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PAGE 1 1 DESIGN AND COMPARISON OF INTELLIGENT AUTO ADJUSTING MECHANISMS By AASAWARI DESHPANDE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MAS TER OF SCIENCE UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Aasawari G. Deshpande PAGE 3 3 To my parents and friends PAGE 4 4 ACKNOWLEDGMENTS I would like to thank Dr. John Schueller and Dr. Wiens (members of my committee) for overseeing the thesis. My special thank s go to Dr. Carl Crane III, my committee chairman, for guiding me throughout my work, and for his support and dedication. He not only helped me with my work but also was very encouraging and understanding. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 7 ABSTRACT ................................ ................................ ................................ ..................... 8 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ...... 9 Literature Survey ................................ ................................ ................................ ...... 9 Project Objective: ................................ ................................ ................................ .... 15 2 PR OBLEM FORMULATION ................................ ................................ ................... 19 Nomenclature ................................ ................................ ................................ ......... 19 Simple Tripod: ................................ ................................ ................................ .. 19 Tensegrit y Mechanism: ................................ ................................ .................... 20 General Assumptions ................................ ................................ .............................. 21 Problem Statement ................................ ................................ ................................ 22 Give n Data ................................ ................................ ................................ ....... 22 Simple Tripod ................................ ................................ ................................ ... 22 Tensegrity Mechanism ................................ ................................ ..................... 22 To Find: ................................ ................................ ................................ ................... 23 3 ANALYSIS OF THE SIMPLE TRIPOD ................................ ................................ .... 24 Defining Coordinate Systems ................................ ................................ ................. 24 Third Coordinate System: ................................ ................................ ................. 25 Second Coordinate System ................................ ................................ .............. 25 First Coordinate System ................................ ................................ ................... 26 Constant Mechanism Parameters: ................................ ................................ .......... 26 Coordinates of Points P 1 P 2 and P 3 in Second Coordinate System ................. 27 Coordin ates of Point P 1 in Second Coordinate System ................................ .... 27 Coordinates of Point P 2 in Second Coordinate System: ................................ ... 28 Coordinates of Point P 3 in Second Coordinate System: ................................ ... 30 Calculation of Transformation Matrix: ................................ ................................ ..... 30 Coordinates of Laser Line in First Coordinate System ................................ ............ 32 Concept and Calculations for Infinitesimal Twist ................................ ..................... 33 Numerical Example ................................ ................................ ................................ 36 PAGE 6 6 4 ANALYSIS OF THE TENSEGRITY MECHANISM ................................ ................. 39 Initial Transformation Matrix ................................ ................................ .................... 39 Point Transformations ................................ ................................ ............................. 40 The Concept of Desired Transformation Matrix ................................ ...................... 42 Desired Transfo rmation Matrix without Considering Conditions for Equilibrium of Mechanism ................................ ................................ ............. 42 Conditions for Equilibrium of Mechanism ................................ ......................... 44 Calcula tion of Desired Transformation Matrix without Force Balance ..................... 45 Desired Plcker Coordinates of Line along Laser Pointer: ............................... 45 Desir ed Orientation of XY plane of Top Coordinate System ............................ 46 Transformation Matrix without Applying Equilibrium Conditions ....................... 47 Calculat ion of Final Desired Transformation Matrix ................................ ................ 50 Calculation of Change in Struts and Ties Length ................................ .................... 52 Numerical Example ................................ ................................ ................................ 55 5 FUTURE WORK ................................ ................................ ................................ ..... 61 APPENDIX: MAPLE PROGRAM FOR TRIPOD: ................................ .......................... 62 LIST OF REFERENCES ................................ ................................ ............................... 76 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 77 PAGE 7 7 LIST OF FIGURES Figure page 1 1 Tensegrity Mechanisms ................................ ................................ ...................... 12 1 2 Simple tripod with laser pointer ................................ ................................ ........... 12 1 3 Three Strut Tensegrity Mechanism ................................ ................................ ..... 13 1 4 Simple tripod mechanism ................................ ................................ ................... 16 1 5 Three Strut Tensegrity Mechanism ................................ ................................ ..... 17 2 1 Nomenclature for simple tripod ................................ ................................ ........... 19 2 2 Nomenclature for tensegrity mechanism ................................ ............................ 20 3 2 Top view (XY plane of top coordinate system) ................................ ................... 29 4 1 Ball joint between laser pointer and robotic arm ................................ ................. 43 PAGE 8 8 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Require ments for the Degree of Master of Science DESIGN AND COMPARISON OF INTELLEGENT AUTO ADJUSTING MECHANISMS By Aasawari G. Deshpande December 2009 Chair: Carl Crane III Major: Mechanical Engineering The task of self adjustment of a platform mounted with a laser pointer to point to a desired location on the ground is to be satisfied. Two different types of mechanisms are presented to perform this task and a comparative study between them is presented. A simple tripod with three degrees of freedom with var iable leg lengths is first studied in order to perform the task. It is found that a tripod is very easy to model mathematically. An infinitesimal twist in order to orient the laser pointer towards the desired location is calculated. It was found that the w orkspace of allowable orientations of the laser pointer line was very limited. A SolidWorks model of the same mechanism was checked to confirm the mathematical model. It was found that the mechanism covers only a small area on ground even after adjusting l eg lengths to their maximum extent. A tensegrity mechanism was considered next. The mechanism is studied with regards to geometric and force balance constraints. The solution presents the change in lengths of the ties and struts to orient the laser pointe r to point to the desired location on the ground. After checking equilibrium conditions, the tensegrity mechanism seems to work without a problem satisfying the desired goal. PAGE 9 9 CHAPTER 1 INTRODUCTION An (Improvised Explosive Device (IED) can be almost any thing with any type of injury by using explosives alone or in combination with toxic chemicals, biological toxins, or radiological material. In 2007 during Afganistan War, 230 IED incidences were reported killing over 12 members of coalition forces. This number increased to 49 reported. The death rate for IEDs was 90 troops in 915 IEDs in 2007. As these explosives can be constructed in wide varieties, it is not possible to deactivate them even after they have been detected. Therefore, space charge is used to blow up these