FORM-FINDING OF A COMPLIANT T-4 TENSEGRITY MECHANISM

Transcription

1 FORM-FINDING OF A COMPLIANT T-4 TENSEGRITY MECHANISM By MELVIN MCCRAY IV A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2017

2 2017 Melvin McCray IV

3 To my mother, Jan, my father, Melvin III, and my two sisters Jame and Tyi

4 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Carl Crane, for his continuous guidance and support throughout my time at the University of Florida. I would also like to thank my committee members Dr. John Schueller, Dr. Curtis Taylor, and Dr. Jorg Peters for their assistance during the writing process of my dissertation. 4

5 TABLE OF CONTENTS ACKNOWLEDGMENTS...4 LIST OF TABLES...7 LIST OF FIGURES...9 ABSTRACT...12 CHAPTER 1 INTRODUCTION...13 page Background...13 Focus LITERATURE REVIEW...22 Form-Controlled Method Research...22 Snelson s Heuristic Method...22 Analytical Method...23 Non-Linear Optimization Method...25 Force-Controlled Method Research...26 Force Density Method...27 Analytical Method...29 Energy Method CURRENT RESEARCH...35 Goal...35 Problem Statement...36 Optimization Process Overview...40 Optimization Algorithms...48 SolidWorks Motion Analysis Numerical Case One...53 SolidWorks Motion Analysis Numerical Case Two...55 Forward Position Analysis Process...56 Forward Position Analysis Numerical Case Overview...86 Forward Position Analysis Numerical Case One...89 Forward Position Analysis Numerical Case Two CONCLUSION NUMERICAL CASE VARIABLES LIST OF REFERENCES

6 BIOGRAPHICAL SKETCH

7 LIST OF TABLES Table page 3-1 Tensegrity Spring Variables Bottom Point Coordinates Top Points at Equilibrium for Asymmetric Tensegrity Top Points Defined Using Rotation Angles Tensegrity Case One Results Tensegrity Case Two Results Case Two Spring Constant Values Twist Angles and Corresponding Foundation Nodes Bottom Point Coordinates for Position Analysis Case Elastic Tie Member Lengths Top Non-Elastic Tie Member Lengths Strut Member Lengths Twist Angles and Corresponding Foundation Nodes Coefficients Ai, Bi, Ci, Di and Ei Real Solutions for the Top Four Point Coordinates Tensegrity Case One Results Tensegrity Case Two Results Case Two Spring Constant Values...92 A-1 Values for x, y, z and w variables A-2 Angles for SolidWorks Method Case One for Interior Point and Global Search A-3 Angles for SolidWorks Method Case One for MultiStart A-4 Angles for SolidWorks Method Case Two for Interior Point and Global Search A-5 Angles for SolidWorks Method Case Two for MultiStart

8 A-6 Angles for Position Analysis Case One for Interior Point and Global Search A-7 Angles for Position Analysis Case One for MultiStart A-8 Angles for Position Analysis Case Two for Interior Point and Global Search A-9 Angles for Position Analysis Case Two for MultiStart

9 LIST OF FIGURES Figure page 1-1 R. Buckminster Fuller s Tensegrity System Patent Two Body Configurations X-Piece made by Snelson Polyhedral Tensegrity Models Kurilpa Bridge in Brisbane, Queensland Australia Patent of Snelson s X-Piece Structure Tensegrity depicting node steps and twist angle General Tensegrity Depiction on x, y, z Plane Triangular Tensegrity Prism Two Dimensional Framework T-4 Tensegrity Structure Rotation angles shown on Tensegrity Sum of Forces at Point O T-4 Tensegrity in Solidworks with Highlighted Elastic Side Ties Initial (left) and Final Orientation (right) after SolidWorks Motion Analysis Run Solidworks Motion Analysis Tensegrity Displayed in MATLAB Tensegrity Side and Top views from Solidworks Motion Analysis Local Minimums of Objective Function Tensegrity from Interior Point Algorithm Case One Tensegrity Side and Top Views from Interior Point Algorithm Case One Tensegrity from Multi-Start Algorithm Case One Tensegrity Side and Top Views from Multi-Start Algorithm Case One Tensegrity from Interior Point Algorithm Case Two

10 3-14 Tensegrity Side and Top Views from Interior Point Algorithm Case Two Tensegrity from Multi-Start Algorithm Case Two Tensegrity Side and Top Views from Multi-Start Algorithm Case Two T-4 Tensegrity Structure with x-y Plane T-4 Tensegrity with Top Point Fold Angle Rotation Translation of Directional Vectors at One Point Spatial Closed-Loop Mechanism with Directional Vectors Spatial Closed Loop Mechanism and Equivalent Spherical Quadrilateral Mechanism Spherical Quadrilateral with Corresponding Twist and Joint Angles Spherical Quadrilateral at Point O Spherical Dyad with Directional Vectors and Twist Angles Spherical Quadrilaterals at Foundation Nodes O1 and P Initial Orientation of Strut Member P Orientation after Translation of Point P Angle Between Two Vectors First and Second Rotations of Strut P Third Rotation about m Axis Final Tensegrity Orientations from Solution 2 (left) and Solution 4 (right) Tensegrity from Interior Point Algorithm Case One Tensegrity Side and Top Views from Interior Point Algorithm Case One Tensegrity from Multi-Start Algorithm Case One Tensegrity Side and Top Views from Multi-Start Algorithm Case One Tensegrity from Interior Point Algorithm Case Two Tensegrity Side and Top Views from Interior Point Algorithm Case Two Tensegrity from Multi-Start Algorithm Case Two

11 3-39 Tensegrity Side and Top Views from Multi-Start Algorithm Case Two

12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FORM-FINDING OF A COMPLIANT T-4 TENSEGRITY MECHANISM Chair: Carl Crane Major: Mechanical Engineering By Melvin McCray IV August 2017 Tensegrity structures are defined as unique mechanisms consisting of strut members, elements in a compressive state, and tie members, elements in a tensile state, arranged in an orientation in which the device is in a stable state of self-equilibrium. The process of arranging these elements in geometrically symmetric and asymmetric orientations while simultaneously satisfying several geometric and force constraints for tensegrity mechanisms is known as formfinding. This study begins with an analysis the two main classifications of tensegrity formfinding, form-controlled and force-controlled, along with their corresponding advantages and disadvantages. A tensegrity mechanism with four struts, known as a T-4 tensegrity mechanism, is then analyzed with respect to two new methods that address the disadvantages of several of the form-controlled and force-controlled processes. Computer aided design simulation and a forward position analysis algorithm are used in conjunction with non-linear constrained optimization to create a mathematical model of the T-4 tensegrity mechanism. Numerical examples of the two newly proposed form-finding methods are given not only to illustrate the advantages these processes have over current methods, but also to show how these methods can be used to aid users in creating accurate and stable tensegrity and tensegrity-based mechanisms. 12

13 CHAPTER 1 INTRODUCTION Background Architecture is a field of vital importance that encompasses the subjects of art, engineering, and many other areas of research. It is defined as the study and overall design of a building, structure or system that unifies its elements into a coherent and functional whole ( Architecture, n.d.). All throughout history, these buildings and structures were constructed with a specific function and design intent, and have assisted people in various aspects of society. The functions of architectural structures ranged from serving as tombs for Egyptian pharaohs, as in the case of the great pyramids of Giza in 2560 B.C. (Clayton & Price, 1990) to housing residents and providing restaurants and shopping centers such as the Shanghai Tower in China, one of the world s tallest buildings (Hewitt, 2015). No matter the size of a structure, whether large or small, one of the most important aspects of architectural design is the support and stability of the structures members. Their ability to support each other and obtain a state of stability depends upon the collection of internal and external forces acting within and on the system. These forces are known as compression forces and tension forces. A compression force is a force that results from the application of power or exertion against an object that causes it to be squashed or compacted ( Compression Force, n.d.). The opposite of a compression force is a tension force, which is a force along the length of a medium that pulls or stretches the member ( Normal, Tension, and Other Examples of Forces, n.d.). How these two types of forces affect the material from which structures are made defines their overall strength and stability. Strength, as it relates to architecture, refers to the ability of a structure s material to resist mechanical forces while in use (Jain, 2016). Measuring a materials tensile strength and compressive strength is significant because it determines the maximum 13

14 amount of tensile and compressive loads the material can experience before succumbing to fracture (Merriam-Webster, 2015). These strength properties directly affect the stability of a structure as well as its overall performance. Author Anthony Pugh researched how materials tensile and compressive strengths played a significant role in architectural projects in the past as well as in modern times. Pugh (1976) highlights that in the building industry up until the midnineteenth century, most materials that were used were effective in resisting compressive forces, but few could withstand even small amounts of tensile stress. This prompted architecture researchers to look into the relationship between a system s tensile and compressive components to maximize both the system s stability and performance. One of the leading researchers in this area was Richard Buckminster Fuller. Richard Buckminster Fuller was an American architect and engineer who explored design principles found in nature and used these discoveries to construct structures that balanced compressive and tension forces in a similar manner ( About Fuller, n.d.). Fuller noticed that in nature there existed a balance between tension and compression (Fuller, 1975). In this balance, compressive components were found to be thicker to prevent fracture, where the tensile components were thinner because they only had to be thick enough to withstand the specified load (Pugh, 1976). This characteristic highlighted the efficiency of the tensile elements. Speaking at Black Mountain College during 1947 and 1948, Fuller talked about this relationship found in nature: [The] universe seems to rely on continuous tension to embrace islanded compression elements. We must find a way to model this structural principle (Edmondson, 1987, p. 251). The principle Fuller was speaking of was one he called tensional integrity, and it was this same principle that lead to his invention of the word tensegrity. Fuller stated, Tensegrity is a 14

15 contraction of tensional integrity and that tensegrity structures can accomplish visibly differentiated tension-compression inter-functioning in the same manner that is accomplished by pneumatic structures (Motro, p. 13). Pneumatic structures in this regard are defined as membrane structures stabilized by compressed air (Kuiper, 2009). Changing the pressure level of the air in these pneumatic structures changes their overall shape the same way outside forces encountering a tensegrity change its original orientation. It is this relationship between a tensegrity structures tension and compression components that make the structure so unique. The definition of the term tensegrity has evolved since Fuller first coined the term in the early 1950s and several of these definitions describe the elements of a tensegrity in various ways. When Fuller created the word, he was defining a structure with tensional integrity where integrity, as it relates to structures, refers to the stability of the system. This system is described in further detail by Fuller (1962) in his granted patent of a tensegrity system shown in Figure 1-1, and is defined as follows: Tensegrity systems are spatial reticulate systems in a state of selfstress, tensioned elements have no rigidity in compression and constitute a continuous set. Compressed elements constitute a discontinuous set. Each node receives one and only one compressed element (Skelton & de Oliveira, 2009). The two types of elements described here are the ties, which are the thinner string-like members that are the ones in tension, and the struts, which are the thicker members in compression. This definition highlights the importance of existing tensioned and compressed elements as well as the existence of a state of self-stress, referring to the absence of external influences, in classifying a structure as a tensegrity system. Engineer and author Robert E. Skelton further expounded on this definition of a tensegrity system. Skelton states that a tensegrity system is composed of any given strings connected to a tensegrity configuration of 15

16 rigid bodies, where a tensegrity configuration is a configuration that can be stabilized by some set of internal tensile members that can be connected between the rigid bodies (Skelton & de Oliveira, 2009). An illustration of this tensegrity system is shown in Figure 1-2(c). One of the most significant properties of the tensegrity system described by both Fuller and Skelton is its stability and state of equilibrium. These tensegrity systems are described as being in a stable state of self-equilibrium, which means the tensegrity is in a state in which the structure returns to its original given configuration after the application of arbitrarily small perturbations anywhere within the configuration (Skelton & de Oliveira, 2009). Many tensegrity structures have been created over the last several decades, and the first model was created by American artist Kenneth Snelson. While Fuller patented the first tensegrity system in 1962, it was Snelson, his student, who created the first tensegrity model at Black Mountain College in 1948 (Pars, n.d.). This model, known as the X-Piece, is shown in Figure 1-3. It was an interconnection of several strut-like members and ties that were arranged in a way where the members stabilized the system. Snelson wrote a letter to Rene Motro, one of the leading authors on tensegrities, describing this interconnection between the wood modules and the tie lines in the X-Piece: By replacing clay weights with additional tension lines to stabilize the modules to one another, which I did, making X, kite-like modules out of plywood. Thus, while forfeiting mobility, I managed to gain something even more exotic, solid elements fixed in space, one-to-another, held together only by tension members. (Snelson, 1990) This piece of art became the solution to Fuller s desire to model the structural principle of tensional integrity and spawned the creation of many structures thereafter that used this same principle. 16

17 It is evident from analyzing the tensegrity systems shown in Fuller s patent as well as Snelson s X-Piece model that a tensegrity structure is not limited to a constrained set of tensioned and compressive members. Tensegrity models are also not limited to a constrained set of nodes, which are the points on the model at which the struts and ties intersect. Therefore, because the number of members and nodes of a tensegrity system can vary, there is an infinite number of possible configurations. Fuller s tensegrity system patent denotes one of the simplest forms of the model with a structure that has three struts, six nodes, and several ties threaded throughout, each of the nodes holding the piece together. More advanced models that were later created were known as polyhedral tensegrities and these are shown in Figure 1-4. These models depict several types of polyhedral tensegrities, from left to right, in the tetrahedral, cubic, and octahedral configurations. An alternative way of identifying these tensegrity structures, for simplification purposes, is by a capital T notation followed by the number of strut members in the system (Whittier, 2002). In the case of the tetrahedral tensegrity, it would be classified as a T-6 structure. The three configurations of the polyhedral tensegrity show that by varying the position and number of nodes, struts, and ties, as well as the lengths of each of the members self-equilibrated models, they can be constructed with differing overall heights, lengths, and functions. A case in architecture that displays how this range of overall height, length, and node count can be used for a specific function in a model with a specific function is the Kurilpa Bridge in Brisbane, Australia in the province of Queensland. This is shown in Figure 1-5. The Kurilpa Bridge is a 470-meter-long bridge constructed in 2009 that oversees the Brisbane River (Bunbury, 2016). It is often referred to as the Tensegrity Bridge because it has numerous strut-like members in compression lying in a network of cable tensile members 17

