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Towards an affordable multi-DOF force feedback motion control input device Yu, Zhang 2000

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Towards an Affordable Multi-DOF Foree Feedback Motion Control Input Device by Yu Zhang M . A . S c , Tsinghua University, Beijing, P.R.China, 1993 B .A.Sc , Tsinghua University, Beijing, P.R.China, 1990 : ' A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in . T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A September 2000 © Yu Zhang, 2000 In presenting this thesis in partial fulfilment of the requirements" for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ^(e-cX{-<c<>t The University of British Columbia Vancouver, Canada Date / £ Cbf. DE-6 (2/88) Abstract A novel multi-degree-of-freedom force feedback motion control input device design has been proposed for 3D human-machine applications. This haptic device utilizes a new electro-mechanical design to achieve a large translation range for each axis of motion, while remaining suitable for mass production at low cost. Parallelogram linkages have been used to obtain displacements along each axis. The current device prototype uses an orthogonal arrangement of three parallelogram linkages to obtain dis-placements along all axes in Cartesian space. Springs have been used to center the device and a slot-and-tab hinge structure has been designed and used as a practical joint. An affordable and compact microelectronic sensor that is based on a grayscale with varying reflectance has been employed in order to sense the end-effector position. The nonlinearity of the sensor has been addressed and linearly compensated. A Lorentz force-based linear actuator design has been proposed. The actuator consists of a stator and a slider. The equivalent magnetic circuit model has been derived to assist the design computations. Experimental results show that the magnetic flux density along the air gap is approximately uniform and that the actuating force, although its level needs to be increased for use in the commercial product, is a linear function of the current applied to the coil windings on the slider. The device kinematics and dynamics have been derived and simulations have been performed to investigate the relationship between joint trajectories and work space actuating forces. ii Contents Abstract ii Table of Contents iii List of Tables vii List of Figures viii Acknowledgments xi 1 Introduction 1 1.1 Motivation and Objectives 2 1.2 Typical Force-Feedback Input Devices 3 1.2.1 Pantograph-based Haptic Devices 3 1.2.2 String-based Haptic Devices 4 1.2.3 P H A N T o M ™ 5 1.2.4 Maglev-based Haptic Devices 6 1.3 Scope and Contributions 11 1.4 Thesis Outline 12 2 The Mechanical Design 14 2.1 Design Goals 14 2.2 The Parallelogram Linkage - A Mechanism for 1-DOF Motion 15 2.3 The 3-DOF Motion Schematics 16 iii CONTENTS iv 2.4 The Centering Mechanism 19 2.5 The Slot-and-Tab Joint Structure 20 2.6 Summary 23 3 The Position Transducer 25 3.1 HOA0149 and its Control Circuit 25 3.2 Grayscale Design and Preliminary Tests 28 3.2.1 The Grayscale 28 3.2.2 Preliminary Tests 28 3.3 The Sensor Calibration and the OSC Algorithm 34 3.3.1 The OSC Algorithm 34 3.3.2 Example 1: Compensation with a Fixed Scaling Factor 37 3.3.3 Example 2: Compensation with a Flexible Scaling Factor 41 3.4 Summary 41 4 The Linear Actuator 45 4.1 Actuator Design Requirements 45 4.2 A Typical Magnetic Circuit 46 4.3 The Actuator Design 49 4.3.1 The Stator 49 4.3.2 The Slider 51 4.4 Design Computations 51 4.4.1 Recoil Line of TRI-NEO-30 52 4.4.2 The Stator Equivalent Circuit Model 53 4.4.3 The Flux and Flux Density Computation 56 4.4.4 The Lorentz Force 58 4.5 The Actuator Design Validation 59 4.5.1 The Experimental Mechanism 59 4.5.2 Actuating Force Computation 61 4.5.3 Test 1: Actuating Force versus Applied D C Current 63 CONTENTS v 4.5.4 Test 2: Uniformity of Magnetic Field along Air Gap 63 4.6 Position Transducer - Actuator Assembly 65 4.7 Summary 67 5 Interface Kinematics and Dynamics 68 5.1 The General Motion Description 68 5.2 The Center of Mass for Each Plate 71 5.3 The Interface Kinematics 75 5.4 The Potential Energy 77 5.5 The Kinetic Energy 79 5.6 The Interface Dynamics 81 5.7 The Velocity and Actuation Jacobians 83 5.7.1 The Velocity Jacobian J(q) 84 5.7.2 The Actuation Jacobian Ja(q) 84 5.8 Dynamic Model Simulations 86 5.8.1 Parameter Measurements and Calculations 87 5.8.2 Inverse Dynamic Model Simulations 87 5.8.3 Forward Dynamic Model Simulations 91 5.9 Summary 94 6 Conclusions and Recommendations 96 6.1 Conclusions 96 6.2 Recommendations for Future Work 99 Bibliography 103 Appendices 107 A The Actuator Design Computations 107 B Experimentally Determined Spring Constants 110 CONTENTS vi C The Installation of Actuators 114 D The Component Cost Breakdown 115 E The Effect of Change of Actuator Dimensions 116 List of Tables 3.1 The M D L P values for example 1 40 3.2 The M D L P values for example 2 41 4.1 Parameters of the cross-shaped arm 62 5.1 The equivalent link parameters for the motion control input device 69 5.2 The maximum values for joint angles. . . 71 5.3 The parameters of the device main plates 87 5.4 The parameters of the springs 87 A - l Parameters for the actuator design computations 109 A - l The component cost breakdown 115 vii List of Figures 1.1 The Hay ward's 5-bar linkage 4 1.2 The schematic of SPIDAR II 5 1.3 The current version of P H A N T o M 6 1.4 Schematic of the Magic Wrist Assembly 7 1.5 Schematic of the Magic Mouse Assembly 8 1.6 Schematic of the U B C Wrist Assembly. 9 1.7 Schematic of the U B C PowerMouse Assembly 11 1.8 Schematic of the C M U Haptic Device Assembly. 12 2.1 The parallelogram schematic and its 3D view 15 2.2 The X-Parallelogram schematic 17 2.3 The X- and Y-Parallelograms 18 2.4 The X- , Y - and Z-Parallelograms 18 2.5 A practical hinge design 20 2.6 The hinge design computation 21 2.7 The designed prototype 24 3.1 HOA0149 and its package dimensions in millimeters 26 3.2 A schematic of the HOA0149 26 3.3 The control circuitry for HOA0149 27 3.4 A n exemplary grayscale generated by specifying linearly increasing gray-level values from 0 to 255 29 vm LIST OF FIGURES ix 3.5 The sensor and grayscale experimental setup 30 3.6 Voltage readings versus displacements for sensor's resolution test 30 3.7 Voltage readings versus distances for different positions 31 3.8 Voltage readings versus positions for different gray patches 32 3.9 Averaged voltage readings versus specified gray-level values 33 3.10 The experimental results for the repeatability test 33 3.11 Results from sensing and compensating the 0th, 3 r d and 6th grayscales for example 1. 38 3.12 Results from sensing and compensating the 9th, 12th and 15th grayscales for example 1 39 3.13 The M D L P values ofAV = Vd-V for example 1 40 3.14 Results from sensing and compensating the 0th, 1st and 2 n d grayscales for example 2. 42 3.15 Results from sensing and compensating the 3rd, 4th and 5*^  grayscales for example 2. 43 3.16 The M D L P values of AV = Vd-V for example 2 44 4.1 A typical magnetic circuit 47 4.2 Two equivalent magnetic circuit models 48 4.3 The stator design 50 4.4 The path of the magnetic flux flowing in the stator 50 4.5 The slider design 51 4.6 The actuator prototype 52 4.7 Dimensions and demagnetization curves of TRI-NEO-30 53 4.8 The stator equivalent magnetic circuit model 54 4.9 Dimensions for reluctance computations 54 4.10 The equivalent model for the left-half sub-circuit 56 4.11 The flux density distribution along the air gap 58 4.12 Illustration of the Lorentz force 59 4.13 The experimental mechanism for the actuator validation 60 4.14 The actuator prototype on an experimental platform 61 4.15 The schematic of the cross-shaped arm 62 4.16 Actuating force versus applied dc current 64 LIST OF FIGURES x 4.17 Magnetic field density versus the length of the air gap 65 4.18 The schematic of position transducer - actuator assembly. 66 4.19 Actual position transducer - actuator assemblies in the device prototype 67 5.1 The equivalent link model of the motion control input device 70 5.2 The equivalent X-axis model 72 5.3 The equivalent y-axis model 73 5.4 The equivalent Z-axis model 74 5.5 The workspace of the haptic device 76 5.6 Simulation results of driving 0\ only 88 5.7 Simulation results of driving #2 o n l y 89 5.8 Simulation results of driving #3 only 90 5.9 Results of forward dynamics simulation for applying the actuating force along the X axis only. 92 5.10 Results of forward dynamics simulation for applying the actuating force along the Y axis only 93 5.11 Results of forward dynamics simulation for applying the actuating force along the Z axis only 94 6.1 A new slot-and-tab design schematic 99 6.2 The detent mechanism schematic 100 B . l The spring constant experiment setup 110 B.2 Experimental result for the X-Spring constant I l l B.3 Experimental result for the Y-Spring constant 112 B.4 Experimental result for the Z-Spring constant 112 C l The X - , Y - and Z-Parallelograms and X - , Y - and Z-Actuators 114 E . l The actuator and corresponding permanent magnet row 116 E.2 Magnetic field density distribution along the air gap for different number of magnets per row 117 Acknowledgments I would like to thank my supervisor, Dr. Chris C. H . Ma, who introduced me into the haptic device design world and constantly supported me throughout this project. This work would not have been possible without his guidance, expertise, encouragement and patience. I would also like to thank my co-supervisor, Dr. Peter D. Lawrence, who has extensive experi-ence in the area of device design. His valuable advice and insight regarding this project have helped me a great deal. Many thanks to the machinists/technicians in the E C E workshop, who revised my mechanical design, machined the components and built the prototype, as well as the corresponding testing devices. Without their expertise, creative ideas and time, I would not have the current device prototype. I would like to thank all of my friends and fellow students, who have made my past years of study and work at U B C productive, cheerful and enjoyable. Their knowledge, care, encouragement, assistance and humor have meant a lot to me. Finally, I would like to thank my parents, who often encouraged me to challenge myself. Without their extended love, I would not have come to the end of this project. xi C h a p t e r 1 Introduction A n "input device" is any interface used to introduce data into a computer. The most common input devices are the keyboard, mouse and track ball. A keyboard enables text entry, and directional cursor movement but no direct position input is allowed. A mouse is used to specify absolute position. It can select an object and control its 2D motion. A track ball is used most often with a laptop computer. The user controls the cursor by rolling the ball with the thumb. A l l these input devices are 2D interfaces. For manipulating and viewing 3D objects, a 3D input device is desirable [41]. Several 3D input devices, including mouse-based 6 degree of freedom (dof or DOF) devices, free-moving isotonic position control devices, desktop isometric and elastic devices, and multi-dof armature-based devices have been investigated in [40,41]. For example, the Magel lan™ 3D Controller made by Logitech is a commercial desktop 6-DOF elastic pointing device with a small motion range [1]. These 2D and 3D interfaces are uni-lateral, passive devices [21]. They transfer energy from the operator to the computer while only vision and/or auditory feedback is directed to the user [12,39]. In the interaction between human and computer via input devices, especially when dealing with tasks that involve contact, one would expect the relationship between the hand and the controlled machine to be bi-directional [12]. The interface would not only receive mechanical inputs from the hand, but would also deliver stimuli to the force and displacement receptors of the hand. This 1 1.1 Motivation and Objectives 2 tactile and kinesthetic information is very useful to an operator who manipulates an object [3]. While a passive uni-lateral input device can't achieve this goal, a force-feedback or haptic device can [21]. According to the Merriam-Webster dictionary, the word "haptic" refers to "relating to or based on the sense of touch". A haptic device by providing haptic feedback allows a user to interact with a computer. Haptic feedback has two cognitive senses: (i) the tactile sense giving an awareness of surface, and (ii) the kinesthetic sense providing information on body position and movement. Thus, a haptic device is a machine that is controlled by the human hand and can be programmed to give the human operator a sensation of forces associated with various arbitrary maneuvers [12]. Haptic or force-feedback input devices can be widely used in the areas of C A D design, tele-operation, training, medical simulation and even entertainment. For example, such a device can be used as a hand (master) controller that gives the operator a force or load feeling when he is remotely maneuvering a mass, just as if he were working on the site [20,22]. Over the past decades, research in this area has led to a number of device products. The present research is concerned with the design of a new haptic input device. 1 . 1 Motivation and Objectives Haptic device designs vary from planar 2-DOF to spatial 6-DOF with a specified motion range. As explained before, a planar 2-DOF haptic device is not suitable for 3D applications no matter how large the workspace is. Most desktop 6-DOF haptic devices have very limited translation and/or rotation ranges. The three translations and three rotations provided by these 6-DOF devices are not fully decoupled. This is due to the complex electro-mechanical design of the device. For applications where only pure, larger translations along three axes in Cartesian space are required, a 6-DOF haptic device is not necessary. The cost for making a haptic device depends mostly upon the selected sensors, actuators and the mechanical design. Position sensors have been widely used in haptic devices to detect the end-effector positions and/or orientations. Due to the required resolution, the cost of these sensors 1.2 Typical Force-Feedback Input Devices 3 varies tremendously, from a few dollars to hundreds of dollars apiece. In most cases, however, high-cost and high-fidelity sensors may not be necessary [10]. Most current haptic devices are usually research-based and not designed for mass production. As long as the cost is not a big problem, any haptic device could be designed and built at will . For a product that is to be attractive to consumers, not only performance but also cost are of particular importance! The force feedback of a haptic device comes from installed actuators. Simply, a brushed dc motor can be used as an actuator, however, nonlinearity and friction may render the motor control non-trivial. Most haptic devices are driven directly, thus a linear actuator that is easy to control is highly desirable. In summary, an affordable force-feedback device that provides three larger, axial translations in Cartesian space would be welcomed by the consumer market. The design of such an interface is the goal of this research project. The main objectives of the thesis are: (i) to design a mechanical device based on ease of use, suitability for mass production and achieving three orthogonal translations in Cartesian space with large motion ranges; (ii) to select an affordable, but effective position sensor; (iii) to design a linear actuator for force-feedback; and (iv) to derive and simulate kinematic and dynamic models of the device. 1.2 Typical Force-Feedback Input Devices A force-feedback input device has bi-directionality as a distinguishing feature. It "reads and writes" to and from the human hand [12]. In the following, the electro-mechanical design of some typical force-feedback input devices including: pantograph-based devices, string-based devices, PHANToM™ and maglev-based devices, are reviewed. 1.2.1 Pantograph-based Haptic Devices The first pantograph-based haptic interface was built by Ramstein and Hayward at McGi l l 1.2 Typical Force-Feedback Input Devices 4 University [11]. It is a 2-DOF planar five-bar linkage device with a workspace of 100x160mm2. Two 2-bar serial linkages, the left and right chains, are connected to a ground-mounted bar. Their distal ends are connected to a handle. Two grounded dc motors are installed on the ground-mounted bar, to drive the handle in a plane. Two optical encoders are installed co-axially with each of the two motors in order to measure the two active joint angles. The two chains are identical, thus providing a well conditioned and symmetrical work space for the pantograph. The Hayward's 5-bar pantograph is shown in Figure 1.1. This picture is reproduced from http://www. cim.mcgill.ca/~haptic/devices/pantograph.html. Figure 1.1: The Hayward's 5-bar linkage. Based on the pantograph, a number of similar haptic devices have been designed and built, such as the UBC 3-DOF twin-pantograph haptic mouse [35] and 5-DOF twin-pantograph haptic pen [31], the 6-DOF parallel platform master hand controller [23], the 6-DOF desktop force display [17] and the 6-DOF master arm [25]. 