CHI 2016 paper "Modeling the Steering Time Difference between Narrowing and Widening Tunnels" Videos are deleted.

1. Modeling the Steering Time Difference
between Narrowing and Widening Tunnels
Shota Yamanaka (Meiji University & JSPS)
Homei Miyashita (Meiji University)
10, May, 2016
Meiji University Japan Society for the
Promotion of Science
Session: Quantify efficiency of input method 1

2. Demo Video of the Experimental Task
Narrowing tunnel vs. Widening tunnel
2

3. Demo Video of the Experimental Task
Narrowing tunnel vs. Widening tunnel
3

4. Demo Video of the Experimental Task
Movement time comparison
4

5. Demo Video of the Experimental Task
Movement time comparison
5

6. Observation and Goals
6
(MT & ID)
Movement time (MT) for a Narrowing Tunnel is longer than a Widening Tunnel
MTNT > MTWT
Their indexes of difficulty (IDs) of the steering law will be:
IDNT > IDWT
Narrowing Widening
(1) To prove that there is a time difference
(2) To model the ID for the time difference
Our goals

7. Background
7

8. Steering Law [Accot and Zhai 1997]
Steering law: 𝑀𝑇 = 𝑎 + 𝑏
𝐴
𝑊
• a and b: empirically determined constants
•
𝐴
𝑊
is called Index of Difficulty (ID)
e.g., a narrower or longer path has a higher ID
that requires a longer MT
8
When navigating a tunnel of width W and amplitude A,
the movement time MT has a linear relationship to A/W
W
A

9. 𝐼𝐷 =
𝐴
𝑊
is Held to Constant-width Tunnels
9
Constant-width circle
[Accot and Zhai 1999, 2001]
A
WW
A
Constant-width straight tunnel
[Accot and Zhai 1997, 1999, 2001]
𝐼𝐷 =
𝐴
𝑊
𝐼𝐷 =
𝐴
𝑊

10. Steering Law: Various Devices/Environments
10
Racing wheel controller
for driving simulator [Zhai+ 2004]
Direct-input stylus
[Kulikov+ 2005]
Mouse, touchpad, trackpoint, trackball, indirect-input stylus
[Accot+ 1999]
3D controller for ball+tube and
ring+thread tasks [Casiez+ 2004]

11. Other Tunnel Shapes: Different ID formulae
11
Narrowing straight tunnel
[Accot and Zhai 1997]
Widening spiral tunnel
[Accot and Zhai 1997]
𝐼𝐷NT =
𝐴
𝑊𝑅 − 𝑊𝐿
ln
𝑊𝑅
𝑊𝐿
𝐼𝐷ST =
2𝜋
2𝜋 𝑛+1
𝜃 + 𝜔 6 + 9 𝜃 + 𝜔 4
𝜃 + 2𝜋 + 𝜔 3 − 𝜃 + 𝜔 3
𝑑𝜃
n : the number of turns
θ : current position (in angle)
ω : width-increasing parameter
WL : left width (start side)
WR : right width (end side)
WL
A
WR

12. Consistency of Narrowing and Constant-width Tunnels
12
WL
A
WR
When WL → WR , IDNT is matched to IDConstant
lim
𝑊 𝐿→𝑊 𝑅
𝐴
𝑊𝑅 − 𝑊𝐿
ln
𝑊𝑅
𝑊𝐿
= lim
𝑟→1
𝐴
𝑥 − 𝑥𝑟
ln
𝑥
𝑟𝑥
where 𝑊𝑅 = 𝑥, 𝑊𝐿 = 𝑟 × 𝑊𝑅
=
𝐴
𝑥
lim
𝑟→1
1
1 − 𝑟
ln
1
𝑟
=
𝐴
𝑥
× 1
=
𝐴
𝑊𝑅
IDNT includes IDConstant
WL
A
WR
Shrink

13. Deriving the ID Difference
13
Narrowing Widening

14. Observation of the Pilot Study
Tunnel type seems to be a significant reason behind the MT difference
14
① For narrowing, left/right directions did not affect the MT
② For widening, left/right directions did not affect the MT
③ Narrowing/widening type always affected the MT
Narrowing, to left Widening, to rightNarrowing, to right
(MT) (MT)
Widening, to left
(MT)
(Narrowing or Widening)
≈≈
① ②
③
①ｰ③ will be checked also
in the main experiment