18 oriented so that the system remains in a state of equilibrium. While the Kurilpa Bridge is not considered a true tensegrity structure, due to several struts being attached at the same node, there are large sections that display tensegrity-like properties. In these sections there exist struts and cables that have several functions. The struts and cables can help to resist wind loads and earthquake loads and they laterally restrain the tops of the primary and secondary bridge masts which prevent the masts from buckling ( Kurilpa bridge by Cox Rayner architects, 2011). Focus Fuller s patented tensegrity system, Snelson s X-Piece, and Brisbane s Kurilpa Bridge provide a glimpse into the infinite range of possible tensegrity models that can be created by changing the orientation of the compressive and tensile members of the system. The study of what these orientations are and how they can be created has been researched for many years and falls into a category known as form-finding, which is a process where a tensegrity system is created by satisfying both geometric and stress constraints. For the structure to be considered a tensegrity system, the geometry of the model must adhere to the characterization of either a class 1 or a class k tensegrity system. A class 1 tensegrity system is a tensegrity configuration that has no contacts among any of its rigid bodies, which means only one strut will be allowed to be present at each node, and a class k tensegrity system has as many as k rigid bodies in contact with each other (Skelton & de Oliveira, 2009). The stress constraints refer to pre-stressability properties of the system. Regarding these properties, the internal forces in the strut members must be in compression, the internal forces in the tie members must be in tension, and the summation of the forces at each of the nodes of the tensegrity must be equal to zero when the system is not in contact with any outside forces. The forces in each of the members are shown to abide by these properties by having the compressive force values show as negative numbers and the tensile force values as positive numbers. 18

19 Although both geometric and stress constraints should be considered in the construction of a tensegrity system, form-finding processes are initially driven either by geometry or force equilibrium parameters. Consequently, they can be put into two categories: form-controlled or kinematic methods and force-controlled or statical methods. Three significant form-controlled methods include Kenneth Snelson s heuristic method, an analytical method, and a non-linear programming method. Three major force-controlled methods include the force density method, an analytical method, and an energy method. The form controlled methods have proved to be ideal when the configuration of the tensegrity system is mostly known, while the force-controlled methods are more suitable when new configurations are being explored. These kinematic and static methods will be analyzed in the following literature review section. Afterwards, an alternative method tying in facets from both the form-controlled and force-controlled methods via non-linear optimization will be presented along with a numerical example applying this method to the T-4 tensegrity structure. 19

20 Figure 1-1. R. Buckminster Fuller s Tensegrity System Patent, Reproduced from Fuller (1962) Figure 1-2. Two Body Configurations, Reproduced from Skelton and de Oliveira (2009) Figure 1-3. X-Piece made by Snelson, Reproduced from Pars (n.d.) 20

21 Figure 1-4. Polyhedral Tensegrity Models, Reproduced from Zhang, Li, Cao, Feng, and Gao (2012) Figure 1-5. Kurilpa Bridge in Brisbane, Queensland Australia, Reproduced from Corp (2006) 21

22 CHAPTER 2 LITERATURE REVIEW Form-Controlled Method Research Form-controlled processes were the first kinds of methods used to construct tensegrity systems. These kinematic methods are defined as processes where a tensegrity system is created by changing the orientation of the structures tensile and compressive components until the structure is stable and fits the geometric properties of a class 1 or class k tensegrity system. These ends are achieved in the following ways: (a) through a purely trial-and-error approach experimenting with the orientation of the systems struts and ties, (b) by keeping the length of the strut components constant while the tie lengths are minimized, or (c) by keeping the length of the tie components constant while the strut lengths are maximized. Snelson s Heuristic Method The first method ever employed to create a tensegrity system was an experimental one used by Snelson in 1948 (Gough, 2005). When using this method, a tensegrity is built by starting with a set of rigid compressive components and adding a network of ties around them. In that way, each node of the structure consists of one strut and is self-stabilizing in the absence of outside forces. How this method is implemented in practice can be seen in Snelson s patent of the X-piece tensegrity structure. Snelson created this structure by starting with two separate X- shaped modules that were built by crossing two compression members at an intermediary point. The tie elements were then connected to each of the four corners of the X-shaped component creating four edges on each module. One tensile edge was removed from each module, labeled 23 and 23 in Figure 2-1, and the two modules were attached by adding tensile components labeled 24, 24, 26 and 26 to create a rhombus of cables between the two pieces (Motro, 2003). This arrangement of 22

23 components forms a self-supporting unit because of the outer ends of the compressive members being pulled towards adjacent ends by tension members, which resulted in a continuous tension network (Snelson, 1965). The choice to eliminate two of the tensile edges from the structure and connect both modules through a rhombus of cables is key in the construction of this tensegrity. This choice allows for a system to be created that is not only self-stabilizing but also falls under the category of a class 1 tensegrity because at each node, only one compressive member is present in a network of surrounding ties that are in tension. Although this method proved to be beneficial for experienced sculptors such as Snelson, key decisions necessary to build tensegrity sculptures this way are instinctive. Therefore, a specific algorithm cannot be extrapolated to be used as a frame of reference for others. Analytical Method Another form-controlled method widely used in the building of tensegrity systems is known as the analytical method. In this method, a tensegrity configuration is found by either keeping the length of the strut components constant while varying the length of the ties or keeping the length of the tie components constant while varying the lengths of the struts. Mathematicians Robert Connelly and Maria Terrell explored this method when they analyzed the case of form finding for a symmetrical tensegrity structure when the lengths of the cables were kept constant while the lengths of the struts were maximized (Connelly & Terrell, 1995). Variables were used to describe the tensegrity system so that general equations could be formulated. The value n denotes the number of vertices on the top or bottom platform, with a total of 2n vertices altogether due to symmetry, the j and k variables represent the node steps for the bottom and top of the structure, respectively, and θ is the twist angle that shows how much each strut is rotated from an initial vertical position. This twist angle and the top of the tensegrity with 23

24 five node points labeled pa, pb, pc, pd and pi are shown in Figure 2-2 where the element between points pa and pi is a strut and the element between points pb and pi is a tie. For simplicity, the two circles that define the top and bottom of the tensegrity structure are unit circles, with a radius of 1 and a height in the z direction of -1. Therefore, the top five node points have the following coordinates in the x, y, and z plane: pi = (1, 0, 0) pa = (cos θ, sin θ, -1) pb = (cos (θ- 2πj n pc = (cos ( 2πk n pd = (cos ( 2πk n 2πj ), sin (θ- ), -1) n 2πk ), sin ( ), 0) n 2πk ), -sin ( ), 0) n Pellegrino used these same coordinates and defined the length of both the struts and the ties in a more generalized form by using the variable R to define the radius of the bottom and top circles, H to define the vertical distance between the bottom and top of the structure, v as the number of polygon edges, and j as an integer smaller than v as follows: Ltie 2 = 2R 2 *(1-cos θ) + H 2 (2-1) Lstrut 2 = 2R 2 * [1- cos (θ+ 2πj v )] + H2 (2-2) Using both Equations 2-1 and 2-2 and keeping all the cables lengths constant, the length of the struts will be at its maximum value when the twist angle theta is as follows: θ = π * ( 1 2 j v ) (2-3) 24

25 Variables such as theta and the twist angle between the ends of the struts are the same in all members of the structure due to the symmetrical nature of the system. However, when the tensegrity is not symmetrical, defining the model becomes much more difficult as each member must be defined by several additional variables. This case creates many equations that make the solving for a tensegrity configuration impractical via the analytical method. Non-Linear Optimization Method The non-linear optimization method is an approach that was used by Pellegrino in which a mathematical minimization problem was constructed to either maximize the lengths of the struts or minimize the lengths of the ties of a structure with symmetrical elements subject to a set of constraints (Pellegrino, 1986). A constrained optimization problem minimizes an objective function f(x) and constrains it, subject to the following set of equality and inequality constraints h(x) = 0 and g(x) 0 (Bonnans et al., 2006). In the case of finding a tensegrity configuration of a symmetrical model in which the lengths of the ties remain constant, the length of the struts can be maximized by minimizing the negative length of one of the struts. This problem was analyzed for a numerical case where a symmetrical triangular tensegrity prism was modeled so that the bottom three nodes 4, 5 and 6, shown in Figure 2-4, were fixed and the negative length of strut 1 was minimized (Tibert & Pellegrino, 2011). No inequality constraints need to be defined for this problem, and therefore the negative strut length is minimized subject to an equality constraints set for the lengths of the struts and the ties. The optimization problem is then set up in the following way: Minimize f(x) = -Lstrut1 2 Subject to: L tie(n) 2 - a =0 and L strut(j+1) 2 -L strut(j) 2 =0 25

26 Here N goes from 1 to 6, labeling the 6 ties on the top and side of the tensegrity, J goes from 1 to 2 so that each of the three struts are represented and a is the length of the nine ties. This nonlinear constrained optimization problem can be solved by using minimization algorithms found in MATLAB software. The version of MATLAB used for all computations is the MATLAB R2016a edition. This non-linear optimization method is a useful technique to use on tensegrity structures with a relatively low number of elements and high degrees of symmetry. However, it is not practical in the cases with many elements or models that are not symmetric because an increasingly large number of constraint equations and variables must be produced to define the system. Each of the three form-controlled methods show that a set of tensile and compressive elements can be oriented to satisfy the geometric properties of a tensegrity system. However, not only are these methods only feasible for systems with a low number of elements and a high degree of symmetry, but also, except for Snelson s heuristic method, there is no way of enforcing the force constraints that are required for all tensegrity structures. These constraints are merely listed as assumptions in which the user assumes the elements conform to the properties of prestressability. Force-Controlled Method Research The second category of form-finding is known as force-controlled methods. These processes focus on ensuring that the properties of pre-stressability are satisfied in addition to adhering to the geometric constraints for a tensegrity. By solving for a tensegrity configuration, the user guarantees that all of the strut members are in compression, the ties are in tension, and the model conforms to the definition of a class 1 or a class k tensegrity system. By analyzing several widely used statical methods it is evident that while these processes are able to ensure force and geometry constraints, the more complicated the design becomes, the more difficult it is 26

27 to derive an algorithm to solve for a tensegrity configuration. The computational time needed for computers to solve complex form-finding problems greatly increases as the elements of a tensegrity increase. The problems do not always converge to a solution. However, for smaller scale tensegrity problems, force-controlled methods can prove to be very useful and efficient. Force Density Method The force density method is a widely known force-controlled process that defines a tensegrity system through variables called force densities or tension coefficients. These tension coefficients are a ratio of force per unit length and can be used to simplify the equilibrium equations at each node of the structure (Megson, 2014). Mathematicians Linkwitz and Schek first defined the summation of forces at each node through the following non-linear force equilibrium Equation 2-4 (Schek, 1974) where i and j subscripts denote each end of the structures element, fix denotes the forces in the x direction at node i, tij is the axial force along the element with endpoints i and j, and xi and xj are the x components of the coordinates at node i and j, respectively (Linkwitz, 1999). t ij j (x l i x j ) = f ix (2-4) ij A variable qij is defined as the tension coefficient along the element with endpoints i and j in the following way: q ij = t ij l ij (2-5) 27

28 Equation 2-4 was reduced from a non-linear equation to a linear one. This process was repeated for the equilibrium equations at each node in the y and z directions as well. To create a general and complete form finding problem describing a structure that has b elements and n nodes, the equilibrium equations in the x direction, which are also written in a similar fashion for the y and z directions, can be written in matrix form as follows: C T QCx = fx (2-6) In this equation, C is called the incidence matrix with b n dimensions that define how the elements of the structure are connected. Elements in this matrix from an element k that have the nodal endpoint i and j are defined with values of -1 and 1 to represent a member in compression and tension, respectively. These variables are defined as follows: c ki = -1 c kj = 1 (2-7) Here i is less than j. Q is defined as a diagonal matrix containing b number of force densities defined by Equation 2-5, x is an n 1 vector containing the x coordinates of the nodes, and fx is the n 1 vector containing the external forces in the x direction. Because the initial state of a tensegrity is defined in a way in which no outside forces are in contact with it, the right side of Equation 2-6 is set to zero. Vassart and Motro manipulated the left side of the equilibrium equation and assigned it to an n n matrix D, known as the force density matrix, resulting in the following: D x = 0 (2-8) 28

