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ANALYSIS OF TENSEGRITYBASED PARALLEL PLATFORM DEVICES By MATTHEW Q. MARSHALL A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003 ACKNOWLEDGMENTS I would like to thank my graduate committee for their help and oversight. They are Dr. Gloria Wiens, Dr. Gary Matthew, and Dr. Carl Crane III. I would especially like to thank Dr. Crane for his creativity and dedication. Thanks also go to Mrs. Rebecca Hoover for her help throughout my career as a graduate student at the University of Florida. TABLE OF CONTENTS A C K N O W L E D G M E N T S ......... ..................................................................................... ii ABSTRACT .............. ..................... .......... .............. iv CHAPTER 1 IN T R O D U C T IO N ............................................................................. ................ .. 1 2 TENSEGRITYBASED 33 PARALLEL PLATFORMS........................................ 7 Defining the Vertices in a Local Coordinate System .................................................. 7 Coordinate Transform nations ........ .................................. ................... .............. 9 P lu ck er L in e C oordin ates .......................................................................................... 10 Screw Coordinates and W renches ......................................................... ... 11 Obtaining the Unitized Plticker Coordinates for the Legs..................................... 12 Obtaining Spring Elongation Values............... ... .................................. 13 O btain Leg and C able Lengths ............................................... ............ ...... .. 16 On the Requirement for an External Wrench .................................................... 17 N um erical E xam ples................................ .............. ........................ .................. 20 3 TENSEGRITYBASED PARALLEL PLATFORM ............................................. 32 A application of a Seventh L eg ............................ .................................................... 32 N um erical Exam ples.................................................... .. .. .......... .. .............. 33 4 C O N C L U SIO N S .... ...................................................... ........ .. ............. 42 REFERENCES ................... ............................ .. 44 BIO GRAPH ICAL SK ETCH ......................................................... ... .............. 46 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ANALYSIS OF TENSEGRITYBASED PARALLEL PLATFORM DEVICES By Matthew Q. Marshall May 2003 Chairman: Dr. Carl Crane Major Department: Mechanical Engineering A parallelplatform device that is based on tensegrity is studied in this paper. A tensegrity structure is one that is tensionally continuous and compressionally discontinuous. It comprises several noncompliant struts, which are in compression along with ties, both elastic and inelastic, which are in tension. The device studied in this thesis replaces the noncompliant members of a tensegrity structure with prismatic actuators, and each elastic member with a cablespring combination in series. The length of the cable is adjustable. These changes are made so that the device can have six degrees of freedom, i.e. the length of the three prismatic actuators and the length of the three cables that are in series with the springs. The study performed in this thesis shows that the external wrench can act along any screw that is independent of five of the device legs. This study also shows that the device's compliance characteristics can be varied while maintaining its position and orientation. A reverse analysis of the device's position and orientation, along with its internal potential energy, is performed. The effect of a seventh leg, another v prismatic actuator, is also analyzed and found to satisfactorily implement the needed external wrench. The position of the seventh leg needs to be variable in order for it and five other legs to remain linearly independent. CHAPTER INTRODUCTION A 33 inparallel platform device consists of two rigid platforms connected by six noncompliant legs. The legs are connected to the platforms with ballandsocket joints. The length of each leg is variable. If the bottom platform is connected to ground, then the top platform retains six degrees of freedom; the freedoms to translate along the x, y, and zaxes, and to rotate about each of them. A 33 inparallel platform is pictured below in Figure 11. Figure 11. 33 Parallel Platform Device A device of this type is immobile for a given set of leg lengths. If the legs were elastic then an external force could displace the platform. Its harmonic frequency would also be affected by altering the elasticity of its legs. A tensegrity structure is a loose example of a 33 inparallel platform with fixed leg lengths, some of the legs being elastic. Buckminster Fuller assigned a meaning to the word tensegrity. The word refers to the phenomena that all objects in the universe exert a pull on each other and thus the universe is tensionally continuous. The universe is also compressionally discontinuous [4]. A tensegrity structure is an illustration of this phenomenon on a mansized scale. A tensegrity structure is pictured in Figure 12, below. This structure is a triangular tensegrity prism. A tensegrity structure consists of multiple members, some of which are solely in tension; the others are in compression. None of the compressional members come in contact with another compressional member. However, the members in tension are different in this respect. Any vertex of the structure can be connected to another point on the structure by tracing a line along tensional members. This is evidence of the structure's tensional continuity. The members labeled "S" in Figure 12 are compressional. The remaining members are ties, and thus can only be in tension. This structure, without the application of an external force, can only be in equilibrium at two positions where one is simply a mirror image of the other reflected through the base plane. A condition on this position will be discussed in the next chapter. Figure 12. Tensegrity Structure A parallel prism is the root of a tensegrity prism. The parallel prism has ties that are parallel and have fixed lengths. A tensegrity prism is created by taking a parallel prism and rotating the top plane about its central, perpendicular axis by an angle a, and then inserting noncompliant members connecting