Università degli Studi di Pisa
Scuola Superiore Sant’Anna
Master of Science in Embedded Computing Systems
Software control architecture for a
gearshift haptic simulator
Candidate: Luca Fancellu
Supervisors: Prof. Antonio Frisoli Ing. Massimiliano Solazzi Ing. Domenico Chiaradia
Abstract xi
1 Motivation and objectives 1
2 Introduction 3
2.1 Haptic term . . . 3
2.2 Theory of permanent magnet DC motor . . . 6
2.2.1 Magnetic field and current relation . . . 6
2.2.2 Introduction to Brushless DC motor . . . 12
2.2.3 Torque control . . . 14
2.3 Haptic interface control strategies . . . 15
2.4 The Haptic interface: gearshift simulator . . . 17
2.5 Mechanical description of a car gearbox transmission . . . 19
3 State of the art 25 3.1 Force reflecting haptic devices . . . 25
3.1.1 Virtual wall . . . 25
3.1.2 Passivity . . . 27
3.1.3 Time Domain Passivity Approach TDPA . . . 30
3.1.4 Memory Based Passivition Approach MBPA . . . 31
3.2 A virtual gearshift application . . . 32
3.3 The gear engagement process in manual transmission . . . 33
3.4 Haptic gearshift research work . . . 35
4 System setup 39 4.1 Hardware . . . 39
4.1.1 First simulator . . . 39
4.1.1.1 Hardware components . . . 40
CONTENTS iii
4.1.1.2 Dynamics characterization . . . 45
4.1.1.3 Viscous-stiffness characterization . . . 46
4.1.1.4 Limitations . . . 47
4.1.2 Second simulator . . . 48
4.1.2.1 Evaluation of best commercial motor . . . 48
4.1.2.2 Hardware components . . . 50
4.2 Software and low level control . . . 54
4.2.1 First simulator low level control . . . 54
4.2.2 Second simulator low level control . . . 57
4.2.2.1 CANOpen over EtherCAT profile . . . 59
4.2.2.2 Master-slave EtherCAT communication state machine . . 61
4.2.2.3 CANOpen state machine DS402 . . . 62
4.2.2.4 Torque ripple . . . 64
4.2.2.5 Velocity signal . . . 68
4.2.3 Motor driver torque reference characterization . . . 74
5 Simulation and control logic modeling 77 5.1 Gearshift model configuration . . . 77
5.2 Relation between force curves and gearshift components . . . 80
5.3 Analysis of static engagement gearshift curve . . . 82
5.3.1 Force curve structure . . . 82
5.3.2 Force analysis and specifications . . . 84
5.4 Analysis of selection gearshift curve . . . 87
5.4.1 Force curve structure . . . 88
5.4.2 Force analysis and specifications . . . 95
5.5 Curve fitting methods . . . 96
5.5.1 Piecewise linear functions . . . 97
5.5.2 Piecewise cubic spline . . . 98
5.5.3 Chosen method and motivations . . . 99
5.6 Control law for profile rendering . . . 99
5.7 Haptic device control stability . . . 99
5.7.1 Fixed dumping coefficient . . . 100
5.7.2 Adaptive dumping coefficient . . . 100
5.7.3 Modified memory based passivation approach . . . 100
6.1 Automatic piecewise linear fitting . . . 107
6.1.1 Best fit point number . . . 110
6.2 Extraction of gearshift simulator parameters . . . 111
6.2.1 Engagement curve profiles . . . 111
6.2.1.1 Fundamental parameters . . . 112
6.2.1.2 Automatic extraction . . . 113
6.2.2 Selection curve profiles . . . 116
6.2.2.1 Fundamental parameters . . . 116
6.2.2.2 Automatic extraction . . . 119
6.2.3 Virtual grid . . . 123
6.2.3.1 Fundamental parameters . . . 123
6.2.3.2 Automatic extraction . . . 124
6.3 High level control structure . . . 125
6.3.1 Engagement axis state machine . . . 126
6.3.2 Selection axis state machine . . . 128
6.3.3 Selected gear method . . . 129
6.3.4 Force control structure for the engagement axis . . . 132
6.3.5 Force control structure for the selection axis . . . 134
6.3.6 Fixed dumping implementation for stability control . . . 138
6.3.7 Adaptive dumping implementation for stability control . . . 138
6.3.8 Modified memory based passivation implementation . . . 139
7 Results 141 7.1 Fixed dumping modeling results . . . 141
7.2 Adaptive dumping modeling results . . . 142
7.3 Modified memory based approach results . . . 142
8 Conclusion 151
8.1 Future works . . . 151
List of Tables
4.1 Elmo Violin 10/100 specification. . . 424.2 Maxon RE motor specification. . . 43
4.3 TMS320F28335 main specification. . . 44
4.4 Kollmorgen driver specification. . . 52
4.5 Kollmorgen AKM23 motor specification. . . 53
4.6 Kollmorgen CoE fixed PDO mappings. . . 59
4.7 Kollmorgen CoE fixed PDO mappings. . . 60
4.8 Ethercat allowed protocol for each state. . . 61
4.9 Ethercat allowed protocol for each state. . . 62
4.10 Ethercat allowed protocol for each state. . . 63
4.11 Status word bit and DS402 state relation. . . 64
4.12 Status word bit and DS402 state relation. . . 64
4.13 Characterization of torque input value coefficient. . . 76
5.1 Static engagement gear features. . . 87
5.2 Static engagement gear axis specification. . . 87
5.3 Selection axis features. . . 96
5.4 Selection axis specification. . . 96
5.5 Viscous coefficient for each engaged gear. . . 105
6.1 Best fit point number for each gearshift. . . 110
6.2 Description of the statex signal value. . . 133
2.1 User interaction with the haptic gearshift simulator . . . 5
2.2 DC motor components . . . 6
2.3 The Lorentz force effects on the windings . . . 8
2.4 Torque as theta changes . . . 9
2.5 Torque ripple. . . 9
2.6 Typical brushless motor structure. . . 12
2.7 Control strategies for haptic interfaces. . . 15
2.8 The gearshift lever . . . 17
2.9 The gearshift simulator kinematic scheme. . . 18
2.10 Two types of unsynchronized manual gearbox transmission. . . 20
2.11 The synchromesh gearbox. . . 21
2.12 Details about components with synchronizer. . . 22
2.13 Selector forks mechanism. . . 23
2.14 Forks interlocking device. . . 24
2.15 Stop device for the rod. . . 24
3.1 Common system setup for a virtual wall. (Colgate et al., 1993) . . . 26
3.2 Hybrid loop (continuous/discrete). . . 27
3.3 Virtual spring force/position plot. (Colgate et. al., 1994) . . . 28
3.4 Condition to achieve passivity. . . 29
3.5 Schematic network representation of the haptic system with a passivity controller (the block labeled α). . . . 30
3.6 The position versus force graph (command force in dotted line) of an interaction with a spring-like VE (solid line). . . 31
3.7 Acquired position and force vs time. (Frisoli et al. 2001) . . . 33
3.8 Force vs position plot of entire maneuver. (Frisoli et al. 2001) . . . 34
3.9 Garcia–Canseco et al. research on gearshift. . . 35
LIST OF FIGURES vii
3.10 Frisoli, Avizzano, Bergamasco gearshift simulator setup. . . 36
3.11 The Wingman Force Feedback Joystick . . . 36
3.12 Shift-pattern of the automatic gearshift (Chiaradia et al., 2016) . . . 37
3.13 Force profile with hysteresis behavior (Chiaradia et al., 2016) . . . 38
4.1 First simulator mechanic part. . . 40
4.2 Details about first simulator joint pulley. . . 41
4.3 The Elmo Violin current amplifier. . . 41
4.4 Maxon DC motor. . . 42
4.5 Encoder HEDL 5540. . . 43
4.6 Texas Instrument Delfino TMS320F28335. . . 45
4.7 Dynamics characterization - Bode plot of magnitude. . . 46
4.8 Dynamics characterization - Bode plot of phase. . . 46
4.9 Viscous-stiffness characterization first simulator. . . 46
4.10 Force rendering comparison between Kollmorgen and Maxon engine. Dy-namic gear engagement. . . 49
4.11 Force rendering comparison between Kollmorgen and Maxon engine. Static gear engagement. . . 49
4.12 Second simulator mechanical part. . . 50
4.13 Details about second simulator joint pulley. . . 51
4.14 The Kollmorgen AKD driver. . . 51
4.15 The Kollmorgen AKM23 motor. . . 52
