CHAPTER 3 MODELLING, SIMULATION AND KINEMATIC STUDY OF 3 - DOF PARALLEL MANIPULATORS

Transcription

1 49 CHAPTER 3 MODELLING, SIMULATION AND KINEMATIC STUDY OF 3 - DOF PARALLEL MANIPULATORS 3.1 INTRODUCTION Most existing PKM can be classified into two main families. The PKM of the first family generally called as hexapods. They have a Stewart- Gought parallel kinematic architecture. Many prototypes and commercial hexapod PKM already exist like the CMW300, the TORNADO 2000, the MIKROMAT 6X, the hexapod OKUMA, the hexapod G500. In this first family, a hybrid architecture with a 2-axis wrist mounted in series to a 3-DOF tripod positioning structure (the TRICEPT from Neos Robotics) also found. The second family of PKM has been more recently investigated. In this category the HEXAGLIDE which features six parallel and coplanar linear joints is found. The HexaM is another example with non coplanar linear joints. A 3-axis translational version of the hexaglide is the TRIGLIDE (Mikron), which has three coplanar and parallel linear joints. Another 3-axis translational PKM is proposed by the ISW Uni Stuttgart with the LINAPOD. This PKM has three vertical (non coplanar) linear joints. The URANE SX and the QUICK-STEP are 3-axis PKM with three non coplanar horizontal linear joints. The SPRINT Z3 is a 3-axis PKM with one degree of translation and two degrees of rotations. A hybrid parallel/serial PKM with three parallel inclined linear joints and a two-axis wrist is the GEORGE V (Philippe Wenger and Damien Chablat 2002). Dan Zhang et al (2006) addressed that

2 50 since machining operation requires five axes at most, new configurations with less than six parallel axes would be more appropriate. Development on new configurations is mainly on three axes PKMs. Examples include Tri-Glide and Tripod. From the literatures, the existing PKM structures Tripod and Tri-Glide are considered for the study. In conventional drilling machine, drilling an angular hole requires special fixtures for each specific angle. In this present work, the Tri-Glide and Tripod parallel manipulators mechanisms have been modified to an angular drilling machine for performing angular drilling operation. The mobile platform of the Tri-Glide and Tripod were used as worktable to attain the required angular tilt. This chapter deals with two types of 3-DOF PMs for the kinematic study. In the first type, the lead screws are kept or placed horizontally (Glide type), and in the second type the lead screws are kept vertically (POD type). This study is carried out to find the parallel configuration, which will give a better MP tilt for the smaller linear displacement of the nut, a larger work volume and better singular positions. At first, the architectural description and mobility of the PMs are briefly given. Secondly, the kinematic analyses with single link movement of the PMs are illustrated. Later, the work volume by the Pappus-Guldinus theorem is determined, and the singular positions are simulated. Finally, a study is carried out based on the experimental, analytical and simulation results. 3.2 ARCHITECTURAL DESCRIPTION Figures 3.1 and 3.2 show the 3-PRS Tripod and Tri-Glide PMs, which depict the various names of the linkage assembly. These mechanisms typically consist of a circular plate, referred to as the MP. This MP is connected to a base platform through links. The link is connected to a revolute

3 51 joint at the bottom end, and a spherical joint at the other end. The revolute joint is attached to the nut, which is mounted on the guide way. The guide way consists of a lead screw and two guide rods. The mobility or motion of the MP is accomplished by the screw and nut pairs on the guide ways. Figure 3.1 Tripod PM Figure 3.2 Tri-Glide PM

4 MOBILITY EQUATION The degrees of freedom of the PM are mainly dependent on the number of links which connect the MP and the base platform. In this work, the links are connected by spherical joints to the MP at one end, and the other end is connected by pin joints to the nuts. Nuts are mounted on the lead screws which are actuated by the stepper motors. The mobility of the mechanism is calculated by Equation (1.2). DOF = 6(8-9-1) + 15 = 3 (3.1) = 6 for the spatial PM, F i = 1 for the Revolute joints and Prismatic joints and F i = 3 for the Spherical joints. For the proposed mechanisms, N=8 (3 links, 3 nuts, 1 MP and 1 BP), J=9 and F i = 15. Therefore, the given mechanisms have 3-DOF. The 3-DOF of the PMs are 1) the rotation about the x axis, 2) rotation about the y axis and 3) translation along the z axis. 3.4 KINEMATICS OF SINGLE LINK MOVEMENT The fundamental problem of robot kinematics deals with mapping between vectors in two spaces, viz., joint space ( ) and Cartesian space (X), where represents the position and orientation of the manipulator in the kinematic analysis. a) MP b) BP Figure 3.3 Geometrical representation of the MP and BP of the two PMs

5 53 Two coordinate systems (global and local) are used to describe the position of the mobile platform of the manipulator, as shown in Figures 3.3. The kinematics also considers the motion conversion in the spherical joints (S 1, S 2 and S 3 ) and the pin joints (P 1, P 2 and P 3 ). The ball joints are used to rotate the mobile platform in any specified direction, and the pin joint connects the lower end of the link. The forward kinematics refers to the computation of the position or motion of each link as a function of the joint variables. The kinematic equation for finding the angle of the mobile platform is to be found, in terms of the link length, joint angles and the radius of the mobile platform Kinematics of Tripod Since the ball joints are placed at the vertices of an equilateral triangle, the Cartesian position or the origin of the X, Y and Z frame is essentially the centroid of the triangle. The spherical joints 1 and 3 are considered as fixed when the link connected with spherical joint 2 is actuated. Figure 3.4 Single link movement of Tripod

