figure 8.2: Vibration Reduction at 25 Hz (Manual) Vibration 11 Hz Vibration Magnitude (RMS) Time (seconds) Figure 8.3: Vibration Reduction at 11 Hz (Manual) 69

85 Results and Discussion Vibration 9 Hz Vibration Magnitude (V) Time (seconds) Figure 8.4: Vibration Reduction at 9 Hz (Manual) Vibration 8 Hz Vibration Magnitude (V) Time (seconds) Figure 8.5: Vibration Reduction at 8 Hz (Manual) 70

86 Results and Discussion At 8 Hz, approximately 50 % vibration reduction was obtained as the bar moves along from α = 35 to α = 55. The results for vibration at 8 Hz is shown in Figure 8.5. Figure 8.4 shows vibration reduction in the structure at 9 Hz. At this frequency, approximately 33 % reduction was achieved. The vibration was reduced as the bar moved along from α = 35 to α = Adaptive Passive Controller The results obtained using manual control were used to verify the robustness of the controller. The experiments with the controller were conducted for four different frequencies, namely, at the same frequencies which were at 8 Hz, 9 Hz, 11 Hz and 25 Hz. These frequencies of excitation varied slightly by about 1 Hz as the DC motor with the imbalance mass drifted. Therefore, some variations in magnitude of vibration were also accounted in when the controller was implemented. The experiment for each frequency was conducted three times to show the consistent values of vibration reduction. These results were shown in Figures 8.6 to 8.9. The initial data collected at the start of each of the graphs for vibration response oscillates in large amplitudes as this corresponds to the movement of the servomotors which rotate clockwise and counter-clockwise from one end to other end. The reason behind this was that initial error of the RMS vibration signal was quite large as the value of unit delay that acted as a memory of the previous RMS vibration signal was initially set to 0. The structure was found to have different patterns of vibration response for diverse shaking frequencies. Figure 8.6 shows vibration response at 25 Hz. At this frequency, the vibration response due to the controller was found to be quite flat. This was consistent with the response in manual mode. The bars rotated slightly toward the region of α = 35. At higher frequencies than 25 Hz, the vibration reduction was not achieved and the magnitude of the signal remained the same. At 11 Hz, the results showed an increased vibration as seen in Figure 8.7. This problem was possibly due to the fact that the taps in LMS block function were trapped in the local minima in the α = 55 o region. Based on analysis in vibration response for this frequency in manual mode, 71

87 Results and Discussion the vibration reduction was expected as the bar moved from α = 55 to α = 35. In the α = 55 o region, the signal consisted of many small oscillations. In order to fix this problem, a new combination of LMS parameters and sampling periods could be applied. From Figure 8.8, at 9 Hz, which was resonance frequency for the structure at α = 35, the vibration is attenuated when the bar move along to region α = 55. As seen in vibration response for manual control, there were two decrease regions. α = 55 demonstrated the lowest vibration signal and in controller mode, the bar moved towards α = 55. Figure 8.9 shows the vibration reduction at 8 Hz. This frequency corresponds to the resonance frequency for the structure with α = 55. It was found that the vibration was attenuated when the bar move along to region α = 55. Figure 8.9 showed a 50 % reduction of vibration in RMS. It was found that the structure adapted quite well at this frequency, while the amplitude of vibration was found to change slightly due to imbalance motor drifting. The reduction time of 200 seconds was quite long and finding optimum parameters for LMS block function with combination to the sampling period was time consuming. At this stage, the current LMS parameters and sampling period was not an optimum design, as these values did not work at frequencies higher than 10 Hz. However, the controller was found to work better at resonance frequencies of 8 and 9 Hz. Some evident of working controller were that the bars moved in clockwise and counter-clockwise before they settled down slowly. The error and taps moved in opposite direction to each other while their values converged to one value. These even demonstrated at 11 Hz. The structure was also found to interact when external shaking force was applied to it. For example, at instance when the structure reduced the vibration and someone shake the structure at comparable shaking frequency with the excitation, the structure adapted to it by moving in opposite direction and as soon as the shaking force was pulled out, the structure continued to reduce vibration in original direction. 72

88 Results and Discussion Vibration 25 Hz in Controler Mode Test1 Test2 Test3 Vibration Magnitude (RMS) Time (seconds) Figure 8.6: Vibration Reduction at 25 Hz (Controller) 1 Vibration Response in Controler Mode Vibration Magnitude (RMS) Test1 Test2 Test Time (seconds) Figure 8.7: Vibration Reduction at 11 Hz (Controller) 73

89 Results and Discussion Vibration Response in Controller Mode Test1 Test2 Test3 Vibration Magnitude (RMS) Time (seconds) Figure 8.8: Vibration Reduction at 9 Hz (Controller) Vibration Response in Controller Mode Test1 Test2 Test3 Vibration Magnitude (RMS) Time( seconds) Figure 8.9: Vibration Reduction at 8 Hz (Controller) 74

90 9 Conclusion The aim of the project which was to design and build a tensegrity structure and minimise vibration induced into the structure was achieved. A 3-bar 2-stage tensegrity was chosen as the structure while an adaptive passive controller was utilized to reconfigure the structure depending on the vibration induced into the structure. Experiments were conducted on the structure which was done to investigate the properties and characteristics of the structure. These experiments included investigation of the effect of changing string lengths, the effect of string pretensioning to the structure, the capability for the structure to reduce vibration at particular configuration and frequency response measurements. Several experiment results were found inconclusive due to the complexity of the structure. As such, mathematical calculations were done to determine the appropriate string lengths in the structure in order to change the geometry and hence the properties of the structure. This was achieved by using symmetrical reconfiguration. The tensegrity controller was divided into three main sections; the servomotor controller, the motion controller for the tensegrity structure and the vibration controller. The adaptive passive controller using the Least Mean Square (LMS) algorithm was applied to the tensegrity structure in order to adaptively change the lengths of the strings in the structure to minimise vibration. 75

