Applied Science, Faculty of
Electrical and Computer Engineering, Department of
DSpace
UBCV
Chen, Jennie Jia Yi
Master of Applied Science - MASc
University of British Columbia
The Steering Law has long been a fundamental model in predicting movement time for tasks involving navigating through constrained paths such as menu selections and virtual environments. The base model and past research on its extensions primarily focused on simple geometries such as straight lines and simple or compound circular arcs. While useful in simple user interfaces, they do not capture the complexities of real-world tasks where curvatures can vary arbitrarily. This limits the generalizability of the model to envi- ronments where users interact with more irregular paths. This thesis revisits and extends the Steering Law by introducing the total curvature pa- rameter K into the equation to account for the overall curviness characteristic of a path. Drawing on an information-theoretic interpretation of human motor behavior, we propose an extension to the Steering Law where K contributes as logarithmic noise in the movement time prediction equation: MT= a + b L + c·log2(K + 1) + d L·K To validate this extension, we conducted a mouse-steering experiment (N= 20) on fixed- width paths characterized by 3 levels of length and 3 levels of curviness. Width was held constant to isolate curvature effects. Results demonstrate that the introduction of K significantly improves model fitness for movement time prediction over traditional models. We further identify a significant in- teraction between length and curviness, which led to improved performance when incor- porated into the model. We also show that previous models developed for circular arc trajectories, when reformulated using the K parameter, generalize well to our experimental data. These findings support the hypothesis that curvature introduces a form of perceptual or motor noise that increases task difficulty in human steering behavior. Beyond empirical validation, this thesis presents a detailed account of the theoretical framing and experimental reasoning through a series of pilot studies and design explo- ration that informed the final model design. By integrating curvature in complex curved paths, this work advances our understanding of movement in curved paths and supports potential applications in fields like speech motor control and virtual navigation.
Revisiting the Steering Law: Curvature, Complexity, andthe Path to a New Predictive ModelbyJennie Jia Yi ChenB.Eng., McGill University, 2021A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Electrical and Computer Engineering)The University of British Columbia(Vancouver)April 2025© Jennie Jia Yi Chen, 2025The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Revisiting the Steering Law: Curvature, Complexity, and the Path to a New Predic-tive Modelsubmitted by Jennie Jia Yi Chen in partial fulfillment of the requirements for the degreeof Master of Applied Science in Electrical and Computer Engineering.Examining Committee:Sidney S. Fels, Professor, Electrical and Computer Engineering, UBCSupervisorSeptimiu E. Salcudean, Professor, Electrical and Computer Engineering, UBCSupervisory Committee MemberDongwook Yoon, Professor, Computer Science, UBCSupervisory Committee MemberiiAbstractThe Steering Law has long been a fundamental model in predicting movement time fortasks involving navigating through constrained paths such as menu selections and virtualenvironments. The base model and past research on its extensions primarily focused onsimple geometries such as straight lines and simple or compound circular arcs. Whileuseful in simple user interfaces, they do not capture the complexities of real-world taskswhere curvatures can vary arbitrarily. This limits the generalizability of the model to envi-ronments where users interact with more irregular paths.This thesis revisits and extends the Steering Law by introducing the total curvature pa-rameter K into the equation to account for the overall curviness characteristic of a path.Drawing on an information-theoretic interpretation of human motor behavior, we proposean extension to the Steering Law where K contributes as logarithmic noise in the movementtime prediction equation: MT = a+b L+ c · log2(K+1)+d L ·KTo validate this extension, we conducted a mouse-steering experiment (N = 20) on fixed-width paths characterized by 3 levels of length and 3 levels of curviness. Width was heldconstant to isolate curvature effects.Results demonstrate that the introduction of K significantly improves model fitness formovement time prediction over traditional models. We further identify a significant in-teraction between length and curviness, which led to improved performance when incor-porated into the model. We also show that previous models developed for circular arctrajectories, when reformulated using the K parameter, generalize well to our experimentaldata. These findings support the hypothesis that curvature introduces a form of perceptualor motor noise that increases task difficulty in human steering behavior.Beyond empirical validation, this thesis presents a detailed account of the theoreticalframing and experimental reasoning through a series of pilot studies and design explo-ration that informed the final model design. By integrating curvature in complex curvedpaths, this work advances our understanding of movement in curved paths and supportspotential applications in fields like speech motor control and virtual navigation.iiiLay SummaryThe Steering Law is a well-known model used to predict how long it takes people to movethrough narrow paths when navigating user interfaces like when navigating to a specificoption in a drop-down menu. However, the model assumes simple paths like smooth linesor circular curves, which do not fully represent the complex paths we encounter in real-world applications.This thesis expands on the traditional Steering Law by adding a new factor to accountfor the overall curviness of the path. Through a user study, we show that by including thiscurviness factor, the movement time prediction is more accurate. Our findings could helpimprove the design of user interfaces, virtual environments, and understand human musclecontrol for complex actions like speech production.ivPrefaceThe research study conducted in this thesis was performed at the Human-CommunicationTechnologies (HCT) laboratory at the University of British Columbia (UBC). The studywas approved by the UBC Research Ethics Board (certificate number H24-01850).A significant portion of this thesis has been published in [6] as:Jennie J.Y. Chen and Sidney S. Fels. 2025. Curves Ahead: Enhancing the Steering Lawfor Complex Curved Trajectories. In CHI Conference on Human Factors in ComputingSystems (CHI ’25), April 26-May 1, 2025, Yokohama, Japan. ACM, New York, NY, USA,12 pages. https://doi.org/10.1145/3706598.3713102I (Jennie Chen) was the lead investigator in this research and was responsible for con-ceptualizing the study, designing, implementing and conducting the experiment, analyzingthe data, and preparing the manuscript. Dr. Sidney Fels provided supervision and valu-able insights in formulating research questions, experiment design decisions and feedbackon the manuscript. Fellow colleagues of the HCT lab contributed by participating in pilotstudies, sharing their feedback on the experiment design, and offering their insights duringthe analysis and interpretation.In addition, throughout the writing of this thesis, I used ChatGPT to assist in refiningmy writing in terms of structure, grammar and style. All interpretations, analyses andconclusions are my own.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Fitts’ and Steering Law Models . . . . . . . . . . . . . . . . . . . . . . . 52.2 Steering Law Extensions and Evaluations . . . . . . . . . . . . . . . . . 72.2.1 Path Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Path Width and Length . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Effective Parameters and Accounting for Error . . . . . . . . . . 132.2.4 Compound Paths and Tasks . . . . . . . . . . . . . . . . . . . . . 142.3 Factors Affecting Fitts’ and Steering Tasks . . . . . . . . . . . . . . . . . 152.3.1 Scale Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16vi2.3.2 Operational Bias Effects . . . . . . . . . . . . . . . . . . . . . . 182.3.3 Directional Effects . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.4 Other Influential Factors . . . . . . . . . . . . . . . . . . . . . . 192.4 Applications of Fitts’ and Steering Law . . . . . . . . . . . . . . . . . . 202.4.1 Input Device Evaluation . . . . . . . . . . . . . . . . . . . . . . 202.4.2 Lasso Operations and Group Selection . . . . . . . . . . . . . . . 202.4.3 Virtual Reality and Virtual Driving . . . . . . . . . . . . . . . . . 212.4.4 Speech Production . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Experimental Design and Rationale . . . . . . . . . . . . . . . . . . . . . . 233.1 Experiment Design Summary . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 Steering Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.2 Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.3 Experimental Protocol . . . . . . . . . . . . . . . . . . . . . . . 243.1.4 Setup and Data Acquisition . . . . . . . . . . . . . . . . . . . . . 293.1.5 Software Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.6 Curve Representation and Generation . . . . . . . . . . . . . . . 323.2 Design Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Curviness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.2 Length (L) and Width (W ) . . . . . . . . . . . . . . . . . . . . . 383.2.3 Curve Representation and Construction . . . . . . . . . . . . . . 413.2.4 User Interface Elements . . . . . . . . . . . . . . . . . . . . . . 433.2.5 Experiment Protocol . . . . . . . . . . . . . . . . . . . . . . . . 463.2.6 Participant-Related Issues . . . . . . . . . . . . . . . . . . . . . 473.2.7 Number of Participants . . . . . . . . . . . . . . . . . . . . . . . 543.2.8 Error Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 Results, Modeling and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 564.1 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.1.1 Movement Time (MT ) . . . . . . . . . . . . . . . . . . . . . . . 574.1.2 Out of Path Movement (OPM) . . . . . . . . . . . . . . . . . . . 584.1.3 Average Speed (Vavg) . . . . . . . . . . . . . . . . . . . . . . . . 594.1.4 Learning Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Model Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.1 Proposed Models . . . . . . . . . . . . . . . . . . . . . . . . . . 61vii4.2.2 Comparison Models from Prior Work . . . . . . . . . . . . . . . 624.3 Model Fitting and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.1 Regression Results . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.2 Interpretation of Model Fit . . . . . . . . . . . . . . . . . . . . . 644.3.3 Comparison Models Fit . . . . . . . . . . . . . . . . . . . . . . . 664.3.4 Model Simplification . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Future Work and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 705.1 Limitations and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 705.1.1 Width and Path Shape Effects . . . . . . . . . . . . . . . . . . . 705.1.2 Effective Parameters . . . . . . . . . . . . . . . . . . . . . . . . 715.1.3 Scale and Generalization . . . . . . . . . . . . . . . . . . . . . . 715.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74A Pilot Studies and Helper Applications . . . . . . . . . . . . . . . . . . . . . 80A.1 Curve Visualizer Applications . . . . . . . . . . . . . . . . . . . . . . . 80A.2 Pilot Round 3 Landing Page Images . . . . . . . . . . . . . . . . . . . . 81B User Study Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.1 Introduction Speech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.2 Background Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86B.3 Consent Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87B.4 Video Transcript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89viiiList of TablesTable 3.1 Lengths distribution per L . . . . . . . . . . . . . . . . . . . . . . . . 29Table 3.2 Total curvature distribution per K . . . . . . . . . . . . . . . . . . . . 29Table 3.3 Lengths and total curvatures of each of the 9 trial types. . . . . . . . . 29Table 3.4 Summary of all 8 pilot rounds and changes made to the experiment design. 33Table 3.5 Parameters of the trial types used in Rounds 6 and 7 of the pilot experi-ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Table 3.6 Parameters of the trial types used in Round 8 of the pilot experiment. . 38Table 3.7 Trial statistics of the tasks used in Round 2 of the pilot experiment. . . 40Table 3.8 Trial statistics of the tasks used in Round 3 of the pilot experiment. . . 49Table 4.1 Movement time [ms] for each of the 9 trial types. . . . . . . . . . . . . 58Table 4.2 Out of path movement [%] measure for each of the 9 trial types. . . . . 58Table 4.3 Average speed [px/ms] for each of the 9 trial types. . . . . . . . . . . . 60Table 4.4 Regression coefficients (with 95% confidence intervals), adjusted r2 andAIC values for models predicting MT . Note that *, **, and *** indicatep< 0.05, p< 0.001, and p< 0.0001 respectively. . . . . . . . . . . . 65Table 4.5 Average Root Mean Square Error values computed across 5-fold crossvalidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67ixList of FiguresFigure 1.1 A children’s game akin to a steering task. The goal is to move a metalloop from one end of a wire to the other without the two componentstouching each other. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Different types of paths with the same width and length. A) Straight;B) Circular arc; C) Sinusoidal; D) Path with corner. . . . . . . . . . . 3Figure 2.1 The original Fitts’ Law task, where users move a stylus between twoplates. The width and distance between the two target plates is variedbetween different trials. . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 2.2 Intuition behind the derivation of the Steering Law from Fitts’ Law:A straight tunnel of length L can be approximated as a sequence of Nevenly spaced goals. . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.3 Paths types with different corner angles used in the experiment in [25]. 8Figure 2.4 The steering task in [19] where users use a pen as input device to movea virtual ball through various Three-Dimensional (3D) tunnels. . . . . 8Figure 2.5 An example of a path type in the experiment in [18]. S1, S2 and S3designate sections of the tunnel of constant curvature, while J1 and J2identify the joints between S sections where there is an inflection inpath direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.6 An example of a compound path composed of N=6 arcs of differentangles (α), lengths, and smoothly varying widths. The direction arrowindicates the orientation of the overall movement task. . . . . . . . . 10Figure 2.7 Examples of the steering tasks in [34] with different radii of curvature.The white area designates the steering task path, while the blue andgreen areas designate the staring and ending zones, respectively. . . . 11Figure 2.8 The Narrowing and Widening conditions in [32], where the widthsmoothly varies from start to finish. . . . . . . . . . . . . . . . . . . 12xFigure 2.9 The Narrowing and Widening conditions in [35], where there is a dis-crete jump in the width constraint. . . . . . . . . . . . . . . . . . . . 12Figure 2.10 A straight tunnel used in steering tasks, overlaid with a hypotheticalheatmap of users’ cursor trajectories. The distribution of hits perpen-dicular to the length of the tunnel is modeled as Gaussian. The pinklines delineate We, which encompasses 96% of cursor hits. . . . . . . 13Figure 2.11 The combined steering and pointing task used in [7]. The grey zone in-dicates the steering tunnel, while the green zone indicates the clickingtarget. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.12 The combined steering and pointing task used in [11]. The green zonesdesignate the clicking targets with varying sizes WT for different trialtypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.13 The relative tablet display sizes in the experiment in [3]. All aspectratios are maintained. The scale factor S is inversely proportional toboth the horizontal and vertical dimensions between displays. . . . . . 16Figure 2.14 Visualization of the visual angle metric in a Fitts’ task, which is mea-sured as the perceived angle between the two targets from the partici-pant’s point of view. The visual angle depends on the distance betweenthe two targets as well as their distance from the observer. . . . . . . . 17Figure 2.15 A lasso operation in a drawing program. . . . . . . . . . . . . . . . . 21Figure 3.1 An example of the steering task as demonstrated through the first tu-torial trial encountered by participants. a) Idle, awaiting participantaction; b) Subject has clicked on the start button and is steering to theright flag button; c) Subject has reached and is clicking on the rightflag button; d) Subject is steering to the left flag button; e) Subject hasreached and clicked the left flag button, ending the first trial. . . . . . 25Figure 3.2 Tunnel tasks for each of the 9 trial types. . . . . . . . . . . . . . . . . 26Figure 3.3 An example of a tutorial trial. A red error circle is shown to indicatethe location where the participant exited the boundaries of the tunnel. 26Figure 3.4 The experimental procedure flowchart. . . . . . . . . . . . . . . . . . 27Figure 3.5 The two possible orientations of trial ‘L0-K0’. A) Original orientation;B) Flipped orientation. . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 3.6 Hardware setup for the experiment and mouse sensitivity settings. . . 30Figure 3.7 Visualization of the types used in Rounds 6 and 7 of the pilot experiment. 36xiFigure 3.8 Box plots of the movement times obtained from Round 6 and 7 of pilottesting, organized by a) Length; and b) Energy. A clear trend is visiblefor the 3 levels of length, but not for energy. . . . . . . . . . . . . . . 