15. Revisiting ID for a Narrowing Straight Tunnel [Accot and Zhai 1997]
Navigating a narrowing tunnel can be converted to
navigating the infinite number of constant-width infinitesimal-length tunnels
𝐼𝐷NT =
0
𝐴
𝑑𝑥
𝑊 𝑥
=
0
𝐴
𝑑𝑥
𝑊𝐿 +
𝑥
𝐴
𝑊𝑅 − 𝑊𝐿
=
𝐴
𝑊𝑅 − 𝑊𝐿
ln
𝑊𝑅
𝑊𝐿
W
A
𝐼𝐷 =
𝐴
𝑊
WR
Start line
End line
WL
A
W(x)
dx
x
constant-width linear tunnel
15

16. ID for a Widening Straight Tunnel
• Integration does not take account of the left/right direction
• The same calculation of IDNT can be used to derive IDWT
𝐼𝐷NT = 𝐼𝐷WT =
𝐴
𝑊𝑅 − 𝑊𝐿
ln
𝑊𝑅
𝑊𝐿
WR
Start line
End line
WL
A
W(x)
dx
x
WR
End line
Start line
WL
A
W(x)
dx
x
Narrowing direction Widening direction
16
Our first derivation
This does not reflect our observation:
MTNT > MTWT

17. Our Hypothesis of the Time Difference
Users perform a limited number of movement corrections
Users repeatedly determine the current movement (distance×angle)
with getting visual feedback
WR
Start line
End line
WL
A
W(x)
dx
x
The conventional steering law calculates dx → 0,
which means that movements are continuous
(infinite number of re-aiming)
17

18. Difficulty of One Movement
Our model
Acceptable slippage in y-axis is affected by the goal-side width
The current strategy is limited by the width at a little forward
Narrowing tunnel: users cannot use
the wider (start) side efficiently
Widening tunnel: users can use the
full width of the wider (end) side
18
Speed-down Speed-up

19. Deriving IDNT Based on Our Hypothesis
For simplicity, we assume that there are three movement corrections at
regular distance intervals
WR
Start line
End line
WL
A/3 A/3 A/3
e.g.) ID for ① is
𝐴/3
𝑊1
=
𝐴/3
(2𝑊 𝐿+𝑊 𝑅)/3
=
𝐴
2𝑊 𝐿+𝑊 𝑅
𝐼𝐷NT(3) =
𝐴
2𝑊𝐿 + 𝑊𝑅
+
𝐴
𝑊𝐿 + 2𝑊𝑅
+
𝐴
3𝑊𝑅
Narrowing direction
19
W1 W2 W3
① ② ③
① ② ③

20. Deriving IDWT Based on Our Hypothesis
As the same manner, IDWT(3) can be derived:
WR
End line
Start line
WL
A/3 A/3 A/3
Widening direction
20
W1 W2 W3
𝐼𝐷WT(3) =
𝐴
3𝑊𝐿
+
𝐴
2𝑊𝐿 + 𝑊𝑅
+
𝐴
𝑊𝐿 + 2𝑊𝑅

21. Deriving the ID Difference (IDGap)
𝐼𝐷 )Gap(3 = 𝐼𝐷 )NT(3 − 𝐼𝐷WT 3
=
𝐴
3𝑊𝑅
−
𝐴
3𝑊𝐿
=
𝐴(𝑊𝐿 − 𝑊𝑅)
3𝑊𝐿 𝑊𝑅
=
𝐴
2𝑊𝐿 + 𝑊𝑅
+
𝐴
𝑊𝐿 + 2𝑊𝑅
+
𝐴
3𝑊𝑅
−
𝐴
3𝑊𝐿
+
𝐴
2𝑊𝐿 + 𝑊𝑅
+
𝐴
𝑊𝐿 + 2𝑊𝑅
21
WR
Start line
End line
WL
A/3 A/3 A/3
Narrowing direction
W1 W2 W3
① ② ③
WR
End line
Start line
WL
A/3 A/3 A/3
Widening direction
W1 W2 W3
3 → N to generalize

22. Generalizing IDGap
Users’ strategies may be affected by some conditions:
• Tunnel parameters: A, WL, WR, and the degree of change of W
• Current width: one movement becomes shorter under a narrower W
• Current speed: the lower speed is, the more re-thinking occurs in a certain distance
22
WRWL
A
N
A
N
A
N
A
N
A
N
WRWL
𝑎% 𝑏% 𝑐% 𝑑% 𝑒%
𝐼𝐷Gap(𝑁)=
𝐴(𝑊𝐿 − 𝑊𝑅)
𝑁𝑊𝐿 𝑊𝑅
If users perform movement corrections N times at regular distance intervals,