91 Conclusion Some experiments for obtaining the vibration response were conducted to test the functionality of the controller. These experiments were performed using two methods. The first method was running the LMS algorithm and noting the vibration response while the second method was changing the position of the 6 servomotors manually and obtaining the vibration response. The results using both methods were compared and analysed. The result from this controller was the reduction of vibration signal of about 50% at certain driving frequencies. However, the robustness of this controller was not verified, as at random frequencies, with exception to the resonance frequency of the structure, vibration was not reduced. Further analysis of this controller with some modification in algorithm parameters and sampling period for obtaining the acceleration signal is still required. At this stage, the controller for the tensegrity structure can only successfully reduce vibration at the resonance frequency of the structure. Through this project, the authors have gained a better understanding of tensegrity structures and the complexities involved in designing and building the structure. The understanding involved with implementation of the controller to the structure also served as a more practical approach for dealing with real time applications. 76

92 10 Recommendations This project has many possibilities for improvement. The main focus for this project in the future would include reducing more vibration in the tensegrity structure. This can be done by increasing the change, improvements in pulley design and a more robust controller. A large change in would result in a larger shift in resonance frequencies. This would mean that the configuration of the structure needs to change more than the current =20 change. In this project, the restrictions imposed on the vertical string restricted the change in. However, by controlling the vertical strings, a larger could be achieved. Controlling additional strings requires extra motors, which means an addition of another two servomotors would be required to control the vertical strings individually, one servomotor to control the vertical strings in each stage. However, ideally, the number of motors can be reduced to three with an appropriate pulley system; one for controlling the saddle strings, one for the diagonal strings and one for the vertical strings. This method would require each motor to control six strings. The control of each type of string by one motor simplifies the controller and also ensures that the structure rotates symmetrically. However, more research on the amount of torque required by the strings has to be conducted to ensure that the motors would be capable of on spooling six strings at once. 77

93 Recommendations Increase in would also mean a larger change in the string lengths. A motor which can turn more than 180 which still has the holding torque capability of the servomotors would be desirable. One possible solution is to use a DC motor coupled with a gearbox to provide the necessary torque. However, this solution is more expensive than the current motor configuration. In this project, circular pulleys were used and it was found that the transition between the two configurations were not exact to the required string length change. For smoother transition during the symmetrical reconfiguration, a cam profile could be designed for the pulleys to accurately follow the string length path produced in MATLAB. However, more research needs to be done on symmetrical reconfiguration to ensure that the pulley design can accommodate for small or big changes in. The controller for the tensegrity structure could also be more robust. The tensegrity structure should be able to adapt to vibrations in both the horizontal and vertical directions for a wider range of frequencies. The capabilities of the controller can also be extended for reducing vibration of random noise, white noise or pink noise as this would be more relevant to real life applications. 78

94 References Calladine, C. R. (1978), Buckminster Fuller s Tensegrity Structures and Clerk Maxwell s rules for the construction of stiff frames, International Journal of Solids and Structures, v. 14, pp Chalam, V. (1978), Adaptive Control Systems: Techniques and Applications, M., M. Dekker, New York. Clark, R.L., Saunders, W., Gibbs, G. (1998). Adaptive Structures: Dynamics & Control, John Wiley & Sons Inc., New York. Crowder, R. M., (1995), Electric Drives & their Controls, Oxford Science Publication. Dorf, R., Bishop, R. (2000) Modern Control Systems, Upper Saddle River, N.J. Prentice Hall E.I. du Pont de Nemours and Company (2000) Available from: [Accessed on 2 May 2003] Engdahl, T. (2000) Available from: home2.planetinternet.be/gronsijn/ham/schemes/reserves.htm. Fletcher, V. (2003), Sculpture Garden: Kenneth Snelson, Hirschhorn: Musuem and Sculpture Garden. Available from: gardens/snelson.html [Accessed on 3 April 2003] Funakubo, H. (1991), Actuator For Control, Gordon & Breach Science Publishers, New York. Futaba Corporation of America (2000) Available from: [Accessed on 24 April 2003] 79

95 References Furuya, H. (1992). Concept of Deployable Tensegrity Structures in Space Application, International Journal of Solids and Structures, v. 7, no. 2, pp Inman, D. (1989), Vibration with Control Measurement and Stability, Englewood Cliffs, N.J : Prentice Hall International. Landau, G., Lazzano, R. (1999). Adaptive Control, London ; New York : Springer. Motro, R. (1992), Tensegrity Systems: the state of the art, International Journal of Space Structures, v. 7, no.2, pp Motro, R., Najari, S., Jouamma, P. (1986), Static and Dynamic Analysis of Tensegrrity Systems. Proceedings of the ASCE International Symposium on Shell and Spatial Structures 38, pp Pellegrino, S. (1990), Analysis of presetressed Mechanisms, International Journal of Solids and Structures. v. 26, no. 12, pp Pinaud, J.P. (2003), String configurations. Available from <jpinaud@mae.ucsd.edu> [Accessed on 2 April 2003] Skelton, R.E., Helton, J. W., Adhikari, R., Pinaud J. P. and Chan, W. (2001) An Introduction to the Mechanics of Tensegrity Structures System, Proceedings of the 40 th IEEE Conference on Decision and Control, December 2001, Orlando, Florida USA. Skelton, R.E., Helton, J. W., Adhikari, R., Pinaud, J. P. and Chan, W. (2001a). An Introduction to the Mechanics of Tensegrity Structures. In Handbook on Mechanical Systems Design, Electrical Engineering CRC Press LLC. Skelton, R. (1999). Design: The Absentee System Theory, presentation. Snyder, S.D. ( 2000), Active Noise Control Primer, Springer-Verlag. Snyder, S.D. and Hansen, C.H. (2000), Active Control of Noise and Vibration, E&F Spon. Sultan, C., Corless, M., Skelton, R. (2002), Symmetrical Reconfiguration of Tensegrity Structures, International Journal of Solids and Structures, n. 29, pp