6 54 Equations (3.2) to (3.9) are formulated based on the arrangement of the single link movement of the Tripod PM, as shown in Figure 3.4. L = 1.5r (3.2) X = L + L Cos (3.3) D = p n (3.4) H = Z D (3.5) From the triangle AOE in Figure 3.4, Cos = (( )( ) ) (3.6) From the triangle AEC in Figure 3.4, Cos = (( ) (3.7) = = tan (3.8) = + (3.9) The input parameters of the single link movement are link length (L), nut displacement (D), mobile platform radius (r), initial angle between link and base platform ( ). The output parameter is mobile platform tilt ( ) Kinematics of Tri-Glide From Figure 3.5, the kinematics of the single link movement of the Tri-Glide PM is formulated for the nut movement of towards centre (Tri- Glide-A) and the nut movement of away from the centre (Tri-Glide-B) for the MP tilts are calculated from Equations (3.10) to (3.14). Nut movement towards centre, From the triangle AEC in Figure 3.5a,

7 55 Cos = ( ) (( ( ) ) (3.10) From the triangle AOF in Figure 3.5a, Cos = ( ) (( ( ) ) ( ) (( ( ) ) (3.11) (a) Towards the centre (b) Away from the centre Figure 3.5 Single link movement of the Tri-Glide PM

8 56 Nut movement away from centre, From the triangle AOG in Figure 3.5b, Cos = ( ) (( ( ) ) (3.12) From the triangle AOF in Figure 3.5b, Cos = ( ) (( ) ( ) ) (3.13) = (3.14) Kinematic Synthesis of the PM The synthesis of the mechanism is the design or criteria of the mechanism to produce a desired output motion for a given input motion. Merlet (2005) stated that dimensional synthesis is to determine the length of the links, the axis and location of the joints, etc. The word dimension will have the broad sense of any parameter that will influence the robot behavior and is needed for the manufacturing of the robot. In this thesis, the dimensional synthesis is taken into account, for the determination of the suitable dimensions of the mechanism, by a logical approach. The greater mobile platform tilt for smaller displacement of nut is considered as an important factor for the proposed study. The geometrical parameters are taken by logical approach. The link lengths are considered to be 200, 300,400 and 500 mm. Similarly, the dimensions of the radius of the MP are considered to be 60, 70, and 100 mm. The initial angle between MP and link are considered as 65º, 70º, 75º and 80º. The simulations of the PMs are carried out using the ADAMS package. Based on the simulation results, the synthesis of the mechanisms is carried out.

9 Kinematic Modelling and Simulation of the PM The various parts of two PMs are modeled. The parts are the lead screws, links, spherical joints (ball & socket joint), base plate etc. The ball and socket joint is selected to withstand the load and the compactness for light weight. Similarly, the lead screw is also designed to withstand the force acting on it. Proper bearings are provided for support by considering the friction and the force acting on them. The pay load is calculated from the thrust force of the drill by the Equation (3.15), Thrust force (T) = k d f 0.7 (3.15) Where, k = 84.7 for steel (Rao 2011), d = Diameter of the drill in mm, f = Feed rate in mm / rev. T = 84.7 x 4 x (0.1) 0.7 = 67.6 N The calculated thrust force is considered for pay load calculations. The PMs are modeled by considering a payload of 150N with a factor of safety more than 2. Considering these specifications as the target parameters, the PMs are modeled and simulated. 3.5 WORK VOLUME ANALYSIS Work space analysis is a specified problem in Direct Kinematics, and it can be conveniently solved by formulating input output equations in a suitable form for easy repetitive calculations, which are needed to compute all the reachable positions and the orientations of the MP. The work volume can be considered as the union of the unit volumes, which have the same manifold geometry. For a PM, the work volume is limited only because of the bounded

10 58 range of the linear actuators, the mechanical limits on the passive joints, and the interference between the links. The workspace of the mechanism was studied, using different methods, e.g., geometric and numerical approaches. But most of them are related to the position workspace, which is a part of the workspace. In fact, the workspace can be divided into the position workspace and the orientation workspace for a manipulator with rotational capability. The problem of the determination of the workspace, in terms of the volume of the PMs, is dealt here. The spherical joints connected to the MP and the circumferential points of the MP are considered here, for finding the work volume Analytical Approach The analytical approach is carried out to determine the work volume of the PMs by considering the MP geometry. Knowing the MP radius (r) and its angle of tilt ( ), the work volume is determined using the Pappus- Guldinus theorem. The theorem states that the volume of the body of revolution is equal to the generating area times the distance travelled by the centroid area while the body is being generated. Figure 3.6 Schematic representation of MP for work volume analysis

11 59 a) MP b) I region c) II region Figure 3.7 Regions of surface area of the MP The circle ABCD of Figure 3.6 represents the MP of the PMs. In this approach, the total surface is divided into two separate regions to calculate their area and the centroid as shown in Figure 3.7. The work volume profiles of different parts of the MP are shown in Figures 3.8 and 3.9. Figure 3.8 Work volume of I region Figure 3.9 Work volume of II region Using the Pappus-Guldinus theorem, the work volume is calculated from the following Equations (3.16) to (3.22). Consider the region ABCD, Area 1 (OEAFO) = (3.16) Area 2 (OEGBO) = (3.17)