29 Here each variable in D is defined by qij or qik, or zeros, depending on whether the element was one in compression or tension (Vassart & Motro, 1999). A system with all tension components and fixed bottom nodes was found by Schek to yield a force density matrix that was positive-definite and therefore invertible, making the finding of a solution for Equation 2-8 feasible (Schek, 1974). However, problems of form finding for tensegrity structures arise because not only are the bottom nodes not fixed, but also the force density matrix is semidefinite. To find an orientation for the tensegrity that makes it super-stable, the nullity N of the matrix D has to be equal to d+1 where d is the dimension of space (Tibert & Pellegrino, 2011). Super-stability in this sense was described by Connelly as the strongest type of pre-stress stability where the force density matrix is both positive semi-definite as well as at maximum rank (Connelly, 1999). Satisfying this condition for pre-stress stability by altering the force densities is either done through a trial and error process, analytically, or through an iterative method (Vassart & Motro, 1999). The force-density method has been found to be ideal for finding new configurations of tensegrity structures, but not as practical a method for structures with a large number of nodes or structures that do not display symmetry (Schenk, 2006). Analytical Method The analytical method is another force-controlled process that can take advantage of the force density concept. By using force densities and setting up summation of force equations at the nodes of the tensegrity structure, linear equilibrium equations can be constructed. Connelly and Terrell used this method of form finding to analyze tensegrity structures that were rotationally symmetric in which the twist angle for each type of element was the same. In analyzing the structure shown in Figure 2-3 where all members are connected at node i, the summation of forces in both the y and z directions yield the following equations: 29

30 qi,ah + qi,bh = 0 (2-9) qi,arsinθ + qi,brsin(θ + 2πj v ) = 0 (2-10) H represents the distance between the two specified points taken from the y and z components of the coordinates previously defined in the kinematical analytical method. It should be noted that Equation 2-9 is a simplified form of the equilibrium equation at node i because qi,c and qi,d are equivalent as a result of the symmetrical geometry of the tensegrity. Equations 2-9 and 2-10 are combined into one equation and the twist angle θ is found. Connelly and Terrell found that the only solution for this equation where all the cables are in tension is the following: θ = π ( 1 2 j v ) (2-11) Energy Method Connelly (1982) proposed a force-controlled method that analyzed the total energy in a rigid framework to find a system in a stable tensegrity configuration. In this method, an energy function is defined for the purpose of describing a tensegrity system where the goal is to minimize the total energy in the rigid framework. The framework being referenced is labeled as G and is defined as an abstract finite graph made up of nodal points, pi and pj, and edges that are either a cable, strut, or a rod. Each of these edges represents an element of the structure and is assigned a stress variable w, which is equivalent to the tension coefficients described previously, such that the following equation holds true: j w ij (p i p j ) = 0 (2-12) 30

31 It should be noted that this relationship is identical to the force equilibrium equation shown in Equation 2-4 with the force on the right side equal to zero. The energy of this framework is mathematically modeled through a function E(p) shown in the following two equations: E(p) = 1 w 2 2 ij ij(p i p j ) (2-13) E(p) = 1 2 pt Ω p (2-14) Here Equation 2-14 is the quadratic form of the energy function in which p is a column vector that contains the x, y, and z coordinates of each point and Ω is the stress matrix that is also equivalent to the force density matrix. For the rigid framework in the orientation p to satisfy the conditions for super stability, the energy function E(p) has to have a local minimum at that orientation and contain a set of force densities so that the stress matrix Ω has a nullity of the order d+1. Connelly (1982) analyzed the two-dimensional framework shown in Figure 2-5. In this case, to find a super stable tensegrity orientation, the stress matrix has to be found with a nullity of 3 that is positive semi-definite. The resulting stress matrix was found to be the following (Connelly, 1982) where the stress in the cables are 1 and the stress in the struts are -1: Ω = (2-15) This solution satisfies both constraints for super-stability. Therefore, a tensegrity configuration has been found that minimized the energy of the system. The previous example as well as other 31

32 cases of symmetrical tensegrities have been analyzed and it can be concluded that although solutions are feasible for structures with few elements having the same tension coefficients, larger more complicated structures that are irregular in shape make this energy method an infeasible one. 32

33 Figure 2-1. Patent of Snelson s X-Piece Structure, Reproduced from (Snelson, 1965) Figure 2-2. Tensegrity depicting node steps and twist angle. Reproduced from Connelly and Terrell (1995) Figure 2-3. General Tensegrity Depiction on x, y, z Plane, Reproduced from (Tibert & Pellegrino, 2011) 33

34 Figure 2-4. Triangular Tensegrity Prism, Reproduced from (Tibert & Pellegrino, 2011) Figure 2-5. Two Dimensional Framework, Reproduced from Connelly (1982) 34

35 CHAPTER 3 CURRENT RESEARCH Goal The two previously reviewed categories of form-controlled and force-controlled tensegrity form finding show that several advantages and disadvantages exist in every process. In the case of the form-controlled methods there are many possible geometric configurations that can be found. Still, success in finding a tensegrity configuration is highly dependent upon the user initially knowing most of the geometric parameters of the structure. Also, except for Snelson s heuristic method, form-controlled methods often have no way of directly controlling the variation of the forces in each element. Therefore, the properties of pre-stressability are only assumed and not enforced. In the case of the force-controlled methods, the forces in each element are controlled throughout the whole form finding process and therefore the properties of pre-stressability are always satisfied. However, these methods are regarded as feasible when they are applied to simplistic or geometrically symmetric tensegrity structures. These structures contain compressive and tensile elements with the same properties and geometrical orientation and therefore satisfying equilibrium at only one node is assumed to satisfy the conditions of equilibrium for the entire model. When the geometry of the tensegrity structure is irregular the number of equilibrium equations needed to define the model increases significantly, it becomes more difficult to define and constrain the lengths of the elements during the form-finding process and knowledge of a starting position that is close to equilibrium becomes essential. The disadvantages previously discussed place significant limits on the type of tensegrity structure that can be feasibly produced via form-finding. While many other methods of kinematic and static form-finding exist, many of them exhibit the same shortcomings. Our research focused on creating a method that improves on these shortcomings where a user has the ability to create 35

36 symmetric and non-symmetric tensegrity models. Simultaneously, the user has control over all the compressive and tensile elements force and length properties while satisfying both the geometric and pre-stressability conditions of a class k tensegrity. This process is defined as a force-driven non-linear optimization method, and it is outlined as a constrained minimization problem. The setup of this algorithm is like the non-linear programming method described in the literature review section. However, the axial forces in each member of the structure are now explicitly defined. Also, instead of focusing on maximizing or minimizing the length of a specified strut or tie, the sum of the forces at each node in the x, y, and z direction is minimized subject to several constraints set on the lengths and properties of each member. This minimization is carried out using MATLAB s optimization solvers to find the equilibrium positions of a T-4 tensegrity structure. Not only will these equilibrium positions will be solved for but also the effects that constraints and minimization algorithms have on the solution as well as computation time will be analyzed. Problem Statement The configuration of the tensegrity that is analyzed here is shown in Figure 3-1. The top and bottom of the tensegrity consists of a total of eight points. The bottom of the device is labeled as points O1, P1, Q1, and R1. The top of the device is labeled as points O2, P2, Q2, and R2. There are a total of twelve ties that include four elastic ties and eight non-elastic ties. The elastic ties fall along points P1O2, Q1P2, R1Q2, and O1R2. The other 8 ties are non-elastic and make up the bottom and top of the tensegrity. For this analysis, it is assumed that the four base points, O1, P1, Q1, and R1 are fixed. The problem at hand is to determine the equilibrium configuration of the mechanism when given the base point locations, the strut and tie lengths, and the spring constants and free lengths of the elastic ties. Each of the eight points illustrated on the tensegrity 36

37 in Figure 3-1 represents components with x, y, and z coordinates. Because only the top four points are allowed to vary, the tensegrity configuration that is closest to equilibrium is defined by the twelve coordinates of these four points. A large scale iteration approach is undertaken to obtain the equilibrium configuration. To increase efficiency in solving this problem, the twelve output variables can be reduced to eight by representing each of the top four coordinate points by rotation angles, for the length of each strut is known. These rotation angles are defined in the following manner: each of the four struts is aligned along the x axis and then rotated about the z axis and the y axis, respectively, as the bottom four points remain constant. The rotation angle about the z axis is defined as alpha and the rotation about the modified y axis is defined as beta. Figure 3-2 illustrates how this rotation is done with strut O. The bottom point O1 remains fixed and the top point is allowed to move as the rotation angles vary. The transformation of any point in one coordinate system to a reference coordinate system can be found when the relative position and orientation of the pair of coordinate systems are known (Crane & Duffy, 1998). Compound transformations define instances where multiple translations and rotations can take place to define the initial and final coordinates of a given point. Equation 3-1 defines this transformation for the top four points as follows: A P1 = TTr*Tα*Tβ* TTr* * B P1 (3-1) A P1 and B P1 represent the end points of the strut members. B P1 is the variable that defines the top four points before the rotations and A P1 defines the top four points seen in Figure 3-1 after the rotations. In Equation 3-1, TTr, TTr*, Tα, and Tβ are 4 4 transformation matrices. TTr and TTr* represent the translation that occurs to go from the origin to each of the bottom four points on the 37

38 tensegrity. In Figure 3-1, the origin is defined as the point O1. The two transformation matrices that will be used are defined as follows: TTr =[ B x B y ] (3-2) B z TTr*= [ B x B y ] (3-3) B z Here the variables Bx, By, and Bz are the x, y, and z coordinates of the bottom four points. Because the origin lies at O1, this matrix will be equal to the identity matrix for the first point. The second two matrices Tα and Tβ are defined as follows: cos (α) sin (α) 0 0 sin (α) cos (α) 0 0 Tα = [ ] (3-4) cos (β) 0 sin (β) Tβ = [ ] (3-5) sin (β) 0 cos (β) This last matrix identified as Tβ is defined as the rotation matrix about the negative y axis as the positive y axis is defined as going from point O1 to point R1. Now given any two points along each end of the struts, the rotation angles about the y and z axis can be solved for. 38

39 The configuration of the tensegrity that is the closest to equilibrium is solved for two cases: one where the forces in the elastic ties are constants defined by constant k values and the other where all four k values can change throughout the form-finding process. These two cases are chosen for the following two purposes: to show tensegrity equilibrium configurations can be found in cases where the user decides to explicitly define a set of properties of the tensegrity as well as cases where these properties are not given, and to show if restricting the properties of the tensegrity s elements has any effect on how stable the resulting equilibrium orientation is. The problem statement for case one is presented as follows: Given: 1. O1, P1, Q1, and R1 (bottom four coordinate points) 2. L1, L2, L3, and L4 (length of the four strut members) 3. k1, LO1 ( spring constant and free length for elastic tie 1) 4. k2, LO2 ( spring constant and free length for elastic tie 2) 5. k3, LO3 ( spring constant and free length for elastic tie 3) 6. k4, LO4 ( spring constant and free length for elastic tie 4) 7. O2, P2, Q2, and R2 (initial guess for the top four coordinate points) Find: The tensegrity configuration that is the closest to equilibrium. For case two Given: 1. O1, P1, Q1, and R1 (bottom four coordinate points) 2. L1, L2, L3, and L4 (length of the four strut members) 3. O2, P2, Q2, and R2 (initial for the top four coordinate points) Find: The tensegrity configuration that is the closest to equilibrium 39

40 Optimization Process Overview The solution for the stated problem was solved for using a combination of systems of equations involving the forces of the tensegrity members, the given point coordinates, the rotation angles, and mathematical optimization solving for local and global minimum values. The members of the tensegrity who length cannot change include the struts and the non-elastic ties. The lengths of the struts are defined by first stating the numerical coordinates of the bottom four points and the top four points. The bottom four points will remain constant throughout the problem as the top four points will be allowed to change as long as each of the strut lengths remain constant. The forces in the struts and non-elastic ties are labeled as constant variables and the numerical values for each of the forces in the elastic ties are entered using the following equation for force in a spring: Fspring = k*(l-lo) (3-6) Then, the sum of forces at each of the top coordinate points of the tensegrity is taken. A simplified form of the summation equation used is the following: f x Forces at Node j = Fi* f y (3-7) f z In this equation, the variable defined as j denotes each of the top points of the tensegrity, Fi is the magnitude of the force in each member, and the vector with components fx, fy, and fz is the unit vector along each member of the tensegrity. Because the summation of forces at each of the top points will be zero when the tensegrity is in equilibrium, Equation 3-7 will be set to a value of 40

41 zero. Figure 3-3 illustrates this sum at the top point O2 of the tensegrity. The sum of forces at the other three top points is defined in the same manner. The force in the top tie is in tension, and therefore the force vectors are moving away from each other along the same axis the tie lies on, which satisfies the first two conditions for equilibrium where internal forces must be opposite in sense and collinearity (Seely & Ensign, 1921). The next property that needs to be satisfied for the mechanism to be in equilibrium is that the forces in each of the members have to be equal in magnitude (Hibbler, 2013). The struts and elastic ties internal forces are defined as uniform and thus satisfy this last condition of equilibrium. Therefore, if the internal forces of the top ties are equal in magnitude, this condition for equilibrium will be satisfied. To ensure the internal forces of the top ties are equal in magnitude, a constrained function was devised to minimize the sum of the forces at each of the top nodes. Simultaneously, the difference between the internal forces in each of the top ties was kept equal to zero, which was done through the use of constraints via mathematical optimization. This mathematical minimization problem was created in MATLAB by first defining an objective function f(x) where x is a design variable that defines the orientation of the tensegrity subject to both equality and inequality constraints set on the system. The design vector for case one will differ in structure from case two. Case one will output a design vector that includes the eight rotation angles that describe the orientation of the tensegrity, and both case two and case three will output a design vector that includes the eight rotation angles as well as four k values associated with the four elastic ties. The problem is set up in the following way for the irregular tensegrity cases one and two: 41