4.16 Controller computer for the application. . . 53
4.17 Interconnection between controller pc and electronics hardware. . . 54
4.18 First simulator. Blocks that implements the communication logic. . . 55
4.19 First simulator. Kinematic block. . . 56
4.20 First simulator. Encoder block. . . 56
4.21 First simulator. Gravity compensation block. . . 56
4.22 First simulator. Actuation block. . . 57
4.23 The Ethernet IEEE802.3 frame carrying EtherCAT frame. . . 58
4.24 The EtherCAT state machine. . . 62
4.25 The EtherCAT state machine. . . 63
4.26 Velocity loop to obtain ripple torque. . . 66
4.27 Position control loop parameters tuning. . . 66
4.28 Ripple torque recorded. Plot position-ripple. . . 67
4.30 Encoder noise. . . 68
4.31 Velocity signal issues. . . 69
4.32 Position signal visual inspection. . . 70
4.33 Velocity antispike filter applied. . . 72
4.34 Velocity signal filtered with FOAW. Window size [20,25] . . . 73
4.35 Characterization setup. . . 74
4.36 Characterization PD control loop. . . 75
4.37 Torque trend to counteract a mass of 50 grams. . . 76
5.1 The C510 gearbox configuration. . . 78
5.2 The C514 gearbox configuration. . . 78
5.3 The 635 gearbox configuration. . . 79
5.4 Position-Force plot of a first gear static engagement. . . 80
5.5 Position-Force plot of a first gear dynamic engagement. . . 81
5.6 Position-Force 3d plot of a first gear dynamic engagement. . . 82
5.7 Position-Force 3d plot of C635 static gear engagement. . . 84
5.8 Position-Force plot of third gear static engagement. . . 85
5.9 lever length and axis stroke relation. . . 86
5.10 Position-Force 3d plot of C635 selection axis. . . 88
5.11 Position-Force plot of C635 crossgate phase. . . 89
5.12 First split method for force curve profile in the crossgate phase. . . 90
5.13 Second split method for force curve profile in the crossgate phase. . . 91
5.14 Compare between crossgate N and crossgate R on a C635 gearbox. . . 92
5.15 Engaged force curve profile. . . 93
5.16 Plot of first engagement selection forces with backlash profile forces. . . . 93
5.17 Engagement force curve profile. . . 94
5.18 Control law for profile rendering. . . 99
5.19 The position versus force graph (command force in dotted line) of an interaction with a virtual wall (solid line) when a M-MBPA is used. . . . 101
5.20 Engagement force curve profile and velocity. . . 102
5.21 Example of data points representing force profile and velocity profile. . . . 103
5.22 Extension of force/velocity profile to the same position domain. . . 104
5.23 Delta velocity - delta force plot with linear fitting. . . 104
5.24 First-second engagement force curve profile without friction. . . 105
6.1 Matlab GUI for the engagement profile fitting. . . 108
6.2 Experimental results for best number of knot for the fitting function. . . . 111
6.3 The static engagement curve profile divided in phases. . . 112
6.4 Position-time during static engagement. . . 113
6.5 Position-time during static engagement, finding stable point. . . 114
6.6 Position-force during static engagement, stable points. . . 115
6.7 Selection force profiles. . . 116
6.8 Position-force plot of crossgate registration. . . 119
6.9 Crossgate position trend over time. . . 120
6.10 Crossgate position-force plot of one repetition. . . 121
6.11 Stable position points in the engaged selection phase. . . 122
6.12 Virtual grid parameters extraction from gearbox shiftpattern. . . 123
6.13 Engagement axis state machine. . . 126
6.14 Selection axis state machine. . . 128
6.15 Force control structure blocks for the engagement axis. . . 132
6.16 Force control structure block of one gear. . . 133
6.17 Force control structure for the selection axis. . . 134
6.18 Force control structure for the crossgate selection region. . . 136
6.19 Crossgate selection region block internal structure. . . 137
6.20 The M-MBPA algorithm. . . 140
7.3 Rendering of a manual gearshift maneuver using M-MBPA to guarantee the physical stop contact stability. . . 144
7.1 Fixed dumping results. Engagement axis. . . 145
7.2 Fixed dumping results. Selection axis. . . 146
7.4 Variable dumping results. Selection axis. . . 146
7.5 Variable dumping results. Engagement axis. . . 147
7.6 Instable contact with the virtual wall using virtual coupling. . . 148
7.7 Comparison between the TDPA, the proposed M-MBPA and the classical damping-based method during a contact with a virtual wall. . . 149
7.8 Comparison between the TDPA, the proposed M-MBPA and the classical damping-based method during a contact with a virtual wall. . . 150
Abstract
The work done for this thesis had the purpose of realizing an haptic simulator for a car gearshift whose aim was to replicate the real force profile behavior of several commercial cars mounting different gearbox models. The force profiles were obtained recording the forces of real gearshift engagement on those cars. The process has involved different phases, the choice of electronic components for the mechanical interface and the design of the software control architecture. A first analysis of the recorded forces was performed to draw the specification for the hardware components. A second analysis of the data led to the drafting of a model for the simulation and the creation of algorithms to process the force profiles. After that the high level control software for the force profile simulation was developed using the hardware at our disposal, despite the fact that it cannot fit all the specifications this choice has speeded up the development process. The high level application’s work has involved the study of some physical phenomena that can ruin the operator’s feeling, some control strategy were performed to overcome those issues and the results were compared against the recorded profiles to ensure the match between force recorded and force rendered by the interface.
Once obtained the new hardware the development of a low level software control interface was started, highlighting some limitations that have been overcome creating some filter algorithms. The low level software interface will be used in future works with the high level software application created for the simulation.
Chapter 1
Motivation and objectives
Automotive manufacturers are under pressure to reduce time to market and optimize products to higher levels of performance and reliability. A much higher number of components are being developed in the form of virtual prototypes in which engineer-ing simulation software are used to predict performance prior to constructengineer-ing physical prototypes. Engineers can quickly explore the performance of thousands of design al-ternatives without investing the time and money required to build physical prototypes. The ability to explore a wide range of design alternatives leads to improvements in per-formance and design quality. Yet the time required to bring the product to market is usually reduced substantially because virtual prototypes can be produced much faster than physical prototypes. The basic components of product design cycle are design, drafting and validation. The design, drafting are electronically generated and validation is both electronic and physical building of a product. The development cycle is time consuming process, mainly due to conceptual and prototyping stages. These two activ-ities are time consuming and play a vital role in product development cycle.
