1. Bibliographic Information

1.1. Title

Predictive multiuser redirected walking using artificial potential fields

1.2. Authors

• Christian Hirt (ETH Zurich, Innovation Center Virtual Reality)
• Noah Isaak (ETH Zurich, Innovation Center Virtual Reality)
• Christian Holz (ETH Zurich, Sensing, Interaction & Perception Lab)
• Andreas Kunz (ETH Zurich, Innovation Center Virtual Reality)

1.3. Journal/Conference

Frontiers in Virtual Reality

• Reputation: This is a reputable open-access journal dedicated to virtual reality research. It covers a broad range of topics from haptics to locomotion and is considered a significant venue for specialized VR contributions.

1.4. Publication Year

2024 (Published on August 08, 2024)

1.5. Abstract

This paper addresses the challenge of navigating large virtual environments (VEs) within limited physical spaces using Redirected Walking (RDW). While real walking is the most intuitive locomotion method in VR, physical constraints often limit its use. The authors propose two novel predictive RDW systems designed for multi-user scenarios in unconstrained virtual spaces. These systems utilize Artificial Potential Fields (APFs) for path planning and employ two distinct trajectory prediction methods based on clothoids (geometric curves).

• Method 1 (PredRed LPP): Uses a lemniscate (figure-eight) shape to generate candidate paths.
• Method 2 (PredRed FPP): Uses a forward path prediction based on derivatives of the user's motion. Validating these against standard reactive approaches (Steer-to-Center, Steer-to-Gradient) in a study with 150 participants, the results show that the new predictive systems generally outperform existing methods, reducing the number of interruptions (resets) required.

1.6. Original Source Link

Predictive multiuser redirected walking using artificial potential fields (Open Access, Officially Published)

2. Executive Summary

2.1. Background & Motivation

• The Core Problem: "Real walking" is the gold standard for VR locomotion due to its high immersion and low motion sickness. However, the physical room size is almost always smaller than the virtual world users want to explore.
• Existing Solutions & Gaps: Redirected Walking (RDW) solves this by subtly steering users away from physical walls while they think they are walking straight.
  • Gap 1: Most advanced RDW methods (Predictive RDW) rely on predefined "skeleton graphs" (maps of where users can walk), limiting them to constrained layouts like corridors. They struggle in open, unstructured spaces.
  • Gap 2: Most systems are designed for single users. Handling multiple users in the same physical space adds dynamic obstacles (other people) that static maps cannot handle effectively.
• Innovation Point: This paper introduces a general-purpose predictive framework that combines Artificial Potential Fields (APFs) (which handle dynamic obstacles like other users naturally) with short-term path prediction based on clothoid curves, removing the need for pre-baked maps or skeleton graphs.

2.2. Main Contributions & Findings

1. Two Novel Prediction Algorithms:

   • LPP (Lemniscate Path Prediction): Generates a bundle of possible paths within a lemniscate shape and selects the best one using scene awareness.
   • FPP (Forward Path Prediction): Analytically calculates a single future path by solving differential equations of the user's current motion derivatives.

2. Integrated Steering System: A cost-based planning system using Non-Harmonic APFs. This allows the system to treat walls and other users as repulsive forces, dynamically calculating the safest direction to steer the user.

3. Large-Scale Validation: A user study with 150 participants (75 pairs) exploring diverse virtual rooms.

4. Key Finding: The PredRed LPP system statistically outperformed traditional reactive methods (like Steer-to-Center) and the simpler predictive FPP in terms of the distance users could walk without being interrupted (reset).

   The following figure (Figure 1 from the original paper) illustrates the concept of guiding multiple users in a shared physical space using these potential fields.

