DESIGN AND CONTROL OF A THREE DEGREE-OF-FREEDOM PLANAR PARALLEL ROBOT. A Thesis Presented to. The Faculty ofthe. Fritz J. and Dolores H.

Transcription

1 7 / ~'!:('I DESIGN AND CONTROL OF A THREE DEGREE-OF-FREEDOM PLANAR PARALLEL ROBOT A Thesis Presented to The Faculty ofthe Fritz J. and Dolores H.Russ College of Engineering and Technology Ohio University In partial Fulfillment of the Requirements for the Degree Master of Science by Atul Ravindra Joshi August 2003

2 iii TABLE OF CONTENTS LIST OF FIGURES v CHAPTERl: INTRODUCTION Introduction of Planar Parallel Robots Literature Review Objective of the thesis 5 CHAPTER2: PARALLEL ROBOT KINEMATICS Pose Kinematics Inverse Pose Kinematics Forward Pose Kinematics Rate Kinematics Workspace Analysis 20 CHAPTER3: PARALLEL ROBOT HARDWARE IMPLEMENTATION Design And Construction Of3-RPR Planar Parallel Robot Control Of3-RPR Planar Parallel Robot 34 CHAPTER4: EXPERIMENTATION AND RESULTS 41 CHAPTERS: CONCLUSIONS 50 S.1 Concluding Statement Future Work 51

3 IV REFERENCES 53 APPENDIX A: LVDT CALLIBRATION 56 APPENDIX B: OPERATION PROCEDURE 61 APPENDIX C: INVERSE POSE KINEMATICS CODE 64 ABSTRACT

4 v LIST OF FIGURES Figure Page RPR Diagram RPR Hardware RPR Planar Parallel Manipulator Kinematics Diagram RPR Manipulator Hardware Inverse Pose Kinematics Velocity Kinematics Angle Nomenclatures Example of the 3-RPR Reachable Workspace Workspace analysis by geometrical method Summary of the 3-RPR Hardware Configuration RPR Planar Parallel Robot Hardware External MultiQ Boards, Solenoid Valves, and Power Supply Simulink/Wincon Interface to the 3-RPR Planar Parallel Robot RPR Planar Parallel Robot Control Hardware Power Supply For 3-RPR Planar Parallel Robot Oil-Less Air Compressor For 3-RPR Planar Parallel Robot External MultiQ Board Solenoid Valve 33

5 vi 3.10 Cartesian Pose Control Mode for 3-RPR Planar Parallel Robot Cartesian Rate Control Mode for 3-RPR Planar Parallel Robot Prismatic link Leg Length Control Block Diagram Simulink block diagram for implementing the coordinated Cartesian control for the 3-RPR planar parallel robot Prismatic link length actual sensed (mm) versus Time (sec) for Case Prismatic link length actual sensed (mm) versus Time (sec) for Case Prismatic link length actual sensed (mm) versus Time (sec) for Case3 47

6 1 Chapter 1 : INTRODUCTION The design and control ofa three degrees-of-freedom (dot) planar parallel robot is presented in this thesis. This chapter presents an introduction to the parallel manipulators, a literature review of the related research and the objective of the thesis. 1.1 Introduction of Planar Parallel Robot This section of the chapter deals with the introduction of planar parallel robots, especially the 3-RPR planar parallel robots. Parallel manipulators are robots that consist of separate serial chains that connect the fixed link to the end-effector link. The planar parallel robots refer to the class of manipulators, which are planar and actuated in parallel. The 3-RPR as shown in Figure 1.1, refers to the three serial chains, each having three dof. Each of these serial chains has a revolute joint (referred to as "R"), a prismatic joint (referred to as "P") and again a revolute joint (referred to as "R") connected in the respective order. These three serial chains of the configuration RPR with three dof each connect the fixed link (a triangular base in this thesis) to the end effector link (a moving triangular link in this thesis). Now with this configuration, one joint per chain is actuated (the prismatic joint is actuated in this thesis) and the remaining two joints are passive. Figure 1.2 shows the 3-RPR hardware developed in this thesis.

7 2 Regarding the potential advantages of parallel robots over serial robots it is found that parallel robots have a better stiffness and accuracy as compared to serial robots. Parallel robots are lighter in weight as compared with serial robots. In spite of having lower weights, parallel robots have a greater load bearing capacity than serial robots. Parallel robots have higher velocities and accelerations, and less powerful actuators as compared with serial robots. The only major drawback of the parallel robots over serial robots is that parallel robots have a reduced workspace as compared with serial robots. Figure RPR Diagram

8 3 Figure RPR Hardware 1.2 Literature Review This section presents the literature review for this thesis. Topics such as parallel robots; especially the three-dof planar parallel robots will be discussed in terms of how other authors have proceeded. The prominent subtopics, which will be covered, are the pose kinematics, the velocity kinematics, the workspace analysis and the control implementation of planar parallel robots. Parallel robotic devices were proposed by MacCallion and Pham (1979). Some configurations have been built and controlled (e.g. Sumpter and Soni, 1985). Numerous works analyze kinematics, dynamics, workspace and control of parallel manipulators (see Williams, 1988 and references therein). Hunt (1983) conducted preliminary studies

9 4 of various parallel robot configurations. Cox and Tesar (1981) compared the relative merits of serial and parallel robots. Aradyfio and Qiao (1985) examined the inverse kinematics solutions for three different three dof planar parallel robots. Williams and Reinholtz (1988a and 1988b) studied the dynamics and workspace for a number of parallel manipulators. Shirkhodaie and Soni (1987), Gosselin and Angeles (1988), and Pennock and Kassner (1990) each present a kinematic study of one planar parallel robot. Gosselin et al. (1996) presents the position, workspace, and velocity kinematics of one planar parallel robot. Recently, more general approaches have been presented. Daniali et al. (1995) present an in-depth study of actuation schemes, velocity relationships, and singular conditions for general planar parallel robots. Gosselin (1996) presents general parallel computation algorithms for kinematics and dynamics of planar and spatial parallel robots. Merlet (1996) solved the forward pose kinematics problem for a broad class of planar parallel robots. Williams and Shelley (1997) solved the inverse pose and velocity kinematics problem for all possible planar parallel manipulators. Williams and Joshi (1999) designed and built a three-dof planar parallel robot and proposed its control implementation scheme (the 3-RPR of this thesis). When the author was working on the research, much of the theory regarding the kinematics, workspace analysis, algorithms for the control architecture had been done

10 5 for planar parallel robots by various other authors. Therefore, the focus of the current work is hardware design, implementation and control. 1.3 Objective of the thesis This section of the chapter presents the objective of this thesis. The objective of this work is to implement in hardware a 3-RPR planar parallel robot design and to implement control using pneumatic actuators. The 3-RPR planar parallel robot has been built and has been controlled in real time using a personal computer. When the author started this research, the main emphasis was to make use of the available resources for designing, building and implementing control for the 3-RPR planar parallel robot. Thus when the author began working on the thesis the main emphasis was given on the following considerations: 1. Use of compressed air as an actuation medium. 2. To use the existing sensors, actuating devices, actuators and the power supply to form the electromechanical support system for the robot. 3. To implement the control using a personal computer with the MultiQ boards from the Quanser Consulting (1999); to develop and run the programs using Matlab/Simulink.