explosives. A shape charge creates plasma from molten metal and transforms it into a forceful jet stream. The goal here is to develop a mechanism that can aim a projectile towards a user specified point on the target IED The mechanisms used and compared in this research were designed and tested for this goal and the results ar e presented. The two mechanisms chosen for the comparison are a tripod and a three three tensegrity mechanism. Literature Survey A tripod is a word generally used for a three legged object. It comes from a ripod is used more often for its stability while moving up and down as well as sideways. A very common use of a simple tripod is a stand for machine guns. It absorbs the recoil from the gun and helps soldiers to stabilize the gun while firing. This tripod is more of a foldable structure where as it can PAGE 10 10 be used as a moving foldable mechanism with some basic modifications. As in case of the tripod stands used for photography, they can be folded to fit into a small rectangular box. They are mostly very rigid a nd heavy. This is the reason they are used where stability is more important. Another mechanism chose for comparison is a tensegrity mechanism. They are a more complex mechanism and need a lot of study to understand their working. This chapter will presen t more about tensegrity mechanisms and their comparison with a simple tripod. Tensegrity is a blend of words tension and integrity (Edmonson, 1987 and Fuller, 1975). Tensegrity structures refer to balanced light weight components in tension and compressi on. They are different from a truss structure in a way that all tensile elements of the mechanism are replaced by strings (R.E.Skelton, University of California). The in case of tensegrity mechanisms do not touch each other and are connected by ties subjected to tension. This type of design is known as Class I tensegrity composed of a set of three or more elongated compression struts within a network of Special attention should be given to mechanical stability and static construction while designing the tensegrity mechanism. It is assumed that all the components of thes e mechanisms are either in pure tension or in pure compression (Pallegrino and Calladine, 1985). If the mechanism is subjected to buckling or yielding, it may fail. If PAGE 11 11 subjected to unbalanced tensile and compressive loads, the mechanism may not be in equil ibrium. One of the advantages of a tensegrity structure is that it can be made with elastic ties. Hence, the structure can be folded into a small set of struts and when released, it will regain its original shape. The shape of a tensegrity mechanism is v ery unique. It is formed by rotating a parallel prism about its axis by a specific angle. The mechanisms formed by clockwise rotations are known as left handed tensegrity mechanisms and those rotated counter clockwise are right handed ones. The angle of ro tation is very specific and has been determined by Kenner to be (1.1) rotat ion is ( /2), regardless of n. Examples of tensegrity mechanisms are as shown in Figure 1 1. PAGE 12 12 Figure 1 1 Tensegrity Mechanisms In this figure, the struts are represented by blue lines and the ties are represented by red ties Figure 1 2 Simple tripod with laser pointer PAGE 13 13 A tripod is a simple mechanism comparatively. It has a very small top surface on which a laser pointer is mounted as shown in the figure. Three base points form one more triangular surface on the ground cal led base platform. For both the tripod and the tensegrity mechanism, it is assumed that the distance between two vertices of the base will remain constant. A detailed mathematical analysis needs to be done to study this mechanism to perform the required ta sk of aiming a projectile towards the target improvised explosive device. Figure 1 3 Three Strut Tensegrity Mechanism A common tensegrity mechanism consists of a top surface and a base surface of the same shape connected by struts and ties. Many stud ies have been done to determine the equilibrium position of a tensegrity mechanism. During these studies, two parameters are determined related to these mechanisms. One of the parameters is the effect of external forces and moments. The other is the positi on of top plane with PAGE 14 14 respect to base. Kenner, Yin and Tobie studied the relationship between top and base platform. Length of struts and ties is an important parameter to calculate equilibrium position. Many people have derived the equations to find the le ngth of struts and ties for the equilibrium position of a tensegrity mechanism. Most recently, University of Florida, Vikas determined all the equilibrium positions of a parallel three strut tensegrity mechanism. All these studies have been done when the top and base surfaces are parallel to each other as shown in figure 1 1. This is not possible all the time. There are many applications that require different orientations of the top platform with respect to the base, as shown in Figure 1 3 Also, whenever any operation is performed or an external force/moment is applied, the top platform never stays parallel to the base. The relationship between orientations of the top platform with respect to the base is studied in this research and one of the solutions is presented. This solution is then compared with that of a simple tripod and the results are analyzed. To obtain the change in orientation in a specific pre determined manner, many mechanical adjustments were needed for a tensegrity mechanism. One of t he options is to make struts from pneumatic cylinders to adjust their length and change their orientation. The other option is to have a prismatic joint inside each strut connected to a tie. This mechanism allows all adjustments in the length of each tie a nd the strut resulting in reduced tensile force in the ties and change in orientation of the top platform. This research combines the study of tensegrity mechanisms and parallel platforms to form a structure which has adjustable ties and struts. PAGE 15 15 Ties are constructed from compliant and non compliant parts. The non compliant parts are subjected to a change in length. A reverse displacement analysis of such a mechanism has already been studied by Tran, 2002, University of Florida. His work presents the set o f equations to calculate the change in length of non compliant part. This can only be applied if the final orientation of the top platform with respect to the base is already known and also the orientation satisfies specific linearity constraints. This res earch presents the way to find out that orientation and hence, use a part of compliant element of each tie and change in length of the strut. Project Objective: Both the tripod and the tensegri ty mechanism have their own advantages and disadvantages. In both the mechanisms, positioning of the top platform to a required orientation is studied. There are many applications found where positioning the top platform with respect to the base is conside red to be a very useful tool. One of the applications taken into consideration for this research is the aiming of a shape charge projectile at a user specified point on an improvised explosive device (IED). A robot will be used to place the mechanism over the target IED. A laser pointer will be mounted on the top platform. Its directional axis will be parallel to the line of action of the shape charge projectile which is also attached to the top platform. The laser pointer shows the initial aiming point. The objective is to move this point to the desired aiming point. There are many pros and cons of both the mechanisms. Following is the list of pros for a simple tripod: 1. It is a very simple mechanism. Mathematically modeling a simple tripod is very easy. PAGE 16 16 2. It is very stable mechanism due to its heavy and strong legs. Hence, stability is not a concern. 3. It can be folded to fit in a box and is easy to carry. Following is the list of pros for a tensegrity mechanism: 1. Elastic elements can be used for the ties. The potential energy that is contained in these elements can be used to change the shape of the mechanism from the stowed to the deployed shape. This potential energy can also be used to change the orientation of the top platform to obtain the desired aim ing point. 2. The mechanism can be folded to fit all the struts into a small rectangular box without deforming its original shape and becomes easy to carry due to less volume. Figure 1 4 Simple tripod mechanism The tripod taken into consideration for this research is as shown in Figure 1 4. Following are some of the characteristics of this mechanism: 1. Three legs of the tripod are connected to the top platform by revolute joints. The axes of these joints are parallel to three edges of the top platform. PAGE 17 17 2. Three legs are made up of two concentric cylinders. These two cylinders are connected by a prismatic joint with each other. This changes length of each leg and helps orienting the laser pointer mounted on top platform. 