12 60 Area 3 (OBDO) = (3.18) Area 4 (BCD) = Area 1- Area 3 (3.19) Work volume = Surface area generated Distance traveled by the centroid Work volume 1(BAD) = ( ) (3.20) Work volume 2 (BCD) = ( ) (3.21) Total work volume = Work volume 1 + Work volume 2 (3.22) Normalization of Work volume Normalization is the process to transform different scales and units among various criteria into common measurable units to allow comparisons across the work volume. Assume f ij to be of the evaluation value R of link length j under work volume i, then an element r ij of the normalized evaluation value R can be calculated from the Equation (3.23). r ij f J j 1 ij f 2 ij j = 1,2,3,.,J, i = 1,2,3,.,n. (3.23) Methodology ADAMS is used to simulate the PMs. Once the simulation is done, the positions of the joints on the MP are noted, with respect to the simulation time. The simulations of the PMs are done and shown in Figures 3.10 a to f. The positions of the spherical joints connected to the MP and the circumferential points of the MP are noted, with respect to the simulation time. The noted points are given as the input to the MATLAB software, for

13 61 getting the work envelope, and later to the AUTOCAD package for determining the work volume. a b c d e f Figure 3.10 Simulation of the MP circumferential points a) Tripod Link 1 Movement b) Tripod Link 2 Movement c) Tripod Link 3 Movement d) Tri-Glide Link 1 Movement e) Tri-Glide Link 2 Movement f) Tri-Glide Link 3 Movement 3.6 SINGULARITY ANALYSIS Singularity is a significant problem in parallel mechanisms, because it leads to loss of controllability and degradation of the stiffness of the system. The Figures 3.11 and 3.12 shows the schematic diagram of Type I and Type II Singularity positions. Type I Singularity position shows the posture of the link 1 and the mobile platform being in the same plane. Similarly, the posture of link 2, link

14 62 3 and the mobile platform being in the same plane in the Type II Singularity position (Xiang Cheng et al 2004). Figure 3.11 Singular Poses - Type I Singularity Figure 3.12 Singular Poses - Type II Singularity The proposed mechanisms are simulated to find the various singularities, and the positions of the PMs are as shown in Figures 3.13a to f.

15 63 Figures 3.13a and 3.13d show the first kind (Type I) of singularity of the PRS PMs, when the MP has 180º with any one of the links. Similarly, the MP is in line with the other two links (L 2, L 3 ) while link (L 1 ) is kept constant. a b c d e f Figure 3.13 Three kinds of singularity positions a) Tripod Type I singularity b) Tripod Type II singularity c) Tripod Type III singularity d) Tri-Glide Type I singularity e) Tri-Glide Type II singularity f) Tri-Glide Type III singularity The second kind (Type II) of singularity of the mechanism is obtained as shown in Figures 3.13b and 3.13e. The third kind (Type III) of singularity is obtained when all the links are actuated in different positions to reach the first or second type of singularity position as shown in Figure 3.13c and 3.13f. The above mentioned three kinds of singularities are taken in to consideration for the comparative study of the two PMs.

16 EXPERIMENTAL MODELS The experimental models of the Tripod and the Tri-Glide PMs are built, and are shown in Figures 3.14 and The models are made of the same geometrical size of the simulation models. Each link is actuated at various positions to get various MP tilts. Figure 3.14 One Link arrangement of Tri-Glide PM Figure 3.15 Prototype model of Tri-Glide PM

17 65 Figure 3.16 One Link arrangement of Tripod PM Figure 3.17 Prototype model of Tripod PM

18 METHODS USED IN THE POSITION ANALYSIS Experiment An experiment was carried out for the position analysis by actuating one of the links at a time. A laser torch is mounted on the center C of the MP, and the laser light is projected on the vertical screen at a point A, which is at a predetermined distance AC from the point of the laser source. When the MP is tilted by the actuation of the link, the laser beam gets deflected to some other point B on the vertical screen. The angle of tilt of the MP is measured from the orientation of the source. From Figure 3.18, the MP tilt angle ( ) is measured from the Equation (3.24). = tan -1 (AB / AC) (3.24) Figure 3.18 Calculation of angle of tilt of the MP Figure 3.19 and 3.20 show the experimental setup of laser torch on PMs. The Laser torch is placed on the MP in such a way that the laser torch axis is aligned with one of the joint axis from the MP. The experimental procedures of actual experiments are shown in Figure 3.21 to In initial position all the links are placed in same distance in such a way that the nut positions are same in the lead screws. Similarly, at the final position any one or two links are actuated and the other link or links are kept constant. Due to

19 67 the various positions of nut, the MP is tilted. The tilt is measured from the positions of laser source. Figure 3.19 Experimental setup with laser torch Tripod PM Figure 3.20 Experimental setup with laser torch - Tri-Glide PM

20 68 Figure 3.21 Positioning of laser torch Figure 3.22 Initial position of PM with laser torch In Figure 3.24, the link1 is kept constant and the links 2, 3 are actuated. In initial position all the links are placed in same distance in such a way that the nut positions are same in the lead screws.

21 69 Figure 3.23 Laser source on screen at initial position Figure 3.24 Final position of PM with laser torch

22 70 Figure 3.25 Laser source on screen at final position Figure 3.26 Laser source positions on screen

23 Analytical Software programs were written in C and JAVA languages for finding the displacement of the nut for the PMs. The programs were written based on the kinematic equations formulated. The tilt of the MP obtained by actuating the links was obtained from the programs. In the analytical method, in order to verify the displacement of the nut, the angle of tilt of the MP is given as the input. Once the input is given, the program calculates the linear displacement of the nut and the number of rotations of the screw, which will be useful for giving the pulses to the stepper motor Measuring the MP tilt using the Accelerometer To measure the angle of tilt of the MP, an accelerometer which works on 2.6 V to 5 V power supply is used, and it is directly interfaced to the ADC of a microcontroller. This module is used to sense the motion or the tilt in 3 axes. The accelerometer senses tilt angle which is manifestation of acceleration. The reference frame of the accelerometer and the coordinate frame of the mobile platform are aligned with each other and hence the tilt angle obtained from the accelerometer corresponds to the orientation of the mobile platform. For the present study the ADXL 335 and ADXL 203 accelerometers are used and its specifications are given in Appendix 1. The only drawback of the accelerometer is that it gives outputs only in the form of voltages. The angle at which the MP is directly tilted is not measured directly. This is why the calibration of the accelerometer module is very much essential.

