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PAGE 1 ANALYSIS OF THREE DEGREE OF FREEDOM6 6 TENSEGRITY PLATFORM By ANTOIN LENARD BAKER A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMEN T OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005 PAGE 2 Copyright 2005 By Antoin Lenard Baker PAGE 3 iii ACKNOWLEDGEMENTS I would like to thank Dr. Carl Crane fo r being the chairperson of my committee and providing his technical expe rtise during my time as a mast ers degree student. With his indepth knowledge, we have overcome many obstacles that hindered the solution of this analysis. In addition, I would also like to thank Dr. John Schueller, Dr. John Ziegert and Dr. Brian Mann for serving on my comm ittee and providing me with valuable insight. I thank my parents, Clinton and Lorraine, for their love and support. Last but not least, I thank my brother for keeping my head on straight and keeping me grounded. PAGE 4 iv TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.iii LIST OF FIGURES..v LIST OF TABLESvii ABSTRACT.....viii CHAPTER 1 INTRODUCTION.1 2 DETERMING THE ORIENTATION OF TOP PLATFORM......7 Defining the Equations of a Generic Spherical Four Bar. Defining the Equations of th e Three Spherical Four Bars... Relationships Between and .19 Solution of the Three Spherical Equations...21 Numerical Conformation of Results.25 Orientation of Top Pl atform Using Quaternions..28 3 MOTION ANALYSIS USI NG RECIPROCAL SCREWS..33 4 CONCLUSION .37 APPENDIX REFERENCES..41 BIOGRAPHICAL SKETCH.42 PAGE 5 v LIST OF FIGURES Figure page 11: Air Balloon and Geodesic Domes as Tensegrity Examples 12: 3x3 Platform........3 13: 6x3 PlatformAlso Called a Stewart Platform.....3 14: General 6x6 Platform.......4 15: 6x6 Tensegrity Platform...5 21: Model Used for Anal ysis of 6x6 Platform...8 22: Angle 1 .10 23: Angle 2 .....11 24: Angle 3 ......11 25: Spherical Four Bar Mechanism... 26: First Spherical Quadrilateral... 27: Second Spheri cal Quadrilateral.. ....17 28: Third Spherical Quadrilateral.. 29: Visual Repr esentation of angle 2 ..........20 210: Definition of Point E1....30 211: Orientation from Analysis Results 1..31 212: Orientation from Analysis Results 2......31 213: Orientation from Analysis Results 3......32 PAGE 6 vi 214: Orientation from Analysis Results 4......32 31: 6x6 Platform Supporting External Wrench......33 PAGE 7 vii LIST OF TABLES Table page 21: Relationship Between Generic Sphe rical Quadrilateral to Three Cases...19 22: Leg Lengths...26 23: Coefficients of Eq uation (2.18) through (2.20).27 24: Results of Analysis PAGE 8 viii Abstract of Thesis Presente d to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree Master of Science ANALYSIS OF THREE DEGREE OF FREEDOM 6 6 TENSEGRITY PLATFORM By Antoin Lenard Baker May 2005 Chairperson: Dr. Carl Crane Major Department: Mechanical Engineering The mechanism studied in this paper is a three degree of freedom 6 6 tensegrity mechanism. A tensegrity structure is one that balances internal (pre stressed) forces of tension and compression. These structures have the unique property of stabilizing themselves if subjected to certain types of di sturbances. The structure analyzed in this paper consists of two rigid bodi es (platforms) connected by a total of six members. Three of the members are noncomplia nt constantlength struts and the other three members consist of springs. For typical parallel mechanisms, if the bottom platform is connected to ground and the top platform is connected to the base by six leg connectors, the top platform will have six degrees of freedom relative to the bottom platform. However, because three of the six members connecti ng two platforms are noncompliant constantlength struts, the top platform has only three degrees of freedom. The orientation of the top platform is not known rela tive to the bottom platform. PAGE 9 ix The primary contribution of this thesis is the analysis of the three degree of freedom tensegrity platform. Specifically, gi ven the location of the connector points on the base and top platforms, the lengths of the three noncompliant constantlength struts, and the desired location of a point embedded in the top platform measured with respect to a coordinate system attached to the base, all possible orientations of the top platform are determined. The thesis presents the devel opment of three equations in three variables that are based on the analysis of three sphe rical four bar mechanisms that are present within the tensegrity mechanism. These thr ee equations are then manipulated in order to obtain a single