3. The mechanism is designed in a special way The revolute joints have a brake on them. Whenever the leg length is changing, the respective revolute joint is allowed to rotate and the other two act as fixed joints. 4. As all three legs of the mechanism are far away from each other, they have enough spa ce between them to point the laser pointer to the required orientation. Figure 1 5 Three Strut Tensegrity Mechanism The tensegrity mechanism taken into consideration for this re search is as shown in Figure 1 5 Following are some of the characteristic s of this mechanism: 1. with the base are made up of two elements; compliant springs and non compliant strings. 2. The struts are designed to be curved hollow cylindrical rods. This gives more space for laser pointer to point to the required location. 3. Prismatic joints are used to connect the strut with side ties. This helps to change the length of non compliant part of the tie which changes orientation of top platform. PAGE 18 18 4. Ties for ming the top platform are completely non compliant and have high tensile strength. 5. The base platform is formed by the surface of soil. It is assumed that the mechanism stands on top of the soil. The connection between the struts and ground may be modeled b y spherical joints. Ties connecting the bottom of three struts are also non compliant and are made up of same material as those in pt. 4. This thesis presents the mathematical model of this new type of special tensegrity mechanism and the modified simple t ripod to perform the required task. Further chapters will present the mathematical equations to calculate the required orientation of the top platform and the change in length of legs in case of simple tripod and the change in tie length and strut length i n the tensegrity mechanism. PAGE 19 19 CHAPTER 2 PROBLEM FORMULATION The simple tripod and tensegrity mechanisms discussed in the last chapter have many parameters that can be adjusted to change the orientation of the top platform. To understand the workings of bot h mechanisms, it is necessary to understand th e nomenclature for both of them. Nomenclature Simple Tripod: The tripod is very easy to analyze in a way that it does not have a long list of different parameters and variables. There are only three values; i. e. the leg lengths, which can be changed to change orientation of top platform. Each leg of a tripod is numbered as 1, 2 and 3. The following nomenclature is used: 1. Length of each leg is named as L1, L2 and L3. 2. Vertices of base are named as P 1 P 2 and P 3 and that of top platform are named as P 4 P 5 and P 6 Figure 2 1 Nomenclature for simple tripod PAGE 20 20 There are two coordinate systems; the base coordinate system and the top coordinate system. The base coordinate system is named as 1 and the top coordinate system is named as 2. The origin of the base coordinate system is located at vertex P 1 and that of the top coordinate system is located at vertex P 4 of top platform. The transformation matrix can be calculated using simple geometry. Tensegrity Mechanism: The tensegrity mechanism is a very complex mechanism considering its stability issues. There are many parameters and variables that can be changed to change the orientation of the top platform. The nomenclature used for this mechanism is discussed here. T he three struts are named A, B and C and the respective ties are numbered as 1, 2 and 3. The compliant and non compliant parts of each tie are given the similar nomenclature. Tie 1 connects the end point of strut A with top vertex of strut B. The two ends of the tie are vertex P 1 and P 4 as shown in figure 2 1. All the vertices, ties and struts are also shown in figure 2 2. Figure 2 2 Nomenclature for tensegrity mechanism L A L B L C Length of each strut K 1 K 2 K 3 Spring constants l01, l02, l03 Free length of springs L1, L2, L3 Actual length of each sp ring l1, l2, l3 Length of non compliant part of each tie P 1 P 2 P 3 Vertices of the base platform P 4 P 5 P 6 Vertices of the top platform PAGE 21 21 Tie 2 connects vertex P 2 and P 5 and tie 3 connects P 3 and P 6 The following nomenclature is used: 1. The free lengths of three springs and spring constants are represented by l01, l02, l03 and K 1 K 2 K 3 respectively. 2. Length of three variable length compressive struts is represented by L A L B and L C respectively. 3. Len gth of each spring in tension i.e. actual length of each spring is written as L1, L2 and L3, respectively. 4. Length of non compliant part of each tie is given as l1, l2 and l3, respectively. 5. Total length of side ties is given by l 1total, l 2total and l 3total There are two coordinate systems located on top and base platforms respectively. The origin of base coordinate system is located at one of the vertices of base, P 1 The X axis of the base coordinate system is assumed to be along the line joining points P 1 and P 3 The origin of the top coordinate system is assumed at the laser pointer with its Z axis pointed in the same and opposite direction as that of laser pointer. The X axis and Y axis of top coordinate system are assumed to be parallel to that of bas General Assumptions For the purpose of this research, it is assumed that the robot is used to detect landmines and keeping the trip od at correct location. It also provides the basic data required for the calculations. The tripod is kept at correct location by this robot. It is also assumed that the tripod stands still on uneven soil surface and the joint between soil and leg behave as spherical joints. PAGE 22 22 Assuming all the above data is known and the coordinate systems defined, the problem statement can be specified. P roblem S tatement Given Data initial para meters of the mechanism. Simple Tripo d For a simple tripod, very less number of parameters is already known. They can be listed as follows: Coordinates of points P1, P2 and P3 in coordinate system 1; P11, P21 and P31 Coordinates of points P4, P5 and P6 in coordinate system 2; P42, P42 and P62 Initial lengths of three legs; L1, L1 and L3 Coordinates of desired point of intersection of laser pointer with XY plane of coordinate system 1; PdesiredB Tensegrity Mechanism As discussed earlier about the complexity of a tensegrity mechanism, there are many parameters in the mechanism which can be used to orient the top platform with respect to b ase. The given data are: 1. Initial position of origin of top coordinate system represented in base coordinate system; P0T B (i)* 2. Coordinates of points P 1 P 2 and P 3 in base coordinate system; PB1, PB2, PB3 (i)** 3. Coordinates of points P 4 P 5 and P 6 in top coor dinate system; PT4, PT5, PT6 (i)** 4. Initial lengths of struts; L A L B L C (i)* PAGE 23 23 5. Initial lengths of compliant and non compliant parts of ties; L1, L2, L3 and l1, l2, l3 respectively (i)* 6. Coordinates of desired point of intersection of laser pointer with XY Pl ane of base coordinate system, P desiredB (r)** To Find: To achieve the required goal, following data needs to be calculated: 1. Desired orientation of top platform with respect to base; T BT 2. Length of three legs ( for simple tripod) 3. Length of three variable struts; LA, LB, LC (for tensegrity mechanism) 4. Length of three non compliant parts of the string; l1, l2, l3 (for tensegrity mechanism) The change in length of a tie, strut, or leg will be very difficult to actuate in an exact and continuous manner. Hence, for this research, it was decided to have the change in length at specific intervals. It was decided to use a clicking type of mechanism that will click and change the length by specific number of intervals. Therefore, after the exact change in length has been determined, the number of clicks is decided. This data is sent to an inexpensive microprocessor mounted on the mechanism which will give a signal to change the orientation of the top platform. This is also discussed in further chapters. [(i): known due to same initial construction of a mechanism : constant value parameter ** : variable] PAGE 24 24 CHAPTER 3 ANALYSIS OF THE S IMPLE TRIPOD A simple tripod is easier to study and adjust than a tensegrity mechanism. Once the required position of the top platform is known, the only task is to calculate the change in leg length. In this research, the concept of a twist is used to calculate the desired orientation of the top platform. A small infinitesimal twist can be appli ed to top platform that will move the laser pointer towards the desired point by adjusting the leg lengths. The calculations to get this twist are presented in this chapter. Defining Coordinate Systems Three different