1.2.2 String-based Haptic Devices The first string-based haptic device was the SPace Interface Device for Artificial Reality, or SPIDAR, developed by Ishii and Sato at the Precision and Intelligence Laboratory of the Tokyo Institute of Technology [19]. It is a 3-DOF wire-driven interface that measures the 3D motion of an operator's finger tip [28], 1.2 Typical Force-Feedback Input Devices 5 The S P I D A R consists of a cap, into which one inserts the index finger. The cap is held by four strings from the corners of a cube frame. The string is wound around a pulley, to which an electrical motor is attached. The string tension is controlled by the motor. The finger motion is measured using rotary encoders mounted to each motor. A stereoscopic vision system provides visual feedback. Through red and green glasses, the operator sees a 3D wireframe object. When the finger reaches a position occupied by the object, the wire is restrained and the finger motion is restricted. The user feels as if he has touched the object. As an extension of SPIDAR, SPIDAR II uses two finger-caps, each of them held by four strings from four separate corners of a cube frame. It can be used for virtual pick-and-place tasks [15,16]. Figure 1.2 depicts the schematic of SPIDAR II. This drawing is reproduced from [15]. Other string-based devices have also been presented [38]. Figure 1.2: The schematic of S P I D A R II. 1.2.3 P H A N T o M ™ The P H A N T o M is a multi-DOF (three active and three passive) desktop force-reflecting inter-face developed by Massie and Salisbury at the Massachusetts Institute of Technology [24]. It tracks finger motion and can exert a controlled force on the finger tip, creating compelling illusions of interaction with solid objects. 1.2 Typical Force-Feedback Input Devices 6 The current version of P H A N T o M is shown in Figure 1.3. This picture is reproduced from http://www. sensable.com/products/premium.htm. Figure 1.3: The current version of P H A N T o M . The x, y and z finger tip coordinates are recorded by three encoders, while three decoupled brushed dc motors control the x, y and z forces exerted upon the user. Motor torques are transmit-ted through pre-tensioned cables to a stiff, lightweight aluminum linkage. A passive 3-DOF gimbal is attached to a thimble at the end of the linkage. Since the three passive rotational axes of the gimbal coincide at a point, no torque is applied on the user, but allows any orientation of the finger tip. The P H A N T o M is commercially produced by SensAble Technologies, Inc. and has gained acceptance in the haptic research community. For example, see Mor [26] for an arthroscopic surgery simulation using P H A N T o M . 1.2.4 Maglev-based Haptic Devices Maglev is an abbreviation of magnetic levitation. Such levitation is generated by a Lorentz force acting on a current-carrying linear conductor in a static magnetic field [14]. Well known maglev devices include the Magic Wrist, Magic Mouse, UBC Wrist, UBC PowerMouse, and CMU Haptic Device. 1.2 Typical Force-Feedback Input Devices 7 1. The Magic Wrist The Magic Wrist was the first maglev-based haptic device that was designed and built by the I B M Research Division at Thomas J . Watson Research Center [13]. It consists of a stator and a flotor. The stator is a rigid support structure with mounted magnet assemblies and three narrow-beam LEDs . The flotor is a hexagonal box structure containing position-sensing photodiodes or position-sensing device (PSD) and flat copper coils that are nested between the stator's inner and outer rings. This flotor is levitated by Lorentz forces that are generated by driving controlled currents through the coils in the magnetic fields. A ball grip, the end-effector, is attached on top of the flotor. The position and orientation of the flotor are calculated as follows: a triplet of narrow, coplanar and radial light beams generated by the L E D s impinge on the three two-dimensional lateral effect PSDs. The centroids of the light spots projected on the active areas of the PSDs are obtained by measuring the current through the PSD's electrodes. The Magic Wrist allows a fine motion range of ± 5 m m for translation and ±4° for rotation. It can lift a weight of 20N in addition to the 9.4N weight of its flotor body. The maximum torque about the vertical axis is 1.7Nm [6]. Figure 1.4 depicts the Magic Wrist schematic. This drawing is reproduced from [6]. Figure 1.4: Schematic of the Magic Wrist Assembly. 1.2 Typical Force-Feedback Input Devices 8 2. The Magic Mouse The Magic Mouse is a 2-DOF haptic interface, designed and built by Kelley and Salcudean [21]. Its main structure is a stationary unit, incorporating a lightweight 2-DOF moving coil plate. The coil plate consists of a hand-wound voice-coil that is sandwiched between two sheets of aluminum. A suspension system, made up of two perpendicular linear ball slides and a single low-friction Teflon leg, constrains the plate to translational motion within a plane. A n infrared L E D has been hidden inside the handle. The L E D light is directed onto a two-dimensional position sensitive detector (PSD) mounted onto the base plate. The position of the Magic Mouse handle is detected by the PSD. The Magic Mouse has a motion range of 17x17mm2. Two permanent magnet stators have been anchored above each coil by an L-shaped aluminum plate, while another two matching permanent magnet stators have been anchored to the base plate below the coils. Thus, two electromagnetic flat coil actuators are formed to provide force feedback to the user. The Magic Mouse schematic is depicted in Figure 1.5. This drawing is reproduced from [21]. handle with LED inside moving coil plate coil plate support leg stator supports permanent magnet stator assembly ball slides permanent magnet stator assembly position sensitive detector Figure 1.5: Schematic of the Magic Mouse Assembly. 3. The U B C Wrist The U B C Wrist was designed and built in the Robotics and Control Laboratory at the University of British Columbia [32]. Similar to the Magic Wrist, it consists of a stator and a flotor in parallel, and uses the same optical position sensors and similar actuators. 1.2 Typical Force-Feedback Input Devices 9 The UBC Wrist fits within a small cylinder. Three horizontal and three vertical flat coils are imbedded in the flotor. Each coil fits in the gap of a matching magnetic pair attached to the stator. The flotor's horizontal plate has holes to allow supporting posts to hold the stator as well as the magnetic pairs. The wrist flotor has a motion range of roughly ±4.5mm for translation and ±6° for rotation. The UBC Wrist is substantially smaller than the Magic Wrist, since it uses a star configuration for all the flat coils. No center volume is left unused! Figure 1.6 depicts the schematic of the UBC Wrist. This drawing is reproduced from [32]. Flotor Top I2D Mounting Columns Vertical Magnet Assembly PSD Mounts Vertical Coil Horizontal Magnet Assembly Rotor Base Horizontal Coil Support Post Stator Base Figure 1.6: Schematic of the UBC Wrist Assembly. 1.2 Typical Force-Feedback Input Devices 10 4. The U B C PowerMouse The U B C PowerMouse is another haptic device built at the U B C Robotics and Control Labo-ratory [29]. Compared to other maglev devices, it has utilized a novel geometry and a novel optical sensor. The PowerMouse is a desktop mouse-like device. A disk-shaped handle with two buttons is attached to a cubic flotor with the flat coils of six Lorentz actuators embedded in its faces. Twenty-four permanent magnets on the stator generate the six magnetic fields crossing the coils. The stator is attached by three mounting posts to a plastic base. A printed circuit board, located at the base of the stator, carries the position sensors, power electronics and a microcontroller. The wide magnetic gaps of the actuators allow spatial fiotor motion with a translation of ± 3 m m and a rotation of ±5° from a nominal center. The optical sensing system is designed to detect flotor motion with respect to the stator. The system uses three LED-generated infrared light planes projected in sequence onto three linear one-dimensional position sensing diodes, one tenth of the price of the two-dimensional sensor used in the Magic and U B C Wrist and mounted as an equilateral triangle on the circuit board. Each light plane crosses two of these diodes. Thus, six light-plane intersections with the diodes are obtained, allowing for the solution of the handle location. The PowerMouse actuator design maximizes the force-to-power-consumption ratio. The device provides a peak force of 34N and a maximum continuous force of 16N. The PowerMouse schematic is depicted in Figure 1.7. This drawing is reproduced from [30]. 5. The C M U Haptic Device The C M U Haptic. Device was designed and built in the Microdynamic System Laboratory at Carnegie Mellon University [5,18]. It features a large decoupled motion range and comfortable form for hand manipulation. The device flotor is a hemispherical shell with a handle at center. This shape results in decoupled translation and rotation, since the flotor can be rotated about its center without colliding with the stator. 1.3 Scope and Contributions 11 Figure 1.7: Schematic of the U B C PowerMouse Assembly. The main flotor body is a thin hemispherical aluminum shell with large oval cutouts for the actuator coils. The coils wound from ribbon wire on spherical forms fit together in a densely packed configuration to maximize the flotor area used to generate actuating forces. The free space around the flotor, the magnet assemblies, and the position sensing system in the stator have been completely redesigned to give the desired motion range and also to conform to the spheric shape. The C M U Haptic Device has a motion range ±25mm for translation and ±20° for rotation. The maximum force is 60N and the maximum torque 3Nm. Figure 1.8 depicts the schematic of the C M U Haptic Device assembly. This drawing is reproduced from [18]. 1.3 Scope and Contributions The scope of the thesis project was to design a low-cost large workspace motion control input device. This includes the mechanical design of the device, the selection and testing of the position sensor, and the design and validation of the linear actuator. 1.4 Thesis Outline 12 coil (6) puter manget assembly (6) position sensor (3) Figure 1.8: Schematic of the C M U Haptic Device Assembly. Due to the limitations of the available facilities, the motion control input device prototype is not yet complete. The thesis contributions are: (i) completion of the device mechanical design; (ii) completion of the selection, testing, and linear compensation of the position sensor; (iii) completion of the design and validation of the linear actuator; (iv) completion of the installation of the sensor, actuator, and other accessories into the device prototype, (v) completion of the derivation and simulation of the device kinematics and dynamics. Following is the outline of the thesis: Chapter 1, Introduction: This" chapter introduces the background of the new motion control input device design. The need for a large translational motion range and low cost motivate the present research. The objectives of the research and an outline of the thesis are given thereafter. Chapter 2, The Mechanical Design: The detailed mechanical design of this haptic device is presented, including the design of a parallelogram linkage mechanism, a centering mechanism and a slot-and-tab hinge structure as a joint. 1.4 Thesis Outline 1.4 Thesis Outline 13 Chapter 3, The Position Transducer: The design of a position transducer used for the device is presented. A n infrared reflective opto-sensor was chosen and tested for the position sensing. The nonlinearity between voltage output and measured position is addressed and calibrated using a One Shot Compensation algorithm. A flexible scaling factor is introduced to improve the convergence rate of this algorithm. Chapter 4, The Linear Actuator: The design of a linear actuator is presented. The actuator's magnetic circuit model is derived and necessary design computations are performed. A n experi-mental platform was built for testing the magnetic field uniformity along the length of the air gap, as well as the dependency between the actuating Lorentz force and the applied dc current. Chapter 5, Interface Kinematics and Dynamics: The kinematics and dynamics of the motion control input device are derived. Simulations of the inverse and direct dynamics are performed and motion coupling is investigated through these simulations. Chapter 6, Conclusions and Recommendations: Research results, contributions and recommen-dations for future work are summarized in this final chapter. Chapter 2 The Mechanical Design This chapter describes the mechanical design of the motion control input device. The resulting prototype aims at achieving low friction 3-DOF translation with a large motion range in Cartesian space, at an affordable price. The design goals are discussed first. Then, the l - D O F motion mechanism, the parallelogram linkage, is detailed. Based on this structure, motion along all three axes is achieved by appropri-ately designing and arranging three parallelograms, the X - , Y - and Z-Parallelograms. Finally, the centering mechanism and the machinable joint design are presented. Necessary calculations are performed to assist these designs. 2.1 Design Goals This haptic device is designed to achieve the following goals: 1. It should provide 3-DOF translation in Cartesian space with a motion range of [—25.4,+25.4]mm along each axis. 2. It should have a zero/center position to which the input device should return when there is no force applied to it. This position should also be used to calibrate the device before usage. 14 2.2 The Parallelogram Linkage - A Mechanism for 1-DOF Motion 15 There must be a centering mechanism in order to achieve this zero posit ion. 3. It should have installed posit ion transducers to sense the handle posit ion in Cartesian space and actuators to give the operator a feeling of the contact and/or load. 4. It should be designed to have as li t t le friction as possible, i.e. a l l joints should be designed carefully. 2.2 The Parallelogram Linkage - A Mechanism for 1-DOF Motion Large motion along each axis is achieved by a parallelogram linkage. Such a linkage is depicted in Figure 2.1(a) and its corresponding 3D view is shown in Figure 2.1(b). Direction (a) The parallelogram linkage schematic. (b) A 3D view of the parallelogram linkage. Figure 2.1: The parallelogram schematic and its 3D view. The parallelogram linkage comprises four sides, as shown i n Figure 2.1(a): • one fixed side, segment A D , the edges of which, A and D , form the fixed joints. • one translating side, segment B C , which translates relative to the fixed side A D . 2.3 The 3-DOF Motion Schematics 16 • two rotating sides, segments A B and D C with identical length /, that connect the fixed side to the translating side, by rotating about the fixed joints A and D, in the plane of the parallelogram. The displacements of the translating side B C can be decomposed into two orthogonal compo-nents: (i) dp, parallel to the fixed side A D , and (ii) dc, perpendicular to A D . Both dp and dc are in the same plane of the parallelogram. From Figure 2.1(a), dp and dc can be expressed by following equations: dp = /sinfi, (2.1) dc = / - / c o s e , (2.2) where 9 is the rotation angle for segments A B and D C . Then, the relationship between dc and dp can be solved as: Q dc — dptan - . (2-3) According to the design goals, the maximum value for dp would be: If 9 is small, dc will be much smaller than dp. For example, if 9 — 0.2593rad or 14.86°, then dc = 3.31mm <C dPmax = 25.4mm. Thus, the displacement of B C is predominantly in the direction parallel to the fixed side A D . Perpendicular direction of motion, dc, is comparatively small and can be neglected. 2.3 The 3-DOF Motion Schematics The input device uses an arrangement of three orthogonal parallelograms, the X - , Y - and Z-2.3 The 3-DOF Motion Schematics 17 Parallelograms, to restrict displacement to translation in Cartesian space. The major difference between them lies in the size of plates, the position and orientation of the hinge. Specifically, motion along the X axis is provided by the X-Parallelogram, which forms the base for the other two parallelograms. Figure 2.2 illustrates the X-Parallelogram schematic. Figure 2.2: The X-Parallelogram schematic. Note in Figure 2.2, the partial hinge structure on the X-TranslatingPlate is used for hinging two rotating plates of the Y-Parallelogram. Motion along the Y axis is achieved by hinging two rotating plates of the Y-Parallelogram onto the translating plate of the X-Parallelogram, as depicted in Figure 2.3. Note that a vertical plate, Y-VerticalPlate, welded perpendicularly to the translating plate of the Y-Parallelogram, forms the fixed plate of the Z-Parallelogram. For motion along the Z axis, the two rotating plates of the Z-Parallelogram are hinged onto the vertical plate of the Y-Parallelogram. Figure 2.4 illustrates this assembly. These three parallelograms are designed such that the X-Parallelogram forms the base on which the Y-Parallelogram hinges and the Z-Parallelogram hinges on the Y-Parallelogram. The device 2.3 The 3-DOF Motion Schematics 18 Figure 2.3: The X - and Y-Parallelograms. Figure 2.4: The X - , Y - and Z-Parallelograms. 