24 Calibration of accelerometer The calibration of the accelerometer is carried out by considering the output voltages, that the accelerometer module gives for certain specific standard angles. By knowing these voltages, the necessary angle of tilt of the MP can be found. The following steps are used for the calibration of the accelerometer, 1) A sine bar is first placed on a flat surface plate. The sine bar is used to get the different angles with the help of the gauge blocks and the corresponding voltage values of the accelerometer are obtained. 2) At first, the sine bar is tilted at zero degrees when placed over the surface plate. Now the accelerometer is kept over the sine bar as shown in Figure 3.27, and the readings are noted when the sine bar is tilted to various angles. Figure 3.27 Calibration of the accelerometer 3) With the help of the gauge blocks, different angles are taken, the readings are tabulated and the corresponding voltages are tabulated. 4) The circuit connections are made as per the diagram shown in Figure Two multimeters are used to find the MP tilt in the X direction and in the Y direction.

25 73 Figure 3.28 Circuit diagram of the accelerometer ADAMS The manipulator models are constructed in ADAMS by building the physical attributes of the elements, or the parts in the mechanical systems that have rigid bodies, point masses, flexible bodies and constraints. The working models are created using ADAMS, as shown in Figure The models are simulated for the nut displacement of 50 mm and the angles of tilt of the MP about the X and Y axes are obtained from the simulation graphs. a b Figure 3.29 Various constraints of ADAMS models

26 74 The detailed description of Adams model is shown in Figures 3.30 and The Figure 3.30 shows the various constrains associated with base and lead screw, the description of joints are as follows, Fixed Joint 1 Between base platform and Channel section Fixed joint 2 Between Plate 1 and Channel section Rotational motion Between Lead screw and stepper motor Fixed joint 3 Between Plate 2 and Guide rod 1 Fixed joint 4 Between Plate 2 and Guide rod 2 Cylindrical joint 1 Between Nut and Guide rod 1 Cylindrical joint 2 Between Nut and Guide rod 2 Figure 3.30 Various constraints associated with base and lead screw Similarly, the Figure 3.31 shows the various constrains associated with base and lead screw, the description of joints are as follows, Fixed Joint 5 Between Plate1and Channel section

27 75 Pin joint Between the Nut and Link Screw joint Between Nut and Lead Screw Spherical joint Between Link and Mobile Platform Fixed Joint 6 Between Plate 3 and Guide rod 1 Fixed Joint 7 Between Plate 3 and Guide rod 2 Figure 3.31 Various constraints associated with link and MP Similarly, the other link associated joints are made in Tripod and Tri-Glide PMs. The necessary nut movement is actuated by rotational joint in terms of angle of rotation with respect to time. The model is simulated based on the nut displacement.

28 SIMULATION STUDY OF 3- PRS, 3-PRR AND 3-PUS PMs In the present work, simulation study on three Tripod PMs (3- PRS, 3-PRR and 3-PUS PMs) is carried out to find which Tripod configuration is better to achieve better singular positions and the MP tilt (by considering the Transmission angle between MP and the link). The PUS PM is considered with PRR and PRS PMs here because the base location of actuators of PUS PM lead to: (i) reduction of the (motor) weight carried by the legs; (ii) elimination of the actuation transmission requirement (through intermediary joints as in the case of the Stewart-Gough platform); and (iii) most importantly absorption of reaction-forces by the ground. By selecting the base actuated joint to be prismatic, the proximal links are not subjected to the bending moments and the corresponding stresses (Madhusudanan Sathia Narayanan 2010). In this study, first, the Transmission angle and singularity positions are briefly described and next, the simulation results are compared based on the geometrical parameters, tilt, singularity and driving torque for the input motion of the mechanism Transmission Angle Transmission angle is the angle between the coupler link and the output link of the four bar mechanism. The Transmission angle is an important index evaluating the quality of motion / force transmission. It helps to decide the best among a family of possible mechanisms for the most effective force transmission. Though a good Transmission angle is not a cure to all for every design problem, for many mechanical applications it can guarantee the performance of a linkage at high speed without unfavorable vibrations. The study of link mechanisms shows that Transmission angle is

29 77 significant not only as an indicator of good force and motion transmission but also a prime factor in the linkage sensitivity to small design parameter errors. For the purpose of high speed, high accuracy and high quality of motion transmission, the most widely accepted design limits for the Transmission angle are (45º, 135º) or (40º, 140º). Additionally, the Transmission angle does not consider the dynamic forces due to velocity and accelerations. For this reason, it is widely used in the kinematic synthesis. A planar four bar mechanism is a single closed loop system. A parallel robot is a multi closed loop mechanism. Usually, a fully parallel robot has more or less the characteristic of a planar four bar mechanism. Wang Jinsong et al (2009) suggested that the design concept of the four bar mechanism using Transmission angle should be used in the design of parallel robot. They also proposed the local and global transmission indices in the design of a special 3-DOF parallel robot, which is kinematically considered as the combination of two planar mechanisms. Chang (1988) explained that for a general RSSR-linkage, the axes of the two R-pairs do not intersect. When these two axes do intersect, the RSSR-linkage becomes a spherical four-bar. From the Figure 3.32, the PM is considered as 4 bar linkage when single link is actuated. In this study, the parallel manipulator is selected for angular machining application and the term transmission angle is related to quality motion of the system is considered. Hence, the transmission angle is introduced in the four bar mechanism and to evaluate the performance of the system.