37Figure 3.9 Box plots of the movement times obtained from Round 8 of pilot test-ing, organized by a) L; and b) K. A clear trend is visible for both metrics. 38Figure 3.10 The 5 trial types used in Round 1 of varying widths. . . . . . . . . . . 39Figure 3.11 Plots of the a) Number of path exits; and b) Movement time (ms); withrespect to different path widths from Round 1 of pilot testing. . . . . . 39Figure 3.12 Box plot of the movement times obtained from Round 2 of pilot testing,organized by length. . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.13 Representation of simple circular arc path and a compound path ofcircular arcs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.14 G0/G1 continuitity matching: Tangent vector−→T at the end of curve Aand at the start of curve B are matched. . . . . . . . . . . . . . . . . 42Figure 3.15 Scatter plot of the movement times obtained from Round 3 of pilottesting. The grey dashed lines mark the block boundaries. Each colordesignates a different participant. . . . . . . . . . . . . . . . . . . . . 49Figure 3.16 Scatter plot of the pseudo-speed obtained from Round 3 of pilot testing. 50Figure 3.17 Scatter plot of the pseudo-speeds obtained from Round 4 of pilot testing. 51Figure 3.18 The power law of practice fitted on the tutorial movement time data foreach participant in Round 4 of pilot testing. . . . . . . . . . . . . . . 51Figure 3.19 Box plot comparing the movement times obtained from the experimentphases of Round 3 (no tutorial practice) and Round 4 (after 36 tutorialtrials as practice) of pilot testing. . . . . . . . . . . . . . . . . . . . . 52Figure 3.20 The power law of practice fitted on the movement time for the partici-pant in Round 5 of pilot testing. . . . . . . . . . . . . . . . . . . . . 53Figure 4.1 Plots of the mean total movement time for each of the 9 trial types.Note that the boxes present the Interquartile Range (IQR) and median,while whiskers extend to 1.5 * IQR. Each color in the strip plot repre-sents a different participant. . . . . . . . . . . . . . . . . . . . . . . . 57Figure 4.2 Plots of the out of path movement measure for each of the 9 trial types. 59Figure 4.3 Plots of the average speed for each of the 9 trial types. . . . . . . . . 60Figure 4.4 Visualization of model fitness of MT (or log10(MT ) for Equation 4.8)plotted against ID for each model. . . . . . . . . . . . . . . . . . . . 69xiiFigure 5.1 Heatmap of trajectory frequencies computed over 300 repetitions foreach of the L2 trial types. . . . . . . . . . . . . . . . . . . . . . . . . 72Figure B.1 Steering experiment background survey questions. . . . . . . . . . . 86Figure B.2 Steering experiment consent form, page 1. . . . . . . . . . . . . . . . 87Figure B.3 Steering experiment consent form, page 2. . . . . . . . . . . . . . . . 88xiiiGlossary1D One-Dimensional2D Two-Dimensional3D Three-DimensionalAIC Akaike Information CriterionANOVA Analysis of VarianceCSV Comma-Separated ValuesCV Coefficient of VariationDPI Dots Per InchGUI Graphical User InterfaceHCI Human-Computer InteractionIQR Interquartile RangeOPM Out of Path MovementRMSE Root Mean Squared ErrorSVG Scalable Vector GraphicsUI User InterfacexivAcknowledgmentsThis thesis marks the end of a long and curvy journey—one I should perhaps attempt tocompute the index of difficulty for another time.Through the ups and downs, one constant has been the guidance and patience of my super-visor, Dr. Sidney Fels. Beyond his support, I’m especially grateful for his encouragementto question assumptions and explore unconventional directions, which taught me how toapproach problems with curiosity and creativity.To everyone else who helped carry me through this journey, thank you.To my HCT labmates, without you, this whole research journey would have felt incrediblylonely. To Deb, thank you for being my mentor since Day 1. To Hamid and Nima, thankyou for your warm greetings and conversations every time I walked into x427. To Kaseyaand Sahar, thank you for being my CHI trailblazers and discussion partners. To Ursa, thankyou for our shared coffees and impromptu yapping sessions across monitors.To my friends back home, malgre´ les fuseaux horaires qui nous se´parent, merci de tou-jours me rappeler d’ou` je viens. To the friendos, especially Shirley and Zheng, thank youfor the video calls, memes, and check-ins that anchored me when I felt far away fromeverything familiar.To all of the friends I made in Vancouver, your friendship transformed this place into ahome. To my fellow ECEGSA execs, thank you for making our year on the council trulyspecial. To Marie, Elena and Anna, thank you for your thoughtfulness and for alwayschecking in during hard times. To Daisy and Erick, thank you for the trips and weekendadventures. My brain chemistry is forever altered by that spontaneous run and ocean dip ona warm September evening, right when I needed it the most. To the tennis gang, how canI even begin to express how much I will miss our Thursday evenings and long weekendroadtrips? To Reva, thank you for being such a kind soul and for your enthusiasm. ToRoozbeh, thank you for your trust and for always being there when I need support. ToMalinda, thank you for seeing through me so clearly, for always making time to hear to myproblems and for pushing me when I needed it most. I don’t know if I would’ve submittedthat draft without you, and for that, I’m endlessly grateful.xvTo Yueh, my partner in crime, thank you for being my anchor and my constant. I willforever be thankful that you chose to accompany me on this long journey despite all itstwists and uncertainties.To Enzo and Lena, the best feline research assistants, thank you for causing the perfectamount of chaos.And to my parents, thank you for your unwavering belief in me, even when my workprobably feels like a mystery to you. Your unconditional support and love have carried mefarther than you realize. I hope this thesis makes you proud.xviChapter 1IntroductionConsider a children’s game where the objective is to guide a metal loop along a curvy wirewithout touching it as shown in Figure 1.1. The difficulty of this task is directly influencedby the length of the path, the size of the loop, and the bends in the wire, but how muchmore difficult does it get if we straightened the wire or added an extra bend? The game,while simple, illustrates a broader problem in human performance modeling.In Human-Computer Interaction (HCI) research, two models are commonly used in move-ment time prediction and task difficulty evaluation. The most prominent one, Fitts’ Law,describes the movement time taken to perform reciprocal pointing tasks based on the widthof the targets and the distance between them [8]. One formulation of Fitts’ Law is derivedfrom Shannon’s Theorem 17 in information theory [26]. In this analogy, information istransmitted through the human channel, and movements are assigned an index of difficulty(ID, given in bits) based on the task’s parameters [20]. The amplitude of the movementneeded to perform a pointing task is analogous to signal power, and the width constraint ofthe pointing target is analogous to noise power, limiting the movement time - the informa-tion capacity of the human motor system.The second human performance model, the Steering Law, proposed by Accot and Zhai,extends Fitts’ Law to model trajectory-based tasks where users must steer through a con-strained tunnel [1]. Once again, the tunnel’s length and width constraints are used to deter-mine the index of difficulty of the task.In HCI research, Fitts’ Law and Steering Law are often applied to User Interface (UI)design such as navigating through drop-down menus and designing button interactions,evaluating the performance of input devices [2, 27, 38] and comparing the performance ofuser groups [30, 43]. There are also proposed models that combine the two fundamentallaws for compound tasks that involve targeted steering actions [7, 11, 15]. However, asinteraction tasks become more complex, the limitations of these fundamental models be-1Figure 1.1: A children’s game akin to a steering task. The goal is to move a metalloop from one end of a wire to the other without the two components touchingeach other.come apparent. For instance, all of the paths illustrated in Figure 1.2 would be assignedthe same ID by the Steering Law as they have the same width and length. Therefore, itis expected that all four paths would take the same amount of time to steer through, butintuitively, we know that there would be some variation due to their differing shapes.There have been many proposed extensions to expand the Steering Law for more complextunnel shapes, such as widening/narrowing widths [32], tunnels with corners [25], curvedpaths [18, 23, 34, 39], paths with sequential segments of varying widths [18, 35, 37], etc.Specifically in the curved paths case, previous research established that curvature plays asignificant role in steering tasks [22, 39]. However, these studies are limited to scenariosinvolving constant curvature, such as circular paths or compound paths with circular arcs.While these findings are valuable in demonstrating the effects of curvature, they may notaccurately predict user performance in more complex scenarios where curvature can varyarbitrarily within a single path, such as Two-Dimensional (2D) graphics segmentation or2Figure 1.2: Different types of paths with the same width and length. A) Straight; B)Circular arc; C) Sinusoidal; D) Path with corner.image labeling tasks. The ability to predict the movement time and evaluate the ID ofarbitrarily curved paths would also be helpful in determining level difficulties in videogames such as Osu! [10], Trombone Champ [28], or even physical ones such as the wireand loop game.In our case, we are interested in applying the Steering Law in areas outside of UI re-search. Although they are typically evaluated on hand-based interactions, human perfor-mance models have valuable potential in explaining the complexities of other motor controltasks such as vocal articular trajectories and tongue movements during speech production[14, 17].Motivated by the need to enhance the predictive ability of steering models for a broaderrange of tasks such as navigation through arbitrarily curved paths, this study incorporatesthe overall curviness of a tunnel task as a critical factor. It is important to note that whilewidth is a traditional factor in movement models, our current work does not examine itseffects and instead focuses instead on the implications of curvature. Our study design holdsthe width constant across all conditions to isolate the effect of curviness. Similar to howthe width constraint is analogous to noise in Fitts’ Law’s physical interpretation of the taskparameters, we hypothesize that the overall curviness of a path is also a source of noise,meaning that it would introduce variability in the overall movement time.31.1 Research QuestionsThis thesis addresses the following questions:1. How do we model the overall curviness of a steering task path?2. How do we account for the overall curviness in the movement time prediction?3. How does the predictive performance of curviness-augmented models compare tothe base Steering Law formulation?1.2 ContributionsThe contributions of this work are:1. We include a new factor, the total curvature parameter K, into the Steering Law toaccount for varying curvatures, creating models that are more applicable to a broaderrange of steering tasks;2. We validated model variations and compared their fitness with models derived fromprevious work in a mouse-steering experiment with 20 participants on tunnel pathsof varying lengths and curviness levels. We used a 3 L x 3 K x 15 repetitions x 20participants design providing 2700 trials.Our results showed that the introduction of K significantly improves the SteeringLaw model movement time prediction;3. We developed a methodological framework by iteratively refining the experimentprotocol through pilot testing, carefully designing task structure, tutorial and inter-face to address common biases, participant experience, and improve upon limitationsin prior steering experiments.1.3 Thesis StructureChapter 2 reviews related work on the Fitts’ Law and Steering Law, as well as their var-ious extensions. Chapter 3 outlines the experimental design, including path generation,experiment protocol, and data collection. It also presents the design rationale for the ex-periment, detailing each decision back by pilot studies and past research insights. Chapter4 presents the experiment results, the incorporation of total curvature into the Steering Lawand the modeling details and analysis. Chapter 5 discusses the limitations and concludeswith directions for future work.4Chapter 2Related WorkIn this chapter, we discuss the foundation model and key extensions to the Steering Law,with a focus on prior research on complex steering tasks. We begin by introducing Fitts’Law, which is used in pointing tasks, and its direct extension, the Steering Law for steeringtasks. Next, we review various refinements to the Steering Law that have been proposedto extend it to handle paths of varying width, length, curvature, and complexity. We alsodiscuss effective parameters and methods for incorporating user error and movement vari-ability.In addition to spatial constraints, we explore studies on factors that influence steeringperformance such as visual scale, operational biases, movement direction, and temporalconstraints, which motivate the experimental decisions in our own research. Finally, weoutline practical applications of Fitts’ and Steering Law to highlight the relevance of thesehuman movement models in the field of human-computer interaction and beyond.2.1 Fitts’ and Steering Law ModelsFitts’ Law [8], originally introduced by Paul Fitts in 1954, describes the fundamental trade-off between speed and accuracy in human movement tasks. Fitts conceptualized reciprocalOne-Dimensional (1D) target selection tasks as an information-processing task and pro-posed a linear relationship between movement time (MT ) and task difficulty, quantified asthe index of difficulty (ID). This relationship is given by:MT = a+b · ID (2.1)ID= log2(2LW)(2.2)5where a and b are empirically determined constants, L is the distance between the twotargets and W is the width of the targets. Figure 2.1 demonstrates the experiment setup forthe pointing task. It was later extended for 2D target tasks with different approach anglesand target heights [4, 21]. Its robustness across different tasks and input devices has madeit a foundational model in HCI research and usability evaluations.Figure 2.1: The original Fitts’ Law task, where users move a stylus between twoplates. The width and distance between the two target plates is varied betweendifferent trials.While Fitts’ Law focuses on discrete pointing tasks, it does not explicitly address contin-uous constrained movements. By framing 2D steering tasks as a series of small pointingtasks along the length of a path, Accot and Zhai (1997) applied Fitts’ Law to computethe cumulative effect of small target acquisition tasks and obtained the Steering Law [1].Figure 2.2 shows the intuition behind this modeling.The general form is expressed by the following equation for a path C of width W (s) atpath length s:ID=∫CdsW (s)(2.3)In cases where path width is constant (W (s) =W ), the ID can be rewritten as:ID=LW(2.4)The ID of a steering task is also related to MT in the same linear relationship as Fitts’Law in Equation 2.1:MT = a+b · LW(2.5)6Figure 2.2: Intuition behind the derivation of the Steering Law from Fitts’ Law: Astraight tunnel of length L can be approximated as a sequence of N evenlyspaced goals.In Accot and Zhai’s original work, the Steering Law was validated for three differenttypes of paths: a straight tunnel of constant width, a straight tunnel of decreasing width,and a spiral tunnel of increasing width [1].In a later work, it was verified on circular paths of constant curvature [2]. However, thedata was analyzed separately from straight paths and the authors note that circular pathswere significantly harder than straight paths, indicating that curviness may have a factor toplay in the overall movement time.2.2 Steering Law Extensions and EvaluationsWhile the Steering Law provided a foundational model for continuous, constrained move-ments, subsequent research extended this model to address additional spatial constraintsthat introduce path complexity.2.2.1 Path CurvaturePastel’s work (2006) extended the Steering Law to paths with corners, a form of pathcomplexity involving a change in curvature that deviates from the traditional straight anduniform paths [25]. A mouse-steering experiment was conducted on straight paths (base-line condition) and paths with three corner angles (45°, 90° and 135°), shown in Figure2.3.Additionally, two path widths and six path lengths were tested. The study modeled the7Figure 2.3: Paths types with different corner angles used in the experiment in [25].task as a coupled motion of steering through a straight path and pointing at the corner, andfound that the corner angle had a significant effect on the overall movement time, with 90°being the most difficult as it requires a sharp movement deceleration and redirection. Aninteresting finding from this study was that narrower paths resulted in a corner-aiming be-havior, where participants would slow down significantly at the corners to maintain precisecontrol in order to stay within the boundaries of the path. As for wider paths, participantstended to cut corners by taking shorter, more curved trajectories. However, since the studyfocuses on discrete corner angles, the applicability to paths with continuous curvature vari-ations remains unexplored.Liu et al. (2010) investigated the effects of path curvature and orientation in Three-Dimensional (3D) interactive steering tasks through experiments where participants used apen input device to navigate a virtual ball through paths of varying but fixed length, width,curvature and orientation [19], as demonstrated in Figure 2.4.Figure 2.4: The steering task in [19] where users use a pen as input device to move avirtual ball through