For those reasons the group FCA (Fiat Chrysler Automobiles), that is a leader in the automotive market, is pushing the development of a virtual prototyping environment. This project aims to realize an haptic simulator for manual gearshift that can give hints or conduct ergonomic studies with human in the loop.
Improvement of gearshift quality of manual transmissions has become more prevalent over the past few years as the refinement of other vehicle to driver interfaces has in-creased. Gearshift quality is made up of several different areas, including gate definition, shift effort, second load, and vehicle response. Gate definition requires careful design of both the internal and external selector mechanism, paying attention to the compliance, static load, backlash, friction, cross gate loads and gear positions. Shift effort requires
careful sizing of the synchronizer cones, cone angle, friction material and balancing this with the transmission reflected inertia, clutch inertia and transmission drag.
Since the engineers knows which parameters could be changed to realize a better final user experience, this simulator is a challenging and innovative project that can transform the developing process related to the car design.
The realization of an haptic interface is always a challenging engineering process, it comprise a union of numerous field like robotics, electronics, computer science and more over the UX science (User eXperience science).
The main objective of this work is the replication of the gearshift force feedback behavior of multiple gearshift models. As a starting point we have at our disposal, thanks to the CRF (Centro Ricerche FIAT) and FCA (Fiat Chrysler Automobiles), the experimental data measured on many commercial cars. Those data comprises the forces and positions measured at the lever knob of all gearshift maneuver, including static and dynamic en-gagement. The difference between the two is that the engagement process is performed with the engine switched off for the first, instead for the dynamic engagement, the engine is on and the wheels are rotating at a certain speed. This difference involves different behavior in the force response of the system.
Here an accurate project planning has led to a considerable time reduction of the de-velopment process, the activities involved in this work included several team and were drawn to realize different objectives:
• Analysis of experimental data to realize a simulation model.
• Drawing of the specification for mechanical and electrical components for the haptic interface.
• Implementation of the software control architecture that realizes the simulation of static engagement force profiles.
• Analysis of several different techniques for the haptic force feedback stability. • Validation of the experimental results.
This work is only a starting point for new projects that could realize the simulation of many gearshift behavior. The software architecture is written taking in consideration future improvements. It opens the door for the hardware in the loop concept, where a mathematical model of a gearshift could drive the haptic interface giving more power to the design process of this car component.
Chapter 2
Introduction
Humans explore objects in many steps. Typically, we first visually scan the environment with our eyes to locate the position of the object. We then touch the object to feel its general shape. Finally, a more careful manual exploration is made to investigate the surface and material characteristics of the object.
2.1
Haptic term
The term ”haptic” comes from a Greek word hapteshtai and it refers to the sense of touch. The human ability to recognize objects simply by touching them can be categorized into two group: tactile and kinesthetic. those cues are received throughout the body.
• Tactile: it is related to the touch or to the sense of touch. The sense by which pressure or traction acting on the skin is recognized as touch. The sense by which the properties of bodies are determined by contact. For example when feeling the roughness of a surface.
• Kinesthetic: it is related to the sense of perception of a movement. Often used to describe the sensory system in the human body that records limb positions, forces and joint torques. Examples of kinesthetic cues would be contours, shapes, and sensations like the weight of the glass filled with water, the resistance of the piano keys, or the impact of hitting a tennis racquet’s sweet spot.
Manual sensing of shape, softness, and texture of an object occurs through tactile and kinesthetic sensory systems that respond to spatial and temporal distribution of forces on the hand. Recent advances in virtual reality and robotics enable the human tactual
system to be stimulated in a controlled manner through force feedback devices, also known as haptic interfaces.
A haptic interface is a device that enables manual interaction with virtual environments or teleoperated remote systems. They are employed for tasks that are usually per-formed using hands in the real world. Force-reflecting haptic devices generate computer-controlled forces to convey to the user a sense of natural feel of the virtual environment and objects within it. In this regard, haptic force rendering can be defined as the process of displaying computer controlled forces on the device to make the user sense the tactual feel of virtual objects. The general process of communicating information back to the user using force signals is referred to as haptic feedback.
How can an haptic interface make the user feel the sense of touch? This kind of interfaces are formed by links and joints which are partially or totally provided with sensors and actuators.
To better understand the concept of haptic feedback, we can make an example of a 1 DoF interface with a link having attached a knob and a prismatic joint that permits movements only backward and forward, a sensor that compute the link displacement and a DC motor that can actuate the link movements in its path. By sending appropri-ate electric signal to the motor, it is possible to exert a force at the knob. The concept of haptic rendering is really simple, once a working frequency is decided, at each time-sample the computer use the information provided by the displacement sensor and it compares the measure with the software logic to provide a force reference to the motor driver, then a current flows in the motor to provide a torque with a mainly proportional relation.
The figure2.1represents a schematic view of a human-machine interaction adopted for the gearshift lever. The input of the haptic interface consists of the operator acting on the lever knob (end effector), which is monitored by the sensing system. After evalua-tion of this informaevalua-tion by the computer, the output of the haptic interface is a reacevalua-tion to the operator through the actuation system. The end effector input and output can together be regarded as interaction. The interaction between operator and the virtual environment through the haptic interface is a combination of force and displacement. The interaction between user and the interface can be seen as a bidirectional transfer of energy, since force times displacement represents mechanical work. If not properly controlled, this energy transfer may have a destabilizing effect on the control and on the user feel. Although the basic principles behind haptics are simple, there are significant technical challenges, such as the construction of the physical devices, real-time compu-tation of the reaction forces and force control in the human-mechanical hybrid system.
2.1. HAPTIC TERM 5
Figure 2.1: User interaction with the haptic gearshift simulator
Bi-directionality is the single most distinguishing feature of haptic interfaces, when com-pared with other machine interfaces. A haptic device must be designed to "read and write" to and from the human hand (or other contact part). The "read" part has been extensively explored, since there are many types of devices already existent (joysticks, pointing devices, etc.). The "write" part is comparatively more difficult to achieve, the function of the haptic interface is to be capable of recreating mechanical phenomena of perceptual relevance and functional importance.
2.2
Theory of permanent magnet DC motor
The aim of this part is to present a briefly introduction to the theory behind the behavior of the actuators used for this thesis work. A DC motor is an actuator that can produce rotational displacement and a rotary torque or linear displacement and an aligned force.
Figure 2.2: DC motor components
The structure of a permanent magnet DC motor is shown in figure2.2. The external part is called stator and is composed by permanent magnet built with rare earth that create a magnetic flux. The internal part is called rotor and is formed by copper wind-ings.
An electric motor converts the electric energy into mechanical energy using the attrac-tion and repulsion forces of the magnetic field.
2.2.1 Magnetic field and current relation
A magnetic field can be generated by a flowing current. The effect of a magnetic field may be experimented, for instance, by positioning a magnet close to a wire where a current is flowing, since can be observed that the magnet experiences a force which is due to the magnetic field generated by the current flowing in the wire. If the magnet is moved around the wire, the force changes depending on the positions assumed by the magnet. In particular, it can be observed that the magnetic field decreases as the distance from the wire increases and increases as the current increases.