30 78 a) Front view b) Top view Figure 3.32 Single link movement of PM a b c d Figure 3.33 Synthesis of mechanism using transmission angle ( )

31 79 The transmission angle is an important parameter to evaluate the quality of motion/force transmission, for this reason, Local Transmission Index (LTI) is used in the design of parallel manipulator. The presented mechanisms are synthesized for the transmission angles of 135º to obtain high quality of motion/force transmission as per LTI. Figures 3.33a and d shows the simulation of 3-PRR and 3-PUS parallel manipulator single link movement, similarly Figures 3.33b and c shows two link movement at equal vertical displacement of the nut Simulation of Singularity Positions of PMs The proposed mechanisms are simulated to find various singularity positions as explained in section 3.6. Figure 3.34a shows the Type I singularity of the PUS and PRS PMs. Similarly, Type II singularity of the mechanism is shown in Figure 3.34b for 3-PRR mechanism. The above mentioned two kind of singularity is taken in to account for the study. a) 3- PUS and 3-PRS b) 3-PRR Figure 3.34 Singularity positions of tripod PMs

32 STRUCTURAL ANALYSIS Neugebauer et al (2006) summarized that a FEM tool was required for to optimize the PM structure. The main aim of this structural analysis is to analyze the structure of the Tripod, Tri-Glide and 3-PRR parallel manipulators for angular drilling applications. The Parallel manipulator structures are analyzed based on characteristics like stress, deformation and moment reactions in PMs joints. The geometrical parameters of the link structure is taken as 10 mm,12 mm,14 mm,16 mm and18mm diameter for solid structure. For hollow structure, 18 mm is taken as outer diameter and inner diameters are varied as 11mm, 12 mm, 13 mm, 14 mm and 15 mm. Similarly, taking 18 mm as inner diameter the outer diameters are varied as 21mm, 22 mm, 23 mm, 24 mm, and 25 mm. Figure 3.35 ANSYS model of Tripod PM

33 81 Figure 3.36 ANSYS model of Tri-Glide PM Figure 3.37 ANSYS model of 3-PRR PM To reduce the weight of the mild steel (MS) structure, the link is replaced by the aluminum (Al) alloy. Figures 3.35 to 3.37 show the ANSYS simulated models of the Tripod, Tri-Glide and 3-PRR PMs using BEAM 189 element. The mechanism is modeled by considering the geometrical parameters of link length 200 mm, mobile platform radius 90 mm and initial angle between the link and base platform is 70º. A load of 150N is applied at the centre of the mobile platform.

34 82 The Materials used in this structural analysis are mild steel and aluminum alloy. Table 3.1 shows the properties of the two materials. Table 3.1 Material property of mild steel and aluminium alloy Properties Mild Steel Aluminium Alloy Young's Modulus N/mm N/mm 2 Poisson's Ratio Density 7.85 e-006 kg/mm³ 2.7e-006 kg/mm³ 3.11 RESULTS AND DISCUSSION In the kinematic study of the two PMs, the dimensional synthesis and the work volume were carried out, to evaluate the influence of the geometrical parameters (MP radius, Link length and the initial angle between link and the base platform). Similarly, the simulation of the singularity positions and the experiments on the position analysis were conducted, to evaluate the singular configurations and the tilt of the MP of the two PMs Dimensional Synthesis The link lengths (L), the initial angle between the link and the base ), and the radius of the MP (r) are considered to be the important parameters, which play a vital role in increasing or decreasing the tilt of the MP of the two PMs, and are compared here. The same geometrical parameters are used for the purpose of comparing these two configurations. The geometrical parameters for the two models are as follows, the MP radius is varied from 60 to 100 mm (60, 70, 80, 90 and100), the link length is varied from 200 to 500 mm (200, 300, 400 and 500), and is varied from 65 to 80 (65, 70, 75 and 80 ).

35 83 Table 3.2 shows the theoretical results of the MP tilt for the Tri- Glide and Tripod PMs. From the results presented, it is observed that when the MP radius is 60 mm, the Tripod PM has a maximum MP tilt of (for = 80 and L = 500 mm), and a minimum of (for = 65 and L = 200 mm) for the nut displacement of 50 mm. The Tri-Glide (B) PM has a maximum MP tilt of (for = 65 and L = 200 mm), and a minimum of 7.47 (for = 80 and L = 500 mm). Similarly, the Tri-Glide (A) PM has a maximum MP tilt of (for = 65 and L = 500 mm) and a minimum of 1.56 (for = 80 and L = 200 mm). Table 3.2 MP tilt for MP radius of 60 mm and nut displacement of 50 mm Initial angle ( ) In Deg Link Length (L) in mm Tri-Glide (Towards center (A)) Angle of Tilt of MP ( ) in Deg Tri-Glide (Away from center (B)) Tripod (C)