42 Minimize f(x) = N j L 2 tj l tj = 0 T jk T jk = 0 τ Case one subject to: l < β ij < τ u γ l < T jk < γ u γ l < L ij < γ u { γ l < S ij < γ u L 2 tj l tj = 0 T jk T jk = 0 σ l < k ij < σ u Case two subject to: τ l < β v < τ u γ l < T jk < γ u γ l < L ij < γ u { γ l < S ij < γ u Here, the variables τ l and τ u represent the lower and upper limits that the beta angle is allowed to vary in between, γ l and γ u are the lower and upper limits of the force magnitudes in each member, l tj is the length in each of the top ties, Ltj is the square root of the sum of squares between the end point coordinates of the top ties, the force magnitudes in the elastic tie members, non-elastic tie members, and the strut members are noted as Lij, Tjk, and T jk and Sij, respectively where the number of bottom points of the tensegrity are defined as i, going from 1 to 4 in the T-4 tensegrity case and the number of top points are defined as j and the number of the second node in the top tie is k, going from j+1 to n where n is the total number of top nodes. N j is the summation of the forces at each top node j. Lastly σ l and σ u define the lower and upper limits of the spring constant variable. Placing these lower and upper limits on these variables ensures that during the form-finding process, the numerical value of zero is not assigned to any constant and 42

43 it also ensures that the force magnitude in the compressive elements are negative and the force magnitudes in the tensile elements are positive. Because the function defines the position of the tensegrity in terms of the sines and cosines of the rotation angles, it is characterized as a non-linear optimization problem. Several algorithmic solvers can be chosen from the MATLAB optimization toolbox to minimize the objective function, and they will be discussed in further detail in the following optimization algorithm section. This software is used to ensure that the difference between the internal forces in the top ties of the tensegrity is equal to zero and therefore satisfies the third condition for equilibrium. The internal forces in the top ties are labeled as FtopO2P2a, FtopO2P2b, FtopR2O2a, FtopR2O2b, FtopP2Q2a, FtopP2Q2b, FtopQ2R2a, and FtopQ2R2b. The subscript notation denotes the two points at each end of the top tie. Figure 3-3 shows how the internal forces in the top tie between points O2 and P2 are oriented. The force in the elastic tie going from point P1 to O2 is defined using the following variables: kpo = 1 lbf/in, LOPO = 1 in (3-8) o2x p1x VPO = o2y p1y, LPO = norm (VPO) (3-9) o2z p1z Now the force in the elastic tie going from point P1 to O2 is noted as follows: FsideP1O2 = kpo * (LPO LOPO) (3-10) 43

44 The forces in the three-remaining side elastic ties are defined in the same way and are labeled as FsideQ1P2, FsideR1Q2, and FsideO1R2. The variables used to define each of those forces include the following: kqp = 2 lbf/in, LOQP = 1 in (3-11) krq = 1.8 lbf/in, LORQ = 1 in (3-12) kor = 1.3 lbf/in, LOOR =.5 in (3-13) The forces in each of the four struts are constants denoted using similar notation and are labeled FstrutO, FstrutP, FstrutQ and FstrutR. Now all the tensegrity s forces and corresponding variables are defined and Equation 3-7 is used to define the forces in the struts and elastic ties in terms of the given point coordinates and the forces in the side elastic ties that have been defined as constants. By using the force components in the x, y, and z coordinates, there will be three equations and three unknowns at each of the top four points of the tensegrity. The three unknowns will include the two forces in the top ties and the force in the strut at each point. The system of equations, an arrangement of Equation 3-7 with the tensegrity variables, can be written in general form shown in Equations 3-14 through c s L ij (px j px i ) U ij c s L ij (py j py i ) U ij + c s T jk (qx k qx j ) V jk + c s T jk (qy k qy j ) V jk + c s S ij (rx j rx i ) W ij + c s S ij (ry j ry i ) W ij + c s T jk (tx k tx j ) Y jk + c s T jk (ty k ty j ) Y jk = 0 (3-14) = 0 (3-15) c s L ij (pz j pz i ) U ij + c s T jk (qz k qz j ) V jk + c s S ij (rz j rz i ) W ij + c s T jk (tz k tz j ) Y jk = 0 (3-16) 44

45 In this system of equations, the points px, py, and pz are the x, y, and z coordinates of the points that are located at each end of the specified elastic tie, and the fraction term in each equation is taken from the directional cosines of the corresponding line (Crane, Rico, & Duffy, 2012). The points that represent the ends of the non-elastic ties and the strut members are denoted in the same way using the variables q, r, and t. The vectors Uij, Vjk, Wij, and Yjk represent the forces in the elastic ties, non-elastic ties, and the strut members at each top node. It can be seen from Equations 3-14 to 3-16 as well as Figure 3-3 that each top node will have two non-elastic ties, one elastic tie, and one strut represented at each nodes system of equations. Lastly cs is a constant defined as either 1 or -1 depending on if the force vector is in the same or opposite directions of the unit vector along each member. In this system of equations, the three unknowns are the two non-elastic tie force magnitudes and the strut force magnitude. At the top node O2 these magnitudes are listed as FtopO2P2a, FtopR2O2a, and FstrutO. These three equations and three unknowns are then solved for using MATLAB and expressed in terms of the top coordinates of the tensegrity and the variable FsideP1O2, which has already been defined in terms of given numerical values. These steps are repeated for the system of equations associated with the remaining three top points P2, Q2, and R2. Now all of the forces in the struts and the top ties are expressed in terms of the top coordinates of the tensegrity and the forces in the side ties. Defining the forces in this way ensures that the summation at each of the top points of the tensegrity is zero and satisfies the fourth and last condition for equilibrium. To define an ideal starting position for the coordinates of the top points, the motion analysis in the 3D CAD software SolidWorks is used. The version used is the SolidWorks 2016 x64 Edition. Figure 3-4 represents a tensegrity model in SolidWorks that is used to get the 45

46 orientation of a four-strut tensegrity in equilibrium. In this model, the members are fixed to each of the eight points on the bottom and top of the tensegrity. Numerical values for the k constants and free lengths are entered into the motion analysis toolbox. These values are in Table 3-1. Table 3-1. Tensegrity Spring Variables Spring Member k LO (in) (lbf/in) P1O2 1 1 Q1P2 2 1 R1Q O1R The coordinates for the bottom four points of the tensegrity are presented in Table 3-2 in units of inches: Table 3-2: Bottom Point Coordinates Bottom Points Coordinates O1 [0,0,0] P1 [9,0,0] Q1 [6,7,0] R1 [-1,8,0] The values for the lengths of the struts O, P, Q, and R are 12 in, 14 in, 12 in, and 10 in, respectively. The lengths of the top ties going from R2O2, O2P2, P2Q2, and Q2R2 are 4 in, 5 in, 5 in, and 6 in respectively. After numerical values were entered in for the top four points, a damper was selected for each of the four elastic ties so ensure that motion of the springs would eventually stop during the motion analysis. The motion study in Solidworks was then run and the values for the top four points of the tensegrity in equilibrium were as follows: 46

47 Table 3-3. Top Points at Equilibrium for Asymmetric Tensegrity Top Points Coordinates O2 [ , , ] P2 [.92101, , ] Q2 [ , , ] R2 [.56521, , ] Now that all variables associated with the tensegrity have been defined in terms of constants and the x, y, and z coordinates of the top and bottom points, the coordinates are then written in terms of the rotation angles alpha and beta, as shown in Table 3-4. Table 3-4. Top Points Defined Using Rotation Angles Point Coordinate Variable in terms of rotation angles o2x 12*cos(α1)*cos(β1) o2y 12*cos( β1)*sin( α1) o2z 12*sin( β1) p2x 14*cos( α2)*cos( β2) + 9 p2y 14*cos( β2)*sin( α2) p2z 14*sin( β2) q2x 12*cos( α3)*cos( β3) + 6 q2y 12*cos( β3)*sin( α3) + 7 q2z 12*sin( β3) r2x 10*cos( α4)*cos( β4) 1 r2y 10*cos( β4)*sin( α4) + 8 r2z 10*sin( β4) 47

48 The variables in terms of the rotation angles shown in Table 3-4 are set equal to the top coordinate values from Table 3-3, and the four systems of equations are solved to get the values for the alpha and beta angles. These rotation angle values are used to define the tensegrity at the starting point of the optimization process for all three cases. This starting orientation is shown in Figures 3-6 and 3-7 where the red members are the elastic ties, the blue members are the struts, the yellow members are the non-elastic top ties, and the black members are the non-elastic bottom ties. The angles α1 and β1 used to define the tensegrity correspond to point O2, α2 and β2 correspond to point P2, α3 and β3 correspond to point Q2, and α4 and β4 correspond to point R2. For the purpose of the design vector in the MATLAB function, α1, β1, α2, β2, α3, β3, α4, and β4 correspond to x1 through x8, respectively. Optimization Algorithms To solve the minimization problem outlined in the process section, an optimization algorithm that will drive the corresponding function to zero must be selected. In the previously outlined three cases, the summation of forces at each top node will be driven to zero by the following MATLAB methods: the interior point algorithm, the global search algorithm, and the multi-start algorithm. The interior point algorithm finds a local minimum of the objective function and the global search and multi-start algorithms find global minimum values. A local minimum of a function is defined as a point where the function value is smaller or equal to the values at neighboring points, but perhaps a greater value than a distant point where a global minimum is a point smaller than or equal to the function value at all other feasible points (MathWorks, 1994a). 48

49 Two local minimums for an objective function f(x) are shown in Figure 3-8 as two black dots that lie on the x axis. In the grey section of the function, a local minimum is found that is the lowest out of all the neighboring values in its section, but not lower than the minimum found in the green dotted section. These sections are called basins of attraction, and they illustrate the importance of selecting the most ideal starting point when running a local minimum solver. Basins of attraction are a set of initial points that lead to the same local minimum value. Figure 3-7 shows that functions often yield several basins of attraction where if the user-defined starting point xo is not defined as a point in the correct basin, an inaccurate solution will be found. Such inaccuracy can cause problems in running a local minimization solver, such as the interior point algorithm, when the most ideal initial point is not known. The interior point algorithm is a method that can be used to solve a nonlinear constrained optimization problem that has both equality and inequality constraints. The general form of this problem is written as follows: minimize f(x) subject to ge(x) = 0 and gi(x) 0 In this problem, ge(x) and gi(x) represent the equality and inequality functions, respectively (Byrd, Gilbert, & Nocedal, 2006). The two tasks of minimizing the objective function, f(x), and satisfying the equality and inequality constraints are conflicting goals that prove to be computationally impractical (Hauser, 2007). The first difficulty lies in the presence of the inequality constraints defined by gi(x) 0. These inequality constraints can be removed by introducing a slack variable s that converts all inequality constraints into non-negativity equality constraints, which can be seen in the following conversion (Vanderbei, n.d.). 49

50 minimize f(x) minimize f(x) subject to gi x + s = 0 subject to gi x 0 where s 0 Restricting the values of the slack variable s to zero or a number greater than zero allows for the original inequality relationship to still hold true. However, we now have a non-negativity constraint associated with the slack variable. To remove this added inequality, a barrier term is introduced to create a new problem that retains the non-negativity property of the slack variable. The new minimization problem is now an approximation of the original minimization problem. This approximated problem can be outlined as follows (MathWorks, 1994a). minimize f(x,s) = minimize f(x) - ln s i subject to gex 0 and gix + s = 0 i In this problem, the variable μ is known as a penalty parameter where μ>0, and as μ converges to zero, the minimum of the approximated function fμ(x,s) approaches the minimum of the objective function f(x). The term being subtracted from the f(x) in the preceding problem is known as the barrier function. The equality constraints from that function shown in this problem can now be incorporated into a minimization problem by using the penalty parameter to create a resulting function known as a merit function, which is a function that simultaneously satisfies the two goals of minimizing the object function and satisfying the constraints (Willcox, 2010) and can be written in the following way: f( J(x), gi(x), ge(x) ) where J(x) is the original objective function. At each iteration, the interior point algorithm attempts to minimize the merit function by taking a step, also known as changing the variables x and s. This step is defined as the 50

51 direction ( x, s). The first step that is taken at each iteration is called a Newton step and is taken directly from Newton s method in which a Hessian matrix is used as a basis for defining the direction taken at each iteration to minimize a corresponding function. In our case, this Hessian matrix is composed of the functions f(x), gi(x), ge(x) as well as the Lagrange multipliers of both constrained functions (MathWorks, 1994b). If the Newton step does not successfully reduce the merit function, then a new direction is taken using the conjugate gradient method in which gradients, also known as the collection of all of a function s partial derivatives ( The Gradient, n.d.) as well as the Jacobian matrices of gi(x) and ge(x), are used to define the direction ( x, s). The functions in this case are f(x), gi(x), and ge(x). After the merit function has been successfully reduced after the first iteration, subsequent iterations as this process are repeated until a local minimum is found. These two methods are used to find the minimum of the specified function with respect to a user defined tolerance. Afterwards, the resulting vector x that corresponds to the local minimum function value is defined. An objective function often has multiple local minimums, and the one that corresponds to the lowest function value is considered a global minimum. As a result, local minimums produced by local minimization solvers can also be equivalent to a global minimum that is found. In MATLAB software, there exist several algorithms that compare numerous local minimums to find a global minimum. Two of these algorithms to be analyzed are named global search and multi-start. Global search is an algorithm that employs both the scatter search method and the interior point method to find a global minimum. The scatter search method is an approach that operates on a set of points known as reference points that are defined as ideal solutions obtained from 51