various 3D tunnels.They found that the curvature and orientation of the circular arc path significantly in-8fluenced the overall steering time. This is in contradiction to the original Steering Lawformulation, which only depends on the length and width parameters Specifically for thecurvature effect, they found that a model of the form:log10(MT ) = a+b[log10(LW)+ c κ](2.6)where κ is the curvature of the circular arc provided the best fit to the experiment data. 1In another study, Liu and Liere (2010) extended their experiment to include scenarioswhere path properties change dynamically during a task by joining semicircular arcs ofdifferent widths and curvatures to form a complex tunnel shape [18] (Figure 2.5).Figure 2.5: An example of a path type in the experiment in [18]. S1, S2 and S3designate sections of the tunnel of constant curvature, while J1 and J2 identifythe joints between S sections where there is an inflection in path direction.They compared the average steering velocities for each of the tunnel shapes, as well as foreach type of tunnel section (J-joints and S-semicircles) within each tunnel shape. Althoughthey did not propose a model, they found that the average velocity decreases when navi-gating through J sections and that curvature is highly correlated with the average velocitythrough S sections. The sectioning and speed analysis approach they employed was usefulin this case since the complex tunnel shape was composed of simple circular arcs. Thisapproach, however, may not extend well to scenarios where curvature is not uniform.Drawing inspiration from the 2/3 Power Law relationship in curved trajectories, Nanceland Lank (2017) proposed an augmented model to accommodate paths of varying curvature1The logarithm used by Liu et al. was base 10. We keep the same form here for consistency.9and width [23] of the form:MT = a+b′∫CdsW (s) ·R(s)1/3 (2.7)where R(s) is path C’s radius of curvature at path length s. Similar to Liu and Liere’sexperiment, their steering task shape consisted of compound paths of joint circular arcs, asshown in Figure 2.6.Figure 2.6: An example of a compound path composed of N=6 arcs of different an-gles (α), lengths, and smoothly varying widths. The direction arrow indicatesthe orientation of the overall movement task.In their experiment however, each arc had smoothly varying widths, different arc anglesand orientations. The experiment was conducted on two input modalities: touch (fingeron tablet) and mouse. In their analysis, they binned the continuous data based on twopath properties: 1) Path width (W (s)) into 10 equal-width bins spanning 3 to 13 mm; and2) Radius of curvature (R(s)) into 3 bins spanning 6 to 41 mm. This resulted in a totalof 30 bins. For each bin, the average movement speed, movement time and error ratewere computed. Linear regression was performed on the data which showed that the baseSteering Law fails to predict movement time when curvature varies and that their proposedmodel significantly improves prediction accuracy. It remains to be seen how the modelperforms for paths of arbitrary curvature.Yamanaka and Miyashita (2019) investigated how the path curvature of simple circulararc paths affects movement time in pen-steering tasks [34] (Figure 2.7).10Figure 2.7: Examples of the steering tasks in [34] with different radii of curvature.The white area designates the steering task path, while the blue and green areasdesignate the staring and ending zones, respectively.The conditions tested were 3 levels of path width, 8 levels of path curvature, and 2 levelsof path length. Similar to past studies, they found that the base steering law model does notgeneralize well to curved paths and that curvature significantly affects movement speed.They also found interactions between path width and curvature, where for sharper curves,larger widths had little effect on movement speed, whereas gentler curves led to a largereffect of width on steering speeds. They fitted several regression models to predict theaverage speed and movement time based on width, length and the radius of curvature ofthe arc, and proposed a model of the form:MT = a+bLW + c 1R +dWR(2.8)Although the model showed good fit to the data, it was only validated on simple circulararcs.While previous extensions have expanded the Steering Law to accommodate diverse pathtypes, they do not capture the steering behavior in paths with non-uniform, arbitrary cur-vatures, which our research aims to address directly.2.2.2 Path Width and LengthYamanaka and Miyashita (2016) investigate whether the navigation time through lineartunnels of varying width is affected by the narrowing or widening of the width and estab-lish a model for the relationship between ID and MT in both of these cases [32]. Theyconducted a pen-steering experiment where participants navigated paths of narrowing andwidening widths, shown in Figure 2.8, recording the movement time as they steered fromone end of the tunnel to the other.The authors found that the widening direction was faster and less error-prone than nar-11Figure 2.8: The Narrowing and Widening conditions in [32], where the widthsmoothly varies from start to finish.rowing and proposed a correction factor to account for the difficulty between the two di-rections.Yamanaka et al. (2017) explore extending the Steering Law to linear paths composed oftwo segments of different widths (W1 and W2) [35], as shown in Figure 2.9.Figure 2.9: The Narrowing and Widening conditions in [35], where there is a discretejump in the width constraint.They performed pen-steering experiments on three different conditions: widening (W1 <W2), narrowing (W2 >W1) and constant (W1 =W2). Through a speed profile analysis,their results show that participants anticipate the width of the next path. Specifically innarrowing cases, participants would slow down before the W2 segment. However, theauthors found that the naive approach of summing the ID for each segment fits well to thedata despite not modeling this anticipatory behavior. They also propose a more complexSteering-Crossing-Steering model which yields a slightly better fit that includes a crossingterm for the transition movement between the two segments.122.2.3 Effective Parameters and Accounting for ErrorKulikov et al. (2005) investigated how the Steering Law can be improved by incorporatingerror tolerance and accounting for actual user behavior [16]. They argue that in prior mod-els, studies typically discard trials where users make errors (i.e., exiting the defined path),which does not reflect real world scenarios where users can make mistakes. Additionally,the base Steering Law model assumes that users fully utilize the available width of the tun-nel, but that in practice, this would depend on the constraints of the task. They proposedthe use of effective width We, inspired by Fitts’ Law research, applied to the Steering Lawformulation to account for the variability in the user’s movement. We is computed as:We = 4.133 ·σ (2.9)where σ represents the standard deviation of the cursor trajectory perpendicular to the path.Figure 2.10 demonstrates the intuition behindWe, where the green shaded area represents aheatmap of users’ cursor trajectory. Basically, We provides a description of a participant’sFigure 2.10: A straight tunnel used in steering tasks, overlaid with a hypotheticalheatmap of users’ cursor trajectories. The distribution of hits perpendicular tothe length of the tunnel is modeled as Gaussian. The pink lines delineate We,which encompasses 96% of cursor hits.steering behavior for a trial’s spatial constraints.They also proposed Out of Path Movement (OPM) as a new metric to capture steeringerrors. OPM is computed as the percentage of cursor sampled outside of the tunnel bound-ary.A pen and mouse-steering experiment was conducted where participants were instructedto steer through the tunnel from start to finish as quickly and as accurately as possible. Twotypes of constraints were evaluated: None (no visible tunnel boundaries) and Lines (visibletunnel boundaries). Two movement directions were evaluated (right and left), as well as13five path widths and five path lengths. Their results showed that using We consistentlyimproves movement time predictions. They also found that as expected, users do not fullyutilize the entire path width available, meaning that users tend to steer along a tightertrajectory.Yamanaka (2019) explored error-accepting delays in steering tasks as another way toaccount for real user interaction behavior instead of directly discarding trials where usersleave the designated path [31]. In the experiment, users were allowed to leave the path forup to a short interval of Tdelay before it is counted as an error. They found that allowinglimited deviations improved model accuracy and realism.Kasahara et al. (2024) investigated how using effective parameters can improve the cal-culation of throughput (TP) in a pen-steering experiment on three types of paths (linear,circular and sine-wave) where participants were subject to different speed-accuracy biases(accurate, neutral and fast) [12]. In addition to effective width, they also evaluated effec-tive length Le, which is defined as the actual distance traveled by the user, as opposed toL which is the nominal path length. They found that using We led to more stable TP inall speed-accuracy biases conditions. In circular paths, they found that users tend to cutcorners by staying close to the inner side of the track, which led to Le improving the modelfit and TP stability. Their work shows that the use of effective parameters in performancemodeling stabilizes TP computation by accounting for speed-accuracy biases and providesa more robust measure for applications such as input device evaluation.Overall, strategies that account for actual user steering behavior such as using effectiveparameters or allowing for a small error interval prove to be useful in smoothing out speed-accuracy biases as well as improve model accuracy.2.2.4 Compound Paths and TasksThe previously discussed works by Pastel (2006) [25] and Yamanaka et al. (2017) [35] aretwo examples of compound paths composed of straight segments.Additionally, Dennerlein et al. (2000) conducted a combined steering and targeting ex-periment [7]. Half of the participants were asked to perform a pure steering task through astraight tunnel, while the other half would traverse the tunnel then stop in a small clickableregion. The combination task is shown in Figure 2.11.Participants repeated the tasks with both regular and force-feedback mice. They foundthat the haptic feedback reduced the number of errors and movement times. More impor-tantly, they found that the steering sub-task contributed more to the overall movement timethan the pointing sub-task.Hong and Ryu (2007) introduce a combined model that integrates the Steering Law and14Figure 2.11: The combined steering and pointing task used in [7]. The grey zoneindicates the steering tunnel, while the green zone indicates the clicking target.Fitts’ law to account for real-world tasks that combine both steering and targeting [11].They conducted a mouse-steering experiment where participants steer a cursor through apath and click on a target at the end. 3 path widths, 3 target widths and 4 path lengths weretested. The task is shown in Figure 2.12.Figure 2.12: The combined steering and pointing task used in [11]. The green zonesdesignate the clicking targets with varying sizes WT for different trial types.The authors found that the movement speed follows a three-phase pattern: Acceleration(cursor speeds up at the start, similar to targeting), constant speed (cursor maintains steadyvelocity in the middle of the path), and deceleration (cursor slows down before reachingtarget). They found that for wider and shorter paths, the movement time follows Fitts’ law,whereas for narrower and longer paths, the movement time is closer to the Steering Law’sprediction. Their findings further motivate the need to constrict width and ensure that thepath length is sufficiently long in our experiments in order to minimize the contributionsfrom pointing.2.3 Factors Affecting Fitts’ and Steering TasksPointing and Steering performance is sensitive to a range of factors beyond spatial con-straints like length, width and curvature. Understanding these additional factors is impor-15tant for precise experimental protocol design and controlling for potential biases.2.3.1 Scale EffectsMovement scale refers to the physical amplitude of movement when performing a task.Accot and Zhai (2001) investigated how movement scale affects performance in trajectory-based tasks. They conducted a steering experiment with linear and circular paths as the taskat five different movement scales (S= 1,2,4,8,16). The relative sizes of the display ratiosare shown in Figure 2.13.Figure 2.13: The relative tablet display sizes in the experiment in [3]. All aspectratios are maintained. The scale factor S is inversely proportional to both thehorizontal and vertical dimensions between displays.They found that extremely large or small physical movements degrade performance,forming a U-shaped function between performance and scale. Smaller scales require moremotor precision while at larger scales, users must move their entire arm which introducesbiomechanical constraints as it requires different muscles.Their findings have two implications for our study design: 1) While the Steering Law stillholds within each movement scale, for steering experiments, it is important to use a fixedscale that is in a comfortable range for users. 2) In practical applications such a website,users would subconsciously zoom in or out of the interface to stay at the bottom of the U-shaped performance-scale function. Since this is not possible in our steering experiment,we should alternatively offer users the freedom to adjust their viewing distance from the16screen to allow them to perform at an optimal level.Kovacs et al. (2008) explored how visual display properties affect the speed-accuracytradeoff in Fitts’ tasks [13]. The experiment evaluated three different visual angles (5.1°,7.4°and 13.3°) and levels of ID. Visual angle is a measure of how large an object appearsto the participant based on its physical size and distance to the observer. This can bevisualized in Figure 2.14. A larger visual angle corresponds to a larger perceived display.Figure 2.14: Visualization of the visual angle metric in a Fitts’ task, which is mea-sured as the perceived angle between the two targets from the participant’spoint of view. The visual angle depends on the distance between the two tar-gets as well as their distance from the observer.The authors found that larger visual angles significantly reduced the movement time forhigher ID tasks. This implies that visual display characteristics can affect a participant’smotor control strategies, which in turn can affect the movement time in Fitts’ tasks. Thesefindings imply that the visual angle is an effect that should be controlled for in motorcontrol experiments.In an extension to their work on narrowing/widening linear tunnels, Yamanaka and Miya-shita (2016) test how scale affects the time difference between the two directions [33]. Theyconducted a direct-input pen tablet experiment across five different scales (S= 1,2,4,9,12)and for narrowing and widening conditions. Direct-input means that at each scale, both thevisual scale (the displayed path) as well as the motor scale (physical movement) changetogether. They found that smaller scales (S = 9,12 resulted in longer MT due to the smallpath width requiring more precise control, while larger scales S = 1,2 resulted in shorterMT . This contrasts with Accot and Zhai’s (2001) findings of a U-shaped performancescale function [3]. However, Yamanaka and Miyashita’s work was done on a direct-inputdevice, while in Accot and Zhai’s setup, the display size was fixed while the input scale17varied. Despite the difference in outcomes, the observation that performance changes withrespect to different movement scales highlights the necessity of maintaining consistent andcomfortable visual-motor scales in steering experiments.2.3.2 Operational Bias EffectsOperational bias refers to a participant’s prioritization of speed versus accuracy. This couldbe influenced by the wording of the task instructions, such as emphasizing on the impor-tance of maintaining a lower error rate, for example.Zhou and Ren (2010) investigate the effects of subjective biases in steering tasks [41].In a pen-steering experiment through straight and circular tunnels of various widths andlengths, participants were subject to five different bias conditions: extremely accurate,accurate, neutral, fast, and extremely fast. The standard deviation of movement increasesas users move faster. They also found that for circular steering, the effect of bias is strongerthan in linear steering. In a subsequent study, Zhou (2012) found that the Steering Lawdoes not hold well across different speed-accuracy biases [40]. These findings suggestthat speed-accuracy tradeoffs must be controlled in steering experiments to ensure faircomparisons.Kasahara et al. (2024) addressed these limitations in their study, proposing the use of ef-fective parameters to smooth the effect of speed-accuracy biases [12]. In their experiment,participants were asked to perform a series of pen-steering tasks in linear, circular and sine-wave paths under three different biases: 1) Accurate (prioritizing accuracy over speed); 2)Neutral (balance of accuracy and speed); and 3) Fast (prioritize speed over accuracy). Theyfound that the Fast condition resulted in the highest error rate and smallest movement timeacross all tasks. Specifically, in circular and sine-wave paths, the gap between conditionsin error rate and movement time was higher compared to the linear case. This implies thatsteering tasks in non-linear paths were more sensitive to subjective biases, highlighting theimportance of reducing these biases in our experiments.2.3.3 Directional EffectsIn their work on introducing effective parameters to the Steering Law, Kulikov et al. (2005)also investigated the effects of the movement direction on steering tasks [16]. They