36 84 From Table 3.3, it is observed that, when the MP radius is 70 mm, the Tripod PM has a maximum MP tilt of (for = 80 and L = 500 mm) and a minimum of (for = 65 and L = 200 mm). The Tri-Glide (B) PM has a maximum MP tilt of (for = 65 and L = 200 mm) and a minimum of 6.38 (for = 80, L = 500 mm). Similarly, the Tri-Glide (A) PM has a maximum MP tilt of (for = 65 and L = 200 mm) and a minimum of 1.34 (for = 80 and L = 200 mm). Table 3.3 MP tilt for MP radius of 70 mm and nut displacement of 50 mm Initial angle ( )In Deg Link Length (L) in mm Angle of Tilt of MP ( ) in Deg Tri-Glide (Towards center (A)) Tri-Glide (Away from center (B)) Tripod (C)

37 85 From Table 3.4, it is observed that, when the MP radius is 80 mm, the Tripod PM has a maximum MP tilt of (for = 80 and L = 500 mm) and a minimum of (for = 65 and L = 200 mm). The Tri-Glide (B) PM has a maximum MP tilt of (for = 65 and L = 200 mm) and a minimum of 5.57 (for = 80, L = 500 mm). Similarly, the Tri-Glide (A) PM has a maximum MP tilt of 9.38 (for = 65 and L = 200 mm) and a minimum of 1.17 (for = 80 and L = 200 mm). Table 3.4 MP tilt for MP radius of 80 mm and nut displacement of 50 mm Initial angle ( ) In Deg Link Length (L) in mm Angle of Tilt of MP ( ) in Deg Tri-Glide (Towards center (A)) Tri-Glide (Away from center (B)) Tripod (C)

38 86 From Table 3.5, it is observed that, when the MP radius is 90 mm, the Tripod PM has a maximum MP tilt of (for = 80 and L = 500 mm) and a minimum of (for = 65 and L = 200 mm). The Tri-Glide (B) PM has a maximum MP tilt of (for = 65 and L = 200 mm) and a minimum of 4.94 (for = 80, L = 500 mm). Similarly, the Tri-Glide (A) PM has a maximum MP tilt of 8.36 (for = 65 and L = 200 mm) and a minimum of 1.04 (for = 80 and L = 200 mm). Table 3.5 MP tilt for MP radius of 90 mm and nut displacement of 50 mm Initial angle ( ) In Deg Link Length (L) in mm Angle of Tilt of MP ( ) in Deg Tri-Glide (Towards center (A)) Tri-Glide (Away from center (B)) Tripod (C)

39 87 From Table 3.6, it is observed that, when the MP radius is 100 mm, the Tripod PM has a maximum MP tilt of (for = 80 and L = 500 mm) and a minimum of (for = 65 and L = 200 mm). The Tri-Glide (B) PM has a maximum MP tilt of (for = 65 and L = 200 mm) and a minimum of 4.44 (for = 80, L = 500 mm). Similarly, the Tri-Glide (A) PM has a maximum MP tilt of 7.53 (for = 65 and L = 200 mm) and a minimum of 0.94 (for = 80 and L = 200 mm). Table 3.6 MP tilt for MP radius of 100 mm and nut displacement of 50 mm Initial Angle of Tilt of MP ( ) in Deg Link Length angle Tri-Glide Tri-Glide (L) ) in (Towards center (Away from Tripod (C) in mm Deg (A)) center (B))

40 Influence of the link length on the MP tilt The dimensional synthesis was carried out by the analytical method, and the results are shown in Figures 3.38 to For the Tripod PM, the dimensional synthesis results show that by increasing the link length from 200 mm to 500 mm, the angle of tilt of the MP was found to increase by 0.54 % (for r = 60 mm and = 65 ). Similarly, 0.53% for = 70, 0.55% for = 75 and 0.59% for = 80. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to increase by 7.08 % (from = 65 to = 80 ) and 7.14 % for the MP radius of 60 mm. Similarly, by increasing the link length from 200 mm to 500 mm for the Tripod PM, the angle of tilt of the MP was found to increase by 0.22 % (for r = 100 mm and = 65 ). Similarly, 0.22% for = 70, 0.21% for = 75 and 0.21% for = 80. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to increase by 4.35 % (from = 65 to = 80 ) and 4.34 % for the MP radius of 100 mm. For the Tri-Glide PM (A), by increasing the link length from 200 mm to 500 mm, the angle of tilt of the MP was found to increase by % (for r = 60 mm and = 65 ). Similarly, 25.68% for = 70, 35.80% for = 75 and 60.60% for = 80. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to decrease by % (from = 65 to = 80 ) and % for the MP radius of 60 mm. Similarly, by increasing the link length from 200 mm to 500 mm for the Tri-Glide PM (A), the angle of tilt of the MP was found to increase by % (for r = 100 mm and = 65 ). Similarly, 26.12% for = 70, 35.96% for = 75 and 60.50% for = 80. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to decrease by % (from = 65 to = 80 ) and % for the MP radius of 100 mm.

41 89 For the Tri-Glide PM (B), by increasing the link length from 200 mm to 500 mm, the angle of tilt of the MP was found to decrease by % (for r = 60 mm and = 65 ). Similarly, 26.96% for = 70, 26.26% for = 75 and 29.39% for = 80. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to decrease by % (from = 65 to = 80 ) and % for the MP radius of 60 mm. Similarly, by increasing the link length from 200 mm to 500 mm for the Tri-Glide PM (B), the angle of tilt of the MP was found to decrease by % (for r = 100 mm and = 65 ). Similarly, 23.44% for = 70, 24.55% for = 75 and 28.38% for = 80. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to decrease by % (from = 65 to = 80 ) and % for the MP radius of 100 mm Influence of the MP radius on the MP tilt For the Tripod PM, by increasing the radius of MP from 60 mm to 100 mm, the angle of tilt of the MP was found to decrease by 38.67% (for L = 200 mm and = 65 ), 39.23% for = 70, 39.82% for = 75, 40.43% for = 80. Similarly, by increasing the radius of the MP from 50 mm to 100 mm, the angle of tilt of the MP was found to decrease by 38.87% (for L = 500 mm and = 65 ). Similarly, 39.42% for = 70, 40% for = 75 and 40.66% for = 80. For the Tri-Glide PM (A), by increasing the radius of the MP from 60 mm to 100 mm, the angle of tilt of the MP was found to decrease by 39.76% (for L = 200 mm and = 65 ), 40.02% for = 70, 40.09% for = 75, 39.74% for = 80. Similarly, by increasing the radius of the MP from 60 mm to 100 mm, the angle of tilt of the MP was found to decrease by 39.47% (for L = 500 mm and = 65 ). Similarly, 39.66% for = 70, 39.94% for = 75 and 39.89% for = 80.