52 previous solution trials (Rego & Alidaee, 2005). Solutions are characterized as ideal based on several underlying criteria, one of which includes diversity within the set of reference points. In the case of the global search algorithm, the information taken from previous solution trials to create the reference points refers to the data stored from the local minimization run by way of the interior point method. In the first step of global search, the MATLAB function called fmincon takes the input of a user defined initial point xo and finds a local minimum of the objective function and stores this information in an output vector x (MathWorks, 1994). The start and end points as well as the other points evaluated during the optimization are used to create a basin of attraction like the ones depicted in Figure 3-8. After this information is used by the scatter search algorithm to create a set of reference points, the algorithm continuously produces combinations of the reference points, by way of linear combinations, for the purpose of creating new points. These points are analyzed at the end of the optimization run and the set of points with the lowest function value is given. The multi-start algorithm also generates a set of starting points; however it is not based on any previous computations. This global optimization algorithm creates a set of uniformly distributed starting points and can use either a constrained or unconstrained local minimization solver to find the output vector x with the lowest function value. A drawback of using this algorithm is that because the set of starting points are uniformly distributed, initial points within the same basin of attraction are often selected. Therefore, the same local minimum is calculated multiple times, making the multi-start process a thorough but a sometimes-inefficient method. These three optimization algorithms will be run for each of the two cases for two different methods. The first method will involve using the SolidWorks motion analysis package 52

53 for an initial point to input into the optimization problem, and the second method will use a forward position analysis to define the initial point to input into the optimization problem. After the results are collected, they will be analyzed in terms of their resulting function values and computation time. The tic command and the toc function are tools used from the MATLAB software that starts and stops a digital stopwatch timer. By putting the tic command at the beginning of the command line prompt, entering in the command to run the specified solver, and placing toc at the end of the prompt, the total time it takes for each algorithm to find the function s minimum is recorded and displayed along with the corresponding function value. SolidWorks Motion Analysis Numerical Case One Numerical case one presents the results for the tensegrity orientation where the spring constant values in each of the elastic ties are given constant values throughout the optimization process. The four values given are taken from the output of the Solidworks equilibrium case and are shown in Table 3-1. The beta angles are allowed to vary within the range, going from 35 to 135 degrees, and the alpha angles are given no restriction. This limitation given to the beta angles ensures that no solution is given where the beta values are 0 or 180 degrees where the strut members would be lying flat along the x-y plane. Constraints are set for the forces in each of the compressive and tensile elements to ensure that their magnitudes are negative and positive, respectively, and that no force is allowed to be zero. The optimization process is then run using the interior point, global search, and multi-start algorithms to obtain a solution where the summation of forces at each of the top four nodes is as close to zero as possible. This configuration will represent the tensegrity orientation that is the most stable. The results from the interior point algorithm are shown in Figures 3-9 and

54 The interior point algorithm reached a solution very fast, for it only took seconds to obtain a local minimum. The corresponding function value came in very close to zero at *10-14 lbf, resulting in a very stable tensegrity configuration. Case one was then run using the global search algorithm under the same constraint conditions. This run yielded the same function value results as the interior point algorithm. This means that out of the total set of local minimums generated by global search, none were lower than the local minimum from the interior point algorithm. Thus, the local minimum was also declared the global minimum. As a result of having to evaluate numerous reference points, the computational time increased drastically to seconds. The multi-start algorithm was then used and an initial number of 40 starting points was selected. This solver composes a list of randomly selected starting value sets, within certain bounds, in addition to the user defined vector xo to find the local minimum associated with each set. This process yielded the best results in terms of the resulting function value at *10-14 lbf with a slightly higher computational time than the global search algorithm at seconds. The resulting tensegrity orientation can be seen in Figures 3-11 and 3-12, and the comparison between the solutions of each algorithm can be seen in Table 3-5. Table 3-5. Tensegrity Case One Results Optimization Function Value (lbf) Computation Time (s) Algorithm Interior Point * Global Search * Multi-Start *

55 SolidWorks Motion Analysis Numerical Case Two Numerical case two illustrates the form-finding method for a tensegrity where the spring constant values as well as the top four nodal points are allowed to change throughout the optimization process. Constraints are set on the k values so that they are only allowed to vary between the values of 0.5 and 10 lbf/in, which is done to eliminate the instance where a numerical value for k equaling zero is chosen. All other constraints set for case one remain the same for case two. The results for the interior point algorithm are shown in Figures 3-13 and The interior point algorithm reached a solution of *10-14 lbf for its lowest function value with a very fast computation time of only seconds. This algorithm produced a solution that is also very close to zero and is a very stable tensegrity configuration. Now case two was run using the global search algorithm, and once again the same minimum function value was found, as in the case with the interior point algorithm. As a result of analyzing more trial points, the computation time was significantly longer because it took seconds for a solution to be found. Lastly, the multi-start algorithm was used to determine a global minimum and it generated the best results with a solution of *10-14 lbf. The computation time was slightly faster than the global search algorithm, and it took seconds to find the solution. Figures 3-15 and 3-16 show several views of the resulting tensegrity configuration when using multi-start, and the comparison between the solutions of each algorithm is shown in Table 3-6. The values for each of the k constants in each elastic tie member after the optimization run are all defined in Table

56 Table 3-6. Tensegrity Case Two Results Optimization Algorithm Function Value (lbf) Computation Time (s) Interior Point * Global Search * Multi-Start * Table 3-7. Case Two Spring Constant Values Spring Member k (lbf/in) P1O Q1P R1Q O1R Forward Position Analysis Process The force driven non-linear optimization method has proven that it obtains several significant advantages than did the previously outlined form-finding methods. This new method is able to satisfy simultaneously the geometry and pre-stressability constraints of tensegrity structures, regardless of symmetry. That is a characteristic that most form-controlled methods do not have, and as a result, their applications are greatly limited. The fact that the force-driven nonlinear optimization method has shown to always converge to a solution with relatively short computation times gives it a significant advantage over the previous force-controlled methods. However, even in light of these solutions, several disadvantages can be pointed out while the force-driven method is dissected into two main components. In the first component, a tensegrity orientation was found, using the motion analysis toolbox in SolidWorks, that satisfies the geometrical constraints of each element, as well as the properties of the elastic ties, which include the free lengths and the spring constant values. In the 56

57 second component, this tensegrity orientation was defined as the starting point in a non-linear optimization problem. This problem used a series of algorithms to produce several final tensegrity orientations that all satisfied the previously defined geometrical constraints as well as the newly introduced force constraints. While the user is able to define the orientation of the tensegrity in the beginning of the first component of the force-driven method, the motion analysis software in SolidWorks is solely responsible for returning one geometrically constrained final orientation. Outside of defining the length of the top ties, this method not only offers the user no choice in the orientation of the top four points in the final configuration, but it also does not define a solution set of these geometrically constrained orientations. A way of addressing the issue of defining the solution set for the geometrically constrained tensegrity problem is to analyze the elements of the mechanism initially as a serial manipulator, transform it into a spatial closed-loop mechanism, and create the resulting equivalent spherical mechanism for the purpose of conducting a forward position analysis. A serial manipulator is defined as an unclosed or open movable polygon consisting of a series and joints (Crane & Duffy, 1998). The ground coordinate system has one end of the manipulator fixed to it, and the other open end is labeled an end effector that is free to move about in space. In the case of the T-4 tensegrity mechanism, we can analyze different parts as its own geometrical structure. Using each of the four foundation nodes as one end attached to a fixed coordinate system initially, the elements of the tensegrity can be defined through distances and twist angles. These numerical values will define the current position of each element of the tensegrity. Because the current position of each element can be defined, a forward position analysis of each closed-loop mechanism within the tensegrity can be conducted, which is 57

58 important because our goal is to define the solution set of the tensegrity s top point coordinates. A forward position analysis of the tensegrity s closed loop mechanisms will determine these locations. The T-4 tensegrity mechanism to be subject to the forward position analysis is shown in Figure The ground coordinate system is shown attached to strut O with the origin defined as point O1. All four foundation nodes, O1, P1, Q1, and R1, are fixed with no z coordinates. As a result, all four nodes are lying on the x-y plane. The only points allowed to change during the position analysis are the top four points: O2, P2, Q2, and R2. The problem statement for the forward position analysis of this mechanism is the following: Given: 1. O1, P1, Q1, and R1 (bottom four coordinate points) 2. L1, L2, L3, and L4 (length of the four strut members) 3. LO1R1, LP1O2, LQ1P2, LR1Q2 (total length for the four elastic ties) 4. LO2P2, LP2Q2, LQ2R2, LR2O2 (length of the four top non-elastic ties) Find: All equilibrium configurations of the tensegrity defined by the top four point coordinates. After the solution set of the top coordinate points has been calculated, the user will be able to choose the desired tensegrity orientation to be used as the initial point for the non-linear optimization problem. Just as in all other form-controlled methods, the first step in analyzing a tensegrity mechanism is to create a mathematical model that describes each element of the structure. The T- 4 tensegrity will be defined by not only coordinates and lengths, which can be simply illustrated, but also by several triangles and spherical quadrilaterals that are more complex. Figure

59 shows that a tensegrity can be split into a series of four triangles where the endpoints of each triangle consists of two foundation nodes and one corresponding top node. The top point in each of the four triangles is defined as the coordinate of the top node after each triangle is rotated by a specified axis defined by the two foundation nodes on the x-y plane. This angle of rotation is called the triangle s fold-angle or joint angle and will be denoted as theta for the rest of the position analysis. The locations of the top nodes before this rotation will be defined as O2*, P2*, Q2*, and R2*. The final solution set for the coordinates of all four top points will include the points that are constrained by the user-defined distances of the top nonelastic ties. Next, the set of elements that converge at each one of the tensegrity s foundation nodes will be analyzed as a spherical quadrilateral, which is in the family of spherical closed-loop mechanisms. Spherical closed-loop mechanisms are formed from a spatial closed-loop mechanism where joint axis unit vectors, labeled Sk for k=1...n where n is the number of corresponding joints, and twist angles, labeled αij where i and j are the values of the adjacent links, define the orientation of a mechanism created by constructing a hypothetical closure link for the corresponding serial manipulator (Crane & Duffy, 1998). This spherical mechanism is formed by first manipulating the joint axis unit vectors. Because the joint axis vectors are defined as directional vectors, they all can be translated as long as their directions remain constant. Therefore, we can keep each vector s original direction and translate them so that each one intersects at a common point and points outward. For the case of the spherical quadrilateral, this translation is illustrated in Figure 3-19 and Figure Here one can see the directional vectors S1 through S4 rearranged from their positions in the closed loop mechanism through a translation to a common intersection point with their 59

60 original directions. Now a unit n-hypersphere, also known as a unit n-sphere, is drawn centered at the intersection of the directional vectors. This unit hypersphere is a sphere with radius r, where n is referred to as the dimension of the sphere (Weisstein, n.d.). Therefore, a twodimensional sphere is a circle, and a three-dimensional sphere is known as the usual sphere (Coxeter, 1973). On this unit n-sphere, with the center defined as point O, joints are placed at the intersection of the Sk vectors and the surface of the unit n-sphere. These joints are then connected by placing spherical links in between adjacent directional vectors. The final spherical mechanism that is created as a result of these joints and spherical links can be seen in Figure This mechanism illustrates a model known as a spherical quadrilateral because there are four spherical links with four Sk vectors and four joint angle values. The last variable to define is the twist angle, which is the angle between each of the adjacent Sk vectors. These variables now fully define the spherical quadrilateral centered at point O. This method will be used now to complete the mathematical model of the T-4 tensegrity by defining a total of four spherical quadrilaterals, each centered at the four foundation nodes of the structure. Figure 3-22 shows a spherical quadrilateral with all four joint angles shown, θ1 to θ4, as well as the corresponding directional vectors S1 to S4, the link length vectors a12 to a41, and the twist angles α12 to α41 with the intersection point in the center of the sphere defined as the foundation node. This general spherical quadrilateral will now be used to define the orientation of all the elements of the T-4 tensegrity through these vectors and angles. The spherical quadrilateral centered at point O1 of the tensegrity mechanism is shown in Figure In this figure, the vectors S1 and S4 are shown drawn along the bottom non-elastic ties connecting the points O1 and P1 and O1 and R1. The vector S2 is drawn along the strut O connecting point O1 and O2, and S3 is drawn along the elastic tie connecting the points O1 and R2. 60

61 There are four spherical quadrilateral links shown in between the adjacent directional vectors that are labeled as α12 to α41. It can be noted that these twist angles are also the variables that define one of the angles from each of the four triangles shown in Figure For example, the twist angle α12 is the angle in between the points O2, O1, and P1 and the twist angle α23 is the angle in between the points R2, O1, and R1, respectively. The angles α34 and α41 are defined in the same manner, which means because the lengths of each member of the tensegrity, and subsequently the length of each side of every triangle are known, every twist angle is also known. The only variables shown in the spherical quadrilateral left to determine are the joint angles, which are also known as each triangle s fold angle. Joint angle θi is measured in a right hand sense about the vector Si starting from the spherical link αij and ending at the spherical link αij. To create a mathematical expression relating each joint angle to its corresponding twist angles and directional vectors, a spherical dyad is analyzed and illustrated in Figure A spherical dyad illustrates a portion of the spherical quadrilateral where one joint angle is defined by the relation that angle has to its surrounding directional vectors and twist angles. In this case, the angle is θj. Because the end goal is to use these rotation angles to define the top four tensegrity points, and all the twist angles are known, we will need to create mathematical expressions with only the theta angles as the unknown values. In the case of the spherical quadrilateral, we have the vectors S1 to S4, α12 to α41, and θ1 to θ4 present at each foundation node. The directional vectors labeled in the dyad are defined where Si is initially aligned along the z axis pointed outward from the origin point O at a ground coordinate system. This vector is defined as 1 S1, where the superscript 1 denotes the ground coordinate system 1, with the coordinates [0, 0, 1] T. The remaining vectors 1 S2, 1 S3, and 1 S4 from 61