2.4 The Centering Mechanism 19 handle is then attached to the translating plate of the Z-Parallelogram. 2.4 The Centering Mechanism The three parallelogram linkages provide translations predominantly along each axis in Carte-sian space; however, without an appropriate centering mechanism, the plates would lean or fall to one side when there is no actuating force applied. Another reason why the parallelogram linkages should be centered is that the input device needs a zero position that can be used for calibration. A third reason for this is the offset actuator force required to compensate for gravity. The centering force means that zero force needs to be applied at the "home/center" position. When centered, each parallelogram linkage must be in its neutral position, i.e. when each side is perpendicular to its neighbour. Coupled piano springs are used as the centering mechanism. For the X-Parallelogram, four identical springs formed into two pairs are hooked between the X -TranslatingPlates and the device case. During motion along the X axis, one pair of springs compress whereas the other pair extend. Forces from both springs will bring the X-Parallelogram to its zero position. For the Y-Parallelogram, two identical piano springs are hooked between the two Y-RotatingPlates and the center of the X-TranslatingPlate of the X-Parallelogram. Similarly, forces from the springs will bring the Y-Parallelogram back to its zero position. For the Z-Parallelogram, two identical piano springs are mounted in parallel between the two Z-RotatingPlates of the Z-Parallelogram. Unlike the springs used in the X - and Y-Parallelograms, these two springs have initial deformations that balance the gravity along Z axis. So the two Z-RotatingPlates are initially perpendicular to the Z-TranslatingPlate as well as the Y-VerticalPlate. The elastic constants of these springs are derived through experiments, as shown in Appendix B . 2.5 The Slot-and-Tab Joint Structure 20 2.5 The Slot-and-Tab Joint Structure In Figure 2.4, hinges are used to connect one plate to the other. However, due to the limitations of facilities at hand, such moulded structure is difficult to machine. Instead, a slot-and-tab hinge structure is proposed to achieve a practical joint used in the haptic device. Figure 2.5 shows such a slot-and-tab hinge. If two plates A and B need to be hinged together, a slot is cut in plate A and a tab is machined on plate B . B y inserting plate B into plate A , a hinge is obtained. T h e contact between plate A and B is a "line" which can change due to the thickness of the plate - there are two pivot points. However, for a slot-and-tab hinge between two th in plates, insignificant friction occurs and can be neglected. The plate w i th a tab is the translating plate, whereas the plate w i th a slot is the rotating plate. The slot wid th is equal to the tab wid th , thus no side displacement between two plates occurs. The computat ion of the slot height is depicted in Figure 2.6. Here, three plates w i t h the identical thickness Zn, plate 1, 2 and 3 are required to be hinged together. T w o slot-and-tab structures are Plate A Figure 2.5: A practical hinge design. 2.5 The Slot-and-Tab Joint Structure 21 designed: (i) the upper slot-and-tab hinge between plates 1 and 2, and (ii) the lower slot-and-tab hinge between plates 2 and 3. The height of the two slots, hi, and h2, are computed as follows. Position with no rotation Lower Tab Position After rotating Figure 2.6: The hinge design computation. When plate 2 is rotating counter-clockwise, its pivot points are O and B. However, the pivot points are changed into O' and B' when plate 2 is rotating clockwise. Without loss of generality, plate 2 rotates counter-clockwise about the origin O by an angle of 9. Plate 1 stays at its original position while plate 3 is displaced. For the upper slot, the distance from the lowest point C on 2.5 The Slot-and-Tab Joint Structure 22 plate 2 to the surface of plate 1, x i , satisfies: xi = /iicos0 - l0 - Znsin.0, 0 G [-0max,8max], (2.5) where 0 m a x is the maximum rotation angle for plate 2. Height hi is chosen such that xi > 0 when 0 = 0rraax> thus guaranteeing that plate 2 will never touch plate 1. For the lower slot, the distance between the lowest point A' on plate 2 and the surface of plate 3, X2, satisfies: x2 = h2cos9 -l0- l0sm9, 0 e [ - 0 m a x , 0max]. (2.6) Similarly, h2 is chosen such that x2 > 0 when 0 = 0 m a x . This guarantees that plate 2 will never touch plate 3. 0max is computed via: 0max = Sin X—y—, (2.7) where dmax = 25.4mm, or half of the translational range along one axis, and Z, the length of plate 2, is determined by the parallelogram linkage design. Substituting 6max for 0 in (2.5) and (2.6), and also letting xi and x2 be: xi > 0, x2 > 0. Then, hicos6max -l0- losm9max > 0, (2.8) h2cosOmax -IQ- l0sin9max > 0. (2.9) 2.6 Summary 23 The resulting constrains on slot heights hi and hi are: cost^. -'max ^ > ~ c ^ 0 • ( 2 > 1 1 ) CUO1/771/17* 2.6 Summary In this chapter, the mechanical design of the motion control input device has been detailed. A parallelogram linkage is used as the basic mechanism to generate a displacement along one axis. Three parallelogram linkages have been coupled such that one is mounted on top of the other. Piano springs have been used for centering the mechanism. A slot-and-tab hinge structure has been selected to implement a machinable low friction joint. The width of the slot on one plate is equal to that of the tab on the other plate. The height of the slot is solely dependent upon the plate thickness and the maximum rotation angle. A prototype of the interface mechanism has been built, as shown in Figure 2.7. A l l of the plates have been machined and assembled to configure the three parallelogram linkages. Spaces are reserved for the installation of the position transducer and the actuator along each axis in Cartesian space. 2.6 Summary 21 C h a p t e r 3 The Position Transducer This chapter describes the design of a position transducer for the haptic device. The position of its end-effector is prescribed in Cartesian space and is sensed along each axis by a microelectronic sensor. This sensed information is converted into a voltage with range of [0, 5]V. It is desirable that the position transducer achieves linearity between end-effector position and transducer's voltage output. The transducer consists of: (i) a position sensor; and (ii) electronic circuitry for converting sensed information into a voltage signal. In the following, the characteristics of the HOA0149 sensor are presented and its control circuit is designed. Then the grayscale with varying reflectance as the object for HOA0149 to sense, is detailed. Finally, compensation of the sensor's nonlinearity is carried out, and an algorithm called the One Shot Compensation is introduced. 3.1 HOA0149 and its Control Circuit A variety of proprioceptive sensors such as potentiometers, encoders and resolvers, can be used as position sensors [33]. However, the objectives of this project require that the sensor be affordable, have good performance, be compact and light weight so that it may easily be installed within the interface mechanism. It should not introduce additional dynamics. Moreover, the sensor should 25 3.1 HOA0149 and its Control Circuit 26 provide a voltage output within a suitable range. According to these criteria, an optoelectronic sensor, the Honeywell HOA0149, was chosen for position sensing. This microelectronic sensor consists of (i) a transmitter - an infrared emitting diode (IRED) and (ii) a receiver - a focused NPN silicon phototransistor. These components are contained in a low profile as shown in Figure 3.1, and a schematic of the HOA0149 is depicted in Figure 3.2. Emitter Cathode r' . T r A n o d e Figure 3.1: HOA0149 and its package dimensions in millimeters. Anode O Convcrgir Optica] Collector Emitter TRANSMITTER RECEIVER Figure 3.2: A schematic of the HOA0149. Usually, HOA0149 is used to detect the presence of an object within its field of view. When 3.1 HOA0149 and its Control Circuit 27 powered by the control circuit, its transmitter projects a beam of infrared light along its optical axes. If there is no object at the point at which the two optical axes converge, then there is no reflection and the receiver's phototransistor works in the cutoff (OFF) region: no current is passing from the collector to the emitter. On the other hand, if there is an object on the converging optical axes, the receiver receives reflected light from the object's surface. The phototransistor works in the active (ON) region: current IE in Figure 3.2 is occurred and flowing from the collector to the emitter. Hence, the collector-emitter current IE indicates whether or not an object is in front of the sensor. In this project, the use of HOA0149 has been extended by letting the receiver's phototransistor always work in its linear region. By varying the reflectance of the object surface, varied outputs are obtained. The magnitude of the phototransistor's emitter current IE depends solely on the reflectance of the object surface: the brighter the surface, the more reflection and the larger the current. Figure 3.3 shows the control circuitry designed for HOA0149 that is used in this project [34]. Voltage Regulator Power Supply - r HOA0149 Transmitter Receiver R. T I Figure 3.3: The control circuitry for HOA0149. Since the power source is easily influenced by disturbances, a voltage regulator is used to apply a constant voltage to the HOA0149. Thus, different values for Ip are obtained by varying the value of the forward resistor, Rp-It is not very convenient to use the emitter current as receiver output. By connecting a load resistor, RL, between emitter and ground, this current is converted into a load voltage, VL-3.2 Grayscale Design and Preliminary Tests 28 The capacitors and resistors used in Figure 3.3 are tuned as follows: C\ = 100/x/, Ci = 0-lpf, C 3 = 0.001/i/, RF = 115X1, i ? L = 1.925ifeft. 3.2 Grayscale Design and Preliminary Tests As discussed previously, a surface with varying reflectance is required. When this object is placed at the point of convergence of the transmitter and receiver axes and sensed by a moving HOA0149, different voltage readings corresponding to varying reflectance are obtained. If the reflectance versus position relationship is known, the voltage reading versus position can be derived. The design of such a surface, called a grayscale, is addressed in this section. Moreover, three initial tests on the position sensor and on the grayscale are performed. 3.2.1 The Grayscale A black surface has minimum light reflectance while a white surface has maximum light re-flectance. In between black and white are shades of gray with reflectance ranging between the minimum and maximum values. A grayscale is a surface with varying reflectance expressed by black, shades of gray and white. The grayscale was generated using M a t l a b ™ functions to associate varying reflectance with gray-level values ranging from 0 (black) to 255 (white) and printed onto a mask. A n exemplary grayscale generated by specifying linearly increasing gray-level values is shown in Figure 3.4. The specified gray-level values are plotted in the top figure while the corresponding grayscale image is shown in the bottom figure. The grayscale reflectance clearly increases as the gray-level values increase. 3.2.2 Preliminary Tests In this section, the performance of the HOA0149 is investigated. A Newport™ Positioning Platform is used in the experimental setup. This platform provides a 25mm translational range 3.2 Grayscale Design and Preliminary Tests 29 Figure 3.4: A n exemplary grayscale generated by specifying linearly increas-ing gray-level values from 0 to 255. along a horizontal axis. The HOA0149 is fixed onto this platform such that it moves along the horizontal axis. The grayscale is attached to a stationary L-shaped platform facing the sensor. A box covers the sensor and the grayscale in order to eliminate disturbances from other light sources, such as fluorescent light. This experimental setup is illustrated in Figure 3.5. 1. The Sensor Resolutuion Before the preliminary tests, the resolution of the sensor is investigated using the oscilloscope. A grayscale is generated in M a t l a b ™ by specifying gray-level values to be linearly increasing from 6 to 255 and printed onto a mask. Not only the dc component but also the ac component of the sensor outputs have been recorded and plotted in Figure 3.6. As Figure 3.6 shows, the peak-to-peak value from the envelope of the ac component is approx-imately 6mV. Thus, the resulant resolution of the sensor is found to be 0.044mm. 3.2 Grayscale Design and Preliminary Tests 30 Figure 3.5: The sensor and grayscale experimental setup. Lower Envelope Displacement [mm] Figure 3.6: Voltage readings versus displacements for sensor's resolution test. 3.2 Grayscale Design and Preliminary Tests 31 2. The Optimal Distance between the HOA0149 and the Grayscale According to the specification, the distance between the HOA0149 and the grayscale plays an important role in the sensing. Therefore, this effect is investigated now. The same grayscale used in the previous sensor resolution test is sensed again by the HOA0149. Voltage readings are taken at positions of 1.8mm, 5.0mm, 10.0mm, 15.0mm, 20.0mm, 25.0mm and 26.7mm when the distances are varying from 0 to 18.0mm. The results from sensing this grayscale are plotted in Figure 3.7. 3.5 3 2.5 2 CD D ) S > 1.5 0.5 h L . . . . . . . . . I 1 1 1 . ! . . . . ' \ ' ^ P= 1.8mm P= 5.0mm P=10.0mm P=15.0mm P=20.0mm P=25.0mm . P=26.7mm ; / \ \ i • / : Y \ ' ' v- < I l l \ \ \ Mil \ \ \ . ..... ';'/ ~ \ V X ' ' / ••' '•• \ I'l / \ \ \ x 2 4 6 8 10 12 14 16 18 Distance d [mm] Figure 3.7: Voltage readings versus distances for different positions. As Figure 3.7 shows, the optimal distance between the HOA0149 and the grayscale for the testing setup is 4.5mm. At this distance, the sensor will have a maximum output. Thus, in the following tests, the distance between the sensor and the grayscale is always set to be this optimal value. 3. Linearity Test This test investigates the linearity between voltage readings and various specified gray-level values. Several uniform gray patches were generated and sensed by the HOA0149. The gray-level 3.2 Grayscale Design and Preliminary Tests 32 values that were chosen for these patches were 0, 30, 60, 90, 120, 150, 180, 210, 240 and 255, respectively. The experimental results are plotted together in Figure 3.8. 4r 10 15 20 Positions [mm] Figure 3.8: Voltage readings versus positions for different gray patches. As seen from Figure 3.8, the voltage readings sensed from these gray patches increase as the gray-level values increase from 0 to 255. For positions less than 3.9mm, the voltage readings are not stable. This is because that not all the grayscale is within the field view of the HOA0149. Hence, the effective position range, or the stable reading range, is chosen as [3.9, 26.7]mm. The averaged voltage reading over the effective position range for each gray patch can be computed and plotted with respect to the chosen gray-level value in Figure 3.9. It is clear that this averaged voltage value is not a linear function of the gray-level value. This is due to nonlinearities such as the sensor itself, the printer and the quality of the paper on which the grayscale is printed. 4. Repeatability Test This test investigates the repeatability of position sensing using the HOA0149. The grayscale used here is generated by specifying linearly increasing gray-level values from 6 to 255. The same grayscale is sensed four times. Al l voltage readings are plotted in Figure 3.10. 3.2 Grayscale Design and Preliminary Tests 33 0 50 100 150 200 250 Color Value Figure 3.9: Averaged voltage readings versus specified gray-level values. Positions [mm] Figure 3.10: The experimental results for the repeatability test. 3.3 The Sensor Calibration and the OSC Algorithm 34 As Figure 3.10 illustrates, the voltage readings are almost the same for all tests on the same grayscale. The errors between these four tests are very small, below 5mV. This shows that the sensing of a grayscale is highly repeatable. 