Estimating horizontal movement performance of patient beds and the impact on emergency evacuation time
Jaeyoung Kwak, Michael H. Lees, Wentong Cai, Ahmad Reza Pourghaderi, Marcus E.H. Ong
EEstimating horizontal movement performance of patient beds and theimpact on emergency evacuation time
Jaeyoung Kwak a , ∗ , Michael H. Lees b , Wentong Cai c , Ahmad Reza Pourghaderi d,f,a and MarcusE.H. Ong e,f a Complexity Institute, Nanyang Technological University, 61 Nanyang Drive, Singapore 637335, Singapore b Informatics Institute, University of Amsterdam, Science Park 904, Amsterdam 1098XH, The Netherlands c School of Computer Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore d Health Systems Research Center (HSRC), Singapore Health Services, 31 Third Hospital Avenue, Singapore 168753, Singapore e Department of Emergency Medicine, Singapore General Hospital, Outram Road, Singapore 169608, Singapore f Health Services and Systems Research (HSSR), Duke-NUS Medical School, 8 College Road, Singapore 169857, Singapore
A R T I C L E I N F O
Keywords :Emergency evacuationPatient bedTurning movementFatigue effectMovement duration
A B S T R A C T
Emergency evacuation of patients from a hospital can be challenging in the event of a fire. Most emer-gency evacuation studies are based on the assumption that pedestrians are ambulant and can egressby themselves. However, this is often not the case during emergency evacuations in healthcare facil-ities such as hospitals and nursing homes. To investigate emergency evacuations in such healthcarefacilities, we performed a series of controlled experiments to study the dynamics of patient beds inhorizontal movement. We considered a patient bed because it is one of the commonly used devicesto transport patients within healthcare facilities. Through a series of controlled experiments, we ex-amined the change of velocity in corner turning movements and speed reductions in multiple tripsbetween both ends of a straight corridor. Based on the experimental results, we then developed amathematical model of total evacuation time prediction for a patient bed horizontally moving in ahealthcare facility. Factoring uncertainty in the horizontal movement, we produced the probabilitydistribution of movement duration and estimated the probability that an evacuation can be safely per-formed within certain amount of time. In addition, we predicted that the evacuation time would belonger than the prediction results from an existing model which assumes constant movement speed.Our results from the model demonstrated good agreement with our experimental results.
1. Introduction
Pedestrian emergency evacuation has been one of thecentral topics in the field of fire safety engineering. In case oflife threatening incidents such as fire and hazardous chem-ical spills, a well-prepared emergency evacuation plan canefficiently move occupants to the assembly points with min-imum amount of time and ensure the safety of evacuees.Based on the degree of required aid during egress, occupantscan be categorized into ambulant and non-ambulant occu-pants. Ambulant occupants can egress to the place of safetywithout help from other occupants. On the other hand, non-ambulant occupants need help when they move, especiallyin case of an emergency. The non-ambulant occupants canbe further categorized into subcategories such as bedriddenoccupants and wheelchair users depending on the device thatthey are using.Emergency evacuation of ambulant pedestrians has beeninvestigated in experimental studies to understand pedestrianmovement during emergency evacuations. Numerous exper-iments have been conducted in horizontal evacuation scenar-ios in which pedestrians egress within the same floor. For in-stance, researchers have studied evacuations through bottle-necks to understand the relationship between the bottleneckwidth and pedestrian flow, which is critical to the total evac-uation time in room evacuations [1, 2, 3]. A considerable ∗ Corresponding author: Jaeyoung Kwak ([email protected])
ORCID (s): number of experiments have been performed to characterizepedestrian vertical movement through stairs such as down-ward moving speed and flow rate in a stairwell of a high-risebuilding [4] and the influence of stair slope on such pedes-trian flow characteristics [5]. The pedestrian flow charac-teristics have been studied for harsh moving conditions in-cluding crawling in a room [6] and in a corridor [7], and theexistence of earthquake-induced falling debris [8].In most existing emergency evacuation studies, it is as-sumed that pedestrians are able to walk and egress by them-selves. However, this is often not the case during emergencyevacuations especially in healthcare facilities such as hospi-tals and nursing homes. Healthcare facilities accommodateconsiderable numbers of non-ambulant patients who havelimited mobility and need the help of medical staff with evac-uation devices during the evacuation process. To prepareemergency evacuation plans considering the non-ambulantpedestrians, it is necessary to understand the performance ofevacuation devices such as their movement speed and move-ment dynamics.Previous experimental studies of non-ambulant pedes-trian evacuations analyzed video footage of the experimentsto measure travel time between different reference points. Bydoing that, average movement speed was measured for dif-ferent sections of an evacuation route such as a straight cor-ridor and a stairwell. For example, Rubadiri et al. [9] per-formed a horizontal evacuation exercise of manual and elec-tric wheelchair users. They measured the evacuation per-
