Proceedings of the 8th International Conference on Pedestrian and Evacuation Dynamics (PED2016) Hefei, China – Oct 17 – 21, 2016 Paper No. 15
Predicting Optimal Trajectories for Constrained Pedestrian Turning Manoeuvres Charitha Dias1, Majid Sarvi2, Miho Iryo-Asano3 1,3
Institute of Industrial Science, the University of Tokyo Meguro-ku, Tokyo 153-8505, Japan
[email protected];
[email protected] 2 Department of Infrastructure Engineering, the University of Melbourne Melbourne, Victoria 3010, Australia
[email protected] Abstract - Reproducing realistic walking trajectories associated with complex geometries and floor layouts, such as turning configurations, in microscopic simulation models is beneficial in many applications. Estimating reliable evacuation times for complex scenarios, 3-D visualization of realistic microscopic movements and virtual reality applications (for example, in driving simulators) are few of those important applications. However, it is uncertain that existing approaches capture the complete picture of turning manoeuvres with regards to walking paths, speed profiles and acceleration-deceleration patterns. In this paper, minimum-jerk theory (‘jerk’ is defined as the time derivative of acceleration) is utilized to model trajectories through different turning configurations. Normal speed walking (average free-flow speed ≈ 1.4 m s-1) through 90°, 135° and 180° turning configurations are considered in this paper. Modelled trajectories are compared with trajectories collected through controlled experiments qualitatively and quantitatively. This comparison reveals a favourable match between modelled and experimental trajectories. This indicates that the minimum jerk theory can be utilized to model trajectories through turning configurations under different constrained boundary conditions. Output of this study could be beneficial not only for validating existing microscopic simulation models but also for building new simulations based on more realistic representation of pedestrians’ movement mechanisms through complex geometrical settings. Keywords: Turning movements, Minimum-jerk theory, Human movement mechanisms, Optimal
trajectories, Pedestrian behaviour modelling
1. Introduction Currently, microscopic pedestrian simulation tools are widely being utilized in practice such as in optimizing public building designs [1]. However, reliability and precision of those software tools, which are based on well-known microscopic models, are still questionable [2-3]. In order to enhance the reliability of predictions by a model and for model calibration and validation purposes reliable empirical data are required. Such empirical data should be collected under a variety of situations. That is because, microscopic pedestrian behavioural patterns might significantly differ from one situation to another particularly in case of complex scenarios. Understanding pedestrian movement mechanisms associated with complex settings is important before modelling and simulating pedestrian movements realistically. In the context of modelling microscopic pedestrian dynamics associated with rounding corners, several attempts can be found in the literature. The simplest approach is to locate one or several intermediate points to guide the desired movements around corners [4-5]. Desired path of a simulated pedestrian is planned through these intermediate destinations. Such simplified approaches may reproduce unrealistic trajectories and speed patterns as demonstrated in [6]. Instead of guiding points, Chraibi et al. [7] modelled desired direction using guiding lines combined with update rules. These update rules have basically considered the occupancy of guiding lines. Heuristic based methods [8-9] and rule based methods [10-11] have also been implemented by several researchers. Various behavioural phenomena, for ex., pedestrians’ desire to walk closer to the inner corner, have been considered in these studies. Almost all these approaches have captured
15-1
microscopic pedestrian movements through turning configurations qualitatively (i.e., path level). However, it is uncertain whether they have captured microscopic behavioural dynamics quantitatively (i.e., with regards to trajectories, speed and acceleration patterns). Although several studies have presented empirical paths for turning pedestrians (for ex., [4] and [12]) no quantitative examination for speed and acceleration patterns have been conducted. Thus, it is questionable that the movement mechanisms of pedestrians related to walking through complex geometrical configurations have been adequately understood. Reproducing realistic trajectories qualitatively and quantitatively, particularly associated with complex geometries such as turning, in microscopic pedestrian models is beneficial in many ways. For example, to accurately represent the bottleneck effect and for realistically displaying pedestrian movements in 3-D visualizations particularly in virtual reality applications. Considering these benefits, this paper describes a method using minimum-jerk theory to model trajectories of turning pedestrians. Data collected through controlled experiments for different turning angles are utilized to validate the proposed approach. The rest of this paper is structured as follows: The next section will discuss the theoretical background of minimum-jerk theory. Then the methods, i.e., controlled experiments and modelling approach are briefly described. This is followed by a qualitative and quantitative comparison of modelled and experimental trajectories. Lastly, conclusions and recommendations for further studies are presented.
