Spring-Mass-Damper Model
Below we provide the derivation of the application of Fitts’ law to a second-order system model as proposed by Langolf et al. [115, 116] and further investigated by Jagacinski etal [114]. The derivation below provides the intermediate steps between equations (4.8) and (4.9). A−1 2W = A−A e−ζωnt p 1−ζ2 ! 1 2W = A e−ζωnt p 1−ζ2 ! W 2A = e−ζωnt p 1−ζ2 ! q 1−ζ2 W 2A = e −ζωnt ln q 1−ζ2 + ln W 2A = −ζωnt 1 −ζωn ln q 1−ζ2 + 1 ζωn ln 2A W = t 1 −ζωn ln q 1−ζ2 + ln 2 ζωn log2 2A W = t
Appendix A: Appendix: Additional system model derivations 167
By letting a = −ζω1
n ln( p
1−ζ2) and b = ζωln 2
n, we get the traditional Fitts’
Law form t =a+blog2 2A W
A.2
DES-plant Event Tables and DES-controller
State Tables
This section includes the remaining details of DES-plant event and DES-controller symbol definintions. The DES-plant event symbols are given in Table A.1. The de- scription of hypersurface and camera motion combination triggering the event are partitioned by axial component. The first block of symbols describes horizontal mo- tion events. The second block describes vertical motion events. The third block describes depth motion events.
The DES-controller state symbol definitions for the 3-dimensional l2-model are provided in Table A.2. There are eleven symbols: nine of which relate to some required tracking motion; one forOn-Target , and one indicating loss of the target The DES-controller output symbol definitions for the 3-dimensional l2-model are provided in Table A.3. The symbols are mapped from the DES-controller state through the DES-controller output function,φ(˜s). Due to the horizontal axis priority scheme, there ˜r2 and ˜r3 have multiple mappings through the output function.
Appendix A: Appendix: Additional system model derivations 168
Table A.1: DES Plant Event ( ˜X) symbol table for the 3-dimensional l2-model Symbol Description of Plant Event Symbols Hypersurface Camera
Move- ment
˜
x1 On-Targetin horizontal (x1(t)) h1(x)>0 left
˜
x2 Off-Targetin horizontal (x1(t)) h2(x)>0 right
˜
x3 On-Targetin horizontal (x1(t)) h3(x)>0 right
˜
x4 Off-Targetin horizontal (x1(t)) h4(x)>0 left
˜
x5 On-Targetin horizontal (x1(t)) h1(x)>0 left
˜
x6 Off-Targetin horizontal (x1(t)) h2(x)>0 right
˜
x7 On-Targetin horizontal (x1(t)) h3(x)>0 right
˜
x8 Off-Targetin horizontal (x1(t)) h4(x)>0 left
˜
x9 On-Targetin horizontal (x1(t)) h1(x)>0 left
˜
x10 Off-Targetin horizontal (x1(t)) h2(x)>0 right ˜
x11 On-Targetin horizontal (x1(t)) h3(x)>0 right
˜
x12 Off-Targetin horizontal (x1(t)) h4(x)>0 left
εi silent events hi(x)<0, i= 1. . .4 ˜
x13 On-Targetin vertical (x2(t)) h5(x)>0 down
˜
x14 Off-Targetin vertical (x2(t)) h6(x)>0 up
˜
x15 On-Targetin vertical (x2(t)) h7(x)>0 up
˜
x16 Off-Targetin vertical (x2(t)) h8(x)>0 down
˜
x17 On-Targetin vertical (x2(t)) h5(x)>0 down
˜
x18 Off-Targetin vertical (x2(t)) h6(x)>0 up
˜
x19 On-Targetin vertical (x2(t)) h7(x)>0 up
˜
x20 Off-Targetin vertical (x2(t)) h8(x)>0 down
˜
x21 On-Targetin vertical (x2(t)) h5(x)>0 down
˜
x22 Off-Targetin vertical (x2(t)) h6(x)>0 up
˜
x23 On-Targetin vertical (x2(t)) h7(x)>0 up
˜
x24 Off-Targetin vertical (x2(t)) h8(x)>0 down
εi silent events hi(x)<0, i= 5. . .8 ˜
x25 On-Targetin depth (x3(t)) h9(x)>0 forward
Appendix A: Appendix: Additional system model derivations 169
Table A.2: DES-controller state ( ˜S) symbol table for the 3-dimensional l2-model Symbol Description of Controller State Symbols
˜ s1 Off-Target-Depth ˜ s2 Off-Target-Left ˜ s3 Off-Target-Right ˜ s4 Off-Target-Below ˜ s5 Off-Target-Above ˜
s6 Off-Target-Left-Below (horizontal+vertical) ˜
s7 Off-Target-Right-Below (horizontal+vertical) ˜
s8 Off-Target-Left-Above (horizontal+vertical) ˜
s9 Off-Target-Right-Above (horizontal+vertical) ˜
s10 On-Target(horizontal+veritcal+depth) ˜
s11 No-Target
Table A.3: DES-controller output ( ˜R) symbol table for the 3-dimensional l2-model
Symbol Description of Controller Output Symbols φ(˜s1) = ˜r1 Move-Forward φ(˜s2) = ˜r2 Move-Left φ(˜s3) = ˜r3 Move-Right φ(˜s4) = ˜r4 Move-Down φ(˜s5) = ˜r5 Move-Up φ(˜s6) = ˜r2 φ(˜s7) = ˜r3 φ(˜s8) = ˜r2 φ(˜s9) = ˜r3
φ(˜s10) = ˜r6 Stop: successful acquisition of target φ(˜s11) = ˜r7 Halt: no target present
170
Curriculum Vitae
NAME: Duane J. Jacques, B.Sc., B.E.Sc., M.E.Sc. PLACE OF BIRTH: Nairobi, Kenya
YEAR OF BIRTH: 1973
POST-SECONDARY University of Western Ontario EDUCATION 2003 M.E.Sc.