11 6 Chapter 2: PARALLEL ROBOT KINEMATICS This chapter presents the kinematics equations and solutions for the 3-RPR planar parallel manipulator. This chapter is divided into three sections: 1. The Pose Kinematics section, which deals with describing the inverse pose kinematics and forward pose kinematics. 2. The Rate Kinematics section, which describes the forward velocity kinematics, inverse velocity kinematics and the Jacobian matrix. 3. The Workspace analysis section, which discusses the computation of workspace for the 3-RPR planar parallel manipulator. 2.1 Pose Kinematics This section presents the symbolic inverse and forward pose kinematics solutions for the 3-RPR planar parallel manipulator. In order to pursue the kinematics analysis, abbreviations and symbols are used to define the geometry of the 3-RPR planar parallel manipulator. Most of the theoretical analysis in this section has been derived and presented by Dr Robert L. Williams II (Williams and Joshi, 1999). The kinematics diagram describing the geometry and configuration of the 3-RPR planar parallel manipulator is given in Fig There are three grounded passive revolute joints located on the base triangle at Ai' (i =1,2,3 ) and there are three moving passive revolute joints that are located on the moving triangular end effector link, at [.i'

12 7 (i = 1,2,3). The active prismatic joint variables indicate the total lengths L i, between the passive revolute joints. The frame {H} depicts the moving frame, positioned at the triangle centroid and the frame {B} depicts the fixed frame at the base. The Cartesian pose variables are described by the triangular end effector link pose array x ={x y f/jy. \ \ \ \ y Al / L1 {B} I.. X '-----1~ X / / / / C1 I e~l~ I.. I \ C2 \ -\ L2 \ Figure RPR Planar Parallel Manipulator Kinematics Diagram The intermediate joint angles B; (THETA i, where i=1,2,3) are passive and, are not required for hardware control, but which may be calculated for computer simulation and/or velocity and dynamics calculations (Williams and Joshi, 1999).

13 8 Using the Grubler mobility equation (Williams and Joshi, 1999), it is found that this device has three degrees-of-freedom, by counting eight rigid links connected by nine one-dofjoints (Williams and Joshi, 1999). As shown in the figure 2.2 each prismatic joint is an actively controlled pneumatic cylinder and each grounded passive revolute joint, located on the base triangle at Ai' where i =1,2,3 is the spigot connected to the base table through a shoulder bolt. Each of the three moving passive revolute joints, which are located on the moving triangular end effector link at ~i' is shoulder bolt connecting each of the prismatic link to each of the vertex of the triangular end effector link. Figure RPR Manipulator Hardware

14 Inverse Pose Kinematics This subsection describes the symbolic inverse pose kinematics solution for the 3-RPR planar parallel manipulator. The inverse pose kinematics problem is stated as follows: Given the desired Cartesian pose variables X = {x Y }T, calculate the required prismatic joint lengths L = {Lt L2 ~}T. This problem is required for robot control, calculating the actuator lengths given the Cartesian pose variables of the triangular end effector link. Generally for parallel manipulators the inverse kinematics problem solution is straightforward, while the forward kinematics problem solution is difficult, which is opposite the case for serial manipulators. For the 3-RPR manipulator, given the pose X = {x Y }T, we first calculate the moving revolute locations {2; and then the inverse pose solution is simply finding the vector lengths L; between C i and Ai' where i =1,2,3. For each RPR leg, the following vector loop closure equation may be written (see Fig 2.3) : i = 1,2,3 (1)

15 10 Ai Figure 2.3 Inverse Pose Kinematics where: 1. B C i is the position vector of Cj,with respect to the base frame {B}. 2. B E H = {:} is the position vector of moving frame {H}, with respect to the base frame te). 3. JR =[c - S ] is the rotation matrix of moving frame {H}, with respect to the base s c frame {B}. c =cos and s =sin. 4. H C i is the position vector of C,,with respect to the base frame fh}. 5. B Ai is the position vector of Ai,with respect to the base frame {B}. 6. e'" =coso + i sin 0

16 11 Knowing the manipulator geometry and calculating B~i =B E H +//R H Ci from the given X = {x y ~}T, the solution is found using the Euclidean norm in (2): L 1 =IIBC._B -1-1A II (2) where i = 1,2,3. The intermediate passive joint angles B; are given by: (3) where i = 1,2,3. The Inverse Pose kinematics problem is straightforward and has a unique solution. This solution is found for each of the three RPR legs independently Forward Pose Kinematics This subsection describes the forward pose kinematics solution for the 3-RPR planar parallel manipulator. The forward pose kinematics problem is stated as follows: Given the current prismatic joint lengths L = {Lt L2 ~}T, calculate the Cartesian pose variables X ={x y ~}T. This solution is required for off-line simulation and sensor-based control. This problem requires the solution of coupled nonlinear equations. The same vector loop closure equations (1) apply, and are rewritten in (4): i = 1,2,3 (4)

17 12 L.CO.} {X} [Ct/J - St/J]{Hc.} {BAu} { 4s~ = y + s c He: - B~y i =1,2,3 (5) where: 1. HCix is the horizontal component of the position vector of HC;,with respect to the moving frame {H}. HC iy is the vertical component of the position vector of HC;,with respect to the moving frame {H}. 2. BAu is the is the horizontal component of the position vector of A;, with respect to the base frame {B}. B ~Y is the vertical component of the position vector of A;,with respect to the base frame {B}. Equation (5) includes all unknown Cartesian pose variables, essentially x = {x Y }T and one passive unknown variable e;, for each i =1,2,3. In order to solve this problem, numerical Newton-Raphson method (Williams and Joshi, 1999) is used. Since 8; is a passive variable not required for control, we can square and add the x and y component equations in (5) to eliminate 8; as shown in equation (6): where ;=1,2,3. (6)