J. Kwak et al .: Preprint submitted to Elsevier
Page 1 of 14 a r X i v : . [ phy s i c s . s o c - ph ] J u l atient Bed Horizontal Movement Performance Estimation formance index as a ratio of the wheelchair user movementspeed without assistance to the movement speed of ambulantpedestrians. Based on the measured evacuation performanceindex, they predicted evacuation time of the wheelchair usersand then compared it with the actual measurement of thewheelchair users. Strating [10] collected movement speeddata of bedridden occupants in Dutch healthcare facilities.He reported an evacuation speed range from 0.54 to 1.34 m/sin Dutch hospitals and from 0.25 to 1.30 m/s in Dutch nurs-ing homes. Hunt et al. [11] measured horizontal and verti-cal movement speed of evacuation devices including stretch-ers, evacuation chairs, carry chairs, and rescue sheets. Theyobserved that the evacuation chair is the fastest among thetested evacuation devices with an average speed of 1.5 m/sin the horizontal evacuation and of 0.83 m/s in the verti-cal evacuation. Based on the measured movement speed ofthose evacuation devices, they also estimated the total evac-uation time of non-ambulant patients in a high-rise hospi-tal building. While those studies focused on measuring theaverage movement speed of non-ambulant pedestrian evac-uation devices, change of velocity during the movement hasnot yet been studied in detail.In pedestrian flow dynamics, it has been reported thatpedestrians continuously moving with heavy load tend tohave reduced movement speed due to fatigue [12]. Further-more, several studies investigated speed profiles of pedes-trians moving near obstacles in front of an exit [13], cor-ners [14], and merging areas [15]. Such structural elementsof pedestrian facilities often act as a constraint on the ef-ficiency of pedestrian flow seemingly because pedestriansneed to change their moving direction and speed.Similar to the case of pedestrian flow dynamics, the move-ment of an evacuation device becomes slower when the de-vice is moving in corners and areas of merging flow andinteraction with other devices coming from different direc-tions. In addition, there are often far more bedridden pa-tients than medical staff in healthcare facilities. It is ap-parent that, in emergency evacuations, every medical staffneeds to make multiple trips between the location of bedrid-den patients and the place of safety. One can expect thatthe medical staff experience some level of exhaustion whenthey evacuate all the bedridden patients and it is likely thattheir movement becomes slower as they travel longer dis-tance. Such a slow-moving evacuation device affects theflow of subsequent evacuation devices and pedestrians, es-pecially with limited passing opportunities in a narrow corri-dor. Consequently, understanding these non-linear dynam-ics of evacuation devices enables us to better estimate theevacuation performance of the devices with taking into ac-count their speed profile and change of moving directions.Among various evacuation devices, we considered themovement of a patient bed because it is one of the most com-monly used devices to transport patients from one place tothe another in healthcare facilities. In this study, we per-formed a series of controlled experiments to investigate thedynamics of patient beds in horizontal movement. Throughthe controlled experiments, we examined the change of ve- locity in corner turning movements and speed reductionsin multiple trips between both ends of a straight corridor.Based on the experimental results, we then developed a move-ment duration prediction model and then applied the modelfor a patient bed horizontally moving in a healthcare facil-ity. Incorporating uncertainty in the horizontal movement,we predicted probability that an evacuation can be safely per-formed within certain amount of time. The experiment setupand data collection methods are described in Section 2. Asshown in Section 3, we analyze the dynamics of patient bedmovement, develop a movement duration prediction model,and then apply the model for a patient bed horizontally mov-ing in a healthcare facility. We summarizes the results withconcluding remarks in Section 4.