96 Appendix A Experiments 79

97 Appendix A A.1 Setting String Lengths According to Skelton et. al. (2001), a beer-barrel type structure is stiffer than an hour-glass structure. This would mean that vibration applied to the structure can be noticeably suppressed if the geometry changes occur from an hour-glass shape to a beer-barrel shape. Hence, it was decided that the two configurations would have from -5 to 10 and from 35 to 60 Several methods were used to try to set the lengths to ensure that the structure was symmetric throughout and the angles could be appropriate angles could be obtained. From the different methods used, the procedure below was found to be the most suitable way for reconfiguration of the structure. A.1.1 Procedure for Setting Lengths of Tensegrity Structure: 1) Initially, set the lengths for one pair of saddle strings. The tensegrity structure has: 6 saddle strings 6 diagonal strings 6 vertical strings The saddle strings are divided to three red strings and three green strings while the diagonal strings are divided to three blue strings (1 st stage) and three white strings (2 nd stage). One pair of saddle strings consists of one red string and one green string. 2) Adjust the length of the saddle string (red string), then set the lengths of the other two red saddle strings until an optimum length is found. Optimum length here would mean when all three strings can be pulled/slackened to a particular length and the strings are able to apply a reasonable amount of tension to support the structure. The length of the strings was measured from the ball joint to the top centre of the bars. This method of string measurement was kept consistent to ensure that there were no variations in measurement. 3) Continue and adjust the green strings until an optimum length is found. The red strings will probably have to be readjusted to accommodate for an optimum length for the green strings. In accommodating for the readjustment of lengths for the red and green strings, it must be ensured that all the red strings are of the same length and the same applies to all the green strings. 80

98 Appendix A 4) Adjust the blue and white strings using steps 2 and 3 but starting with the blue string first then the white string. 5) Throughout the procedure to set the lengths of the structure, steps 2 till 4 will have to be repeated till the structure remains in tension and is symmetric. 6) The angles, and of the structure can be determined. A.1.2 Hour-Glass Structure For + (hour-glass structure) whereby the bars are inside the base, the lengths of strings found are shown in Table A.1. Table A.1: Corresponding Lengths of Strings for Hour-Glass Structure String Colour Type of String Lengths (cm) Red Saddle (S) 19.0 Green Saddle (S) 20.0 Blue Diagonal (D) 26.0 White Diagonal (D) 29.0 Black (2 nd Stage) Vertical (V) 28.5 Black (1 st Stage) Vertical (V) 27.0 Overall height of structure = 50 cm Angles at each corner: A: = 10 and = 35 B: = 10 and = 35 C: = 12 and = 35 A.1.3 Beer Barrel Structure In order to obtain a wider range of resonance frequencies for the structure, the structure was set to a beer-barrel shape. For - (beer-barrel) structure whereby the bars are outside the base, the lengths of the strings found using the procedure in A1.1 are shown in Table A.2. 81

99 Appendix A Table A.2: Corresponding Lengths of Strings for Beer-Barrel Structure String Colour Type of String Lengths (cm) Red Saddle (S) 29.0 Green Saddle (S) 22.0 Blue Diagonal (D) 24.0 White Diagonal (D) 23.0 Black (2 nd Stage) Vertical (V) 27.5 Black (1 st Stage) Vertical (V) 23.0 Overall height of structure = 38.0 cm A.1.4 Discussion Several attempts were made to obtain a value of -. From the attempts made through trial and error, it was found that in order to change the structure from an hour-glass shape to a beer-barrel shape, the vertical strings (black strings) have to be shortened by a large amount. The black string in the 1 st stage is decreased by 1cm to 27.5cm and the 2 nd stage black string is decreased by 4cm to 23cm. Refer to Table A.3 for the change in lengths of strings for the beer-barrel structure compared to the hour-glass structure. Angles at each corner: A: = -5 and = 35 B: = -5 and = 35 C: = -5 and = 35 Table A.3: Change in Lengths of Strings between Beer-Barrel and Hour-Glass Structure String Colour String Lengths (cm) Change in Length Beer-Barrel Hour-Glass (cm) Structure Structure Red Green Blue White Black (1 st stage) Black (2 nd stage) Structure height

100 Appendix A A.2 Frequency Response Frequency response tests were conducted on the structure to determine the resonance frequency of the structure and to determine the best position of the shaker/imbalance motor and the highest amplitude which can be obtained from the vibration. A.2.1 Hour Glass Structure (=10) The tensegrity was set to an hour-glass structure using the lengths found from Table A1. Horizontal Direction Table A4 shows the results obtained for the hour-glass structure when the top plate was hit with the impact hammer in the horizontal direction. Table A.4: Results for the Frequency Response in Horizontal Direction for Hour-Glass Structure Hammer impact Side Test Resonance Frequency Hz) Amplitude (db) Hammer impact Corner Test Resonance Frequency Hz) Amplitude (db) Figure A.1 shows the Bode magnitude and coherence plots for the hour-glass type structure for tests conducted when structure was vibrated at the side of the top plate. Figure A2 shows the plots for the structure when vibrated at the corner of the top plate. For these plots, as long as coherence is above 0.75, the data in the Bode diagram displaying the magnitude can be taken as accurate readings. 83