42 90 For the Tri-Glide PM (B), by increasing the radius of the MP from 60 mm to 100 mm, the angle of tilt of the MP was found to decrease by 51.96% (for L = 200 mm and = 65 ), 44.82% for = 70, 42.49% for = 75, 41.39% for = 80. Similarly, by increasing the radius of the MP from 50 mm to 100 mm, the angle of tilt of the MP was found to decrease by 43.78% (for L = 500 mm and = 65 ). Similarly, 42.16% for = 70, 41.16% for = 75 and 40.56% for = Influence of the initial angle on the MP tilt For the Tripod PM, by increasing the initial angle between the link and the base platform from 65 to 80, the angle of tilt of the MP was found to increase by 7.08% (for L = 200 mm and r = 60 mm), 6.07% for r = 70 mm, 5.34% for r = 80 mm, 4.77% for r = 90 mm and 4.35% for r = 100 mm. Similarly, by increasing the initial angle between the link and the base platform from 65 to 80, the angle of tilt of the MP was found to increase by 7.14% (for L = 500 mm and r = 60 mm), 6.12% for r = 70 mm, 5.37% for r = 80 mm, 4.76% for r = 90 mm and 4.34% for r = 100 mm. For the Tri-Glide PM (A), by increasing the initial angle between the link and the base platform from 65 to 80, the angle of tilt of the MP was found to decrease by 84.25% (for L = 200 mm and r = 60 mm), 84.25% for r = 70 mm, 84.29% for r = 80 mm, 84.31% for r = 90 mm and 84.25% for r = 100 mm. Similarly, by increasing the initial angle between the link and the base platform from 65 to 80, the angle of tilt of the MP was found to decrease by 68.17% (for L = 500 mm and r = 60 mm), 68.19% for r = 70 mm, 68.33% for r = 80 mm, 68.42% for r = 90 mm and 68.39% for r = 100 mm. For the Tri-Glide PM (B), by increasing the initial angle between the link and the base platform from 65 to 80, the angle of tilt of the MP was found to decrease by 65.18% (for L = 200 mm and r = 60 mm), 61.39% for r = 70 mm, 59.52% for r = 80 mm, 58.33% for r = 90 mm and 57.53% for r = 100 mm. Similarly, by increasing the initial angle between the link and the base platform from 65 to 80, the angle of tilt of the MP was found to decrease by 61.75% (for L = 500 mm and r = 60 mm), 60.88% for r = 70 mm, 60.29% for r = 80 mm, 59.87% for r = 90 mm and 59.56% for r = 100 mm.

43 91 a) MP = 60 mm b) MP = 70 mm c) MP = 80 mm d) MP = 90 mm e) MP = 100 mm Figure 3.38 Influence of link lengths = 65 over MP tilt for various MP radii and

44 92 a) MP = 60 mm b) MP = 70 mm c) MP = 80 mm d) MP = 90 mm e) MP = 100 mm Figure 3.39 Influence of link lengths = 70 over MP tilt for various MP radii and

45 93 a) MP = 60 mm b) MP = 70 mm c) MP = 80 mm d) MP = 90 mm e) MP = 100 mm Figure 3.40 Influence of link lengths = 75 over MP tilt for various MP radii and

46 94 a) MP = 60 mm b) MP = 70 mm c) MP = 80 mm d) MP = 90 mm d) MP = 100 mm Figure 3.41 Influence of link lengths = 80 over MP tilt for various MP radii and

47 Comparison on maximum and minimum tilt of the MP The kinematic study on the single link movement of the two PMs structure is based on the MP tilt, by considering the geometrical parameters, like the link length, MP radius and initial angle between the link and the base platform. Table 3.7 shows the results of the maximum and minimum MP tilt of the Tri-Glide (A), Tri-Glide (B) and Tripod PMs. By considering the nut displacement of 50 mm, for the Tripod PM, it is observed that the maximum angle of tilt of the MP is found to be 31.80, when the link length is 500 mm with = 80, and a minimum of is obtained when the link length is 200 mm with = 65. Table 3.7 MP tilt comparison for various geometrical parameters for the nut displacement of 50 mm MP radius in mm The angle of the MP tilt in Tri-Glide (A) Tri-Glide (B) Tripod (C) Max (L=500 mm, =65 ) Min (L=200 mm, =80 ) Max (L=200 mm, =65 ) Min (L=500 mm, =80 ) Max (L=500 mm, =80 ) Min (L=200 mm, =65 ) Similarly, for the Tri-Glide (A) PM, it is observed that a maximum angle of tilt of the MP of is obtained when the link length is 500 mm with = 65 and a minimum of 0.94 is obtained when the link length is 200 mm with = 80. For the Tri-Glide (B) PM, it is observed that a maximum angle of tilt of the MP is obtained, when the link length is 200 mm