the erstwhile endpoints of the diagonals of the planes made of the parallel ties [6]. For a triangular parallel prism there are three such planes, thus there are three noncompliant members in a triangular tensegrity prism. Tensegrity prisms, like parallel prisms, can be of any degree polygon. Kenner [5] found the rotation angle, a, for the general tensegrity prism as a = 2 (1.1) 2 n where n is the number of sides in the polygon of the upper or lower plane. A different property of tensegrity prisms is calculated to be independent of the value of n. Knight et al. [6] show that, due to the arrangement of its members; a triangular tensegrity structure has instantaneous mobility. This is a characteristic of all tensegrity prisms. Tensegrity prisms share another interesting characteristic; they are deployable from a bundled position. Duffy et al. [3] analyze the deployable characteristics of elastic tensegrity prisms, as do Tibert [11] and Stern [10]. Tensegrity prisms have this attribute because the potential energy stored in their elastic members is greater when bundled than when in the position of Figure 12. Indeed, this figure illustrates a position of minimum potential energy. The allure of tensegrity structures encompasses more than their characteristics of motion. Burkhart [2] fabricates and analyzes domes using triangular tensegrity prisms. The triangular tensegrity prisms that he uses have smaller tops than bottoms. Others have studied the uses of tensegrity prisms when the top and bottom are the same size, but the lengths of the side ties are variable. Tran [12] discusses a device comprising three compliant ties and three non compliant struts, all of which have adjustable lengths. Each compliant leg consists of a nonelastic cable in parallel with an elastic member, a spring. These legs can be called side ties. This device is a triangular tensegrity prism with variable lengths. Oppenheim [8] also deals with adjustablememberlength tensegrity prisms. The aim of these devices is to allow a tensegrity structure to achieve varied positions. These devices can have different stiffness values for identical postures because of the variable nature of the compliant member lengths. Skelton [9] discusses using tensegrity structures with variable member lengths in wing construction. Doing this would allow the wing to have adjustable stiffness or shape. The thesis presented here evaluates a method of calculating the conditions necessary to attain variable configurations of a triangulartensegrityprism based device. Tran states that because the ties of its top and bottom are noncompliant, the device is effectively a parallel mechanism. Lee [7] suggests a parallel mechanism in which the joints are offset along the platform sides. A parallel mechanism with three compliant legs and offset joints is pictured in Figure 13. This mechanism serves as the model for analysis in this paper. The top and bottom ties of a tensegrity structure are, in this device, replaced with platforms. If three ties that form a triangle are kept in tension with forces acting only at the corners of the triangle, then the shape of the triangle cannot change. This is true regardless of any change in the positions of the corners relative to each other. It follows that as long as the struts are in compression and the side ties in tension, and the struts and side ties meet at only three points per platform, then a platform would behave the same as the three ties found in the top and bottom of the tensegrity structure in Figure 12. The device in Figure 13, however, does not meet all the above criteria. i_,I 105 S\ platform side lengths: /,, Ib t strut lengths: L;, L2, L3 C5j L5 compliant lengths: L4, L5, L6 Cable lengths Ic4, Ic5, Ic6 spring free lengths 104, 105, 106 spring deformations: 64, S5, 86, spring constants: k4, ks, k6 Figure 13. TensegrityBased Parallel Platform Device The joints in Figure 13 are not connected at only three points per platform. Since the device is modeled after the special 33 Parallel Platform studied by Lee, the joints are separated along one side of the triangle. This is done to increase the quality index of the robot at home position [13]. For the analysis in this paper, the placement of the joints on the platforms will not hinder the acquisition of results. Problem Statement. The given information in the problem is listed here: * The lengths of the sides of the top and bottom platforms of the device (It, Ib) * The position and orientation of the top platform relative to the bottom, or the transformation matrix (7) * The spring free lengths (lol, lo2, los) * The spring constants (kl, k2, k3) * The potential energy stored in the springs (U) * The screw along which an external wrench acts ($,t) The need for an external wrench will be discussed in chapter two, section eight. The values sought are listed below: * The length of each strut (L1, L2, Ls) * The length of cable in each compliant leg (1c4, 1c5, lc6) * The spring deformations (64, 65, 66) For a reverse analysis of a parallelplatform device, it is desired to find the six leg lengths for a given position and orientation. In order to position a device like this, the length of the cable portion of each compliant leg would be required. Thus the knowledge of the total length of each compliant leg is useless without the value of the spring deformation for each leg. The spring deformations are accordingly part of the desired information. CHAPTER 2 REVERSE ANALYSIS SOLUTION FOR TENSEGRITYBASED 33 PARALLEL PLATFORMS This chapter will present a solution for the leg lengths of a tensegritybased 33 parallel platform device when given its position and orientation. This is also known as a reverse analysis solution. The reverse analysis presented here will differ from that of the nontensegritybased device because it deals with three compliant legs and because the potential energy in these members is part of the information given in the problem statement. 