sixteenth de gree polynomial in one of the variables. Unique corresponding values for the remaining two va riables are then obtai ned and thus it is shown that a maximum of sixteen possi ble orientation solutions can exist. The thesis also presents a static force/to rque analysis of the device in order to determine the external wrench that must be applied to the top platform to maintain equilibrium when the point embedded in the t op platform is positioned as desired. Twists that are reciprocal to this wrench are iden tified as these represent motion of the top platform upon which the extern al wrench performs no work. PAGE 10 1 CHAPTER1 INTRODUCTION The word tensegrity is a combination of the words tension and integrity Tensegrity describes a structural relationshi p principle in which structural shape is guaranteed by the closed, continuous, tensiona l behaviors of the system and not by the discontinuous compressional member behaviors. Tensegrity provide s the ability of a structure to yield increasingl y without ultimately breaking.1 Houses can and do collapse when a su fficient horizontal (shearing) force is applied above the foundation. The walls act as levers, creating torque. The house deals with this torque by destroying itself. Beams and cables have strength in compression and tension respectively, but both ar e poor in resisting torque.2 Because torque is the cause of failure in many structures, the design challenge was to come up with a structure that can s upport loads without generating torque. This design criterion is how tensegrity structures we re born. Tensegrity st ructures are able to support a load in the vertical and horizontal directions without cr eating any torque or moment throughout the structure. Tensegrity is the case when compression and tension are said to be in balance with each other. When these two forces are in balance, the struct ure is said to be optimally strong. An example of tensegrity that everyone is familiar with includes the muscularskeletal system. For this case the muscles are in tension, while the skeletal system is in compression. Other examples in clude air balloons or automobile tires. The PAGE 11 2 air molecules provide a discontinuous pus h while the rubber provides continuous tension.3 A final example of a tensegrity st ructure includes geodesic domes. These structures are shown in Figure 11. Figure 11. Air Balloon and Geodesic Dome as Tensegrity Examples This paper will present an analysis of the geometric properties of platforms which incorporate tensegrity principles. A platform is described as any device that has multiple legs connecting a moving (top) platform to a bottom (base) platform.4 It was only recently that the forward position solution of a 3 3 platform was formulated by Duffy and Griffis.5 In this analysis, all positions and orientations of the top platform are determined based on given lengths of the six leg connectors. This 3 3 platform (see Figure 12) was the simplest of the geometri es to solve. The formulation of these solutions yielded and eighth degree polynomial in the square of one defining parameter. Later, this solution technique was applied to a 3 6 platform (see Figure 13).6 A PAGE 12 3 forward displacement analysis of a general 6 6 platform (see Figure 14) showed that the solution was in the form of a 40th degree polynomial.7 Figure 12. 3 3 Platform Figure 13. 3 6 PlatformAlso Called a Stewart Platform PAGE 13 4 Figure 14. General 6 6 Platform The tensegrity structure analyzed in this paper consists of a special6 6 platform. This geometry was designed by Griffis a nd Duffy to include the benefits of a 3 3 platform and a general 6 6 platform.8 This special platform makes the analysis comparable to a 3 3 platform while eliminating m echanical interference associated with the 3 3 platform. The 6 6 platform analyzed in this paper consists of two rigid bodies connected together by three constant length noncom pliant struts and three compliant ties that each consists of a spring in series with a noncompliant tie where the length of the noncompliant tie can be cont rolled. The legs ar e connected to the platforms with ball and socket joints. For th is analysis, the bottom platform is connected to the ground and the top platform has three degrees of freedom relative to the bottom platform. The special 6 6 tensegrity platform is shown in Figure 15. PAGE 14 5 Figure 15. 