3 .3 The Sensor Calibration and the OSC Algorithm The initial tests have shown that the voltage output is not a linear function of the gray-level value. Hence, if the gray-level values vary linearly with the device end-effector position, the rela-tionship between voltage reading and position is not linear. However, a linear position transducer is our objective. To achieve this objective, the nonlinearity between voltage readings and gray-level values must be compensated. Various methods can be used to compensate for this nonlinearity. One technique compensates by using a lookup table in software. Whenever a new voltage reading is obtained, the corresponding position value is found by referring to the table. The basis for doing this is the repeatability of position sensing. However, this method would rely on the computer since it requires a lot of resources from the computer. Another option consists in modifying the grayscale itself in terms of the difference between voltage readings and desired linear voltages. The resultant compensated grayscale is independent from other resources such as the computer. This forms what we call the One Shot Compensation (OSC) algorithm. 3.3.1 The O S C A lgo r i t hm The OSC algorithm can be summarized as follows: 1. Choose position values to construct a vector P: N (3.1) 3.3 The Sensor Calibration and the OSC Algorithm 35 where N is the total number of positions. 2. Specify a vector C of linearly increasing gray-level values corresponding to the position vector P: Ci 6 [CV^n, Cmax\, i = 1,... , N. (3-2) Use vector C to generate a grayscale in Matlab with increasing reflectance. 3. Sense the grayscale and construct the vector V from voltage readings: V e [Vmin,Vmax], i = l,...,N, (3.3) where Vmin is the minimum value and Vmax is the maximum value. Plot results in the following two graphs: • Graph C versus P: gray-level value versus position. • Graph V versus P: voltage reading versus position. 4. Draw a straight line between points (Pmin,Vmin) and (Pmax,Vmax)- This line forms the vector Vd, the objective of the compensation, which can be expressed as: Vdt = ^ Z V » i n \ (Pi ~ Pmin) + Vmin, i = l,...,N. (3.4) (,-• max * min) 5. Compute the voltage difference vector A U : A U = Vd- V. (3.5) 6. Compute the Maximum Deviation from Linearity in percentage (MDLP) for A U . max | AVi \ MDLPAV = i = 1 ' - ' N x 100%. (3.6) max Vi i=l,...,N 3.3 The Sensor Calibration and the OSC Algorithm 36 If the value is smaller than the prescribed threshold, skip the rest of OSC steps and terminate the compensation. Otherwise, 7. Compute the ratio vector R: Ri = ^T, i = l,...,N. (3.7) 8. Compute the gray-level compensation vector A C , A Q = CiRi, i = l,...,N. (3.8) Compute the compensated gray-level vector Cd, Cd = C + AC. (3.9) 10. Based on compensated gray-level values Cd, generate and print a new compensated grayscale. Go back to Step 3. Direct application of the ratio vector R may result in overcompensation. This is analogous to a feedback system with an overly high loop gain. To avoid this problem, a new dimensionless variable, called the Scaling Factor S, is introduced in the algorithm, where: 5 €[0,1]. (3.10) Then Equation (3.8) becomes: ACi = CiRiS, i = l,...,N. (3.11) If S = 0, there is no compensation. If S = 1, then there will be full gray-level compensation as before, since A C ; = C; -Ri. For S € (0,1), there will be partial gray-level compensation. Therefore, the scaling factor gives one degree of freedom to adjust the compensation and controls the speed of convergence. 3.3 The Sensor Calibration and the OSC Algorithm 37 Two examples of applying the OSC algorithm are presented in the following sections. The experimental setup is the same as before and the forward current of the transmitter, Ip, has been adjusted to 30.0mA. Initial gray-level values for both examples are specified linearly increasing from 6 to 255. The only difference of these two examples is that the scaling factor in the first example is chosen to be a small constant whereas in the second example it is chosen manually according to the previous compensation result. 3.3.2 Example 1: Compensation with a Fixed Scaling Factor In this example, the scaling factor S is chosen to be a small constant in order to reduce and smooth the gray-level compensation for each OSC iteration: S = 10%. The compensation converges after sixteen iterations. Due to length considerations, only portion of the compensation results are plotted in Figure 3.11 and Figure 3.12. The voltage-position rela-tionships are plotted on the left side, while the relationships between previous and new compensated gray-level values are lotted on the right side. The V versus P graphs show that the voltage difference magnitude satisfies | A V i | > |AV2J > . . . > I AVi6 | . This means that nonlinearities are significant during initial stages. The corresponding gray-level compensations for these stages are larger as well, i.e., |AC7i | > | AC2I > . . . > |AC7i6|. The M D L P values for all iterations are summarized in Table 3.1 and are also plotted in Fig-ure 3.13. After sixteen compensation iterations, the voltage reading is almost a linear function of position. However, as shown in Figure 3.13, the M D L P value for the 9th compensation iteration is larger than that of the 8th. Also, the M D L P values for the 12th and 13th compensation iterations are larger than that of the 11th. These may be due to printer and paper nonlinearities. For example, if the paper quality is not uniform, even if the gray-level values are the same, the printed grayscales are different. 3.3 The Sensor Calibration and the OSC Algorithm 38 (a) Measured voltages V\ from sensing the 0 t h original grayscale and computed linearized voltages V^i-5 10 15 20 25 Position [mm] (c) Measured voltages V4 from sensing the 3 r d compensated grayscale and com-puted linearized voltages Vdn-5 10 15 20 25 Position [mm] (e) Measured voltages Vj from sensing the 6th compensated grayscale and com-puted linearized voltages Vdj. (b) Gray-level values C\ specified for the 0th original and Cdi for the 1 s t compen-sated grayscale. 5 1 0 1 5 2 0 2 5 Position [mm] (d) Gray-level values C3(=Cd2) for the 3 r d and Cdi for the 4th compensated grayscale. 5 1 0 1 5 2 0 2 5 Position [mm] (f) Gray-level values C V ( = C < J 6 ) for the 6 t h and C d 7 for the 7 t h compensated grayscale. Figure 3.11: Results from sensing and compensating the 0th, 3rd and 6' grayscales for example 1. 3.3 The Sensor Calibration and the OSC Algorithm 39 3 2.5 5 10 15 20 25 Position [mm] (a) Measured voltages Vio from sensing the 9th compensated grayscale and com-puted linearized voltages Vdio-5 10 15 20 25 Position [mm] (c) Measured voltages V13 from sens-ing the 12th compensated grayscale and computed linearized voltages Vdi3-5 10 15 20 25 Position [mm] (e) Measured voltages Vi6 from sens-ing the 15 t h compensated grayscale and computed linearized voltages Vdw. J 150 50 5 10 15 20 25 Position [mm] (b) Gray-level values C\o(=Cd§) for the 9th and Cdw for the 10 t h compensated grayscale. Position [mm] (d) Gray-level values C\z{=Cd\2) for the 12 t h and Cdl3 for the 13 t h compensated grayscale. 2501 200 5 10 15 20 25 Position [mm] (f) Gray-level values C\e(=Cdib) for the 15"1 and Cdw for the 16 t h compensated grayscale. Figure 3.12: Results from sensing and compensating the 9th, 12th and 15' grayscales for example 1. 3.3 The Sensor Calibration and the OSC Algorithm 40 Table 3.1: The M D L P values for example 1. Number of Iterations i max| AVi |[V] MDLP&Vi [%] 1 0.5562 15.8101 2 0.5088 14.6207 3 0.4746 13.5755 4 0.4326 12.4525 5 0.4171 11.8226 6 0.3345 9.5653 7 0.2388 6.9177 8 0.2197 6.3848 9 0.2635 7.6710 10 0.1937 5.65054 11 0.1161 3.3077 12 0.1157 3.3497 13 0.1196 3.4859 14 0.0862 2.4906 15 0.0833 2.4229 16 0.0634 1.8522 Number of Iterations Figure 3.13: The M D L P values of AV = Vd - V for example 1. 3.4 Summary 41 3.3.3 Example 2: Compensation with a Flexible Scaling Factor In this example, a new initial grayscale is generated similarly by specifing gray-level values to be linearly increasing from 6 to 255. The scaling factor for each step is chosen visually, depending on the result of the previous iteration. If an overcompensated result is obtained, the scaling factor for the current step should decrease and vise versa. The initial value for S is chosen: Si = 100%. The compensation converges after six iterations. Compensation results are illustrated in Fig-ure 3.14 and Figure 3.15. Again, V, Vd versus P relationships are plotted on the left side, while C , Cd versus P relationships are plotted on the right side. V versus P graphs show that | A V i | > | A V ^ | > . . . > | A V ^ | . Hence, the gray-level compensation satisfies | A C i | > I A C 2 I > . . . > |ACe | . After five iterations, color compensation becomes very small and a practically linear voltage reading is achieved. Table 3.2 summarizes the M D L P value for each OSC iteration. Also, the statistical results are plotted in Figure 3.16. Table 3.2: The M D L P values for examp le 2. Number of Iterations i max| AVt |[V] MDLPAVi [%] Scaling Factor Si 1 0.2582 7.2325 100% 2 0.1740 4.8293 100% 3 0.0967 2.7378 100% 4 0.1079 3.0515 50% 5 0.0717 2.0186 50% 6 0.0556 1.5751 50% 3.4 Summary The design of the position transducer has been presented in this chapter. The transducer consists of a sensor with its control circuit and a grayscale. In accordance with the project objectives, 3.4 Summary 42 5 10 15 20 25 5 10 15 20 25 Position [mm] Position [mm] (a) Measured voltages V\ from sensing (b) Gray-level values C\ for the 0th orig-the 0th original grayscale and computed inal and Cdi for the 1 s t compensated linearized voltages Vdi- grayscale when Si = 100%. 5 10 15 20 25 5 10 15 20 25 Position [mm] Position [mm] (c) Measured voltages V2 from sensing (d) Gray-level values C l2(=Cd1) for the the 1 s t compensated grayscale and com- 1 s t and Cdi for the 2 n d compensated puted linearized voltages Vdi- grayscale when S2 = 100%. Position [mm] Position [mm] (e) Measured voltages V3 from sensing the 2 n d compensated grayscale and com-puted linearized voltages Vdz-(f) Gray-level values C%{-=Cdi) for the 2 n d and Cdz for the 3 r d compensated grayscale when 53 = 100%. Figure 3.14: Results from sensing and compensating the 0 , 1 s t and 2' grayscales for example 2. 3.4 Summary 43 5 10 15 20 25 Position [mm] (a) Measured voltages V4 from sensing the 3 r d compensated grayscale and com-puted linearized voltages Vd4-5 10 15 20 25 Position [mm] (c) Measured voltages V 5 from sensing the 4 t h compensated grayscale and com-puted linearized voltages V d 5 . 5 10 15 20 25 Position [mm] (e) Measured voltages Ve from sensing the 5th compensated grayscale and com-puted linearized voltages Vd6-2501 200 100 50 5 10 15 20 25 Position [mm] (b) Gray-level values C^(=Cd3) for the 3 r d and Cdi for the 4 t h compensated grayscale when 54 = 50%. 5 10 15 20 25 Position [mm] (d) Gray-level values Cs(=Cdi) for the 4th and Cds for the 5 t h compensated grayscale when S5 = 50%. Position [mm] (f) Gray-level values C§(=Cds) for the 5th and Cd6 for the 6 t h compensated grayscale when 56 = 50%. Figure 3.15: Results from sensing and compensating the 3 r d , 4th and 51 grayscales for example 2. 3.4 Summary 44 , i 1 . 1 . 0 1 2 3 4 5 Number of Iterations Figure 3.16: The M D L P values of AV = Vd - V for example 2. Honeywell's infrared reflective sensor HOA0149 has been chosen as the position sensor. A grayscale is generated in Matlab-™, printed onto a mask and used as an object with varying reflectance for the HOA0149 to sense. The performance of the sensor on the grayscale has been tested. The nonlinearity between voltage reading and position has been addressed and calibrated using the OSC algorithm. The effectiveness of the OSC algorithm has been illustrated by two examples. A flexible scaling factor was introduced to improve the convergence rate of the OSC algorithm. Some other preliminary tests on the sensor have been performed, such as how fast the sensor responds to a reflectance change and to determine the effective reflectance sensing area. However, due to the limitations in the experimental instruments, these tests have not been completed. Hence, the results obtained are not discussed in this thesis. C h a p t e r 4 The Linear Actuator For controlling the end-effector motion in Cartesian space, a linear actuator has been designed for each axis. The actuator input is a current passing through a moving coil placed in a constant magnetic field. Its output is a force that is parallel to one axis. Actuator design is based on a permanent magnetic system, and the actuating force is a Lorentz force. In this chapter, the actuator design is presented. A typical permanent magnetic system with an air gap is reviewed. Then, the actuator mechanical design is presented and its characteristics such as the flux density along the air gap and the generated Lorentz force, are computed. Finally, the actuator prototype is validated through two experiments. 4.1 Actuator Design Requirements For a haptic device, actuators drive the end-effector within its work space. A n actuation force is generated to give an operator the feeling of force or load. This is called "force feedback" or "haptic feedback". The actuator design is based on energy conversion from electrical to mechanical. For the actuators used in the motion control input device, the requirements are as follows: 45 4.2 A Typical Magnetic Circuit 46 1. The motion range of an actuator should be no less than [-25.4, 25.4]mm since the end-effector of the input device can translate within that range along each axis in Cartesian space. 2. The actuator body should be stationary whereas its moving part should be installed such that it does not touch the body and therefore not generate friction. 3. The actuator should be easy to control. The actuating force should vary linearly with the control signal. 4. The actuator should be compact so as to easily fit within the device. 5. The actuator should consume as little power as possible since three actuators are required for translation along the X, Y and Z axes. A permanent magnetic system best fits these requirements, since a permanent magnet can produce flux in an air gap without exciting coils and, thus no electric power is dissipated [8]. It can also be small so that an actuator with limited dimensions is feasible. Furthermore, a current-carrying conductor exposed to a uniform magnetic field produced by a permanent magnet generates a force which varies linearly with the current. These features initiate the actuator design. 4.2 A Typical Magnetic Circuit A typical magnetic circuit consists of a permanent magnet, an air gap and two pieces of soft iron [7,9], as depicted in Figure 4.1. The magnetic flux path of the whole system, the dashed line in the figure, is generated by the magnet and is completed by the two soft iron pieces and an air gap. The permanent magnet is the only source of energy in the system. This energy is dissipated by the air gap, the soft iron and the magnet itself. The magnet's performance depends solely on its residual flux (Bri) and recoil permeability (prec), which are determined by the recoil line on the magnet's demagnetization curve [2,4,27]. 4.2 A Typical Magnetic Circuit 47 Magnetic Flux Path Soft Iron Pieces with Length I m Permanent Magnet A s Air Gap Figure 4.1: A typical magnetic circuit. The reluctances of the magnet, Rm, the air gap, Rg, and the soft iron, Rs, are defined as [2,4,27]: Rn ly. L P^sAs iM)Ag (4.1) (4.2) (4.3) where, • the lengths of the magnet, the air gap, the soft iron in the flux path are lm, lg and ls respectively. Usually the length of each element is computed by choosing its center line [9,36]. For example, the length of the soft iron in the path, ls, could be found by: Is — L lm lg. (4.4) where L is the total length of the flux path. • the corresponding cross-sectional areas of the magnet, the air gap, the soft iron are Am, Ag and As respectively, as depicted in Figure 4.1; 4.2 A Typical Magnetic Circuit 48 the permeability of the air gap (free space) is HQ, and Mo = Aix • 10 H / m ; the permeability of the soft iron is /z s, and it is very high, e.g. \is = 5000^0-The analysis of such magnetic circuit is complex, therefore, an approximating model is most often used [36]. Two equivalent magnetic circuit models, depicted in Figure 4.2, can be used here. (a) The Flux Source Model R R R (b) The MMF Source Model Figure 4.2: Two equivalent magnetic circuit models. In the flux source model, Figure 4.2(a), <j>ri is the total flux produced by the permanent magnet where <\>T\ = Br\Am. (f>g is the flux flowing through the air gap. In the mmf source model, Figure 4.2(b), Fm = faiRm is the corresponding magnetomotive force (mmf) [9]. These two models are analogous to the current and voltage source models commonly encountered in electric circuit analysis with reluctance corresponding to resistance. Hence, all electric circuit 4.3 The Actuator Design 49 theorems can be applied here. For example, cf)g, the magnetic flux in the air gap, is computed via: ^9 = o , d ^ l - (4.5) For the magnetic circuit as depicted in Figure 4.1, it is the magnetic field along the air gap that is of great interest to users. Since 4>r\ = BriAm and 4>g = BgAg, the magnetic flux density in the air gap, Bg, is found by: £ _ Tim Am g ^ 9 Rm + Rs + Rg Ag 4.3 The Actuator Design A stationary part, the stator, and a moving part, the slider, are the two main actuator compo-nents. Their design schematics are presented in the following section. 