# Modeling and Simulation of Pedestrian Movement Planning Around Corners

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## Abstract

**:**

## 1. Introduction

## 2. Model Framework

#### 2.1. Problem Description

_{m}in Figure 1b). Then, they regain their desired walking speed (v

_{f}in Figure 1b) at a final target point (point f in Figure 1a). This assumption is in line with authors’ previous studies that demonstrated that a turning maneuver occurred in a region called the “turning region”, and the turn initiation and end points can be spatially invariant [16]. Other previous studies also experimentally confirmed the spatial invariance of turn initiation points [17,18]. Such points can be considered as starting and final target points. Speeds at those points can be considered as the normal average walking speed of a healthy human with no accelerations or decelerations (i.e., 0 m/s

^{2}). Using such assumptions and assuming that the movement time between initial and final target points are known, pedestrians’ turn negotiation maneuvers could be described by the minimum-jerk concept. The minimum-jerk principle describes smoothness in human arm movements [19], and this concept is described in detail in Section 2.2. However, movement time (t

_{f}) is generally unknown. If coordinates, speed, and acceleration at an intermediate location are known, such information can be used to obtain an estimate for t

_{f}. If we assume that a pedestrian who is planning a turning maneuver is targeting a point on the middle of the corner (Figure 1a), the tangential speed of that point can be estimated based on the instantaneous radius of that point based on the one-thirds power law. The one-thirds power law explains the inverse correlation between the tangential speed and the instantaneous speed as a general feature of human movements. The one-thirds power law is explained in detail in Section 2.3.

#### 2.2. Minimum-Jerk Concept

_{f}) is the time integration of the square of jerk, which can be formulated as:

#### 2.3. One-Thirds Power Law Concept

#### 2.4. Model Formulation

^{2}.

^{2}.

## 3. Model Verification

#### 3.1. Estimating Instantaneous Radius of the Path

#### 3.2. Power Law Parameters

^{(2/3)}s

^{−1}(approximately), respectively. In this study, the instantaneous radii at the intermediate point were varied approximately between 0.5 m and 2 m. Thus, K = 1.00 m

^{(2/3)}s

^{−1}was used in this study to estimate speeds at the intermediate point.

#### 3.3. Results

^{2}. A Monte Carlo simulation was conducted with 100 random seeds, and the outputs, i.e., paths, speed, and acceleration profiles, for 90°, 135°, and 180° turning angles were compared with empirical data, as shown in Figure 4.

_{f}values resulted because of the range of v

_{i}and v

_{f}values used in simultaneous equations.

_{f}or t

_{m}< 0). Thus, the global minimum was carefully chosen by setting appropriate initial values.

_{f}was required in addition to location, speed, and acceleration vectors at initial and final locations. Thus, at this stage, we used the location and the speed information at the intermediate location only for the estimation of t

_{f}.

## 4. Sensitivity Analysis

#### 4.1. Entry and Exit Acceleration

^{2}to 0.8 m/s

^{2}. Resulting trajectories for 90° and 180° turning cases are shown in Figure 5.

#### 4.2. Exit Location

_{f}= 2.00 m for 90° case and y

_{f}= −1.50 m for 180° case). Accelerations at initial and final locations were set as 0 m/s

^{2}. Based on such settings, simultaneous equations were solved for the same entry location but for a range of different exit locations. Resulting trajectories are shown in Figure 6.

_{f}and y

_{f}. Nevertheless, the results obtained here are logical. The smaller the minimum radius of the path is, the smaller the minimum speed at the intermediate location will be.

#### 4.3. Radius of the Walking Path

^{2}. Resulting paths, speed, and acceleration profiles for 90° turning are compared in Figure 7b–d, respectively.

^{2}and 0.56 m/s

^{2}.

## 5. Conclusions and Further Studies

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Smoothed paths with fourth order polynomials; (

**b**) Estimated instantaneous radii along the path.

**Figure 3.**Sensitivity of the power law relationship for different K on; (

**a**) x-y space; (

**b**) ln(x)-ln(y) space.

**Figure 4.**Comparison of model outputs with experiment data: (

**a**) paths for 90° turning; (

**b**) speed profiles for 90° turning; (

**c**) paths for 135° turning; (

**d**) speed profiles for 135° turning; (

**e**) paths for 180° turning; (

**f**) speed profiles for 180° turning.

**Figure 5.**Sensitivity of the model to the entry and the exit accelerations: (

**a**) paths for 90° turning; (

**b**) paths for 180° turning; (

**c**) speed profiles for 90° turning; (

**d**) speed profiles for 180° turning; (

**e**) acceleration profiles for 90° turning; (

**f**) acceleration profiles for 180° turning.

**Figure 6.**Sensitivity of the model to the exit location: (

**a**) paths for 90° turning; (

**b**) paths for 180° turning; (

**c**) speed profiles for 90° turning; (

**d**) speed profiles for 180° turning; (

**e**) acceleration profiles for 90° turning; (

**f**) acceleration profiles for 180° turning.

**Figure 7.**(

**a**)-Estimated instantaneous radii along paths; (

**b**) comparison of paths; (

**c**) comparison of speed profiles; (

**d**) comparison of acceleration profiles.

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**MDPI and ACS Style**

Dias, C.; Abdullah, M.; Sarvi, M.; Lovreglio, R.; Alhajyaseen, W. Modeling and Simulation of Pedestrian Movement Planning Around Corners. *Sustainability* **2019**, *11*, 5501.
https://doi.org/10.3390/su11195501

**AMA Style**

Dias C, Abdullah M, Sarvi M, Lovreglio R, Alhajyaseen W. Modeling and Simulation of Pedestrian Movement Planning Around Corners. *Sustainability*. 2019; 11(19):5501.
https://doi.org/10.3390/su11195501

**Chicago/Turabian Style**

Dias, Charitha, Muhammad Abdullah, Majid Sarvi, Ruggiero Lovreglio, and Wael Alhajyaseen. 2019. "Modeling and Simulation of Pedestrian Movement Planning Around Corners" *Sustainability* 11, no. 19: 5501.
https://doi.org/10.3390/su11195501