23. Our model IDGap
Replacing the number of equal partitions N with a free weight k
that reflects the experimental conditions and tunnel parameters
23
𝐼𝐷Gap(𝑘)=
𝐴(𝑊𝐿 − 𝑊𝑅)
𝑘𝑊𝐿 𝑊𝑅
Our model:
IDGap(k)
Narrowing Widening
Consistency of our model and the constant-width model: When WL → WR, IDGap(k) → 0
“If the width becomes constant, the time difference between MTNT and MTWT becomes 0”
✔

24. Experiment
24

25. Experiment (Goals, Task)
Goals: testing (1) the MT difference, and (2) the validity of IDGap
Task: Navigating non-constant-width straight tunnels
Start area End area
Start line
End line
WL
WRA
Direction of movement
Path
Outside of path region
Cursor
25

26. Experiment (Design)
A 300, 600 pixels
(= 61.2, 122.4 mm)
WL & WR 11, 31, 51 pixels
( = 2.2, 6.3, 10.4 mm)
Dir Left, Right
2 (A) ×6 (W) × 2 (Dir) = 24 conditions = 1 block
Only WL ≠ WR (not constant-width) conditions were selected
3 (WL) × 3 (WR) － 3 (WL = WR) = 6 (W)
Tunnel type (narrowing/widening) was defined by the combination of {WL , WR , Dir}
26

27. Experiment (Device, Participant, Procedure, Data)
Device: direct-input 13.3-inch pen-tablet
Wacom Cintiq 12WX, 261.1 × 163.2 mm, 1280 × 800 pixels
Participant: eleven local university students (within-participant)
11 males, all right-handed, Mage = 21.9 years, SDage = 2.27 years
Each participant performed 1 warm-up and 5 actual blocks
24 conditions × 5 blocks × 11 participants = 1320 trials
Collected data
MT, error rate, time-stamped cursor trajectory
27

28. General Results (repeated measures ANOVA and the Bonferroni post hoc test)
main effects: ID (F5, 50 = 29.449, p < .001) and
tunnel type (F1, 10 = 23.667, p < .01)
post hoc test: widening is faster than narrowing
(p < .01; 826 ms vs. 1233 ms)
28
0
0.05
0.1
0.15
0.2
0 10 20 30 40
Errorrate
ID [bits]
y = 80.4x - 154
R² = 0.961
y = 55.9x - 138
R² = 0.993
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]
Widening
Narrowing
Accot’s ID model Accot’s ID model
main effects: ID (F5, 50 = 4.204, p < .01) and
tunnel type (F1, 10 = 12.111, p < .01)
post hoc test: widening produces less errors
than narrowing (p < .01; 1.05% vs. 6.78%)
Left/right directions had no significant effects on MT (F1, 10 = 0.083, p = .780) and error rate (F1, 10 = 0.040, p = .846)
Widening
Narrowing
Error rateMovement time

29. Without the separation, steering law shows a bad fit
The regression expression predicts nothing
Model Fitness of Conventional Steering Law
Steering law shows good fits for the both tunnel types
Users can predict MTNT and MTWT at high
accuracy when the tunnel types are separated
29
y = 80.4x - 154
R² = 0.961
y = 55.9x - 138
R² = 0.993
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]
y = 68.1x - 146
R² = 0.826
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]
✔
The center line between the two tunnel data
✘
Conventional model does not take account of this
Widening
Narrowing
Accot’s ID model
Accot’s ID model

30. Converting IDNT to “IDWT + IDGap”
𝐼𝐷Gap = 𝐼𝐷NT − 𝐼𝐷WT =
𝐴(𝑊𝐿 − 𝑊𝑅)
𝑘𝑊𝐿 𝑊𝑅
30
Original results of regression expression:
Narrowing: 𝑀𝑇 = −154 + 80.4 × 𝐼𝐷 ・・・①
Widening: 𝑀𝑇 = −138 + 55.9 × 𝐼𝐷 ・・・②
By using IDGap model,
regression expression for narrowing is onto ②
Narrowing: 𝑀𝑇 = −138 + 55.9 × 𝐼𝐷 +
𝐴(𝑊 𝐿−𝑊 𝑅)
𝑘𝑊 𝐿 𝑊 𝑅
y = 80.4x - 154
R² = 0.961
y = 55.9x - 138
R² = 0.993
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]
y = 55.3x - 113
R² = 0.991
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]
Our IDGap model
with k = 3.14
Widening
Narrowing
Accot’s ID model

31. Predicting MTNT from MTWT
Measurement:
step 1) measure MTWT at 6 IDs
step 2) measure MTNT at the lowest ID
step 3) calculate k from MTWT and MTNT
at the lowest ID (→ k = 2.22)
31
y = 55.9x - 138
R² = 0.993
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]
Widening
Narrowing
y = 55.9x - 138
R² = 0.993
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]
Prediction:
step 4) predict the regression line
of MTNT using IDGap(2.22)
step 5) predict MTNT at the other five IDs

