The document summarizes three experiments investigating whether a general steering law model can accurately predict movement time for sequential linear path segments. The experiments found that: 1) Movement time was accurately predicted by the sum of the index of difficulties of the two path segments, even when path lengths and widths differed. 2) Speed profiles were affected by path shape (narrowing, widening, constant width), with slower speeds in the first segment of joined paths. 3) A revised model that included a crossing index of difficulty term for the path joint improved prediction accuracy by accounting for deceleration in the first segment.

1. Steering Through
Sequential Linear Path Segments
Shota Yamanaka (Meiji University and JSPS)
Wolfgang Stuerzlinger (Simon Fraser University)
Homei Miyashita (Meiji University)
Japan Society for the
Promotion of Science
Simon Fraser University
May 8, 2017
<<video_joined>>
Yahoo Japan Corporation
Meiji University

2. Application: Lasso Operations
Steering through various paths, corners, curves, and joints between them
2
Path joint

3. 𝑀𝑇 = 𝑎 + 𝑏
𝐴
𝑊
The Steering Law [Accot+, CHI '97, '99]
3
W
A
A
W
Movement time
Path length
Path width
Simple form,
constant-width paths
General form,
arbitrary paths
s
W(s)
𝑀𝑇 = 𝑎 + 𝑏
𝐶
𝑑𝑠
𝑊 𝑠

4. Derived Steering Models
4
Narrowing path
[Accot+, CHI '97]
Spiral path
[Accot+, CHI '97]
WL
A
WR
A
WRWL
Widening path
[Yamanaka+, CHI '16]

5. Revised Steering Model
Corner requires pointing motion
[Pastel, CHI '06]
5
W
A/2
A/2
𝑀𝑇 = 𝑎 + 𝑏
𝐴
𝑊
+ 𝑐 log2
𝐴/2
𝑊
+ 1
Fitts’ IDSteering-ID
W
A
𝑀𝑇 = 𝑎 + 𝑏 𝐴
Open-loop behavior in very wide paths
[Thibbotuwawa+, Ergonomics '12]

6. Steering Law Path Shape Variety
Does the steering law generally hold for any path shapes?
If not, how should the model be revised for each shape?
6
W
A
W
A
W
A/2
A/2
WL
A
WR
A
WRWL
A
W

7. Research Goal
• Can general steering law model time for sequential path segments?
7
𝑀𝑇 = 𝑎 + 𝑏
𝐶
𝑑𝑠
𝑊 𝑠
= 𝑎 + 𝑏
𝐴1
𝑊1
+
𝐴2
𝑊2
General model: “MT is linear to
the sum of the two ID values”
ID of path1 ID of path2
W1
A2 A2
W2
W1
A2 A2
W2
Path1 Path2
Path1 Path2

8. Exp. 1: Traditional Steering Task, as Baseline
• Instruction
“Pass through the path as quickly and accurately as possible.”
• 13 Participants
• Length A: 2 levels
480 and 640 pixels
• Width W: 4 levels
15, 23, 33, and 45 pixels
8
Avoid moving out of the path
<<video_single>>

9. Exp. 1: Results (Model Fitness, Speed)
R2 > 0.998
9
y = 47.666x - 57.524
R² = 0.9984
0
700
1400
2100
0 10 20 30 40 50
MovementTime[ms]
ID = A/W [bits]
• Higher speed in wider path
• Gradual acceleration after pen tip
contacted surface
0
0.4
0.8
1.2
1.6
2
960 1120 1280 1440 1600
Speedonthex-axis
Cursor position on the x-axis [pixels]
0 640480320160
W = 15 pixels
W = 23 pixels
W = 33 pixels
W = 45 pixels
𝑀𝑇 = 𝑎 + 𝑏
𝐴
𝑊
Speedonthex-axis[pixels/ms]

10. Exp. 2: Sequential Path Segments
Navigating two path segments (as used in Exp. 1)
Other conditions same as Exp. 1
Rule: A1 = A2, and W1 ≠ W2 (= always a joint between segments)
10
W1
A1 A2
W2
W1
A1 A2
W2
Path 1
Path 2
Path 1
Path 2
<<video_joined>>

11. Exp. 2: Results (Model Fitness)
R2 > 0.985
11
𝑀𝑇 = 𝑎 + 𝑏
𝐴1
𝑊1
+
𝐴2
𝑊2
y = 55.508x - 137.28
R² = 0.9856
0
1000
2000
3000
4000
0 20 40 60 80
Movementtime[ms]
ID = A1/W1 + A2/W2 [bits]
Steering law still holds even for
a pair of joined path segments

12. Exp. 2: Speed Analysis (Narrowing Path)
Path1
Deceleration in advance
of joint
Path2
Gradual acceleration.
similar to single path segment
12
0.0
0.2
0.4
0.6
0.8
1.0
1.2
800 1040 1280 1520 17600 240 480 720 960
Cursor position on the x-axis [pixels]
Speedonthex-axis[pixels/ms]
23
480 480
15
Path1
Path2