4.3.1 The Stator The stator is composed of a trunk, permanent magnets and two air gaps. The stator trunk is constructed from five highly permeable soft iron bars. Four of them form a rectangular shape while a fifth, the center bar, is placed lengthwise down the center of the stator. Two rows of fourteen neodymium-iron-boron permanent magnets are attached on the inner sides of the two long side iron bars, and thus two long thin air gaps are formed between the center bar and long side iron bars in the trunk. The stator is depicted in Figure 4.3. A l l magnets face the center bar with same polarities, such that a flux path is formed through the stator. This flux path can be decomposed into the following sub-paths: upper-left, upper-right, lower-left and lower-right, as shown in Figure 4.4. 4.3 The Actuator Design 50 Figure 4.3: The stator design. Upper-Left Flux Sub-Path Permanent Magnets Upper-Right Flux Sub-Path Rear Air Gap 7 1 2 i i i i i / i i i i ~ r N l l l l l T Lower-Left Flux Sub-Path Front Air Gap Lower-Right Flux Sub-Path Figure 4.4: The path of the magnetic flux flowing in the stator. 4.4 Design Computations 51 4.3.2 The Slider The slider is composed of two aluminum brackets, a copper core and a coil winding. The brackets fix the slider onto a movable plate. Connecting these two brackets is the copper core, on which the coil winding is tightly wound, as shown in Figure 4.5. When a current-carrying unconstrained conductor is placed in a magnetic field, it moves due to a Lorentz force. Thus, one way of assembling the slider and the stator is to install the slider inside the stator along the center bar as shown in Figure 4.6. Appendix C illustrates how to install the three actuators onto the 3-DOF parallelogram linkages for the haptic device. Figure 4.5: The slider design. 4.4 D e s i g n C o m p u t a t i o n s Three key issues are discussed in this section: (i) the recoil line of the permanent magnet, (ii) the induced flux density along the air gap, and (iii) the Lorentz force exerted upon the current-carrying winding in the air gap. 4.4 Design Computations 52 Figure 4.6: The actuator prototype. 4.4.1 Recoil Line of TRI-NEO-30 The chosen permanent magnets are TRI-NEO-30 (neodymium-iron-boron). They are small, as depicted in Figure 4.7(a), however, their energy product BmaxHmax is quite large, as up to 27/30MGOe. The demagnetization curves, under various working temperatures, are shown in Figure 4.7(b). Since the actuator works at room temperature, the demagnetization curve at 20°C is used to approximate the recoil line: Bm = 1.15 + 1.43 • 10-eHm, (4.7) where the residual flux, Br\, is 1.15T and the recoil permeability, prec, is 1.43 • 10~ 6 H/m. Thus, for such a permanent magnet with length lm and cross-sectional area Am, its reluctance 4.4 Design Computations 53 North 2.54 (a) TRI-NEO-30 dimen-sions in millimeters. 0 T R I - N E O - 3 0 D A T A S H E E T NEODYMIUM-IRON B O R O N DEMAGNETIZATION CURVES BH max (MGOe) 40 30 20 10 (b) Demagnetization curves of TRI-NEO-30. (= Figure 4.7: Dimensions and demagnetization curves of TRI-NEO-30. could be computed by: p _ m m 1.43 • 1 0 - 6 A , (4.8) 4.4.2 The Stator Equivalent Circuit Model Since a magnet is equivalent to a flux source in parallel with its reluctance, all twenty-eight permanent magnets used in one actuator are equivalent to twenty-eight flux sources: <t>r,\, 4>r,2, • • •, 4>r,2&i in parallel with their corresponding reluctances Rmti, Rm,2i • • • > Rm2,8-The soft iron and two air gaps can be segmented into small pieces according to the permanent magnet's dimension. Thus, twenty-eight air gap segments with reluctance Rgti, Rg,2, • • •, Rg,2S, a n d forty-five soft iron segments with reluctance R^i, Ri^, • • • > Ri,45 have been formed. The equivalent circuit model for the stator is then depicted in Figure 4.8 and the corresponding dimensions are 4.4 Design Computations 54 illustrated in Figure 4.9. R « R u R u R i ,7 A R w B R,,, R U 3 Ri,M R l l 5 Figure 4.8: The stator equivalent magnetic circuit model. Permanent Short Side 1 ^ r ^ S. Magnets Iron Bar 1 i 1 c D E G / \ F / L o n g M a e iron B a r r m . 1 1 \ 2 \ 3 \ 4 \ 5 : 6 V 7 S ! 9 ! 20 ! 7 i I 12 j /3 j 14 1 A \ C B \ Center Iron Bar 2S 27 j 26 j 25 i 24 j 23 j 2^ 21 \ 20 \ 19 \ 18 \ 17 \ 16 \ 15 Long Side Iron Bar 1 1 1 1 1 1 1 * (i (I Figure 4.9: Dimensions for reluctance computations. According to the actuator design, lr = 7.5mm, bm = 5.08mm, lc = 7.5mm, lm = 2.54mm, lg = 2.5mm, and IQ = 3.75mm. If the fringing flux is neglected, the cross-sectional areas of the magnet, the air gap segments and the soft iron segments are the same, Am = Ag = At. 4.4 Design Computations 55 A l l magnets have the same demagnetization curve, and their cross-sectional area are all the same, e.g. Am = 10.16 * 5.08mm2, thus each of them produces the same flux: <f>r,k = BrlAm, * = 1,...,28, (4.9) and has the same reluctance: i W = —H-> *: = I,--- ,28. (4.10) Prec-^m The reluctance of each air gap segment is Rg,k = -^r, fc = l , . . . ,28. (4.11) The soft iron segments have different reluctances since they have different lengths. Without loss of generality, the lengths of the soft iron segments have been chosen as shown in Figure 4.9. Therefore, (1) for segments with a length of lo, i.e. segment AB: RQ = (4.12) Pzs±i (2) for segments with a length of l^\ = \bm + ^lr, i.e. segment BC: Ri,3l = #i,45 = ~~T> (4-13) (3) for segments with a length of Z^i = lo + lg + lm + lc + \lr + \bm, i.e. segment CDEF: Ri,l = #1,15 = #i,16 = #i,28 = % \ ) (4-14) P'iAi 4.4 Design Computations 56 (4) for segments with a length of kt2 = bm + ^lc, i.e. segment EGH: Ri,2 = • • • = Ri,U = Ri,32 = • • • = Ri,U — Ri,V7 = . . . = R%,29 = ~~4~- (4-15) Hence, the equivalent flux source model is symmetrical with respect to the vertical and horizon-tal center lines, the dash-dotted lines, as shown in Figure 4.8. The entire magnetic circuit model for the stator can be decomposed into four identical sub-circuits: the upper-left, the lower-left, the upper-right and the lower-right sub-circuits. In the following computation, the upper-left and lower-left sub-circuits are combined into a left-half sub-circuit. This is due to the fact that all the flux from those two rows of permanent magnets are flowing through the center iron bar. Figure 4.10 illustrates the equivalent model of the left-half sub-circuit. Figure 4.10: The equivalent model for the left-half sub-circuit. 4.4.3 The Flux and Flux Density Computation Kirchhoff's law is applied to compute the mmf at each labeled node: F\, . . . , F13, F29, ..., F35, F 4 3 , . . . , F50, and F 7 3 , . . . , F 8 5 in Figure 4.10. Then, the magnetic flux, . . . , <j>gj, ^ 2 2 , • • • > 4>g,2&, and the flux density, JB S ) i , . . . . , Bgj, Bg$2, • • •, -Bg,28) for each small air gap segment are 4.4 Design Computations 1 57 derived (See Appendix A) . For example, the magnetic flux 4>a^ is: i<29 — F44 4>g,l = E > and the corresponding flux density B9ti is: g _ <frgJ}_ _ #29 ~ #44 Ag Rg^Ag As discussed before, the upper-left and upper-right sub-circuits are symmetrical. Thus, the flux densities for air gap segments labeled 1,2, . . . , 7, 8, 9, . . . , 14 satisfy the following relationships: #9,1 = #9,14, (4.18) #9,2 = #9,13, (4.19) #9,3 = #3,12, (4.20) #9,4 = 59,11> (4.21) #9,5 = #3,10, (4.22) #9,6 = #3,9, (4.23) #9,7 = #3,8- (4.24) Let d be the displacement from the air gap segment to the center line as shown in Figure 4.9. The computed flux densities #3,1, . . . , #3,7, and #3,8, • • •, #g,i4 as functions of d, are plotted in Figure 4.11. As Figure 4.11 shows, the magnetic flux densities on both ends of the air gap are slightly greater than those of the center. This is due to the fact that the magnetic flux takes the shortest path. More flux is flowing through both left and right soft iron ends. The averaged magnetic flux density in the air gap (Bg) is 0.53T. (4.16) (4.17) 4.4 Design Computations 58 -30 -20 -10 0 10 Displacement d [mm] 20 30 Figure 4.11: The flux density distribution along the air gap. 4.4.4 The Lorentz Force Figure 4.12(a) illustrates the magnetic field that is present in the air gap, with respect to the current applied to the coil windings. Lorentz forces exert upon segments AD and BC of the windings, to push the core away from reader (see Figure 4.12(b)). The Lorentz force for an iV-turn coil winding that is conducting a current, 7, in a uniform magnetic field Bg is: F = NIBgLeff, (4.25) where the effective length of the winding that is exposed to the magnetic field is I*e/f- For our actuator prototype, Leff is 0.010mm, determined by the structure of the stator and the slider. Note that Equation (4.25) is valid only if the direction of the current / is perpendicular to that of the magnetic field Bg. 4.5 The Actuator Design Validation 59 P e r m a n e n t M a g n e t s N - T u r n W i n d i n g C o r e P e r m a n e n t M a g n e t s (a) Side view. L. K t , , t O O Segment A D Segment B C • Current flowing out of the paper x Current flowing into the paper (b) Top view. Figure 4.12: Illustration of the Lorentz force. 4.5 T h e A c t u a t o r D e s i g n V a l i d a t i o n In this section, the actuator design is validated through two experiments performed using a specially designed testing mechanism. The first experiment investigates the linearity of the Lorentz force with respect to an applied dc current. The second experiment investigates the distribution of the magnetic flux density along the length of the air gap. A nearly uniform magnetic field is expected. 4.5.1 The Experimental Mechanism A n experimental mechanism, consisting of a base plate, a supporting pillar, a cross-shaped rotating arm and two plastic weighing cups, is designed for validating the actuator prototype. The supporting pillar has been screwed onto the base plate. The center of the cross-shaped arm is connected to the tip of the pillar by a pair of bearings. The entire cross-shaped arm is supported by the pillar and can rotate about a pivot point that is formed by the two bearings. The two cups are used for adjusting weights on the cross-shaped arm. The weight difference between the two cups drives the cross-shaped arm to rotate clockwise or counter-clockwise. 4.5 The Actuator Design Validation 60 A pointer is attached to the vertical part of the cross-shaped arm and a scale plate is attached to the pillar. Thus, when the cross-shaped arm is rotating about the pivot point, the angle and direction of the rotation can be read from the scale. Figure 4.13 illustrates the experimental mechanism. The stator has been screwed onto the base plate, and the slider is attached to the vertical end of the cross-shaped arm, as illustrated in Figure 4.14. Therefore, when the slider is actuated by a dc current, it pushes the cross-shaped arm to rotate around the pivot point. Again, the angle and rotation direction can be obtained from the scale reading. By design, the rotational range of the cross-shaped arm is [—7.3°, 8.3°], or [—15.5,17.6]mm. This experimental mechanism can be used to validate the actuator design since the Lorentz force can be balanced by adjusting the weights in two cups at each location within the motion range of the cross-shaped arm. If the dc current applied to the coil windings and the weights in the cups are known, and the rotating angle is read, the Lorentz force as well as the magnetic field density at any location can be computed. The relationship between the Lorentz force and the weights in the cups, the rotating angle is derived in the next sub-section. Figure 4.13: The experimental mechanism for the actuator validation. 4.5 The Actuator Design Validation 61 Figure 4.14: The actuator prototype on an experimental platform. 4.5.2 A c t u a t i n g F o r c e C o m p u t a t i o n A schematic of the cross-shaped arm and all forces exerting upon it are depicted in Figure 4.15. The vertical part of the cross-shaped arm is composed of segments PO and CO, whereas its horizontal part is composed of segments LO and RO. The corresponding masses of these segments are shown at the center of mass (COM). A coordinate system is chosen with its origin O at the pivot point. The parameters of the cross-shaped arm are measured and the associated torques are computed symbolically. These results are summarized in Table 4.1. The resultant torque about the pivot point is: r = ^mpgaps'm9 + ^migaicos9 + WigaicosO — \msgacsu\8 —mcgacsm9 + Faaccos9 — ^mrgarcos9 — Wrgarcos9. (4.26) For mp W 0, a/ = ar = a and m; = mr, Equation (4.26) becomes: r = Wigacos9 — -msgacsin9 — mcgacsin9 + Faaccos9 — Wrgacos9. (4-27) 4.5 The Actuator Design Validation 62 A Y I Figure 4.15: The schematic of the cross-shaped arm. Table 4.1: Parameters of the cross-shaped arm. Symbol Explanation Value Unit Torque [Nm] mp mass of segment PO wO kg ^mpgapsm9 ap length of PO 116.0 mm mi mass of segment LO 0.040 kg \migaicos0 wt weights added to the left cup kg WigaicosO ai length of LO 136.0 mm ms mass of segment CO 0.0447 kg — \msgacsm6 mc mass of the slider 0.0074 kg —mcgacsm6 ac length of CO 122.0 mm Fa actuating force N Faaccos6 mr mass of segment RO 0.040 kg — ^mrgarcos6 Wr weights added to the right cup N — Wrgarcos6 ar length of RO 136.0 mm 4.5 The Actuator Design Validation 63 B y setting J ^ r = 0 in Equation (4.27), the actuating force, Fa, becomes: Fa = (Wr - Wt)g— + hms + 2m c ) 5 tan 9. (4.28) ac 2 4.5.3 Test 1: Actuating Force versus Applied D C Current In this experiment, the force-current dependency is investigated. For convenience and also to eliminate possible non-uniformity in the air gap, the equilibrium is chosen at 9 = 0. By setting 6 — 0 in Equation (4.28), the actuating force Fa becomes: Fa = (Wr - W , )<A (4.29) dc Thus, the flux density at 9 — 0 is computed from Equations (4.29) and (4.25): B|«=Tnhr, <4-30> When a dc current is applied to the slider winding and the swinging arm is driven by the slider to a position where 9^0, the weights in the two cups are adjusted so that the swinging arm is forced to back to 9 = 0. The values of W\ and Wr are used to compute Fa. Four tests have been performed, the averaged results of which are plotted in Figure 4.16. As Figure 4.16 shows, the force varies linearly with the current. The slope of the line (F-I) is 1.444N/A. Since the winding has N = 260 turns, the magnetic flux density at 9 = 0 is 0.555T. This experimental result is roughly consistent with the design computations where Bgj = 0.5333T, as illustrated in the previous section. 4.5.4 Test 2: Uniformity of Magnetic Field along A i r Gap In this experiment, a constant dc current is applied; therefore, the actuation force should be a constant, provided that Bg is constant. 4.5 The Actuator Design Validation 64 CD O -0 .3 -0.2 -0.1 0 0.1 Current I [A] Figure 4.16: Actuating force versus applied dc current. From Equations (4.25) and (4.28), the flux density along the air gap is computed as: B = kw{Wr - Wj) + M a n ( 9 , (4.31) where kw = j^[affi a n d kg = \rjfjpjjf are both constants if the current / is constant. The experiments are carried out by adjusting the weights in the two cups so that measurements can be collected at several positions along the length of the air gap. Two tests are performed. First, the swinging arm is driven to rotate counter-clockwise (6 > 0) by a negative current: J_ = -224.6mA, kw_ = -0.01873T/g, ke_ = -0.4998T, (4.32) and Fa is along the —X axis. 4.6 Position Transducer - Actuator Assembly 65 Second, the swinging arm is rotated clockwise (6 < 0) by a positive current: i+ = 262.2mA, kw+ = 0.01604T/g, ke+ = 0.4281T. In this case, Fa is along the +X axis. The results from these two experiments are combined and plotted in Figure 4.17. 0.6r 0.59 0.58 0.57 E-0.56 'co § < Q = 0.54 L L 0.53 0.52 0.51 0.5 — l + = 262.2mA _e_ I* =-224.6mA -10 -5 0 5 Displacement [mm] 10 15 (4.33) Figure 4.17: Magnetic field density versus the length of the air gap. As Figure 4.17 shows, the magnetic field along the air gap is almost uniform, as predicted. 4.6 Pos i t ion T r a n s d u c e r - A c t u a t o r A s s e m b l y As mentioned in Chapter 2, spaces are reserved for installing the position transducer and the actuator into the device. Due to the fact that they are all related to the motion along one axis, 4.6 Position Transducer - Actuator Assembly 66 they can be assembled into one integrated assembly, called position transducer - actuator assembly. The stator is fixed with respect to one direction of motion, and originally the slider is screwed onto the translating plate along that direction. In order to install the position transducer together with the actuator, two new mounting brackets are designed to attach this assembly onto the trans-lating plate. The slider, however, is now mounted onto these two new brackets rather than the translating plate. The HOA0149 sensor is mounted onto one of the new brackets, facing towards the outside of the stator trunk on which a compensated grayscale is glued. The other new bracket is available for other component installation in the future. The schematic of the assembly is depicted in Figure 4.18, while Figure 4.19 shows the resultant assemblies installed along the X, Y and Z axes in the device prototype. Permanent Magnets Iron Bar Figure 4.18: The schematic of position transducer - actuator assembly. 