foundthat steering from left to right resulted in faster movements as well as higher out of pathmovement.In the work by Liu et al. (2010) on investigating the Steering Law in 3D paths, they alsolooked into the effect of handedness and found that right-handed participants were faster18when steering to the right, and vice-versa for left-handed participants [19].The findings from these two works motivate the need for reciprocal movement testing,particularly if participant handedness is not controlled for.2.3.4 Other Influential FactorsThe previously discussed experiment by Dennerlein et al. (2000) on combined steering andpointing tasks also demonstrated that haptic feedback can provide performance benefitsfor both pure steering tasks and combined tasks [7]. This emphasizes potential benefits ofincorporating multimodal feedback.A study by Zhou et al. (2011) assessed how aging affects performance in steering tasks[43]. Users were tasked to perform a series of pen-steering experiments through circulartunnels of varying width and length, and under different speed-accuracy biases. The au-thors found that older users exhibit greater movement variability and lower responsivenessto bias instructions. This is in accordance with the assumption that age-related motor skilldeclines lead to greater performance differences.In addition, another factor that can affect the Steering Law’s prediction accuracy involvestemporal constraints. Zhou et al. (2009) conducted an experiment where participants arerequired to complete the task in a fixed amount of time [42]. The study tested 5 levels ofwidth, 3 levels of length, 5 levels of movement time goals, and 3 levels of temporal errortolerance. Prior to each trial, the participants were informed of the movement time goal,as well as the acceptable time range. They found that temporal constraints significantlyaffect steering performance, and that the base Steering Law model is not sufficient whenmovement time is fixed in a paced task. They also found that users adjust their movementaccuracy based on the imposed time constraint. This study has implications on the applica-bility of the Steering Law to other functions that are time-constrained, such as for modelingspeech articulation. In vowel production, the tongue moves through a constrained trajec-tory in the vocal tract, similar to how a stylus navigates a tunnel. However, since speechoccurs at a natural pace, the base Steering Law may need some additional extensions toaccount for the temporal constraints of articulation.Together, these additional factors demonstrate the wide range of potential influences onsteering performance. Recognizing and managing these factors in experimental design arecrucial for accurate movement behavior modeling.192.4 Applications of Fitts’ and Steering LawFitts’ and Steering Laws have been applied widely across various HCI domains due totheir predictive ability and robustness. In this section, we summarize several practicalapplications of the performance models to illustrate their relevance and the importance ofcontinued research in this field.2.4.1 Input Device EvaluationOne of the earliest and most prevalent applications of Fitts’ and Steering Law researchis in the quantitative evaluation and comparison of different input devices. In Accot andZhai’s (1999) work, they conducted a series of experiments on five different input devices:mouse, stylus & tablet, trackball, touchpad, and isometric joystick [2]. Participants wereinstructed to steer a cursor through constrained linear and circular tunnels of varying widthsand lengths. They found that all tested devices followed the Steering Law with high ac-curacy, which confirms that the Steering Law can be used alongside Fitts’ law for deviceevaluation. They also found that for trajectory-based applications, the stylus and mouseinputs are the best performing devices. These evaluations provide an objective method forcomparing novel input devices and informing device design decisions.2.4.2 Lasso Operations and Group SelectionThe Steering Law and its extensions have many applications in Graphical User Interface(GUI) design. Figure 2.15 demonstrates an example of a lasso operation in a drawingapplication where the user is selecting the pear graphic in the middle of the other elements.In order to select the pear, the user must navigate through a path constrained in width (thespace between the pear and the apple/orange), curviness (defined by the shape of the pear),and length (defined by the movement amplitude). In such an operation, the movement timewould be difficult to predict with the base Steering Law.Yamanaka et al. (2018) investigate how users move through multiple successive con-straints, specifically focusing on straight segments through obstacles with different condi-tions [36]. Their findings show that steering through successive objects follows differentmovement behaviors depending on object density and spacing.In a subsequent study, Yamanaka et al. (2022) refine their work on lasso selection model-ing by introducing path segmentation methods [37]. Participants were tasked to use a stylusto follow a near-rectangular, closed-loop path composed of segments of varying width.However, in both studies, the authors focused only on straight paths or straight segmentslinked by a 90 degree corner, while lasso selection typically involves curved trajectories20Figure 2.15: A lasso operation in a drawing program.around objects. This further motivates the need to understand how curvature affects move-ment time to enhance the Steering Law predictions and improve selection accuracy forcomplex path geometries.2.4.3 Virtual Reality and Virtual DrivingZhai and Woltjer (2003) demonstrated that the Steering Law can be applied to tasks be-yond hand movements, such as driving in a virtual driving simulator. They conducted anexperiment on both straight and circular roads (either turning left or right). Participantscould control the direction, braking, as well as the acceleration of the vehicle. They hadfound that the Steering Law was applicable to driving tasks, which demonstrates the wideapplicability of the law with other types of interfaces.2.4.4 Speech ProductionSpeech production tasks share similarity with steering tasks in that they both involve con-tinuous, constrained movements along prescribed trajectories. Speech motor control re-quires coordination from across multiple articulators and is limited by biomechanical con-straints.Lammert et al. (2018) investigated whether Fitts’ law applies to human speech produc-tion, analyzing speed-accuracy tradeoffs in articulation using real-time MRI data [17]. Us-ing individual phonemes as articulatory targets, they conceptualize continuous speech as asequence of target-reaching tasks. Their results indicate that certain task categories exhibitclearer speed-accuracy tradeoffs than others. Some speakers also exhibit strong adherence21to Fitts’ law, while others do not.Kuberski (2021) also evaluated the application of Fitts’ law to speech articulation, partic-ularly in the case of repetitive speech [14]. They conducted an experiment where partic-ipants repeated two types of syllables at 8 metronome-controlled rates (30-570 beats perminute). The two syllables were the articulatory targets, with one being at the tongue tip([ta]) and the other at the tongue back ([ka]). They found that for faster rates (≥150-300beats per minute), MT follows a linear relationship with ID. For slower rates, there is noconsistent relationship between MT and ID.These two studies provide evidence that articulatory movements exhibit speed-accuracytradeoffs similar to limb movements, but with different constraints. Given that speecharticulation involves continuous trajectory-based movements rather than discrete target ac-quisitions, the applicability of the Steering Law remains an open question.2.5 Chapter SummaryThis chapter has provided a review of foundational research related to the Steering Law.We reviewed its extensions, focusing on studies that modeled more complex scenarios in-volving path curvature, variable widths, and effective parameters to reflect user variability.We then discussed additional factors influencing steering performance, including scaleeffects, operational biases, directionality, and other task-specific considerations. Recog-nizing these factors aids us in designing an experiment that accurately captures movementbehavior and controls for unwanted factors.Finally, we summarized practical applications of Fitts’ and Steering Laws, such as inputdevice evaluation, GUI design, virtual navigation, and speech production. These applica-tions highlight the utility of human movement models and reveal research gaps.22Chapter 3Experimental Design and RationaleTo evaluate the impact of curvature on steering performance, we designed a controlledexperiment based on prior studies and iterative pilot testing.In the first part of this chapter, we present an overview of the experiment design, whichincludes the steering task, participant demographic information, experimental protocol,hardware and software setup, curve representation and construction, as well as the outcomemeasures used to evaluate the performance. This overview provides a snapshot of the coremethodology and the data we aimed to collect.In the second part, we present the justification for each of the key design elements ofthe experiment. These design choices were backed by past research as well as resultsobtained from a series of eight pilot testing rounds prior to conducting the experiment.While not part of the final dataset, these pilot findings provided important insights anditerative feedback.Throughout these discussions, we highlight the lessons learned from pilot testing. Thisiterative approach ensured that the final study setup is grounded in both past literature aswell as empirical findings from our preliminary trials to arrive at a cohesive experimentalsetup that provides a methodologically sound framework for analysis.3.1 Experiment Design Summary3.1.1 Steering TaskParticipants were asked to move the provided mouse to steer a cursor as quickly and asaccurately as possible through the displayed tunnels of varying conditions. For each trial,the subject performed a series of actions: click the start button, steer the cursor through thetunnel to reach the other side, click the flag button, steer the cursor back through the tunnel23to reach the starting point, then click the end button. The action was reciprocal to accountfor potential directional effects and the steering direction was from left to right, then rightto left. The start and end buttons were always at the same location and spaced 1300 pxapart. This series of actions is shown in Figure 3.1 which demonstrates the first tutorialtrial shown to participants. A visual example of all of the trial types from the experimentis demonstrated in Figure 3.2.3.1.2 Participants20 participants (9 women, 11 men) were recruited from the University of British Columbia(UBC) Vancouver campus for the experiment. All participants had normal or corrected-to-normal vision. 4 participants were in the 19-24 age group and 16 participants were inthe 25-34 age group. In the background survey, 18 participants reported that they usedthe mouse daily, 1 participant used the mouse a few times a week, and 1 participant afew times a month. All participants reported that they typically used the mouse usingtheir right hand and subsequently performed the experiment using their right hand. Allparticipants successfully passed a tutorial that was performed at the start of the experimentto evaluate their speed and accuracy, as described in 3.1.3. The experiment was conductedunder approval from the university’s behavioral research ethics board (Certificate Number:H24-01850). All user study materials are included in Appendix B.3.1.3 Experimental ProtocolThe study was divided into 2 phases: the tutorial phase and the experiment phase. In the tu-torial phase, participants were introduced to the same 9 types of paths that they would laterencounter in the experiment phase. The sequence of the tutorial tasks was pre-randomizedand standardized across all participants. Error circles were displayed on screen wheneverthe subject’s cursor exited the tunnel’s boundaries, as shown in Figure 3.3.The participants’ steering speed and accuracy were logged for a moving window of 8trials. The tutorial phase ended when a participant reached the practice threshold, or ifthey performed 30 trials without reaching this criterion. In the latter case, they would notmove onto the experiment phase. The practice threshold was defined by maintaining aconsistent cursor speed with a coefficient of variation below 0.15 and achieving an averageerror rate of less than 2 cursor exits per trial across the moving window.The experiment phase was divided into 27 blocks of 5 trials each, with a 15-second breakbetween blocks. The subject’s cursor position and movement times were logged from themoment they clicked the start button until they completed the round-trip for each trial. If24Figure3.1:Anexampleofthesteeringtaskasdemonstratedthroughthefirsttutorialtrialencounteredbyparticipants.a)Idle,awaitingparticipantaction;b)Subjecthasclickedonthestartbuttonandissteeringtotherightflagbutton;c)Subjecthasreachedandisclickingontherightflagbutton;d)Subjectissteeringtotheleftflagbutton;e)Subjecthasreachedandclickedtheleftflagbutton,endingthefirsttrial.25Figure 3.2: Tunnel tasks for each of the 9 trial types.Figure 3.3: An example of a tutorial trial. A red error circle is shown to indicate thelocation where the participant exited the boundaries of the tunnel.a participant exited the defined path, the error was logged, but the participant was allowedto proceed without visual interruption or restarting. The experiment procedure flowchart isshown in Figure 3.4.To account for the overall curviness of a path, we introduce the total curvature parameterK, computed as:K =∫C|κ(s)|ds (3.1)where κ(s) represents the instantaneous curvature at any point s along path C.A within-subjects study design with repeated measures was used. Independent variableswere total curvature K and length L, with 3 levels each. Tables 3.1 and 3.2 show a summaryof the parameters for each trial.26Figure 3.4: The experimental procedure flowchart.27For each combination of factors, subjects performed 15 repetitions which were dividedinto 3 blocks of 5 trials. In total, each participant performed 135 trials and took approxi-mately 1 hour for the entire experiment.The sequence of blocks for the experiment was initially randomized and subsequentlyreversed for half of the participants to mitigate the learning effects throughout the ex-periment. Additionally, the orientation of the trials was pre-randomized. A flipped trialinvolves mirroring the curve across the horizontal axis, preventing participants from antic-ipating the initial direction of the path at the start of each trial. Figure 3.5 shows the twopossible orientations of trial ‘L0-K0’ in its original and flipped orientations. We fixed thewidth of all trials at 50 px to focus on the curvature effect.Figure 3.5: The two possible orientations of trial ‘L0-K0’. A) Original orientation;B) Flipped orientation.It should be noted that the lengths of the curves within each L level vary slightly due to themethods used in their construction, as detailed in 3.1.6. Table 3.3 summarizes the actuallengths and total curvatures of each of the trial types. In the next chapter, L and K are usedfor Analysis of Variance (ANOVA), while the actual values of L and K are employed in thelinear regression model fitting process.28Table 3.1: Lengths distribution per LL Mean (px) Std. (px)0 1500.10 2.251 1882.33 2.232 2319.75 16.18Table 3.2: Total curvature distribution per KK Mean (px) Std. (px)0 10.00 0.001 16.00 0.002 22.00 0.00Table 3.3: Lengths and total curvatures of each of the 9 trial types.Trial ID Length (px) Total CurvatureL0-K0 1502 10L0-K1 1498 16L0-K2 1500 22L1-K0 1885 10L1-K1 1880 16L1-K2 1882 22L2-K0 2303 10L2-K1 2322 16L2-K2 2335 223.1.4 Setup and Data AcquisitionThe experiment was conducted on a Macbook Pro computer with the output displayed ona 13.3”, 2560-by-1600 pixel display. Participants used a 3200 Dots Per Inch (DPI) MSIClutch GM08 mouse without additional weights as the sole input device to control thecursor. Hardware acceleration was disabled for the mouse control, and the sensitivity wasfixed for all participants at the 5th notch (from the left of the scrollbar) in default Applesettings. Mouse movements were polled and interpolated using cubic splines during post-processing to a rate of 200 Hz. The setup is shown in Figure 3.6.29Figure 3.6: Hardware setup for the experiment and mouse sensitivity settings.3.1.5 Software SetupThe experiment evaluation software was developed to introduce the experiment, recordparticipant data and to provide real-time event handling for detecting user actions.When participants began the experiment, they were prompted to enter their participantidentification, which is logged by the system. Then, the landing page is displayed, whichshows the explanatory video and text. The participant would then be prompted to click abutton to proceed to the tutorial. At this point, the experiment evaluation began.Data CollectionAt the end of each trial, collected data was formatted and logged into Comma-SeparatedValues (CSV) files. The software was implemented in JavaScript (Node.js) and relied onevent-driven data logging.During each trial, event listeners track specific user interactions on the drawn Scalable30Vector Graphics (SVG) elements, which include:• Clicking start/end buttons;• Entering and exiting the tunnel boundaries;• Mouse movements along the tunnel path;A client-server architecture was used. Trial data was first captured on the client-side,stored in temporary variables, then formatted and transmitted to the server for storage intoparticipant-specific CSV logs, which were stored in the following directory:trial_logs/${participantId}/Each file contains different types of trial data:• Tutorial Parameters:– File name: tutorial params.csv– Format: ‘Trial ID,Error Count,Half Time (ms),Total Time (ms),Flip,Error Mean,Time Mean (ms),Time