2. Optimal Trajectories and Minimum-Jerk Theory Minimum-jerk concept has initially been utilized in neuroscience domain to study optimality characteristics of skilled human arm movements. Flash and Hogan [13] verified that smoothness of a skilled human planar arm movement, i.e., reaching, writing and drawing movements, can be evaluated as a function of jerk (jerk is defined as the time derivative of acceleration). As they noted the objective function that should be minimized to obtain the smoothest trajectory for moving hand from an initial position to a final position in a given time T (T = tf – ti) is the time integration of the square of jerk that can be given as: 𝑡𝑓
2
2
1 𝑑3 𝑥 𝑑3 𝑦 J = ∫ (( 3 ) + ( 3 ) ) 𝑑𝑡 2 𝑑𝑡 𝑑𝑡
(1)
𝑡𝑖
Later, this theory was utilized to model more complex scenarios such as 2-dimentional robot arm movements [14] and human reaching and catching movements in 3-dimentiolanal space [15]. Further, a recent study revealed that turning manoeuvres of vehicles at intersections can be explained with minimum jerk approach [16]. Applying the same theory, Pham et al. [17] demonstrated that whole body movement share some common features with hand movements. That is, as they described, goal-oriented human walking trajectories can also be described with minimum-jerk concept. Through an experimental study, Dias et al. [18] described several behavioural characteristics that are specific to walking through turning configurations. As they pointed out, pedestrians perform the turning manoeuvre within a fixed region (described as “turning region”). Within this turning region there is a deceleration phase followed by an acceleration phase and the minimum speed is occurred in the vicinity of the middle of the corner. In this study we hypothesize that turning manoeuvres performed by individuals within this turning region are skilled tasks and therefore, can be described with minimum-jerk approach similar to reaching or drawing or goal-oriented movements.
3. Methods 3.1. Trajectory Data Dias et al. [18] conducted a series of solo (one person at a time) walking experiments under different conditions, i.e. for different turning angles (45°, 60°, 90°, 135°, 180°) and under 3 desired speed levels (i.e. normal speed walking, fast speed walking and slow speed running). Out of those data collected through these experiments, trajectories for normal speed walking through 90°, 135° and 180° turning configurations were utilized in this study.
15-2
3.2. Modelling As described in previous Section 2 the objective function to be minimized to obtain the minimum-jerk trajectory is the time integration of squared jerk that is given in Equation 1. Note that the jerk is equivalent to the rate of change of acceleration (or force) and therefore, a minimum jerk trajectory is a trajectory that minimizes the rate of change of forces when changing initial walking direction gradually. Solution for minimization of Equation 1 can be obtained as a system of fifth order polynomials as given in Equations 2 and 3. Detailed derivations of this system of equation from equation 1, can be found in [13].
𝑥(𝑡) = 𝑎0 + 𝑎1 𝑡 + 𝑎2 𝑡 2 + 𝑎3 𝑡 3 + 𝑎4 𝑡 4 + 𝑎5 𝑡 5 𝑦(𝑡) = 𝑏0 + 𝑏1 𝑡 + 𝑏2 𝑡 2 + 𝑏3 𝑡 3 + 𝑏4 𝑡 4 + 𝑏5 𝑡 5
(2) (3)
Where; 𝑎𝑗 and 𝑏𝑗 (j = {0, ..., 5}) are constants Equations 2 and 3 show that there are 12 unknowns and therefore 12 boundary conditions are required to solve this system of equations. In this study, ti and tf (in Equation 1) are defined as the entry time to the turning region and the exit time from the turning region respectively. For convenience, ti was set as zero. Further, initial position (turn initiation point) was set as (0, 0) and the coordinates of the final positions (turn completion point) for each turning angle case were set relative to the origin, based on the corridor geometry based on the findings of [18]. Approaching speed at the turn initiation point and receding speed at the turn completion point were set as 1.4 m/s as there were no statistically significant difference between them as reported in [19]. Acceleration at initial point and final point was considered as 0 m/s 2 as it was assumed, based on [18], that the deceleration and acceleration phases are occurred within the turning region. Movement time from turn initiation point to turn completion point (T = tf – ti) is required to solve the above system of equations. In Section 4 we consider that T is known and the average T for each angle case was obtained from experimental trajectories. Later in Section 5 we show that T can be estimated as a function of approaching speed and turning angle. For the above boundary conditions and average T obtained from experimental trajectories, average minimum-jerk trajectories (paths, speed and acceleration profiles) were obtained based on Equation 2 and 3 for each turning angle. These minimum-jerk trajectories are compared with trajectories obtained from experiments as discussed in the next section.