2. Experiment setup
To collect patient bed movement data, we carried out aseries of controlled experiments in September 2019 at theSingapore General Hospital, Singapore. In total, 4 males and4 females aged between 25 to 35 without movement impair-ments took part in the experiments. In order to move a pa-tient bed, the handlers were grouped into pairs of same gen-der handlers: male-male and female-female handlers. Addi-tionally, one male (27 years old, 189 cm, 87 kg) was lyingon the patient bed during its movement in order to substitutefor a real non-ambulant patient. Before starting the experi-ment, the handlers had an orientation session and conducteda few warm-up trials to make them proficient enough in ma-neuvering the patient bed. In the experiment, we asked allthe participants to move the patient bed as fast as possiblewhile making sure their safety.Figure 1 shows the sketches of the experiment setup: a ° angled corner and a straight corridor. In the ° angledcorner setup, each handler pair was maneuvering a patientbed in the intersection of 2 m × J. Kwak et al .: Preprint submitted to Elsevier
Page 2 of 14atient Bed Horizontal Movement Performance Estimation (a) (b)
Figure 1:
Schematic representation of experiment setup: (a) a ° angled corner and (b) a straight corridor. In (a), Areas 1and 3 indicate 3 m long 2 m wide straight corridor sections and Area 2 shows 2 m-by-2 m intersection. Blue rectangles depictpatient beds and yellow arrows show their moving direction. Red circles and red arrows indicate the locations of cameras andtheir pointing direction, respectively. (a) (b) (c) (d) Figure 2:
Schematic representation of handler positions with the direction of travel: (a) left-turning movement from the top tothe right branches in the ° angled corner setup, (b) right-turning movement from the right to the top branches in the ° angledcorner setup, (c) rightward movement in the straight corridor setup, and (d) leftward movement in the straight corridor setup.Blue circles depict handlers and yellow arrows show their moving direction. on tripods which were set on the top of tables. In the ° an-gled corner setup, we set two cameras near the corner toclosely observe the turning movement of the patient bed. Inthe straight corridor setup, two cameras were used to recordthe bed movement from each end of the corridor. From videofootage of the ° angled corner setup, we extracted patientbed movement trajectories using T-analyst, a semi-automatictrajectory extraction software developed from Lund Univer-sity, Sweden [17]. The software has been applied in cyclisttraffic safety analysis [18] and pedestrian flow analysis [19].The conversion from video coordinates to the real world co- ordinates was performed by T-calibration, a calibration soft-ware accompanied by T-analyst. In T-calibration software,we placed calibration points and its real-world coordinatesand then performed TSAI-calibration algorithm [20]. Afterthe calibration process, we manually annotated pedestrianpositions for every 10 frames on average in the video footagewith T-analyst software. Next, T-analyst software then ex-tracted pedestrian trajectories. Details of pedestrian trajec-tory extraction process can be found from T-analyst manualavailable from its webpage [17].Table 1 shows the basic movement characteristics of han- J. Kwak et al .: Preprint submitted to Elsevier
Page 3 of 14atient Bed Horizontal Movement Performance Estimation
Table 1
Basic movement characteristics of handler groups in the corner area of 𝑥 ≤ and 𝑦 ≤ . Handler Gender Movement Orientation (rad) No. of Distance Duration Average Enteringgroups direction starting target trajectories (m) (s) speed (m/s) speed (m/s)LT_1 Female Left turn 1.57 0 12 7.59 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 2
Basic movement characteristics of handler groups in straightcorridor experiment.