Standard Deviation,Time Constant of Variation\n’• Experiment Parameters:– File name: params.csv– Format: ‘Trial ID,Error Count,Half Time (ms),Total Time (ms),Flip\n’• Experiment Cursor Data:– File name: data.csv– Format: ‘New Trial (*),Trial ID,(*)\nTimestamp(ms),Position X,Position Y\n’Note that unlike tutorial trials, where only aggregated parameters are logged, cursor posi-tions are recorded for the experiment trials for post-processing.Data Post-ProcessingSince raw cursor data points depend on browser event polling, movements were interpo-lated using cubic splines to achieve uniform sampling at 200 Hz. While the number of pathexits were recorded in real-time through event listeners, we are also interested in the OPM31metric, introduced by Kulikov et al. [16]. This is calculated as the percentage of sampledpoints during a participant’s trial that fall outside the boundaries of the drawn tunnel as:OPM =MTOuto fBoundsMTTotal·100% (3.2)In post-processing, for each interpolated cursor position, we check its minimum distanceto the tunnel centerline. If the distance is larger than 25 px (half the tunnel width), thecursor position would be outside of the tunnel boundaries. We can then use this to computeOPM for later analysis.3.1.6 Curve Representation and GenerationThe experiment paths are represented by the following equations:x(s) = sy(s) =acc∑i=0sin(AM[i] ·φ · s)φ =np ·2πxmax(3.3)where:• a = amplitude• c = number of sinusoidal functions used in the construction of the curve (1≤ c≤ 3)• AM = list of c angle multipliers• np = number of periods• xmax = maximum value of xA custom curve visualizer software was developed to visualize the effects of manipulatingeach of the parameters used in the curve’s construction.To find potential candidate paths, we perform a grid search over the parameters c, AM andnp and numerically find the value of a for a desired K using the Newton-Raphson method.xmax is fixed at 1300 px for all paths. If a exists for a set of parameters, we compute itslength L. If it is within range of our target L levels, it is appended to a list of potentialcurve candidates. Another application was developed to cycle through different generatedcurves in this list. The interfaces of these helper applications are shown in Appendix A.32Table 3.4: Summary of all 8 pilot rounds and changes made to the experiment design.Rd. #Part. Tutorial Experiment UI Elements & In-structionsHardware1 1 N/A • Simple sine wave• 5 W (10–90 px)• Not reciprocal• Errors logged• MT measuredleft to rightButtons use onlycolor• Logitech M100mouse• MacBook Prodisplay2 3 N/A • 5 blocks of L(2000–6000 px)• Unbalancedblocks• Reciprocal task• Fixed order• No orientationflips• No breaksInstructions addedas text at topNo change3 2 N/A • Same L blocks asRd.2• Balanced blocks(12 trials perblock)• Reciprocal• Flipped orienta-tions• Landing pagewith images• Icons on buttons• Break remindersand enforcementsNo change4 3 36 repetitions ofL= 2500 curvesSame trials as Rd.3 No change No change5 1 72 repetitions ofL= 2500 curvesNone No change No change6 3 Pass/fail tutorialcriteria added3L × 3E × 12 rep. No change No change7 3 No change 3L × 3E × 15 rep. Instructional videoaddedMSI Clutch GM08mouse8 2 No change 3L × 3K × 15 rep. No change No change3.2 Design JustificationThe experiment design involved many decisions that underwent multiple iterations andtesting. We conducted 8 rounds of pilot testing to refine our experimental setup, addressingfactors such as user interface elements, tutorial design, instruction clarity, randomization,experimental protocol, hardware changes, etc. Table 3.4 presents a summary of the setup33for each round of pilot tests.In the following subsections, we present the rationale behind each design decision, backedby the results obtained from our pilot tests as well as insights from past research in the field.3.2.1 CurvinessDefining a curviness metric was essential for our experiment. Initially, we considered us-ing the energy E of the curve as a curviness measure, under the assumption that higherfrequencies introduce more rapid directional changes in the curve, making paths subjec-tively “curvier”. However, our mathematical analysis and pilot study results, showed nosignificant differences in movement times across varying E levels. We then explored otherrepresentations and introduce the total curvature K. From our pilot studies, we found thatK is a more suitable measurement of curviness and that as hypothesized, it leads to statis-tically significant differences in participants’ steering behavior.Energy EFor a finite sine signal defined by:y(s) = A · sin(2πTs)Its energy over 1 period T is calculated as:E =A2 ·T2As the amplitude A increases, the height of the curve described by y(s) also increases,which is in accordance with the idea that E becomes larger as the path becomes curvier.However, as the frequency f = 1/T of the signal increases, say, by double, T is halved.If we compute the new energy of the function from 0 to 2T, it would still have the sameenergy as before, despite the fact that the curve would be subjectively curvier due to theextra bends. Consequently, we discarded E as a viable metric.Total Curvature (K)Since past studies with circular arcs have used the radius of curvature R to represent curvi-ness, we explore adapting this for sinusoidal paths. Approximating a sinusoidal path as aseries of circular arcs is one approach, but would introduce a lot of approximation errors.Instead, we opted for total curvature K. As presented in Equation 3.1, we introduce K to34account for the overall curviness of a path:K =∫C|κ(s)|ds (3.4)For 2D parametric plane curves, the instantaneous curvature at point s along the curve isgiven by:κ(s) =|x′(s) · y′′(s)− y′(s) · x′′(s)|(x′(s)2+ y′(s)2)3/2(3.5)The parametric equation of a circle is:x(s) = R(s) · cos(s)y(s) = R(s) · sin(s)where R(s) is the radius of curvature at path length s, which is constant over the entire arcby definition. Therefore, its instantaneous curvature can be written as:κ(s) =1R(s)The total curvature K over an arc of length L is then given by:K =∫C|κ(s)|ds= κ ·L= LRThis means that for circular paths, the total curvature is directly proportional to the arclength and inversely proportional to the radius. A sharper curve (smaller R) results in ahigher total curvature.In our case, our paths defined in Equation 3.3. Combined with Equations 3.1 and 3.5,the total curvature K of our sinusoidal paths can be computed numerically. K effectivelycaptures the cumulative curvature over the entire path, making it a suitable curviness pa-rameter.Pilot Studies to Explore E and KIn pilot testing Rounds 6 and 7, we explored using E to define curviness, while in Round 8,we revised the protocol to use K instead. In all 3 rounds, the experimental setup consistedof a 3 × 3 factorial design.In Round 6 and 7, path length L (1300 px, 2100 px, 2900 px) was crossed with energylevels E (2e6, 8e6, 1.4e7). For Round 6, each participant completed 12 repetitions percondition, divided into 2 blocks, totaling 18 blocks of trials. In Round 7, the same trial35Figure 3.7: Visualization of the types used in Rounds 6 and 7 of the pilot experiment.types were used, but the protocol was adjusted to 15 repetitions per trial type, divided into2 blocks, for a total of 27 blocks. The block order was pre-randomized but remained thesame across participants to maintain consistency. Figure 3.7 shows the trials used in thisthese two pilot experiments and Table 3.5 summarizes the corresponding parameters.Analyzing the data gathered, we observed that the energy parameter was not a reliablemeasure of curviness. Figure 3.8 illustrates the combined movement time results fromRound 6 and Round 7 across all trial types.A 2-way ANOVA shows that there is a statistically significant effect of L on the movementtime (F = 47.752, p < 0.001), indicating that longer paths resulted in longer movementtimes as expected. However, the results did not show a statistically significant effect of E(p> 0.05), suggesting that it was not an appropriate metric for curviness in this context.Given these findings, we revisited the total curvature K. We conducted Round 8 of pilotstudies to assess whether K provided a more reliable measure of curviness and how itinfluenced movement behavior. For this pilot, three levels of path length L were crossedwith three levels of total curvature K, each repeated 15 times per condition, resulting in 936Table 3.5: Parameters of the trial types used in Rounds 6 and 7 of the pilot experi-ment.Trial ID a np c AM L K E xmaxL0-E0 55.47 1 1 [1] 1300.0 1.036 2e6 1277.7L1-E0 173.81 2 3 [1, 2, 3] 2100.0 19.487 2e6 1284.4L2-E0 188.69 3 3 [1, 1, 3] 2900.0 32.901 2e6 1291.8L0-E1 110.94 1 1 [1] 1300.0 1.890 8e6 1221.2L1-E1 261.57 1 3 [1, 3, 3.5] 2100.0 10.569 8e6 1294.5L2-E1 265.50 1 3 [1, 3.5, 3.5] 2900.0 13.261 8e6 1299.9L0-E2 146.76 1 1 [1] 1300.0 2.314 1.4e7 1174.7L1-E2 254.20 1 2 [1, 3] 2100.0 10.420 1.4e7 1292.9L2-E2 348.07 1 3 [1, 2.5, 5] 2900.0 18.243 1.4e7 1285.8Figure 3.8: Box plots of the movement times obtained from Round 6 and 7 of pilottesting, organized by a) Length; and b) Energy. A clear trend is visible for the3 levels of length, but not for energy.trial types and a total of 135 trials per participant. Same as in Round 7, each condition wassplit into 3 blocks of 5 trials, and the order of the blocks was pre-randomized but kept thesame for all participants. Two pilot participants were recruited to complete this phase oftesting.Table 3.6 summarizes the parameters used in this pilot study. Figure 3.9 presents themovement time results for each trial type, grouped first by L then by K.A 2-way ANOVA confirmed that there is a large and statistically significant effect of L(F = 31.055, p < 0.001), as well as K (F = 5.264, p < 0.05). The results validate thatK is a more suitable curviness metric than E, as it demonstrates a consistent influence37Table 3.6: Parameters of the trial types used in Round 8 of the pilot experiment.Trial ID a np c AM L K E xmaxL0-K0 64.90 3 2 [1, 1.5] 1502.20 10.0 3.8e5 1300L0-K1 56.27 3 3 [1, 1.5, 2.5] 1497.72 16.0 1.6e5 1300L0-K2 36.45 3 2 [1, 3] 1500.37 22.0 9.6e4 1300L1-K0 157.95 1 3 [1, 3, 3] 1884.63 10.0 3.0e6 1300L1-K1 178.65 1 3 [1, 2, 5] 1880.17 16.0 3.9e6 1300L1-K2 125.18 2 3 [1, 1.5, 3.5] 1882.19 22.0 1.1e6 1300L2-K0 262.23 2 2 [1, 1.5] 2302.62 10.0 9.3e6 1300L2-K1 223.69 1 3 [1, 4, 5] 2321.84 16.0 5.2e6 1300L2-K2 155.05 2 3 [1, 3, 3.5] 2334.78 22.0 1.4e6 1300Figure 3.9: Box plots of the movement times obtained from Round 8 of pilot testing,organized by a) L; and b) K. A clear trend is visible for both metrics.on movement behavior, making it a more appropriate independent variable for the finalexperimental design.3.2.2 Length (L) and Width (W )In the original Steering Law, length and width are important parameters. Ideally, we wouldconduct an experiment with L, W and K as the independent factors, but incorporating allthree parameters would overcomplicate the experimental design. Therefore, we choose tofix W and solely focus on the effects and interactions of L and K.38Figure 3.10: The 5 trial types used in Round 1 of varying widths.Figure 3.11: Plots of the a) Number of path exits; and b) Movement time (ms); withrespect to different path widths from Round 1 of pilot testing.Choice of Fixed WidthIt has been demonstrated in the past that ‘corner-cutting’ is a common behavior whennavigating through corners. Users take advantage of the increased width tolerance to cutthrough a sharp corner, decreasing the overall movement time as the width of the path in-creases [25]. In our case, the peaks of the sine wave are visually similar to corners. There-fore, we restrict the width of the tunnel to a value that restricts corner-cutting behavior, butthat was not too small as to induce too many steering errors.To determine an appropriate width, we conducted a simple pilot experiment on a singleparticipant with a sine wave curve and 5 different widths (10, 30, 50, 70, 90 px), as il-lustrated in Figure 3.10. The recorded movement time and error rates from this pilot areshown in Figure 3.11.From the results, 50 px was selected as the fixed width constraint for the experiment. Thisdecision was based on the following observations:• Lower widths (10 px and 30 px) induced excessive errors, making the task unneces-39Table 3.7: Trial statistics of the tasks used in Round 2 of the pilot experiment.Block # Trials Avg. Length [px]1 13 2000.092 12 3000.513 8 4000.144 7 5000.295 5 6001.12sarily difficult.• Higher widths (70 px and 90 px) resulted in zero errors on average, suggesting thatthe task became too easy and participants could steer through the tunnel with mini-mal control.• 50 px provided an optimal balance: it induced some errors, ensuring participants hadto apply motor control strategies, but did not cause excessive difficulty.Additionally, the participant provided qualitative feedback that narrower widths causedhand fatigue due to the need for a tight grip on the mouse. This introduced discomfortand could negatively affect participant performance over longer trials. For these reasons,we fixed W = 50 px for the remainder of the experiment to maintain a controlled level ofdifficulty while ensuring participant comfort and meaningful performance variability.By ignoring width, we are indeed restricting the possible ranges of applications, so fu-ture work is required to examine the effects of width and curvature interactions on themovement time.L RangePast research has demonstrated that in combined steering-pointing movements, such asscenarios where participants must steer through a tunnel to reach an endpoint target, theSteering Law provided a better fit than Fitts’ Law to the movement time data as the lengthof the path increased [11]. This implies that in our study design, the length of the pathmust be sufficiently long so that users maintain a controlled steering behavior throughoutthe task.We tested path lengths ranging from approximately 2000 to 6000 px in Round 2 of pilottesting, which involved 3 participants performing 5 blocks of experiments, each corre-sponding to a different path length L. Table 3.7 summarizes the number of trials per blockand their corresponding average path lengths.40Figure 3.12: Box plot of the movement times obtained from Round 2 of pilot testing,organized by length.Figure 3.12 shows the total time required to execute 1 full round-trip of the experimentfor different path lengths L. The data suggests that there is a trend where movement timeincreases as path length increases, which aligns with expectations based on prior researchin steering and pointing tasks.3.2.3 Curve Representation and ConstructionIn previous studies on extending the Steering Law to account for varying curvature, theexperiment paths either consist of simple circular arcs or compound circular arcs. Figure3.13 presents these shapes.Using circular arcs is computationally straightforward, allowing easy calculations for cur-vature and length, but fails to model realistic steering scenarios with non-uniform curva-ture. We also considered Be´zier curves, commonly used in computer graphics and ani-mations due to their smooth interpolation. However, one main issue with both compoundcircular paths and compound Be´zier paths is continuity. It is easy to ensure G0 and G1continuity by matching the tangent at the end of an arc and the start of the next (see Figure3.14) by ensuring that the end point of curve A and start point of curve B are at the samelocation, as well as ensure that they have the same tangent vector at these points.However, higher levels of continuity dictate that the instantaneous curvature (G2) andcurvature rate of change (G3) would have to be the same at these points. This is not possibleto achieve for compound circular arcs and extremely difficult for compound Be´ziers due to41Figure 3.13: Representation of simple circular arc path and a compound path of cir-cular arcs.Figure 3.14: G0/G1 continuitity matching: Tangent vector −→T at the end of curve Aand at the start of curve B are matched.42the complexity of the equations required.Another method to construct arbitrary curves would be to use sinusoidal functions. Sinu-soids are by definition smooth curves as they are continuous and differentiable. In a recentexperiment, Kasahara et al. employed a sine-wave shape as one of the tunnel types in anexperiment to refine throughput using effective parameters [12]. They found that the orig-inal Steering Law held up well with the sine wave shape and identified that the additionaleffect of curvature should be investigated for this shape. Based on these findings, we se-lected sinusoidal functions for our curve construction to leverage their inherent smoothnessand differentiability while maintaining practical computational feasibility.Curve SelectionTo identify possible candidates for each of the trial types, we conducted a grid search overthe sinusoidal function parameters c, AM and np, and the targeted curviness level K. Foreach combination of these parameters, we solve for the corresponding amplitude a, com-pute the total length of the resulting curve, and record the parameter sets that producecurves in range with the desired lengths. Finally, through visual inspection of the resultingcurves and pilot trials, we select the most appropriate curve for each L and K combination,specifically choosing those that minimize corner-cutting behavior and prioritizing match-ing the lengths of the curves within each designated category to ensure consistency acrosstrials.In the case of ‘L2-K2’, alternative curves generated during the search process for this trialtype include those with lengths of 2246, 2421 and 2441 px. These were significantly out-side the target range for the L2 category, with ‘L2-K0’ and ‘L2-K1’ having lengths of 2303and 2322 px, respectively. Given the importance of length consistency in our experimentaldesign, curves that did not meet this criterion were excluded from consideration, regardlessof their other characteristics.3.2.4 User Interface ElementsFor our experiment, we opted for a minimalist interface to avoid distractions, which iscomposed of 2 buttons on either side of the experiment path and a text box at the top toindicate system