   该图像是插图，展示了多个用户在物理重叠的环境中进行的高效行走引导，依据非谐人工势场实现。图中可见一名用户在虚拟环境中行走，周围是各类设备，表明此设置旨在优化用户在有限空间内探索虚拟场景的体验。

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

To understand this paper, a novice needs to grasp three core concepts:

1. Redirected Walking (RDW) & Gains:

   • Concept: The VR system tracks the user's real movement and slightly rotates or translates the virtual camera. This tricks the user into walking a curved physical path while perceiving a straight virtual path.
   • Gains: These are the "intensities" of the trick.
     • Curvature Gain: Bending the user's path (e.g., user walks a circle, sees a straight line).
     • Rotation Gain: Rotating the virtual world faster/slower than the user's head turn.
   • Reset: When RDW fails and a user is about to hit a wall, the system pauses and forces the user to spin around (reorient) to face a safe direction. This is called a "Reset" and breaks immersion; fewer resets are better.

2. Artificial Potential Fields (APF):

   • Concept: Originally from robotics. Imagine the physical room as a landscape.
   • Repulsive Force: Walls and obstacles (or other users) are "mountains" (high potential).
   • Attractive Force: Safe open spaces are "valleys" (low potential).
   • Result: A "ball" (the user) naturally rolls away from mountains toward valleys. The RDW algorithm steers the user along this "force vector."

3. Clothoids (Euler Spirals):

   • Concept: A curve where the curvature changes linearly with the curve's length.
   • Why used here? They perfectly model natural human walking paths (transitions from straight to turning) better than simple circles or lines.

3.2. Previous Works & Technological Evolution

• Reactive RDW: The earliest and simplest form (e.g., Steer-to-Center (S2C)). It looks at the user's current position and steers them toward the center of the room.
  • Limitation: It doesn't know where the user is going, often steering them wrongly if they are about to turn.
• Predictive RDW: Uses knowledge of the future path to steer more intelligently.
  • Skeleton Graphs (Peck et al., 2012; Zmuda et al., 2013): Pre-mapping the virtual world into nodes and paths. Good for corridors, bad for open fields.
• APF in RDW: Introduced by Thomas and Rosenberg (2019) and Bachmann et al. (2019). These were primarily reactive (Steer-to-Gradient), treating the user as a particle pushed by wall forces.

3.3. Differentiation Analysis

• Vs. Skeleton Graphs: This paper's method does not require a pre-built graph of the virtual world. It predicts paths dynamically using curve geometry (clothoids), making it usable in "unconstrained" open virtual spaces.
• Vs. Standard APF: Previous APF methods were reactive (pushing the user away from the nearest wall). This paper makes APF predictive: it projects a future path into the potential field and calculates the "cost" of that entire path to decide the best steering action.

4. Methodology

4.1. Principles

The core principle is to minimize a cost function over a predicted future trajectory.

1. Predict where the user will walk in the next few seconds ($T_{pred}$).
2. Simulate applying different RDW gains (steering actions) to this path ($T_{red}$).
3. Evaluate these redirected paths against the physical environment (walls, other users) using an Artificial Potential Field (APF).
4. Select the action that keeps the user in the safest ("lowest potential") areas.

4.2. Artificial Potential Field (APF) Implementation

The authors choose Non-harmonic APFs over Harmonic or Wavefront versions because they are computationally faster (3.0 ms vs 16.5 ms).

The Redirection Vector ($\vec{F}_{red}$) represents the "force" pushing the user away from danger. It is the sum of repulsive forces from all obstacles $i$ (walls, other users).

The total force vector is calculated as: $ \vec { F } _ { red } = \sum _ { i \in O } \vec { F } _ { rep , i } $

The force from a single obstacle $i$ is modeled using a Gaussian-like decay function: $ \vec { F } _ { rep , i } = \left{ \begin{array} { l l } { a _ { o } \cdot exp \left( - b _ { o } \cdot d _ { o } ^ { 2 } \right) \cdot \vec { e } _ { o } , } & { \mathrm { ~ if ~ } d _ { o } \leq d _ { d } } \ { 0 , } & { \mathrm { ~ else } } \end{array} \right. $

Symbol Explanation:

• $a_o$: Maximum repulsive force (height of the "potential mountain").
• $b_o$: Distribution width (how steep the slope is).
• $d_o$: Distance between the user ($p_{eval}$) and the obstacle ($p_i$).
• $d_d$: Cut-off distance (forces beyond this range are ignored).
• $\vec{e}_o$: The direction vector pointing away from the obstacle.