18 13 where C ix represents H C;x ' C;y represents H C;y,Au represents BAu and A y represents BA y for notational convenience. Also c = cos and s = sin. Also: B 3 =C. + C +. 1: +. 1: - T~ IX ly L ~X.L 'iy "-i Newton-Raphson (NR) method is used to solve the nonlinear equation (6) numerically. NR method is discussed briefly as follows: Given a function Fi(X) (i =1,2,3,...n) such that X = {XI,X2,X3,...Xn.}T then ( i = 1,2,3,...n.) (7) where: 1. Fi(X + zx) =0 and OX is the correction vector. ( i =1,2,3,...n.) J is Newton-Raphson Jacobian matrix Since the value of 2 OXj is negligibly small, equation (7) can be written as: ( i = 1,2,3,...n.) (8)

19 14 Rearranging the terms in equation (9) results in the following: J (X)&. =-Fi(X) (i =1,2,3,...n.) where" k" is the present step. Solving the equation (9) for &.k results in the following: (10) where" k+]" is the next immediate step. Iterating equation (10) until the following condition is achieved: where E refers to smallest tolerance. For the 3-RPR planar parallel robot: 1. X={X,y,<p}T. 2. Fi = -L~ + x 2 + y2 + 2x(C ij rctjj - CjystjJ - Au) + 2y(C ixstjj + C;yctjJ - Aiy) + B1ctjJ + B2stjJ + B 3 =0 3. (i = 1,2,3.) J(X)=[ 8F~~)]= 8Fl 8Fl 8Fl & & 5 8F2 8F2 8F2 - - & ~ 5 8F3 8F3 8F3 & ~ 5

20 15 where: PI 1. & = 2x + 2C;xctjJ - 2C;ystjJ - 2~X PI 2. ~ =2y+2CixCtjJ-2CiystjJ-2~y PI 3. - =2x(-CixS - C;yC ) + 2y(C;Xc - C;yS )+ 2(C;X~X + CiY~Y)S + 2(C;y~X - C;X~Y)C 5 The NR method has the following limitations: 1. For implementing the NR method for the forward pose kinematics problem, a good initial state guess i.e. X, ={r«, yo,<l>o}t is required. 2. The NR method can only find one solution out of multiple solutions for the forward pose kinematics problem. The forementioned NR method limitations are not a problem in practice since the current known pose is an excellent initial guess for the next control cycle pose and generally the one solution calculated (from theory, there are six multiple solutions, Williams and Joshi (1999)) is the proper one. 2.2 Rate Kinematics This section presents the symbolic rate kinematics solution for the 3-RPR planar parallel manipulator. This section will cover the Jacobian matrix, inverse velocity kinematics and forward velocity kinematics. The theoretical analysis in this section has been derived and presented by Williams and Joshi (1999).

21 16 Inverse velocity kinematics is used for resolved-rate control and forward velocity kinematics is used for simulation (Williams and Joshi, 1999). The vector loop closure equations are rewritten in order to derive the analyses for velocity kinematics (see Fig 2.4): [3 Ai Figure 2.4 Velocity Kinematics i = 1,2,3 (11) B P =B A. + L-ej(}i + I.ej((}i+PJ where i = 1,2,3 and j = H -H -1 1 HI where in addition to the previously-defined terms: 1. lhi is the length from C j to the origin of {H} (directed opposite to H C i ).

22 17 2. f3; is the variable angle, related to, from the end of the L, direction to the current direction of lhi (see Fig 2.5): Al L2 Figure 2.5 Angle Nomenclatures The inverse velocity problem is stated: Given the manipulator configuration and the desired Cartesian rates X = {x y r)r'calculate the required prismatic joint rates L = {~ ~ L, y. The velocity kinematics method for all planar parallel manipulators is presented in Williams and Shelley (1997); the method is briefly summarized here and the results are given for the 3-RPR. Taking a time derivative of (11) yields a velocity equations (12a, 12b, 12c, 12d) for RPR leg i only that can be arranged as a 3x3 Jacobian matrix [J ] as shown in equation (13). The Jacobian matrix maps the leg i joint rates

23 18 ~ =- Lsf). 8. +i.,f). -IH.s(f).+ R.)(8. + R.) I I I I I I I PI I PI (12a) Y =L.ef).8.+i.sf). +IH e(f) )(8.+ R.) I I I I I I I PI I PI (12b) (12c) jfa (). () e =cos.+ ] SIn. I I je j fa = j COS 0i - sin 0i wherej=h (12d) x = [J] [Pi] where i=i,2,3. (13) The above equation (14) can also be represented as: x (14) where i=1,2,3 and [1] is the manipulator Jacobian matrix, described by the following expression, derived from equation (12). - LiS (); -IHiS( (); + fi) C(); -IHiS((); + fi) [J]= t.c«+ lhic(f}; + fi) Sf}; IHiC(fJz + fi) (15) Since the active joint rates i; are of primary importance, equation (15) is inverted. Then, the L; row only is extracted; this is repeated for all legs i == 1,2,3. The overall Jacobian relationship for the 3-RPR planar parallel manipulator results in equation (16):

24 19 COl SOl IHlSfJl C02 S02 IH2sfJ2 C03 S03 IH3sfJ3 x (16) This equation (16) can also be expressed as: L=MX (17) where M is the 3x3 inverse of the manipulator Jacobian matrix. The above expression (17) gives the solution for the inverse velocity kinematics problem. The forward velocity kinematics problem solution can be obtained by inverting the matrix expression (17), which results in the following expression: (18) The 3-RPR planar parallel manipulator is subjected to singularities for forward velocity kinematics problem when one row or column in the manipulator jacobian matrix (see equation (16» is a linear combination of other columns or rows or when: det[m] =O. The 3-RPR planar parallel manipulator is not subjected to singularities for inverse velocity kinematics problem.

25 Workspace Analysis This section describes the workspace analysis done for the 3-RPR planar parallel manipulator.as mentioned in the introduction chapter, the major disadvantage of planar parallel manipulators is that they have a limited workspace as compared with serial manipulators. This limited workspace disadvantage of planar parallel manipulators is due to the closed link structure of the planar parallel manipulators. When the author started his work on the research, there were many theories being proposed by other authors, for computation of the workspace of planar parallel manipulators (e.g. Williams 1988). The reachable workspace of a manipulator is defined (Williams and Reinholtz 1988) as the volume in space whose boundary is the limit of the manipulator hand reach, (the centroid of the triangular end effector link in this thesis) regardless of the hand orientation. If a point is reachable with only one specific hand orientation, it is within the reachable workspace. For planar manipulators, the above definitions apply when "volume" is replaced by "area" (see figure 2.6). The dexterous workspace is generally a subset of the reachable workspace, as it is the maximum area attainable by the manipulator hand in all possible hand orientations. The author has not dealt with dexterous workspace of 3-RPR planar parallel robot, since it doesn't exist (it is a null set). The shaded region in figure 2.6 shows an example of the reachable workspace. Lmax refers to the maximum prismatic link length, Lmin refers to the minimum prismatic link length and Lh refers to the altitude of the triangular end effector link..