32. Predicting MTNT from MTWT
32
y = 0.833x + 104
R² = 0.971
0
1000
2000
3000
0 1000 2000 3000
Observedtime[ms]
Predicted time [ms]
Prediction accuracy
The correlation coefficient between
predicted MTNT and observed MTNT shows good fit
Usefulness of IDGap model:
Prediction of MTNT at high IDs
Measurement of MTNT requires a long time and many errors
y = 55.9x - 138
R² = 0.993
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]

33. Speed Profiles: Velocity × Elapsed Time
Some characteristic peaks appear at regular time intervals, but sometimes not
33
0
1
2
3
4
8
48
88
128
168
208
248
288
328
368
408
448
488
528
568
608
648
688
728
768
808
848
888
928
968
1008
1048
1088
1128
Velocity[pixels/ms]
Elapsed Time [ms]
0 160 320 480 640 800 960 1120
Narrowing
Widening
Single raw speed profiles for narrowing and widening directions by participant B
A = 600, WL = 11, WR = 51

34. Speed Profiles: Velocity × Progress in the X-axis
Narrowing has more peaks (speed-ups/downs) than widening
34
0
1
2
3
4
1 101 201 301 401 501
Velocity[pixels/ms]
Progress in the x-axis [pixels]
Narrowing
Widening
0 100 200 300 400 500 600
A = 600, WL = 11, WR = 51
0
1
2
3
4
(Start) (End)
Single raw speed profiles for narrowing and widening directions by participant B
Movement corrections are affected by the tunnel type

35. Speed Profile of a Narrowing Tunnel
Movement corrections are affected by the current width
They are more often and low peaks where the current width is narrow
35
0
1
2
3
4
1 101 201 301 401 501
Velocity[pixels/ms]
Progress in the x-axis [pixels]
Narrowing
0 100 200 300 400 500 600
0
1
2
3
4
Wide = less & high peaks
Narrow = many & low peaks
(Start) (End)
WRWL
𝑎% 𝑏% 𝑐% 𝑑% 𝑒%

36. 0
0.3
0.6
0.9
1.2
1.5
1 101 201 301 401 501
Velocity[pixels/ms]
Average Speed Profiles
Speed profiles of all the strokes filtered by a seven-point simple moving average
“Turnovers” of the speed appeared at 25ｰ30% of the tunnel length A
36
0 100 200 300 400 500 600
Narrowing
Widening
A = 600
0
0.3
0.6
0.9
1.2
1.5
1 51 101 151 201 251
Velocity[pixels/ms]
0 100 200 300
1.5
1.2
0.9
0.6
0.3
0
Narrowing
Widening
A = 300
Progress in the x-axis [pixels] Progress in the x-axis [pixels]
(Start) (End)
1.5
1.2
0.9
0.6
0.3
0
(Start) (End)
Velocity[pixels/ms]
Velocity[pixels/ms]
This is in contrast of the conventional steering law’s local form

37. Local Form of Steering Law
37
“The speed at any point is proportional
to the permitted variability at that point” 𝑣 𝑠 =
𝑊 𝑠
𝜏
s
W(s)
velocity at the
current position
empirically determined
time constant
width of the current position
This suggests “turnovers” would appear
at 50% of A (where their widths are the same)
Another finding: Local form is also affected by the tunnel type:
whether the tunnel width will be narrowing or widening

38. Validity of IDGap
Can IDGap always be applied to narrowing and widening tunnels?
38
Steering law is held to circular tunnels
[Accot and Zhai 1999]
Steering law is held to A = 3.7－237 mm
[Accot and Zhai 2001]
Our model was tested under a limited condition (tablet size, tunnel parameters, etc.)
How about other sizes?How about other shapes?