status.Endpoint ButtonsInitially, endpoint buttons were colored as green and red for simplicity. However, it wasraised during a group discussion that this may prove difficult for green-red colorblind par-43ticipants to follow the instructions. To mitigate this, a triangle start icon was added to thegreen button and a flag icon was added to the red button’s active state. During Round 3of pilot testing, a colorblind participant provided verbal feedback that the icons make theprocedure easy to understand, as they would simply have to “follow the icon” to determinethe next appropriate action.Audio and Visual Feedback of Error and Trial StatusIn past studies, visual and auditory cues have been used to signal different phases of a trialas well as to indicate user errors. For example, in [2, 3, 32], the color of the cursor’s tra-jectory indicated the phase of the experiment. In [19], the color of the target ball indicatedif the user is inside of the boundaries of the task and if the cursor was in contact with thetarget, while in [31], a beep sound was played to provide feedback on errors. However,to simplify the interface, we decided against using color changes and auditory feedback toindicate trial status and error feedback. These decisions are also made in consideration ofusers who may be colorblind or hard of hearing.In the tutorial phase, we introduced red error circles to indicate when a participant exitedthe trial during Round 4 of pilot testing. This decision was backed by our observationsduring pilot Round 2, where participants’ movement patterns suggested a bias toward speedover accuracy. Many participants exited the path entirely and simply followed its generalshape rather than making an effort to stay within its boundaries. The error circles providea visual indication that is not color or sound-dependent.Task Instructions Placement and ClarityA landing page with instructions was introduced in Round 3 of pilot testing to addressissues that arose during previous rounds. In the initial version of the interface, the in-structions would be presented in the form of text that appeared at the top of the page whileparticipants performed the steering task. Round 2 of testing revealed several issues with theplacement and clarity of the instructions. Since participants were focused on the task, theyoften ignored the instructions at the top of the page. This revealed that we needed a betterway to display the information that would not interrupt the participant’s concentration, butthat would still clearly indicate the steps.Also, during this round, some mistakenly attempted to click and drag the mouse instead ofclicking, releasing, and steering through the tunnel, indicating a need for clearer guidanceon task execution.Another issue arose at the endpoints of the tunnel. Some participants were unsure whetherthey needed to follow the path on the return trip after reaching the right endpoint, which44led to inconsistent pausing behaviors.Overall, these observations led us to develop a landing page that would be displayedprior to the start of the steering tasks to provide detailed instructions. The first versionof this landing page consisted of an image carousel illustrating the different steps of theexperiment, followed by written instructions to reinforce comprehension. The images areprovided in Appendix A.However, the data obtained from Round 3 following the introduction of a landing pagesuggested that instruction clarity still remained a concern. We noted that performance var-ied widely in the first block, which indicated that some participants may not have fullyunderstood the task mechanics from the landing page alone. This aligns with our obser-vations made during the experiment, where a participant needed to ask questions to theperson conducting the experiment, but appeared to adapt to the task as they progressedthrough the trials. This highlights a potential limitation in the current instruction format,suggesting that verbal and visual instructions alone may not be sufficient.Having the person conducting the experiment answer questions or provide verbal guid-ance also had the potential to introduce unintended variability in instruction delivery. Forexample, participants often received reminders such as ”try not to dwell too long at theendpoints”. Past studies on Steering Law extensions found that the operational biases havea significant effect on participant performance [12, 40, 41]. Therefore, it is crucial for theinstruction delivery to be consistent across all participants in order to prevent them frombeing biased more towards speed or towards accuracy.Following the introduction of the tutorial phase in Round 4, detailed in 3.2.6, which par-tially addressed the instruction clarity and bias issues, we discovered another point of con-fusion for the participants. Participants were accustomed to seeing error circles during thetutorial phase, but were confused that they were not showing up during the experimentphase. This omission was initially designed to minimize distractions, but caused confusionand potentially distracted participants during the trials. This indicated to us that the transi-tion from the tutorial was not adequately explained and underscored the need to clarify thedistinction between the tutorial and experiment phases in future iterations.To eliminate the issues with inconsistency, potential experimenter bias and points ofconfusion, we introduced a pre-recorded instructional video that summarized all writteninstructions participants needed to know before beginning the experiment. This changeensured that all participants received uniform instructions, reducing the likelihood of unin-tentional experimenter influence. Furthermore, leveraging the advantages of video, it waseasier to demonstrate the task instructions via visual demonstrations rather than verbally.A transcript of the video script can be found in Appendix B. Participants reported that the45video was informative, and the experimenter refrained from answering questions duringthe main trials, maintaining experimental integrity.3.2.5 Experiment ProtocolMovement ReciprocitySince Fitts’ Law was developed for reciprocal tasks, we designed the steering experimentto also be reciprocal. This decision was made from Round 2 of pilot testing and onwards.Previous research has found that the handedness of a participant can influence performanceand direction preference in a steering task experiment [16, 19]. The reciprocal movementdesign choice allows the experiment to be balanced for handedness and account for thedifferences in muscle coordination for movements from left to right and vice-versa.Endpoint Targeting MovementIn our experiment, participants are required to click the endpoint buttons when they reacheither end of the tunnel, which adds an additional pointing task component. However, pastresearch demonstrated that in combined steering-pointing scenarios, the steering subtaskdominates the total movement time [7]. In our case, the length of the experiment path isat least 30 times its width, which should ensure that the steering behavior is maintainedthroughout the task. Additionally, requiring an explicit endpoint click prevents uncon-trolled ballistic movements toward the end of the trial, as well as provides an accuratemeasurement of the trial duration.Trial Order and BalancingIn Rounds 2 to 4, participants progressed through the trials in ascending order of L withoutany randomization scheme. Specifically in the case of Round 4 where we investigatedlearning trends and observed that participants’ speed was increasing over the course ofthe experiment, this lack of randomization interfered with our interpretation of the results.Since trials were presented in ascending order of length (from 2000 px to 6000 px), itwas unclear whether this trend was solely due to learning or if movement time was simplycorrelated with path length. This observation highlighted the need for a trial randomizationscheme in future iterations to better isolate these effects.As detailed in 3.2.1, we introduced trial randomization in Round 6. This involved dividingthe n repetitions of a trial type into blocks of m balanced trials, then randomizing the orderof the blocks. For Round 6, n= 12 and m= 6 for a total of 18 blocks, while for Round 7,46n = 15 and m = 5 for a total of 27 blocks. Increasing the total number of blocks allowedfor greater trial order variability, reducing the potential for learning effects that could biasperformance trends.Due to the size of the experiment, true counterbalancing was not feasible. This wouldinvolve testing each unique permutation of the 27 blocks, which would require more par-ticipants than the number of people in the world. For this reason, the pre-randomized orderof the blocks was reversed for half of the participants in the final experiment design tomitigate any order effects.Trial Orientation FlippingIn the original experiment setup, trials would maintain the same initial orientation, alwaysstarting in the upward direction. In Round 2 of pilot testing, one participant noted that therepeated pattern of the trials made the task too predictable since all paths begun in the sameupward-start orientation. To reduce repetition and increase variation in subsequent tests,some of the trials were flipped horizontally, ensuring that participants would not be able topredict the initial direction of the next trial. The flipped trials were pre-chosen randomlyand kept the same for all participants.3.2.6 Participant-Related IssuesFatigueAs mentioned in 3.2.5, movement reciprocity was introduced in Round 2. This changerequired participants to continuously navigate through the tunnels, repeating the processuntil the end of the experiment. However, due to the duration of the overall experiment,this proved to be very tiring for participants.Following its introduction, in Round 2 of testing, we noticed that participants wouldsometimes pause at the endpoints to take a small break before continuing. This introducederrors in the movement time measurement, since participants were expected to continu-ously move across the tunnel without stopping.As a result, we changed the protocol in Round 3 so that participants then began by clickingthe green start button on the left, navigating through the tunnel to the red button on theright, and clicking it. Once clicked, the red button would turn green, and the flag iconwould disappear, signaling the transition. Simultaneously, the left-side button would turnred with a flag icon, prompting participants to return and click on it. Once the participantnavigated back to the the left side and clicked on the red button with the flag icon, this47concludes one round-trip of the experiment trial. The new protocol improved clarity in themovement sequence and allowed participants to pause before initiating the next trial.Another change to the protocol in Round 3 to address participant fatigue was the intro-duction of enforced breaks in-between blocks. At the end of each block, the tunnel andbuttons disappeared from view, and the text at the top displayed a countdown, as well as aprompt for the participant to rest.ComfortFor this study, the laptop display size was chosen for its common form factor and usage ineveryday tasks, capturing natural user behavior.Following comments from participants during Round 2 of pilot testing, where some re-ported discomfort due to the chair and table height, we changed the instructions to allowparticipants to adjust their seat height and viewing angle of the screen.While outside the objective of this study, future work could extend our findings by control-ling effective screen sizes and viewing angles to establish the movement time relationshipbetween curviness, range of display sizes and viewing conditions as these are known to im-pact movement times [13]. Further, this type of study could determine whether large visualangles are associated with more efficient motor control strategy adoption when increasedcurviness makes the task more difficult, similar to [13]’s finding for Fitts’ type tasks.Another issue that arose during multiple rounds of pilot testing was the uneven and ir-regular movement of the mouse, which made precise steering more difficult. Some testersreported that they noticed jittering of the cursor during steering, which could be due to thelow DPI (100) of the initial mouse used in testing and a low-quality sensor that does nottrack slow movements properly.During Round 6 of pilot tests, one participant failed to reach the tutorial passing thresh-old. They reported discomfort with the mouse sensitivity and ergonomics, suggesting thatimprovements to the input device and workstation setup were necessary to ensure that taskdifficulty stemmed from the experimental conditions rather than physical constraints.To mitigate this, we added a large mousepad to cover the entire movement area, as wellas changed the mouse for a higher DPI and quality one. Subsequent pilot testing validatedthat the new hardware setup enabled much smoother movement and reduced ergonomicdiscomfort.However, in order to maintain uniformity, we do not allow participants to change themouse sensitivity. As noted in [3], movement scale significantly influences steering per-formance. We therefore maintain the same scale across all participants.48Table 3.8: Trial statistics of the tasks used in Round 3 of the pilot experiment.Block # Trials Avg. Length [px]1 12 2000.092 12 3000.013 12 4000.184 12 5000.165 12 5999.37Figure 3.15: Scatter plot of the movement times obtained from Round 3 of pilot test-ing. The grey dashed lines mark the block boundaries. Each color designatesa different participant.PracticeIn Round 3 of the experiment, we recruited two participants to perform five blocks of trials,each corresponding to a different path length L. Table 3.8 shows the number of trials perblock, as well as the average length for each block.Pilot results revealed irregular movement speeds in one of the participants throughout theexperiment. Figure 3.15 illustrates the total movement times for each trial, while Figure3.16 presents the pseudo-speed metric–—computed as two times the path length dividedby the total time taken per trial.Grey dashed lines in both figures delineate the separation between each block of the exper-iment. From the data, we can see that the participant labeled in pink struggled significantlyin the first block, as indicated by the prolonged movement times and lower pseudo-speed49Figure 3.16: Scatter plot of the pseudo-speed obtained from Round 3 of pilot testing.values compared to the second participant. However, starting from the second block, theirperformance improved, suggesting a learning effect over time.To address this, we developed a tutorial which consisted of 36 repetitions of L = 2500px paths. This addition aimed to familiarize participants with the task mechanics beforedata collection began. If a participant exited the designated path during the tutorial, a rederror circle would appear to indicate where they had deviated, reinforcing the importanceof staying within the path constraints.This round included three participants, whose performance was analyzed to assess theeffectiveness of the new tutorial. Figure 3.17 presents the pseudo-speed measurementsover the course of the experiment.The dashed grey line in the graph marks the transition from the tutorial to the experi-ment phase. The data suggests that participants increased their speed as the experimentprogressed, indicating a potential learning effect.To further investigate learning trends, we computed the Power Law of Practice, plotted inFigure 3.18, to analyze participant improvements throughout the tutorial. The Power Lawof Practice is defined asMT (n) = a · (n+1)−b (3.6)where n is the number of trials completed, a represents the initial performance level andb reflects the learning rate specific for each trial type and participant [24].The results indicate that participants continued to improve in speed throughout the tu-50Figure 3.17: Scatter plot of the pseudo-speeds obtained from Round 4 of pilot testing.Figure 3.18: The power law of practice fitted on the tutorial movement time data foreach participant in Round 4 of pilot testing.torial, suggesting that 36 repetitions were not sufficient for reaching a stable expert-levelperformance. Ideally, participants should achieve a plateau where movement times remainconsistent, signaling that further learning is minimal. Since this was not the case, additionalrefinements to the tutorial may be necessary, such as increasing the number of repetitions.To evaluate whether the tutorial had a measurable impact on performance, we compared51Figure 3.19: Box plot comparing the movement times obtained from the experimentphases of Round 3 (no tutorial practice) and Round 4 (after 36 tutorial trialsas practice) of pilot testing.the results from the experiment phases of Round 3 and Round 4, shown in Figure 3.19.The comparison suggests that Round 4 exhibited a clearer linear trend in movement time,indicating that the tutorial phase was at least partially effective in stabilizing performance.Although participants had not reached expert-level proficiency, the overall results alignedmore closely with expectations compared to Round 3.To further investigate the learning effect observed in previous pilots, we doubled the num-ber of trials in the tutorial phase, increasing it from 36 to 72 trials in Round 5 of pilot test-ing. Since the primary objective of this test was to analyze the learning curve rather thanmeasure performance in an experimental setting, the experiment phase was excluded fromthis study. A single participant was recruited to complete this round.Figure 3.20 presents the participant’s movement time progression over the course of thetutorial, fitted to the Power Law of Practice, which describes how performance improveswith repeated practice.Despite reaching 72 trials, the data suggests that the participant continued to improve,with movement times still decreasing throughout the tutorial. This indicates that the learn-ing process had not yet plateaued, meaning that even after extensive practice, the partici-pant was still refining their ability to complete