26 21 Figure 2.6 Example of the 3-RPR Reachable Workspace The computation of the workspace for the 3-RPR planar parallel robot is done as follows: 1. The position ofgrounded passive revolute joints, located on the base triangle at AI, A2 and A3 is given. 2. With AI, A2 and A3 as centers, draw circle of radius Lmin -Lh is drawn for each of the centers, followed by drawing another concentric circle of radius Lmax-iLh for each of the centers. 3. The area of intersection of the circles is the reachable workspace.

27 22 The author has made use of the geometric method approach, for computation of the workspace of a 3-RPR planar parallel robot with three degree of freedom, which was proposed by Dr. R.L.Williams II. This method has been useful in determining the 3-RPR workspace and also, to design the manipulator parameters to maximize the workspace. Figure 2.7 shows a sample reachable workspace for the 3-RPR planar Parallel robot. x Figure 2.7 Workspace analysis by geometrical method. Since the author was making use of the existing actuators, the parameters Lmin -Lh and Lmax-iLh were given. The positions of the grounded passive revolute joints, located on the base triangle, at AI, A2 and A3 are decided so that there is no interference with the trajectory followed by the centroid of the triangular end effector link. The triangle

28 23 formed by the positions of the grounded passive revolute joints, located on the base, (AI, A2 and A3) can be either an equilateral triangle or an isosceles triangle or can be any general triangle, depending on the positions of AI, A2 and A3. For the experimentation the author has located the passive revolute joints (AI, A2 and A3) on the base such that an equilateral triangle is formed. This effort is not complete, but the hardware has been built to allow for different ground revolute locations.d.i to evaluate workspace results in the future, constrained by the end -effector triangular link (which we can also change) and the prismatic joint limits. The workspace area of the 3-RPR planar parallel manipulator is dependent on the following parameters: 1. The distance between the three grounded passive revolute joints located on the base triangle at AI, A2 and A3. 2. The parameters Lmax, Lmin and Lh.

29 24 Chapter 3: PARALLEL ROBOT HARDWARE IMPLEMENTATION This chapter deals with the hardware implementation of the 3-RPR planar parallel robot. The hardware implementation is presented in two sections: the design and construction of the 3-RPR planar parallel robot and the control implementation of the 3-RPR planar parallel robot. The 3-RPR has been built at Ohio University and the real-time control has been implemented using a personal computer (PC). 3.1 Design And Construction Of3-RPR Planar Parallel Robot This section discusses the design and construction of the 3-RPR planar parallel robot hardware. The most important design specification/consideration was to make use of existing actuators, sensors, control elements, and Input/Output (I/O) boards (since the hardware budget for this project was small). Also, since most robotic projects at Ohio University make use of DC servomotors, it was desired to make use pneumatic power for a change of pace (Williams and Joshi, 1999). This dictated the use of existing pneumatic air cylinders (single-acting, spring-loaded to return to minimum joint length), linear variable displacement transducers (LVDTs) for length sensing, solenoid valves to regulate air flow to the cylinders, and a PC based control system using Quanser MultiQ I/O boards. The components (already existing in the robotics lab) that were used for design and construction of the 3-RPR planar parallel robot are as follows:

30 25 1. Pneumatic air cylinders: Three identical pneumatic air cylinders are used for actuating the robot. These three air cylinders form the three prismatic joints for the robot. The air cylinders are single acting and spring-loaded to return to the minimum joint length. These air cylinders are manufactured by Bimba, Inc. The specifications of each cylinder is as follows: - * Bore size=3/4 ". * Stroke length=4". * Maximum operating pressure=250 pounds per square inch (psi). * Spring force: In relaxed position of the piston=3 pounds. In extended position of the piston=6 pounds. * Inlet port diameterel/s NPT, female. 2. Linear variable displacement transducer (LVDT): Three identical LVDTs are used for sensing the displacement of the air cylinder pistons. These LVDTs are used as sensors to give feedback to the data acquisition system (Quanser MultiQ boards). These LVDTs give an output voltage proportional to the actuator displacement. The rated output voltage range (in ideal conditions) for each LVDT is 5volts to Ovolts.Thus when the author started building the hardware, he had to calibrate the three LVDTs for their actual rated outputs. The input supply voltage for each LVDT is ±15 volts. The LVDT calibration procedure and the graphs are given in Appendix A. The LVDTs are manufactured by Faster Instruments.

31 26 3. Solenoid Valves: Three identical solenoid valves are used. Each solenoid valve gives a pressure output (pressurized air) proportional to the input 0-10 volt ramping command signal. The regulated pressure output from each solenoid valve is given to each air cylinder. The input to the solenoid valve is a 0-10 volt ramping command signal commanded by the PC. The input supply voltage to each solenoid valve can be between +13volts to +24volts (during experimentation, the input supply voltage to each solenoid valve is maintained +15volts). Each solenoid valve receives a regulated pressurized air supply from an oil-less compressor, through a filter of 20 microns. The three solenoid valves are manufactured by Proportional Air Control, Inc. 4. MultiQ 110 board: The MultiQ board is a data acquisition and control board. The MultiQ board is connected to the PC that facilitates the reading of input signals and generation of output signals using Wincon software (version3.0.2c), developed by Quanser Consulting (Wincon software is compatible with Matlab5.2 and Simulink 2.2 and works in the Windows98/NT environment). The MultiQ board has several features that include 8 analog inputs, 8 analog outputs, and 8 digital inputs, 8 digital outputs, 3 real-time clocks, and up to 8 quadrature encoder inputs. (Quanser Consulting, 1999). The author has made use of only two features of the MultiQ board, the analog inputs and analog outputs. The analog output can only supply -5 volts to +5 volts and the analog input can only receive -5 volts to +5volts. The analog outputs command the input ramping voltage to the solenoid valves and the analog inputs receive the voltage feedback from the LVDTs. MultiQ board is manufactured by Quanser Consulting. 5. Matlab, Simulink and Wincon software: Matlab is a mathematical tool developed by

32 27 MathWorks, Inc. Simulink is an add-on graphical interface for Matlab where a program is represented by a block diagram. When the simulink block diagram is run, the code (in C++) is automatically called to generate outputs. Simulink provides various standard block libraries, which contain all the blocks that can be used for creating block diagrams. Wincon is software developed by Quanser Consulting, and is compatible with Matlab and Simulink. Wincon contains an additional library of functions (in the form of blocks) that facilitates Simulink to send the robot analog commands and receive analog feedback via the MultiQ boards. Hardware components that were procured included pneumatic plumbing elements, Delrin plastic for the hand triangle, link extensions, and ground fixtures, plus bolts for the passive revolute joints. The summary of the 3-RPR planar parallel robot configuration is as shown in figure 3.1. The blocks represented by Matlab, Simulink, Wincon and internal MultiQ Board reside in the control PC. This control PC in turn connects the External MultiQ Board, which in turn connects to the electromechanical hardware (3-RPR manipulator, connected to the solenoid valves actuating devices and LVDT sensors.