4.3. Prediction Algorithms

The paper proposes two methods to generate the predicted path $T_{pred}$. Both use smoothing on the user's head tracking data (Double-exponential smoothing) to remove jitter before prediction.

4.3.1. Method 1: Lemniscate Path Prediction (LPP)

Concept: Humans rarely walk in random squiggles; they usually turn left, right, or go straight. This method generates a "bouquet" of possible paths shaped like a Lemniscate (Figure-8) and picks the best one.

Step 1: Generate the Lemniscate Shape. The boundary of possible paths is defined by the parametric equations of a Lemniscate: $ \begin{array} { l } { \displaystyle { x \left( t \right) = A _ { Lem } \cdot \frac { sin \left( t \right) } { 1 + cos ^ { 2 } \left( t \right) } } } \ { \displaystyle { y \left( t \right) = A _ { Lem } \cdot \frac { sin \left( t \right) cos \left( t \right) } { 1 + cos ^ { 2 } \left( t \right) } } , } \end{array} $

• $A_{Lem}$: A constant defining the size (horizon) of the prediction (set to 3 meters).
• $t$: Running variable $\in (0, \pi)$ to generate the loop.

Step 2: Discretize and Check Collisions. The system generates multiple clothoid curves ending at points along this Lemniscate. It then performs Scene Awareness: it checks if any of these curves hit a virtual wall. If a path hits a virtual wall, it is discarded.

Step 3: Select the Best Path. From the remaining valid paths, the system compares them to the user's past trajectory (stored in a buffer) using a mean-squared error. The path that best continues the user's current motion trend is selected as $T_{pred}$.

The following figure (Figure 2 from the original paper) visualizes this process: determining valid paths (white) versus colliding paths (discarded) within the Lemniscate shape.

该图像是一个示意图，展示了在LPP中运用场景感知的过程。图中红色路径为先前行走的轨迹，黄色为真实轨迹，绿色为虚拟障碍物的边缘。图(A)中，场景感知早期移除不合适的轨迹，仅保留有效的白色轨迹，而左侧的对称预测因碰撞而被丢弃。图(B)展示了从有限有效轨迹中识别出的最佳预测，预测的起点（紫色）稍微位于过去。

4.3.2. Method 2: Forward Path Prediction (FPP)

Concept: Instead of guessing multiple paths, calculate the single most likely path mathematically based on the physics of the user's current movement (velocity, turning rate, change in turning rate).

Step 1: The Non-holonomic System. Human walking is modeled as a dynamic system: $ \left( \begin{array} { c } { { \dot { x } _ { T } } } \ { { \dot { y } _ { T } } } \ { { \dot { \varphi } _ { T } } } \ { { \dot { \kappa } _ { T } } } \end{array} \right) = \left( \begin{array} { c } { { cos \ \varphi _ { T } } } \ { { sin \ \varphi _ { T } } } \ { { \kappa _ { T } } } \ { { 0 } } \end{array} \right) u _ { 1 } + \left( \begin{array} { c } { { 0 } } \ { { 0 } } \ { { 0 } } \ { { 1 } } \end{array} \right) u _ { 2 } $

Symbol Explanation:

• $x_T, y_T$: User position.
• $\varphi_T$: User heading (direction).
• $\kappa_T$: Curvature of the path.
• $u_1$: Linear velocity ($\nu_T$).
• $u_2$: Rate of change of curvature ($\dot{\kappa}_T$).