coordinate systems are considered for this mechanism. The mechanism is as shown in Figure 3 1 : Figure 3 1 Coordinate systems for the simple tripod Following are the three coordinate systems: PAGE 25 25 Third Coordinate System: This coordinate system moves with the top platform. Its origin is a poin t on the top platform. The laser line is assumed to be passing through the origin of third coordinate system. Therefore, it can be said that this system represents the orientation of the laser line. The Z axis of this coordinate system is along the laser l ine pointing towards the ground. If checked at the initial position of the mechanism, then the Z axis points vertically downwards. The XY plane of this coordinate system is the top platform itself. It is assumed that the two axes, X and Y are parallel with those of the second coordinate system. Second Coordinate System This coordinate system is also mounted on the top platform. This represents the orientation of the top platform. The origin is assumed at one of the vertices of the top platform; P 4 The X a xis of this coordinate system is assumed along the line joining the two vertices P 4 and P 5 The Z axis of this system is assumed to be perpendicular to the plane of the top platform and parallel to that of first co ordinate system initially. Orientation of the Y axis can be easily calculated as it is perpendicular to both the X and Z axes. The coordinates of all three vertices of the top platform are assumed to be known in this coordinate system. As shown in the F igure 3 1 the revolute joints are along the sides of the top platform and not at the vertices. The mechanism is constructed in such a way that these revolute joint s will lie at the will be same as that of each side of the top platform. Let d 1 be the distance of the first revolute joint from point P 4 (origin of second coordinate system). Therefore, d 1 can be calculated using the following equation: d 1 = distance ( P 4 to P 5 )/2 (3.1) PAGE 26 26 Similarly, let d 2 and d 3 be distances of mid points of joints 2 and 3 from vertices P 5 and P 4 respectively. Th ese distances can be calculates as: d 2 = distance ( P 5 to P 6 )/2 (3.2) d 3 = distance ( P 4 to P 6 )/2 (3.3) First Coordinate System This coordinate system represents the firm ground on which the mechanism stands. The origin of this coordinat e system is assumed at one of the vertices of the base, P 1 The X axis of the system is assumed along the line joining vertices P 1 and P 2 The Z axis points vertically upwards. The Y axis can be calculated in a similar way as that in case of second coordi nate system. The main step is to get the accurate transformation matrix of the third coordinate system with respect to the first coordinate system. This initial transformation matrix will be the same for all the mechanisms. Constant Mechanism Parameters: Listed below are the parameters of the mechanism which will never change as they depend on the initial construction of the mechanism. 1. Transformation matrix be tween coordinate system 3 and 2 2. Angle between leg 1 and a line parallel to Z axis and passing thr ough center of revolute joint 1 3. Angle between leg 2 and a line parallel to Z axis and passing through center of revolute joint 2 4. Angle between leg 3 and a line parallel to Z axis and passing through center of revolute joint 3 5. Distances between vertices of the top platform and the center of the respective revolute joints PAGE 27 27 6. Coordinates of points P 1 P 2 and P 3 in the first coordinate system and coordinates of points P 5 and P 6 in the second coordinate system ram. These parameters are known throughout the program and except for the three angles associated with the revolute joints, all others are constant throughout the program. Coordinates of P oints P 1 P 2 and P 3 in Second Coordinate System It is a little com plicated to understand and calculate the coordinates of all three points at the same time as the transformation matrix that related the first and third coordinate systems is unknown. Therefore, a single point is considered each time with separate cases. Co ordinates of Point P 1 in Second Coordinate System For calculating the coordinates of this point, one more coordinate system is assumed along leg L1. This coordinate system as its Z and Y axes parallel to second coordinate system but the X axis along the l 1 will be the angle between X axis of this new coordinate system and Z axis of second coordinate system. The origin of this new coordinate system is assumed at the center of revolute joint. Therefore, the transfor mation matrix can be calculated as: (3.4) Coordinates of point P 1 in this new coordinate system will be given as: (3.5) PAGE 28 28 where L1 is the length of leg 1. From above two equations, point P 1 is known in new coordinate system and tra nsformation matrix between new coordinate system and second coordinate system is known. Therefore, following equation can be used to calculate coordinates of point P 1 in second coordinate system. (3.6) These are coordinates of point P 1 in seco nd coordinate system. Coordinates of Point P 2 in Second Coordinate System: The calculations for th is part are similar to those for P 1 with little variation. The temporary coordinate system is assumed along leg 2 and its origin is assumed at center of revo lute joint 2. Similar to previous part, the transformation matrix can be calculated. The same equation is used but, in this case, that equation gives a transformation matrix that will move the temporary coordinate system to vertex P 5 instead of the origin of second coordinate system. This can be given as: (3.7) This transformation matrix will bring the origin of temporary coordinate system at vertex P 5 Therefore, the origin of temporary coordinate system is in XY plane of second co ordinate syst em. Now, it is easy to break the steps of calculating transformation matrix as rotation and translation PAGE 29 29 Figure 3 2 Top view (XY plane of top coordinate system) From the figure, the angle between side of the top platform (P 5 to P 6 ) and X axis of second coordinate system can be calculated using the equation: (3.7) This equation will give the angle of rotation for the temporary coordinate system about Z axis and the translation part is jus the X coordinate of point P 5 (obtained from the figure). Therefore, the transformation matrix is: (3.8) Similar to point P 1 coordinates of point P 2 in temporary coordinate system will be: (3.9) Therefore, the coordinates of point P 2 can be given by following equation: PAGE 30 30 (3.10) Coor dinates of Point P 3 in Second Coordinate System: This part follows the exact steps as that in calculations for P 2 As there is no difference except for the calculation of angle of rotation, the equations and figure are as given below: (3.11) (3.12) (3.13) (3.14) Calculation of Transformation Matrix: Any transformation matrix can be represented in a specific manner. It can be represented as four separate parts making first three elements of four columns. The first column i s X axis of a coordinate system represented in another coordinate system. In this case, the X axis of first coordinate system is represented in second coordinate PAGE 31 31 system. This becomes first three elements of first column of a transformation matrix. The elem ents of second and third column are Y and Z axes of first coordinate system represented in second coordinate system. The last and fourth column is coordinates of origin of first coordinate system in second. For using this method, the things required can be calculated as the coordinates of all three vertices of base are already known in second coordinate system. 