33 28 INTERNAL.-. MultiQ BOARD SOLENOID VALVES 3 RPR PLANAR PARALLEL MANIPULATOR Figure 3.1 Summary of the 3-RPR Hardware Configuration. The three RPR legs are identical, with three independent pneumatic (air) cylinders and each LVDT coupled in parallel across each air cylinder. An oil-less air compressor is the pneumatic power source, providing 120 psi, regulated to a constant 60 psi that is delivered through a three-way hose-coupling manifold to each solenoid valve to power the cylinders. The 3-RPR robot hardware is shown in figure 3.2. In figure 3.2 the three pneumatic cylinders are on the bottom while the three LVDTs are mounted parallel to the cylinders on top.

34 29 Figure RPR Planar Parallel Robot Hardware The external MultiQ boards (only the right board is used), the three solenoid valves, and the electric power supply for the solenoid valves and LVDTs are shown (left-to-right) in figure 3.3. Figure 3.4 shows the PC screen during experimental length control of one pneumatic cylinder actuator. The background window shows the Simulink model controlling the 3-RPR hardware. The lower small window on the right is the Wincon window. The small window directly above that is the real-time numerical display of the LVDTs measurement of this pneumatic cylinder length (currently -6.21mm). The graph window on the left displays in real-time the time history of this controlled length (of the air cylinder piston).

35 30 Figure 3.3 External MultiQ Boards, Solenoid Valves, and Power Supply Figure 3.4 Simulink/Wincon Interface to the 3-RPR Planar Parallel Robot.

36 31 Figure 3.5 describes the 3-RPR planar parallel robot control hardware system. This system describes the mechatronic design that couples with the control architecture for the 3-RPR planarparallel robot hardware (to be discussed in Section3.2). Figure RPR Planar Parallel Robot Control Hardware. An electric power supply (see Figs 3.3 and 3.6) supplies ±15 volts to the LVDTs and +15 volts to the solenoid valves. An oil-less compressor (see Fig 3.7) supplies regulated pressurized air (60 psi) to the three solenoid valves, through a three-way hose-coupling manifold. Three pneumatic cylinder/lvdt units are connected in parallel.

37 32 Figure 3.6 Power Supply For 3-RPR Planar Parallel Robot. Figure 3.7 Oil-Less Air Compressor For 3-RPR Planar Parallel Robot.

38 33 Three analog inputs and three analog outputs on the external MultiQ board (see Fig 3.8) are used, one for each pneumatic cylinder/lvdt combination. (A pair of analog input and analog output is represented by one channel on the external MultiQ board). Each cylinder receives air pressure by commanding voltage (from an external MultiQ board analog output) to each solenoid valve (see Fig 3.9) Figure 3.8 External MultiQ Board Figure 3.9 Solenoid Valve The resulting displacement of the cylinder piston is detected by the LVDT sensor that is sent to the PC (via an external MultiQ board analog input). The external MultiQ board interfaces to an internal MultiQ board. The PC reads the feedback LVDT analog voltage value and commands appropriate voltage values to each solenoid valve via the Simulink-based controller. The Simulink block diagram for the system is discussed in Section 3.2. which presents the control details for the 3-RPR planar parallel robot.

39 Control Of 3-RPR Planar Parallel Robot This section deals with the real-time control architecture for the 3-RPR planar parallel robot. The 3-RPR planar parallel robot can be controlled in joint mode and in Cartesian mode (inverse pose control or resolved-rate control using inverse velocity). The joint control mode deals with control of the prismatic joint lengths by controlling the piston displacements. This joint control can be either open-loop joint control or closed- loop joint control with LVDT feedback. The Cartesian control modes (inverse pose control or resolved-rate control) requires closed-loop control of all three pneumatic cylinders (actuators) simultaneously. The author has not implemented the Cartesian control modes on the 3-RPR planar parallel robot; this should be accomplished in future work. The proposed control routine for the Cartesian control modes (see Figs 3.10 and 3.11) is discussed briefly. Considering inverse pose control (see Fig 3.10) X c= {x y Y is the commanded Cartesian pose, which is the input to the "Inverse Pose Kinematics" block. The inverse pose kinematics solution calculates L ci(i=1,2,3); the commanded leg length (displacements of the cylinder piston) for the three prismatic actuators. Lc i (i=1,2,3) is the input to the "Length Control" block (to be discussed later in the section). The output from the "Length Control" block is L Ai(i=1,2,3); the actual length (displacement) of the cylinder piston. L Ai(i=1,2,3) is the input to the "Forward Pose Kinematics" block. The forward pose kinematics solution calculates X Ai(i=1,2,3); the actual Cartesian pose from L Ai(i=1,2,3). This last block is required for simulation

40 35 only i.e. the real-world hardware provides the actual motion simulated by forward pose kinematics in Figure Considering resolved rate control (see Fig 3.10), in addition to the previously described... terms, Xc ={x, y,m}t commanded Cartesian velocity that is the input to the "Inverse Velocity Kinematics" block. The inverse velocity kinematics solution calculates LCi ; the commanded actuator length rates. IntegratingL, yields Lei (i=1,2,3); the commanded leg lengths (displacements of the cylinder piston) for the prismatic actuators. The remaining control routine for the resolved rate control implements the inverse pose control routine (described earlier). LC1 Length Control LA1 Xc Inverse Pose Lc LC2 LA2 LA Length Control Forward Pose X A KinematiCS KinematiCS LC3 Length Control LA3 INVERSE POSE CONTROL Figure 3.10 Cartesian Pose Control Mode for 3-RPR Planar Parallel Robot.

41 36 Xc Inverse Velocity Kirerrati RESCLVIDRA1Ea:NIR<L Figure 3.11 Cartesian Rate Control Mode for 3-RPR Planar Parallel Robot. For control of the 3-RPR planar parallel robot the required kinematics solutions (discussed in chapter 2) for control are implemented in Matlab's Simulink graphical interface. The output of the Simulink model interacts (commands analog output voltage signals to the solenoid valves and receives analog input voltage signals from the LVDTs) with the real-world hardware via the Quanser Wincon software and internal and external MultiQ boards. In open-loop case, (Williams and Joshi, 1999) the Simulink model commands desired voltage values to the solenoid values that operate the pneumatic cylinders without LVDT feedback. In closed-loop joint control mode, the Simulink model commands voltage values to the solenoid values based on values calculated by the on-line controller for achieving the three desired prismatic link lengths. Solenoid voltage commands go out and the LVDT voltage readings come into the PC via the external MQ3 MultiQ board and internal MultiQ board, which interface with Simulink via Wincon software.