4. Results Modelled and experimental trajectories (walking paths, instantaneous walking directions, speed profiles and acceleration profiles) were compared as depicted in Figure 1 and Figure 2. Absolute errors for location, direction, speed and acceleration estimates were calculated according to Equation 4 and mean absolute errors (MAE) for those estimates are tabulated in Table 1. 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑒𝑟𝑟𝑜𝑟 = |𝑆⃗𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑,𝑡 − 𝑆⃗𝐴𝑐𝑡𝑢𝑎𝑙,𝑡 |
(4)
Where; ⃗S⃗Estimated,t = Estimated state vector at time t ⃗S⃗Actual,t = Actual (experimental average) state vector at time t Table 1: Mean absolute errors of estimates.
Turning angle 90° 135° 180°
Path (m) 0.064 0.075 0.074
Direction (°) 1.8 3.0 6.8
15-3
Speed (m/s) 0.018 0.025 0.029
Acceleration (m/s2) 0.081 0.065 0.087
0.5
Experimental paths Minimum-Jerk path
0 -0.5
0
0.5
1
1.5
2
2.5
135
X (m) -0.5
Y (m)
Instantaneous walking direction (degrees)
180
Experimental average
-1
-1.5
-2
90
45
Minimum-Jerk Experimental average 0
0.0 (d)
(a)
-2.5 0.5
Minimum-Jerk path 0
0.5
1
1.5
2
2.5 X (m)
Y (m)
-0.5
-1
-1.5
90
45 Minimum-Jerk Experimental average 0 0.0
(b) Experimental average Minimum-Jerk path
0 0
0.5
1
1.5
2
2.5 X (m)
-0.5
-1
-1.5
(e)
1.0
2.0
3.0
4.0
Time (seconds)
180
Experimental paths
Instantaneous walking direction (degrees)
-2.5 0.5
Y (m)
4.0
135
-2
-0.5
3.0
180
Instantaneous walking direction (degrees)
Experimental average
-0.5
2.0 Time (seconds)
Experimental paths
0
1.0
135
90
45
Minimum-Jerk Experimental average 0
-2
0.0
(c)
(f)
1.0
2.0
3.0
4.0
Time (seconds)
-2.5
Fig. 1: Comparison of walking paths and instantaneous walking directions for: (a) and (d) – 90°; (b) and (e) – 135°; (c) and (f) – 180° turning movements (instantaneous walking direction is the angle between instantaneous walking path and the x-axis measured clockwise direction).
15-4
1.2
0.4
0.8 0.4
Minimum-Jerk
(a)
Experimental average
0.0 1.0
2.0 Time (seconds)
3.0
0.0 0.0
Experimental average
1.2
0.4
Minimum-Jerk
(b) 0.0
Experimental average
1.0
2.0 Time (seconds)
3.0
0.0
0.0
Experimental average
0.4
0.8
Minimum-Jerk Experimental average
0.0
1.0
2.0 Time (seconds)
3.0
3.0 4.0 Time (seconds)
Minimum-Jerk
4.0
1.2
(c)
2.0
(e)
-0.8
0.8
0.0
1.0
-0.4
1.6
0.4
Minimum-Jerk
4.0
0.8
3.0 4.0 Time (seconds)
(d)
0.8
0.0
2.0
-0.4
1.6
0.4
1.0
-0.8
Acceleration (m/s2)
Speed (m/s)
0.0
Speed (m/s)
Acceleration (m/s2)
0.8
Acceleration (m/s2)
Speed (m/s)
1.6
0.0 0.0
-0.4
1.0
2.0
3.0 4.0 Time (seconds)
(f) -0.8
4.0
Minimum-Jerk
Experimental average
Fig. 2: Comparison of speed and acceleration profiles for: (a) and (d) – 90°; (b) and (e) – 135°; (c) and (f) – 180° turning movements.