6 6 Tensegrity Platform If the three struts for this mechanism we re able to change in length, the position and orientation of the top platform could be determined when given the six leg connector lengths as was done by Duffy and Griffis.5 This device would have six degrees of freedom. However, for the case presented in this paper, the struts are noncompliant and are of constant length. This reduces the syst em to a three degree of freedom device. Chapter 2 discusses how the orientations of th e top platform are determined with respect to a coordinate system that is connected to the ground when given th e desired position of a single point in the top platform. All orie ntations that satisf y the length condition associated with the constant length struts are determined. PAGE 15 6 Another important aspect of this pape r revolves around the concept of reciprocal screws. For any position and orientation of th e top platform, there ex ists certain rotations and/or translations (twists) that the top platform can be moved along to eliminate any work being performed the external force/moment (wrench) that holds the top platform in static equilibrium. This analysis will be pe rformed using the concept of reciprocal screws and will be discussed in Chapter 3. Although beyond the scope of this thesis, th e future goal of this analysis is to determine if the point embedded in the top pl atform can be moved to a desired goal point along a path where the instantane ous twist of motion is always instantaneously reciprocal to the applied external wrench that maintains static equili brium. By moving along this path, the energy required to move the top platform would be minimized. PAGE 16 7 CHAPTER 2 ORIENTATION OF TOP PLATFORM This chapter presents the solution of how to find all possible orientations of the top platform with respect the bottom platform when a point embedded in the top platform is positioned at a desired location. The model us ed for this analysis is shown in Figure 21. Points B1, B2, B3, and T1, T2, T3 correspond to the centers of the spherical joints at the bottom and top ends of the th ree constant length noncompliant struts which are numbered 2, 5, and 8 in the figure. Point P is a point that is embedded in the top platform. Two coordinate systems are defined for this problem. The first is attached to the base platform with its origin at point B1 and X axis through B2. Point B3 lies in the XY plane such that the Z axis is defined as pa rallel to the cr oss product of th e vector along the X axis with the vector from B1 to B3. This first coordinate system is shown in the figure. The second coordinate sy stem is attached to the top platform. The origin is at point T1 and its X axis passes through point T2. Point T3 is in the XY plane and the direction of the Z axis is de fined in a similar manner as in the prior case. The precise problem statement is now presented as follows: PAGE 17 8 Figure 21. Model Used for Analysis of 6 6 Platform B1 B2 B3 T1 T2 T3 P 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x1y1 z1 PAGE 18 9 Given all dimensions of the top and bottom platforms, i.e. the lengths L10, L11, L12, L13, L14, and L15, where the notion Li refers to the length of bar i, the lengths of the three constant length noncompliant struts, L2, L5, and L8, the coordinates of point P in the 2nd (top) coordinate system, which implies that the lengths L3, L6, and L9 are known, the coordinates of point P in the 1st (base) coordinate system, which implies that the lengths L1, L4, and L7 are known Find all possible orientations of the top plat form, which can be represented by the 3 rotation matrix R1 2. It is important to note that based on the pr oblem statement, the lengths of all fifteen line segments shown in Figure 21 are known. Al so it is helpful to visualize the problem as that of having two tetrahed rons, one defined by points B1, B2, B3, and P and the other by points T1, T2, T3, and P, that share the common point P. The problem can be thought of as that of determining all the possible relative orientations of the two tetrahedrons such that the three distance constraint s associated with the constant length struts are satisfied. PAGE 19 10 As stated in the given information, the lo cation of point P in the top platform is known in both the first and second coordinate system. However, the orientation of the top platform with respect to the bottom platform is not known. The analysis begins by identifying three ma jor planes that are fixed with respect to the ground coordinate system. These three planes consists of lin es (1712), (4711), and (1410). Because these planes all ha ve vertices at points known in the first coordinate system, the equations of thes e planes can be readily determined. Three new angles will now be defined, 1, 2, and 3. 