the task efficiently.In total, the 72 trials took an average of 20.27 seconds per trial, resulting in approxi-mately 24.3 minutes of total task time, excluding any breaks taken between trials. While52Figure 3.20: The power law of practice fitted on the movement time for the partici-pant in Round 5 of pilot testing.this confirms that additional repetitions contribute to skill improvement, it also highlightsthe practical limitations of extended tutorial phases. Increasing the number of trials furtherwould likely lead to participant fatigue, discomfort, or frustration, which could introduceundesirable changes in speed-accuracy bias, such as a higher rate of errors due to dimin-ished focus. These findings reinforce the idea that while a longer tutorial improves taskproficiency, there is a trade-off between learning and participant fatigue.Based on these results, we revised the tutorial strategy to focus on performance consis-tency rather than sheer repetition. Instead of requiring participants to complete a fixednumber of trials, we introduced performance-based passing criteria, evaluating both steer-ing behavior consistency and speed-accuracy balance.To measure this, we implemented the Coefficient of Variation (CV), calculated as:CV =σµwhere σ is the standard deviation and µ is the mean of the movement time MT over amoving window of 8 trials.Reanalyzing pilot data from Rounds 3 and 4, we used the participant who had struggled asa benchmark for ‘not passing’, in contrast to the other participants who demonstrated moreconsistent steering behavior. Based on this comparison, we established a passing thresholdof CV < 0.15, ensuring that only participants with stable movement times and controlledsteering variability would proceed to the main experiment. Additionally, participants were53required to maintain an error rate below two errors per window to proceed to the experimentphase. If a participant failed to meet these criteria after a predefined number of trials, theirdata would be discarded.In Round 6 of testing, one participant failed to meet the tutorial criteria and was excludedfrom the main experiment. The excluded participant demonstrated a high degree of caution,frequently pausing to adjust the mouse, which led to inconsistent steering and prolongedtrial completion times. This result suggests that the tutorial was effective in identifyingparticipants who might not be suitable for the experiment, ensuring that only those with astable steering approach proceeded to data collection.3.2.7 Number of ParticipantsAfter analyzing the data from the final pilot experiment (Round 8), we performed a poweranalysis focusing on the effect of K on movement time. Based on the observed effect sizes,we found that a sample size of 20 participants would yield a power of approximately 80%under our experimental design, assuming the effect of K matches that in our pilot study.This implies that with 20 participants, we have a strong likelihood of detecting meaningfuldifferences in user performance attributable to changes in total curvature.Our decision to target 20 participants per condition balances two practical considerations:minimizing participant burden, and ensuring a robust chance of observing the effect ifit is truly present. Currently, with our experimental design, participants take on averageapproximately an hour to go through both the tutorial and the experiment phases, which isan appropriate amount of time for a human trial. If the final effect size in the main study issmaller than observed in the pilot, additional participants may be required.3.2.8 Error HandlingIn Fitts and Steering Law experiments, participants would inevitably make mistakes duringtrials by accidentally exiting the path.In Round 2 of pilot testing, we noticed that certain participants were biased towards speedas opposed to accuracy, exiting the path entirely and following its general shape insteadof staying within the boundaries. This inaccurate, shape-following behavior highlighted alimitation in the error metric used at this stage, as it relied solely on the number of exits. Atthe end of each block, participants would see the average number of path exits per trial, butin an extreme case, a participant could potentially exit the path once at the start and re-enteronly at the endpoint, which would only result in a single path exit. This would minimizetheir logged errors while not actually adhering to the intended movement constraints and54create a false sense that they are still following the task instructions. This behavior wasnoted prior to the introduction of error circles as visual feedback in Round 4, as detailed in3.2.4.In past studies, researchers used several ways of dealing with error trials. In the originalSteering Law study, the data obtained from trials where a user steering error occurredwere simply discarded [1]. This is not ideal, since an error trial represents valuable data.Also, discarding certain trials could artificially improve model fit by excluding harder,more error-prone trial types. For instance, in our case, it could indicate that the task is toodifficult to follow or too curvy.Motivated by the need to find a better error metric that reflects actual participant perfor-mance, we turn to the OPM metric, which is the percentage of cursor samples outside ofthe tunnel path, rather than counting discrete exits. Essentially, OPM evaluates how a par-ticipant satisfies the accuracy requirement of the steering task, which is set by the widthconstraint. OPM would be computed in post-processing and used in the analysis.3.3 Chapter SummaryIn this chapter, we first introduced the final experimental setup. Then, we described howeach critical design decision was informed based on observations through multiple roundsof pilot studies and prior literature. The iterative process led to a structured experimentsetup and protocol, enabling us to draw confident conclusions about steering performancein curved paths.55Chapter 4Results, Modeling and DiscussionThis chapter presents the results of the user study, as well as proposes and evaluates severalmodels that incorporate total curvature K to improve predictions of movement time forsteering tasks along curved paths of constant width. We build on the experimental designand findings introduced previously, as well as prior work on the Steering Law. We begin byintroducing and analyzing the experimental results, which include the movement time, outof path movement and average speed measures. We also examine the participant learningeffects, specifically confirming its presence and assessing the consistency of learning trendsacross different trial types.We then propose new models that treat curvature as either an additive or logarithmic noiseterm. We also propose two additional models that include an additional interaction termbetween L and K, motivated by our empirical results. In addition to our proposed formu-lations, to benchmark our models against prior work, we adapt several existing extensionsof the Steering Law that were originally derived for circular paths, reformulating them interms of total curvature K to allow for direct comparison under our experimental condi-tions.To assess model performance, we employ statistical metrics such as adjusted R2, AkaikeInformation Criterion (AIC), and cross-validation Root Mean Squared Error (RMSE). Thesemetrics serve as a point of comparison between the models in terms of their performanceand account for the differences in each model’s complexity. In addition to the quantitativeresults, we interpret and discuss the findings and their implications.564.1 Experiment Results4.1.1 Movement Time (MT )We first perform repeated measures ANOVA with Greenhouse-Geisser correction for themovement time.Figure 4.1 illustrates the movement time for each trial averaged across n = 15 repe-titions. From the ANOVA results, we observed statistically significant main effects ofL (F = 216.55; p < 0.001;η2p = 0.402) and K (F = 29.13; p < 0.001;η2p = 0.049).We also found statistically significant but small interaction effects for L x K (F = 3.95;p< 0.05;η2p = 0.008).Table 4.1 shows a summary of the measurements for each of the trial types. As thecurviness of a path increases, the movement time also tends to increase. The only exceptionis in trial ‘L2-K2’, which has an unusual section as we discuss in the Limitations sectionin Chapter 5. This indicates that the steering task becomes increasingly more difficult themore curvy the task trajectory is, similar to findings from previous works [19, 22, 34, 39].Figure 4.1: Plots of the mean total movement time for each of the 9 trial types. Notethat the boxes present the Interquartile Range (IQR) and median, while whiskersextend to 1.5 * IQR. Each color in the strip plot represents a different partici-pant.57Table 4.1: Movement time [ms] for each of the 9 trial types.Trial ID Mean Std.L0-K0 11932.85 2724.46L0-K1 13084.73 2439.85L0-K2 14809.09 3480.69L1-K0 15867.06 3521.73L1-K1 17152.55 4598.70L1-K2 18038.69 3587.48L2-K0 20092.95 4214.74L2-K1 21704.08 5663.71L2-K2 21432.13 4601.69Table 4.2: Out of path movement [%] measure for each of the 9 trial types.Trial ID Mean Std.L0-K0 0.0021 0.0024L0-K1 0.0013 0.0020L0-K2 0.0005 0.0007L1-K0 0.0028 0.0031L1-K1 0.0027 0.0020L1-K2 0.0019 0.0021L2-K0 0.0052 0.0042L2-K1 0.0034 0.0029L2-K2 0.0018 0.00174.1.2 Out of Path Movement (OPM)To examine the effects of the path parameters on the error rate, we use the OPM, a mea-sure introduced by Kulikov et al. [16]. This is calculated as the percentage of sampledpoints during a participant’s trial that fall outside the boundaries of the drawn tunnel. Wefound statistically significant main effects of L (F = 23.062; p < 0.001;η2p = 0.112) andK (F = 12.639; p< 0.001;η2p = 0.096). We did not find statistically significant interactioneffects for L x K. Figure 4.2 shows the boxplot of the results, overlaid with the individualparticipant data. The raw data is shown in Table 4.2.58Figure 4.2: Plots of the out of path movement measure for each of the 9 trial types.4.1.3 Average Speed (Vavg)The average speed is calculated as the amount of distance traveled by the cursor divided bythe total time taken to complete the trial (in px/ms). The results are shown in Figure 4.3.The raw data is shown in Table 4.3. We did not find statistically significant main effectsfor L, but found a large, significant effect for K (F = 74.074; p < 0.001;η2p = 0.083).We also found a small but statistically significant interaction effect for L x K (F = 7.299;p< 0.01;η2p = 0.017). We can note that as K increases, the average speed decreases. Thisis in accordance with past studies [22, 34] and also the intuition that as a path gets curvierand more difficult, users slow down.4.1.4 Learning EffectsConsistent with expectations, post-hoc analyses revealed a small but significant learningtrend within each block (F = 2.988; p< 0.05;η2p = 0.002) and a large, persistent learningeffect over the course of the experiment (F = 11.285; p< 0.001;η2p = 0.209).To further investigate the learning process, we applied regression analysis for each trialtype and participant to estimate the coefficients of the power law of practice, defined as:MT (n) = a · (n+ 1)−b, where n is the number of trials completed, a represents the initialperformance level and b reflects the learning rate specific for each trial type and partici-59Figure 4.3: Plots of the average speed for each of the 9 trial types.Table 4.3: Average speed [px/ms] for each of the 9 trial types.Trial ID Mean Std.L0-K0 0.261 0.058L0-K1 0.232 0.041L0-K2 0.205 0.049L1-K0 0.250 0.052L1-K1 0.235 0.063L1-K2 0.215 0.041L2-K0 0.242 0.043L2-K1 0.229 0.057L2-K2 0.224 0.046pant [24]. Our results indicated no statistically significant difference (p > 0.05) betweenthe learning rates (b) across the trial types, indicating that all trial types were learned atcomparable rates.604.2 Model Derivation4.2.1 Proposed ModelsWe return to the information theory basis of Fitts’ Law and the signal and noise analogyand hypothesize that either a simple model of the form:MT = a+b L+ c K (4.1)where K is an additive noise, or a case where the noise addition is logarithmic:MT = a+b L+ c · log2(K+1) (4.2)would improve on the model fitness of the original Steering Law.In the case of a straight path where K = 0, both of these equations would be equivalent tothe base Steering Law formulation (Equation 2.5).Additionally, the presence of small interaction effects detected between length and curvi-ness on the movement time from the ANOVA results leads us to the addition of an extrainteraction factor L ·K to Equations 4.1 and 4.2, creating the new models:MT = a+b L+ c K+d L ·K (4.3)MT = a+b L+ c · log2(K+1)+d L ·K (4.4)It is unclear what the information theoretic analogy for this interaction term would be.However, in a previous work by Yamanaka and Miyashita, they found that the inclusionof a significant interaction factor between the width and the reciprocal of the radius ofcurvature of a circular path resulted in higher model fitness than alternative models withoutit [34]. Thus, we also investigate the effect of adding this significant interaction.It should be noted from Equations 4.1 and 4.2 that we have omitted the W parameter. Acommon behavior in steering through sharp corners identified in [25] is ‘corner-cutting’,which is the tendency of users to navigate wide corners by taking a shortcut, reducing thetotal movement time. In our experiment, we use a small, constant width for all conditionsto reduce corner-cutting. Further, to constrain the scope of our experiment, we are notexamining the potential interaction of width and curvature, leaving that to future work. In-stead, our study focuses on length and curvature. The corner-cutting behavior is discussedin the Limitations section in Chapter 5 as well.614.2.2 Comparison Models from Prior WorkIn our analysis, we compare the fitness of our model with the base Steering Law model, aswell as past proposed models adapted to be compatible with our experiment paths. Thesemodels are reformulated by removing the width parameter, as it is kept constant in our ex-periments. In addition, past work on extending the Steering Law to circular paths typicallyuse the radius of curvature R. To be used in our comparisons, we adjust the past modelsas if they were formulated for paths of arbitrary curvature rather than constant curvature.All models derived in this section are summarized in Table 4.4 along with the model fittingresults.Base ModelThe base Steering Law model by Accot and Zhai [1] was presented in Equation 2.5 andgiven as:MT = a+b · LWRemoving the effect of width, asW is a constant that would be absorbed into the regressioncoefficients, we obtain:MT = a+b ·L (4.5)This serves as the baseline for comparison throughout this chapter.Nancel and Lank ModelWe first compare our proposed models with the fitness of Nancel and Lank’s model [23],presented previously in Equation 2.7:MT = a+b′∫CdsW (s) ·R(s)1/3Since the radius of curvature R is the reciprocal of the instantaneous curvature κ , we canuse κ(s) = 1/R(s). Furthermore, since we are not examining the effect of width, the W (s)term becomes a constant that can be absorbed into the regression coefficients. Equation2.7 then becomes:MT = a+b L∫Cκ(s)1/3ds (4.6)62Yamanaka and Miyashita ModelThe model proposed by Yamanaka and Miyashita [34] was presented previously in Equa-tion 2.8:MT = a+bLW + c 1R +dWRThis model was derived for circular arc paths, so we cannot directly compare it to ourproposed models. We will instead adjust their equation as if it was formulated for a pathof arbitrary curvature.Presented previously in Equation 3.1, the total curvature K can be computed as:K =∫C|κ(s)|dsSince κ(s) = 1R for a circular arc, we obtain K =LR .Rearranging Equation 2.8 by absorbing the W parameter into the regression constantsand introducing K, we obtain the modified version of Yamanaka and Miyashita’s model,adapted for constant width paths and extended for arbitrary curviness:MT = a+bL2L+ c K(4.7)Liu et al. ModelSimilarly for the model proposed by Liu et al. [19], we modify Equation 2.6:log10(MT ) = a+b[log10(LW)+ c κ]to absorb the width term into the regression intercept constant. We then use circular arcrelationships to include the total curvature parameter and convert it to MT form from theoriginal log10(MT ) formulation to obtain:MT = 10a+b·log10(L)+cKL (4.8)4.3 Model Fitting and AnalysisWe perform regression analysis and use statistical metrics to evaluate the performance ofthe proposed and comparison models. The evaluation aims to first determine if models aug-mented with K improve the movement time prediction ability compared to the base model.63By comparing our proposed models, we also determine the form of contribution that thecurviness term takes in the movement time prediction equation. We also aim to answerwhether the addition of an interaction term between length and curvature enhance modelaccuracy. Finally, we assess how well existing models from prior work, reformulated usingK, generalize to our task setting involving curved 2D paths.4.3.1 Regression ResultsTo evaluate model performance, we performed linear regression using the average move-ment time across the 9 trial types. Table 4.4 describes the linear regression coefficients,adjusted r2 values, and their Akaike Information Criterion (AIC) values for the MT mod-els [5]. Similar to other recent studies on Steering Law extensions comparing models[12, 31, 34], we use AIC as the comparison metric as it balances the model fit and com-plexity. A lower AIC value indicates a better model. To check the fitness visually, Figure4.4 shows the fitted models for MT = a+ b · ID on the average values for each trial type(N f it = 9). As well, the data points for each trial type and participant, averaged across 15repetitions, are plotted in the same graph (Ntotal = 180).From the linear regression results in Table 4.4, we can see that the models described byEquations 4.3 and 4.4 show the best fits in terms of AIC and adjusted r2. While thesetwo models contain the most free parameters, raising potential concerns about overfitting,AIC and adjusted r2 are criteria designed to penalize complexity. To address any residualconcerns, we