4.7 Summary 67 Figure 4.19: Actual position transducer - actuator assemblies in the device prototype. 4.7 Summary This chapter has presented the design of a linear actuator prototype. The actuator consists of a slider and a stator. TRI-NEO-30 permanent magnets are installed on the stator trunk and produce the flux flowing into the air gap. A n A^-turn core winding is wound around the slider, which is placed in the air gap. When a dc-current flows through the winding, a Lorentz force is generated and pushes the slider. The actuator magnetic circuit model has been derived and the design computations have been performed using this model. A n experimental mechanism has been designed and built for evaluating the actuator prototype. The results show that the force is a linear function of the applied current, and the magnetic field along the air gap is nearly uniform. Finally, the position transducer and actuator for motion along one axis have been grouped into an integrated assembly by virtue of the actuator design. C h a p t e r 5 Interface Kinematics and Dynamics In this chapter, motion along the X, Y and Z axes in Cartesian space is described. The center of mass of each plate is computed and the mechanism kinematics are derived. Then, the device joint space dynamics are computed using the Lagrangian formulation. Finally, inverse and forward dynamics simulations are performed. 5.1 T h e G e n e r a l M o t i o n D e s c r i p t i o n The general motion of the device in Cartesian space can be decomposed into motions along the X, Y and Z axes: 1. for motion along the X axis, the two X-RotatingPlates, and X-TranslatingPlate onto which the Y-Actuator is installed are involved. Under actuation, both X-RotatingPlates rotate by an angle of 9\ about an axis parallel to the Y axis, whereas X-TranslatingPlate translates along the X axis. 2. for motion along the Y axis, the two Y-RotatingPlates, and Y-TranslatingPlate onto which the Z-Actuator is installed are involved. Under actuation, two Y-RotatingPlates rotate about an axis parallel to the X axis, by an angle of 82. Meanwhile, Y-TranslatingPlate translates 68 5.1 The General Motion Description 69 along the Y axis. Since Y-Plates are hinged onto the X-Plat , motion along the Y axis is coupled with motion along the X axis. 3. for motion along the Z axis, two Z-RotatingPlates, and Z-TranslatingPlate to which a handle is attached are involved. Both Z-RotatingPlates are designed such that Z-TranslatingPlate translates along the Z axis when they rotate about an axis parallel to the Y axis by an angle of (93. Since Z-Top and Z-Bottom are hinged onto Y-Plat , motion along the Z axis is coupled with motion along the Y axis. Hence, it is also coupled with motion along the X axis. Five springs are installed in the device: two X-Springs with elastic constants kx\ and kX2 for motion along the X axis, two Y-Springs with elastic constants ky\ and ky2 for motion along the Y axis and one Z-Spring with elastic constant kz for motion along the Z axis. These springs bring the moving plates to the center position if there is no actuating or hand force. As mentioned in Chapter 2, the X-Springs and Y-Springs are installed with one end hooked onto a moving plate and the other end hooked onto a plate stationary with respect to that direction of motion. For motion along the Z axis, Z-Spring is hooked onto the two Z-RotatingPlates. A l l plates are rigid, thus each of them is equivalent to a simple link. The equivalent link model for the motion control input device is illustrated in Figure 5.1, and Table 5.1 lists its equivalent link parameters. Table 5.1: The equivalent link parameters for the motion control input device. Plate Equivalent Link Properties Mass Inertia COM Length COM to Pivot Point Front X-Plates(X-RotatingPlates) mxi 1x1 Cxi ^x Xxl^x Rear X-Plates(X-RotatingPlates) mX2 1x2 CX2 lx Xx2^x X-Plat (X-TranslatingPlate) + Y- Actuator mx3 Cx3 1x3 Left Y-Plates(Y-RotatingPlates) myl Iyl Cyl ly Xylly Right Y-Plates(Y-RotatingPlates) my2 Iy2 Cy2 ly Y-Plat (Y-TranslatingPlate)+Z- Actuator rriys Cy3 ly3 Z-Bottom(Z-RotatingPlates) mzi hi Czl lz Xzllz Z-Top (Z-RotatingPlates) mz2 h2 cz2 lz Xz2h Z-Side(Z-TranslatingPlate)+Handle mZ3 C 2 3 lz3 5.1 The General Motion Description 70 5.2 The Center of Mass for Each Plate 71 The joint variables are chosen as: q=[0lM]T. (5.1) and @k £ [ @k,maxi 8k,max], k — 1,2,3, (5-2) where Qk,max is determined by the maximum motion range of the translating plate and the length of each rotating plate. According to the specification, the maximum motion range of the translating plate along one axis, dmax, is specified to be 25.4mm. If the lengths of X-RotatingPlates, Y-RotatingPlates and Z-RotatingPlates are lx, ly and lz respectively, then 9i^max, 6<i,max and ^3,max c a n be computed by: l,max = s i n - 1 ( dmax \ lx ) = s i n- l /25 .4 \ v ix ) 2,max = s i n - 1 / dmax \ ly ) = s i n- l /25 .4 \ \ ly ) 3,max = s i n - 1 / dmax V lz ) = s i n-The results of the maximum rotating angles are summarized in Table 5.2. Table 5.2: 1 "'he maximum values for joint angles. Joint Angle Length [mm] Maximum Value [rad] 01 lx = 99.06 = 0.2593 92 ly = 117.39 62,max = 0.2181 #3 lz = 64.5 03,max = 0.4048 5.2 The Center of Mass for Each Plate The input device motion can be decomposed into motion along three orthogonal axes. The equivalent single axis model is separated from the general link model to facilitate analysis. 5.2 The Center of Mass for Each Plate 72 1. The Equivalent X-axis Model Figure 5.2: The equivalent X-axis model. Using Figure 5.2, the C O M positions for the equivalent X-axis model are computed as: 1 - Ko)lx3 - A x iZ x s in0i Vcxl = 0 2-cxl lx(l - Axicos6>i) Xcx2 — A x oZx3 — A X 2 ^ s i n ^ i Vcx2 = 0 Zcx2 ^(1 - A x 2 cos0i ) %cx3 —Zxsin 6>i Vcx3 0 ZcxZ lx(l - cos^i) (5.4) (5.5) (5.6) 5.2 The Center of Mass for Each Plate 73 2. The Equivalent Y-axis Model V c O I a v E Figure 5.3: The equivalent Y-axis model. Using Figure 5.3, the C O M positions for the equivalent Y-axis model are computed as: —/xsin 6\ —ay — AyiZysin#2 lx(l — cos(9i) + \yilyC0s62 —lxsinOi &y — A^j/Sin 62 lx(l - COSt^l) + \y2lyCOs62 -lxsin Oi Vcyl zcy\ %cy2 Vcy2 Zcy2 %cy3 VcyZ = Zcy3 —lyS'm 62 lx(l - COS(9i) + lyCOS02 + (Xy3 ~ >^yQ)ly3 (5.7) (5.8) (5.9) Figure 5.4: The equivalent Z-axis model. 5.3 The Interface Kinematics 75 Using Figure 5.4, the C O M positions for the_equivalent Z-axis model are computed as: Vczl = —lxsin8i + \zilzcos$3 —lysin92 1 (5.10) zcz\ — cos Oi) + lycos 82 — A z iZ z s in 8s — \yoly3 XCz2 Vcz2 = —lxsm9i + A22Z zcos#3 —lysm02 (5.11) Zcz2 _ 1x0-— cos 9i) + lycos 82 — A Z2Z zsin 83 + (1 - Xy0)ly3 _ %cz3 Vcz3 = -lxsin9i + lzhcos 93 —lySin 82 (5.12) ZCz3 — cos 8\) + lycos 82 — Izhsin #3 — AyO j^/3 + A Z 3/ Z 3 5.3 T h e Interface K i n e m a t i c s The kinematics relates joint variables and to end-effector variables. Forward kinematics maps joint positions q to end-effector positions in work space X. Let the end-effector position be: X=\px,py,pz]T. (5.13) From Figures 5.2, 5.3 and 5.4, the forward kinematics is: Px -lxsmOi + lzhCos83 x = Py = —Zysin 82 (5.14) Pz lx(l - cos81) + lycos82 - Izhsin83 - Xy0ly3 + lz3 Thus, the workspace of the haptic device can be computed by specifying the values for 61, 82 and 83, and the result is plotted in Figure 5.5. 5.3 The Interface Kinematics 76 0.49 0.03 0.09 -0.03 0.03 Figure 5.5: The workspace of the haptic device. The velocity mapping from joint velocity q to end-effector velocity X and acceleration mapping from joint acceleration q to end-effector acceleration X are derived by computing the first and second time derivatives of Equation (5.14). Inverse kinematics maps end-effector positions to joint positions. From Equation (5.14), end-effector position X is mapped to joint space as: 0 2 = arctan2(-pj / , - p2) € [-0.2181,0.2181]rad, 01 = arcsin(^ + / *1"f - aretan2(zi,p g) 6 [-0.2593,0.2593]rad, 0 3 = a r c s m ( ^ + p ' , + 2 z ' ~ f ) - arctan 2(p g, zx) € [-0.4048,0.4048]rad, (5.15) where «| = lx + Z ycos0 2 — Xyolys + / Z 3 — p 2 after 0 2 is solved first. 5.4 The Potential Energy 77 5.4 T h e Potent ia l E n e r g y The potential energy has two components: (i) gravitational potential energy, and (ii) elastic potential energy. 1. The gravitational potential energy for a plate with mass mi is: Vg,mi = -misF'r^g^i. (5.16) where g = [0, 0, — g]T is the acceleration due to gravity, and z ^ 0 g ,mt i s t n e displacement from the center of mass Cmi to the corresponding pivot point. According to Equation (5.16), the gravitational potential energy of the device can be obtained by (i) computing the gravitational potential energies of the X-Parallelogram, V9jX, the Y-Parallelogram, VgiV, and Z-Parallelogram, V ^ j 2 , and (ii) summing these results into Vg: Vg = Vg:X + Vg:y + V9tZ, (5.17) where: Vg<x = {TnxiXxl+mX2XX2 + mx3)glx(l-cos9i), (5.18) Vg,y ~ '(rnyi+rny2 + rny3)glx(l - cos 8i) -{rnylXyi + my2Xy2 + my3)gly(l - cos 62), (5.19) VgtZ = (mzl + mZ2 + mz3)glx(l-cos91)-(mzi + mz2 + mz3)gly(l-cose2) -(mzlXzilz + mz2Xz2lz + mz3lzh)g sin (93. (5.20) Note that all the gravitational potential energies are computed with respect to the same reference point so that they can be summed together. 5.4 The Potential Energy 78 2. The elastic potential energy for a spring with constant ki and deformation Ski is: 1 Ve,ki = ^hSki2, (5.21) where 5ki is the deformation of the spring. The deformations for all the springs used in the haptic device are computed and summarized as follows: • for the two X-Springs: o~xi = Val + bl + 2a x 6 x sin 9\ - lsx0, (5.22) 6x2 = \Ja2x + bx- 2a x 6 x sin 6>i - lsx0, (5.23) where Z s xn is the free length of the spring, for the two Y-Springs: Syi = ^Jay + b^ + 2aybysm 92 - hyo, (5-24) 6y2 = \Jo^ + by~ 2aybysin92 - lsyo, (5.25) where lsyo is the free length of the spring. • for the Z-Spring: 5Z = \jaz2 + bz2 + 2a 26 2sin 83 - lsz0, (5.26) where ISZQ is the free length of the spring. Thus, the elastic potential energy of the device is computed as: Ve = Vejx + VetV + VetZ, (5.27) 5.5 The Kinetic Energy 79 where: Ve,x = '^kx\(y/ax + bl + 2axbxsm.9i - lsx0)2 + ^ kx2{y/al + b\ - 2axbxsin61 - lsxQ)2, (5.28) Ve,y = \ k y ^ \ J a y + by + 2a y 6 y sin0 2 - ^ o ) 2 y ^ y + ^ - 2a v b w sin0 2 - k*o) 2, (5.29) Ve,z = \kz{yja?z + b2 + 2azbzsm93 - lsz0)2. (5.30) Note that VetX, Ve^y and VeiZ are the elastic potential energies of the X-Parallelogram, Y -Parallelogram and Z-Parallelogram respectively. 5.5 T h e K i n e t i c E n e r g y The kinetic energy for a plate with mass and inertia Ij is: Ti — ^ m i V c m i o V c m i o + —l^mi^Ii^lcmi^o, (5.31) where Vcmi,o l s ^ s C O M linear velocity, and Ucmit0 is the plate angular velocity. According to Equation (5.31), the X-Plates and X-Plat kinetic energies are: Txl = l-{mxl\2xll2x + Ixl)92, (5.32) T x 2 = \{mx2\2x2l2x + Ix2)9l (5.33) Tx3 = \mx3l2x9\. (5.34) 5.5 The Kinetic Energy 80 The Y-Plates and Y-Plat kinetic energies are: Tyi = ^myill^l + ^(mylxlill + ^1)^2 _ m y i Z x A y i i y s i n 0 i s i n 0 2 0 i 0 2 , . (5.35) Ty2 = — m ^ ^ ^ i + ^{myZ^il + Iy2)&2 ~ ra^xAy2^sin #ism 026>i6>2, (5.36) ?2/3 = ^ 1 / 3 ^ ^ 1 + ^ m y 3 ^ 2 _ m y 3y ysin6>isin0 2(9'it9 2. (5.37) The Z-Top, Z-Bottom and Z-Side kinetic energies are: T*i = \mzll2x62 + l-rnzll2yel + \ ( m z l \ 2 z l l 2 z + Izl)62 lysm ^isin t92#it92 + rnzilyXz\lzsa\ t92cos t93t92t93 - m ^ i x A ^ i ^ s i n ^ i - e3)6\e3, (5.38) T 2 2 = i m a 2 ^ ? + i m Z 2 ^ 1 + i ( m a 2 A 2 2 Z 2 + / 8 2 ) ^ - m ^ U j / S i n 6>isin 020102 + m ^ A ^ s i n 02cos 0 3 0 2 0 3 - m ^ A ^ s i n ^ i - 03)0103, (5.39) T 2 3 = ^ m z 3 ^ 0 ? + \ ^ y B \ + \mz3l2J\ -mz3lxlySin 0isin 020102 + mz3lylzhsm 02cos 0 3 0 2 0 3 - m 2 3 / x U s i n ( 0 i - 0 3)0'i0 3. (5.40) The kinetic energy for the M C I D is then given by: 3 3 3 i—l i = l i = l 5.6 The Interface Dynamics 81 5.6 T h e Interface D y n a m i c s Set M n = rrixiXl^l + IxX + mx2\2x2l2x + 1x2 + mx3l2x +mylll + my2ll + my3l2x + mzll2x + mz2l2x + mz3l2x, (5.42) M 1 2 = (my\\y\ + my2\y2 + my3 + mzi + mz2 + mz3)lxly, (5.43) M13 - [mzl\zllz + mz2Xz2lz + mz3lzh)lx, (5.44) M 2 2 = JWyiAjx /J + Iyi + my2\2y2l2y + Iy2 + my3l2y + mzll2y + mz2l2y + mz3l2y) (5.45) M 2 3 = (rnzl\zilz + mz2\z2lz + mz3lzh)ly, (5.46) M 3 3 = m ^ A y i 2 / / + 7 zj +mz2\z22l2 + Iz2 + mz3lzh2, (5.47) G i = ( m x i A x i + m x 2 A x 2 + m l 3 + m y i + m y 2 + m y 3 + m 2 i + m 2 2 + ^ 3 ) ^ 5 , (5.48) G2 = (m^iAj,! + my2\y2 + my3 + mz\ + mz2 + mz3)lyg, (5.49) G3 = ( m 2 l A 2 i ^ + r n 2 2 A z 2 Z z + mz3lzh)g, (5.50) p i = a x + 6X, gi = a x b x , (5.51) p2 = + q2 = ayby, (5.52) p 3 = a 2 + 6 2, q3 = azbz, (5.53) fcx = fcii = fcx2, ( 5 - 5 4 ) fcj, = fcyi = ky2. (5.55) The Lagrangian of the haptic device, L , is defined as [33,37]: L = T-(Vg + Ve). and the Lagrangian equation is expressed by: d dL dL , , „ „ ^ 7 ^ 7 r - ^ Z ~ = T f c ' K = 1,2,3. dt d6k d9k (5.56) 5.6 The Interface Dynamics 82 Thus, the joint space dynamic model of the device is derived as: D(q)q + C(q,q)q + G(q) + S(q)=T, where: (1) the inertia matrix D(q) is: D{q) = Mn - M i 2 s i n 0 i s i n 0 2 - M i 3 s i n ( 0 i - 0 3) -Mi2sin0isin02 M 2 2 M2 3 sin 02cos 0 3 -M 1 3 s in(0! - 0 3) M 2 3 s i n 02cos 0 3 M 3 3 and D(q) is a symmetrical 3-by-3 matrix; (2) the centrifugal and coriolis terms are: C(q,q) = 0 - M 1 2 s i n 0 1 c o s 0 2 0 2 Mi 3 cos(0 i - 0 3 )0 3 - M 1 2 C O S 0i sin 0 2 0i 0 — M 2 3 s i n 0 2sin 0 3 0 3 -Mi 3 cos(0i - 03)0'i M 2 3 c o s 02cos 0 3 0 2 0 and D(q) — 2C(q,q) is a skew-symmetrical 3-by-3 matrix; (3) the gravitational force vector is: G{q) = Gis in0 i —G2sin 0 2 - G 3 c o s 0 3 (5.57) (5.58) (5.59) (5.60) 5.7 The Velocity and Actuation Jacobians 83 (4) the spring force vector is: S(q) = l. _ ( y/y\+2q\sm8\—lsxo _ -y/pi — 2<?isin9\-l SxO\ n n„ Q K x q i [ - Vpi+2<?1sine91 V P l -2gisinei ) c o s a l i / y / P 2 + 2 g 2 s i n f l 2 - ^ y O _ \Zp2-2g2sin02-^^0 \ a VP2+2c/2sint)2 v / P 2 - 2 g 2 s i n d 2 ) C O S U 2 k g V - p 3 + 2 9 3 s i n ^ ^ o C Q S g \/p3+2q3Sint93 0 (5.61) (5) the joint torques are: r(g) = r„ - J T (c?)F e , (5.62) where JT(q) is the transposed velocity Jacobian. The actuating torque vector, r 0 , is defined as: Ta = [Tal,Ta2,Ta3]T, (5.63) and the contact force vector, Fe, is defined as: Te — [fexj fey) fez\ (5.64) 5.7 T h e Ve loc i ty a n d A c t u a t i o n Jacob ians Two Jacobian matrices are required for the kinematics and dynamics of the haptic device: (i) the velocity Jacobian J(q), and (ii) the actuation Jacobian Ja(q). The velocity Jacobian maps joint velocities q to end-effector velocities X. Since the contact force, Fe, is acting upon the end-effector, the velocity Jacobian also maps the work space contact force Fe to joint torques. The actuation Jacobian maps the work space actuating force, Fa, to actuating joint torques ra. Since the actuator's slider has been screwed onto one translating plate such'as X-Plat , Y-Plat , and Z-Side, the actuation Jacobian is derived based on the C O M velocities of these plates. 5.7The Velocity and Actuation Jacobians 84 5.7.1 The Veloc i ty Jacobian J(q) The end-effector velocity X: X = [vx,vy,vz]T, (5.65) is derived by taking the first time derivative of Equation (5.14): X = J(q)q, (5.66) where J(q), the velocity Jacobian, is given by: —lxcosQi 0 — lzhs'mds 0 —lyCOS62 0 ^ s i n ^ i — lysm02 —lzhCOs9% (5.67) The determinant of the velocity Jacobian, J(q), is: det[J(q)] = -lxlylzhcos(0i - 0 3)cos0 2- (5.68) Due to the effective ranges of joint variables, det[J(q)] is never zero. Thus, the velocity Jacobian J(q) is never singular and its inverse J~l(q) always exists. 5.7.2 The Ac tua t i on Jacobian Ja(q) The velocities X C x 3 of the X-Plat , Xcy3 of the Y-Plate, and Xcz3 of the Z-Side are derived by computing the first time derivatives of Equations (5.6), (5.9) and (5.12), respectively: 5.7 The Velocity and Actuation Jacobians 85 cx3 — XCy3 — -Z x COS di X, cz3 — —^cost9i 0 0 0 0 0 1, Z asin0i 0 0 0 0 0 —lycos92 0 lxsvn0i — Zysin#2 0 -Z xcos Oi 0 -Zzhsin t93 0 —lycos92 0 Zxsin(9i —Zysin02 —lzhcos0s (5.69) (5.70) (5.71) Thus, T a i , r a 2 , r a 3 , the joint torques generated by the X-Actuator, Y-Actuator and Z-Actuator respectively, are computed as: Tai = Ta2 T~a3 = — lxCOs6ifa:X 0 0 0 -lyCOs82fa,y 0 ' Zxsin 9ifa,z -lyS'm 92fa,z -IzhCOS 83fajZ (5.72) (5.73) (5.74) 5.8 Dynamic Model Simulations 86 and the joint actuating torques are summed into: (5.75) i = i where Fa = [fax, fay, faz]T is the actuating force and Ja(q), the actuation Jacobian, is given by: Ja(q) = -lxcos6i 0 0 0 —lycos92 0 Zxsin#i — Zj,sin#2 —IzhCosO^ (5.76) The determinant of the actuation Jacobian, Ja(q), is then: det[Ja(q)] = -Ixlylzhcos9icos92cos0z. (5.77) Due to the effective ranges of joint variables, det[Ja(q)] is never zero. Thus, the actuation Jacobian Ja(q) is never singular and its inverse J~1(q) always exists. Inserting Equations (5.67) and (5.76) into the dynamic model (5.57), the dynamics of the motion control input device become: D(q)q + C(q, q)q + G(q) + S(q) = JT(q)Fa - JT(q)Fe. (5.78) 5.8 D y n a m i c M o d e l S imulat ions In this section, simulations of the dynamic model are performed. The relationship between the specified joint positions and resultant actuating forces, and the relationship between the applied actuating forces and resultant joint positions and end-effector positions are illustrated. 