101 Appendix A Figure A.1: Frequency Response of the Structure in Horizontal Direction at the Centre (Hour-Glass) Figure A.2: Frequency Response of the Structure in Horizontal Direction at the Corner (Hour-Glass) 84

102 Appendix A The results obtained to determine the resonance frequency of the structure from all the tests shown in Figure A.1 and A.2 can be used as the coherence of the data obtained are above 0.8. From Figure A.1, it can be seen that the resonance frequency of the structure is about 9 Hz and it vibrates at amplitude of about 12dB. From Figure A.2, the resonance frequency of the structure is about 10 Hz and it vibrates at amplitude of 14 db. There is more vibration when the structure is applied with vibration at the corner of the structure. However, the difference in amplitude of vibration in the structure at the corner and centre of the top plate of the structure only differs by a small amount which is about 2dB. Vertical Direction Table A.5 shows the results when the impact hammer was hit in the vertical direction. Figure A.3 and A.4 shows the Bode magnitude and coherence plots for tests conducted when the structure was vibrated at the side and corner of the top plate respectively. Table A.5: Results for Frequency Response in Vertical Direction for Hour-Glass Structure Hammer impact Side Test Resonance Frequency Hz) Amplitude (db) Hammer impact Corner Test Resonance Frequency Hz) Amplitude (db) From Figure A.3, it can be seen that the resonance frequency of the structure is about 10 Hz and it vibrates at amplitude of about 4dB. From Figure A.4, the resonance frequency of the structure is about 10 Hz and it vibrates at amplitude of 3 db. There is less vibration when the structure is applied with vibration at the corner of the structure. Overall, there is more vibration if the structure is applied with horizontal vibration. 85

103 Appendix A Figure A.3: Frequency Response of the Structure in Vertical Direction at the Side (Hour-Glass) Figure A.4: Frequency Response of the Structure in Vertical Direction at the Corner (Hour-Glass) 86

104 Appendix A A.2.2 Beer-Barrel Structure The frequency response experiment was repeated for the beer-barrel structure. Horizontal Direction Table A.6 shows the results obtained for the structure when the impact hammer was hit in the horizontal direction. Figure A.5 and A.6 shows the Bode magnitude and coherence plots for tests conducted at the side and corner of the top plate respectively. Table A.6: Results for Frequency Response in Horizontal Direction for Beer-Barrel Structure Hammer impact Side Test Resonance Frequency Hz) Amplitude (db) Hammer impact Corner Test Resonance Frequency Hz) Amplitude (db) Figure A.5: Frequency Response of the Structure in Horizontal Direction at the Side (Beer-Barrel) 87

105 Appendix A Figure A.6: Frequency Response of the Structure in Horizontal Direction at the Corner (Beer-Barrel) From Figure A.5, it can be seen that the resonance frequency of the structure is about 12 Hz and it vibrates at an amplitude of about 11dB. From Figure A.6, the resonance frequency of the structure is about 13 Hz and it vibrates at an amplitude of 11.5 db. Vertical Direction Table A.7 shows the results in the vertical direction. Figure A.7 and A.8 shows the Bode magnitude and coherence plots for tests conducted when the top plate was hit with an impact hammer at the side and corner of the top plate respectively. Table A.7: Results for Frequency Response in Vertical Direction for Beer-Barrel Structure Hammer impact Side Test Resonance Frequency Hz) Amplitude (db) Hammer impact Corner Test Resonance Frequency Hz) Amplitude (db)

106 Appendix A Figure A.7: Frequency Response of the Structure in Vertical Direction at the Side (Beer-Barrel) Figure A.8: Frequency Response of the Structure in Vertical Direction at the Corner (Beer-Barrel) 89

107 Appendix A From Figure A.7, it can be seen that the resonance frequency of the structure is about 11 Hz and it vibrates at an amplitude of about 14dB. In this plot, only the results from test 3 are shown because the coherence at the point of resonance frequency for tests 1 and 2 are less than 0.7. From Figure A8, the resonance frequency of the structure is about 10.5 Hz and it vibrates at an amplitude of 13 db. In this plot, only results from test 1 and 2 are taken as the coherence for test 3 is less than 0.7. There is slightly less vibration (1dB less) when the structure is applied with vibration at the corner of the structure. However, overall, there is more vibration if the structure when vibration is applied in the vertical direction. A.3 String Pretension This aim of this experiment was to pretension the structure at a particular configuration and see the resulting effect in reducing vibration. The configuration was set to = -5 and = 35. A shaker, weigh approximately 800 g was placed on the top plate act as the external load The structure was set to a beer-barrel shaped structure with = -5 and = 35 with the associated string lengths determined in previous experiments Table A.2. This experiment was conducted by applying a set amount of tension on all the strings except the vertical strings which were held constant. Then, vibration was applied to the structure and the amplitude of the structure was recorded. The strings in the structure was then applied with another set amount of tension, and vibration was applied again and the amplitude of the structure was recorded. The shaker was used to inducing vibration in the structure in a vertical direction at 11 Hz. A.3.1 Calibration To measure the tension in each of the strings, the gauges must provide some means of control, small and easy access for them. With aid from Mr. George Osborne, the group devised two spring gauges and calibrated them with RS Portable force indicators. The calibration results are shown in Figure A.9. 90

108 Appendix A The red and blue lines show the calibration for the two spring gauges. The average was taken for both gauges and the equation finalized on is Equation A.1 where T is the tension in the string measured in Newton. T = x (length in mm) (A.1) y = x y = 0.551x 20 Tension (N) Delta x (mm) Figure A.9: Calibration of Spring Gauges A.3.2 Tension at 20mm The tensions of the string were measured at each of the three corners of the structure. Table A.8 shows the original lengths of the strings and the change in length when each of the tensions in the string were set at 20mm. This was equivalent to: T = x 20mm = Newtons 91