62 the same coordinate system will have to be defined to create the mathematical equations with θ1 to θ4 as the only unknowns. The vector 1 S2 can be defined with the following equation: 1 S2 = R 2 1 * 2 S2 (3-17) In this equation, R 2 1 is defined as a matrix resulting from a series of rotations where the directional vector 2 S2 is rotated about the x-axis by the angle α12 and then about the z-axis by the angle θ2. The rotation matrices are as follows: 1 2R = [ = [ c 12 s 12 ] * [ 0 s 12 c 12 c 2 s 2 0 s 2 c 12 c 2 c 12 s 12 ] s 2 s 12 c 2 s 12 c 12 c 2 s 2 0 s 2 c ] (3-18) In the preceding rotation matrices, ci = cos (θi), si = sin (θi), cij = cos (αij), and sij = sin (αij). The resulting value of 1 S2 = [0,-s12, c12] T. A variation of Equation 3-18 will be used to solve for the remaining vectors 1 S3 and 1 S4. We now have the following relationship for 1 S3 in Equation 3-19: 1 S3 = R 2 1 * 2 S3 (3-19) In this equation, the 2 S3 term can be defined by exchanging the α12 angle used to define 1 S2 with the new directional vector s corresponding α23 angle, which results in 2 S3 = [0, -s23, c23] T where the x, y, and z coordinates can be re-written as [X 1, Y 1, Z ] T 1 for simplification. Now we have the following coordinates for 1 S3, shown in Equation

63 1 S3 = [ s 23 s 2 (s 12 c 23 + c 12 s 23 c 2 ) (c 12 c 23 s 12 s 23 c 2 ) ] (3-20) For Equation 3-20, the three coordinates of 1 S3 will be re-written again as follows [X 2, Y 2, ] Z T 2. Lastly, we solve for the vector 1 S4 with the following Equation S4 = R 2 1 * 2 S4 (3-21) We now define the vector 2 S4 by using the same angle exchanging method that was used for 2 S3. The α12, α23, and θ2 angles used to define the three coordinates of the 1 S3 vector are exchanged for the α23, α34, and θ3 angles used to define the new coordinates of the corresponding 2 S4 vector, which results in the following two equations. 2 S4 = [ s 34 s 3 (s 23 c 34 + c 23 s 34 c 3 ) (c 23 c 34 s 23 s 34 c 3 ) X 3 ] = [ Y 3 ] (3-22) Z 3 1 S4 = [ (X 3 c 2 ) (Y 3 s 2 ) c 12 (X 3 s 2 + Y 3 c 2 ) s 12 Z 3 s 12 (X 3 s 2 + Y 3 c 2 ) + c 12 Z 3 X 4 X 32 ] = [ Y 4 ] = [ Y 32 ] (3-23) Z 32 Z 4 Now the general expression for the three coordinates of the directional vectors can be defined by replacing the subscripts in the preceding vector equation with i, j, and k. These three coordinates are the mathematical expressions needed to relate the directional vectors to the fold angles. Because all the twist angles are given and the only fold angles that affect the rotation of the set of four triangles of the tensegrity shown in Figure 3-18 are the input and output angles θ1 and θ4, 63

64 only equations with both θ1 and θ4 as unknowns should be analyzed. Converting the subscripts of the coordinates in Equation 3-23 results in the following: [ X kj Y kj Z kj ] = [ (X k c j ) (Y k s j ) c ij (X k s j + Y k c j ) s ij Z k s ij (X k s j + Y k c j ) + c ij Z k ] (3-24) In the preceding equation, the values for j=i+1 and k=j+1. We can now define the spherical sine and cosine laws for each of these three coordinates using subscripts 1 and 4 as k and j because we want the expressions from Equation 3-24 to be in terms of θ1 and θ4. The spherical sine and cosine laws are shown in Equation [ X 41 Y 41 Z 41 ] = [ s 23 s 2 s 23 c 2 ] (3-25) c 23 The first two coordinates contain θ2 and so will choose only the third coordinate to use in conjunction with the third coordinate from Equation 3-24 where Zkj = Z41. Now we have the following relationships shown in Equation 3-26: Z41 = s12*( X4*s1 + Y4*c1 ) + c1*z4 = c23 (3-26) In this equation, the terms X4, Y4 and Z4 are defined in the following equations: X4 = s 34 s 4 (3-27) 64

65 Y4 = (s 41 c 34 + c 41 s 34 c 4 ) (3-28) Z4 = (c 41 c 34 s 41 s 34 c 4 ) (3-29) Now the only two unknowns we have are θ1 and θ4. These angles are listed as unknowns for each of the four spherical quadrilaterals at every foundation node. As a result, the theta angles will be re-written to denote their corresponding nodes with θ1o and θ4o defining the fold angles at point O1. A simplification that can be noted that allows for all θ4 values to be expressed as θ1 values occurs when the spherical links located at adjacent spherical quadrilaterals lie in the same plane. For example, the fold angle θ1o is defined as the angle between the planes defined by α41o and α12o, and θ4p is defined as the angle between the planes defined by α34p and α41p. Because both α41p and α41o both lie in the same x-y plane, and the links α12o and α34p both lie in the plane defined by the three endpoints of a triangle O1, P1, and O2, their corresponding fold angles are equivalent, which means θ1o = θ4p. The same condition applies at each of the other remaining adjacent point sets P and Q, Q, and R, and R and O. Therefore, θ1p = θ4q, θ1q = θ4r, and θ1r = θ4o. These equivalent fold angles can be seen when analyzing the spherical quadrilaterals shown at points O1 and P1 illustrated in Figure The theta angles are not solved for directly, but rather through the use of tan-half angles by introducing the variable xi for i=1,2,3,4, which will be later defined as x, y, z and w for simplification purposes, to solve for all possible values of theta. After the tan-half angles are solved for by defining solutions for xi, the corresponding values for theta will be known as well. The fold angle θi will be defined through the following relationship: xi = tan ( θ i 2 ) (3-30) 65

66 Using this relationship between the term xi andθ i, the sine and cosine of each fold angle can be defined in the following way: s i = 2 x i 1+x i 2 c i = 1 x 2 i 1+x (3-31) 2 i Now these si and ci terms can be substituted into Equation 3-26 so that the only variables that need to be solved for are the xi terms. After this substitution, the denominators from Equation 3-31 with the corresponding subscripts of i=1, 4 are multiplied by Equation 3-26 to result in the following simplified equation: A1 x4 2 x1 2 + B1 x4 2 + C1 x1 2 + D1 x4 x1 + E1 = 0 (3-32) In the preceding equation, the variables A1, B1, C1, D1 and E1 are defined as follows: A1 = c12 (c34 c41+ s34 s41) + s12 (c34 s41 s34 c41) c23 (3-33) B1 = c12 (c34 c41 + s34 s41) + s12 (s34 c41 c34 s41) c23 (3-34) C1 = c12 (c34 c41 s34 s41) + s12 (c34 s41 + s34 c41) c23 (3-35) D1 = 4 s12 s34 (3-36) E1 = c12 (c34 c41 s34 s41) s12 (c34 s41 + s34 c41) c23 (3-37) These relationships involving the twist angles and fold angles depicted by the preceding equations are applicable to all four of the tensegrity s spherical quadrilaterals that are centered at each foundation node. Therefore, the spherical quadrilaterals at each of the other nodes act in the same way as the spherical quadrilaterals shown in Figure The link angles at each of the four spherical quadrilaterals can be defined as the angle taken from the three corresponding 66

67 endpoints of each of the triangles shown in Figure Each set containing these three endpoints and its equivalent twist angle is listed in Table 3-8. Table 3-8. Twist Angles and Corresponding Foundation Nodes Foundation Node O1 O2-O1-P1 R2-O1-O2 R1-O1-R2 R1-O1-P1 P1 P2-P1-Q1 O2-P1-P2 O2-P1-O1 O1-P1-Q1 Q1 Q2-Q1-R1 P2-Q1-Q2 P2-Q1-P1 P1-Q1-R1 R1 R2-R1-O1 Q2-R1-R2 Q2-R1-Q1 Q1-R1-O1 Because all of the lengths of the tensegrity s elements are known, the lengths of the triangles shown in Figure 3-18 are also known and therefore the twist angles shown in Table 3-8 are all known values as well. The terms x1 and x4 shown in Equation 3-32 will now be re-written to account for their corresponding spherical quadrilateral s center point. Because the center points are the four foundation nodes of the tensegrity we will account for each node by adding it to the subscript of each corresponding x term. Therefore, the terms x1 and x4 located at the O1 foundation node will be labeled as x1o and x4o. The same process is repeated for each of the other three spherical quadrilateral centers. We now can re-write Equation 3-32 to account for the four different foundation nodes, and the four resulting equations are as follows: A1 x4o 2 x1o 2 + B1 x4o 2 + C1 x1o 2 + D1 x4o x1o + E1 = 0 (3-38) A2 x4p 2 x1p 2 + B2 x4p 2 + C2 x1p 2 + D2 x4p x1p + E2 = 0 (3-39) A3 x4q 2 x1q 2 + B3 x4q 2 + C3 x1q 2 + D3 x4q x1q + E3 = 0 (3-40) A4 x4r 2 x1r 2 + B4 x4r 2 + C4 x1r 2 + D4 x4r x1r + E4 = 0 (3-41) 67

68 In these four equations, the values for A1 to E4 all remain the same from Equations 3-33 to To simplify the preceding Equations 3-38 to 3-41, the xin terms, with N = corresponding foundation node, will be re-written as the terms x, y, w, and z. Considering the equivalent fold angles mentioned previously, we are left with the following: x = x1r = x4o (3-42) y = x1o = x4p (3-43) z = x1p = x4q (3-44) w = x1q = x4r (3-45) We can now write the simplified versions of Equations 3-38 to 3-41 with the variables x, y, z, and w in the following equations. A1 x 2 y 2 + B1 x 2 + C1 y 2 + D1 x y + E1 = 0 (3-46) A2 y 2 z 2 + B2 y 2 + C2 z 2 + D2 y z + E2 = 0 (3-47) A3 z 2 w 2 + B3 z 2 +C3 w 2 + D3 z w + E3 = 0 (3-48) A4 w 2 x 2 + B4 w 2 + C4 x 2 + D4 w x + E4 = 0 (3-49) Equations 3-46 to 3-49 represent a system of equations where we have four unknowns in x, y, z, and w with four equations. Solving for all solution sets for these variables will enable all corresponding values of the fold angles to be solved for as well. Although there are four equations and four unknowns, they are in the form of a quadratic polynomial with the terms x 2, y 2, z 2, and w 2 all present. As a result, these variables of degree two 68

69 have to be eliminated to solve for the x, y, z, and w terms. Methods that involve this elimination will be outlined in the following process. First to solve for the variables x, y, z, and w, we will solve for two variables at a time. First x and z together and then y and w together. Sets of the x and z variables that solve the corresponding equations will be listed out in a table format at the end. These sets of values will be used to solve for the remaining corresponding sets of y and w. Sylvester s dialytic method and Bezout s method are used to solve for the variables x, y, z, and w. Sylvester s dialytic method of elimination, also known as Sylvester s solution method, is a process where the equation f(x) = 0 is successively multiplied by the variables x m-1,..., x, 1 to yield m equations and g(x) = 0 is successively multiplied by the variables x n-1,..., x, 1 to yield a total of m equations (Kapur, 1995). For our case, the equations we are trying to solve for are Equations 3-46 to 3-49, which represent the four spherical quadrilaterals at each foundation node. The spherical cosine law at each of these spherical quadrilaterals is expanded using the tan-half angles of the fold angles. We can take these four multivariate fourth order polynomial equations and reduce them down to just two by solving for y 2 and w 2 in terms of x and z, and then substituting them into Equations 3-47 and First, we factor out the y variable and the w variable from Equations 3-46 and 3-49, respectively. The system of equations can now be rewritten as follows: (A1 x 2 + C1) y 2 + (D1 x) y + (B1 x 2 + E1) = 0 (3-50) A2 y 2 z 2 + B2 y 2 + C2 z 2 + D2 y z + E2 = 0 (3-51) A3 z 2 w 2 + B3 z 2 + C3 w 2 + D3 z w + E3 = 0 (3-52) (A4 x 2 + B4) w 2 + (D4 x) w + (C4 x 2 + E4) = 0 (3-53) 69

70 Then, the Maple 13 program is used to solve for terms y 2 and w 2 to give the following: y (D x) y (B x E ) A1x C1 (3-54) w (D x) w (C x E ) A4x B4 (3-55) Equations 3-54 and 3-55 are now substituted into Equations 3-51 and 3-52 and are further simplified by taking the denominators of the preceding two equations and multiplying the first one with Equation 3-51 and the second with Equation 3-52, which results in the following two equations: 2 2 A2D1 x z (A1D2x C1D 2)z B2D1x y 2 2 (A1C2 A2B 1)x A2E1 C1C 2 z 0 2 (A1E2 B1B 2)x C1E 2 B2E1 2 2 A3D4 x z (A4D3x B4D 3)z C3D4x w 2 2 (A4B3 A3C 4)x A3E4 B3B4 z 0 2 (A4E3 C3C 4)x B4E3 C3E4 (3-56) (3-57) Now we can see that this simplification yields just two equations that include y and w only, instead of y 2 and w 2. Because there is only one y term in Equation 3-56 and one w term in Equation 3-57, each can easily be solved for by placing all other terms to the opposite side of the equation, resulting in the following expressions: 70