Step 2: Calculate Derivatives via Finite Differences. To solve the system, we need the derivatives of the user's position ($\dot{x}, \ddot{x}, x^{(3)}$). The paper uses Central Finite Differences (CFD) on the last 8 data points ($t$ to t-8). The coefficients ($C_{CFD}$) for the derivatives are: $ \begin{array} { l } { { C _ { CFD , d 1 } = \bigg ( \frac { 1 } { 2 8 0 } - \frac { 4 } { 1 0 5 } \frac { 1 } { 5 } - \frac { 4 } { 5 } 0 \frac { 4 } { 5 } - \frac { 1 } { 5 } \frac { 4 } { 1 0 5 } - \frac { 1 } { 2 8 0 } \bigg ) } } \ { { C _ { CFD , d 2 } = \bigg ( - \frac { 1 } { 5 6 0 } \frac { 8 } { 3 1 5 } - \frac { 1 } { 5 } \frac { 8 } { 5 } - \frac { 2 0 5 } { 7 2 } \frac { 8 } { 5 } - \frac { 1 } { 5 } \frac { 8 } { 3 1 5 } - \frac { 1 } { 5 6 0 } \bigg ) } } \ { { C _ { CFD , d 3 } = \bigg ( - \frac { 7 } { 2 4 0 } \frac { 3 } { 1 0 } - \frac { 1 6 9 } { 1 2 0 } \frac { 6 1 } { 3 0 } 0 - \frac { 6 1 } { 3 0 } \frac { 1 6 9 } { 1 2 0 } - \frac { 3 } { 1 0 } \frac { 7 } { 2 4 0 } \bigg ) } } \end{array} $ These coefficients are dot-multiplied with the position history vectors ($\vec{x}, \vec{y}$) to get the derivatives at time t-4: $ \dot { x } _ { t - 4 } = C _ { CFD , d 1 } \cdot \vec { x } \quad \text{(and so on for y and higher orders)} $

Step 3: Compute Curvature and Clothoid Parameters. Using the derivatives calculated above, the current curvature $\kappa$ is found: $ \kappa = \frac { \dot { x } \ddot { y } + \ddot { x } \dot { y } } { \left( \dot { x } ^ { 2 } + \dot { y } ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } $ And its derivative (change rate) $\dot{\kappa}$: $ \dot { \kappa } = \frac { 2 \left( \dot { x } ^ { 2 } + \dot { y } ^ { 2 } \right) \left( x ^ { ( 3 ) } \dot { y } + 2 \ddot { x } \ddot { y } + y ^ { ( 3 ) } \dot { x } \right) - 6 \left( \ddot { x } \dot { y } + \dot { x } \ddot { y } \right) \left( \dot { x } \ddot { x } + \dot { y } \ddot { y } \right) } { 2 \left( \dot { x } ^ { 2 } + \dot { y } ^ { 2 } \right) ^ { \frac { 5 } { 2 } } } $ Finally, the Clothoid constant $A_{Cl}$ is derived from the curvature change rate: $ A _ { Cl } = \frac 1 { u _ { 2 } } = \frac 1 \Dot { \Dot { \kappa } } $ This uniquely defines a single clothoid path extending into the future.

The following figure (Figure 3 from the original paper) illustrates the single path generated by FPP.

该图像是示意图，展示了多用户重定向行走的预测系统。图中A和B部分分别描绘了不同用户在虚拟环境中行走轨迹的变化，采用了不同颜色表示用户的实际步态和视觉感知路径，模型通过平滑的曲线生成引导用户行走。

4.4. Predictive Redirection & Cost Function

The Planning Loop:

1. Take the predicted path $T_{pred}$ (from LPP or FPP).
2. Generate a set of candidate redirected paths $T_{red}$ by applying different gains (Rotation, Translation, Curvature) from an action set $U$.
3. Calculate the cost $J_{total}$ for each candidate.
4. Pick the gain $\pi_{optimal}$ that minimizes cost: $ \pi _ { optimal } = \underset { \forall \pi \in U } { \mathrm { argmin } } \ J _ { total } \left( \pi \right) $

The Cost Function: The total cost is a sum of costs for each point $i$ along the path, weighted by a discount factor $\alpha = 0.8$ (uncertainty increases with time): $ J _ { total } = \sum _ { i = 1 } ^ { N } \alpha ^ { i } \cdot J _ { i } $

The cost for a single point $J_i$ is composed of four parts: $ J _ { i } = J _ { APF , i } + J _ { Heading , i } + J _ { Reset , i } + J _ { Gain , i } $