42 37 Figure 3.12 (Williams and Joshi, 1999) shows the closed-loop feedback control block diagram for achieving the desired commanded pneumatic cylinder lengths. The block diagram is for one air cylinder; all three air cylinders use the same block diagram, independently but simultaneously. In figure 3.12, Lc is the commanded leg length (displacement of the cylinder piston) for the given prismatic actuator. L E is the length error. V is the commanded solenoid voltage to be applied, calculated by the proportional-integral-derivative (PID) controller. LA is the actual length (displacement) of the cylinder piston, resulting from this commanded voltage (and the ensuing dynamic response of the entire system, i.e. three actuators operating simultaneously in the coupled robot). Ls is the sensed value of this actual length, as read by the LVDT feedback. The system dynamics have not been modeled; the PID gains are tuned experimentally. LENGTH COMMANDED L c LENGTH ERROR VOLTAGE L E V f ~ CONTROLLER SYSTEM DYNAMICS LENGTH SENSED L s L.V.D.T. LENGTH ACTUAL LA Figure 3.12 Prismatic Link Leg Length Control Block Diagram

43 38 This control architecture describes the coordinated Cartesian control of the 3 RPR planar parallel robot via linearized independent (but simultaneous) prismatic joint (link length) control. Implementing the coordinated Cartesian control for the 3-RPR planar parallel robot is facilitated using Simulink. The Simulink block diagram is built, based on the control architecture described earlier. Figure 3.13 shows the Simulink block diagram for implementing the coordinated Cartesian control for the 3-RPR planar parallel robot via linearized independent prismatic joint control. The block diagram is for one air cylinder; all three air cylinders use the same block diagram, independently but simultaneously. In figure 3.13, the "Analog Input I" block is the device driver for the Analog input section of the Quanser Consulting external MultiQ I/O boards. The output from the "Analog Input I" block is the feedback voltage from LVDT#l.This feedback voltage value is passed as an input to the "Calibration Block for LVDT#1", which is a general expression block; it uses the feedback voltage value as the input variable and outputs the displacement of the air cylinder piston. The user must enter the commanded leg length. The author has derived the displacement vs. voltage functions for each LVDT by calibrating the LVDTs (see Appendix A). The output from "Calibration Block for LVDT#1" block is passed to the "Sum" block. The second input to the "Sum" block is from "Minimum Link Length" block. The "Minimum Link Length" block includes the minimum prismatic link length (which is equal to 245 mm). The "Sum" block adds both the inputs and outputs LsI; the actual prismatic link length sensed by the

44 39 LVDT. LsI is monitored by the user through a scope block (indicated by "LsI" block). [JJ~I Length Commanded Lc1 Length Actual Sensed Ls1 Ls1 Length Error Le1 Sum for Unit #1 Sum Length Error Le ~~ Quanser Consulting o MQ3DAC +~---I +~--.. Minimum Link Length PID Controller1 f(u) Calibration Block for L.V.D.T.#1 Analog Output 1 Quanser Consulting MQ3 ADC Analog Input 1 Figure 3.13 Simulink block diagram for implementing the coordinated Cartesian control for the 3-RPR planar parallel robot.

45 40 The scope block is the output, which is a plot versus time. The L s is passed as an input to the "Sum for unit#l" block. The second input to the "Sum for unit#l" block is from the "Length Commanded Let" block. The "Length Commanded Let" block allows the user to enter the commanded prismatic leg length time history. The user must enter a value greater than the minimum prismatic link length (245 mm). The "Sum for unit#l" block calculates the difference between the two inputs and outputs "Let"; prismatic link length error. Let is monitored using a display block represented by "Length Error" block. Let is passed to the PID controller block (represented by "PID Controller" block. The PID controller block allows the user to tune the PID gains (which are determined by the author by experimentation; to be discussed in chapter 4). PID tuning is used to adjust the controller to achieve the desired system performance. The PID controller reduces the prismatic link length error Let. The output from the "PID Controller 1"block is passed to the "Analog Output 1" block. The "Analog Output 1" block the device driver for the Analog output section of the Quanser Consulting external MultiQ I/O boards. This sends the appropriate voltage value, as calculated by the PID controller, to the solenoid valve to actuate the cylinder. Figure 3.13 appears to be open, but the real-world hardware closes the loop between the "Analog Output 1" block and "Analog Input 1" block.

46 41 Chapter 4 : EXPERIMENTATION AND RESULTS This chapter presents the results of operating the 3-RPR planar parallel manipulator under coordinated Cartesian control via linearized independent (but simultaneous) prismatic joint (link length) control. The Simulink block diagram that is discussed in the previous chapter (see Fig 3.13) is used for implementing the coordinated Cartesian control. During experimentation the user has to enter the desired prismatic leg length in the form of step input in the "Length Commanded Lei" block (where i=1,2,3) of the Simulink block diagram (see Fig3.13 in the Chapter 3). The user has to enter a value greater than the minimum prismatic link length (which is equal to 245 mm). The actual prismatic link length sensed by the LVDT is Lsi (i=1,2,3), monitored by the user through a scope block (indicated by "Lsi" block in the Simulink block diagram). The length output is plotted versus time. The PID gains were tuned experimentally for the control of the 3-RPR planar parallel robot. A discussion regarding the plots is presented at the end of this chapter. For all the scopes: 1. "T1" is the time to reach the commanded length from the time the command was made. 2. "T2" is the time to reach the commanded length from the time when the piston started moving.

47 42 Tl, and T2 have been presented in each graph. The PID characteristics for: 1. Prismatic joint (cylinder) #1 and prismatic joint (cylinder) #2 are as follows: P=2, 1=0.25 and D = Prismatic joint (cylinder) #1 are as follows: P=2, 1=0.25 and D =0.3. CASE1: The first case for experimentation deals with closed-loop control of individual prismatic link lengths with relatively small commanded lengths. The first run deals with the user entering Lcl=275 mm, Lc2 =0 mm and Lc3=0 mm. The second run deals with the user entering Lcl=O mm, Lc2=275 mm and Lc3=0 mm.the third run deals with the user entering Lcl=O mm, Lc2=0 mm and Lc3=275 mm. In all three runs, the active prismatic length starts at the minimum length, 245 mm. Figure 4.1 displays the following scopes: 1. LsI vs. time from the first run. 2. Ls2 vs. time from the second run. 3. Ls3 vs. time from the third run.