1 is defined as the angle between planes formed by the line s 13 and 17 (see Figure 22). 2 is defined as the angle between planes formed by the lines 97 and 17 (see Figure 23). 3 is defined as the angle between planes formed by th e lines 46 and 47 (see Figure 24). Figure 22. Angle 1 ; Figure on right shows view looking along line 1 from point P to point B1 1 P B1 PAGE 20 11 Figure 23. Angle 2 ; Figure on right shows view looking along line 7 from point P to point B3 Figure 24. Angle 3 ; Figure on right shows view looking along line 4 from point P to point B2 P B3 2 P B2 3 PAGE 21 12 For the analysis it is possible to mode l the system as three spherical four bar mechanisms. These spherical four bar mechanisms are composed of lines (1397), (7964) and (4631) (see Figure 21). When a sphere of unit radi us is centered at point P, then the four planes passing through the unit circ le cut it into four ar cs. These four arcs form the spherical quadrilateral that will be us ed for this analysis. Also by choosing the spherical four bars in this fashion, the output angle of one spherical four bar mechanism will be the input angle of anothe r causing the equations of the three spherical four bars to be interrelated. 2.1 Defining the Equations of a Generic Spherical Four Bar A generic spherical four bar mechanism is shown if Figure 25. Imagine the a unit sphere is drawn with its center originati ng at an aribitrary point O. The unit vectors Si will meet this sphere at a series of point s, i=1,2,3,4 as shown in Figure 25. Links can be drawn on the unit sphere to join adjacent points, 12, 23, 34 and 41. For these links the angle between them is defined as ij For example, the angle between S1 and S2 is defined as 12. The angle j is defined as the angle between the links ij and jk 10. PAGE 22 13 Figure 25. Spherical Four Bar Mechanism Sets of sine, sinecosine, and cosine laws can be gene rated for the spherical four bar mechanism which contain the link angles, s, and the joint angles, s. One such spherical cosine law th at relates the angles 4 and 1, but does not contain the angles 2 or 3 can be written as Z41 = c23 (2.1) where Z41 is defined as Z41 = s12 (X4s1+Y4c1) + c12Z4 (2.2) The terms X4, Y4, and Z4 are defined as X4 = s34 s4 Y4 = (s41c34 + c41s34c4), Z4 = c41c34 s41s34 c4 (2.3) and where the terms si, ci, sij, and c12 are defined as PAGE 23 14 si = sin( i), ci = cos( i), sij = sin( ij), cij = cos( ij) (2.4) The following tanhalfangle trigonometric iden tity formulas are used to transform the spherical cosine law from a transcende ntal to an algebraic equation. 2 21 1i i ix x c 21 2i i ix x s (2.5) where 2 tani ix Substituting equations (2.3) through (2.5) into (2.2) and collecting terms gives 05 1 4 4 2 4 3 2 1 2 2 1 2 4 1 H x x H x H x H x x H (2.6) where 23 12 34 41 34 41 12 34 41 34 41 5 34 12 4 23 12 34 41 34 41 12 34 41 34 41 3 23 12 34 41 34 41 12 34 41 34 41 2 23 12 34 41 34 41 12 34 41 34 41 14 c c s s c c s s c c s H s s H c c s s c c s s c c s H c c s s c c s s c c s H c c s s c c s c s s c H (2.7) Equation (2.6) will be applied to the thr ee spherical four bar mechanisms defined previously. All the link angles, s, for all three of the spheri cal four bar mechanisms can PAGE 24 15 be determined in terms of the known line segment lengths shown in Figure 21. For example, given a triangle of sides (L1, L2 and L3), the cosine law of a planar triangle returns the interior angle opposite of the side of interest. For example the interior angle opposite of side L3 can be obtained as 2 1 2 3 2 2 2 12 arccos L L L L L (2.8) Therefore, for each of the three spheri cal four bar mechanisms, an equation corresponding to (2.6) can be wr itten where the coefficients H1 through H5 can be determined. 2.2 Defining the Equations of the Three Spherical Four Bars Now that the equation that defines a ge neric spherical quadrilateral has been presented, the goal is to now obtain the specifi c equations for the three spherical four bars present in the mechanism. Figure 26 show s the first spherical quadrilateral that is defined by lines (1793). By using an appropr iate exchange of subscripts in (2.6), the following equation which rela tes the defined angles 1 and 2 is obtained 05 1 2 4 2 1 3 2 2 2 2 1 2 2 1 a x x a x a x a x x a (2.10) PAGE 25 16 Figure 26. First S pherical Quadrilateral where 93 79 31 71 31 71 79 31 71 31 71 5 31 79 4 93 79 31 71 31 71 79 31 71 31 71 3 93 79 31 71 31 71 79 31 71 31 71 2 93 79 31 71 31 71 79 31 71 31 71 14 c c s s c c s s c c s a s s a c c s s c c s s c c s a c c s s c c s s c c s a c c s s c c s c s s c a (2.11) All the terms in (2.11) are expres sed in terms of known parameters. The same procedure is used to deve lop an equation relating the input/output relation for the second spherical quadrilateral (see Figure 27) whic h is defined by lines (7469). However for this case, the input angle is defined by the angle 2 (later it will be shown how the angle 2 can be expressed in terms of 2). By using an appropriate exchange of subscripts in (2.6) th e expression that relates the angles 2 and 3 can be written as Equation (2.12). 