1. APF Cost ($J_{APF, i}$): The magnitude of the repulsive force at that point. High near walls/users. $ J _ { APF , i } = | \vec { F } _ { red } | $
2. Heading Cost ($J_{Heading, i}$): Penalizes walking against the safety force. If the user faces the wall, cost is high. $ J _ { Heading , i } = h _ { 0 } \cdot \frac { 1 } { 2 } \cdot \Bigg ( 1 - \frac { \vec { F } _ { red } } { | \vec { F } _ { red } | } \cdot \frac { \vec { \theta } _ { i } } { | \vec { \theta } _ { i } | } \Bigg ) $
   • $\vec{\theta}_i$: User's heading vector.
   • $h_0$: Design parameter (set to 1).
3. Reset Cost ($J_{Reset, i}$): A massive penalty if the path goes outside the tracking area. $ J _ { Reset , i } = \left{ \begin{array} { l l } { 1 0 0 0 , } & { \mathrm { ~ if ~ outside ~ tracking ~ room ~ boundaries } } \ { 0 , } & { \mathrm { ~ otherwise } } \end{array} \right. $
4. Gain Cost ($J_{Gain, i}$): Set to 0 in this implementation (assuming all valid gains are equally acceptable).

5. Experimental Setup

5.1. Study Design & Participants

• Participants: 150 individuals (75 pairs). Mixed gender (45 females), average age 25.5 years.

• Task: Pairs of users explored separate virtual environments (colocated in the same physical room) collecting stars. They could not see each other in VR, but the system prevented collisions via Resets.

• Hardware: HTC Vive Pro, backpack PC setup.

• Environment: 7 diverse virtual rooms (Hallways, Mazes, Open Forests) to test different constraints.

• Safety: The only safety measure was a "Reset" prompt (spin in place) triggered when users got too close to walls or each other.

  The following figure (Figure 4 from the original paper) shows the safety prompt and the star collection task.

  该图像是图示，包含两个部分。部分A展示了提示信息“<< Spin In Place”，用于指示用户在原地旋转的方向；部分B则显示了环境中的一个黄色星形收集物，供参与者寻找和收集。

The map layout of the 7 rooms is shown below (Figure 5 from the original paper).

该图像是一个示意图，展示了一个虚拟环境的地图布局。不同颜色的区域标识了不同的房间，其中包括红色、黄色和蓝色区域。该布局可能与多用户重定向行走的方法相关，旨在为参与者提供无缝的探索体验。

5.2. Evaluation Metrics

The primary metric for performance is Mean Distance Between Resets ($d_{Reset}$).

1. Conceptual Definition: This measures how far, on average, a user can walk uninterrupted before the system fails to steer them and forces a stop-and-spin (reset). A higher distance is better.
2. Mathematical Logic: $ d_{Reset} = \frac{\text{Total Distance Walked}}{\text{Number of Resets}} $
   • Note: The paper does not provide the explicit formula but describes it as normalizing resets over total walking distance.
3. Subjective Metrics:
   • Simulator Sickness Questionnaire (SSQ): Measures nausea/discomfort.
   • Presence: How "there" the user felt.

5.3. Baselines (Conditions)

The study compared 5 conditions:

1. PredRed LPP (Proposed): Using Lemniscate prediction.
2. PredRed FPP (Proposed): Using Forward path prediction.
3. S2G (Steer-to-Gradient): Reactive APF. Steers based on the local gradient of the potential field.
4. S2C (Steer-to-Center): Reactive. Always steers the user toward the center of the physical room.
5. NULL: No redirection. Users walk 1:1, only stopped by resets when hitting walls.

6. Results & Analysis

6.1. Core Results Analysis

The experimental results demonstrate that the proposed predictive methods generally provide a better walking experience (fewer interruptions) than traditional methods.

• Winner: PredRed LPP achieved the highest mean distance between resets ($5.75 \pm 0.26$ m).

• Runner-up: PredRed FPP ($5.60 \pm 0.27$ m) and S2G ($5.56 \pm 0.26$ m) performed similarly.

• Poor Performers: S2C ($5.42 \pm 0.20$ m) and NULL ($5.28 \pm 0.39$ m) were significantly worse.