2.2. THEORY OF PERMANENT MAGNET DC MOTOR 7 infinitesimal segments of wire d~l, where the current i flows. Each segment induces a magnetic field at any point in the surrounding space. In particular, the Biot and Savart law states that if ~r is the vector connecting the wire segment d~l to a generic point p in
the space, the contribution d ~B of the magnetic field in the point p due to the segment d~l has this relation:
d ~B = id~l × r
|r|3
This equation provides a tool for computing the magnetic field associated to any con-ductor where a current is flowing. In particular, by suitably integrating it for the case of a solenoid, it turns out that, if the corner effects are neglibigle (namely, the solenoid is long enough), the field inside the windings is constant and parallel to the axis of the solenoid, and its magnitude is given by the relation:
| ~B| = µ lN i
where µ is the magnetic permeability of the dielectric inside the solenoid, N is the number of turns of the wire and l is the length of the wiring.
Let’s consider now a surface S crossed by a magnetic field ~B, the magnetic flux φ flowing
through the surface is defined as the integral along the surface on the normal component of the magnetic field (~n is the perpendicular of the surface):
φ = Z
S
~ B˙~ndS
Given the hypothesis that the magnetic field is uniform, crossing a flat surface with an angle β and area A, the relation can be simplified into:
φ = | ~B|Acos(β)
Since the cross section of the solenoid has constant area, the flux flowing through a generic surface normal to the solenoid axis is constant along the whole solenoid. Denoting by A the constant area of the cross section of the solenoid, since the magnetic field is parallel to the solenoid axis so β=0, the flux may be computed in this way:
φ = K0N i
So a moving charge generates a magnetic field, but what about a moving charge that passes through an external magnetic field (Figure2.3). The Lorentz force equation can
be helpful to understand the mechanical aspect of this electromagnetic phenomenon. The equation describe the force experienced by a moving charge in an electromagnetic field where q is a charge moving with a speed ~v in the space:
~
Fq = q[ ~E + ~v × ~B]
The lorentz law applied to a physical body is derived through integration of the elemen-tary forces, for instance if we have a straight line conductor of length l, the resulting force is proportional to the wire current:
F = Il × B
Figure 2.3: The Lorentz force effects on the windings
Since the coil is square, the current i flows in opposite directions on the two sides of the coil; thus, the two forces F generate a torque τ exerted at the center of the coil that is dependent on the angular position θ of the coil with respect to the magnetic field. Being the length of an edge of the square d, then
T = 2| ~F |d
2sinθ = | ~F |dsinθ Since F = Il × B then:
T = | ~B|ildsinθ
We notice that the torque highly depends on the rotor coil angular position θ, if the coil turns of an angle π, the torque exerted has the same amplitude but opposite sign (Figure2.4).
2.2. THEORY OF PERMANENT MAGNET DC MOTOR 9
Figure 2.4: Torque as theta changes
If we want a constant torque for any position θ, the solution that can be adopted is to insert on the rotor shaft a commutator constituted by two segments connected to the rotor windings and brushes that slide between the segments as the rotor turns. In such a configuration, the sign of the current flowing in the coil changes at each half revolution of the shaft. Now that the torque has the same sign there is another problem, that it is highly dependent on the rotor position, a possible solution is to increase the number of coils in the rotor and the segments of the commutator. It turns out that, if N independent windings are located on the rotor, the commutator will have 2N segments and the torque profile will be constituted by 2N half sinusoids overlapped in one single period (Figure2.5). In such a way the torque ripple may be decreased as much as needed
Figure 2.5: Torque ripple.
by increasing N and the residual ripple may be assumed to be filtered by the mechanical system, so that the resulting torque may be approximated as
The rotor conductors carry current supplied from an external source and are located in an exogenous magnetic field. Forces are experienced by the coil and a torque is exerted on the rotor shaft. However, due to the rotation of the coil an electro-motive force is generated (an induced voltage in the rotor windings) that may be computed by means of the Faraday’s law of induction. This law states that if there is a variation of the flux φc flowing in the internal surface of a closed wire, then an electro-motive force e is induced in the wire, according to this equation:
e = −dφc dt
The Faraday’s Law tells us that the potential induced by the permanent magnets on the coil is proportional to the derivative of the flux:
e = −d
dt(| ~B|N πr
2cos(θ(t)))
= | ~B|N πr2θ(t)sin(θ(t))˙
= | ~B|N πr2ω(t)sin(θ(t))
where we have substitute β with the angle ω over time and ω denotes the angular speed of the shaft.
In case of negligible ripple we can write e = | ~B|N πr2ω(t). That in the case of a constant
rotor speed produces sinusoidal flow on the motor coils and consequently a regular voltage at coils terminal which can be expressed as:
Vω = Kωωf (θ)
where f(θ) keeps into account the ripple effects produced by the magnetic flux on the coil circuits. The combined effect of the brushes contacts with the geometry is difficult to estimate, however, given that the electromagnetic conversion does not accumulate energy, we can determine this result by using the energy conservation principle. In case of negligible ripple, the motor can be considered as a torque source whose value is proportional to the current that flows within the coils:
τem = Kemi
The equations that constitutes a model of the DC motor are valid under these as-sumptions:
2.2. THEORY OF PERMANENT MAGNET DC MOTOR 11 • The assumption that the magnetic circuit is linear (it is just an approximation
since there is some flux dispersion inside the motor)
• The assumption that the mechanical friction is only linear in the motor speed, only viscous friction is assumed to be present.
The resulting equations from the following system:
V = Ri + Ldidt + KΩΩ ImdΩdt = Kemi + τext
2.2.2 Introduction to Brushless DC motor
A BLDC motor accomplishes commutation electronically using rotor position feedback to determine when to switch the current. The structure is shown in Figure 2.6. Feed-back usually entails an attached Hall sensor or a rotary encoder. The stator windings work in conjunction with permanent magnets on the rotor to generate a nearly uniform flux density in the air gap. This permits the stator coils to be driven by a constant DC voltage (hence the name brushless DC), which simply switches from one stator coil to the next to generate an AC voltage waveform with a trapezoidal shape. A typical brushless motor has permanent magnets which rotate around a fixed armature, eliminat-ing problems associated with connecteliminat-ing current to the moveliminat-ing armature. An electronic controller replaces the brush/commutator assembly of the brushed DC motor, which continually switches the phase to the windings to keep the motor turning. The con-troller performs similar timed power distribution by using a solid-state circuit rather than the brush/commutator system.
Figure 2.6: Typical brushless motor structure.
Brushless motors offer several advantages over brushed DC motors, including high torque to weight ratio, more torque per watt (increased efficiency), increased reliability, reduced noise, longer lifetime (no brush and commutator erosion), elimination of ioniz-ing sparks from the commutator, and overall reduction of electromagnetic interference (EMI). With no windings on the rotor, they are not subjected to centrifugal forces, and because the windings are supported by the housing, they can be cooled by conduction, requiring no airflow inside the motor for cooling. This in turn means that the motor’s internals can be entirely enclosed and protected from dirt or other foreign matter.