University of Western Ontario
1998 B.E.Sc. (Computer Engineering) University of Western Ontario
1996 B.Sc. (Computer Science)
HONOURS AND PRECARN Scholars Program 2008: National Graduate Scholarship
AWARDS: Faculty of Engineering 2007: Conference Travel Grant Award
PRECARN Scholars Program 2006: National Graduate Scholarship
Faculty of Engineering 2006: Graduate Thesis Research Award
Faculty of Engineering 2005: Conference Travel Grant Award
University of Western Ontario 2002/2003: President’s Graduate Entrance Scholarship (declined due to deferred enrolment)
Sigma Alpha Mu Foundation 2002: Samuel Miller Gradu- ate Scholarship
Sigma Alpha Mu Foundation 2001: Samuel Miller Gradu- ate Scholarship
Faculty of Engineering 1998: Undergraduate Thesis Project (Semi-Finalist) Certificate of Merit
Appendix A: Appendix: Additional system model derivations 171
RELATED Research Associate
EXPERIENCE RoadLAB, Department of Computer Science University of Western Ontario
2013 - present RELATED Lecturer:
EXPERIENCE ES1050 Introduction to Design and Innovation ECE3375B Microprocessors and Microcomputers
Department of Electrical and Computer Engineering University of Western Ontario
2008 - 2012
Graduate Teaching and Research Assistant University of Western Ontario
2000 - 2007
Research Engineer
Mines Research, INCO Ltd. 1999-2001
Publications
Journal (Refereed):
D.J.Jacques, K.A.McIsaac, “An Assistive Device for the Visually Impaired to Per- form Guided Reaching Tasks employing a Hybrid Control Scheme”,IEEE Transactions on Robotics, (In Progress)
D.J.Jacques, K.A.McIsaac, “Comparison of a Deterministic Finite Automata State Transition Metric to Fitts Law Performance in a Guided Reaching Task”,IEEE
Transactions on System, Man and Cybernetics,(In Progress)
Technical Reports:
R.Rodrigo, D.J.Jacques, K.A.McIsaac and J.Samarabandu, “An Object Tracking and Visual Servoing System for the Visually Impaired”, 14th Annual Canadian
Conference on Intelligent Systems, Ottawa, Canada, June 2004.
D.J.Jacques and D. Makrakis, “MALTA Project: Technical Report”, PRE- CARN Inc., Ottawa, Canada, Sept. 2001.
Appendix A: Appendix: Additional system model derivations 172
Dissertations:
D.J.Jacques, “Modeling and Analysis of Network Traffic for a Teleoperated Load- Haul-Dump Scooptram”, M.E.Sc. Thesis, Faculty of Graduate Studies, University of Western Ontario, London, ON, Canada, Jan.2003.
Conference:
D.J.Jacques and K.A.McIsaac “Using feature scale to estimate object distance in a
reaching task”,17th Annual Canadian Conference on Intelligent Systems, Windsor, Canada, May 2008, Poster Presentation.
D.J.Jacques, R.Rodrigo, K.A.McIsaac and J.Samarabandu, “An application framework for measuring the performance of a visual servo control of a reaching task for the visually impaired”, IEEE International Conference on System, Man and Cybernetics, October 2007, Montreal, Canada, p894-901
D.J.Jacquesand K.A.McIsaac, “A Prototype Wearable Assistive Device for the Visually Impaired”,15th Annual Canadian Conference on Intelligent Systems, Victoria, Canada, May 2006, Poster Presentation.
D.J.Jacques, R.Rodrigo, K.A.McIsaac and J.Samarabandu, “An Object Track- ing and Visual Servoing System for the Visually Impaired”,IEEE International
Conference on Robotics and Automation, Barcelona, Spain, April 2005,
p3510-3515.
P.Cunningham, T.Charsley and D.J.Jacques, “Development of an Automated 3D Toping System at INCO Limited”,Proceedings from 5th ISSMA Conference (Telemin1), Sudbury, Canada, June 1999, CD-ROM
D.J.Jacques, “Packet Traffic Behaviour of a MODBUS PLC Application in a Networked Environment”,Proceedings of the 2002 Electrical & Computer
Engineering Graduate Research Symposium, University of Western On-
tario, Editors: T.E.Doyle, W.Greason, Z.Kucerovsky, London, ON, Canada, May 2002, p44-48.
D.J.Jacques and D. Makrakis, “Modelling and Analysis of Network Traf- fic Using Self-Similar Processes”,Proceedings of the 2001 Electrical &
Appendix A: Appendix: Additional system model derivations 173
Computer Engineering Graduate Research Symposium, University of West-
ern Ontario, Editors: T.E.Doyle, W.Greason, Z.Kucerovsky, London, ON, Canada, May 2001, p126-128.