  This indicates that incorporating prediction (especially the robust LPP method) allows the steering algorithm to make "smarter" decisions that pay off over time, rather than just reacting to the immediate wall distance.

The following figure (Figure 6 from the original paper) displays the distribution of $d_{Reset}$ for each condition.

该图像是图表，展示了不同重定向器的重置间距 $d_{Reset}$ 的性能表现。各组数据以小提琴图形式呈现，包含PredRed LPP、PredRed FPP、S2G、S2C和Null。图中显示的重置间距值范围从4.0到6.5米。

6.2. Detailed Data Presentation

The following are the results from Table 4 of the original paper, showing the exact numerical performance:

The statistical significance of these differences was tested using a t-test. The following are the results from Table 5 of the original paper. Green cells (simulated here by bold text with p < 0.05) indicate a statistically significant difference.

Analysis:

• PredRed LPP is significantly better than all other methods ($p < 0.05$).
• PredRed FPP is better than S2C and NULL, but not significantly different from the reactive S2G ($p = 0.312$). This suggests that the simpler analytical prediction of FPP might not add enough value over a good reactive system in this specific setup.

6.3. Room-Specific Analysis

The performance varied by room architecture.

• PredRed LPP excelled in the "Hallway" and open spaces, likely because its scene awareness successfully pruned invalid paths.

• PredRed FPP struggled in curved guided environments (like "Blue01"), as its single-path prediction cannot anticipate turns forced by virtual walls.

• S2C performed well in the straight hallway but failed miserably in complex rooms, confirming it is not suitable for general purpose exploration.

  The heatmap below (Figure 7 from the original paper) visualizes performance across different rooms.

  该图像是一个热图，显示了不同地点的重置距离。在图中，颜色从黄色（最佳）到蓝色（最差）变化，矩阵的各个单元格包含不同的距离值，这些值与不同的预测重定向行走（PredRed LPP, PredRed FPP等）方法相关联。

7. Conclusion & Reflections

7.1. Conclusion Summary

This paper successfully introduces two predictive Redirected Walking algorithms (PredRed LPP and FPP) integrated with non-harmonic Artificial Potential Fields.

1. LPP (Lemniscate prediction) proved to be the superior method, offering statistically significant improvements in walking distance between resets compared to all other tested methods. Its ability to filter paths based on scene geometry (scene awareness) is a key strength.
2. FPP (Forward prediction) offers a computationally lighter predictive alternative that outperforms basic reactive methods but doesn't quite match LPP or significantly beat sophisticated reactive APF (S2G).
3. The framework successfully supports multi-user environments by treating other users as dynamic obstacles in the potential field.

7.2. Limitations & Future Work

• User Modeling: The current system treats the second user as a simple "dumb" obstacle.
  • Future Work: Implement Priority Systems (one user gives way to another) or Cooperative Prediction (sharing predicted paths between users to plan jointly).
• State Space Limits: To save computation, the system only applies one redirection gain for the whole predicted path.
  • Future Work: Segment the path to apply different gains at different stages of the curve.
• Predictive Resets: Currently, resets happen only when a collision is imminent.
  • Future Work: Trigger resets early if the prediction shows the user is heading into a "dead end" (corner), preventing a worse situation later.

7.3. Personal Insights & Critique

• Integration of Math and Intuition: The paper brilliantly combines the "intuitive" geometry of Lemniscates with the "rigorous" physics of differential equations (FPP). It's a great example of how different mathematical models can solve the same problem (prediction) with different trade-offs.
• The Power of Scene Awareness: The success of LPP over FPP highlights that in VR, knowing where the walls are (Scene Awareness) is often more valuable than knowing exactly how the user is moving (Kinematics). FPP's "blind" prediction failed to account for the fact that users must turn when they see a virtual wall.
• Applicability: The use of APFs is highly transferable. This method could be applied to drone navigation or robot path planning in crowded environments, not just VR. The "Smooth + Predict + Cost Function" pipeline is a classic robotics pattern effectively applied here.