Handler Gender speed (m/s)groups min. max. averageGroup_1 Female 0.94 1.20 1.03Group_2 Female 1.34 1.64 1.47Group_3 Male 0.98 1.13 1.04Group_4 Male 1.31 1.70 1.53 dler groups in the turning movement experiment. We col-lected 10 to 15 trajectories from each handler group and intotal 𝑁 = 100 trajectories were collected. The distance trav-eled and the movement duration were measured for each tra-jectory in the area of 𝑥 ≤ and 𝑦 ≤ and then averaged foreach handler group. The average speed was computed by di-viding the distance traveled by the movement duration. Theentering speed was measured when the patient bed is enter-ing the area of 𝑥 ≤ and 𝑦 ≤ . Handler groups RT_2 andRT_4 showed shorter movement duration than other han-dler groups seemingly because they entered the corner withhigher entering speed and experienced smaller speed drop inthe course of their movement.In the straight corridor setup, based on the study of Luo etal. [12], we assumed that the fatigue experienced by the han-dlers is affected by the distance that they transported the pa-tient bed. The handlers were making multiple trips betweenone end to the other end of the corridor. We measured thetime that the handlers took moving between the two endsof the corridor using the video footage. In doing that, wemanually marked the transition period in which the handlersstopped and changed their position. After identifying thetransition period, we obtained the travel time in each one-way trip and then calculated the average speed for each trip.The initial speed was measured from the first one-way tripin which the handlers moved first 21 meters in the straightcorridor setup. Our approach of measuring the speed is anal-ogous to the method presented by Chen et al. [21]. The av-erage speed was measured by dividing the distance traveled(882 m) by the total travel time that the handlers moving thepatient bed. Table 2 summarizes the basic movement charac- y ( m ) x (m) Figure 3: 𝑁 = 100 individual trajectories collected from thecorner turning experiment setup teristics of handler group in the straight corridor experiment.Handler groups Group_2 and Group_4 showed higher speedbut there is not significant difference between male and fe-male. The differences between the groups in the same gen-der (Group_1 vs. Group_2, and Group_3 vs. Group_4 inTable 2) seemingly attribute to the degree of risk aversionset by different groups. Although we asked all the partici-pants to move as fast as possible, some of them might wantto more focus on moving safely without getting injured, sothey were not necessarily in a rush.
3. Results and analysis
Figure 3 shows individual trajectories collected from thecorner turning experiment setup. To understand the changeof velocity in the course of turning movement, we firstly ob-tained the average path of all the handler groups moving inboth movement directions. Similar to Hicheur et al. [22], weresampled the collected trajectories into 𝑁 𝑓 = 60 equal in-tervals and then averaged the resampled trajectories for each J. Kwak et al .: Preprint submitted to Elsevier