2.2. THEORY OF PERMANENT MAGNET DC MOTOR 13 Electronics Commutation Principle
As said before in brushless motor the commutator device is not present, the current must be switched by some external logic. There are in principle three main group:
• Single-Phase BLDC Motor
BLDC commutation relies on feedback on the rotor position to decide when to energize the corresponding switches to generate the biggest torque. The easiest way to accurately detect position is to use a position sensor. The most popular position sensor device is Hall sensor. Most BLDC motors have Hall sensors embedded into the stator on the non-driving end of the motor. The applied voltage, switching frequency, and the PWM duty cycle are three key parameters to determine the speed and the torque of the motor.
• Three-Phase BLDC Motor
A three-phase BLDC motor requires three Hall sensors to detect the rotor’s po-sition. Based on the physical position of the Hall sensors, there are two types of output: a 60◦ phase shift and a 120◦ phase shift. Combining these three Hall sen-sor signals can determine the exact commutation sequence. One signal cycle may not correspond to a complete mechanical revolution. The number of signal cycles to complete a mechanical rotation is determined by the number of rotor pole pairs. Every rotor pole pair requires one signal cycle in one mechanical rotation. So, the number of signal cycles is equal to the rotor pole pairs.
• Sensorless BLDC Motor
The BLDC sensorless driver monitors the BEMF signals instead of the position detected by Hall sensors to commutate the signal. The sensor signal changes state when the voltage polarity of the BEMF crosses from positive to negative or from negative to positive. The BEMF zero-crossings provides precise position data for commutation. However, as BEMF is proportional to the speed of rotation, this implies that the motor requires a minimum speed for precise feedback. So under very low speed conditions such as start-up additional detectors such as open loop or BEMF amplifiers—are required to control the motor. The sensorless commutation can simplify the motor structure and lower the motor cost. Applications in dusty or oily environments that require only occasional cleaning, or where the motor is generally inaccessible, benefit from sensorless commutation.
2.2.3 Torque control
We suppose that the DC motor will be operated by a current driver, so the first equation may be neglected since the drive supply a constant current in the motor coils.
Here the mechanical equation:
Im
dΩ
dt = Kemi + τext
The control variable is the input current i, while the control output may be the motor torque.
We will consider the inertia as a disturbance and the current as the control variable. The torque provided by the motor is the reaction to the external torque and coincides with the torque sensed: τs = τm = −τext, the reference torque is τr and the nominal motor torque constant is the same of the effective motor torque constant: Kem ≈ ˆKem.
Here a proportional-integral controller:
i = Ki
Z
(τr− τs)dt − Kp(τs) + τr ˆ
Kem
By replacing the control strategy in the motor equation and indicating with τm the
external torque measured by the torque sensor we have:
Im dΩ dt = Kem(Ki Z (τr− τs)dt − Kp(τs)) + Kem ˆ Kem τr− τm
that leads to:
Tm= s + KemKi sKemKp+ KemKi Tr+ s sKemKp+ KemKi Λ(J, θ)
The limit theorem leads us to Tm= Tr, while at higher frequencies we allow an attenu-ation of the controlled force so that we achieve a sufficient damping to not conflict with other dynamic on the system.
A pure integral controller can be used sometimes:
i = Ki Z (τr− τs)dt + τr ˆ Kem
Replacing in the motor equation:
Im dΩ dt = Kem Ki Z (τr− τM)dt + τr− τm
2.3. HAPTIC INTERFACE CONTROL STRATEGIES 15 Setting the error as e = τr− τs we can write our controller:
e = s
s + KemKi
Im٨
2.3
Haptic interface control strategies
Haptic interfaces can be used in many ways, but their basic functions are mainly two. The first is to measure forces, positions, and their time-derivatives at the operator’s hand (or other body locations) while the second function is to display forces and positions back to the operator. There exist two fundamental strategies for controlling haptic interfaces [1]. When position is in input to the control loop and forces are in output to the operator, we can call the system as a force-feedback control. Alternatively, the device can use position-feedback control in which forces applied by the operator are sensed by the device and positions are in output of the haptic interface. According to these two different control strategies, haptic interfaces are categorized into impedance controlled and admittance controlled devices. Figure2.7shows both methods used to interact with a virtual environment.
(a) Admittance control.
(b) Impedance control.
Figure 2.7: Control strategies for haptic interfaces.
Impedance controlled devices use force-feedback control (Figure2.7b), so the opera-tor moves the haptic interface, and the device will react with a force according to certain law (for example when an object is met in a virtual environment). The operator will inevitably feel the mass and the friction of the actual device, but these can be made very small by careful mechanical design and they can be compensated using some techniques. Impedance devices are by nature lightly built and highly back drivable (meaning that the operator can move the end effector effortlessly when no virtual objects are met). They are typically cable driven by high performance DC motors. Admittance controlled
devices use position-feedback control (Figure 2.7a), so the operator exerts a force on the haptic interface, and the device will react with the proper position displacement. Admittance control allows considerable freedom in the mechanical design of the inter-face, because backlash and tip inertia can be practically eliminated [2]. As a result, the mechanism can be quite robust, and capable of displaying high stiffness and high forces. Impedance and admittance control strategies are dual not only in their system structure, but also in their performance. The impedance controlled device is typically lightweight, backlash free, and renders low mass ([3] [2]). Consequently, performance is lacking in the region of higher forces, high mass, and high stiffness. Adding complex end effectors is also a problem. Admittance controlled interfaces on the other hand, are capable of rendering very high stiffness and minimal friction. They are very suitable for larger workspace and for carrying complex end effectors with many degrees of free-dom. Moreover, because forces are sensed rather than computed in real time, admittance control has the advantage of reduced modeling computation load and avoiding discon-tinuities force issues. However, admittance controlled interfaces are often not capable of rendering low mass. The minimal mass rendered at the end effector (the minimal mass the operator feels during the simulation) is called the minimal tip inertia. The haptic interface used to develop this project, is an impedance controlled device. The reason for this choice is that in order to mimic the gearshift feel of a manual operated transmission, it is expected that forces and stiffness need to be rendered with respect to variable end effector mass, which is typically low, which an admittance controlled device is not capable of. Moreover, during gear shifting, the operator performs quick movements, which result in great accelerations of the gear knob. Due to its increased high-level feedback loop bandwidth, it is expected that an impedance controlled device offers a more accurate and more stable simulation.
2.4. THE HAPTIC INTERFACE: GEARSHIFT SIMULATOR 17
2.4
The Haptic interface: gearshift simulator
Haptic devices can be regarded as robots having a very special function or task: interact-ing with humans. This occurs mostly through the hand, but also via other anatomical regions, often, but not always, limbs and extremities. Thus, many "robotic problems" are relevant to haptic interfaces and vice versa.
Figure 2.8: The gearshift lever
The interface is a two degrees of freedom mechanical device with two rotoidal joints driven by two electrical DC motors. The force is transferred from the motor pulley to the joint thanks to a cable transmission. It is possible to select the knob height with a passive prismatic joint simply twist off a locking bolt.
Without actuation the lever can move and rotate without any force resistance, except some negligible friction from the spherical bearings. Each motor is provided with an encoder that measures rotational position and velocity, with these information it is possible to compute the position and velocity of the end effector (the knob). These motors are permanent magnet motor, driven by two current drivers, that are a series of servo amplifiers for DC brush current. They contains hybrid power stages and provides mixed analog and digital input/output ports. Each driver can control the amount of current flowing in the output stage given a current reference. The user experiences a force through the hand when a current flows in the motor, since the current is modulated by the driver and the force is nearly proportional to the current, it is possible to control
the force simply changing the current reference given in the input stage of the current driver.