Table 1 explains that, in general, errors in estimates tend to increase with the increasing turning angle. Path level comparison of modelled and experimental paths (Figure 1) suggests that the minimum jerk theory can qualitatively predict the walking paths through turning configurations for different turning angles. Thus, this could be useful in modelling desired direction of individuals through turning configurations. Although speed and accelerations are slightly overestimated in modelled profiles (Figure 2) the trends, i.e., acceleration and deceleration patterns, are well explained. In this study, speeds and acceleration values for boundary conditions were approximated. For example, initial and final accelerations at turn initiation point and turn completion points were assumed to be zero. However, experimental average profiles show that initial and final accelerations are not exactly zero. Provision of such complete boundary conditions or using higher order models may generate more accurate trajectories. Further, individual variations of boundary conditions and movement times were also not considered in this study. Regardless of these approximations, trajectories could be generated with a reasonable accuracy using the minimum jerk approach.
15-5
5. Variations in Turning Trajectories Properties of trajectories are dependent on boundary conditions, particularly on entry and exit speed as well as on the movement time through the turning region. For the analyses so far discussed in Section 4, it was considered that the movement time (T = tf – ti) through the turning region is known. However, in applications, such as in microscopic simulation models, T is generally unknown. This section will briefly discuss such issues combined with trajectory variations through turning configurations. 5.1. Properties of Movement Time (T) Figure 3 plots the variation of T with respect to the approaching speed and the turning angle. As can be explained from this figure T is basically dependent on the approaching speed to the turning region and turning angle. Therefore, T could be simply modelled as functions of those variables. This provides a reasonable estimate for T as further discussed in Section 5.2. It should be noted that, as described in [19], approaching speed to the turning region does not vary with the turning angle. Or in other words, approaching speeds for different turning angles do not display statistically significant differences among them.
Movement time through turning region (s)
5.0 4.5
180°
4.0
135°
y = -1.8939x + 6.7386 R² = 0.4537
90° 3.5
y = -2.1829x + 6.6074 R² = 0.36
3.0 2.5 y = -2.0045x + 6.0104 R² = 0.8867
2.0 1
1.2
1.4 1.6 Approaching speed (m/s)
1.8
Fig. 3: Variation of movement time through the turning region with approaching speed and turning angle.
5.2. Trajectory Variation Variation of trajectories with respect to approaching speed to the turning region are explored in this section. In this analysis, distributions of approaching speeds to different turning angles were considered and the trajectory variation for slow (5th percentile of approaching speeds) and fast (95th percentile of approaching speeds) walkers were investigated. Movement time for each approaching speed and turning angle was determined based on the linear relationships shown in Figure 3. Exit speed from the turning region was considered to be similar to the approaching speed. Estimated 5th percentile and 95th percentile approaching speeds for all angle cases were [1.2, 1.3] m/s and [1.55, 1.65] m/s respectively. Similar to settings in Section 4, accelerations at the entry and exit boundaries of turning region were set as 0 m/s 2. For such settings and approximations, trajectories were generated for each turning angle case and compared as depicted in Figure 4 and 5. Figure 4 describes that, although deviations are smaller, variation in walking paths tend to increase with increasing turning angle. Regarding speed and acceleration profiles, increase in variations in trajectories with increasing turning angle can be clearly observable in Figure 5.