Page 4 of 14atient Bed Horizontal Movement Performance Estimation rescaled time point ̂𝑡 ∈ [0 , : 𝑥 𝑎𝑣𝑔 ( ̂𝑡 ) = 1 𝑁 𝑁 ∑ 𝑖 =1 𝑥 𝑖 ( ̂𝑡 ) ,𝑦 𝑎𝑣𝑔 ( ̂𝑡 ) = 1 𝑁 𝑁 ∑ 𝑖 =1 𝑦 𝑖 ( ̂𝑡 ) . (1)Here, 𝑁 = 100 is the number of collected trajectories fromhandler groups shown in Table 1. Likewise, we calculatedtrajectory deviation 𝜎 ( ̂𝑡 ) to quantify the difference betweenthe mean trajectory and an individual trajectory 𝑖 : 𝜎 𝑥 ( ̂𝑡 ) = √√√√ 𝑁 𝑁 ∑ 𝑖 =1 ( 𝑥 𝑖 ( ̂𝑡 ) − 𝑥 𝑎𝑣𝑔 ( ̂𝑡 ) ) , (2) 𝜎 𝑦 ( ̂𝑡 ) = √√√√ 𝑁 𝑁 ∑ 𝑖 =1 ( 𝑦 𝑖 ( ̂𝑡 ) − 𝑦 𝑎𝑣𝑔 ( ̂𝑡 ) ) , (3) 𝜎 ( ̂𝑡 ) = √ 𝜎 𝑥 ( ̂𝑡 ) + 𝜎 𝑦 ( ̂𝑡 ) . (4)By making use of the rescaled time points, we identifiedstart and end of turning movement. We measured the angu-lar displacement 𝜃 and its average 𝜃 𝑎𝑣𝑔 : 𝜃 ( ̂𝑡 𝑗 ) = arccos ( ⃗𝑒 ⃗𝑒 𝑖 ‖‖ ⃗𝑒 ‖‖ ‖‖ ⃗𝑒 𝑖 ‖‖ ) ,𝜃 𝑎𝑣𝑔 ( ̂𝑡 𝑗 ) = 1 𝑁 𝑁 ∑ 𝑗 =1 𝜃 ( ̂𝑡 𝑗 ) , (5)where ⃗𝑒 is the initial moving direction which is set as (0,-1) for left-turn movement and (-1, 0) for right-turn move-ment. The moving direction at rescaled time ̂𝑡 𝑗 is given as ⃗𝑒 𝑖 = (Δ 𝑥 ∕Δ 𝑙, Δ 𝑦 ∕Δ 𝑙 ) . Here, Δ 𝑥 and Δ 𝑦 denote the differ-ence between the values of 𝑥 𝑖 and 𝑦 𝑖 at the current rescaledtime points ̂𝑡 𝑗 and the previous rescaled time point ̂𝑡 𝑗 −1 . Thedisplacement between the positions at rescaled time ̂𝑡 𝑗 and ̂𝑡 𝑗 −1 is given as Δ 𝑙 = √ Δ 𝑥 + Δ 𝑦 .Figure 4(a) presents angular displacement of the patientbed against rescaled time points. One can observe that theangular displacement curve is nearly straight between ̂𝑡 =0 . and ̂𝑡 = 0 . , indicating that the patient bed is turningwith a constant angular speed 0.24 rad/ ̂𝑡 . When the patientbed starts and ends the turning movement, either 𝜎 𝑥 ( ̂𝑡 ) or 𝜎 𝑦 ( ̂𝑡 ) is at peak value. Figures 4(b) and 4(c) indicate that thestart and end of turning movement can be identified based onthe components of trajectory deviation, i.e., 𝜎 𝑥 ( ̂𝑡 ) and 𝜎 𝑦 ( ̂𝑡 ) .The identified locations of turning movement start andend are presented with the mean trajectory in Fig. 5. Similarto work of Dias et al. [23], it was observed that the turningmovement started before and ended after Area 2 ( 𝑥 ≤ and 𝑦 ≤ ). (a) a n g u l a r d i s p l a c e m e n t ( r a d ) rescaled time (b) x s d rescaled time (c) y s d rescaled time (d) t r a j . d e v i a t i o n rescaled time Figure 4:
Angular displacement of the patient bed: (a) in-dividual angular displacement (blue circles) and the averagevalue (red solid line). Black dashed lines indicate start (leftdashed line) and end (right dashed line) of turning movement.(b)-(d) the corresponding graphs of 𝜎 𝑥 ( ̂𝑡 ) , 𝜎 𝑦 ( ̂𝑡 ) , and 𝜎 ( ̂𝑡 ) . Itcan be observed that the start and end of turning movementoccur when either 𝜎 𝑥 ( ̂𝑡 ) or 𝜎 𝑦 ( ̂𝑡 ) is at peak value. Figure 6(a) shows examples of individual speed curvesbefore resampling. Due to various initial speed and maneu-vering duration, it is difficult to understand patterns in thespeed curves. We applied the idea of the rescaled time pointsto such speed curves. We resampled the instantaneous speed 𝑣 𝑖 of individual handler groups into 𝑁 𝑓 equal intervals andthen averaged the resampled speed for each rescaled timepoint ̂𝑡 ∈ [0 , : 𝑣 𝑎𝑣𝑔 ( ̂𝑡 ) = 1 𝑁 𝑁 ∑ 𝑖 =1 𝑣 𝑖 ( ̂𝑡 ) . (6) J. Kwak et al .: Preprint submitted to Elsevier
Page 5 of 14atient Bed Horizontal Movement Performance Estimation y ( m ) x (m) Figure 5:
Average path (red solid line) with the locations ofturning movement start and end (blue cross symbols).