1. X axis of first in terms of second coordinate system: Unit vector along line joining P 2 and P 1 (3.15) 2. Z axis of first in terms of seco nd coordinate system: Z axis is a unit vector along the direction obtained by taking cross product of any vector in XY plane with X axis. Let the vector in XY plane be tempvec tempvec 3 and P 1 : (3.16) Equation of Z axis is: (3.17) 1. Y axis of first in terms of second coordinate system: Y axis can be obtained simply by taking a cross product of X axis with Z axis. (3.18) 3. Origin of first in terms of second coordinate system : 4. Origin of first coordinate system is point P 1 which is already known in second coordinate system. PAGE 32 32 As all the parameters of transformation matrix are known, the matrix can be written as: (3.19) where can be given as: (3.20) Coordi nates of Laser L ine in First Coordinate System The laser line is along Z axis of third coordinate system. Therefore, coordinates of laser line in third coordinate system are given as: $ laser = {0, 0, 1; 0, 0, 0} 3.21) As the line passes through orig in, the moment of line about origin will be zero. To calculate coordinates of this line in first coordinate system, transformation matrix of third coordinate system with respect to first one must be known. This can be calculated as: 3 T 1 = 3 T 2 2 T 1 (3.21) Once the transformation matrix is known, the equations to calculate coordinates of (3.22) This equation will give the coordinates of laser line in first coordinate system. Point of Intersection PAGE 33 33 Point of intersection is a point where laser line intersects ground i.e. XY plane of first coordinate system. XY plane of first coordinate system can be given by: [D 0 ; S xy ] = [0; 0, 0, 1] (3.23) The point of intersection lies in this plane. Therefore, it should satisfy following equation: S xy x P + D 0 = 0 (3.24) Also, the point lies in the laser line. Let Plcker Coordinates of line be { S laser ; SoL las er }. In this case, the point of intersection should also satisfy following equation: P x S laser = SoL laser (3.25) Taking cross product with S xy on both sides of the equation above: S xy x ( P x S laser ) = S xy x SoL laser (3.26) Simplifying abo ve equation and substituting the values from equation obtained from plane, ( S xy S laser ) P + D 0 S laser = S xy x SoL laser (3.27) The only unknown in the equation above is P which is point of intersection. Therefore, coordinates of point of intersectio n can be obtained using above equation. Concept and Calculations for Infinitesimal Twist The top platform will move with some velocity while trying to achieve the desired orientation with respect to base. Let us assume that there is a twist which will rep resent this motion of top platform. If the top platform is applied this particular twist, it will change its orientation and point the laser pointer at desired point. PAGE 34 34 linear velo city of any one point on top platform, v. for ease of calculations, lets assume that this process is carried out in several number of small steps. In this case, the twist is very small and will only move the point of intersection closer to desired point of intersection in one step. This infinitesimal twist has six unknowns; vector of angular velocity and a vector for linear velocity. As there are six unknown parameters, six constraint equations are required to determine their values. Following are the six equations chosen: Constraint 1 : the angular velocity vector will be a unit vector. Constraint 2 : The twist is supposed to move point of intersection towards desired point of intersection. Therefore, the linear velocity vector has to lie in the plane formed by laser line and desired point of intersection. In that case, the twist should satisfy following equation: [ v + x P intersect ]. [( P desired P intersect ) x S laser ] = 0 (3.28) Constr ain t 3, 4 and 5: For next constraint equation, a concept of reciprocal screws is used. A reciprocal product of the wrench and twist gives the virtual power of wrench about twist. This wrench is applied to a body about a screw. If the direction of the screw is same as that of a revolute joint, then this wrench will have no effect on the body and the reciprocal product of twist and such wrench will be zero. Similarly, if the screw about which the wrench is applied passes through the center of a spherical join t (i.e. point of intersection of axes of all three revolute joints), then it will have no effect on PAGE 35 35 the motion of the body. This concept can be used to get three more equations for a simple tripod. Consider three wrenches applied to the mechanism. All thre e of them pass through the three vertices of base respectively. This will nullify the effects of wrench on motion of spherical joint. Also, these three wrenches are parallel to the axis of three respective revolute joints joining each leg with top platform Therefore, the reciprocal product of these three wrenches with the twist applied to top platform will be zero. This can be represented mathematically as follows: [ ; v ] T o $ 1b = 0 [ ; v ] T o $ 2b = 0 [ ; v ] T o $ 3b = 0 (3.29) Where $ 1b $ 2b and $ 3 b are the screws along which a wrench is applied. If the reciprocal product is zero, then the magnitude of force does not make any difference in calculations. Therefore, a unit force is assumed in this case. These equations can be simplified to obtain thre e separate equations to calculate three parameters out of six unknowns. Constraint 6: There are only five constraint equations and the sixth equation comes to free choice. For ease, the Z coord not really affect the effect of twist. Using above six equations, the twist can be calculated. Once the twist is known, the change in leg length can also be calculated. Many trials were carried out to calculate a finite twist but it is observed that the twist comes to be infinity. This means that the velocity required to move the laser pointer to point to desired point of intersection is PAGE 36 36 infinity. This result was double checked with a solid works model. It was fo und that the mechanism can only move laser pointer in a very small circular motion no matter how much the leg length changes. Due to this result the mechanism was not pursued after calculation of velocity till the calculations of leg lengths. It was proved mathematically as well as using a solid works simulation model that a simple tripod is not suitable for these kinds of jobs though its mathematical calculations are a lot easier than that of a tensegrity mechanism. Numerical Example For a numerical examp le, the values of basic constant mechanism parameters are assumed. The length of each leg, the coordinates of point P 5 and that of point P 6 will be constant. They are: Desired point of intersection is assumed to be: Transformation matrix betwe en second and third co ordinate system is obtained as: PAGE 37 37 From the equations 3.6, 3.10 and 3.14, the coordinates of points P 1 P 2 and P 3 are calculated in second coordinate system. These coordinates are: The transformation matrix of first coordinate s ystem with respect to second coordinate system can be calculated once the coordinates of vertices of base are known. This transformation matrix was obtained as: Using this transformation matrix, all the points can be expressed in either of the coordinate systems. Desired point of intersection is known in first coordinate system. Therefore, the initial point of intersection is also calculated in first coordinate system. The initial point of intersection is obtained as: PAGE 38 38 Using the six constraint equations a function is written that takes constant mechanism parameters, the transformation matrix and the coordinates of desired point of intersection as input and calculates the twist as output. The twist can be represented as: From the last constraint, N = 0. The other five values are calculated using the function and are obtained as: It is very clear from the above values that the linear velocity required to move the top platform infinitesimally t owards desired point of intersection is almost equal to infinity. This proves that a simple tripod is not a solution of the problems involving orientation of top platform. PAGE 39 39 CHAPTER 4 ANALYSIS OF THE TENSEGRITY MECHANISM This chapter focuses more on findi ng the proper orientation of the top platform with respect to the bottom. The tensegrity mechanism is a very complicated mechanism. For equilibrium of this mechanism without application of any external force, it has to satisfy certain conditions. One of th e conditions is force balance. As the base vertices are considered to be nailed to ground, the top platform has to be tested for balanced forces. There is no possibility of applying an external force in this case. Therefore, the orientation of the top plat form has to be chosen carefully. Initial Transformation Matrix To express any transformation matrix, it is first assumed that the two coordinate systems coincide with each other. One of the systems is then translated and/or rotated to its new position. T his way, calculating the transformation matrix becomes easy. It takes into account each translation and rotation of the coordinate system and any transformation matrix can be represented using one translation followed by three rotations about three axes. As discussed earlier, the axes of top and base coordinate systems are parallel to each other. Assuming the two coordinate systems were coinciding with each other, the new position of the top coordinate system can be explained as simple translation matrix. The transformation matrix is calculated by assuming the top coordinate system has been translated to a point defined in the base coordinate system. As the initial origin of the top coordinate system is known in the base coordinate system, the initial trans formation matrix is given as: PAGE 40 40 (4.1) where P0T B is the origin of the top coordinate system in the base coordinate system. P0T