0 ) cos( ) cos( ) sin( ) sin( ) cos( ) cos(3 2 3 2 3 2 F E D B A (2.12) 3 9 1 7 1 2 PAGE 26 17 Figure 27. Second S pherical Quadrilateral Lastly, this procedure will be used to develop an equation relating the input/output relation for the third spherica l quadrilateral. Th is third spherical quadrilateral (see Figure 28) consists of lines (4136). By substituting these lines into the coefficients of the generic spherical quadril ateral (2.6), the coefficients for the third spherical quadrilate ral are obtained. 3 2 7 9 4 6 PAGE 27 18 Figure 28. Third S pherical Quadrilateral Defining 3 and 1 as the input/output angles. The general equation for the third spherical quadrilateral is shown in Equation (2.13). 0 ) cos( ) cos( ) sin( ) sin( ) cos( ) 3 cos(1 3 1 3 1 F E D B A (2.13) We now have three general equations that relate the input an d output angles for the three spherical quadrilaterals (2.10, 2. 12 and 2.13). To summarize, Table 21 shows the parameter substitutions that were used for the three spherical quadrilaterals. 3 1 3 1 4 6 PAGE 28 19 Table 21. Relationships Between Generi c Spherical Quadrilateral to Three Cases Generic Quadrilateral 4 1 1 4 2 1 3 2 4 3 1st Quadrilateral 1 2 7 1 9 7 3 9 1 3 2nd Quadrilateral 2 3 4 7 6 4 9 6 7 9 3rd Quadrilateral 3 1 1 4 3 1 6 3 4 6 2.3 Relationships Between i and i Now that equations for the three spherica l quadrilaterals are known, a relationship between the angles i and i must be determined. By re visiting Figures 26 through 28 the relationship between i and i can be written as the following. For example, a visual of 2 will be shown in Figure 29. 01 1 1 (2.14) 02 2 2 (2.15) 03 3 3 (2.16) PAGE 29 20 Figure 29. Visual Re presentation of angle2 The angles i can be obtained in terms of known qu antities. Using the properties of dot products and cross products, the anglei can be determined. Shown below is a guide to finding these angles. The angle 2 will be chosen as an example for clarity. 1. Find a Vector that passes through the axis of rotation. For 2 this vector will be (Point PPoint B3). 2. Unitize this Vector. 1 7 9 4 2 2 2 PAGE 30 21 3. Find a vector perpendicular to a plane cont aining the axis of rotation and the first point of interest. For 2 this first point of interest would be Point B2. 4. Unitize this Vector. 5. Find a second vector perpendicular to a plane containing the ax is of rotation and the second point of interest. For 2 this second point of interest would be Point B1. 6. Unitize this Vector. 7. Because all of the vectors are of unit value, the cosine of 2 will be the dot product of 4 & 6. 8. The sine of 2 will be 2 dot (4 6). 9. 2 = atan2 (sin 2, cos 2) 2.4 Solution of the Three Spherical Equations Now that an expressioni has been obtained, it will be substituted into the three spherical equations (2.10), (2. 12) and (2.13). After using th e tanhalf angle formulas and collecting terms, we obtain three equations in the parameters 1, 2, and 3 as follows 01 1 2 2 1 3 2 4 1 5 2 1 6 2 2 7 1 8 2 1 9 A x A x A x A x A x A x A x A x A (2.17) 01 3 2 2 3 3 2 4 3 5 2 3 6 2 2 7 3 8 2 3 9 B x B x B x B x B x B x B x B x B (2.18) 01 1 2 2 1 3 3 4 1 5 2 1 6 2 3 7 1 8 2 1 9 D x D x D x D x D x D x D x D x D (2.19) PAGE 31 22 Bezouts and Sylvesters methods will now be employed to solve these three equations. The coefficients of Equations (2 .17) through (2.19), which are all defined in terms of known quantities, are presented in the Appendix. First, let 7 1 8 2 1 9 1A x A x A L 7 3 8 2 3 9 2B x B x B L 4 1 5 2 1 6 1A x A x A M (2.20) 4 3 5 2 3 6 2B x B x B M 1 1 2 2 1 3 1A x A x A N 1 3 2 2 3 3 2B x B x B N Substituting equations (2.20) into (2.17) and (2.18) yields 01 2 1 2 2 1 N x M x L (2.21) 02 2 2 2 2 2 N x M x L (2.22) The condition equations (2.21) a nd (2.22) have a common root for 2x is the following: 02 2 2 1 1 2 2 1 1 2 2 1 1 N L N L N M N M M L M L (2.23) Expanding equation (2.23) a nd collecting terms gives 00 1 1 2 1 2 3 1 3 4 1 4 3 0 1 1 2 1 2 3 1 3 4 1 4 2 3 1 1 2 1 2 3 1 3 4 1 4 3 3 0 