109 Appendix A Table A.8: Change in Length after Applied Tension 12 Newtons Corner Original length (cm) Length after applied 20mm tension (cm) A B C Change in length (cm) The structure was then vibrated at its resonance frequency. From the frequency response experiment, it was found that the structure resonant frequency was highest in the vertical vibration at a resonance frequency of about 13 to 15 Hz. However, since a shaker was used to apply the vibration, the structure would consequently have added mass to the structure. Equation A.2 was used to define the relationship between mass and frequency of the structure. k (A.2) m where is the natural frequency in rad/s, k is the stiffness of the structure in N/m and m is the mass of the structure in kg. From equation A.2, it can be seen that as mass of the structure increases, frequency of the structure decreases. Thus, the structure was set at a resonance frequency of about 11 Hz. The shaker was placed in the centre of the top base of the structure and it was found that the amplitude of the structure was 56 mv. A.3.3 Tension at 50mm The tension in the structure was then set at 50 mm. This is equivalent to: T = x 20mm 92

110 Appendix A = 28.4 Newtons (This value is approximately doubled from the previous value) Table A.9 shows the change in lengths for each of the strings after the structure was applied with 11 Hz vibration. The structure was vibrated again at a frequency of 11 Hz. The amplitude of the structure was found to have increased by 2 mv to 58 mv. Table A.9: Change in Length after Applied Tension 29 Newtons Corner Previous length (cm) Length after applied 50mm tension (cm) Change in length (cm) A B C A.3.4 Discussion: The stiffness of the structure is not affected by pretensioning. According to Skelton et. al. (1998), pretensioning only delays the onset of slack strings. Pretensioning also can increase the vertical load capacity of the structure. From the results obtained in the experiment, the changes in string lengths are not consistent. The inconsistencies resulted from inconsistencies of fixing the gauges to the string. A.4 Experiments for Vibration Reduction From the experiments conducted previously, it can be concluded that the best way to increase the stiffness of the structure and hence reduce vibrations in the structure is to initially start the structure in an hour-glass shape and then increment and/or decrement the lengths of the saddle and diagonal strings to change the shape to a beer-barrel type structure. 93

111 Appendix A A.4.1 Initial Testing The strings were initially set to the lengths of strings listed in Table A1. These experiments were conducted using trial and error and the lengths of the strings were decreased or increased suitable depending on how much vibration was reduced in the structure. Table A.10 shows the increment and decrement of the blue with white strings and the red with green strings. For example the red string was set at +1cm from the initial length, green at 1cm, blue at +1cm and white at 1cm. The corresponding lengths for the experiment in the second row and second column would be red string = 21.0 cm, green string = 18.0 cm, blue string = 28.0 cm and white string = 27.0 cm. Table A.10: Initial String Lengths String Colour Length (cm) Red 20.0 Green 19.0 Blue 27.0 White 28.0 Table A.11: Amplitude Vibration from Initial Increments/Decrements Blue/White Length Increment/Decrement (cm) 0 +1/-1 +2/-2 Red/Green Length Amplitude of Structure Increment/ 0 Test 1 Test 2 Test 3 Decrement (cm) +1/-1 Test 4 Test 5 Test /-2 Test 7 Test 8 Test From the experiment, the amplitude of the structure was found to be 3.1 V. Table A.11 shows the complete results. From Table A.11, the vibrations in the structure started to decrease then increase by about 1.5 times to 4.48V when the blue string and red strings increased by 2cm while the white and green strings decreased by 2cm. From the condition of the structure, it could be seen that the blue strings were too slack so it was decided that the blue string should be increased by 1cm and the white string decreased by 1cm from the previous configuration. Refer to Table A.12 for the series 94

112 Appendix A of tests conducted with different lengths of blue and white strings and the corresponding vibration amplitude. However in test 1, the vibration in the structure increased even more. The white string in the structure is too slack. Pulling both blue and white strings simultaneously should be considered since each time the blue string is slacked and the white string is pulled, the vibrations in the structure worsens. A second test, test 2 in Table A12 was also conducted with the same lengths for both the blue and white strings. The results of this test were very good as the vibrations reduced by 57% from the initial vibration amplitude of 5.6 V. It was decided then, that the experiment should continue following the procedure in Table A12 with increment and decrement of the red with green strings and the blue with white strings. This is shown in test 3. The experiment was not continued as the vibration in the structure started to worsen. This was because as the experiment continued, the blue string started to slack too much and was unable to tension the structure properly ad thus unable to support the weight of the structure. Diagonal string colour Lengths Table A.12: Amplitude of Vibration with Different Lengths of Strings Test Red Green Blue White Vibration amplitude (V) A.4.2 Discussion From these experiments conducted, it can be summarized that the blue and white strings play an important role in the stiffness of the structure. If the blue string is too slack, the vibration of the structure will increase significantly. There is also much difficulty in changing the shape of the structure due to the restrictions of the black strings (vertical) in the 1 st and 2 nd stage of the structure. The control of the six vertical strings could also be considered to allow more freedom in the structure. The best vibration decrease was obtained when the lengths of both the blue and white strings were the same. However, as both strings will be pulled in opposite directions in 95