B is one of the parameters which depend on the manufacturing and initial positioning of mechanism. This is assumed to b e same for all the mechanisms. Hence, it can be said that P0T B is constant for all the cases as it completely depends on the construction of mechanism. The position of top platform with respect to base is going to be the same initially for every mechanism. Point Transformations Using the initial transformation matrix calculated in previous part, all the points from top coordinate system can be expressed in terms for base coordinate system Also, the equation of line along laser pointer can also be expresse d in the base coordinate system. The coordinates of every point are represented in homogeneous coordinates. In a homogeneous coordinate system, a three dimensional point given by X, Y and Z are represented by four scalar values, that is, x, y, z and w. T he three dimensional and homogeneous coordinates are related by (4.2) Thus, when w=1, the first three components of the homogeneous coordinates of a point are the same as the three dimensional coordinates of the point. By using homogeneous coor dinates, the equation of all the points is given by following generalized form: PAGE 41 41 [X Y Z w] T (4.3) (4.4) P in the above equation is a 3 1 matrix representing three dimensional coordinates of a p oint. The equation of every point is represented in this form in this report. Knowing the transformation matrix, homogeneous coordinates of all the points can be calculated in base coordinate system Vertices of the base already can be stated as constants as they are nailed to the ground and do not change. Their special properties are as follows: 1. Point P 1 2. Point P 2 lies on X axis of base coordinate system. Therefor 3. Point P 3 does not lie on any of the axes of base coordinate system but it lies in XY I nitial position of points P 4 P 5 ad P 6 i.e. ver tices of top platform, in terms of the base coordinate system can be calculated from the initial transformation matrix calculated in previous part and is given by following equations: (4.5) where PB4, PB5 and PB6 are the coordinates of points P 4 P 5 ad P 6 in base coordinate system, respectively. PAGE 42 42 These values are constant for every mechanism as the initial transformation matrix depends on the construction of the mechanism. The construction is assumed to be same for all the mechani sms. With these equations, coordinates of all vertices of base as well as top coordinate system are known in base coordinate system. The coordinates of desired point of intersection of the laser line with the plane defined by the base points are already k nown in the base coordinate system and will be provided by the operator. These will vary according to the position of mechanism with respect to the position of ordnance. The Concept of D esired T ransformation M atrix There are two steps to calculate the desi red transformation matrix. In first step, the transformation matrix is calculated with considering only one constraint i.e. considering that the laser is supposed to point to the desired point. In second step, the equilibrium of the mechanism is considered and the matrix is modified. Desired Transformation Matrix without Considering Conditio ns for Equilibrium of Mechanism For understanding the concept behind this research, let us assume a few changes. Let us assume that the robot does not use mechanism for blowing the land mines. In that case, the robot will use a robotic arm. The motion of the robotic arm will be easy to imagine. Imagine a laser pointer attached the fixed robotic arm by a spherical joint as shown in following figure: PAGE 43 43 Figure 4 1 Ball joi nt between laser pointer and robotic arm The laser pointer has three degrees of freedom when attached to a fixed robotic arm by ball and socket joint; rotation about X axis, rotation about Y axis and rotation about Z axis. In this case, the laser pointer r otates about the mid point of spherical joint. This point of rotation is fixed. Now let us assume that instead of joining only the laser pointer, there is a small triangular plate which is joined to the fixed robotic arm with the laser pointer. This tria ngular plate will have the same motion as that of laser pointer. The fixed point of rotation of both, the triangular plate and the laser pointer, is the midpoint of the triangular plate. This is where the laser pointer is attached to it. This triangular pl ate is the top platform of the mechanism. For the purpose of this research, it is assumed to be the top platform as the motion explained above. The fixed point of rotation is the origin of the top platform. The line along the laser pointer passes through t he origin too. Therefore, one of the points on this line will always be known. PAGE 44 44 The origin of top coordinate system is assumed to be fixed. Hence, the only change in transformation matrix of top with respect to base coordinate system will be rotation. The re will not be any translation of the origin. Also, the laser pointer is fixed to the top platform. The angle between the line along the laser pointer and top platform will always be constant. This angle is assumed to be 90. In technical terms, the abov between the XY plane of the top coordinate system and the laser pointer will always be the XY plane of the top coordinate system can be calculated. The position of the XY plane can be used to calculate the angle of rotation about the X and Y axes of the original top coordinate system to reach the desired top coordinate system position. The concept of compound transformations can be used to calculate the transformation matrix of the base coordinate system with respect to the desired top coordinate system. Conditions for Equilibrium of Mechanism For any tensegrity mechanism to be in equilibrium, the top platform of the mechani sm should be in static equilibrium. As discussed earlier, all struts in a tensegrity mechanism have compressive forces and all the ties have tensile forces. All of these forces act on the top platform and base platform. The forces acting on the base platfo rm do not matter as all the vertices of the base platform are fixed. But, for the top platform to be in equilibrium, the forces acting on top platform must be balanced. In the case of this research, making arrangements for application of any kind of exter nal force is impossible. Therefore, the top platform has to be in static equilibrium PAGE 45 45 without application of an external force. All the forces acting on the platform (compressive + tensile) must be cancelled by each other. The total wrench acting on the to p platform can be written as: (4.6) where f i side ties) and i (i = 1...6) are the Plcker coordinates of the lines of action of each of above equation reduces to (4.7) Ignoring the trivial solution of f i =0, it is ap parent that when there is no external wrench applied to the top platform that equilibrium can only occur if the Plcker coordinates of the six legs are linearly dependant. This implies that an equilibrium solution with no external wrench will exist only fo r certain positions and orientations of the top platform. Calculation of Desired Transformation Matrix without Force Balance There are five basic steps followed to calculate the desired transformation matrix of the top coordinate system with respect to ba se coordinate system. Desired Plcker Coordinates of Line along Laser Pointer : With the origin of the second coordinate system fixed, only one more point must be known to define the desired position and orientation of the laser pointer. The goal is to mov e the laser pointer to the desired point on the ground. Therefore, an equation of desired laser pointer line can be calculated as the line joining the origin of the top coordinate system and the desired point of intersection. Also, as the coordinates of al l PAGE 46 46 the points are expressed with respect to the base coordinate system, the coordinates of this desired line are calculated in the base coordinate system. Therefore, the desired Plcker coordinates of laser pointer line are calculated as: (4.8 ) (4.9) (4.10) where $ laser is the equation of line in which S D is the desired direction of the line along laser pointer and S0l is moment of the desired line about origin of base coordinate system. This value will vary for every ca se depending on the location of desired point of intersection. Desired Orientation of XY plane of Top Coordinate System Once the desired coordinates of the laser line is known, the equation of the desired orientation of the XY plane of the top coordinate system i.e. top platform, can be easily calculated. The only two conditions are, the plane has to pass through the origin and it should be perpendicular to the desired line along the laser pointer. For calculating the equation