Kinematics
The direct and inverse kinematics of the system can be easily calculated [4].
Figure 2.9: The gearshift simulator kinematic scheme.
Since the system has spherical kinematics, the angular velocity of the knob part ~ωp
only is computed and the vectors of the joint axes si,j, by using the same nomenclature
of figure2.9, can be expressed with respect to the frame Oxyz, in terms of the actuated joint angles θ1, θ2 as:
~s1,1 = 1 0 0 ~s2,1= 0 −cos(θ1) −sin(θ1) (2.1) ~s1,2= 0 2 0 ~ s2,2= −cos(θ2) 0 sin(θ2) (2.2)
2.5. MECHANICAL DESCRIPTION OF A CAR GEARBOX TRANSMISSION 19 be directly determined by:
~ s3,1= ~s2,1× ~s2,2 k~s2,1× ~s2,2k = 1 h −c1s2 s1c2 −c1c2 h2= c21+ c22− c21c22 (2.3)
The position of the knob ~p = [pxpypz] is then given by ~p = L~s3,1 where L is the length
of the gearshift lever. We can now easily obtain the joint angles values by solving the inverse kinematics of the gearshift mechanism:
θ1,1= θ1 = arctan(−ppyz) θ1,2= θ2 = arctan(ppxz) θ2,1= arcsin(pLx) θ2,2= arcsin(−py L) (2.4)
2.5
Mechanical description of a car gearbox transmission
In this section we briefly introduce a short description of what there is inside a car gear-box transmission. Components causes the forces that we feel at the gearshift knob with their movements. To better understand what we want to simulate we have to be aware of the real world interactions between those components.
The gearbox
A Gearbox is a mechanical device that is used to provide Speed and Torque conversions from a rotating power source to output shaft. As the speed of the shaft increases, the torque transmitted decreases and vice versa. Multi-speed gearboxes are used in applications which require frequent changes to the speed/torque at the output shaft. Gearboxes work on the principle of meshing of teeth, which result in the transmission of motion and power from the input source to the output.
A gearbox is formed by mounting different gears in appropriate speed ratios to obtain the desired variations in speed. Gearboxes usually have multiple sets of gears that are
placed appropriately to obtain different speed reductions.
(a) Sliding mesh gearbox schematic. (b) Constant mesh gearbox schematic.
Figure 2.10: Two types of unsynchronized manual gearbox transmission.
There are many types of gearboxes: sliding mesh gearbox, constant mesh gearbox and synchromesh gearbox. In a sliding mesh gearbox, the two types of gears are sliding gears and stationary gears. The sliding gears are mounted on splined shafts to enable them to slide along the axis of the shaft to enable meshing with different pairs of gear (Figure2.10a). In a constant mesh gearbox all the gears of the main shaft are in constant mesh with corresponding gears of the counter-shaft. The gears on the main shaft are free to rotate. The dog clutches are provided on main shaft and the gears on the lay shaft are fixed. When the left dog clutch is sliding to the left by means of the selector mechanism, its teeth are engaged with those on the clutch gear and we get the direct gear; the same dog clutch however, when sliding to right, makes contact with the second gear and the second gear engagement is obtained. Similarly movement of the right dog clutch to the left results in low gear and towards right in reverse gear. Usually the helical gears are used in constant mesh gearbox for smooth and noiseless operation (Figure2.10b).
Those two kind of gearbox are called unsynchronized manual transmission gearbox, the gear changing phase is not straightforward as we get used to in our modern car, in fact those gearbox are pretty old and unused anymore. The changing phase is called double clutch technique and consist in this steps:
• The driver must press the clutch and release the gear putting the vehicle into neutral, then release the clutch.
• then he must optimize the engine speed to match with speed of wheels (differential) shaft by pressing or releasing the accelerator pedal.
2.5. MECHANICAL DESCRIPTION OF A CAR GEARBOX TRANSMISSION 21 • Now the driver must press the clutch and rapidly insert the new gear, then release
the clutch.
As said before, in newer car there isn’t the need to perform this actions because another type of gearbox is mounted on: the synchromesh gearbox (Figure2.11).
Figure 2.11: The synchromesh gearbox.
This type of gearbox is similar to the constant mesh type gearbox, but instead of using dog clutches, here synchronizers are used. The modern cars use helical gears and synchronizer devices in gearboxes, that synchronize the rotation of gears that are about to be meshed. All the gears on the main shaft are in constant mesh with the corresponding gears on the lay shaft. The gears on the lay shaft are fixed to it while those on the main shaft are free to rotate as seen in the constant mesh gearbox. In most of the cars however, the synchronizer devices are not fitted to all the gears, they are fitted only on the high gears. On the first and reverse gears ordinary dog clutches are only provided. This is done to reduce the cost. The parts that ultimately are to be engaged are first brought into frictional contact, which equalizes their speed, after which these may be engaged smoothly.
Figure 2.12: Details about components with synchronizer.
When the vehicle is stationary with the engine in motion and the operator shifts in neutral gear position after opening the clutch, the axis of the first gear continues to rotate by inertia, but by shifting the synchronizer sleeve, the generated friction quickly stops the axis and so the engagement can be performed. When the vehicle is moving and you want to put in higher gear, the sleeve of the synchronizer comes into contact with a surface that has a higher peripheral speed: in this case the friction between the surfaces generates a force that brakes the axis of the putted gear and it also brakes the input shaft and the counter shaft. With the axes having the same speed the engagement can be done. When, on the other hand, with a moving vehicle, we want to put lower gear, the synchronizer sleeve comes into contact with a surface which has a lower speed so that the friction on the driven wheel acts as a force to accelerate the counter and input shaft.
2.5. MECHANICAL DESCRIPTION OF A CAR GEARBOX TRANSMISSION 23
Selection forks
A gear-selector mechanism (Figure 2.13a) permits the driver to select and engage indi-vidual gear ratios. The selector mechanism in general consists of three guiding selector rods supported at their ends either in the actual gearbox housing or in the selector cover housing. Selector forks slide with or over these rods and fit over saddle-like grooves machined on the outer sliding-dog clutch hub. When a selector fork is pushed towards one of the constant-mesh gearwheels, it moves the particular dog-clutch hub over the exposed ring of dog teeth on the gearwheel. Once the hub and gearwheel dog teeth are meshed, the lay-shaft cluster gear is coupled to the main output shaft, so that the power-flow path through the gearbox is completed. For the neutral gear, the selector forks are moved to place the outer dog-clutch hubs directly over the inner hubs. This causes each gearwheel on the main shaft to revolve independently.
(a) Selector forks.
(b) Details about selector forks.
Figure 2.13: Selector forks mechanism.
Selection forks interlocking device
The selector rods are located and held in their selected position by spring-loaded balls or plungers and grooves (notches). The spring load forces the balls into their respective grooves when they align (Figure. 2.13b). Each of the two forward-gear selector rods has three grooves and the reverse selector rod has two grooves. When the selector rods and their forks are shifted one way or the other, these grooves eventually align with the spring loaded balls or plungers when a gear position is reached. On the forward-gear selector rods the central groove corresponds to the neutral position and the grooves on either side corresponds to engaged position for first or second or third or fourth gear. The reverse selector has only a neutral and a reverse position To prevent the engagement
of two different gears, every gearbox incorporates some sort of safety interlocking device.