48 43 -..: : : : ~T~\ I I I I I I I I I T , , I I I I I I I I I I I I I I I I I I I I I I I I I I I TI~1.75secs T2~O.20secs yaxis=l s1 I I I r T , , I I I I I I I I I I I I I I I I f I I I, t I I I I t I I I I I I I ~ 't t t I I I I I I I I I I I I t I I, I.. I I I I, 1 I t I I I I TI~1.75secs T2~O.20secs yaxis=l s2 I I I I I I I I I T- -,- -, r I I I I I I I I I, I, I I I I t I I I t I It. I I I I, I I t to' ~ e It' I I I t I I I I I I I I I I t I I I I I I I t I I f.. I I I I I I Tl~1.40secs T2~O.30secs yaxis=l s3 Figure 4.1 Prismatic link length actual sensed (mm) versus Time (sec) for Case 1

49 44 ease2: The second case for experimentation deals with closed-loop control of individual prismatic link lengths with relatively higher commanded lengths. The first run deals with the user entering Lcl=290 mm, Lc2 =0 mm and Lc3=0 mm. The second run deals with the user entering Lcl=O mm, Lc2=290 mm and Lc3=0 mm.the third run deals with the user entering Lcl=O mm, Lc2=0 mm and Lc3=290mm. Figure 4.2 displays the following scopes: 1. LsI vs. time from the first run. 2. L s2 vs. time from the second run. 3. L s3 vs. time from the third run.

50 45 T1;::::1.60secs T2;::::0.02secs y axiselj, T1;::::2.50secs T2;::::0.1Osecs yaxis=l s2 T1;::::1.25secs T2;::::0.1Osecs yaxis=l s3 Figure 4.2 Prismatic link length actual sensed (mm) versus Time (sec) for ease2

51 46 CASE3: The third case for experimentation deals with closed loop control of all the prismatic link lengths simultaneously. This deals with the user entering L ci=275 mm, L c2 =275 mm and L c3=275 mm, starting from the minimum lengths 245 mm. Figure 4.3 displays the following scopes: 1. LsI vs. time. 2. L s2 vs. time. 3. L s3 vs. time.

52 47 TI~1.40secs T2~O.40secs y axiseli, TI~1.20secs T2~O.20secs yaxis=l s2 TI~1.30secs T 2~O.20secs yaxis=l s3 Figure 4.3 Prismatic link length actual sensed (mm) versus Time (sec) for Case3

53 48 From the plots shown in Figs 4.1 through 4.3, the response may be discussed: 1. The settling times for each air cylinder is different. These varying settling times are due to the different frictional forces which each piston has to overcome and also different inertial forces required to overcome the return spring, in order to reach the desired position. 2. The PID gains for air cylinder 1 and air cylinder 2 are identical (P=2, 1=0.25 and D =0.8). The controller characteristics indicate that the D term is essential to maintain stability, (the D term acts as a damper for the system). Since the spring of air cylinder 3 exhibited more stiffness, the D term for air cylinder3 was tuned to 0.3(the remaining controller gains P and 1 were identical to that for air cylinder 1 and air cylinder 2). The P term is essential to reduce the rise time. The 1 term is supposed to reduce the steady state error, but was set to a lower value of 0.25; higher 1 values caused erratic, marginally stable results. 3. With the existing PID gains, the author found out that the system stayed stable, for not more than 5seconds. The author then tried tuning the PID gains for different values, but the system was found to work unstably. 4. For case 1 and case 2 where the individually commanded lengths were relatively lower (L ci=275mm, i=1,2,3), it was found that the system worked well and produced steady responses. For case 2 where the commanded lengths were relatively higher, it was found that the system produced unsteady responses.

54 49 This can be attributed to the fact that the responses of solenoid valves to higher piston displacements (Lei >290mm, i=1,2,3) and very lower piston displacements (L ei<275mm, i=1,2,3) is not proper due to the hardware configuration. The author has performed experiments where he commanded prismatic link lengths more than 290mm and less than 275mm and found out that the system responded poorly and produced unsteady and unstable responses. 5. It was noted that the system appeared to perform better if each of the three air cylinders were actuated at the same time (i.e. with no time delay on any of the three). If the cylinders were actuated at time delay, there were no piston displacements. This is due to the inherent hardware restriction posed by the solenoid valves not to respond to time delay in the input.

55 50 Chapter 5 : CONCLUSION This chapter presents some concluding statements and suggests future work relating to the 3-RPR planar parallel robot. The future work presents potential improvements and recommendations for future 3-RPR planar parallel robot developments. 5.1 Concluding Statements The objective of this work was to implement in hardware a 3-RPR planar parallel robot design and to implement control using pneumatic actuators. The 3-RPR planar parallel robot has been built and has been controlled in real time using a personal computer. The kinematics for the 3-RPR planar parallel robot has been presented in chapter 2. The workspace computation and analysis has also been presented in chapter 2. A geometric method was used (Williams, 1988) to determine the 3-RPR workspace and design the manipulator parameters to maximize the workspace. This effort is not complete, but the hardware has been built to allow for different ground revolute locations to evaluate workspace results in the future, constrained by the hand triangle link (which can be changed) and the prismatic joint limits. The 3-RPR planar parallel robot hardware implementation and real-time control architecture has been presented in chapter 3. The coordinated Cartesian control of the 3-RPR planar parallel robot via linearized

56 51 independent prismatic link length control has been discussed in detail in chapter 3. The Simulink block diagram is built, based on the control architecture (as described in chapter 3). Finally in chapter 3 the control routine for the Cartesian control modes (inverse pose control or resolved rate control) has been proposed. Chapter 4 deals with the results of operating the 3-RPR planar parallel manipulator under coordinated Cartesian control via linearized independent prismatic link length control. Real-time results were then presented for operation of the 3-RPR planar parallel manipulator. Three different cases were presented to demonstrate the control and output from the robot from coordinated Cartesian control via linearized independent prismatic link length control. Appendix A presents the LVDT calibration procedure along with the calibration graphs. Appendix B presents the operation procedure for 3-RPR planar parallel robot. 5.2 Future Work The author's recommendations for future 3-RPR planar parallel robot work are as follows: 1. Hardware improvements may be made to improve accuracy. The author has had some problems with the solenoid valves and the pneumatic cylinders, if each of the three air cylinders were actuated at a time delay (i.e. If the cylinders were actuated at time delay, there were no piston displacements). 2. Workspace for the 3-RPR planar parallel robot can be improved by using an analytical computational approach and then can be verified using the geometric approach.