of the plane, the direction perpendicular to the plane must be known. This direction is the same as the direction of line $ laser Therefore, the equation of the desired orientation of the XY plane can be calculated in the base coordinate system as: (4.11) (4.12 ) (4.13) PAGE 47 47 where S D is same as that in the equation of line. This will vary depending on the desired point of intersection. Transformation Matrix without Applying Equilibrium Conditions Now that the desired position of the XY plane of the top coo rdinate system is known, the transformation matrix of the desired top coordinate system and the initial top coordinate system can be calculated. If the lines of intersection of any plane M with the YZ plane and the XZ plane are known, then it becomes eas y to align its XY plane with plane M. The angle between line of intersection of the YZ plane and the Y axis is the angle of rotation of the coordinate system about the X axis. The angle between the line of intersection of the XZ plane and the X axis is the angle of rotation of the coordinate system about the Y axis. Once the angle of rotation about the X and Y axes are known, the rotation matrices can be calculated. Using the compound transformation, the transformation matrix of desired orientation of the t op coordinate system with respect to the base can be calculated. As the equation of the desired XY plane is known, its intersection with the initial YZ plane and the XZ plane of the top coordinate system can be calculated. Let Y d be the direction of the l ine of intersection of the YZ plane and the desired XY plane and let x be the required angle of rotation about the X axis. Y d can be easily e and Dr. Duffy, section 1.4. According to this section, if the equations of two planes intersecting each other are known, then the equation of the line of intersection of those two planes can be calculated. The Plcker coordinates of the line of intersect ion can be given by: r ( S 1 S S S 2 (4.14) PAGE 48 48 Comparing the above equation with a standard equation of a line, the direction of the line of intersection is given by ( S 1 S 2). Applying this theory to calculate Y d the following equ ation can be written: (4.15) where <1, 0, 0> is the direction perpendicular to the YZ plane i.e. the direction along the initial X axis. The angle between Y d and the x axis can be found using a simple vector scalar product. A scalar product of two vectors is given by: a b =  a . b (4.16) a and b Using the same equation, the angle between the direction vectors of the line of intersection and the x axis can be calculated. x = acos( X Y d ) (4.17) The transformation matrix with this angle of rotation about the X axis is given by: (4.18) transformation m atrix above gives the transformation between initial top coordinate the direction vector S D should be known in the X stage. It can be calculated us ing the following equations: PAGE 49 49 (4.19) (4.20) where S DX is direction vector S D After this rotation, the X axis of the top coordinate system is already in the desired XY plane. Now, consider the line of intersection of the XZ plane and Y axis in a similar way to calculate the angle of rotation about the Y axis. Let X d be the direction of the line of intersection of the XZ plane and the desired XY plane and let y be the required angle of rotation about the Y axis. Following th ese equations, the equation for X d can be given as : (4.21) where <0, 1, 0> is the direction along the new Y axis. With X d known, the angle of rotation about the Y axis can be given by: y = acos( X d Y ) (4.22) The transformation matrix for this rotation is given by: (4.23) Using compound transformations, the final transformation matrix of the initial top coordinate system with the desired top coordinate system can be given by the following equation: (4.24) PAGE 50 50 where, T TX XD Using the same principle of compound transformations, the transformation matrix of the desired top coordinate system with that of base can be calculated. The equation used is same as that used for the compound transformations used for the top coordinate system. It is given as: (4.25) Calculation of Final D esired T ransformation M atrix With the above transformation matrix, the laser pointer is pointing at the desired location on the ground, but it is not necessary the case that the mechanism is stable. For a mechanism to be stable, the determinant of the following matrix should be zero. (4.26) where, S ij (i=1...6, j=1...6) are Plcker coordinates of the lines along the three struts and these six lines are linearly dependant. As discus sed earlier, if the lines are linearly dependant, the top platform is in static equilibrium. For a tensegrity mechanism to satisfy the equilibrium condition listed above, only two parameters can be changed by keeping the laser pointer pointing at the same location. One of the parameters is translation of the top coordinate system along its Z axis and the other is its rotation about the Z axis. It was checked mathematically and PAGE 51 51 noticed that there is no change in the Jacobean matrix with the change in Z coor dinate of the origin of the top coordinate system. The only other option left is the rotation about Z axis of the top coordinate system. A method was documented to calculate the transformation matrix by rotation of the coordinate system about an arbitrar y axis other than the three standard axes of a coordinate system. Using this method, if the vector about which the coordinate system is rotated is known with the angle of rotation, the transformation matrix can be calculated. The rotation matrix given by t his method is: (4.27) where the elements of the matrix are given as: r11 = (m x ) 2 (1 cos (t)) + cos (t) r12 = m x .m y .(1 cos (t)) m z sin (t) r13 = m x .m z .(1 cos (t)) + m y sin (t) r21 = m x .m y (1 cos (t)) + m z sin (t) r22 = (m y ) 2 (1 cos (t)) + cos (t) r23 = m y .m z (1 cos (t)) m x sin (t) r31 = m x .m z (1 cos (t)) m y sin (t) r32 = m y .m z (1 cos (t)) + m x sin (t) r33 = (m z ) 2 (1 cos (t)) + cos (t) In these equations, the following parameters are used: m ix : X coordi nate of vector about which the coordinate system is rotated m y : Y coordinate of vector about which the coordinate system is rotated PAGE 52 52 m z : Z coordinate of vector about which the coordinate system is rotated In case of this research, that vector is a vector a long the laser line. Therefore, m x m y and m z are known. The re is only one unknown value in the equation; the angle of rotation As discussed earlier, there is only one equilibrium condition that the transformation matrix has to satisfy and now there is only one unknown which is the angle of rotation. The total transformation matrix in terms an be calculated using the following equation: T = TR.T BD (4.28) where TR is the transformation matrix in terms of (t). Using this matrix, al l the elements of Jacobean matrix are recalculated and are found in terms of only one unknown variable, (t). The determinant of this matrix has to be equal to zero according to the equilibrium conditions. The solving of this equation to calculate the angle of rotation results in 12 different values. Out of these 12 values, some of them are imaginary and some of them are real. The number of imaginary and real values is not constant. Once the value of angle of rotation is known, the final desired transforma tion matrix can be calculated by substituting that value. Calculation of Change in Struts and Ties Length Once the transformation matrix is known, all the vertices of the top platform in its desired orientation can be calculated in the base coordinate sys tem using the following equations: PAGE 53 53 (4.29) The desired strut length can be very easily calculated as the distance between points (PB1 PB6), (PB3 PB5) and (PB2 PB4) respectively. Similarly, the total length of ties 1, 2 and 3 are obtained as the distance between points (PB1 PB4), (PB3 PB6) and (PB2 PB5). Further, unit vectors alone each of the lines defined by the ties and struts can readily be determined and will be written as S ij where the subscripts i and j refer to the specific mechanis m vertex points that define the vector from point i to point j, with this notation, it is apparent that S ij = S ji (4.30) The analysis proceeds by performing a static force analysis. There are twelve unknown force magnitudes, i.e. the compressive or tensile forces of the three struts, the three compliant ties, three bottom ties and three top ties. Compressive forces in the three struts are written as Fa, Fb and Fc. The tensile forces in the three varying length ties are written as fa, fb and fc. T he tensile forces in base ties between pair of points are written as T12, T23 and T13 and those in top ties are T45, T56 and T46. Force balance equation: Each vertex provides three