Figure 2.14: Forks interlocking device.
Movements of one rod causes balls to lock other rods in neutral position (Figure2.14).
Selection forks stop device
To prevent the engaged gear from being switched off due to vibrations or a slight impact on the gear shift lever by the driver, while also preventing the gear shifting to a minimum of vibration or shock, a stop device is used. The simplest and widespread system for making such a device is to use a ball which is pressed by a spring. It is located in one of the special cavities made in the rod at the position of the neutral or the engaged position (Figure2.15).
Chapter 3
State of the art
This chapter presents a review of some literature about haptic interfaces.
3.1
Force reflecting haptic devices
[5] A haptic interface may be thought of as a device that generates mechanical impedances. "Impedance," here, should be understood to represent a dynamic (history-dependent) re-lationship between velocity and force. For instance, if the haptic interface is intended to represent squeezing of a spring, it must generate a force proportional to displacement. The performance of a haptic interface is often reported in terms of the dynamic range of impedances it may represent. Impedance fidelity (i.e., the closeness of the impedance as implemented to the desired impedance) is also affected by factors such as actuator nonlinearity and sensor resolution.
The paper provides a theoretical analysis of a force-feedback system and a stiff wall implementation to discuss about issues in the control loop and application.
3.1.1 Virtual wall
Wall implementations differ according to hardware and software details. The paper focuses on the most common system implementation which requires a backdriveable manipulandum and discrete time controller. The implementation, shown in Figure 3.1, is piecewise linear.
• x(t): the manipulandum displacement. 25
• ˆx(t): the manipulandum velocity.
• F (t): the force generated by the manipulandum. • xwall: the location of the virtual wall.
• K: virtual stiffness.
• B: virtual damping coefficient.
Figure 3.1: Common system setup for a virtual wall. (Colgate et al., 1993) The velocity sensor has been shown as optional because velocity is often obtained by differentiation of position.
Two really important factors are stiffness and damping coefficient. Stiffnesses (K) of 2000-8000 N/m seem to be sufficient to generate a perception of rigidity for the authors, but a study aimed to find human factors in the design of those haptic interfaces says that the range to perceive a rigid object without visual feedback is around 15000-41000 N/m [6]. Damping (B) must be sufficient to prevent noticeable oscillations. Unfortunately, increasing B too much tends to cause high frequency oscillations that users often report as a feeling of “rumble.”.
The concept of virtual wall is not intended to represent only a very rigid object (contact with a wall), but also soft rigidity that uses lower stiffness.
3.1. FORCE REFLECTING HAPTIC DEVICES 27
Figure 3.2: Hybrid loop (continuous/discrete).
The chosen parameters affects the stability of the simulation, but in the loop there is also the human operator (Figure 3.2). Given the opportunity to explore, users will quickly find ways to set up sustained or growing oscillations by gripping the manipulan-dum lightly or firmly, as necessary. The human operator’s own physical characteristics are involved in the feedback loop in exploring the virtual world, and he changes those parameters dynamically and radically. Lanman reported human elbow stiffness to vary from a minimum of about 1 .4 Nm/rad to a maximum as high as 400 Nm/rad [7]. Can-non and Zahalak’s measurements showed that both the limb’s natural frequency and damping ratio vary with muscle activation [8]. Thus, it appears difficult to use stabil-ity as the basis of a performance criterion unless the full range of human dynamics is considered.
3.1.2 Passivity
A more appropriate basis may be passivity. The ability of humans to set up oscillations is evidence of active walls: because the frequencies of these oscillations are often outside the range of voluntary motion, and because this behavior is not observed with physical walls, it is evident that the energy supply for the oscillations is the virtual wall, not the human. This motivates the use of a performance criterion based on passivity: the inability to act as an energy source.
Passivity is additionally attractive because it is a property of the wall alone.
Let’s analyze a simpler system, a virtual spring. An ideal physical spring is a lossless sys-tem; therefore, if energy is stored in the spring by squeezing, then removed by releasing, precisely as much energy will be removed as was stored.
Figure 3.3: Virtual spring force/position plot. (Colgate et. al., 1994)
Now consider the virtual spring. Because it is implemented in discrete time, the force provided by the spring will not increase smoothly with deflection. Instead, the force will be repeatedly “held” at a constant value until updated. Because of this, the average force during squeezing will be slightly less than for a physical spring of identical stiffness, and the average force during release will be slightly greater. This is illustrated in Figure 3.3. The result is not only that the exquisite balance of stored energy and released energy is lost, but that the spring always acts to store or generate energy, never to dissipate energy. So if a virtual wall is to be passive, it must incorporate some physical dissipation.
[9] Passivity has proved to be a useful tool for studying both the stability and per-formance of haptic displays. A one-port system is passive if the integral of the power extracted over time does not exceed the initial energy stored in the system. For a trans-lational mechanical system, power is the product of force f and velocity ˙x, with the sign
convention that power is positive when energy flows into the system. The initial energy is defined to be zero, resulting in the following inequality:
Z t 0
f (τ ) ˙x(τ )dτ ≥ 0, ∀t ≥ 0 (3.1)
Maintaining passivity places severe restrictions on virtual environment stiffness and damping. A number of techniques have been developed to manage haptic rendering of high impedance environments. The lower bound on impedance is generally limited by the quality of feedback and mechanical design. The upper bound on passive impedance can be limited by noise, sensor quantization, time delay and sampled data effects. A
num-3.1. FORCE REFLECTING HAPTIC DEVICES 29 ber of methods exist to increase the maximum passive impedance of a haptic interface: controllers, physical mechanisms, and electrical mechanisms. The category of controllers includes virtual couplings and passivity observers. Virtual couplings act as mediators between the haptic display and the virtual environment. Passivity observers and pas-sivity controllers function by adjusting the energy present in the system to maintain passivity 3.1. Mechanical methods are generally the most direct: physical dissipation is added to the mechanism to expand the passive impedance range of a haptic display by counteracting the effects of energy leaks. Electrical methods are like physical methods, thay are implemented electrically using analog electronics.
Colgate and Schenkel [10] derive an analytical passivity criterion for a simple 1 degree of freedom (DOF) haptic interface with unilateral constraint such the one discussed above, and a necessary and sufficient condition for passivity of the sampled data system is:
b > T 2 1 1 − cos(ωT )< n 1 − e−jωTHejωT o f or 0 ≤ ω ≤ ωN (3.2)
where b is the physical damping present in the mechanism, T is the sampling rate, H(z) is a pulse transfer function representing the virtual environment, and ωN = πT is the Nyquist frequency.
The implementation above is composed of a virtual spring and damper in mechanical parallel, together with unilateral constraint operator, the velocity estimation is obtained via backward difference differentiation of position, so the transfer function H(z) is:
H(z) = K + Bz − 1
T z (3.3)
with K > 0 that is the virtual stiffness coefficient and B is the virtual dumping coeffi-cient. The relation 3.2can be reduced to:
b > KT
2 − Bcos(ωT ) 0 ≤ ω ≤ ωN (3.4)
The equation can be maximized with ω = ωN, it leads to this condition of stability:
b > KT
2 + B (3.5)
Figure 3.4: Condition to achieve passivity.
This result shows that to achieve the stability of the system, some physical dissipation is essential. It also shows that, given fixed physical and virtual dumping, the maximum
achievable virtual stiffness is proportional to the sampling rate. Further, the achievable virtual dumping is independent of the sampling rate.