5.8 Dynamic Model Simulations 87 5.8.1 Parameter Measurements and Calculations The masses and lengths of the main plates including X-RotatingPlates, X-TranslatingPlate, Y-RotatingPlates, Y-TranslatingPlate, Z-RotatingPlates, Z-TranslatingPlate and the actuators, were measured. The inertia and the C O M position for each rotating plate (X-RotatingPlates, Y -RotatingPlates, Z-RotatingPlates) were computed based on design dimensions. The results are listed in Table 5.3, Table 5.3: The parameters of the device main p^  Plate Mass [kg] Inertia xlO~3[Nm] Length [m] C O M Other [m] Front X-Plates mxX = 0.1062 Ixl = 0.3977 lx = 0.09906 Ki = 0.5 Rear X-Plates mx2 = 0.1062 Ix2 = 0.3977 Xx2 = 0.5 X-Plat+Y-Actuator mx3 = 0.3195 lx3 = 0.1158 Left Y-Plates rriyi = 0.1375 Iyi = 0.7266 ly =0.11739 A y i = 0.4068 Right Y-Plates mV2 = 0.1375 Iy2 = 0.7266 Xy2 = 0.4068 Y-Plat+Z-Actuator my3 = 0.2968 ly3 = 0.09357 Xy3 = 0.48 Xy0ly3 = 0.07348 Z-Bottom mzl = 0.1285 Izl = 0.6442 lz = 0.1176 Xzl = 0.5 lzh = 0.0645 Z-Top mz2 = 0.0767 Iz2 = 0.2552 Xz2 = 0.3684 Z-Side+Handle mz3 = 0.2348 lz3 = 0.19357 Xz3 = 0.48 ates. The elastic constants of X-Springs, Y-Springs and Z-Spring were tested by a series of experiments (See Appendix B) and the final results are summarized in Table 5.4. Table 5.4: The parameters of the springs. Spring Elastic Constant [N/m] Undeformed Length [m] Hooking Positions [m] X-Springs kx = 867 lsx0 = 0.0138 a x = 0.0275, bx = 0.0242 Y-Springs ky = 2011 lsy0 = 0.0280 ay = 0.0686, by = 0.0220 Z-Spring kz = 217 lsz0 = 0.0175 az = 0.0588, bz = 0.0925 5.8.2 Inverse Dynamic Model Simulations In the inverse dynamic model simulation, joint positions, velocities, accelerations are specified, and working space actuating forces are computed, assuming no contact. According to Equation 5.8 Dynamic Model Simulations 88 (5.78), the required actuating forces for a given joint trajectory (q, q and q) are: Fa = J-T(q)[D(q)q + C{q, q)q + G(q) + S(q)\. (5.79) Three simulations were performed. In each of these, only one joint trajectory is specified and driven by a sinusoidal signal at a time, whereas the other two are set to zero. 1. Simulation by driving 9\ only In this simulation, 62 = Oz = 0 and 9\ is specified as: 01 = 0.2593sin(27ri), t = 0 , . . . , 20s. The simulation results are shown in Figure 5.6. 0.5 n 1 1 1 1 1 r 1-•WWVWVWWWWVM 10 12 14 16 18 20 Figure 5.6: Simulation results of driving 0i only. 5.8 Dynamic Model Simulations 89 As Figure 5.6 shows, only actuating forces along the X and Z axes are required to achieve the given trajectory. No actuating force along the Y axis is required. Also, the force generated by the X-Actuator, fax is predominant, compared to the Z-Actuator force, faz. 2. Simulation by driving 92 only In this simulation, 9\ = 63 = 0 and 92 is specified as: 62 = 0.2181sin(27rt), t = 0 , . . . , 20s. The simulation results are shown in Figure 5.7. 0.5 2, 0 -0.5 ~i 1 1 r KAAAAAAAAAAAA^  0 2 4 6 8 10 12 14 16 18 20 1 i 1 1 1 1 1 1 1 - * 0 J l l l l l_ 8 10 12 14 16 18 20 n i i i i i T~ i-MWVVVWVWVWVVW J 1_ 8 10 12 14 16 18 20 8 10 12 14 16 18 20 Figure 5.7: Simulation results of driving 62 only. As Figure 5.7 shows, actuating forces along both Y and Z axes are required to achieve the given trajectory. No actuating force along the X axis is required. Note that the force generated by the Y axis, fay is predominant, compared to the Z-Actuator force, faz. 5.8 Dynamic Model Simulations 90 3. Simulation by driving #3 only In this simulation, 9\ = 62 — 0 and #3 is specified as: 0 3 = 0.4048sin(27ri), t = 0 , . . . , 20s. The simulation results are shown in Figure 5.8. 0 2 4 6 8 10 12 14 16 18 20 0.21 r • i 1 1 1 1 1 1 1 1 T - 1 1 1 — - — i 1 1 1 :—r I 1 I : I I . i I I I i I 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Figure 5.8: Simulation results of driving 0 3 only. As Figure 5.8 shows, actuating forces along both X and Z axes are required to achieve the given trajectory. No actuating force along the Y axis is required. Also, the force generated by the Z-Actuator, faz, is predominant, compared to the X-Actuator force, fax. 5.8 Dynamic Model Simulations 91 5.8.3 Forward Dynamic M o d e l Simulat ions In the forward dynamic model simulation, the actuating forces are specified, and joint positions or end-effector positions are computed, assuming no contact. According to Equation (5.78), for the given actuating force Fa = [fax, fay, faz]T, the joint accelerations are: q = D-\q)[JT{q)Fa - C(q,q)q - G{q) - S{q)]. (5.80) Then, the joint velocities q can be computed by: q = J q(r)dT, initial q(0), (5.81) and the joint positions q are: q = J q(r)dT, initial q{0). (5.82) Three simulations are performed. Only one actuating force is specified as a sinusoidal signal at a time, whereas the other two forces are zero. The initial joint position 17(0) and velocity q(0) are chosen to be zero. 5.8 Dynamic Model Simulations 92 1. Applying actuating force along the X axis only The actuating force generated by the X-Actuator fax is chosen as a sinusoidal input. The simulation results are shown in Figure 5.9. Figure 5.9: Results of forward dynamics simulation for applying the actuat-ing force along the X axis only. As Figure 5.9 shows, applying fax ^ 0 and fay = faz = 0, motions along both X and Z axes are generated. This means that motion along the X axis is coupled with motion along the Z axis. 5.8 Dynamic Model Simulations 93 2. Applying actuating force along the Y axis only The actuating force generated by the Y-Actuator fay is specified as a sinusoidal input. The simulation results are shown in Figure 5.10. 0.5 0 -0.5 0.01 •5- 0 crT-0.01 -0.02 i 0.5 2 0 -0.5 0.5 "D 1 0 -0.5 0 5 10 15 20 25 30 35 40 I I i i 0 5 10 15 20 25 30 35 40 i • 0 5 10 15 20 25 30 35 40 i i i i i i 10 15 20 25 30 35 40 Figure 5.10: Results of forward dynamics simulation for applying the actu-ating force along the Y axis only. As Figure 5.10 shows, a force along the Y axis generates motions along all three axes. This means that motion along the Y axis is coupled with motions along both the X and Z axes. 5.9 Summary 94 3. Applying actuating force along the Z axis only The actuating force generated by the Z-Actuator, faz is specified as a sinusoidal input. The simulation results are shown in Figure 5.11. Figure 5.11: Results of forward dynamics simulation for applying the actu-ating force along the Z axis only. As Figure 5.11 shows, a force applied along the Z axis generates motions along X and Z directions, but not with motion along the Y axis. 5.9 S u m m a r y The derivation of the haptic device kinematics and dynamics have been presented in this chapter. 5.9 Summary 95 The joint variables have been chosen to be three angles: Q\, 92, and 63, formed by the rotation of the three parallelograms. The dynamics have been computed using the Lagrangian formulation, after the potential and kinetic energies of all plates were derived. Two Jacobian matrices were derived in order to map work space actuating and contact forces into corresponding joint torques. Inverse and forward dynamics simulations have been performed to illustrate the relationships between joint trajectories and work space actuating forces. Also, motion coupling issues have been investigated through these simulations. C h a p t e r 6 Conclusions and Recommendations 6.1 C o n c l u s i o n s The work presented in this thesis describes the design of a new haptic device with the following main features: • low component costs (see Appendix D for details), • desktop dimensions - 160mm x 180mm x 260mm, • 3 degree of freedom in Cartesian space, • translation predominantly along each axis, • large motion range of ±25.4mm along each axis, and • force-feedback by actuators for all axes, but the force magnitude needs to be improved. Although the device prototype has not yet been completed, component design, machining, in-stallation and testing, as well as model derivation and simulation have been carried out individually. The thesis contributions and conclusions can be summarized as follows: 1. in the device and component design: 96 6.1 Conclusions 97 • A parallelogram linkage was employed as a basic mechanism for generating displacement predominantly along one axis in Cartesian space. Three parallelogram linkages have been designed and installed such that the X-Parallelogram forms a base on top of which the Y-Parallelogram sits; the Z-Parallelogram in turn, is mounted on top of the Y -Parallelogram. The device uses an arrangement of these orthogonal parallelograms to restrict displacements to three rectangular axes. However, the device motion is not decoupled. • A n arrangement of piano springs was used as a centering mechanism to bring all par-allelograms to the device zero/center position. This position is also used for device calibration. • A slot-and-tab structure was adopted to produce a machinable low friction joint. The width of the slot on one plate should be equal to that of the tab on the other plate, whereas the height of the slot is solely dependent upon the plate thickness and the maximum rotation angle. • A n assembly of the position sensor, grayscale and actuator was integrated for each axis. 2. in the position transducer: • The HOA0149, an affordable and high performance microelectronic infrared reflective sensor, was selected for position sensing. • A grayscale with varying reflectance was generated using M a t l a b ™ for sensing by the HOA0149. • Investigation of the HOA0149 characteristics for grayscale sensing, such as the repeata-bility and linearity, was performed. • A n OSC algorithm was presented to compensate for the sensor's nonlinearity. A flexible scaling factor was introduced in the algorithm to improve the convergence rate. The effectiveness of the algorithm was illustrated and compared by two examples. 3. in the linear actuator: 6.1 Conclusions 98 • A n actuator consisting of a stator and a slider was designed to provide force feedback to the operator. Two rows of fourteen TRI-NEO-30 permanent magnets have been installed on the stator trunk, generating magnetic fields within two long and narrow air gaps. A n Af-turn coil is wound around the slider, which is placed along the center bar. A Lorentz force is generated and exerting upon the slider when a dc current is applied to the winding. • A n equivalent magnetic circuit model was designed to compute the field density along the length of the air gap. • A n experimental platform was built to validate the actuator design. Test results, which closely approximate the theoretical computations, show that the force is a linear function of the applied current and that the magnetic field along the length of the air gap is nearly uniform. Also, the level of actuating force suggests that the current actuator is suitable for generating a force feedback to a finger motion input device rather than a hand-held device. • Further investigation on the effect of change of actuator dimensions has been performed and the corresponding result can be referred to Appendix E . 4. in the device kinematics and dynamics: • The device forward and inverse kinematic equations were derived by separating motion along each axis in Cartesian space. The joint variables were chosen to be three rotational angles and the center of mass for each plate was computed. • The device dynamic equations were derived using the Lagrangian formulation, after the potential and kinetic energies of all plates were computed. The velocity and actuation Jacobian matrices were derived in order to map work space actuating and contact forces into corresponding joint torques. • Simulations on the device inverse and forward dynamic models were performed to illus-trate the relationships between joint trajectories and actuating forces. Motion coupling issues were also investigated and illustrated by the simulations. 6.2 Recommendations for Future Work 99 6.2 R e c o m m e n d a t i o n s for F u t u r e W o r k The component design and validation have been completed in the thesis. However, there is still a lot of work to be done before a workable device prototype is achieved. Recommendations for possible future work are summarized according to (i) device mechanical design, (ii) position transducer and (iii) linear actuator. 1. Device Mechanical Design • The weight of the haptic device should be reduced, while remaining rigidity and stiffness. • The parallelogram linkages as well as slots machined on the main plates should be made by advanced cutting machines rather than by heavy millers, for improved precision. • The slot-and-tab hinge structure could be further improved. The problem with the original design, shown in Figure 2.5, still lies in the friction. When plate A is not rotating, the contact between plates A and B is a surface contact rather than an edge contact. In order to minimize the contact area, both edges of the slot have been sharpened. Thus, plates A and B have an edge contact all the time. Due to the intense labour and because the joint are easily worn out, a new design using pre-made plastic inserts (pieces Si, S2 and 5 3 ) is presented and depicted in Figure 6.1. Plate A Figure 6.1: A new slot-and-tab design schematic. 6.2 Recommendations for Future Work 100 • In order to give the operator a feeling of the interface zero position, a detent mechanism should be added to the mechanical design. A possible design schematic, based on the fact that a mild steel plate is attracted by a permanent magnet, is depicted in Figure 6.2. The distance between the steel plate and magnet, d, can be adjusted to achieve a desirable detent effect. Mild Steel Plate Translating Plate Permanent Magnet Stationary Plate Figure 6.2: The detent mechanism schematic. • The installation of the sensor, grayscale and actuator is still labour intensive, for example, the slider should be aligned without touching the center bar or the permanent magnets. The current installation distance between the slider and the center bar is specified to be O.lin. A better assembly design and alignment technique is required. 2. Position Transducer Design and Compensation • The current sensor testing setup only provides a motion range up to 26.7mm. However, the actual range for the end-effector to move along one axis is specified as ±25.4mm. Thus, this testing setup needs to be expanded to at least accommodate that maximum motion range. • The motion of the translating plate in the parallelogram linkage is illustrated in Fig-ure 2.1(a). It has two orthogonal components, one is parallel (dp) and the other perpen-dicular (dc) to the fixed side. Although the motion along the perpendicular direction is small, compensation of the sensing in that direction should be also performed. A new 6.2 Recommendations for Future Work 101 sensor compensation setup that exactly simulate the motion of the translating plate is required, and the OSC algorithm may be modified accordingly. • Several preliminary tests on the HOA0149 sensor have been performed in the thesis. The size of the sensor infrared light spot on the grayscale has been roughly tested. In order to improve the sensing resolution, special optical instruments are required in the future to precisely determine this size. • During the sensing, the grayscale is fixed while the HOA0149 sensor is moving with the translating plate along one axis in the Cartesian space. No experiment on how fast the sensor would respond to a reflectance change on the grayscale has been carried out. This is of great importance if the position information sensed by the sensor is used for estimating the motion velocity. 