2.1 Defining the Vertices in a Local Coordinate System There are either three or six vertices on each platform, depending on if the joints are separated along the sides of the platforms as in Figure 13 or not. The amount of separation is denoted by the variable a. This variable represents the portion of the platform side, by which the joints are separated. The equations of the six joint coordinates exist in pairs. The joints are labeled in Figure 21. The local coordinates of the first joint in each pair are found by setting a equal to zero. The coordinates of the second joint in each pair are found by using the a given in the platform geometry. The equations for the locations of the platforms' joint pairs are B =T4 = [0 0 0], (2.1a) 1 =TP14= Ccos c. sin 0 (2.1b) B "2 pT p5 0 O], (2.2a) B 42 =P25 =[(1) 0o ], (2.2b) Bp 3=T = 0 sinjl 0 (2.3a) BP53=TP=(l+ (lojs3in 0 (2.3b) The different superscripts in each of Equations (2.1) (2.3) refer to the fact that these point coordinates are measured in local systems. Thus the origin of the bottom and top coordinate systems are at points P, and P4, respectively. Their xaxes run along the lines from P, to P2 and from P4 to P,. The zaxes of the coordinate systems are perpendicular to the platforms. From these equations, and Figure 21, it is apparent that with a equal to zero, P, would be coincident with P14 and likewise for all other pairs. P14 P4 P25 P6 PP5 P53 P1 P61 P42 P2 Figure 21. Tensegrity Mechanism with Points Labeled 2.2 Coordinate Transformations Often, the information for position and orientation of the top platform is given as a transformation matrix. This is a fourbyfour matrix. It provides the values needed to describe the relative position and orientation of two coordinate systems. One way to define the transformation matrix is to multiply four specific matrices in a specific order. For example, the top coordinate system can be thought of as initially aligned with the bottom coordinate system. It is then translated to some point whose coordinates can be written as [x, y, z]. Then it is rotated about the current x axis bya, followed by a rotation of P about the modified y axis, and finally followed by a rotation of y about the modified z axis. For this example, the transformation matrix that relates the top and bottom coordinate systems, BT, can be calculated 1 0 0 x 1 0 0 0 cos@) 0 sin(8) 0 cosy) sin() 0 0 B 0 1 0 y 0 cos) sins) 0 0 1 0 0 sin(v) cosy) 0 0 0 0 1 z 0 sin@() cos@) 0 sin(8) 0 cos() 0 0 0 1 0 o0 0 0 0 0 1 0 0 0 1 o0 0 0o The coordinates of any point that is known in the top coordinate system can be determined in the bottom coordinate system as Bp= BT P, (2.5) where TP1 represents the coordinates of point 1 written in homogeneous coordinates in terms of the top coordinate system as (2.6) PT = Y (2.6) 1 and similarly, BP1 represents the coordinates of point 1 written in homogeneous coordinates with respect to the bottom coordinate system. 2.3 Pliicker Line Coordinates Plucker coordinates are a way to describe a line in space. The term $L will be used to represent the coordinate of a line which consists of six constants arranged in a column matrix as I m n $L = (2.7) p q r The variables 1, m, and n represent respectively the x, y, and z directions of the line, while p, q, and r signify the moments of the line about the x, y, and zaxes respectively. These moment terms can be calculated as the cross product of the coordinates of a point on the line with the direction of the line. In this thesis it is required to know the coordinates of the lines representing the mechanism legs. These lines can be seen as a group of lines, each of which are defined by two points in space. Plucker line coordinates can be used to describe the line connecting two points, Pi and P2, which exist in space. The equations for the Plucker coordinates of this line are written as 1 x, 1 y, 1 zZ /= m n= 2 Y1 2 1 (2.8) (2.8) 1y, z1 Z1 1 x, y,1 p= ,q r= 2 Z2 Z2 2 2 2 where the term xl is the x component of the position vector of point Pi; y2 is the y component of the position vector of point P2, etc. Plucker line coordinates are homogeneous coordinates, i.e. multiplying all six coordinates by a scalar value results in the same line in space. In this application, the Plicker line coordinates will be written such that the direction of the line is a unit vector. This is readily accomplished by dividing all six of the terms by the magnitude of the x, y, and z components of the line as 1 magnitude m magnitude n magnitude (2.9) pL 2) magnitude q magnitude r magnitude where the value of magnitude is given by magnitude = _12 +m2 +2 (2.10) 2.4 Screw Coordinates and Wrenches A screw is a line with a pitch where 'h' is used to denote the pitch. It is important to note that the pitch term has units of length. A screw, $, can be written as 1 m S n (2.11) p+lh q+mh r+nh A screw multiplied by an angular velocity magnitude describes the instantaneous motion of one body relative to another, i.e. a rotation about the line of axis of the screw combined with a translation along the screw axis direction. This is called a twist, T . Further, a screw multiplied by a force magnitude represents a force and moment acting on a body, i.e. a force along the line of action of the screw combined with a moment about the screw axis direction. This is called a wrench, *. It is shown in Ball [1] that any combination of forces and moments acting on a rigid body can be replaced by a single equivalent wrench. Further it is shown that this equivalent wrench can be determined by simply adding the screw coordinates of the individual forces and moments that are acting on the body. This is one of the elegant aspects of screw theory. 2.5 Obtaining the Unitized Pliicker Coordinates for the Legs The Plicker coordinates of the legs are found by using the point coordinates of the joints in Equation (2.8). The local point coordinates for the joints are found using Equation (2.1), (2.2), and (2.3). The global point coordinates for the joints of the bottom platform are given by the three preceding equations, substituting lb for 1o. The coordinates, relative to the origin of the top platform's coordinate system, of the top platform's joints are also given by that equation using I/ instead of 1o. The local coordinates of the top joints must be transformed into coordinates relative to the bottom platform. The local coordinates of the top joints are substituted into Equation (2.5) to yield their positions relative to the base. The value of T is supplied in the problem statement. The availability of the absolute point coordinates for the joints yields line coordinates for the legs. The Plucker coordinates for the legs are obtained by using Equation (2.8). For each strut, the corresponding topplatform joint is used as P2 in that equation, and that strut's joint that connects it to the bottom platform is used for P1. For the side ties this substitution was reversed. The joint on the top platform for each tie was used as Pi and that of the bottom for P2. The reason for doing this is given in the next section. After obtaining the line coordinates by this method, they are still not useful in the description of forces. This is necessary to obtain the spring lengths. In order to be used in force description the Pliicker coordinates of the line along which a force acts must be unitized. The coordinates of the lines coincident with the legs are unitized using Equation (2.9). 