In order to compare different speed curves, we normalizedthe speed curves with entering speed 𝑣 . Figure 6(b) presentsthe result of normalization performed against rescaled time ̂𝑡 for curves in Fig. 6(a). We performed normalization for 𝑁 = 100 individual speed curves and then averaged theirnormalized speed in rescaled time space. Figure 6(c) illus-trates the average of normalized speed curves and its trendline. The trend line is generated by the normalized speedprofile ̂𝑣 ( ̂𝑡 ) .According to minimum jerk principle (Hogan [24] andPham et al. [25]), people tend to minimize jerk (i.e., a third-order derivative of position) while turning around a corner: ∫ [( 𝑑 𝑥𝑑 ̂𝑡 ) + ( 𝑑 𝑦𝑑 ̂𝑡 ) ] 𝑑 ̂𝑡, (7)where 𝑥 and 𝑦 represent position in 𝑥 - and 𝑦 -axis at a rescaledtime point ̂𝑡 . Based on the minimum jerk principle, the nor-malized speed profile ̂𝑣 ( ̂𝑡 ) can be represented by a fourth-order polynomial equation of rescaled time point ̂𝑡 : ̂𝑣 ( ̂𝑡 ) = 𝑎 + 𝑎 ̂𝑡 + 𝑎 ̂𝑡 + 𝑎 ̂𝑡 + 𝑎 ̂𝑡 . (8)The coefficients 𝑎 , 𝑎 , 𝑎 , 𝑎 , and 𝑎 can be determinedfrom the average of normalized speed curves in Fig. 6(c).Table 3 shows the coefficient values. Following the work ofDias et al. [26], we modeled the normalized speed profile ̂𝑣 ( ̂𝑡 ) with the same values in the start and end of the curve: ̂𝑣 (0) = 1 and ̂𝑣 (1) = 1 . As stated in the last paragraph of Section 2, we measuredthe travel time in each one-way trip and then calculated theaverage speed for each trip. The handler group speed in thestraight corridor setup is shown in Fig. 7. As can be seenfrom Fig. 7, one can observe that the handler group move-ment speed decreases as the distance traveled 𝑠 increases,especially for first 300 𝑚 . (a) s p ee d ( m / s ) time (s) (b) n o r m a li z e d s p ee d rescaled time (c) n o r m a li z e d s p ee d rescaled time Figure 6:
Speed profile of handler groups as a function ofrescaled time in the area of 𝑥 ≤ and 𝑦 ≤ : (a) examplesof individual speed curves. (b) the result of normalization per-formed against rescaled time ̂𝑡 for curves in (a). (c) the averageof normalized speed curves (blue cross symbols) with its trendline (red solid line). The trend line begins to decrease and thenincrease when the patient bed is turning around a corner. To further quantify the tendency of decreasing travel speeddue to fatigue effect, we evaluated the fatigue coefficient 𝑓 𝑎 based on the speed reduction with respect to the initial move-ment speed 𝑣 . According to the work of Luo et al. [12], thefatigue coefficient 𝑓 𝑎 is given as: 𝑓 𝑎 = 𝑣 − 𝑣 𝑖 𝑣 . (9)Here, 𝑣 𝑖 indicates the current speed which was calculatedfor each one-way trip from one end to the other end of thestraight corridor (see Fig. 1(b)). As shown in Eq. (9), thefatigue coefficient examines speed reduction ratio, so the fa- J. Kwak et al .: Preprint submitted to Elsevier
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Table 3
Normalized speed profile coefficients
Coefficient Value 𝑎 𝑎 𝑎 -10.0789 𝑎 𝑎 -8.5632 s p ee d ( m / s ) s (m) Figure 7:
The handler group movement speed in the straightcorridor setup. Different symbols represent the speed of differ-ent handler groups: black cross ( × ) for Group_1, blue rectan-gle ( □ ) for Group_2, purple circle ( ○ ) for Group_3, and redtriangle ( △ ) for Group_4. tigue effect can be compared among different handler groupseven if they have different speed. In addition, the conceptof fatigue coefficient is useful when one wants to estimatethe speed at a certain traversed distance based on the initialspeed. Based on Eq. (9), we evaluated the fatigue coefficient 𝑓 𝑎 for each handler group indicated in Table 2. Note thatthe current speed 𝑣 𝑖 and fatigue coefficient 𝑓 𝑎 are given asfunctions of distance traveled 𝑠 , and 𝑣 𝑖 is the average speedcalculated from each one-way trip. Figure 8 shows the rela-tionship between 𝑓 𝑎 and 𝑠 . One can observe that the fatiguecoefficient values tend to rapidly increase in the beginningand then slowly grow. That is compatible with the decreas-ing trend of handler group movement speed shown in Fig. 7.We applied piecewise linear regression [27] to quantifythose two regimes in the fatigue coefficient curve: 𝑓 𝑎 ( 𝑠 ) = { 𝛽 𝑠 if 𝑠 ≤ 𝑠 × 𝛽 𝑠 + 𝛽 ( 𝑠 − 𝑠 𝑥 ) if 𝑠 > 𝑠 × (10)where 𝑠 × is the breakpoint and 𝛽 and 𝛽 are regression coef-ficients. We used nls function in R package to fit the piece-wise linear regression model in Eq. (10). It was estimatedthat the fatigue coefficient 𝑓 𝑎 curve slope changes at break-point 𝑠 × = 264 . m. The slope for the first and secondsegments are estimated as 𝛽 = 4 .