57 52 3. The control routine for the Cartesian control modes (inverse pose control or resolved rate control) that has been proposed in chapter 3 can be implemented in real time. This can be done using the SimulinklMatlab software interfacing the hardware through I/O MultiQ boards.

58 53 REFERENCES R. L. Williams IT and Atul R. Joshi, 1999, "Planar Parallel3-RPR Manipulator". Applied Mechanisms and Robotics Conference, Cincinnati, OH. D.D. Aradyfio and D. Qiao, 1985, "Kinematic Simulation of Novel Robotic Mechanisms having Closed Chains", ASME Paper 85-DET-81. D.J. Cox and D. Tesar, 1981, "The Dynamic Modeling and Command Signal Formulation for Parallel Multi-Parameter Robotic Devices", DOE Report. H.R.M. Daniali, P.J.Z. Zsomber-Murray, and J. Angeles, 1995, "Singularity Analysis of Planar Parallel Manipulators", Mechanism and Machine Theory, Vol. 30, No.5, pp C.M. Gosselin, 1996, "Parallel Computation Algorithms for the Kinematics and Dynamics of Planar and Spatial Parallel Manipulators", Journal ofdynamic Systems, Measurement, and Control, 118 (1): C.M. Gosselin, S. Lemieux, and J.P. Merlet, 1996, "A New Architecture of Planar Three-Degree-of-Freedom Parallel Manipulator", IEEE International Conference on Robotics and Automation, Minneapolis, MN, 4:

59 54 C.M. Gosselin and J. Angeles, 1988, "The Optimum Kinematic Design of a Planar Three-Degree-of-Freedom Parallel Manipulator", ASME Journal of Mechanisms, Trans., and Automation in Design, 110(1): K.H. Hunt, 1983, "Structural Kinematics ofin-parallel-actuated Robot Arms", Journal of Mech., Trans., and Automation in Design, 105(4). H. MacCallion and D.T. Pham, 1979, "The Analysis of a Six-Degree-of-Freedom Workstation for Mechanized Assembly", 5 th World Congress on TMM, Montreal. J.P. Merlet, 1996, "Direct Kinematics of Planar Parallel Manipulators", IEEE International Conference on Robotics and Automation, 4: G.R. Pennock and D.J. Kassner, 1990, "Kinematic Analysis of a Planar Eight-Bar Linkage: Application to a Platform-type Robot", ASME Mechanisms Conference, DE 25: A.H. Shirkhodaie and A.H. Soni, 1987, "Forward and Inverse Synthesis for a Robot with Three Degrees of Freedom", Summer Computer Simulation Conference, Montreal,

60 55 B. Sumpter and A.H. Soni, 1985, "Simulation Algorithm of Oklahoma Crawdad Robot", 9 th Applied Mechanisms Conference, Kansas City, VI.I-VI.3. R.L. Williams II, 1988, "Planar Robotic Mechanisms: Analysis and Configuration Comparison", Ph.D. Dissertation, VPI&SU, Blacksburg, VA. R.L. Williams II and C.F. Reinholtz, 1988a, "Forward Dynamic Analysis and Power Requirement Comparison of Parallel Robotic Mechanisms", 20 th Biennial ASME Mechanisms Conference, Kissimmee FL, DE Vol. 15-3, pp R.L. Williams II and C.F. Reinholtz, 1988b, "Closed-Form Workspace Determination and Optimization for Parallel Robotic Mechanisms", 20 th Biennial ASME Mechanisms Conference, Kissimmee FL, DE 15-3: R.L. Williams II and B.H. Shelley B.H., 1997, "Inverse Kinematics for Planar Parallel Manipulators", 23,d ASME Design Automation Conference, DETC97/DAC-3851, Sacramento, CA, September

61 56 APPENDIX A LVDT CALIBRATION PROCEDURE The LVDT calibration procedure for the 3-RPR planar parallel robot is discussed in this appendix. Three identical LVDTs are used for sensing the displacement of the air cylinder pistons. These LVDTs are used as sensors to give feedback to the data acquisition system (Quanser MultiQ boards). These LVDTs give an output voltage proportional to the actuator displacement. The rated output voltage range (in ideal conditions) for each LVDT is 5volts to Ovolts. Thus, when the author started building the hardware, he had to calibrate the three LVDTs for their actual rated outputs. The input supply voltage for each LVDT is ± 15 volts. The LVDT calibration procedure and the graphs are presented in this Appendix. The LVDTs are manufactured by Faster Instruments. Calibration Procedure: The author used the same calibration procedure for all the three LVDTs. The calibration procedure (see Fig.2) is used for evaluating an expression that depicts the relationship between the LVDT feedback voltage and the LVDT coil displacement. This expression is then used in the Simulink block diagram (as discussed in chapter 3). The calibration procedure for LVDT is as follows: 1. Adjust the power supply to give output voltage ±15 volts. 2. Connect the LVDT to the power supply.

62 57 3. Connect the Digital Multimeter (DMM) probes to the output knobs of the LVDT to record the feedback voltage. 4. Make sure that the groove on the LVDT coil on side A is aligned with the face of the LVDT body (see Fig.2) This position of the LVDT coil is referred as zero position. Record the DMM reading. SIDE B SIDE A LVDT COUPLER LYDT COUPLER Figure 1 LVDT Calibration Procedure 5. Extend the LVDT coil outwards from the LVDT body (by pulling the coupler, attached to the coil via a screw) by la"and record the reading on the DMM. 6. Repeat the procedure stated in step 5, for every extension of the LVDT coil by la" until the groove on the LVDT coil on side B is aligned with the face of the LVDTbody. 7. Plot a graph of feedback Voltage vs. LVDT coil displacement (in millimeters), with feedback voltage on the X-axis and LVDT coil displacement on the Y-axis.

63 58 8. Derive an expression that relates the feedback voltage and LVDT coil displacement (in millimeters). This expression entered in the "Calibration Block" of the Simulink block diagram (as discussed in Chapter 3). The calibration graphs for the three LVDTs (see Fig.2, Fig.3 & Fig.4) are presented in the following pages. It can be seen that these calibration expressions are linear as expected for LVDTs. All three LVDTs have essentially the same calibration function, with only minor differences. CAUBRAllON CHART FORLVDT# t Z W ~ W ~ 00..J 0.. (J) o VOLTAGEFEEDBACK Figure 2 Calibration Chart for LVDT #1

64 59 I ZW ~ W o c(..j D en C Figure 3 Calibration Chart for LVDT #2

65 60... z w ~ w o <...J Q. U) C \Q.TIG:FEBBAQ( Figure 4 Calibration Chart for LVDT #3