3. Linear Actuator Design • The current actuator is still too weak, even though the magnetic field density along the air gap has been increased nearly six times over that of the previous design. In order to generate enough force-feedback to the human hand, the actuating force needs to be increased. Since the Lorentz force is computed by Fa = NIBgLeff, there are four parameters which would affect the force magnitude: (a) the turns of the coil winding, N, determined by the width of the air gap and the thickness of the wire; (b) the applied dc current, / , limited by the maximum current without overheating, for the current wire (AWG33), the maximum current is less than 250mA; (c) the magnetic field density along the air gap, Bg, determined by the layout and dimension of the actuator as well as the choice of permanent magnets; (d) the effective length of the coil winding immersed in the magnetic field, Leff, is dependent upon the dimensions of the permanent magnet and of the actuator. Compared to the previous maglev-based haptic devices [5,13,21,29,32], Bg in our ac-tuator design is almost the same. Thus, a possible solution to increase the actuating 6.2 Recommendations for Future Work 102 force would be (i) selecting an appropriate wire which can afford a high current / , (ii) efficiently packing the wire to achieve a high value of Leff. • We have not got the best materials for the actuator design. Thus a permanent magnet with a high BmaxHmax value, and a soft iron with a high permeability value are still required to generate a high flux density Bg along the air gap. • The dimensions of the stator and slider play an important role in the actuator design. Further investigations on the selection and optimization on the configuration of the stator components are required. In this project, component design and testing have been completed. The next step of the project would be: (i) the improvement of the actuator design. As mentioned previously, there are a few ways to modify the current design. For example, a type of flat wire giving a high packing ratio and conducting a high current could be selected for the slider's coil winding; (ii) the installation of the position transducers into the input device to verify end-effector position sensing and the nonlinearity compensation of the HOA0149 sensor. Bibliography [1] Magellan 3D Controller, User's Manual, Version 4-2. Logitech, Fremont, C A , USA, 1996. [2] Manlio G . Abele. Structures of Permanent Magnets. John Wiley & Sons, Inc, 1993. [3] Bernard D. Adelstein and J . Edward Colgate. Haptic Interfaces for Virtual Environment and Teleoperator Systems - Introduction. In Proceedings of the 1994 ASME Winter Annual Meeting, Dynamic Systems and Control Division, page 263, November 1994. [4] Amitava Basak. Permanent-Magnet DC Linear Motors. Clarendon Press, 1996. [5] P.J . Berkelman, R . L . Hollis, and D. Baraff. Interaction with a Real Time Dynamic Environ-ment Simulation Using a Magnetic Levitation Haptic Interface Device. In Proceedings of 1999 IEEE International Conference on Robotics and Automation, pages 3261-3266, 1999. [6] P.J. Berkelman, R . L . Hollis, and S.E. Salcudean. Interacting with Virtual Environments us-ing a Magnetic Levitation Haptic Interface. In Proceedings of 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems 95. 'Human Robot Interaction and Cooperative Robots', pages 117-122, 1995. [7] Peter Campbell. Permanent Magnet Materials and Their Application. Cambridge University Press, 1994. [8] Jacek F. Gieras and Mitchell Wing. Permanent Magnet Motor Technology. Marcel Dekker, Inc, 1997. [9] Duane C. Hanselman. Brushless Permanent-Magnet Motor Design. McGraw-Hill , Inc., 1994. [10] Vincent Hayward. Toward a Seven Axis Haptic Device. In Proceedings of 1995 IEEE/RSJ International Conference on Intelligent Robotics and Systems 95. 'Human Robot Interaction and Cooperation Robots', pages 133-139, 1995. [11] Vincent Hayward, Jehangir Choksi, Lanvin Gonzalo, and Christophe Ramstein. Design and Multi-Objective Optimization of a Linkage for a Haptic Interface. In Proceedings of the ARK94, 4th Intl. Workshop on Advances in Robot Kinematics, Ljubliana, Slovenia, June 1994. [12] Vincent Hayward and Astley Oliver R. Performance Measures for Haptic Interfaces. In G. G i -ralt and G . Hirzinger, editors, The 7th International Symposium on Robotics Research ISRR-7, pages 195-207. Springer Verlag, 1996. 103 BIBLIOGRAPHY 104 [13] R . L . Hollis, S.E. Salcudean, and A.P . Allan. A Six-Degree-of-Freedom Magnetically Levitated Variable Compliance Fine-Motion Wrist: Design, Modeling, and Control. IEEE Transactions on Robotics and Automation, 7(3):320-332, June 1991. [14] R . L . Hollis and Salcudean S.E. Lorentz Levitation Technology: a New Approach to Fine Motion Robotics, Teleoperation, Haptic Interfaces, and Vibration Isolation. In Proceedings of the 5th International Symposium on Robotics Research, Hidden Valley, P A , USA, October 1993. [15] M . Ishii, M . Nakata, and M . Sato. Networked SPIDAR: A Networked Virtual Environment with Visual, Auditory, and Haptic Interactions. PRESENCE, 3(4):351-359, 1994. [16] Masahiro Ishii and Makoto Sato. A 3D Interface Device with Force Feedback: A Virtual Work Space for Pick-and-Place Tasks. In Proceedings of the 1993 IEEE Annual International Symposium on Virtual Reality, pages 331-335, 1993. [17] Hiroo Iwata and Haruo Noma. Volume Haptization. In Proceedings of the 1993 IEEE Sympo-sium on Research Frontiers in Virtual Reality, pages 16-23, 1993. [18] Berkelman P. J . Tool-Based Haptic Interaction with Dynamic Physical Simulations using Lorentz Magnetic Levitation. PhD thesis, Carnegie Mellon University, June 1999. [19] David Kahaher. Special Report: Virtual Reality in Japan. IEEE Micro, 13(2):66-73, Apr i l 1993. [20] H . Kazerooni. Human/Robot Interaction via the Transfer of Power and Information Signals. In Proceedings of 1989 IEEE Engineering in Medicine and Biology Society 11th Annual Inter-national Conference, pages 908-909, 1989. [21] A . J . Kelley and S.E. Salcudean. On the Development of A Force-Feedback Mouse and Its Integration into A Graphical User Interface. In Proceedings of the 1994 ASME Winter Annual Meeting, Dynamic Systems and Control Division, pages 287-294, November 1994. [22] Tetsuo Kotoku, Kiyoshi Komoriya, and Kazuo Tanie. A Force Display System for Virtual Environments and Its Evaluation. In Proceedings of IEEE International Workshop on Robot and Huamn Communication, pages 246-251, 1992. [23] Gregory L . Long and Curtis L . Collins. A Pantograph Linkage Parallel Platform Master Hand Controller for Force-Reflection. In Proceedings of the 1992 IEEE International Conference on Robotics and Automation, pages 390-395, 1992. [24] Thomas H . Massie and J . Kenneth Salisbury. The P H A N T o M Haptic Interface: A Device for Probing Virtual Objects. In Proceedings of the 1994 ASME Winter Annual Meeting, Symposium on Haptic Interfaces for Virtual Environment and Teleoperator System, pages 295-299, Chicago, Illinois, USA, November 1994. [25] Nobuto Matsuhira, Makoto Asakura, Hiroyuki Bamba, and Michihiro Uenohara. Development of an Advanced Master-Slave Manipulator Using a Pantograph Master A r m and a Redundant BIBLIOGRAPHY 105 Slave Arm. In Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 1653-1658, Yokohama, Japan, July 1993. [26] Andrew B . Mor. 5 D O F Force Feedback Using the 3 D O F Phantom and 2 D O F Device. Technical Report 1643, M I T , Cambridge, Massachusetts, August 1998. [27] Rollin J . Parker. Advances in Permanent Magnetism. John Wiley Sz Sons, Inc, 1990. [28] Masamichi Sakaguchi and Junji Furusho. Development of a 2 D O F Force Display System Using E R Actuators. In Proceedings of the 1999 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pages 707-712, Atlanta, U S A , 1999. [29] S.E. Salcudean and N.R. Parker. 6-DOF Desktop Voice-Coil Joystick. In Proceedings of the 6th Annual Symposium on Haptic Interfaces for Virtual Environments and Teleoperation Sys-tems, International Mechanical Engineering Congress and Expositions, (ASME Winter Annual Meeting), pages 131-138, Dallas, Texas, USA, November 1997. [30] S.E. Salcudean, R. Six, R. Barman, S. Kingdon, I. Chau, D. Murray, and M . Steenburgh. Control Electronics and Hybrid Dynamic System-Based A P I for A 6-DOF Desktop Haptic Interface. In Proceedings of the ASME 1999, Dynamic Systems and Control Division - 1999, pages 407-413, 1999. [31] S.E. Salcudean and L . Stocco. Isotropy and Actuator Optimization in Haptic Interface Design. In Proceedings of the 2000 IEEE International Conference on Robotics and Automation, pages 763-769, San Francisco, C A , USA, Apr i l 2000. [32] S.E. Salcudean, N . M . Wong, and R . L . Hollis. Design and Control of a Force-Reflecting Tele-operation System with Magnetically Levitated Master and Wrist. 1995 IEEE Transactions on Robotics and Automation, ll(6):844-858, December 1995. [33] Lorenzo Sciavicco and Bruno Siciliano. Modeling and Control of Robot Manipulators. The McGraw-Hill Companies, Inc, 1996. [34] Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits. Saunders College Publishing, 1991. [35] Mohammad R. Sirouspous, S.P. DiMaio, S.E. Salcudean, P. Abolmaesumi, and C. Jones. Haptic Interface Control - Design Issues and Experiments with a Planar Device. In Proceedings of the 2000 IEEE International Conference on Robotics and Automation, pages 789-794, San Francisco, C A , USA, Apr i l 2000. [36] G.R. Slemon and A . Straughen. Electric Machines. Addision-Wesley Publishing Company, 1980. [37] Mark W . Spong and M . Vidyasagar. Robot Dynamics & Control. John Wiley & Sons, Inc, 1989. [38] Somsak Walairacht, Yasuharu Koike, and Makoto Sato. String-based Haptic Interface Device for Multi-fingers. In Proceedings of the 2000 IEEE Virtual Reality, page 293, 2000. BIBLIOGRAPHY 106 [39] S.K. Yeung and E . M . Petriu. A Haptic Device with Collision Detection Response For Sculpt-ing Virtual Objects. In Instrumentation and Measurement Technology Conference, 1997. IMTC/97. Proceedings of IEEE Sensing, Processing, Networking, pages 322-328, 1997. [40] Shumin Zhai. Human Performance in Six Degree of Freedom Input Control. PhD thesis, University of Toronto, 1995. [41] Shumin Zhai. User Performance in Relation to 3D Input Device Design. Computer Graphics, 32(4):50-54, November 1998. A p p e n d i x A The Actuator Design Computations Kirchhoff's law is applied at each labeled node: 1,3, . . . , 13, 29, 30, . . . , 35, 43, 44, . . . , 50, 59, 60, . . . , 65, 73, 75, . . . , 85 in Figure 4.10. The fluxes flowing into and out of the node are summed to zero and the following equations are obtained: 1. At node 1: F l ~ F " + ^ + + = 0. (A-l) Ril Rml Ri2 2. At nodes (3, 5, 7, 9, 11): ^2fc+l - F2k-1 , F2fc+l --Ffc+29 F2k+1 - F2k+3 n , 1 _ 0 s 5 h <Pr,fc+i + 5 H 5 = 0, k = 1,... , 5. (A-2) 3. At node 13: ^ V ^ + ^ 7 + ^ F ^ = 0. (A-3) 4. At nodes (29, 30, 31, 32, 33, 34, 35): fcH 5 r- 5 = 0 , fc = l , . . . , 7 . (A-4) 107 108 5. At node 43: . • Fi - F43 F44 - F43 F73 - F43 • — P + — 5 + — 5 = °- ( A " 5 ) 6. At nodes (44, 45, 46, 47, 48, 49): -ffc+42 — -ffc+43 . Fk+28 — -ffc+43 . -ffc+44 ~ -Ffc+43 . Fk+58 ~ -ffc+43 n 7 1 c 5 + 5 + 5 + 5 = 0 ' « = ! , • • • , 6. -n-i,fc+30 Mg,k *H,k+31 -ttg,29-fc (A-6) 7. At node 50: F49 ~ -^ 50 -^ 35 ~ -^ 50 ^65 ~ -^ 50 _ n (K T\ p R P — U. (A-f J -Ki37 -f<cj7 -n-c722 8. At nodes (59, 60, 61, 62, 63, 64, 65): -Ffc+43 - -ffc+58 . , . J2/C+71 - -Pfc+58 n , 1 7 / . Q s + <Pr,29-fc H 5 =0 , K = 1, . . . ,7. (A-8) •n-g,29-fc ^m,29-fc 9. At node 73: J 7 3 - ^ 4 3 , , , -P73 - -P59 -P73 - -F75 n , . m — 5 + 0r28 + — 5 + — 5 0. (A-9) -n-i30 -n-m28 -<M29 10. At nodes (75, 77, 79, 81, 83): ^2fc+73 - -£2fc+71 , , , ^2fc+73 ~ -Pfc+59 . -^2^+73 ~ F2k+75 n , . _ , A 1 n , 5 + <Pr,28-k + 5 + 5 — =0 , k = 1, . . . , 5. (A-10) lH,30-k -Km,28-fc -rti,29-fc 11. A t node 85: + ^ + ^ p ^ = 0 . • ( A - l l ) ^24 -Km22 Among these equations, only Fi, ..., F13, F2g, • • •, F35, F43, ..., F50, F73, . . . , F 8 5 are unknown mmfs, while the permanent magnetic flux and reluctances can be computed using the magnet, air 109 gap and soft iron dimensions listed in Table A - l . Table A - l : Parameters for the actuator design computations. Symbol Definition Value <t>r,k ke[l,28] the flux of each permanent magnet 59.3547 x 10-6[Wb] Rmtk ke[l,28] the reluctance of each magnet 34.4144 x 106[A/Wb] Rg,k ke[l,28] the reluctance of each air gap segment 38.5454 x 106[A/Wb] Ro the reluctance of the soft iron segment 0.01156 x 106[A/Wb] Ri,l = Ri,30 the reluctance of the iron segment 0.06963 x 106[A/Wb] Ri,k k G [2,7] the reluctance of the iron segment 0.02723 x 106[A/Wb] Ritk k G [24,29] the reluctance of the iron segment 0.02723 x 106[A/Wb] Ri,k k G [32,37] the reluctance of the iron segment 0.02723 x 106[A/Wb] Ri,31 the reluctance of the iron segment 0.01940 x 106[A/Wb] Therefore, equations (A- l ) , (A-2), . . . , and ( A - l l ) can be lumped into: PF = (A-12) where P is a 36-by-36 coefficient matrix of inverses of reluctances, <fr is a 36-by-l vector of magnetic fluxes, and F is a 36-by-l vector of unknown mmfs. Then, F results as: F = P _ 1 $ . (A-13) The magnetic flux for each air gap segment shown in Figure ?? can be computed by: ^ 2 8 + f c ~ J < 4 3 + f c Rg,k+Ro ' Fa7-k— Fr2-k Rg,21+k+R<) ' and the corresponding flux density is: Bg,k = <ftg,fc Aa fc = l , . . . ,7, k = 22 , . . . ,28, (A-14) , jfe = 1,... ,7 and k = 22 , . . . ,28. (A-15) A p p e n d i x B Experimentally Determined Spring Constants In order to simulate device motion in Cartesian space, the spring constants for the X-Springs kx, Y-Springs ky and Z-Spring kz need be determined. A schematic of a simple experimental setup is illustrated in Figure B . l . Stopper Tested Spring Pulley Known Weight Figure B . l : The spring constant experiment setup. The spring that is to be tested, is hooked such that one end is attached to a fixed wooden block while the other end is attached to a known weight over an almost frictionless pulley. By changing 110 I l l the weight (force), the spring length changes. The slope of the force-deformation characteristic is the spring constant. L Spring constant for X-Springs kx Two experiments were performed to determine the X-Springs' constants. The experimental force-deformation dependency is plotted in Figure B.2. Matlab's polyfit command is used to fit the experimental data and to compute the slope. 2 4 6 8 1 0 1 2 1 4 1 6 2 4 6 8 1 0 1 2 1 4 1 6 D e f o r m a t i o n [m] x 1 0 ~ 3 D e f o r m a t i o n [m] x 1 Q (a) Test 1: Force versus X-Spring Deformation. (b) Test 2: Force versus X-Spring Deformation. Figure B.2: Experimental result for the X-Spring constant. The slope of the line in Figure B.2(a) is 432.2N/m, whereas the slope of the line in Figure B.2(b) is 434.8N/m. Thus, the average value of them is 433.5N/m. 2. Spring constant for Y-Springs ky Two experiments were performed to determine the Y-Springs' constants. Figure B.3 shows the measured force-deformation relationship. The slope is 997.6N/m in Figure B.3(a) and 1013.4N/m in Figure B.3(b). Thus, the average value of them is 1005.5N/m. 112 3. Spring constant for Z-Springs kz Two experiments were performed to determine the Z-Springs' constants. The measured force-deformation relationship is plotted in Figure B.4. 5 . 5 5 4 . 5 Z. 4 o £ t i . ' 3 2 . 5 2 • O r i g i n a l L i n e - f i t -i : 0.01 0 . 0 1 5 0 . 0 2 0 . 0 2 5 0 . 0 3 0 . 0 3 5 0 . 0 4 0 . 0 4 5 D e f o r m a t i o n [ m l 5 . 5 5 4 . 5 z 4 o £ 3 . 5 3 2 . 5 2 1.5-• O r i g i n a l L i n e - f i t I 1 I ... 0.01 0 . 0 1 5 0 . 0 2 0 . 0 2 5 0 . 0 3 0 . 0 3 5 0 . 0 4 0 . 0 4 5 D e f o r m a t i o n [ m l (a) Test 1: Force versus Z-Spring Deformation. (b) Test 2: Force versus Z-Spring Deformation. Figure B.4: Experimental result for the Z-Spring constant. 113 The slope is 108.2N/m in Figure B.4(a) and 108.8N/m in Figure B.4(b). Thus, the average of them is 108.5N/m. According to the mechanical design, each X-Spring, Y-Spring and Z-Spring consists of two identical springs in parallel. Thus, the resulting spring constants are double the values obtained in the experiments: kx = 433.5 * 2 = 867 N / m , ky = 1005.5 * 2 = 2011 N / m , (B-1) kz = 108.5*2 = 217 N / m . A p p e n d i x C The Installation of Actuators Figure C l shows the X-, Y- and Z-Actuators installed onto the X-, Y- and Z-Parallelograms. X-Parau eiosram Z- Par all el ogi am Z-Actuator Y-Actuator X-Actuator Y-Pai'all rlogram Figure C l : The X-, Y- and Z-Parallelograms and X-, Y- and Z-Actuators. 114 A p p e n d i x D The Component Cost Breakdown The costs of components used in building this new haptic device are summarized in Table A - l . Table A - l : The component cost Dreakdown. Item Unit Price (C$) Quantity Cost (C$) Steel, aluminum sheets for motion plates, actuators, case, etc. 20.00 Piano springs for centering mechanism 1.00 7 7.00 Position sensors HOA0149 1.00 3 3.00 Permanent magnets NdFeB (27MGOe) 1.00 84 84.00 AWG33 Copper wire 3.00 Total cost: 117.00 115 Appendix E The Effect of Change of Actuator Dimensions The effect of change of actuator dimensions is investigated in this appendix. Particularly, the change of actuator dimension is specified as the change of the motion range of slider. According to actuator design, the motion range of slider is solely dependent upon the number of permanent magnets per row. This can be depicted in Figure E . l . For example, the slider motion range for a configuration of fourteen magnets per row is 71.12mm. Permanent Magnet Row r~n o • • I ' ] o 5 o o I I Figure E . l : The actuator and corresponding permanent magnet row. 116 117 Thus, referring to the flux source equivalent model discussed before, a simulation program has been developed in Matlab™ for computing the magnetic field density along the air gap for various numbers of magnets per row. The resultant field density distribution for different numbers of magnets per row is shown in Figure E.2. Note that only one measurement is taken for each magnet and displacements are computed with respect to the actuator center line. Flux Density Distribution for 1 to 20 PMs per Row 0.551 1 1 1 1 1 1 1 1 r 0.545 -0.5051 1 1 1 1 1 1 1 1 1 1 -50 -40 -30 -20 -10 0 10 20 30 40 50 Displacements from the Actuator Center [mm] Figure E.2: Magnetic field density distribution along the air gap for different number of magnets per row. As Figure E.2 shows, the maximum field density is achieved when there is only one permanent magnet per row. As the number of the magnets per row increases, the field density along the air gap decreases. The outer curve in Figure E.2 corresponds to a configuration of twenty permanent magnets per row. 

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