39. Narrowing and Widening Circular Tunnels
Conventional steering law showed good fits without separating the tunnel type
39
y = 122x - 96.9
R² = 0.985
0
1000
2000
3000
4000
5000
0 10 20 30 40
MT[ms]
ID [bits]
y = 126x - 93.3
R² = 0.989
y = 119x - 101
R² = 0.991
0
1000
2000
3000
4000
5000
0 10 20 30 40
MT[ms]
ID [bits]
Narrowing
Widening
Accot’s model
(separated)
Accot’s model
(NOT separated)
IDGap model is not required for circular tunnels
Shota Yamanaka and Homei Miyashita. A Study of the Steering Time Difference between Narrowing and Widening
Circular Tunnels. Information and Media Technologies, 2016. (In Press)
>>> 3x

40. 0
0.1
0.2
0.3
0.4
0.5
0.6
1
27
53
79
105
131
157
183
209
235
261
287
313
339
Velocity[pixels/ms]
Progress in angle [degrees]
0 90 180 270 360
(c) A = 600
Narrowing
Widening
0
0.1
0.2
0.3
0.4
0.5
0.6
1
27
53
79
105
131
157
183
209
235
261
287
313
339
Velocity[pixels/ms]
Progress in angle [degrees]
(b) A = 450
0 90 180 270 360
Narrowing
Widening
0
0.1
0.2
0.3
0.4
0.5
0.6
1
27
53
79
105
131
157
183
209
235
261
287
313
339
Velocity[pixels/ms]
Progress in angle [degrees]
Narrowing
Widening
0 90 180 270 360
0.5
0.4
0.3
0.2
0.1
0
0.6
(a) A = 300
Narrowing and Widening Circular Tunnels
“Turnovers” appeared at 45-50% in all As
40
✘
Shota Yamanaka and Homei Miyashita. A Study of the Steering Time Difference between Narrowing and Widening
Circular Tunnels. Information and Media Technologies, 2016. (In Press)
Users cannot aim
the goal at early phase
Users can aim
the goal at early phase
✔

41. Scale Effects in Narrowing and Widening Tunnels
41
In submission.
1/1 (48×27 cm) 1/2 (24×13 cm) 1/4 (12×6.7 cm) 1/9 (5.2×3.0 cm) 1/12 (4.0×2.2 cm)

42. Scale Effects in Narrowing and Widening Tunnels
There are always the MT difference in each scale
IDGap model always improve the fitness, and k varied from 3 to 6
42
In submission.
separated
not separated
using IDGap
1/1 1/2 1/4 1/9 1/12

43. Future Work
Other devices/environments Other tunnel shapes
43
Where does the MT difference disappear?
Identifying the role of k
We want to know what k mainly reflects (A, WL, WR, device, etc.)
The results of the three experiments show that there is no optimal k value
k must be calculated in each conditions (scale, tunnel shape, device, etc.)
We have tested only a direct-input pen tablet

44. Summary
Additional results
• Circular tunnels did not require the IDGap model (A Study of the Steering Time Difference between
Narrowing and Widening Circular Tunnels, Information and Media Technologies, 2016.)
• The IDGap relationships were observed in very large to very small scales
44
𝐼𝐷Gap(𝑘)=
𝐴(𝑊𝐿 − 𝑊𝑅)
𝑘𝑊𝐿 𝑊𝑅
y = 55.3x - 113
R² = 0.991
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]
Our IDGap model
with k = 3.14
We derived IDGap model based on “a limited number of movement corrections” hypothesis
We confirmed that MTNT was longer than MTWT (p < .01; 1233 ms vs. 826 ms)
The data supported IDGap as a relationship between IDNT and IDWT
y = 80.4x - 154
R² = 0.961
y = 55.9x - 138
R² = 0.993
0
500
1000
1500
2000
2500
0 10 20 30 40
MT[ms]
ID [bits]WT
NT
Accot’s ID model

45. Same ID but Different MTs
View-point of racing games
[Bateman+ 2011]
Tunnel A & W ×Cursor size [Naito+ 2004]
Tunnel with a corner [Pastel 2006]
45
(ID)
(ID)
Cut-off

46. Consistency of IDGap and Accot’s Model
46
Our 𝐼𝐷Gap(𝑘) =
𝐴(𝑊𝐿 − 𝑊𝑅)
𝑘𝑊𝐿 𝑊𝑅
Accot′s 𝐼𝐷Narrowing =
𝐴
𝑊𝑅 − 𝑊𝐿
ln
𝑊𝑅
𝑊𝐿
• When WL → WR, IDGap(k) → 0
If the width becomes constant, the time difference between MTNT and MTWT becomes 0
• When k → 0, IDGap → ∞: not suitable for ID difference
ー no movement correction
ー users decide the cursor movement when start (only one time)
• When k → ∞, IDGap → 0 : confirming Accot and Zhai’s model
ー tunnel type (narrowing or widening) does not affect the degree of difficulty
ー assuming infinite number of (or continuous) movement corrections

47. Scale Effects in Narrowing and Widening Tunnels
47
In submission.
No significant “U-shaped” function of motor/visual scales
middle scale (≈A5) is the best: see Accot and Zhai's scale effects paper at CHI ’01
The smaller scale was, the worse performance became