15-6
0.5
High approaching speed (95th percentile) Low approaching speed (5th percentile) Average approaching speed
0
-0.5
0
0.5
1
1.5
2
2.5 X (m)
Y (m)
-0.5
-1
-1.5
180°
-2
90° 135° -2.5
Fig. 4: Variation of walking paths through different turning angles based on different approaching speeds. 0.8
90°
90° Acceleration (m/s2)
Speed (m/s)
1.6
1.2
0.8
Low approaching speed
Average approaching speed
0.4
0.0 0.0
1.0
2.0
0.0
1.0
2.0 Time (seconds)
4.0
Low approaching speed
-0.4
Average approaching speed
High approaching speed
0.4
3.0
3.0
High approaching speed
-0.8
4.0
Time (seconds) 0.8
135°
135° Acceleration (m/s2)
Speed (m/s)
1.6
1.2
0.8
Low approaching speed
0.4
0.0 0.0
1.0
2.0
Average approaching speed
High approaching speed 0.0
1.0
2.0 Time (seconds)
4.0
Low approaching speed
-0.4
Average approaching speed 0.4
3.0
3.0
High approaching speed
-0.8
4.0
Time (seconds) 0.8
Acceleration (m/s2)
Speed (m/s)
180°
180°
1.6
1.2
0.8
Low approaching speed Average approaching speed High approaching speed
0.4 0.0
1.0
2.0 3.0 Time (seconds)
0.4
0.0 0.0 -0.4
-0.8
1.0
2.0
3.0
4.0
Low approaching speed Average approaching speed High approaching speed
4.0
Time (seconds)
Fig. 5: Variation of speed and acceleration patterns through different turning angles based on different approaching speeds.
15-7
It can be noted that, for a given turning case, although walking paths do not display larger deviations (for ex., consider 90° case), speed and acceleration patterns can be clearly distinguished for different approaching speeds. This observation highlights that individuals might adjust their speed and acceleration patterns as well to negotiate a corner smoothly. Therefore, although the walking path is the same, speed and acceleration behaviours could be considerably different for different approaching and exit conditions. Further, it can be observable in Figure 5 that deceleration and acceleration values are larger for higher approaching speeds and this effect is magnified when the turning is increased. Pedestrians approaching a corner with higher speeds have larger inertia. In order to overcome the effect of this inertia and negotiate the turning movement smoothly, a pedestrian requires a larger deceleration force compared to a pedestrian approaching at a lower speed. Then, for an individual leaving at higher speed (approaching and leaving speeds were considered same in this section), a larger acceleration is required to regain the speed to the desired level. Such behaviours have adequately captured with minimum-jerk approach as described in this study. Findings of this study reveal important information related to mechanisms and optimality characteristics of human movements through complex environments. Minimizing jerk is equivalent to minimizing the rate of change of acceleration (or force). Thus, it can be stated, based on the findings of this study, that human tend to minimize the rate of change of acceleration when navigating through turning configurations. In the future it may be possible to consider such evidences in microscopic pedestrian simulation models to realistically model pedestrian movement characteristics.
6. Conclusions Understanding microscopic behavioural dynamics and mechanisms associated with walking through complex geometrical configurations could be useful particularly for enhancing or developing microscopic simulation models. In this paper, minimum jerk theory was applied to model trajectories for walking through turning configurations at normal walking speed. Through a qualitative and quantitative comparison of predicted and experimental trajectories, it was clarified that the minimum jerk theory can adequately predict microscopic walking characteristics, for ex., walking paths and acceleration-deceleration patterns, through turning configurations under different conditions (for different angles and different approaching speeds). Variation of trajectories were also tested based on different conditions (different approaching speeds and movement times) and verified that the model is responsive and sensitive to such different movement conditions. Further, this study confirmed that people tend to optimize the smoothness of their movements through complex geometrical settings by minimizing the jerk. Such concept, which describe realistic movement mechanisms of people through turning configurations, could be used to model and simulate complex pedestrian movements more realistically. As further investigations, a comprehensive sensitivity analysis should be performed to better understand the uncertainty of predictions via minimum jerk theory. Further, more complex scenarios related to turning configurations could also be explored under a wide variety of boundary conditions.
References [1] S. Hoogendoorn, M. Hauser, and N. Rodrigues, "Applying microscopic pedestrian flow simulation to railway station design evaluation in Lisbon, Portugal," Transportation Research Record: Journal of the Transportation Research Board, vol. 1878, pp. 83–94, 2004. [2] C. Rogsch, W. Klingsch, A. Seyfried and H. Weigel, “Prediction Accuracy of Evacuation Times for High-Rise Buildings and Simple Geometries by Using Different Software-Tools,” in Traffic and Granular Flow ’07, C. Appert-Rolland, F. Chevoir, P. Gondret, Ed. Springer Berlin Heidelberg, 2009, pp. 395-400. [3] D. C. Duives, W. Daamen, and S. P. Hoogendoorn, "State-of-the-art crowd motion simulation models," Transportation Research Part C: Emerging Technologies, vol. 37, pp. 193–209, 2013.