076 × 10 −4 and 𝛽 + 𝛽 = 5 .
84 × 10 −5 , respectively. The slope coefficients 𝛽 and 𝛽 + 𝛽 indicate that, the speed decreases by roughly4.08% and 0.58% on average for each increase of 100 m in f a s (m) Figure 8:
Relationship between the fatigue coefficient 𝑓 𝑎 andthe distance traveled 𝑠 . The measured fatigue coefficient 𝑓 𝑎 from handler groups is denoted by blue circles ( ○ ). The corre-sponding trend line is indicated by red solid line. Black dashedline indicates the breakpoint location 𝑠 × = 264 . m at whichfatigue coefficient 𝑓 𝑎 curve slope changes. Purple solid lineswith cross symbols ( × ) show the boundaries of 80% predictionband with quantiles 0.10 and 0.90. The fatigue coefficientcurve of Luo et al. [12] is denoted by green solid line with tri-angles ( △ ). In their study, 𝑓 𝑎 was measured for pedestrianscarrying heavy items (10 ∼
20 kg) by hand.
Table 4
The fatigue coefficients estimated by piecewise linear regres-sion: the breakpoint and slopes.
Coefficient Value 𝑠 × ± 𝛽 .
076 × 10 −4 𝛽 −3 .
492 × 10 −4 𝛽 + 𝛽 .
840 × 10 −5 Table 5
The slope coefficients estimated by piecewise linear quantileregression for 80% prediction band with quantiles 0.10 and0.90.
Coefficient Quantiles0.1 0.9 𝛽 .
520 × 10 −4 .
476 × 10 −4 𝛽 −1 .
976 × 10 −4 −4 .
714 × 10 −4 𝛽 + 𝛽 .
447 × 10 −5 .
619 × 10 −5 distance traveled for 𝑠 ≤ 𝑠 × and 𝑠 > 𝑠 × , respectively. Table 4depicts the fatigue coefficients estimated by piecewise linearregression including the breakpoint and slopes. In addition,we also applied quantreg function in R package to performpiecewise linear quantile regression based on Eq. (10) in or-der to generate 80% prediction band showing possible up-per and lower boundaries of data points. Table 5 depictsthe slope coefficients estimated by piecewise linear quantileregression for 80% prediction band with quantiles 0.10 and0.90.We compared the value of fatigue coefficient 𝑓 𝑎 obtained J. Kwak et al .: Preprint submitted to Elsevier
Page 7 of 14atient Bed Horizontal Movement Performance Estimation in this study with that one reported in the study of Luo etal. [12]. In their study, the handlers were walking on a cir-cular track while carrying heavy items (10 ∼
20 kg) by hand.For the same initial speed (between 1.25 m/s and 1.75 m/s),at distance of 325 m, the value of 𝑓 𝑎 measured in this study( 𝑓 𝑎 = 0 . ) is lower than the value ( 𝑓 𝑎 = 0 . ) reportedby Luo et al. [12]. This is seemingly because maneuvering apatient bed is physically less demanding than carrying heavyitems by hand while walking. Although the data points ob-tained from this study are scattered in Fig. 8, it still clearlyillustrates the tendency of increasing fatigue coefficient 𝑓 𝑎 as the distance 𝑠 increases. Based on the average normalized speed curves shown inFig. 