1 1 2 1 2 3 1 3 4 1 4 4 3 0 1 1 2 1 2 3 1 3 4 1 4 T x T x T x T x T x S x S x S x S x S x R x R x R x R x R x Q x Q x Q x Q x Q x P x P x P x P x Po (2.24) PAGE 32 23 where the terms Pi, Qi, Ri, Si, and Ti (i=1..4) are expressed in terms of known quantities. For simplification, equation (2 .24) is rewritten as 00 3 1 2 3 2 3 3 3 4 3 4 V x V x V x V x V (2.25) where V4 = P4x1 4 + P3x1 3 + P2x1 2 + P1x1 + P0 V3 = Q4x1 4 + Q3x1 3 + Q2x1 2 + Q1x1 + Q0 V4 = R4x1 4 + R3x1 3 + R2x1 2 + R1x1 + R0 (2.26) V4 = S4x1 4 + S3x1 3 + S2x1 2 + S1x1 + S0 V4 = T4x1 4 + T3x1 3 + T2x1 2 + T1x1 + T0 Equation (2.19) can be wr itten as the following: 00 3 1 2 3 2 W x W x W (2.27) where, 1 1 2 2 1 3 0 4 1 5 2 1 6 1 7 1 8 2 1 9 2D x D x D W D x D x D W D x D x D W (2.28) Multiplying Equation (2.25) by 3x and multiplying Equation (2.27) by 3 3 2 3 3, ,x x x, gives six equations in six unknowns which can be written in matrix format as PAGE 33 24 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 03 2 3 3 3 4 3 5 3 0 1 2 3 4 0 1 2 3 4 0 1 2 0 1 2 0 1 2 0 1 2x x x x x V V V V V V V V V V W W W W W W W W W W W W (2.29) The set of six homogeneous equations will have a solution only if the determinant of the coefficient matrix equals zero. Thus 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 2 3 4 0 1 2 3 4 0 1 2 0 1 2 0 1 2 0 1 2 V V V V V V V V V V W W W W W W W W W W W W (2.30) The terms Vi and Wi are polynomials in the term x1. Expanding this determinant yields a 16th degree polynomial in x1.Each value of 1x obtained by the polynomial can be substituted back into Equation (2. 29) to obtain corresponding values for3x. Once the corresponding values of 3x have been determined for each value of 1x, Equations (2.31) and (2.32) are used to determined the corresponding values of 2x as 2 2 1 1 2 2 1 1 2N L N L N M N M x (2.31) or PAGE 34 25 2 2 1 1 2 2 1 1 2M L M L N L N L x (2.32) 2.5 Numerical Conformation of Results To confirm that the results are correc t, a model was constructed using CAD software (see Figure 21). Afte r this model was constructed, all of the dimensions for the model were measured. The following informa tion was used as input for the numerical case: Given : Point P in 1st Coordinate System: (6.2486, 10, 3.4732) Point P in 2nd Coordinate System: (1.7590, 2, 1.9251) Point B1 in First Coordinate System: (0, 0, 0) Point B2 in First Coordinate System: (10, 0, 0) Point B3 in First Coordinate Syst em: (6.75, 0, 5.6292) Point T1 in Second Coordinate System: (0, 0, 0) Point T2 in Second Coordinate System: (4.9500, 0, 0) Point T3 in Second Coordinate System: (0.2658, 0, 5.5675) Constant Strut Lengths: L2=11.035, L5=9.970, L8=9.455 PAGE 35 26 Table 22 shows the calculated lengths of all fifteen of the line segments shown in Figure 21. Table 23 presents the calculated values for the coefficients of equations (2.18) through (2.20). For this case, four of the sixteen solu tions of equation (2.30) were real. Table 24 presents the calc ulated values for the parameters 1, 2, and 3 for these four cases. The table also shows that one of the solutions matches that of the CAD model. Table 22. Leg Lengths Leg Length 1 12.292 2 11.035 3 3.286 4 11.231 5 9.970 6 4.232 7 10.242 8 9.455 9 4.415 10 10 11 6.500 12 8.789 13 5.573 14 4.954 15 7.278 PAGE 36 27 Table 23. Coefficients of Equations (2.18) through (2.20) 9A 8A 7A 6A 5A 4A 3A 2A 1A 0.1541 0 0.8172 0 3.2010 0 0.6450 0 0.964 9B 8B 7B 6B 5B 4B 3B 2B 1B 0.6988 1.6103 0.5053 0.83190.48031.811 0.8229 1.61030.2352 9D 8D 7D 6D 5D 4D 3D 2D 1D 0.7258 0.7830 0.4451 0.46500.45201.6870 0.3038 1.7045 0.0223 Table 24. Results of Analysis CAD Model Angle (degrees) CAD Model TanHalf Angle Analysis Results (1) Analysis Results (2) Analysis Results (3) Analysis Results (4) 1 100.749 1.2077 1.2077 2.307 4.1559 4.6202 2 210.122 3.716 3.716 0.3637 1.057 2.153 3 243.046 1.630 1.630 1.0044 0.5439 0.1679 PAGE 37 28 2.6 Orientation of Top Platform Using Quaternions Now that the angles 1, 2, and 3 have been determined, a procedure follows that shows how to determine the coordinates of points T1, T2, and T3 in terms of the ground coordinate system. Determining the Coordinate of Point T1 1. Determine the absolute distance from Point B1 to Point P. 1 1 1B P d (2.33) 2. Obtain a unit vector going from Point B1 to Point P. d B P1 1 1 (2.34) 3. If two spheres of radii L2 and L3 are centered on Points B1 and P, respectively, their intersection would be a circle centered on line Also, this circle would lie on a plane perpendicular to line. The distance from point B1 to the center of this circle is given by Equation(2.35) and the radius of this circle is given by Equation (2.36 ). d L L d D22 2 2 3 2 (2.35) 2 2 2 2 3 2 2 2 24 2 1 L L d L d d r (2.36) PAGE 38 29 4. S is defined as a vector perpendicula r to the plane formed by the lines 17. 