3.1.3 Time Domain Passivity Approach TDPA
TDPA (Time Domain Passivity Approach) was first introduced [11] as method for con-trolling a haptic interface system to ensure stable contact with a VE. The method, as described in [12], is depicted in the scheme of Figure 3.5. The TDPA consists of mon-itoring the energy of the system in realtime (by using Passivity-Observers - POs) and dissipating it only when the system presents an active behavior (by using Passivity-Controllers - PCs). Human Operator Haptic Interface Virtual Environment 𝜶 𝑥ℎ 𝑓ℎ 𝑓𝑐 𝑓𝑣𝑒 𝑥𝑐 𝑥𝑣𝑒 + + + − 𝑓𝑃𝐶
Figure 3.5: Schematic network representation of the haptic system with a passivity controller (the block labeled α).
˙
xh is the human operator velocity, ˙xc is the measured haptic interface velocity and
˙
xve is the velocity of the virtual environment. fh is the force exerted by the human
operator, fc is the haptic interface command force, fve is the force from the virtual environment and fP C is dissipative term from the passive controller. The algorithm explained in [12] for a one-port network with impedance causality is:
1. xc(k) = xve(k) is the input; 2. ∆x(k) = xc(k) − xc(k − 1);
3. fve(k) is the output of the one-port network;
4. W (k) = W (k − 1) + fc(k − 1)∆x(k) is the energy output at the step k;
3.1. FORCE REFLECTING HAPTIC DEVICES 31 6. the PC control force to dissipate the produced energy is calculated
fP C(k) = −Wc(k + 1) ∆x(k) , if Wc(k + 1) < 0 0, if Wc(k + 1) ≥ 0 (3.6) 7. fc= fve+ fP C is the output.
3.1.4 Memory Based Passivition Approach MBPA
The MBPA was introduced in [13] by Ryu, it is a model-free based approach to make passive an haptic system and it was derived from a graphical interpretation of energy leak during haptic interaction. In a control system as one introduced in Figure 3.1the continuous domain and discrete domain are interconnected: the continuous angle po-sition became the discrete x∗ as effect of the encoder quantization, while the discrete command feedback force from the VE fve is hold between consecutive samples by a
ZOH. This interconnection produces energy and causes unstable behavior. In the con-tinuous world a spring-like VE is passive (equation 3.1), but it becomes active with the introduction of time discretization and position quantization. In Figure 3.6a the VE (solid line) and the command force (dotted line) during an interaction are shown. The energy stored in the pressing phase is the area below the pressing curve while the energy released during the interaction is the area below the releasing curve. According to this graph, the system produces energy (i.e. the difference between the releasing curve and the pressing one), thus it is active.
x f x f x f VE position VE position VE position
(a) No passivity controller. x f x f x f VE position VE position VE position
(b) Ideal paths with MBPA control.
Figure 3.6: The position versus force graph (command force in dotted line) of an interaction with a spring-like VE (solid line).
The Figure 3.6b shows the idea of the MBPA, that is to bound the releasing curve below the saved pressing curve in the position versus force graph. When the user presses
the VE, the actual command force and measured position are stored in an array at each encoder count detection. The case showed in Figure 3.6b is an ideal case where the release path is the same of the pressing one. This method require to memorize the command force at each encoder count detection (and not to change the command force at each encoder count detection) and this require an hardware that runs at very high frequency and that is able to detect every encoder pulses. In [13] an FPGA that run at 100 kHz was used.
3.2
A virtual gearshift application
Tideman [14] makes some research about virtual reality technology, in particular their work was pointed out to create a Virtual Prototyping Environment (VPE) in which the behavior of a vehicle or part of a vehicle can be evaluated and adjusted to match the driver’s desires. A VPE can be created by utilizing virtual reality interfaces (e.g. haptic devices, stereoscopic displays, 3D sound devices). In this way, the user gets the illusion of having some kind of interaction. A VPE enables evaluation of specific characteristics of a candidate design without having to build a physical prototype.
The paper describes the design and evaluation of a Virtual Gearshift Application, i.e. an application that raises the illusion of manually shifting gears in a passenger car when operated. By further developing this application, it should become possible to define the desired gearshift feel, after which the physical gearbox is designed in such a way that it matches the desires.
Operation of a manual transmission in a passenger car takes place by means of inter-action between the operator’s hand and the user-interface of the transmission, i.e. the gear knob. Moving the gear knob is done by muscle power, applied through the skeletal system and the contact area between the hand and the gear knob. Depending on the resistance force induced by the complete transmission system and applied on the hand through the contact area, the totality of gear knob and hand executes a certain move-ment. So, the interaction is nothing but a relation between motion and forces.
The paper describes how some physical phenomena can be modeled. In their work mea-surements are performed on the test vehicle’s gear lever. Due to practical limitations, these measurements take place while the vehicle is at rest, the engine is not running, and the clutch is disengaged.
The main results of this testing are:
3.3. THE GEAR ENGAGEMENT PROCESS IN MANUAL TRANSMISSION 33 significant variance for different operators;
• During gear shifting, the forces induced on the operator’s hand do not show any variance for different operating speeds;
• During gear shifting, the forces induced on the operator’s hand are mainly induced by the locking mechanism and the synchronization unit;
• As the locking mechanism of the gearbox consists of a spring-loaded ball that moves into a groove when a particular gear is engaged and out of it when a gear is disengaged, this effect may be modeled as compliance;
• As the synchronization unit of the gearbox consists of two conical faces that make frictional contact just before a gear is engaged, this effect may be modeled as Coulomb friction.
The haptic interface selected to develop the Virtual Gearshift Application in their work is a commercially available admittance controlled device (The HapticMaster).
3.3
The gear engagement process in manual transmission
Frisoli, Avizzano, Bergamasco [15] in their study divide the gear engagement process in three different main stages that characterize the particular force response of a gear-shift: the synchronization, the engagement and the impact against the mechanical stop. Since the gearshift is a multi-body dynamical system, each stage is associated to the interaction of different parts in the gear-box.
The pre-synchronizing stage generates a negligible force peak only and can be reason-ably disregarded. During the synchronization, both force and position are held constant for a definite period of time, as shown in figure 3.7 and the force reaches its maximum value. The engagement stage is characterized by an isolated peak force, that generally is lower than the synchronizing force peak. Moreover the magnitude of such pick force is not constant and varies remarkably at every gearshift maneuver. During the engagement the position remains also constant for a short time, as can be revealed by data measured directly at the gear-box.
Figure 3.8: Force vs position plot of entire maneuver. (Frisoli et al. 2001)
In figure3.8there are some repetition of the maneuver, as we can see the synchroniza-tion and the engagement peaks occur at definite posisynchroniza-tions. The recorded data aboard the experimental car show a synchronization peak that lasts 0.3 msec. The engagement peak is instead instantaneous, since it is due to the impact of the gears teeth. This double bump of force, typical of the engagement process, gives the user the usual feel of engaging the gear. The final stage gives raise to the stop impact peak. Since the lever is arrested by an elastic stop system through the external command, the lever position has an overshoot and a following recovery to the equilibrium point, as shown in figure
3.7. The driver, changing into a gear, exerts forces on the lever, determined mainly by the synchronization, the engagement and the final impact with the stop. Therefore a realistic simulation of a gearshift must replicate these three phases.