113 Appendix A order to reduce the torque, the possibility of pulling and slacking both strings will be examined but it may not be able to be carried out due to the restrictions of the motors used for tensioning the strings in the structure. A.4.3 Options for Vibration Reduction of Structure From analysis of results from the previous experiment, several options were chosen which could possibly reduce the vibrations in the structure. The process of incrementing or decrementing the string lengths need to be efficient in reducing vibration These experiments were conducted to see the results of the following options. 1) Pull blue and white strings simultaneously. 2) Increment/decrement red and green strings twice and increment/decrement blue and white strings once and continue incrementing/decrementing the lengths. 3) Increment/decrement blue and white strings twice and increment/decrement red and green strings once and continue incrementing/decrementing the lengths. The strings were set initially to structure with = 10 and = 35 : The corresponding lengths for these strings are shown in Table A.1. Option 1 This option requires both the blue and white strings to be pulled simultaneously. In this experiment, the lengths of the red and green strings remain the same throughout. The results of the experiment are shown in Table A.13. The initial amplitude of the structure is determined. Then the blue and white strings are decremented by 1cm and the amplitude recorded. The blue and white string is decremented again by another 1cm for both strings (2cm from the initial values). The amplitude of vibration is then measured again. However, the experiment stopped at that point as the structure had too much tension and it was not possible to decrement the lengths of the blue and white strings any further. 96

114 Appendix A Blue/White Table A.13: Amplitude of Vibration using Option 1 Length Increment/Decrement (cm) Red/Green 0/0-1/-1-2/-2 Length Amplitude of Structure (Volts) increment/ 0/0 Test 1 Test 2 Test 3 Decrement (cm) Option 1:15 Hz sine wave Test 1 Test 2 Test 3 1 Amplitude (volts) Time (seconds) Figure A.10: Option 1 Figure A.10 shows the vibration amplitude for option 1 showing the comparisons of the three tests. Discussion Amplitude of the structure is successfully decreased by 53%. The blue and white strings could not be reduced by more than 2 cm due to the limitations in the structure, which were restricted by the saddle strings that were not reduced or increased, and also from the restrictions of the fixed vertical strings. The tension in the strings for a reduction in length of 2 cm is relatively high. From the spring gauge reading, tension required was: Tension = x 60mm 97

115 Appendix A = 36 N Although the amplitude of the vibration decreased, there may not be enough torque in the motor to pull the blue and white strings simultaneously if these two strings are paired on one motor which was the intention initially. Option 2 This option requires red and green strings to be incremented and decremented twice respectively and the blue and white strings to be incremented and decremented once respectively. The experiment will continue until there is too much tension in the structure. The results of the experiment are shown in Table A.14. Table A.14: Amplitude of Vibration using Option 2 Blue/White Length Increment/Decrement (cm) Red/Green 0/0 +1/-1 +2/-2 Length Amplitude of Structure (Volts) increment/ 0/0 Test 1 Test 2 Test 3 Decrement 5.2 N/A N/A (cm) +2/-2 Test 4 Test 5 Test 6 N/A 5.2 N/A +4/-4 Test 7 Test 8 Test 9 N/A N/A 6.6 Note: Test 2,3,6 & 8 not applicable for option 2. Figure A.11 shows the vibration amplitude for option 1 showing the comparisons of the three tests. Discussion This option is not feasible since the vibration of the structure increases even more as the lengths of the strings are respectively increased or decreased. There is a 20% increase in vibration so it was decided that this option was not viable for the structure and was not continued 98

116 Appendix A 4 3 Test 1 Test 5 Test Figure A.11: Option 2 Option 3 This option requires blue and white strings to be incremented and decremented twice respectively and the red and green strings to be incremented and decremented once respectively. The experiment will continue until there is too much tension in the structure. The results of the experiment are shown in Table A.15. Table A.15: Amplitude of Vibration using Option 3 Blue/White Length Increment/Decrement (cm) Red/Green 0/0 +2/-2 +4/-4 Length Amplitude of Structure (Volts) increment/ 0/0 Test 1 Test 2 Test 3 Decrement 5.2 N/A N/A (cm) +1/-1 Test 4 Test 5 Test 6 N/A 3.7 N/A +2/-2 Test 7 Test 8 Test 9 N/A N/A 1.9 Note: Test 2,3,6 & 8 not applicable for option 3. 99

117 Appendix A Figure A.12 shows the vibration amplitude for option 1 showing the comparisons of the three tests. 3 Option 3:15 Hz sine wave Test 1 Test 5 Test Amplitude (volts) Time (seconds) Figure A.12: Option 3 Discussion The amplitude of the structure was successfully decreased by 63%. Compared with the two other options for reducing vibrations. This option (option 3) is the best option with a maximum increase/decrease of 4cm for the diagonal strings (blue and white). 100

118 Appendix B Prototype Designs 101

119 Appendix B B.1 Trial 1: 1 Stage with Two Controlled Strings Strings path: Red string: b Blue string: a Figure B.1: 1-Stage tensegrity with 2 controlled strings Refer to Figure B.1.The controlled strings are string a and b. When these strings are pulled at the same rate, point 4 and 6 moved closer to each other while point 5 moved away from point 4 and 6.This configuration does not give a symmetrical geometry movement in the structure. Thus, a 3-bar with three controlled strings would give a more symmetrical movement. 102

120 Appendix B B.2 Trial 2: 1 stage with three controlled strings Strings path: Red string: a Blue string: b Green string: c Figure B.2: 1-stage tensegrity using three strings Figure B.2 shows the second configuration that was designed using three separate strings. The controlled strings are string a, b and c. When these strings are pulled at the same rate, it was found that points 4, 5 and 6 moved closer towards each other. This configuration has a symmetrical geometrical movement. Hence, it had potential to be upgraded to a 2 stage tensegrity structure. 103

121 Appendix B B.3 Trial 3: 2 stage with six controlled strings -Prototype 1 String path: The paths for the six strings are as follows: 1 st stage: nd stage: Figure B.3: Prototype 1 Figure B.3 shows the first prototype that was built using two stages with three bars in each stage. This structure is a continuation from Trial 2. The controlled strings are 7,8,9,13,14 and 15. Strings 7, 8 and 9 controlled the inclination angle of the first stage and subsequently, the overlap distance between the first stage and the second stage. Strings 13,14 and 15 controlled the inclination angle of the 2nd stage. However, this configuration was lacking the controllability of the angle for both stages. Also, points 4, 5 and 6 in the 2nd stage are able to move freely along the saddle string. 104