3 1 1 3 1 1B P B P S (2.37) 5. is defined as a vector perpendicular to the line that lies in the plane formed by the lines 17. S S (2.38) 6. Now define point E1 as shown in Figure 210. r D P PB E1 1 7. To find the point T1 we simply use the quaternion method for rotating the point E1 about line 1 by the angle 1. 2 sin 2 cos1 1q (2.39) 1 1 1) ( q P q PE T (2.40) PAGE 39 30 The same method can be used to find Points T2 and T3. Once the coordinates of points T1, T2, and T3 are known in the fixed coordinate syst em, it is a straightforward task to determine the relative rotation matrix R1 2. Figure 210. Definition of Point E1 For our particular analysis we obtained 4 real orientat ions of the top platform. Therefore, the analysis will have to be repe ated if all four orientations are to be determined. These four orientations ar e shown in Figure 211 through 214. PAGE 40 31 Figure 211. Orientation from Analysis Results 1 Figure 212. Orientation from Analysis Results 2 PAGE 41 32 Figure 213. Orientation from Analysis Results 3 Figure 214. Orientation from Analysis Results 4 PAGE 42 33 CHAPTER 3 MOTION ANALYSIS USING RECIPROCAL SCREWS Now that all of the possible orientat ions of the top platform have been established; the next goal is to find a way to minimize the work required in moving the top platform to another specified location2 timeP. In other words, consider that the 6 6 platform supports an external force/moment combination (wrench) on the top platform (see Figure 31) to maintain static equilibri um. The goal is to find a way to move the top platform while minimizing any work pr oduced by this external wrench. Figure 31. 6 6 Platform Supporting External Wrench PAGE 43 34 Before undertaking any analysis, it is necessa ry to define two terms. The first of the terms are Plcker line coordinates. Plcker line coordinates can be used to describe a line connected by two points 2 2 2 2 1 1 1 1, & z y x P z y x P .9 The equations for the Plcker line coordinates are shown in Equation (3.1). An attractive property of Plcker line coordinate is that they are homogeneous. Homo geneous coordinates imply that if all the quantities shown in Equation (3.1) are multip lied by the same scalar, the coordinates would define the same line in space. 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 1, 1 1 1 1 1 1y x y x r x z x z q z y z y p z z n y y m x x l (3.1) This chapter will also discuss screws. A screw can be thought of as a line with a pitch h. A screw $ can be written as in equation (3.2). Screws have two unique properties: 1. A screw multiplied by an angular velocity magnitude describes the motion of one body relative to another. 2. A screw multiplied by a force becomes a force/moment combination acting on a body. This force/moment combination is called a wrench. Also discussed in this chap ter is the concept of reciprocal screws For this study, a reciprocal screw can be thought of as any motion(s) that will not cause work to be done by an applied force/moment comb ination (wrench). For this particular platform, screws will be determined that are reciprocal to the external wrench shown in Figure 31. PAGE 44 35 } , ; , { $ nh r mh q lh p n m l (3.2) The problem is stated below. Given All Constant Geometrical Parameters Position & Orientation of Top Platform External Wrench Applied to the Top Platform Find Motions (twists) of Top Platform to E liminate Work Caused by External Wrench Outline of Solution The solution of this problem will be outlined from Crane and Duffy.10 If we consider an external wrench acting on the top platform ext extf $; any screw that is reciprocal to this external wrench must satisfy Equation (3.3). 0. . ext recip est recip ext recip recip ext recip ext recip extR N Q M P L R N Q M P L (3.3) The values of ext extR L ...are of known quantities which repr esent the six coordinates of the wrench or twist. Equation (3.3) is a ho mogeneous equation that consists of termsr rR L .... These terms consist of five independent parameters. In order to find the values of PAGE 45 36recip recipR L ..., we simply select five of these values and compute the six value from Equation (3.3). PAGE 46 37 CHAPTER 4 CONCLUSION This study successfully analyzed a three degree of freedom tensegrity platform that incorporates the special 6 6 platform geometry. This analysis shows that if a coordinate system attached