71 (D x) y 2(A x C ) (3-58) (D x) 4(A x C )(B x E ) (3-59) (D x) w 2(A x B ) (3-60) (D x) 4(A x B )(C x E ) (3-61) Here, we have the variables y and w in terms of x, and by substituting Equation 3-58 into Equation 3-56 and multiplying it by the denominator in the y term, the resulting equation is as follows: 2 2 A2D1 x z (A1D2x C1D 2)z B2D1x 1 4 2A 1(A1C 2 A2B 1)x 2 z 2 2 2A 1(2C1C 2 A2E 1) A 2(D1 2B1C 1) x 2C 1(C1C 2 A2E 1) D1 A1D2x C1D2 x z 2A1 A1E2 B1B2 x 2 2 2A 1(2C1E 2 B2E 1) B 2(D1 2B1C1 x 2C1 C1E 2 B2E1 (3-62) By following the same process for the equation involving the w term, we substitute Equation 3-60 into Equation 3-57 and multiply it by the denominator in the w term to get the following resulting equation: 71

72 2 2 A3D4 x z (A4D3x B4D 3)z C3D4x 2 4 2A 4(A 4B3 A3C 4)x 2 z 2 2 A 3( 2B4C4 2A4E4 D 4 ) 4A4B3B 4) x 2B 4(B3B4 A3E 4) 2 4 D3D4 A4x B4 x z 2A4 A4E3 C3C4 x A 4(2B4E3 C3E 4) C 3(D4 2B4C4) x 2B4 B4E3 C3E4 (3-63) Now that the expanded spherical cosine laws at each foundation node have been simplified into the two preceding equations, all of the terms in both Equations 3-62 and 3-63 no longer include y and w, so we only have to get rid of the δ1 and δ2 variables. This process is done with Equation 3-62 by first squaring both sides of the equation and then moving the 1 term to the right side. Afterwards, the Maple 13 program is used again to identify terms to factor out for further simplification. The factor of 4 (A1x 2 + C1) 2 is identified and divided by both sides of the equation, which results in the following equation: L4 z 4 + L3 z 3 + L2 z 2 + L1 z + L0 = 0 (3-64) Here, the variables L0 through L4 are defined as follows: L 0 2 2E1B 2(B1B 4 2 A1E 2) 2 A1E 2 B1B2 x 2 x B2E 2(D1 2B1C 1) 2 B2E1 C1E 2 3 L D D A E B B x B E C E x (3-65) (3-66) 72

73 L A1 C2E2 A1B 1(D2 2E2A2 2B2C 2) 2A2B1 B2 x 2 2 D 1 (A2E2 B2C 2) D 2 (A1E1 B1C 1) 2E1E 2(A1A 2) 2E B (2A B A C ) 2E C (2A C A B ) 2B B C C 2 2A2E 1(B2E1 C1E 2) 2C1C 2(C1E 2 B2E 1) D2 C1E L D D A C A B x A E C C x x 2 (3-67) (3-68) L 4 2 2A1C 2(C1C 4 2 A2E 1) 2 A1C2 A2B1 x 2 2 x A2C 2(D2 2B1C 1) 2A2 B1E 1 2 A2E1 C1C2 (3-69) This process is repeated for Equation 3-63 by first squaring both sides of the equation and then moving the 2 term to the right side. Afterwards, the Maple 13 program is used to identify terms to factor out for simplification, and the factor of 4 (A4x 2 + B4) 2 is identified and divided by both sides of the equation. Now the resulting equation is: M4 z 4 + M3 z 3 + M2 z 2 + M1 z + M0 = 0 (3-70) Here, the variables M0 through M4 are defined in a similar manner as before and labeled as follows: M A E C C x 2A E (B E C E ) x 2C3C 4(C3E4 B4E 3) C3D4 E3 B E C E (3-71) 73

74 3 M D D A E C C x B E C E x (3-72) 2A3C 4(C3C4 A4E 3) 4 2 2x A4C 4(D3 2B3C 3) 2B3E3A4 2 A3E 3(D4 2A4E4 2B4C 4) 2 M 2 4A3C3C4E4 A4E 4(D3 2B3C 3) 4A B B E B C (D 2B C ) B C D 2A3E 4(C3E4 B4E 3) 2 2 B4E 4(D3 2B3C 3) 2B3B4 E M D D A C A B x A E B B x x 2 (3-73) (3-74) M A 3 (C4E 4 4) 2 A3C4 A4B3 x x 2 A3B 3(D4 2A4E4 2B4C 4) 2 A3E4 B3B4 (3-75) Now we can use Sylvester s dialytic method of elimination to create an equation that eliminates the variable z in Equation 3-64 and Equation 3-70 so that all the values of x can be solved for. We will need to define the f(z) and g(z) equations that were changed from f(x) and g(x) in the preceding Sylvester s method definition. These functions will be defined in the following equations: fi(z) = L4 z 4 + L3 z 3 + L2 z 2 + L1 z + L0 = 0 (3-76) gj(z) = M4 z 4 + M3 z 3 + M2 z 2 + M1 z + M0 = 0 (3-77) In both equations, the m and n terms, taken from the exponents in the Sylvester s method definition as the number of total equations, are both defined as m = n = 4 because they are both associated with fourth order polynomial equations. Now we will successively multiply the fi(z) = 74

75 0 equation by z 3, z 2, z and 1 to yield the following four equations where i=1,..., m and j=1,...,n. These equations are as follows: f1 (z) = L4z 7 + L3z 6 + L2z 5 + L1z 4 + L0z 3 = 0 (3-78) f2 (z) = L4z 6 + L3z 5 + L2z 4 + L1z 3 + L0z 2 = 0 (3-79) f3 (z) = L4z 5 + L3z 4 + L2z 3 + L1z 2 + L0z = 0 (3-80) f4 (z) = L4z 4 + L3z 3 + L2z 2 + L1z + L0 = 0 (3-81) Following this same method using the equation gj(z) = 0 and multiplying it successively by z 3, z 2, z, and 1, the four resulting equations are as follows: g1 (z) = M4z 7 + M3z 6 + M2z 5 + M1z 4 + M0z 3 = 0 (3-82) g2 (z) = M4z 6 + M3z 5 + M2z 4 + M1z 3 + M0z 2 = 0 (3-83) g3 (z) = M4z 5 + M3z 4 + M2z 3 + M1z 2 + M0z = 0 (3-84) g4 (z) = M4z 4 + M3z 3 + M2z 2 + M1z + M0 = 0 (3-85) The f(z) and g(z) equations are all put into matrix format where the L and M coefficients, that are all defined in terms of x, are in one matrix, and the z coefficients are all in a column vector, which is defined in the following matrix equation: x z = 0 (3-86) 75

76 Now the solution set for the x variable in the above matrix equation, where z is a column vector with all the z terms found in Equations 3-78 to 3-85, is desired. Expanding the terms inside the matrix brackets of Equation 3-86 yields the following: L4 L3 L2 L1 L0 z M4 M3 M2 M1 M 0 0 z L4 L3 L2 L1 L0 0 z M4 M3 M2 M1 M0 0 z L4 L3 L2 L1 L0 0 0 z M4 M3 M2 M1 M0 0 0 z 0 L L L L L z M4 M3 M2 M1 M (3-87) For a solution set to exist other than the trivial solution of z = 0, the equations represented by the 8x8 matrix must be linearly dependent. Linear dependence is described as a special relationship between a set of vectors 1 2 v,v,., v n that lie in a matrix R m where m is the matrix dimension. This vector set in R m is defined as linearly dependent if the corresponding vector equation c 1 * v1 c 2 *v 2... cn * vn 0 has a solution where at least one value of cn does not equal zero (Olds, 2009). In the case of Equation 3-86, R m = R 8 where x = R 8, and the set of vectors are defined by the L and M terms. Now if a solution exists for the vector z the following relationship is true: x = 0. This matrix equation is shown as follows: 76

77 0 0 0 L L L L L M M M M M L L L L L M M M M M L L L L L M M M M M L L L L L M M M M M (3-88) Because all of the L and M terms are defined in relation to x, solving Equation 3-88 results in a 16 th order polynomial in x 2. Maple 13 is used to solve for the determinant by way of the Determinant(A) command where A is the corresponding matrix, and the solve(f,a) command where f is the polynomial function in terms of x and a is the variable the user wants to solve for, is used to get all 32 solutions for x. Plugging in each value of x into Equation 3-87 we are now able to rearrange the matrix equation to get the vector z on the left side by itself, which is done by first manipulating the eight equations defined by the matrix Equation In these eight equations, the last row of the x matrix can be eliminated to reduce the number of overall equations. The first two equations can be written out as follows: L4z 4 + L3z 3 + L2z 2 + L1z = -L0 (3-89) M4z 4 + M3z 3 + M2z 2 + M1z = -M0 (3-90) 77

78 Only the first seven equations remain and the terms on the right side of Equations 3-89 and 3-90 are placed in the modified matrix equation so that we now have a square matrix created in x on the left-hand side, which results as follows: L4 L3 L2 L1 z L M M M M z M L L L L L 0 0 z L4 L3 L2 L1 L 0 z M4 M3 M2 M1 M0 z L4 L3 L2 L1 L0 0 z M4 M3 M2 M1 M0 0 z (3-91) The newly modified matrix 7x7 matrix will be defined as * x and because all the corresponding values of x are known and the matrix is square, it is an invertible matrix and can be brought to the other side of the equation so that the z variables can be solved for, which is seen in Equation z L4 L3 L2 L1 L0 6 z M4 M3 M2 M 1 M 0 5 z 0 0 L4 L3 L2 L1 L 0 0 z 0 0 M M M M M z 0 L4 L3 L2 L1 L z 0 M4 M3 M2 M1 M0 0 0 z L4 L3 L2 L1 L (3-92) All of the terms from the right side of Equation 3-92 are defined by the 32 values of x previously solved for. As a result, the preceding equation can be solved for the 32 corresponding values of z. 78

79 Bezout s method is used to find the values of y and w. A system of equations is created by starting with all equations that include the y variable, which are written as follows: A1 x 2 y 2 + B1 x 2 + C1 y 2 + D1 x y + E1 = 0 (3-93) A2 y 2 z 2 + B2 y 2 + C2 z 2 + D2 y z + E2 = 0 (3-94) Factoring out the y 2 and y variables yields the following equations: y 2 (A1 x 2 + C1) + y(d1 x) + (B1 x 2 + E1) = 0 (3-95) y 2 (A2 z 2 + B2) + y(d2 z) + (C2 z 2 + E2) = 0 (3-96) Assigning the terms in parenthesis as Pi, Qi, and Ri for i=1, 2 gives the resulting equations: y 2 (A1 x 2 + C1) + y(d1 x) + (B1 x 2 + E1) = 0 (3-97) y 2 (A2 z 2 + B2) + y(d2 z) + (C2 z 2 + E2) = 0 (3-98) The preceding equations can be rearranged to create a system of linear equations by eliminating the y 2 terms to yield the following: y(p1y + Q1) + R1 = 0 (3-99) y(p2y + Q2) + R2 = 0 (3-100) 79

80 If a solution exists for this system of equations, then the equations will be linearly dependent. If this is the case, the solution for y for each equation will be the same and we can write the following: R1 R2 y = P1 y Q 1 P2 y Q 2 (3-101) The next step is to take Equation and expand it first through cross-multiplication and then to rearrange the terms to group the P1 and P2 terms together and the Q1 and Q2 terms together, which results in the following two equations: R *( P y Q ) + R *( Py + Q ) 0 (3-102) y*( PR P R ) + (Q R Q R) 0 (3-103) The expressions in parentheses can be written in terms of a 2x2 matrix determinant with the corresponding variables from Equation The equation written in this way is shown in Equation P R Q R y* + P R Q R = 0 (3-104) The solution for all 32 values of y can now be taken from Equation because the variables in the determinants are given in terms of the previously defined x and y values as well as the tensegrity s twist angles. 80

81 y = Q Q P P 1 2 R R R R (3-105) The solution for the last unknown variable w is also found using Bezout s method. The system of equations that have to be solved for w is as follows: A3 z 2 w 2 + B3 z 2 +C3 w 2 + D3 z w + E3 = 0 (3-106) A4 w 2 x 2 + B4 w 2 + C4 x 2 + D4 w x + E4 = 0 (3-107) Rearranging the equations by grouping the w terms together in the same way as y terms were grouped yields the following equations: w 2 (A3 z 2 + C3) + w(d3z) + (B3z 2 + E3) = 0 (3-108) w 2 (A4 x 2 + B4) + w(d4x) + (C4x 2 + E4) = 0 (3-109) In these equations, the terms grouped in the parenthesis will represent Pi, Qi, and Ri respectively for i=3 for Equation and i=4 for Equation Substituting these six new terms into the previous two equations and rearranging them to have a linear equation in w gives the following: w(p3w + Q3) + R3 = 0 (3-110) w(p4w + Q4) + R4 = 0 (3-111) These two linear equations will have a common solution if the set of equations are linearly dependent. If this is true, the same solution for w has the following relationship: 81

82 R3 R 4 w P 3w Q 3 P 4w Q 4 (3-112) Now we use cross-multiplication and the rearranging of several variables to group the w terms together to obtain the following equations: R *( Pw Q ) + R *( Pw + Q ) 0 (3-113) w *( P R P R ) + (Q R Q R) 0 (3-114) Because the terms in parenthesis found in Equation again equals the determinants for a 2x2 matrix, the equation can be solved for w and written as follows: w = Q Q P P R 3 3 R R R 4 4 (3-115) Once again, the terms in the preceding determinants are all known because they are defined by the given terms x, z, and the twist angles. Therefore, all 32 solutions for w can now be solved for. Now that all of the solutions for x, y, z, and w can now be defined, so can all possible corresponding values of the fold angles θ1 to θ4. The next step is to calculate the values for the top four coordinate points, O2, P2, Q2 and R2, using the translation and rotation matrix formulas for two corresponding points. 82