2.6 Obtaining Spring Elongation Values In this thesis, the deformations of the springs are found by using a force balance equation on the top platform combined with the value of potential energy given in the problem statement. Knowledge of the deformed spring lengths will yield the length of cable paid out in each side tie, part of the information sought in the problem statement. There are seven wrenches acting on the platform. Six result from the legs of the mechanism. The seventh wrench is an externally applied wrench that acts along the screw given in the problem statement. In order for the device to be in equilibrium, the sum of these forces and moments must be equal to zero. The force and moment balance equation for the top platform can be written as fl $L1 + f2 $L2 + f3 $L3 + f4 $L4 + f5 $L5 + f6 $L6 + fext $ext = 0 (2.12) The lines $L1, $L2, and $L3 in Equation (2.12) were taken in the direction going from the bottomplatform joint to the topplatform joint of each leg because of the assumption that those legs would be in compression and thus their forces would act toward the top platform. The lines $L4, $L5, and $L6 were taken in the opposite direction because they can only exert tensional forces. These lines coordinates were then unitized and placed in Equation (2.12). Equation 2.6.1 can be rearranged as fl $L1 + f2 $L2 + f3 $L3 + f5 $L5 + f6 $L6 + fext $ext = f4 $L4 (2.13) and this equation can be written in matrix format as f, f2 f3 1$L1 $L2 $L3 $L5 $L6 $ext 4 $L4 (2.14) 5 f fext where the term [$L1 $L2 ... $ext] is a 6x6 matrix whose columns are the unitized coordinates of the lines along legs 1, 2, 3, 5, and 6 and the unitized screw coordinates of the given external wrench. Both sides are then divided by the force magnitude f4. The result can be written as f2' 1$L1 $L2 $L3 $L5 $L6 $Lext f' L4 (2.15) 6f' fe't ext where f. = (2.16) f4 The ratios of the six forces on the left hand side of Equation (2.14) to f4 are found by multiplying both sides of Equation (2.15) by the inverse of the matrix containing the screw information. This can be written as f2 f 3 =[L1 $L2 L3 $L5 $L6 $Lext L4 (2.17) f5 6f fext The results of Equation (2.17) are combined with the information regarding the potential energy stored in the springs to solve for the forces in the legs and the external force. Knowledge of the forces in the side ties yields the elongation of the springs. The elongation information, combined with the total length of the side ties, gives the amount of cable paid out to each leg. The equation for the total amount of potential energy stored in the three legs (U) is U k42 4+ k53g52 + k662 (2.18) U= k4 (2.18) 2 where 6i is the deformation of the spring along leg i. This equation is combined with the formula for the force in a spring, f =k*8 (2.19) to yield Equation (2.20), which can be written as f2 2 f2 U += 4 5 +6 /2. (2.20) k4 k5 k6/ When this is combined with the formula forfn then the total potential energy stored in the springs is given in terms of f4 and the ratios of the other forces as f2 f2 f2 2 U f= it 4 /2 (2.21) k4 k5 k6 Solving this equation for f4 yields f4= 2U *k4kk S2 2(2.22) k5k6+ k4k6 f5 +k4k5 f6 The values for f5 and f6 obtained from Equation (2.17) are used in Equation (2.22) to solve forf4. This value yields the other six forces that act on the top platform as f, f' f2 f2 f f' S f4 3 (2.23) fe f 5 5 f6 f' fext fext The elongations of the springs are obtained by rearranging Equation (2.19) into the form 8=f (2.24) k 2.7 Obtain Leg and Cable Lengths Once the spring deformations are known, the desired information of the reverse analysis can be found using a threestep process. The first step is to obtain the endpoints of each leg in space. This is done by using Equations (2.1), (2.2), and (2.3) in conjunction with Equation (2.5). The second step is to calculate the equations of the lines representing the legs. The last step is to calculate the forces in the side ties, which yields the spring elongations and the cable lengths in those legs. After the global point vectors are obtained for the joints one equation remains to yield the values of the leg lengths. A leg length is equal to the length of the line segment connecting its two joints. This magnitude is found by using Equation (2.10). This represents the required value for the noncompliant legs, but for the sideties the length of cable was required. This length is found by subtracting the sum of the spring elongation and the spring free length from the total leg length found in the above equation. The equations for the lengths of the struts and of the cable in each side tie are given below as Equation (2.25) and (2.26). m2 2 L,, = 1,2,3 + ,2,3 + ,2,3 (2.25) 4,5,6 22,3+ 122,3 +n1,2,3 04,5,6 4,5,6 (2.26) These equations result in the desired information from the problem statement. The values for the spring free lengths, 056 are given in the problem statement. 2.8 On the Requirement for an External Wrench If there was no external wrench applied to the mechanism, the force balance equation could be written as fl 0 f, 0 [SL1 $L2 $L3 $L4 $L5 L6 0(2.27) f, 0 f, 0 f6 0 There are two situations in which this equation can hold true. One is the trivial case where all of the forces are zero. The other requires at least one of the lines to be linearly dependant on one or more of the other lines, i.e. the rank of the matrix of line coordinates must be less than six. This can only occur for special configurations of the mechanism's legs and it is said that the mechanism position is singular. That fact limits the equilibrium position and orientation of the top platform to those few states where the line coordinates are linearly dependent. Since the force balance equation is only true at those positions, the device will only be static at those positions. If it is desired for the top platform to attain any desired position and orientation, then the external wrench of the problem statement must be applied. In space, some linear combination of six independent forces is required to produce a general wrench. This means that as long as the line coordinates of the legs are linearly independent then any external wrench applied to the mechanism can be equilibrated by the summation of forces in the six legs. For clarity, this concept