122 Appendix B For the construction of the first prototype, the connection ends of the bars were done using i- hooks. From this prototype, it was found that the strings would flow more smoothly if not more than one string was allowed to flow through one i-hook. This concept would be utilized for the next prototype built. B.4 Connection End Design Considerations Figure B.4: Tubing System The tubing system aligned the string configuration between the top and bottom connection in the bar but the movement of the strings was constraint to the diameter of the hole as seen in Figure B.4. The pulley system was then designed. Figure B.5: Pulley system The pulley system design is shown in Figure B.5. This also occurred when using the pulley system as the degree of freedom of the string movement was limited to certain positions in order for the string to stay on the pulley. Both of the joints were smooth but restricted movement. 105

123 Appendix B B.5 Motor Selection Table 4.1 shows the ability of the two motors to fulfill the requirement motion characteristics. Based on those comparisons, the R/C servomotor is chosen. Experiments need to be conducted on the final structure to determine the specification of the servo motor. Table B.1: Comparison between motors Requirement Motion Characteristics R/C Servo Motor Dc Stepping Motor High Torque, Low Speed x x Short, Rapid Repetitive Moves with high dynamic requirements x x Positioning Applications x x Low Speed, High Smoothness x Closed Loop control x Less Expensive x Positioning Accuracy x x Small rotation movement x 106

124 Appendix C Data for Symmetrical Reconfiguration 107

125 Appendix C C.1 MATLAB Code C.1.1 To Determine the Existence of a Configuration %Symmetrical configuration of a 3 bar 2 stage SVD tensegrity structures %Based on Sultan C., Corless M. and Skelton R. E. %Symmetrical reconfiguration of tensegrity structures %published by the Itl. Journal of Solids and Structures 39 (2002) %clear %input l = 0.41 ; b = ; F = *9.81 ; %length of a bar (m) %length between 2 joints at the bottom/above plate (m) %external force in vertical direction (N) %initiate array values for alpha (a), delta (d) and structure height (z) alpha = [ ]; %array of alpha delta = [ ]; %array of delta height = [ ]; %array of structure height A = zeros (3); %set A matrix to zeros initially T = zeros (3,1); %set T matrix to zeros initially fid = fopen('symmetry.out','w') % begin loop for a = 35:5:40 %range of alpha values alpha = [alpha a]; %update array of alpha values a_rad = a*pi/180; %calculate angle in radians for d = 35:5:40 delta = [delta d]; d_rad = d*pi/180; m = l*cos(d_rad); %range of delta velues %update array of delta values %calculate angle in radians %structure height (z) must be more than this value for z = m+0.001:0.1:0.5 height = [height z]; %update array of height values %calculate strings length (Saddle, Vertical, Diagonal) (m) S = sqrt(z^2 + l^2 + 3*l^2*cos(d_rad)^2-4*l*z*cos(d_rad) + b^2/3 2*l*b*sin(d_rad)*cos(a_rad - 30*pi/180)/sqrt(3)); V= sqrt(l^2 + b^2-2*l*b*sin(d_rad)*sin(a_rad + 30*pi/180)); D = sqrt(l^2 + b^2/3 + z^2-2*l*z*cos(d_rad) - 2*l*b*sin(d_rad) *sin(a_rad) /sqrt(3)); %calculate matrix A A(1,1)=(sqrt(3)*(2*z - 3*l*cos(d_rad))*l*sin(d_rad) - l*b*cos(d_rad)*cos(a_rad 30*pi/180))/(sqrt(3)*S); A(2,1)=(l*b*sin(d_rad)*sin(a_rad - 30*pi/180))/(sqrt(3)*S); A(3,1)=(6*z - 12*l*cos(d_rad))/S; A(1,2)=-(l*b*cos(d_rad)*sin(a_rad + 30*pi/180))/V; A(2,2)=-(l*b*sin(d_rad)*cos(a_rad + 30*pi/180))/V; A(3,2)=0; A(1,3)=(sqrt(3)*l*z*sin(d_rad) - l*b*cos(d_rad)*sin(a_rad))/(sqrt(3)*d); A(2,3)=-(l*b*sin(d_rad)*cos(a_rad))/(sqrt(3)*D); A(3,3)=(6*z - 6*l*cos(d_rad))/D; %calculate the determinant of matrix A det(a); %calculate string tension for Saddle, Vertical and Diagonal (N) %T = [Ts ; Tv ; Td] T=inv(A)*[0;0;F]; 108

126 Appendix C fprintf(fid, '%4.0f %4.0f %5.1f %4.1f %3.1f %4.1f %8.2f %8.2f %8.2f %8.2f\n', a, d, z*100, S*100, V*100, D*100, det(a), T(1,1), T(2,1), T(3,1)) end end end st = fclose(fid) Typical Results The following table shows a typical output from the code above Z S V D Det(A) T S T V T D ============================================== where and in degrees; Z, S, V, D in centimeters; T S, T V, T D in Newtons C.1.2 Symmetrical Reconfiguration %symmetrical reconfiguration real time of a 3 bar 2 stage SVD tensegrity structures %Based on Sultan C., Corless M. and Skelton R. E. %Symmetrical reconfiguration of tensegrity structures %published by the Itl. Journal of Solids and Structures 39 (2002) %clear %initial configuration ai= 55 ; %alpha (degrees) di= 39 ; %delta (degrees) %final configuration af= 35 ; %alpha (degrees) df= 44 ; %delta (degrees) %structure geometry z = ; %structure height (m) b= ; %side length of top/bottom triangle plate (m) l= 0.41; %length of bar (m) %External loading in the 3rd/z/vertical axis, +ve up (N) F = *9.81; %end time (secs) t1= 5 ; %initiate time array values time = []; %change degrees to radians aai=ai*pi/180; aaf=af*pi/180; 109