15-8
[4] B. Steffen and A. Seyfried, “Modeling of pedestrian movement around 90 and 180 degree bends,” in Proceedings of the Workshop on Fire Protection and Life Safety in Buildings and Transportation Systems, J.A. Capote, Alvear, D., Ed. Universidad de Cantabria, GIDAI, Santander, 2009. [5] J.-B. Zeng, B. Leng, Z. Xiong, and Z. Qin, “Pedestrian dynamics in a two-dimensional complex scenario using a local view field model, “International Journal of Modern Physics C, vol. 22, no. 08, pp. 775– 803, 2011. [6] C. Dias, O. Ejtemai, M. Sarvi, and M. Burd, "Exploring pedestrian walking through Angled corridors," Transportation Research Procedia, vol. 2, pp. 19–25, 2014. [7] M. Chraibi, M. Freialdenhoven, A. Schadschneider, and A. Seyfried, “Modeling the Desired Direction in a Force-Based Model for Pedestrian Dynamics,” in Traffic and Granular Flow ’11, V. V. Kozlov, A. P. Buslaev, A. S. Bugaevet, Ed. Springer Berlin Heidelberg, 2013, pp. 263-275. [8] X.-X. Jian, S. C. Wong, P. Zhang, K. Choi, H. Li, and X. Zhang, "Perceived cost potential field cellular automata model with an aggregated force field for pedestrian dynamics," Transportation Research Part C: Emerging Technologies, vol. 42, pp. 200–210, 2014. [9] T. Kretz, “The use of dynamic distance potential fields for pedestrian flow around corners,” in First International Conference on Evacuation Modeling and Management (ICEM 09), TU Delft, the Netherlands, 2009. [10] R.-Y. Guo and T.-Q. Tang, "A simulation model for pedestrian flow through walkways with corners," Simulation Modelling Practice and Theory, vol. 21, no. 1, pp. 103–113, 2012. [11] S. Li, X. Li, Y. Qu, and B. Jia, "Block-based floor field model for pedestrian’s walking through corner," Physica A: Statistical Mechanics and its Applications, vol. 432, pp. 337–353, 2015. [12] F. Johansson, D. Duives, W. Daamen, and S. Hoogendoorn, "The many roles of the relaxation time parameter in force based models of pedestrian dynamics," Transportation Research Procedia, vol. 2, pp. 300–308, 2014. [13] T. Flash and N. Hogan, "The coordination of arm movements: an experimentally confirmed mathematical model," The journal of Neuroscience, vol. 5, no. 7, pp. 1688–1703, 1985. [14] Y. Wada and M. Kawato, "A via-point time optimization algorithm for complex sequential trajectory formation," Neural Networks, vol. 17, no. 3, pp. 353–364, 2004. [15] M. Bratt, C. Smith, and H. Christensen, “Minimum jerk based prediction of user actions for a ball catching task,” in Intelligent Robots and Systems, 2007. IROS 2007. IEEE/RSJ International Conference on, 2007, pp. 2710-2716. [16] C. Dias, M. Iryo-Asano, and T. Oguchi, “Concurrent prediction of location, velocity and acceleration profiles for left turning vehicles at signalized intersections,” in Proceedings of 53th Annual Meeting of Infrastructure Planning (JSCE), 2016, Hokkaido, Japan. [17] Q.-C. Pham, H. Hicheur, G. Arechavaleta, J.-P. Laumond, and A. Berthoz, "The formation of trajectories during goal-oriented locomotion in humans. II. A maximum smoothness model," European Journal of Neuroscience, vol. 26, no. 8, pp. 2391–2403, 2007. [18] C. Dias, O. Ejtemai, M. Sarvi, and N. Shiwakoti, "Pedestrian walking characteristics through Angled corridors: An Experimental Study," Transportation Research Record: Journal of the Transportation Research Board, vol. 2421, pp. 41–50, 2014. [19] C. Dias, M. Sarvi, N. Shiwakoti, and O. Ejtemai, “Experimental study on pedestrian walking characteristics through angled corridors,” in Proceedings of 36th Australasian Transport Research Forum (ATRF), 2013, Brisbane, Australia.
15-9