will be illustrated in the plane. The coordinates of a force in the plane can be described by three variables: 1, m, and r. These represent the force's x component, y component, and resulting moment about the zaxis. In the plane, three independent forces are required in some linear combination to produce a general force (the same is true for lines). These three forces, however, cannot create a resultant force of zero magnitude. There are only two cases where the sum of three forces in the plane can equal zero, either the magnitude of each is zero or they intersect at one point. In addition to intersecting at a single point, it is required that the three forces exist in a certain linear combination, but there is no combination of the forces that can yield zero if they do not all intersect at a single point. Three forces, which would result in a zero magnitude force if they intersected at a point, produce a resultant moment about the zaxis when they do not intersect at a single point. That they intersect at a single point indicates that they are linearly dependent. Linear dependence of lines can be expressed thusly. If a line exists in a set, and can be produced by a linear combination of some other lines in the set, then the set is linearly dependent. It follows that there is not a set of four linearly independent lines in the plane (seven in space). All the linear combinations of any two lines in the plane create a planar pencil of lines through the intersection of the two. That is, every possible line that could pass through the intersection point. Thus if three lines pass through a point they must be linearly dependent. Conversely, a linear combination of two lines cannot result in a general line, only one that passes through their point of intersection. An illustration of a planar pencil is given in Figure 22. If three nonzero forces are to have a resultant of zero, then they must intersect in a point. Thus, they must be linearly dependent. If given the lines of action for two forces in the plane and asked to find a third force which, when summed with the other two, will not have any resultant, then the placement of the third force is limited to cases where it is directed through the point of intersection of the two. This applies to the device under consideration in that it is limited to certain configurations in the absence of an external wrench. In order for six forces in space to be dependent, though they need not all intersect at a point, there are still restrictions on their arrangement. These restrictions prohibit the tensegritybased mechanism discussed thus far from having infinite mobility. Infinite mobility is used here in the sense that the mechanism would not be continuously in equilibrium from one arbitrary position and orientation to another. An external wrench is applied so that the forces in the legs can equilibrate with something other than zero. They can equilibrate with a wrench that acts along a general screw when they are in any arrangement other than a singularity position. Figure 22. Planar Pencil of Lines 2.9 Numerical Examples 2.9.1 Example 1 In example one the top and bottom platforms are of equal size. The joints meet at three points per platform. The three spring constants are equal, as are their free lengths. There is no pitch to the screw of action of the external wrench. That screw passes through the center of each platform. For a set of values given below, the solution follows. It = lb = 20.0 cm C = 0.0 k4 = k5 k6 = 20.0 N/cm 1, =10, = 1, =3.0 cm U= 40.0 N cm 0.893 0.325 0.312 8 0.326 0.944 0.051 5 T 0.312 0.056 0.949 18 0 0 0 1 0.516 0.083 0.853 $ext $et 4.924 8.528 2.147 Incidentally, the given transformation matrix describes translations in the x, y, and z directions of 8, 5, and 18 cm respectively. It also denotes a rotation of 3.1 degrees about its new xaxis, then a rotation of 18.2 degrees about its new yaxis. The final process that the top coordinate system undergoes is a rotation of minus 20 degrees about its zaxis. The external screw has no pitch and lies on a line that passes through the center of each platform. Using the given values for It, lb, and a with Equations (2.1), (2.2), and (2.3) yields the point coordinates for the joints of the bottom and top platforms in the bottom and top coordinate systems respectively. Since a is equal to zero in this example the total joints of each platform lie on only three points. The local coordinates of those points are Bp =T14=[0 0 0]cm, BP2=TP =[20 0 0]cm, Bp =TP =[10 17.32 0]cm. The preceding coordinates for points four, five, and six are in the top platform coordinate system. To transform these point coordinates into the base coordinate system, they, along with the given transformation matrix, are used in Equations (2.5) and (2.6) to yield the following global coordinates: BP4 =[8 5 18]cm, BP =[25.851 1.52 11.769]cm, BP6 =[22.559 18.09 13.915]cm. The six joint coordinates in the base system can be used in Equation (2.8) to yield the following Plicker coordinates: 8 5.854 12.554 12 15.854 22.554 5 1.513 0.775 5 18.833 18.096 18 11.769 13.917 18 11.769 13.917 SL2 L3 L4 L5 0 0 241.049 0 203.84 0 0 235.374 139.17 360 117.687 0 0 30.259 209.696 100 462.927 0 Taking the magnitudes of the lines given above gives the total leg lengths. This is accomplished using equation (2.10). Those leg lengths are L1 = 20.322 cm, L2= 13.231 cm, L3 = 18.759 cm, L4 = 22.204 cm, L5 = 27.286 cm, L6 32.091 cm. The line coordinates in this matrix are unitized using Equations (2.9) and (2.10). The unitized coordinates of these lines and the external screw are then used in the force balance equation, Equation (2.12) as 0.394 0.246 0.886 0 0 0 0.442 0.114 0.889 0 17.790 2.287 0.669 0.041 0.742 12.85 7.419 11.179 0.54 0.225 0.811 0 16.214 4.504 0.581 0.69 0.431 7.47 4.313 16.966 0.703 0.564 0.434 0 0 0 0.516 0.083 0.853 4.924 8.528 2.144 fl f2 f3 f4 f5 f6 fext These values are rearranged into the form of Equation (2.14) to yield 0.394 0.246 0.886 0 0 0 0.442 0.114 0.889 0 17.790 2.287 0.669 0.041 0.742 12.85 7.419 11.179 0.581 0.69 0.431 7.47 4.313 16.966 0.703 0.564 0.434 0 0 0 0.516 0.083 0.853 4.924 8.528 2.144 f3 f2 e, f _ext 0.54 0.225 0.811 0 16.214 4.504 Both sides of this equation are multiplied by the inverse of the 6x6 matrix to yield the following values: f/ 1.583 f, 1.821 f3 1.91 f5 1.41 f' 1.386 2.846 fext The value of f4 was found by placing values above and the given spring data in Equation (2.22). The resulting value is f4 = 18.147 N This value is used along with Equation (2.15) to yield values for the remaining forces acting on the top platform. f, 28.73 f2 33.046 f, 34.664 = N. f, 25.584 f6 24.822 fext 51.649 Equation (2.26) can be used to find the amount of cable paid out to each side tie by using the forces in the side ties (f4, f5, and f6), the given spring data, and the leg lengths found above. The results of that process are 1c4 = 18.297 cm, Ic5 23.007 cm, c6 = 27.850 cm. These values complete the reverse analysis solution for a tensegritybased parallel platform device. 