13. Speed Comparison to Exp. 1
In presence of joint, speed
decreased in 1st segment
・911 ms for a single path
・1323 ms for joined paths (45% up)
No notable change in 2nd segment
・1451 ms for a single path
・1444 ms for joined paths
(0.77% down)
Only behavior in 1st segment
affected by joint
13
0.0
0.2
0.4
0.6
0.8
1.0
1.2
800 1040 1280 1520 17600 240 480 720 960
Cursor position on the x-axis [pixels]
Speedonthex-axis[pixels/ms]
23
480 480
15
Exp. 1
Exp. 2
Exp. 1

14. 0
0.2
0.4
0.6
0.8
1
1.2
800 1040 1280 1520 1760
Speed[pixels/ms]
Cursor position on the x-axis [pixels]
Exp. 2: Speed Analysis (Widening Path)
14
0 240 480 720 960
Cursor position on the x-axis [pixels]
Speedonthex-axis[pixels/ms]
23
480 480
15
Speed decreased in 1st segment
1451 ms for single path
1688 ms for joined paths (16% up)
No notable change in 2nd segment
911 ms for single path
890 ms for joined paths (2.3% down)
Only behavior in 1st segment
was affected by joint
Single path: W=15
Single path: W=23
Joined paths: W1=15 & W2=23
Path1
Path2

15. Exp. 3: Other Sequential Path Sequences
15
A1: 150, 250, 400, 600, 800 pixels
A2: 400 pixels
W1 & W2: 15, 23, 39 pixels
The other conditions (participants, devices, etc.) are same as Exp. 1 & 2
W1
A1 A2
W2
Does the steering law hold for more general combinations of segments?
W1 = W2 (no joint)A1 ≠ A2 (the balance of the lengths are different)

16. Exp. 3: Results (Model Fitness)
High fitness: R2 > 0.962
Even for two joined path segments with
same path type, steering law holds
16
𝑀𝑇 = 𝑎 + 𝑏
𝐴1
𝑊1
+
𝐴2
𝑊2
y = 45.84x - 99.836
R² = 0.9626
0
1000
2000
3000
4000
0 30 60 90
MT[ms]
ID = A1/W1 + A2/W2 [bits]
・Ratio of lengths (A1 : A2)
・Narrowing/widening/constant-width shapes

17. 0
0.4
0.8
1.2
1.6
490 790 1090 1390
0
0.4
0.8
1.2
1.6
490 790 1090 1390
0
0.4
0.8
1.2
1.6
490 790 1090 1390
Exp. 3: Speed Analysis (Single Path)
Speed profiles similar to Exp. 1
• Speed higher in wider path
• Speed gradually increased after pen tip contacted surface
17
W = 15 pixels W = 23 pixels W = 39 pixels
0 300 600 900 1200
Cursor position on the x-axis [pixels]
Speedonthex-axis[pixels/ms]
0 300 600 900 1200
Cursor position on the x-axis [pixels]
Speedonthex-axis[pixels/ms]
0 300 600 900 1200
Cursor position on the x-axis [pixels]
Speedonthex-axis[pixels/ms]

18. Exp. 3: Speed Analysis (Joined Paths)
18
0
0.5
1
1.5
AxisTitle
Axis Title
0
0.5
1
1.5
AxisTitle
Axis Title
0 200 400 600 800 1000 12000 200 400 600 800 1000 1200
Cursor position on the x-axis [pixels]
Speedonthex-axis[pixels/ms]
Cursor position on the x-axis [pixels]
0
1.0
1.5
0.5
0
1.0
1.5
0.5
Speedonthex-axis[pixels/ms]
Speed profiles similar to Exp. 2
• 1st segment: slower speed than single path
• 2nd segment: no remarkable difference from single path
Narrowing path
(W1 = 23 > W2 = 15)
Widening path
(W1 = 15 < W2 = 23)

19. Improve prediction accuracy of steering law model through taking
slow-down behavior until joint into account
19
Narrowing path Widening path Constant-width path
0
0.5
1
1.5
480 880 1280 1680
Speedonthex-axis[pixels/ms]
0 400 800 1200
Cursor position on the x-axis [pixels]
0
0.5
1
1.5
480 880 1280 1680
Speedonthex-axis[pixels/ms]
0 400 800 1200
Cursor position on the x-axis [pixels]
0
0.5
1
1.5
490 890 1290
Speedonthex-axis[pixels/ms]
0 400 800 1200
Cursor position on the x-axis [pixels]
Revisions based on Deceleration in 1st Segment
Joints clearly affected users’ behaviors