48 96 with = 65 and a minimum of 4.44 is obtained, when the link length is 500 mm with = 80. From the study, when the variation of the initial angle is 15 ( = 65 to 80 ) and the MP radius is 100 mm, the maximum variation of the angle of tilt of the MP was 0.86 for the Tripod PM. Similarly, under the same conditions, the angle of tilt of the MP was for the Tri-Glide (B) PM and 6.53 for the Tri-Glide (A) PM respectively. From the above results, it is observed that the MP tilt of the Tripod PM is more precise than that of the Tri-Glide (B) PM and Tri-Glide (A) PM, when the link length is increased for a constant MP radius. Similarly, when the MP radius is varied for 40 mm (from r = 60 mm to 100 mm) for the constant initial angle = 80, the maximum variation of the angle of tilt of the MP was 0.62 for the Tri-Glide (A) PM. Similarly, under the same condition, the angle of tilt of the MP was 4.38 for the Tri- Glide (B) PM and for the Tripod PM. From the analytical results, it is observed that the Tri-Glide (B) PM MP tilt was (For r = 60 mm, L = 200 mm and = 65 ). Similarly, under the same condition, the angle of tilt of the MP was 9.91 for the Tri- Glide (A) and for the Tripod PMs Work Volume Analysis In the work volume analysis, the geometrical parameters are taken as important parameters. To analyze the Tripod and Tri-Glide parallel manipulators, the same geometrical parameters are used. The geometrical parameters include the link length, MP radius and initial angle between the link and the base platform.

49 97 Table 3.8 shows the results of the Tri-Glide and Tripod PMs for MP radius 60 mm. From the results presented, it is observed that when the MP radius is 60 mm, the Tripod PM work volume is found to be a maximum of mm 3 (for = 80 and L = 500 mm) and a minimum of mm 3 (for = 65 and L = 200 mm) for the nut displacement of 50 mm. The Tri-Glide (B) PM work volume is found to be a maximum of mm 3 (for = 65 and L = 200 mm) and a minimum of mm 3 (for = 80 and L = 500 mm). Similarly, the Tri-Glide (A) work volume is found to be a maximum of mm 3 (for = 65 and L = 500 mm) and a minimum of mm 3 (for = 80 and L = 200 mm). Table 3.8 Work volume for MP radius of 60 mm for nut displacement 50 mm Initial angle In Deg Link Length in mm Tri-Glide (Towards center (A)) Work volume in mm^3 Tri-Glide (Away from center(b)) Tripod (C)

50 98 From Table 3.9 normalized work volume results, it is observed that the Tripod PM work volume is found to be a maximum of (for = 80 and L = 500 mm) and a minimum of (for = 65 and L = 200 mm). Similarly, Tri-Glide (B) and Tri-Glide (A) PM work volume is found to be a maximum of (for = 65 and L = 200 mm), (for = 65 and L = 500 mm) and a minimum of (for = 80 and L = 500 mm), (for = 80 and L = 200 mm). Table 3.9 Normalized work volume for MP radius of 60 mm for nut displacement 50 mm Work volume Initial Link Length in angle Tri-Glide Tri-Glide mm (Towards (Away from In Deg Tripod (C) center (A)) center(b))

51 99 Table 3.10 shows the results of the Tri-Glide and Tripod PMs for MP radius 60 mm. From the results presented, it is observed that when the MP radius is 60 mm, the Tripod, Tri-Glide (B) and Tri-Glide (A) PMs work volume is found to be a maximum of mm 3 (for = 80 and L = 500 mm), mm 3 (for = 65 and L = 200 mm), mm 3 (for = 65 and L = 500 mm) and a minimum of mm 3 (for = 65 and L = 200 mm), mm 3 (for = 80 and L = 500 mm) and mm 3 (for = 80 and L = 200 mm). Table 3.10 Work volume for MP radius of 70 mm for nut displacement 50 mm Work volume in mm^3 Initial Link Length angle Tri-Glide Tri-Glide in mm In Deg (Towards (Away from Tripod (C) center (A)) center(b))

52 100 From Table 3.11 normalized work volume results, it is observed that when the MP radius is 70 mm, the work volume of the Tripod PM, Tri- Glide (B) and Tri-Glide (A) PMs is found to be a maximum of , (for = 80 and L = 500 mm), (for = 65 and L = 200 mm) and (for = 65 and L = 500 mm) and a minimum of (for = 65 and L = 200 mm), (for = 80 and L = 500 mm) and (for = 80 and L = 200 mm). Table 3.11 Normalized work volume for MP radius of 70 mm for nut displacement 50 mm Initial angle In Deg Link Length in mm Tri-Glide (Towards center (A)) Work volume Tri-Glide (Away from center(b)) Tripod (C)

53 101 From Table 3.12, it is observed that when the MP radius is 80 mm, the work volume of the Tripod PM is found to be a maximum of mm 3 (for = 80 and L = 500 mm) and a minimum of mm 3 (for = 65 and L = 200 mm). Similarly, Tri-Glide (B) and Tri-Glide (A) PM work volume is found to be a maximum of mm 3 (for = 65 and L = 200 mm) and mm 3 (for = 65 and L = 500 mm) and a minimum of mm 3 (for = 80 and L = 500 mm), mm 3 (for = 80 and L = 200 mm). Table 3.12 Work volume for MP radius of 80 mm for nut displacement 50 mm Work volume in mm^3 Initial Link Length angle Tri-Glide Tri-Glide Tripod (C) in mm (Towards (Away from In Deg center (A)) center(b))