2.9.2 Example 2 The given data for this example are identical to those of Example 1 except for the value of the potential energy. U= 87.20 N cm The solution in this example proceeds in the same manner as that of the previous one. The effect of the larger potential energy is not noticeable in the solution until the value for f4 is calculated as f4 = 26.794 N The ratios of the remaining forces that act on the top platform to f4 are equal to those found in Example 1. The magnitudes of the remaining forces are f, 42.42 f, 48.792 f3 51.181 f, 37.774 f, 36.649 fext 76.259 The length of cable paid out to each side tie is found in an identical manner to that found in Example 1. Those lengths are given below. 1c4 = 17.864 cm Ic5 = 22.397 cm 1c6 27.259 cm 2.9.3 Example 3 The supplied information in Example 3 is identical to that in Example 1 except for the screw of action of the external wrench. This unitized screw is given below. 0.032 0.599 0.8 ext 10.209 10.474 8.032 This screw has the value 0.17 cm for its pitch. The data for this screw are combined below with the unitized line coordinates found in Example 1 to make an equation of the form of Equation (2.14) as 0.394 0.442 0.669 0.581 0.703 0.032 f, 0.54 0.246 0.114 0.041 0.69 0.564 0.599 f2 0.225 0.886 0.889 0.742 0.431 0.434 0.8 f3 0.811 0 0 12.85 7.47 0 10.209 f5 0 0 17.790 7.419 4.313 0 10.474 f6 16.214 0 2.287 11.179 16.966 0 8.032 4.504 This equation is solved to yield f, 18.371 f2 12.816 f3 23.074 f 7.91 f 38.488 fet 23.256 fext These yield the value f4= 7.491N . This force magnitude allows for the solution to complete through the application of first Equation (2.15) then Equation (2.26). The amount of cable paid out to each compliant leg is listed below. 1c4 = 18.829 cm 1c5 23.890 cm Ic6 = 27.167 cm 2.9.4 Example 4 In example four the top and bottom platforms are of unequal size. The end joints are fixed at six points per platform. The three spring constants are also unequal. Their free lengths are the same. The screw of action of the external wrench has a pitch of 0.3. For a set of values given below, the solution follows. It= 18.0 cm lb = 22.0 cm rt =0.07 b = 0.16 k4 = 18.0 N/cm k = 23.0 N/cm k6 = 30.0 N/cm o4 = =06 = 3.0 cm U= 150.0 N cm 0.669 0.743 0 5 0.732 0.659 0.174 6 T 0.129 0.116 0.985 16 0 0 0 1 0.389 0.167 0.906 $ ext ex 8.203 11.309 1.773 Using the given values for lt, lb, ot, and ab with Equations (2.1), (2.2), and (2.3) yields the point coordinates for the joints of the bottom and top platforms in the bottom and top coordinate systems respectively as BP =TP4=[0 0 0]cm, B42= [18.48 0 0]cm, Bp =[22 0 0]cm, p53= [12.76 16.004 0]cm, B 3=[11 19.053 0]cm, BP 1=[1.76 3.048 0]cm, TP5 =[16.74 0 0]cm, P,= [18 0 0]cm, P36 =[9.63 14.497 0]cm, T3 =[9 15.588 0]cm, P14= [0.63 1.091 0]cm, To transform point coordinates from the system that is relative to the top platform into the base coordinate system, they, along with the given transformation matrix, are used in Equations (2.5) and (2.6) to yield the following global coordinates: Bp =[5 6 16]cm, BP25=[16.201 6.251 18.16]cm, BP, =[17.044 7.173 18.323]cm, BP36= [0.670 10.601 18.927]cm, BP =[0.562 10.859 18.973]cm. B14 =[4.611 4.82 16.208]cm. The six joint coordinates in the base system can be used in Equation (2.8) to yield the following Plicker coordinates: 4.611 5.799 10.33 13.48 4.284 2.322 4.82 6.251 8.452 6 8.831 7.811 5 $ 16.208 18.16 18.927 16 18.323 18.973 0 0 360611 0 293.241 57.836 0 399525 208199 295.68 233.799 33.392 0 137527 103.842 11088 181248 20.826 Taking the magnitudes of the lines given above gives the total leg lengths. This is accomplished using equation (2.10). Those leg lengths are L1 = 17.527 cm, L2 = 20.062 cm, L3 23.16cm, L4 21.765 cm, L5 = 20.786 cm, L6 = 20.648 cm. The line coordinates in this matrix are unitized using Equations (2.9) and (2.10). The unitized coordinates of these lines and the external screw are then used in the force balance equation, Equation (2.12) as 0.263 0.275 0.925 0 0 0 0.289 0.312 0.905 0 19.914 6.855 0.446 0.365 0.817 15.571 8.99 4.484 0.619 0.276 0.735 0 13.585 5.094 0.206 0.425 0.881 14.108 11.248 8.72 0.112 0.378 0.919 2.801 1.617 1.009 0.389 0.167 0.906 8.203 11.309 1.773 fl f2 f3 f4 f5 f6 fext This data is rearranged into the form of Equation (2.14) to yield 0.394 0.246 0.886 0 0 0 0.442 0.114 0.889 0 17.790 2.287 0.669 0.041 0.742 12.85 7.419 11.179 0.581 0.69 0.431 7.47 4.313 16.966 0.703 0.564 0.434 0 0 0 0.516 0.083 0.853 4.924 8.528 2.144 f2 f3 f5 f ext 0.54 0.225 0.811 0 16.214 4.504 Both sides of this equation are multiplied by the inverse of the 6x6 matrix to yield the following values: f, f3 f5 f6 ext 0.234 0.711 1.998 0.19 0.553 4.309 The value of f4 was found by placing values above and the given spring data in Equation (2.22). The resulting value is f4 = 66.757 N This value is used along with Equation (2.15) to yield values for the remaining forces acting on the top platform. f, 15.596 f2 47.435 f3 133.406 S= N. f, 12.673 f, 36.92 fext 287.642 Equation (2.26) can be used to find the amount of cable paid out to each side tie by using the forces in the side ties (f4, f5, and f6), the given spring data, and the leg lengths found above. The results of that process are 1c4 = 15.056 cm, Ic5 = 17.235 cm, 1c6 = 16.417 cm. These values complete the reverse analysis solution for a tensegritybased parallel platform device. CHAPTER 3 TENSEGRITYBASED PARALLEL PLATFORM WITH SEVENTH LEG 3.1 Application of a Seventh Leg This chapter introduces the concept of adding a seventh leg to the mechanism. The seventh leg consists of a prismatic connector attached to the top and bottom platforms by a Hooke and ballandsocket joint respectively. The purpose of this leg is to apply an "external force" to the platform of the tensegritybased 33 inparallel platform device. The six other legs are to equilibrate with this force. As shown in Chapter two, section nine, a pure applied force is sufficient for the device to obtain a general configuration. For the external wrench to model a pure force, its pitch must equal zero. This means that