83 The process of defining the final top four points with respect to the origin, placed at point O1, has changed since the force-driven non-linear optimization method was detailed in the previous sections. Now the top four points will be defined in terms of the twist and fold angles α and β. The orientation of the top four points are defined in relation to the endpoints of the strut members in the following process. First, each strut is placed with one endpoint lying at the origin and the other endpoint, labeled O2 *, P2 *, Q2 * and R2 * shown in Figure 3-18, is lying on the x-axis. Strut P, with endpoints P1 and P2, will be analyzed as an example. This strut is translated so that the endpoint that was coincident with the origin is now lying on its corresponding foundation node. Because the coordinates of the corresponding foundation node P1 only have an x coordinate, the point along strut P is translated along the x-axis; this distance is defined as LO1P1. The corresponding translation matrix is shown in Equation 3-116, and the movement of the strut is shown in Figure 3-26 and Figure L O1P1 T t (3-116) The endpoint coincident with its corresponding foundation node will now remain fixed for the rest of the problem. The other endpoint is allowed to move and will undergo a series of rotations to get it to its final orientation at point P2. The point P2a * is now rotated about the positive z axis, the magnitude of the angle between the x-axis and the line L P1Q1. Because the coordinates of the endpoints of strut P are known and the coordinate of the foundation node Q1 is known, the angle between the vector 83

84 along strut p and LP1Q1 shown in Equation can be calculated and defined as 1. The corresponding rotation matrix is cos 1 sin sin cos T r1 (3-117) The calculation for this angle is found using the following cosine formula for two vectors a and b shown in Figure 3-28 and Equation (Mykhailo, 2011). cos( ) = ab (3-118) a b Then, point P2b* is rotated about the negative z axis the magnitude of the twist angle α12, which has been previously expressed as the angle between the lines going from P2 to P1 and P1 to Q1. The corresponding rotation matrix is shown in Equation and illustrated in Figure cos 2 sin sin cos T r2 (3-119) The last rotation is about an axis m defined as the line from point P1 to Q1. This line, as it relates to this rotation, will be given the following coordinates: [mx, my, mz] T and the rotation will be the magnitude of the fold angle θ1p, which is illustrated in Figure This rotation matrix is 84

85 defined from the simplified matrix that shows A BR where a rotation of an angle θ occurs about an axis m where m = [mx, my, mz] T. This matrix is shown in Equation m (1 cos ) cos m m (1 cos ) m sin 2 x x y y A 2 BR = mxmy(1 cos ) my (1 cos ) cos mx sin mysin mxsin cos (3-120) The preceding matrix equation can be simplified, because the z coordinate of the m axis is zero, and expanded because v = 1-cos(θ) and the terms s and c stand for sine and cosine of θ to yield the following matrix shown in Equation m (1 cos ) cos m m (1 cos ) m sin 2 x x y y A 2 BR = mxmy(1 cos ) my (1 cos ) cos mx sin mysin mxsin cos (3-121) Factoring out the cos(θ) terms and defining the rotation about the negative m axis shown in Figure 3-30 for a 4x4 rotation matrix, we now have the following Equation cos 1P 1mx mx mxmy 1cos 1P my sin 1P mxmy 1cos 1P cos 1P 1my my mx sin 1P 0 T r3 (3-122) my sin 1P mx sin 1P cos 1P We can now use the equation that defines the translation and rotation of a point that details the movements shown in Equations 3-116, 3-117, and in the following equation: 85

86 P2 = Tt Tr1 Tr2 Tr3 P2initial* (3-123) Now the top point P2 can be solved for. This process is repeated for the other three top points O2, Q2, and R2. Because the four top points of the T-4 tensegrity can now be solved for, the forward position analysis process is complete and can be applied to a numerical tensegrity case. In this case, a tensegrity with an equivalent geometry to the one described in the previous process section will be used in conjunction with the forward position analysis to find stable equilibrium orientations that satisfy the geometrical constraints for a tensegrity. Afterwards, these orientations will be defined as the starting points in the force-driven non-linear optimization method and the final equilibrium orientations. Then, results will be compared to the previous method where the SolidWorks equilibrium case, via the motion analysis toolbox, was used to define the initial guess. Forward Position Analysis Numerical Case Overview Now a numerical case will be solved for and analyzed and the results will be compared to the previously defined SolidWorks equilibrium case method. Because we want to solve for the top four points of the tensegrity mechanism, the coordinates of the foundation nodes and the lengths of all the members have to be defined first. The bottom point coordinates are listed with units of inches in the following table: Table 3-9. Bottom Point Coordinates for Position Analysis Case Bottom Points Coordinates O1 [0,0,0] P1 [9,0,0] Q1 [6,7,0] R1 [-1,8,0] 86

87 Then the total lengths of the elastic ties as well as the top four ties are defined. These are seen in Table 3-10 and Table Table Elastic Tie Member Lengths Elastic Tie Member Total Length (in) P1O2 10 Q1P2 12 R1Q2 9 O1R2 8 Table Top Non-Elastic Tie Member Lengths Top Tie Member Length (in) O2P2 4 P2Q2 5 Q2R2 5 R2O2 6 Finally, we will define the lengths of the four strut members in units of inches in Table Table Strut Member Lengths Strut Member Length (in) O1O2 12 P1P2 14 Q1Q2 12 R1R

88 Because we now have all of the coordinates and lengths as stated values in the given section of the problem statement have been defined, the forward position analysis process can proceed by first calculating the twist angles associated with each of the four spherical quadrilaterals characterized by their respective foundation nodes. These values are listed in Table Table Twist Angles and Corresponding Foundation Nodes Foundation Node O P Q R Because all of the twist angles are now defined and the coefficients Ai, Bi, Ci, Di, and Ei for i=1 to 4 from Equations 3-46 to 3-49 are all in terms of these twist angles, they all can now be solved for. These values are shown in Table Table Coefficients Ai, Bi, Ci, Di and Ei i Ai Bi Ci Di Ei Having all of these coefficients now allow us to solve for the 32 values of x, y, z, and w through the use of Sylvester s dialytic method of elimination and Bezout s method outlined in the previous section. These values are all listed in Table 5-1 in the Appendix section. These values are used to define the fold angles θ, and as a result Equation can now be used to find all possible values of the top four points of the T-4 tensegrity mechanism. Because only the first 88

89 four sets of values for x, y, z and w are real solutions, there will be only 4 sets of real solutions for the top four point coordinates, which are shown in Table Table Real Solutions for the Top Four Point Coordinates Solution O2 P2 Q2 R Looking at the final coordinates given for the first two solution sets and the last two solution sets, it is evident that the geometry of the tensegrities are identical, which is because the z coordinates given in solution sets 2 and 4 are about the negative z axis given for solution sets 1 and 3, respectively. Therefore, there are two unique real solutions, and these are shown in Figure Forward Position Analysis Numerical Case One Numerical case one presents the results for the tensegrity orientation where the spring constant values in each of the elastic ties are given constant values throughout the optimization process. The four values of the top coordinate points are taken from the output of the forward position analysis solution 1 case shown in the left image in Figure Just as in the previous section, the beta angles are allowed to vary within the range going from 35 to 135 degrees, and the alpha angles are given no restriction. This limitation given to the beta angles ensures that no 89

90 solution is given where the beta values are 0 or 180 degrees where the strut members would be lying flat along the x-y plane. Constraints are again set for the forces in each of the compressive and tensile elements to ensure that their magnitudes are negative and positive, respectively, and that no force is allowed to be zero. The optimization process is then run using the interior point, global search, and multi-start algorithms to obtain a solution where the summation of forces at each of the top four nodes is as close to zero as possible. This configuration will represent the tensegrity orientation that is the most stable. The results from the interior point algorithm are shown in Figures 3-32 and The interior point algorithm reached a solution very fast because it only took seconds to obtain a local minimum. The corresponding function value came in very close to zero at *10-14 lbf, resulting in a very stable tensegrity configuration. Case one was then run using the global search algorithm under the same constraint conditions. This run yielded the same function value results as the interior point algorithm, which means that out of the total set of local minimums generated by a global search, none were lower than the local minimum from the interior point algorithm. Therefore, the local minimum was also declared the global minimum. As a result of having to evaluate numerous reference points, the computational time increased drastically to seconds. The multi-start algorithm was then used and an initial number of 40 starting points was selected. This solver composes a list of randomly selected starting value sets, within certain bounds, in addition to the user defined vector xo to find the local minimum associated with each set. This process yielded the best results in terms of the resulting function value at *10-14 lbf and a lower computational time than the global search algorithm at seconds. 90

91 The resulting tensegrity orientation can be seen in Figures 3-34 and 3-35, and the comparison between the solutions of each algorithm can be seen in Table Table Tensegrity Case One Results Optimization Algorithm Function Value (lbf) Computation Time (s) Interior Point * Global Search * Multi-Start * Forward Position Analysis Numerical Case Two Numerical case two illustrates the form-finding method for a tensegrity where the spring constant values as well as the top four nodal points are allowed to change throughout the optimization process. Constraints are set on the k values so that they are only allowed to vary between the values of 0.5 and 10 lbf/in. This process is done to eliminate the instance where a numerical value for k equaling zero is chosen. All other constraints set for case one remain the same for case two. The results for the interior point algorithm are shown in Figures 3-36 and The interior point algorithm reached a solution of *10-13 lbf for its lowest function value with a very fast computation time of only seconds. This algorithm produced a solution that is also very close to zero and thus a very stable tensegrity configuration. Now case two was run using the global search algorithm, and once again the same minimum function value was found as in the case with the interior point algorithm. As a result of analyzing more trial points, the computation time was significantly longer, for it took seconds for a solution to be found. 91

92 Lastly, the multi-start algorithm was used to determine a global minimum and it generated the best results with a solution of *10-13 lbf. The computation time was slightly faster than the global search algorithm and it took seconds to find the solution. Figures 3-38 and 3-39 show several views of the resulting tensegrity configuration when using multi-start, and the comparison between the solutions of each algorithm is shown in Table Table Tensegrity Case Two Results Optimization Algorithm Function Value (lbf) Computation Time (s) Interior Point * Global Search * Multi-Start * The values for each of the k constants in each elastic tie member after the optimization run are all defined in the following table. Table Case Two Spring Constant Values Spring Member k (lbf/in) P1O Q1P R1Q O1R

93 Figure 3-1. T-4 Tensegrity Structure Figure 3-2. Rotation angles shown on Tensegrity Figure 3-3. Sum of Forces at Point O2 93

94 Figure 3-4. T-4 Tensegrity in Solidworks with Highlighted Elastic Side Ties Figure 3-5. Initial (left) and Final Orientation (right) after SolidWorks Motion Analysis Run 94

95 Figure 3-6. Solidworks Motion Analysis Tensegrity Displayed in MATLAB Figure 3-7. Tensegrity Side and Top views from Solidworks Motion Analysis 95

96 Figure 3-8. Local Minimums of Objective Function, reproduced from MathWorks (1994) Figure 3-9. Tensegrity from Interior Point Algorithm Case One Figure Tensegrity Side and Top Views from Interior Point Algorithm Case One 96

97 Figure Tensegrity from Multi-Start Algorithm Case One Figure Tensegrity Side and Top Views from Multi-Start Algorithm Case One 97

98 Figure Tensegrity from Interior Point Algorithm Case Two Figure Tensegrity Side and Top Views from Interior Point Algorithm Case Two 98

99 Figure Tensegrity from Multi-Start Algorithm Case Two Figure Tensegrity Side and Top Views from Multi-Start Algorithm Case Two 99

100 Figure T-4 Tensegrity Structure with x-y Plane Figure T-4 Tensegrity with Top Point Fold Angle Rotation 100

101 Figure Translation of Directional Vectors at One Point Figure Spatial Closed-Loop Mechanism with Directional Vectors 101

102 Figure Spatial Closed Loop Mechanism and Equivalent Spherical Quadrilateral Mechanism Figure Spherical Quadrilateral with Corresponding Twist and Joint Angles 102

103 Figure Spherical Quadrilateral at Point O1 Figure Spherical Dyad with Directional Vectors and Twist Angles, Reproduced from Crane and Duffy (1998) 103

104 Figure Spherical Quadrilaterals at Foundation Nodes O1 and P1 Figure Initial Orientation of Strut Member P Figure Orientation after Translation of Point P1 104

105 Figure Angle Between Two Vectors, Reproduced from Mykhailo (2011) Figure First and Second Rotations of Strut P Figure Third Rotation about m Axis 105

106 Figure Final Tensegrity Orientations from Solution 2 (left) and Solution 4 (right) Figure Tensegrity from Interior Point Algorithm Case One 106

107 Figure Tensegrity Side and Top Views from Interior Point Algorithm Case One Figure Tensegrity from Multi-Start Algorithm Case One 107

108 Figure Tensegrity Side and Top Views from Multi-Start Algorithm Case One Figure Tensegrity from Interior Point Algorithm Case Two 108

109 Figure Tensegrity Side and Top Views from Interior Point Algorithm Case Two Figure Tensegrity from Multi-Start Algorithm Case Two 109

110 Figure Tensegrity Side and Top Views from Multi-Start Algorithm Case Two 110