127 Appendix C ddi=di*pi/180; ddf=df*pi/180; fid = fopen('deploy1.out','w') for t = 0:0.01:t1 time = [time t]; %calculate alpha and delta as a function of time (rad) aa = aai + 30*(aaf - aai)*(t^5/30 - t^4*(t-t1)/6 + t^3*(t-t1)^2/3)/t1^5; dd = ddi + 30*(ddf - ddi)*(t^5/30 - t^4*(t-t1)/6 + t^3*(t-t1)^2/3)/t1^5; %calculate strings length (Saddle, Vertical, Diagonal) (m) S=sqrt(z^2 + l^2 + 3*l^2*cos(dd)^2-4*l*z*cos(dd) + b^2/3-2*l*b*sin(dd)*cos(aa - 30*pi/180)/sqrt(3)); V=sqrt(l^2 + b^2-2*l*b*sin(dd)*sin(aa + 30*pi/180)); D=sqrt(l^2 + b^2/3 + z^2-2*l*z*cos(dd) - 2*l*b*sin(dd)*sin(aa)/sqrt(3)); %calculate matrix A A(1,1)=(sqrt(3)*(2*z - 3*l*cos(dd))*l*sin(dd) - l*b*cos(dd)*cos(aa - 30*pi/180))/(sqrt(3)*S); A(2,1)=(l*b*sin(dd)*sin(aa - 30*pi/180))/(sqrt(3)*S); A(3,1)=(6*z - 12*l*cos(dd))/S; A(1,2)=-(l*b*cos(dd)*sin(aa + 30*pi/180))/V; A(2,2)=-(l*b*sin(dd)*cos(aa + 30*pi/180))/V; A(3,2)=0; A(1,3)=(sqrt(3)*l*z*sin(dd) - l*b*cos(dd)*sin(aa))/(sqrt(3)*d); A(2,3)=-(l*b*sin(dd)*cos(aa))/(sqrt(3)*D); A(3,3)=(6*z - 6*l*cos(dd))/D; %calculate the determinant of matrix A %check that determinant A must not be zero det(a); %calculate string tension for Saddle, Vertical and Diagonal (N) %T = [Ts ; Tv ; Td] T=inv(A)*[0;0;F]; fprintf(fid, '%5.3f %4.1f %4.1f %5.1f %5.3f %3.1f %4.1f %8.2f %8.2f %8.2f %8.2f\n', t, aa*180/pi, dd*180/pi, z*100, S*100, V*100, D*100, det(a), T(1,1), T(2,1), T(3,1)) end st = fclose(fid) 110

128 Appendix C Typical Results Time (s) Z (cm) S (cm) V (cm) D (cm) Det (A) Ts (N) Tv (N) Td (N)

129 Appendix C C.2 Pulley size calculation Several factos that had to be taken into account when determining the appropriate pulley size were: Saddle pulley and diagonal pulley are paired together in the opposite direction from each other. [put pic here] Structure is more robust from external load. Maximum pulley radius available was 46 mm. Change in the saddle string length, ds Change in the diagonal string length, dd = 31 mm = 69 mm The diagonal pulley radius was set to 46 mm. The saddle pulley radius can be calculated by ds = r s x S (C.1) Both of the pulleys were paired together. This means the saddle pulley and the diagonal pulley have the same turn. S = D (C.2) Hence, ds rs r D (C.3) dd r S = 20.7 mm Thus; Diagonal pulley radius Saddle pulley radius = 46 mm = 20.7 mm 112

130 Appendix C C.3 Motor torque calculation The torque required for this motion can be found by Torque = Fr (C.4) where F is the maximum tension recorded and r is the pulley radius. Hence, the torque for saddle is Torque S = 7.9 N x 20.7 mm = Nmm = 1.67 kgcm. The torque for the diagonal is Torque D = 5.9 N x 46.0 mm = Nmm = 2.77 kgcm The pairing of diagonal and saddle pulley would mean that the effective torque is less than the calculated value. However, a safety factor of 5 was applied to the design motor toque. This account for the friction in the joints and allow the possibility of moving the structure beyond =55 which was estimated to have higher tension. Design motor torque = 11.6 kg/cm. 113

131

132 Appendix D Controller 115

133 Appendix D D.1 Data for Lookup Table Time (s) Saddle (cm) Time (s) Saddle (cm) Time (s) Saddle (cm) Time (s) Saddle (cm) Time (s) Saddle (cm)

134 Appendix D Time (s) Saddle (cm) Time (s) Saddle (cm) Time (s) Saddle (cm) Time (s) Saddle (cm) Time (s) Saddle (cm)

135

136 Appendix E Prototype Pictures 119

137 Appendix E E.1 Prototype 1 Figure E.1: Prototype 1 120

138 Appendix E Figure E.2: Top half of Prototype 1 Figure E.3: Bottom half of Prototype 1 Figure E.4: View from top of Prototype 1 121

139 Appendix E E.2 Prototype 2 Figure E.5: Prototype 2 122

140 Appendix E E.2.1 Labelled Sections 1, 2, 3: Two fixed strings 5,7,9: 1 fixed string & 3 sliding strings (one adjustable by 1 cm) Figure E.6: Connection at 1,2,3 for Prototype 2 Figure E.7: Connection at 5,7,9 for Prototype 2 4,6,8: 4 fixed strings 10,11,12: 4 sliding strings & 1 fixed string Figure E.8: Connection at 4,6,8 for Prototype 2 Figure E.9: Connection at 10,11,12 for Prototype 2 123