a force applied by the seventh leg as described above is sufficient for the obtaining of a solution. Though the magnitude of this force is not controllable with a prismatic joint, the length of the seventh leg need only be set and the proper resultant force will arise naturally. The leg lengths for this type of mechanism are found using the method of the previous chapter, substituting the unitized Plticker Coordinates of the seventh leg for the external screw. Placement of the endjoints of the seventh leg is important Chapter two, section nine, demonstrates that solutions for identical configurations (potential energy, spring and platform characteristics, and position and orientation) existed for different screws along which the external wrench acted. This quality of having multiple possible external screws of action for a given configuration implies that the distinction of the external screw of action is not paramount to the solution. Section eight of the same chapter however describes to what extent this distinction is important. The external screw of action cannot be placed in such a way that it is dependent on any five of the legs. This applies to the placement of the seventh leg of the device. If the seventh leg is a slider joint, then in a given position and orientation of the device, the seventh leg must lie along a line such that the determinant of the six by six matrix in Equation (2.13) is not zero. If the determinant is zero, then the analysis cannot be solved in the manner previously discussed, because the matrix cannot be inverted. It is shown in the next section of this chapter that if the determinant equals zero, then either the characteristics of the seventh leg can be changed or the value of a can be changed, and a solution will exist. A computer program was written that would both create depictions of, and provide solutions for, sevenlegged tensegritybased parallel platform devices, provided that they are in nonsingular configurations. If the matrix discussed above is singular, then the device will be rendered, but no solution will be produced. 3.2 Numerical Examples Three renderings of a sevenlegged tensegritybased parallelplatform device, Figure 31, 32, and 33, are shown in 3.2.1, 3.2.2, and 3.2.3. The blueandgrey legs in the figures represent the compliant ties, the blue portions depicting the springs. The greenandyellow legs are thus the struts. The device in these drawings is in the same configuration except that in Figure 32 the seventh leg acts along a screw of nonzero pitch, and in Figure 33 the joints are offset 20% of the platform length. The seventh leg is means of applying the required external wrench to the device. For this reason its properties are represented in the following examples in the same way those of the external force were denoted in the examples of the previous chapter. For example the magnitude of the force acting along the seventh leg is given by the variable fext 3.2.1 Example 1 In this example the device is translated in the three principal directions and not rotated about any axes. The transformation matrix for this process is 100 8 0 1 0 8 T = 0 0 1 16 0 0 0 1 The platform lengths, the joint offset, and the spring data are given here: It = lb = 20.0 cm C = 0.0 k4 = k5 k6 = 20.0 N/cm 10 = 10 = 1, =3.0 cm U= 40.0 N cm The endjoints for the seventh leg of this device are placed in the center of each platform. The point coordinates of these joints, Pext and Px, in a system local to the platform of each joint, are BPexB Tp =[10 5.77 0]cm. The point vector of the top joint is then transformed into global coordinates by using Equation (2.5). The resulting point coordinates are BPext =[18 13.77 16] cm. In this example the force in the seventh leg acts on the top platform along a screw of zero pitch. Thus the equation of this screw, when unitized, is equal to the unitized Plticker coordinates of a line along which the leg lies. Those coordinates are obtained from Equation (2.8) and ca be written as 0.408 0.408 0.816 $ext $ t 4.711 8.165 1.727 Using the given values for It, lb, and a with Equations (2.1), (2.2), and (2.3) yields the point coordinates for the joints of the bottom and top platforms in the bottom and top coordinate systems respectively. Since a is equal to zero in this example the total joints of each platform lie on only three points. The local coordinates of those points are BP =TP4=[0 0 0]cm, BP2=TP =[20 0 0]cm, B =T 6 =[10 17.32 0]cm. The preceding coordinates for points four, five, and six are in the top platform coordinate system. To transform these point coordinates into the base coordinate system they, along with the given transformation matrix, are used in Equations (2.5) and (2.6) to yield Bp = [8 8 16]cm, BP =[28 8 16]cm, Bp =[18 25.32 16]cm. The joint coordinates of legs one through six can be used in Equation (2.8) to yield their Plicker coordinates, which are expressed in matrix form as [$L1 $L2 $L3 $L5 $L6]: 8 8 16 0 320 160 8 8 16 277.128 160 58.564 12 8 16 0 320 160 18 9.32 16 277.128 160 404.974 18 25.32 16 0 0 0 The line coordinates in this matrix are unitized using Equations (2.9) and (2.10). The unitized coordinates are then used in the force balance equation, Equation (2.12) as 0.408 0.408 0.816 0 0 0 0.408 0.408 0.816 0 16.33 8.165 0.408 0.408 0.816 14.142 8.165 2.989 0.557 0.371 0.743 0 14.856 7.428 0.697 0.361 0.62 10.731 6.196 15.682 0.515 0.725 0.458 0 0 0 0.408 0.408 0.816 4.711 8.165 1.727 f2 f3 f4 f5 f6 fext This data is rearranged into the form of Equation (2.14) to yield 0.408 0.408 0.816 0 0 0 0.408 0.408 0.816 0 16.33 8.165 0.408 0.408 0.816 14.142 8.165 2.989 0.697 0.361 0.620 10.731 6.196 15.682 0.515 0.725 0.458 0 0 0 0.408 0.408 0.816 4.711 8.165 1.727 f, f3 f5 f6 fext 0.557 0.371 0.743 0 14.856 7.428 The square matrix above cannot be inverted, thus there is no solution to this equation. The results of inputting this example into the program described earlier are shown below in Figure 31. Figure 31. SevenLegged TensegrityBased Device in Singularity Position If a solution for this setup existed then the black window on the right side of the above figure would show a pictorial representation of the relative sizes of the force magnitudes in the legs. The next figure shows the device in the same position and orientation, but that a solution exists for the input of the next example. 