equations and the sum of these forces in X, Y and Z direction is zero. In v ector form, writing force balance equation at points 1, 2, 4 and 6yields fa Fa + T12 + T13 = 0 fb Fb + T12 + T23 = 0 fa Fc + T45 + T46 = 0 fc Fb + T56 + T46 = 0 (4.31) PAGE 54 54 Equations for all the vertices will give (6X3 = 18) equations and there a re only 12 unknown variables. Therefore, four vertices were arbitrarily chosen. Let S ij be unit vector along i and j, i.e. along the ties and struts, where j>I for i = 1,2,..5 and j = 2, 3..6. The system of equations given above can be written as: fa S 14 Fa S 15 + T12 S 12 + T13 S 13 = 0 fb S 25 Fb S 26 + T12 S 12 + T23 S 23 = 0 fa S 14 Fc S 34 + T45 S 45 + T46 S 46 = 0 fc S 36 Fb S 26 + T56 S 56 + T46 S 46 = 0 (4.32) Since, the position of the structure is known, vectors, S ij can be easily found. They are then broken down into x,y and z components: S ij x, S ij y and S ij z. Therefore, 12 equations are obtained from the set of equations given above. These 12 equations can be written in the form as J v = 0 (4.33) Where, and PAGE 55 55 Solving the 12 equati ons listed above, the values of 12 unknowns are obtained. By applying simple concept for force in a linear spring, the equation for the force in each spring can be written as: fa = K1.x1 fb = K2.x2 fc = K3.x3 (4.34) where x1, x2 and x3 are the ch ange in length of each spring and can be calculated from the equation above. Finally the length of the three variable length tie segments l1, l2 and l3 can be determined from following equations: la = l 1total len1 x1 la = l 2total len2 x2 la = l 3tot al len3 x3 (4.35) where, l 1total l 2total and l 3total is total length of side ties (length of compliant part + length of non compliant part) and len1, len3 and len3 are spring free lengths. Numerical Example In this example, the values of cons tants and inputs are arbitrarily decided. They can be changed before starting the example, but will remain constant for all the similar mechanisms once selected. The three spring constants are considered to be same and free lengths of the springs are equal This means the side ties will experience the same force at the mechanisms initial configuration. Assuming that the initial constant values are given: PAGE 56 56 From all above values, the initial transformation matrix is calculated. PAGE 57 57 Using above transformation matrix, the vertices of top platform are calculated in base coordinate system. Desired orientation of XY plane of top coordinate system is obtained from the equations 4.11, 4.12 and 4.13 as: PAGE 58 58 To calculate transformation matrix f or changing orientation of top coordinate system to match desired position of XY plane, the angle of rotation about X and Y axes is calculated using equations 4.15 and 4.22 The values obtained are: Using these values, the final transformation matrix (without considering equilibrium conditions) is obtained to be: To calculate the Jacobean matrix, a transformation matrix with purely rotational transformation matrix (considering equilibrium constraints) is calculated. Using this transformation matrix, all the vertices are recalculated in base coordinate system. Using these vertices, the six lines along three struts and three side ties are calculated to get a Ja PAGE 59 59 It is observed that some of the values are imaginar y and only some of the values are real. Neglecting the imaginary values and selecting one of the real value, the forces in each strut and side tie are calculated. Using these forces, the values of non complaint parts of side ties are obtained as: PAGE 60 60 The se values are well within the limits. the minimum value of angle of rotation gives minimum change in the length of non compliant part of the side tie. Therefore, out of all the real values, only the minimum value of angle of rotation is selected in the above example and answers are presented. PAGE 61 61 C HAPTER 5 FUTURE WORK This research opens up to new challenges. One of them is to actually manufacture the designed mechanism There are many things that need special consideration while doing so. To name a few material and manufacturing cost, a cheap microprocessor and wireless contacts are some of the important things under consideration. This is on design aspect side of it. In addition to this, the change in length of struts and ties calculated is a real number. It will be very difficult to achieve such a great accuracy in real life. It will be great if the accuracy is mapped as a specific area around the desired point of int ersection depending on the accuracy provided by moving mechanical parts of the mechanism. Cost consideration might be an issue where accuracy is expected. Also, if change in length is adjusted in the form of small clicks of specific distance, then number o f clicks and its effect on the mechanism might be an interesting topic to study. The main concern in this case will be stability of the mechanism. Also, only two mechanisms to achieve the desired goal are discussed in this research. There might be other m echanisms that might be more efficient than the ones work in this topic. PAGE 62 62 APPENDIX MAPLE PROGRAM FOR TR IPOD: > > > Constants: > > Procedures > > > Given > > > > PAGE 63 63 > > > > > > > > > > Step 1 get coordinates of P1, P2, and P3 in the 2nd coordinate system > > PAGE 64 64 > > > > > > > > PAGE 65 65 Step 2 get T2to1 transformation matrix > > > > > > > Step 3 determine coordinates of laser line in coord inate system 1 > > > > > > Step 4 determine intersection point of laser line and base plane in coord inate sys tem 1 > PAGE 66 66 > > Now, given the desired intersection point in coord inate sys tem 1, determine the infinitesimal twist that will move the intersection point towards the goal PAGE 67 67 > > PAGE 68 68 MAPLE PROGRAM FOR TENSEGRITY MECHANISM: > > > > > Given: > > > > > > > PAGE 69 69 > > > > > > > > PAGE 70 70 > P rocedure to find desired transformation matrix S tep 1: original transformation matrix > > > S tep 2: find co ordinates of points P4, P5, P6 in base co ordinate system > > > S tep 3: find the equation of desired position of XY plane in base co ordinate sys tem > > > PAGE 71 71 > > > Step 4: Calculation of transformation matrix between top co ordinate system and desired position of top plate > > > > > > > > > > > > PAGE 72 72 > > S tep5: Calculation of required tran s formation matrix between Base and desired co ordinate system. > Bringing determinant of transformation matrix to '0' with laser intersecting at desired point of intersection > > > > > > PAGE 73 73 > > > > > > > > > > > > PAGE 74 74 > > > > > > > > > > > PAGE 75 75 > > PAGE 76 76 LIST OF REFERENCES 1. Crane C., Duffy J., Kinematic Analysis of Robot Manipulators Cambridge University Press, New York, 1998. 2. Carl D. Crane III, Jose Maria Rico, and Joseph Duffy, Application to Spatial Robot Ma nipulators Cambridge University Press, New York. 3. Doughty S., Mechanics of Machines John Wiley & Sons, Inc., New York, 1988 4. F. Bossens, R.A de Callafon R.E. Skelton, Model Analysis of a Tensegrity Structure University of California, San Diego, 2004. 5. R.E.Skelton Mauricio C. de Oliveira Tensegrity Systems Springer, 2009 6. Crane, C., Bayat, J., and Vikas, V., Kinematic Analysis of a Planar Tensegrity Mech anism with Pre Stressed Springs Advances in Rob ot Kinematics, Springer Verlag, 2008 7. Tran, T., Reverse Displacement Analysis for Tensegrity Structures University of Flor ida 2002 8. Bayat, J., Position Analysis of Planar Tensegrity Structures University of Florida, 2006. 9. Stern, I.P., De velopment of Design Equations for Self Deployable N Strut Tensegrity Systems Master of Science Thesis, University of Florida, Gainesville, 1999 10. Tobie, R.S., A Report on an Inquiry into The Existence, Formation and Representation of Tensile Structures M aster of Industrial Design Thesis, Pratt Institute, New York, 1976. 11. Yin, J.P., An Analysis for the Design of Self Deployable Tensegrity and Reinforced Tensegrity Prisms with Elastic Ties, Report for the Center for Intelligent Machines and Robotics Univer sity of Florida, Gainesville, 2000. 12. Yin, J., Duffy, J., and Crane, C., An Analysis for the Design of Self Deployable Tensegrity and Reinforced Tensegrity Prisms with Elastic Ties International Journal of Robotics and Automation 2002. PAGE 77 77 BIOGRAPHICAL SKETC H Ms. Aasawari Deshpande was born in Mumbai, India in 1986. After completing her high school stud ies in Mumbai, she pursued her u ndergraduate studies at University of Mumbai, India, where she received her Bachelor of Engineering in the field of m echanical e ngineering. She worked for Feed Tech Eng, India and Larson and Toubro, India as an Automation Engineer intern. In August 2007, she came to the University of Florida to pursue her Master of Science degree in m echanical e ngineering. 