20. Approach for Revised Model
Model entering 2nd segment as a crossing task
20
Crossing law [Accot+, CHI '02]: Time to cross a finite-length line:
𝑀𝑇 = 𝑎 + 𝑏 log2
𝐴
𝑊
+ 1
A
W

21. Our Revised Model
21
W1 W
2
A1 A2
5W1A1－5W1 A2
𝑀𝑇 = 𝑎 + 𝑏
𝐴1 − 5𝑊1
𝑊1
+ 𝑐 𝐥𝐨𝐠 𝟐
𝟓𝑾 𝟏
𝑾 𝟐
+ 𝟏 + 𝑑
𝐴2
𝑊2
Steering-ID
of 1st segment
Crossing–ID
of joint
Steering-ID
of 2nd segment
From related work [Senanayake+]

22. Best Model Selection
Other variations: Separate Steering-IDs
22
Model
Exp. 2 Exp. 3
R2 AIC R2 AIC
𝑀𝑇 = 𝑎 + 𝑏
𝐴1
𝑊1
+
𝐴2
𝑊2
0.986 290 0.963 578
𝑀𝑇 = 𝑎 + 𝑏
𝐴1
𝑊1
+ 𝑐
𝐴2
𝑊2
0.992 277 0.963 580
𝑀𝑇 = 𝑎 + 𝑏
𝐴1 − 5𝑊1
𝑊1
+ 𝑐 log2
5𝑊1
𝑊2
+ 1 + 𝑑
𝐴2
𝑊2
0.994 272 0.979 556
𝑀𝑇 = 𝑎 + 𝑏
𝐴1 − 5𝑊1
𝑊1
+
𝐴2
𝑊2
+ 𝑐 log2
5𝑊1
𝑊2
+ 1 0.996 272 0.970 571
General
Proposed
General
(Separated)
Proposed
(Merged)
Indicators for selecting best model: R2 (higher is better), AIC: (lower is better)
*Best values are in green cells

23. Limitations
When joint position is not at center of path,
operational strategies could depend on user
23
Many joined short path segments, or very short path lengths
W1 W2
A1
A2
W2
(A1 + A2) × 6

24. Steering Through Sequential Linear Path Segments
24
• MT accurately predicted by general steering law (= sum of two IDs),
even if path lengths are different (A1 ≠ A2), and widths are same (W1 = W2)
• Speed profiles affected by shape (narrowing/widening/constant)
• In joined path segments, deceleration was observed clearly in 1st segment
• Model fitness improved by inserting crossing-ID at end of 1st segment
Narrowing path Widening path
0
0.5
1
1.5
480 880 1280 1680
Speedonthex-axis[pixels/ms]
0 400 800 1200
Cursor position on the x-axis [pixels]
0
0.5
1
1.5
480 880 1280 1680
Speedonthex-axis[pixels/ms]
0 400 800 1200
Cursor position on the x-axis [pixels]
<<video_joined>>
Contact: syamanak@yahoo-corp.jp

25. Appendix: Revised Model
Movement to enter 2nd segment is modeled by crossing-ID
25
To determine crossing-ID, we have to know distance for crossing operation
(= red line)
① Steering ② Crossing ③ Steering

26. Appendix: Steering Tasks with a Pointing Motion
[Senanayake+, Experimental Brain Research '13]
To click a target at end of a steered path, observed motion changes
at 5W distance from target
(After that point, users seem not to worry about deviating from path)
26
Steering motion Pointing motion
5×W
W
Target

27. Appendix:
Revised Model with Fewer Free Parameters
27
𝑀𝑇 = 𝑎 + 𝑏
𝐴1 − 5𝑊1
𝑊1
+ 𝑐 𝐥𝐨𝐠 𝟐
𝟓𝑾 𝟏
𝑾 𝟐
+ 𝟏 + 𝑑
𝐴2
𝑊2
𝑀𝑇 = 𝑎 + 𝑏
𝐴1 − 5𝑊1
𝑊1
+
𝐴2
𝑊2
+ 𝑐 𝐥𝐨𝐠 𝟐
𝟓𝑾 𝟏
𝑾 𝟐
+ 𝟏
As number of free parameters (a, b, c, and d) increases,
prediction accuracy decreases (leads to “overfitting”)
If we assume that steering motion parameters (b & d) are similar,
we can merge two steering-ID values
steering steeringcrossing

28. Appendix:
General Model with More Free Parameters
Separating two steering motions potentially improves fitness
28
𝑀𝑇 = 𝑎 + 𝑏
𝐴1
𝑊1
+
𝐴2
𝑊2
𝑀𝑇 = 𝑎 + 𝑏
𝐴1
𝑊1
+ 𝑐
𝐴2
𝑊2
(Original)
(Separate)

29. Appendix: Paths Joined at the Top of the Y-axis
Pilot observation with three persons shows different strategies
29
Participant 1
Participant 2
Participant 3
Narrowing path Widening path