to the bottom platform is specified, and a predetermined location of a point attached to the top platform is also specified, then the orientation of the top platform with respect to the coordi nate system of the bottom platform can be determined. This analysis focused on a closed form algebraic solution of the platform. The analysis proceeding by defining three de pendent spherical quadrilaterals. Although iterative methods would have also given th e orientation of the top platform, these methods would leave out critical information. By using iterative procedures, one would never know how many orientations are possible of the top platform for a given position P. For example, a numerical case shown in this thes is identified four po ssible orientations of the top platform for the given point P. Usi ng an iterative procedur e, the solution of the orientation of the top platform could po ssibly converge at any one of the four orientations. The solution of the iterative me thod is almost entirely based on the initial values given to the method. Given this anal ysis is much more ri gorous than iterative methods, the information obtained about the syst em makes up for the additional effort. It was learned that there are sixteen possible orientations of the top platform. This analysis also shows once the orient ation of the top plaform is determined, screws that are reciprocal to the external wrench can be determined. These reciprocal PAGE 47 38 screws form the basis of continued analysis on ways to minimize the energy required to move the top platform. PAGE 48 39 APPENDIX Coefficients of First Spherical Quadrilateral 02 4 6 8 5 1 3 3 4 5 2 7 1 9 A A A A a A a A a A a A a A Coefficients of Second Spherical Quadrilateral U T S Q B R B T U Q S B S Q B R B S Q B U S Q T B R B U Q T S B ) cos( ) cos( ) sin( 2 ) cos( ) cos( ) sin( 2 ) sin( 2 ) cos( 4 ) sin( 2 ) sin( 2 ) cos( ) cos( sin( 2 ) cos( ) cos(2 2 1 2 2 2 2 3 2 2 4 2 5 2 2 6 2 2 7 ) 2 8 2 2 9 where, 9 6 7 9 4 7 6 4 7 9 4 7 6 4 7 9 4 7 6 4 7 9 6 4 7 9 4 7 6 4 c c c c U c s s T s s c S s s R s c s Q PAGE 49 40 Coefficients of Third Spherical Quadrilateral Z X V W Y D W V Y D Z V W X Y D V W X D W V D W X V D Z Y V W X D V Y W D Z Y X V W D ) cos( ) cos( ) cos( ) sin( ) sin( ) cos( ) cos( ) sin( 2 ) sin( ) cos( 2 ) sin( 2 ) cos( ) cos( ) sin( ) sin( ) cos( ) cos( ) cos( ) sin( 2 ) sin( ) cos( 2 ) sin( 2 ) cos( ) cos( 4 ) sin( ) sin( 4 ) sin( ) cos( 2 ) sin( 2 ) cos( ) sin( 2 ) cos( ) cos( ) cos( ) sin( ) sin( ) cos( ) sin( ) cos( 2 ) sin( 2 ) cos( ) sin( 2 ) cos( ) cos( ) cos( ) cos( ) sin( ) sin(3 1 3 1 3 1 1 1 3 1 3 1 2 1 3 1 3 3 1 3 1 3 1 3 3 4 1 3 1 3 5 1 3 3 1 3 6 1 1 3 1 3 3 7 1 3 1 1 3 8 1 3 1 3 1 3 9 where, 6 3 4 6 1 4 3 1 4 6 1 4 3 1 4 6 1 4 3 1 4 6 3 1 4 6 1 4 3 1 c c c c Z c s s Y s s c X s s W s c s V PAGE 50 41 REFERENCES 1. R. Fuller, SynergeticsExplor ations in the Geometry of Thinking, Volumes I & II, New York, Macmillan Publishing Co,1975, 1979. 2. Compression and Tension Are Good; Torques a Killer, http://www.frontiernet.ne t/~imaging/tenseg1.html, 1996. (accessed April 4, 2005). 3. T. Wilken, Tensegrity, http://futurepositive.syneart h.net/stories/storyReader$261,2001 (accessed April 4, 2005). 4. W. Abbasi, S. Ridgeway, P. Adsit, C. Crane, and J. Duffy, Investigation of a Special 66 Para llel Platform for Contour Milling, Florida, CIMARUniversity of Florida, 2000. 5. J. Duffy, and M. Griffis, A Forward Disp lacement Analysis of a Class of Stewart Platforms,Trans. ASME Journal of M echanisms, Transmissions, and Automation in Design. 6 (June 1989),703720 6. D. Stewart, A Platfo rm with Six Degrees of Freedom, London, Proc. Inst. Mech. Engrs. 180 (1965), 371386. 7. M. Raghavan, The Stewart Platform of General Geometry has 40 configurations, Trans. of the ASME, Journal of Mechanical Design. 115 (June 1993), 277282. 8. M. Griffis and J. Duffy, Method and A pparatus for Controlling Geometrically Simple Parallel Mechanisms with Distinctive Connections, US Patent No. 5,179,525, January 12, 1993. 9. M. Marshall, Analysis of Tensegri tyBased Parallel Platform Devices, Florida, CIMARUniversity of Florida, 2003. 10. C. Crane and J. Duffy., Kinematic Analysis of Robot Manipulators, Cambridge University Press, Cambridge, UK, 1998. PAGE 51 42 BIOGRAPHICAL SKETCH Antoin Lenard Baker was born on 1981 in Portsmouth, Virginia. In 2003, he received his Bachelor of Sc ience in Mechanical Engineer ing from the University of Florida. After graduating, he joined the Ar my Reserves while deciding to continue his education in mechanical engineeri ng at the University of Florida. 