3.2.2 Example 2 The relative magnitudes of the forces in the legs appear in the small window. From left to right they represent the force magnitudes fl, f2,f3, f4, f, f, and fex,. A solution exists in this example because the screw of action for the seventh leg is different from that in 3.2.1 Example 1. The information given in the problem description is the same for this example as it is for Example 1, except for the pitch of the external screw, which is equal to 0.67. Figure 32. SevenLegged Device with NonZero Pitch on the Seventh Screw Here the Pluicker coordinates for the line along which the leg lies are the same, but they are not equal to the equation of the external screw of action as shown in the previous example. In this example the unitized screw is found by using Equation (2.12) 0.408 0.408 0.816 4.958 7.891 2.274 This value is used with the unitized line coordinates of the other six legs from Example 1 to yield 0.408 0.408 0.816 0 0 0 0.408 0.408 0.816 0 16.33 8.165 0.408 0.408 0.816 14.142 8.165 2.989 0.697 0.361 0.620 10.731 6.196 15.682 0.515 0.725 0.458 0 0 0 0.408 0.408 0.816 4.985 7.891 2.274 fl f2t f ext 0.557 0.371 0.743 0 14.856 7.428 which is in the form of Equation (2.14). This square matrix can be inverted, and both sides of the equation are multiplied by that inverse to yield f2 f3 f5 f6 fext 7.583 6.927 7.818 1.199 1.622 19.598 This information is used with the spring data in Equation (2.22) to find f4= 17.766 N. The spring elongations and leg lengths can then be found as described in chapter two, section seven and illustrated in chapter two, section nine. 3.2.3 Example 3 The following example is identical to the first example of this chapter except that the joint offset is not zero. Instead S= 0.2 The results of inputting the data of this case into the previously mentioned program are illustrated below in Figure 33. Figure 33. SevenLegged Device with 20% Joint Offset 41 The relative magnitudes of the forces in the seven legs can again be seen in the black window on the right side of the figure above. Their values are f, 80.616 f2 66.933 f3 70.859 f4 = 17.889 N f, 21.337 f, 28.718 fxt 160.997 CHAPTER 4 CONCLUSIONS A reverseanalysis solution method for a device that is similar to a 33 inparallel platform is explored. This device differed from a 33 inparallel platform in that it is based on the principle oftensegrity. This characteristic allows the device to be varied in position and orientation as well with regards to its compliance characteristics. This device also differs in that it requires the application of an external wrench to achieve a general position and orientation. This paper discusses the need for that wrench It is put forward in this paper that, for a given position and orientation and with a given internal potential energy in the springs, there are an infinite number of external wrenches with which the six legs of the mechanism can equilibrate. It is shown that the potential energy can vary while the device remains in a constant position and orientation. The solution method of this paper is used on several tensegritybased devices of varying platform and spring characteristics. This shows that the solution method is applicable for a general tensegritybased 33 parallel platform device. A different parallelplatform device is proposed. It is based on tensegrity and has four noncompliant struts and three compliant ties. The placement of the seventh leg's joints on the platforms is shown there to be of importance to the feasibility of the solution. It is shown that the singularity problem arising from the seventh leg's location can be overcome either by changing the joint offset, or by altering the pitch of the screw of action of the seventh leg. There are facets of this device yet unstudied. Its compliance characteristics pose an interesting puzzle. A beneficial solution would be that of the ideal arrangement of leg end joints. Also, the forward analysis of this device must be considered. Here the mechanism dimensions, spring constants, and spring free lengths would be known together with the length of the noncompliant legs and the length of the noncompliant strings that are in series with the springs. The objective would be to determine all the possible positions and orientation of the top platform along with the associated potential energy at this equilibrium position. REFERENCES 1. Ball, R. A Treatise on the Theory of Screws, Cambridge University Press, New York, 1900 2. Burkhart, R. "A Technology for Designing Tensegrity Domes and Spheres," Tensegrity Solutions. Accessed 27 March 2003, http://www.channell.com/users/bobwb/prospect/prospect.htm#sec:intro 3. Duffy, J., Rooney, J., Knight, B., and Crane, C., "A Review of a Family of Self Deploying Tensegrity Structures with Elastic Ties," The Shock and Vibration Digest, Vol. 32, No. 2, Mar 2000, p. 100106. 4. Fuller, B. "Letter on Tensegrity: Section 1," Buckminster Fuller Institute. Accessed 1 March 2003, http://209.196.135.250/burkhardt/section1.html. 5. Kenner, H. Geodesic Math and How to Use It, University of California Press, Berkeley, 1976. 6. Knight, B., Zhang, Y., Duffy, J., and Crane, C., "On the Line Geometry of a Class of Tensegrity Structures," Sir Robert Stawell Ball 2000 Symposium, University of Cambridge, UK, July 2000. 7. Lee, J., "Investigations of Quality Indices of InParallel Platform Manipulators and Development of Web Based Analysis Tool," Ph.D. dissertation, University of Florida, 2000. 8. Oppenheim, I., "Mechanics of Tensegrity Prisms," study for CarnegieMellon University, Pittsburgh. 9. Skelton, R., "Smart Tensegrity Wings," Accessed 20 March 2003, www.darpa.mil/dso/thrust/matdev/chap/briefings/timchap2000day2/tensegrity_sk elton.pdf 10. Stern, Ian., "Development of Design Equations for SelfDeployable NStrut Tensegrity Systems," MS thesis, University of Florida, 1999. 11. Tibert, G., "Deployable Tensegrity Structrues for Space Applications," Doctoral Thesis, Royal Institute of Technology, Department of Mechanics, Stockholm, Sweden, 2002. 45 12. Tran, T., "Reverse Displacement Analysis for Tensegrity Structures," Master of Science Thesis, University of Florida, 2002. 13. Zhang, Y., Duffy, J., and Crane, C., "The Optimum Quality Index for a Spatial Redundant 48 InParallel Manipulator," Proceedings of the Advances in Robot Kinematics Conference, Piran, Slovenia, June 2000, p. 239248. BIOGRAPHICAL SKETCH Mr. Matthew Quincy Marshall was born in 1978 in DeLand, Florida. In 2001 he received a Bachelor of Science degree in mechanical engineering at the University of Florida. He worked for Exponent Failure Analysis Corp., as an intern. He returned to the University of Florida in August 2001 to garner his Master of Science degree in mechanical engineering. 