perform cross-validation in the next section.4.3.2 Interpretation of Model FitExamining the model fitting results from our proposed models derived from the signal andnoise analogy, they suggest that the curviness of a path may contribute to a logarithmicnoise addition, as indicated by the lower adjusted r2 in Equations 4.2 and 4.4 compared totheir counterparts in Equations 4.1 and 4.3 The regression coefficient for the curviness termalso has a smaller p-value in the logarithmic forms, suggesting a stronger effect. However,the current study’s range of K values may be too narrow to capture the full behavior of thisrelationship. Expanding the range of K in future studies, especially the inclusion of K = 0in straight paths, could provide further insights into the nonlinear noise addition.It should also be noted that the regression coefficients for the interaction term in Equations4.3 and 4.4 are statistically significant (p< 0.05), which is in line with our ANOVA results.This suggests that there is a need to include the K ·L term.64Table4.4:Regressioncoefficients(with95%confidenceintervals),adjustedr2andAICvaluesformodelspredictingMT.Notethat*,**,and***indicatep<0.05,p<0.001,andp<0.0001respectively.Eqn.Model(MT)abcdAdj.r2AIC4.4a+bL+c·log 2(K+1)-22052.8*12.0***5238.3*-0.2*0.99133.4+dL·K[-35672.9,-8432.7][9.4,14.7][1890.6,8586.0][-0.3,-0.0]4.3a+bL+cK+dL·K-9579.8*12.6**537.2*-0.2*0.99134.1[-15679.0,-3480.6][9.5,15.8][175.6,898.8][-0.4,-0.0]4.7a+bL2L+cK-3771.5*9.7***-13.8**0.981139.0[-6375.2,-1167.7][8.5,10.9][-18.7,-8.9]4.2a+bL+c·log 2(K+1)-8693.6*9.5***1930.3*0.981139.0[-12932.9,-4454.3][8.3,10.7][1032.6,2828.1]4.1a+bL+cK-3647.4*9.5***170.1*0.979140.0[-6368.9,-925.9][8.3,10.7][85.8,254.3]4.810a+b·log10(L)+cKL0.21.2***8.1*0.978140.7[-0.4,0.8][1.0,1.4][3.9,12.3]4.5a+bL-980.09.5***0.911152.6[-5768.9,3809.0][7.0,12.0]4.6a+bL∫ Cκ(s)1/3ds-335.00.04**0.793160.1[-7777.8,7107.8][0.0,0.1]654.3.3 Comparison Models FitModels described by Equations 4.1, 4.2, 4.7 and 4.8, also exhibit strong performance interms of AIC and adjusted r2. All four models have AIC values within 2 units of eachother, which indicates that they show comparable fitness. Since they are within 10 units ofAICminimum (133.4), we consider them to be valid representations of the data.Most importantly, all four models incorporate the total curvature term K and outperformthe base Steering Law model (Equation 4.5), N.B., as it does not account for curvaturevariations in any way.They also outperform the model described by Equation 4.6, which uses an alternativeparameter based on the 2/3 Power Law for unrestrained movement [23, 29]. In the orig-inal work by Nancel and Lank, this model was evaluated on compound paths with dis-continuities in the width and curvature constraints not present in our paths, which mayrequire additional adjustments in user behavior to transition between constraints. In addi-tion, the original Power Law relationship between the radius of curvature and movementspeed is applicable to motion segments marked by landmarks such as cuspids and inflectionpoints. In Nancel and Lank’s experiment paths, joints between circular arcs are inflectionpoints, and their binning of radii of curvature adheres to the segmented motion assump-tion. These differences likely contributed to its reduced performance on our experimentalcurves. Overall, these findings strongly suggest that the inclusion of the parameter K im-proves the predictive accuracy of Steering Law models.The models adapted from Yamanaka and Miyashita [34] as well as Liu et al. [19] under-performed slightly compared to our proposed models. Since they were originally designedfor simple or compound uniform circular paths, these models required modifications to fitour specific experimental conditions involving 2D paths of fixed width and arbitrary curva-ture. We suspect that the transformation of these models—by assuming constant curvatureas a specific instance of arbitrary curvature—might have left out terms that could furtherexplain the variation in data observed in our study. Despite these differences, the relativelystrong performance of these adapted models confirms their validity and utility in contextssimilar to our experimental setup.4.3.4 Model SimplificationBy examining the regression coefficients for the models described by Equations 4.6 and4.8, we see that the a constants’ 95% confidence interval ranges include zero and non-significant p-values, which indicates that these models may perform better without thisterm. By refitting the data to a modified version of Equation 4.6 without an intercept66(MT = b L∫C κ(s)1/3ds), we obtain b = 0.0347 [0.032,0.037] with p < 0.0001, adjustedr2 value of 0.819 and an AIC value of 158.1, exhibiting an improvement in model fit-ness. Then, doing the same with Equation 4.8 (MT = 10b log10(L)+cKL ), we obtain b =1.267 [1.258,1.276], c= 8.887 [5.521,12.253] with p< 0.0001, adjusted r2 value of 0.978and AIC value of 139.8. The slight improvement may simply be due to the model using oneless parameter. The improved forms of these two models are used in the cross-validation.4.4 Cross-ValidationWe conduct a form of 5-fold cross-validation to further assess the potential for overfitting,despite the ability of AIC and adjusted r2 measures to balance out model complexity withfitness. During the model fitting process, our dataset comprised of 9 data points, one foreach combination of L and K, with each point obtained from the average movement timeacross all participants and for 15 repetitions of each trial type. For the cross-validationanalysis, each of the models is fitted 5 times, each time using 9 datapoints computed fromthe average of 12 of the 15 repetitions. For each fold, the models are tested on the averageof the last 3 trials left out. The average Root Mean Square Error (RMSE) values across the5-fold validation are shown in Table 4.5.The results indicate that the models incorporating the L ·K interaction term (Equations4.3 and 4.4) consistently have the highest performance, as indicated by the lowest RMSEvalues. Based on the cross-validation results, we can conclude that overfitting is not aconcern in these models, even with the inclusion of additional terms.Table 4.5: Average Root Mean Square Error values computed across 5-fold cross val-idationEqn. Model (MT ) RMSE4.4 a+b L+ c · log2(K+1)+d L ·K372.334.3 a+b L+ c K+d L ·K 381.184.7 a+b L2L+c K 480.764.2 a+b L+ c · log2(K+1) 480.884.1 a+b L+ c K 498.624.8 (without a) 10b·log10(L)+cKL 535.364.5 a+b L 967.334.6 (without a) b L · ∫C κ(s)1/3ds 1435.43674.5 Chapter SummaryWe presented and analyzed the results from our user study, detailing the effects of thelength and curviness parameters on the movement time, out of path movement, and averagespeed. We also assessed the learning effects present throughout the experiment trials. Then,we derived several models of movement time prediction, including comparison models.Overall, models incorporating total curvature K demonstrate the best performance in bothregression and cross-validation. These results support our hypothesis that path curvature,in combination with path length, contribute in predicting movement time in curved steeringtasks.68Figure 4.4: Visualization of model fitness of MT (or log10(MT ) for Equation 4.8)plotted against ID for each model.69Chapter 5Future Work and ConclusionThis chapter reflects on the limitations of our study and outlines several directions for futurework. We then conclude the thesis by summarizing our contributions and the implicationsof incorporating curvature into models of human movement time prediction.5.1 Limitations and Future WorkThis study focused on evaluating the contribution of path curviness to movement time, withseveral design choices made to isolate this effect. However, some of the choices introducelimitations and areas for future work.5.1.1 Width and Path Shape EffectsThe main limitation of our study is the exclusion of the width parameter as one of theindependent variables in our trials. We expect that the corner-cutting behavior can bebetter understood through analyzing tunnels of different widths. For instance, as seen inFigure 4.1 a), the trial ‘L2-K2’ had a lower average movement time than ‘L2-K1’ despitethe increase in curviness. We suspect that this behavior is due to the very small undulationsin the middle portion of the ‘L2-K2’ trial that allow the section to be bypassed as a straighttunnel task. A visual inspection of the plotted heatmaps of the trajectory frequencies ofall 300 repetitions of the L2 trials, shown in Figure 5.1, supports this hypothesis. Theparticipant trajectories do not follow the undulations in the middle section of ‘L2-K2’,suggesting that the large width tolerance of the tunnel allowed participants to ‘cut corners’.We suspect that a narrower tunnel would result in longer movement times.Additionally, the heatmaps highlight that areas of highest curvature (i.e., the sharpest cor-ners) are more brightly colored than the rest. This may correspond to the ‘corner-aiming’behavior identified in [25], which happens in narrow corners. The corner-cutting and aim-70ing effect could be interpreted as participants using a perceived motion model, preemp-tively adjusting their movement to avoid unnecessary corrections, reducing the noise trans-mitted through the human channel (i.e., the curviness of the path) and thereby decreasingthe ID of the overall task. Interestingly, incorporation of predictive movement models andlinguistic and physical constraints for reducing the ID of the complex trajectory movementtasks, like tongue movements in speech, may be a strategy for reducing motor control loadwhile speaking [9].5.1.2 Effective ParametersIn future work, incorporating effective parameters into the models such as We and Le,which account for actual user behavior, could improve the movement time prediction asdemonstrated in previous studies [12, 16, 42]. Specifically, We is computed from the stan-dard deviation σ of movement coordinates perpendicular to the centerline of the tunnel as:We = 4.133 σ .We represents both the utilization rate of the entire width of the tunnel as well as a mea-sure of error. This is especially important in the information-theoretic analogy to under-stand how the variability in noise (width and curviness constraints) perturbs the movement[20]. Understanding howWe interacts with the curviness of the tunnel would provide moreinsight on corner cutting/aiming behaviors.For a future study on the effects of curviness and width, an additional parameter that couldbe used is the curviness of the shortest path through the tunnel between the two endpoints.Kmin could potentially account for sections that can be bypassed as straight tunnels as wellas the curviness of the shortest edge of sharp corners.5.1.3 Scale and GeneralizationAnother limitation of the study is the restricted range of path lengths evaluated. Past re-search suggests that the Steering Law has operational constraints at different scales ofmotion [3]. Future work would involve evaluating the model on both shorter and longerpaths to better understand and decouple the effects of scaling.5.2 ConclusionIn this thesis, we introduced the total curvature parameter K as a significant extension tothe Steering Law to better account for the curviness of constant-width paths in movementtime prediction and human behavior modeling. This work was motivated by the limitationsof the original Steering Law, which predicts the same movement time for paths with equal71Figure 5.1: Heatmap of trajectory frequencies computed over 300 repetitions for eachof the L2 trial types.length and width regardless of their curviness. We hypothesized that total curvature con-tributes to task difficulty in a way that is not captured by existing formulations, since paststudies in this field primarily focused on simple or compound circular paths. We also de-signed an experiment to test this hypothesis using fixed-width sinusoidal tunnels of varyinglevels of length and curviness.Our study demonstrated that the inclusion of K improves model fitness, as indicated by thehigher adjusted r2, as well as lower AIC and RMSE values when fitted and cross-validatedon the data. We suggest the following form for predicting movement time:MT = a+b L+ c · log2(K+1)+d L ·K72which includes a significant interaction term L ·K between path length and curviness.We have also adapted models originally developed for uniform circular paths to be appli-cable on paths of fixed width and varying curvature. By excluding width and incorporatingthe total curvature parameter K, the models derived from prior work by Yamanaka andMiyashita (Equation 4.7) and Liu et al. (Equation 4.8) have demonstrated good fitness,effectively capturing a varied spectrum of curved path scenarios.The extensions to the Steering Law and the inclusion of path curviness can help designersunderstand and optimize user paths in complex interfaces, such as in menu selections, ac-cessibility design, lasso operations, image segmentation tasks, etc. Beyond user interfacedesign, it has applications in areas such as motor control, virtual navigation, game leveldifficulty evaluation, and extends into understanding complex motor tasks like speech pro-duction.73Bibliography[1] J. Accot and S. Zhai. Beyond Fitts’ law: models for trajectory-based HCI tasks. InCHI ’97 extended abstracts on Human factors in computing systems looking to thefuture - CHI ’97, page 250, Atlanta, Georgia, 1997. ACM Press. ISBN978-0-89791-926-5. doi:10.1145/1120212.1120376. URLhttp://portal.acm.org/citation.cfm?doid=1120212.1120376. → pages 1, 6, 7, 55, 62[2] J. Accot and S. Zhai. 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IEEE.doi:10.1109/ICINFA.2011.5949107. → pages 1, 1979Appendix APilot Studies and Helper ApplicationsA.1 Curve Visualizer Applications80A.2 Pilot Round 3 Landing Page Images818283Appendix BUser Study MaterialsB.1 Introduction SpeechHello and welcome! Thank you for participating in our research study. My nameis Jennie, and I’ll be guiding you through today’s session. Your participation isinvaluable to us as we seek to understand more about human-computer interac-tion.Today, you will be participating in a task that involves navigating a cursor througha series of tunnels displayed on a computer screen. The aim of this study is toinvestigate how changes in the path’s parameters affect your ability to control thecursor.Before we begin, please remember that your participation is entirely voluntary.You have already signed a consent form, but if you have any questions or con-cerns, feel free to ask at any time. I’ll be here to assist you with any technicalissues or questions. If you wish to withdraw from the study at any time during thesession, rest assured that any data collected will be destroyed immediately afterthe session.First, please fill out this background survey. You may abstain from answeringany question you are not comfortable with. Please be assured that all the data wecollect today will be de-identified, kept confidential and used solely for researchpurposes. Specifically, the demographics data is collected to gain insight into thegeneralizability of the research. Your personal identifiable information will bedetached from the data and kept confidential. Are there any questions before webegin?84[Wait for survey response]Thank you for filling the survey! Let’s move onto the experiment. You will startwith a brief tutorial to get you accustomed to the task. Once you pass the tutorialstage, you will proceed to the main part of the experiment. The purpose of thetutorial is to see if you are able to complete the task accurately and consistently,so try to keep this in mind! If at any point you need a break, you may do so in-between tasks.Once you enter your participant ID, you will be brought to the next page whereyou will get a written version of what I just told you and of the instructions, aswell as some visual examples of the tasks.85B.2 Background SurveyFigure B.1: Steering experiment background survey questions.86B.3 Consent FormsFigure B.2: Steering experiment consent form, page 1.87Figure B.3: Steering experiment consent form, page 2.88B.4 Video TranscriptStudy TaskWelcome to the experiment! In this task, you will navigate a cursor through aseries of curves as accurately and quickly as possible using the provided mouse.At the start of each trial, you will see a green button with a triangle on the leftside, a gray tunnel, and a red button on the right side. When you are ready tobegin, click the green button to start. Then, steer the cursor through the tunneltowards the red button. The red button will turn into your target with a flag icon.Once you reach it, click it. After clicking the red button, it will turn green, andthe red button with the flag will appear on the left side. Navigate back throughthe tunnel and click the red button to complete the trial. The button will turn backinto the green start button. This is a good time to take a break if needed. Click thegreen button when you are ready to start the next trial.Common Mistakes to Avoid:There are a few common mistakes to avoid. First, you don’t need to hold downthe mouse button while steering; just click the buttons once you reach them. Also,avoid taking breaks during the trial, as the timer runs from the moment you clickthe start button until you complete the round-trip. Take breaks between trialswhen the screen shows the green start button.2 PhasesThe study has two phases: the tutorial phase and the experiment phase. In thetutorial phase, you will perform the same task as in the experiment phase. Red‘error circles’ will appear if you exit the path. Try to minimize errors and maintaina constant speed to pass the tutorial.In the experiment phase, the task is divided into 27 blocks of 5 trials each. Youwon’t see the error circles, but you will receive a summary of mistakes at the endof each block. There will be a 15-second break between blocks. Good luck!89
Thesis/Dissertation
eng
Electrical and Computer Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Fels, Sidney
Revisiting the steering law : curvature, complexity, and the path to a new predictive model
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