6(c), we can predict the movement duration of a patientbed in the corner area (i.e., 𝑥 ≤ and 𝑦 ≤ ). We describethe distance traveled in the corner area 𝑠 𝑐 with the followingequation: 𝑠 𝑐 = 𝑁 𝑡 ∑ 𝑖 =1 𝑣 𝑖 Δ 𝑡 𝑖 = Δ 𝑡 𝑖 𝑁 𝑡 ∑ 𝑖 =1 𝑣 𝑖 , (11)where 𝑁 𝑡 is the number of intervals and 𝑣 𝑖 is the speed atinterval 𝑖 . The interval length Δ 𝑡 𝑖 can be obtained by rear-ranging Eq. (11): Δ 𝑡 𝑖 = 𝑠 𝑐 ∑ 𝑁 𝑡 𝑖 =1 𝑣 𝑖 . (12)The movement duration in the corner area 𝑡 𝑐 is a summationof Δ 𝑡 𝑖 , 𝑡 𝑐 = 𝑁 𝑡 ∑ 𝑖 =1 Δ 𝑡 𝑖 = 𝑁 𝑡 Δ 𝑡 𝑖 . (13)The speed at interval 𝑖 can be obtained from ̂𝑣 𝑖 by multiply-ing entering speed 𝑣 , i.e., 𝑣 𝑖 = 𝑣 ̂𝑣 𝑖 . Here, ̂𝑣 𝑖 is normalizedspeed profile ̂𝑣 at interval 𝑖 . The movement duration in thecorner area 𝑡 𝑐 is given as: 𝑡 𝑐 = 𝑠 𝑐 𝑁 𝑡 𝑣 ∑ 𝑁 𝑡 𝑖 =1 ̂𝑣 𝑖 . (14)The average normalized speed profile ̂𝑣 𝑎𝑣𝑔 is given as: ̂𝑣 𝑎𝑣𝑔 = 1 𝑁 𝑡 𝑁 𝑡 ∑ 𝑖 =1 ̂𝑣 𝑖 (15) Table 6
Probabilistic model input parameters. parameters mean sd min max 𝑠 𝑐 𝑣 and if we approximate Eq. (15) to continuous space by uti-lizing Eq. (8), ̂𝑣 𝑎𝑣𝑔 becomes ̂𝑣 𝑎𝑣𝑔 = ∫ ̂𝑣 𝑖 𝑑 ̂𝑡 = 𝑎 ̂𝑡 + 12 𝑎 ̂𝑡 + 13 𝑎 ̂𝑡 + 14 𝑎 ̂𝑡 + 15 𝑎 ̂𝑡 |||| = ( 𝑎 + 12 𝑎 + 13 𝑎 + 14 𝑎 + 15 𝑎 ) . (16)Accordingly, Eq. (14) should read 𝑡 𝑐 = 𝑠 𝑐 𝑣 ̂𝑣 𝑎𝑣𝑔 . (17)Here, distance traveled in the corner area 𝑠 𝑐 , entering speed 𝑣 , and average normalized speed profile ̂𝑣 𝑎𝑣𝑔 are input pa-rameters. Based on Eq. (16), we can obtain ̂𝑣 𝑎𝑣𝑔 = 0 . .In order to reflect uncertainty in the movement durationof a bedridden patient (see Eq. (17)), we developed a prob-abilistic model based on the input parameters 𝑠 𝑐 and 𝑣 formovement duration prediction. Previous studies [28, 29] notedthat a probabilistic approach is beneficial to estimate evacu-ation time based on an empirical predictive model utilizingexperimental data, especially when there exists notable vari-ability in evacuation parameters such as premovement timeand movement time. In the presented experimental results,one can notice that the difference in basic movement charac-teristics among handlers is considerable, thus our predictionmodel was developed in line with the probabilistic approach.Analogous to previous studies [9, 10, 11], the evacuation inthis study refers to the movement of a bedridden patient mov-ing with a group of handlers to the place of safety.In the development of probabilistic prediction model, weassumed that 𝑠 𝑐 and 𝑣 follow Gaussian distribution. Ta-ble 6 shows the probabilistic model input parameters. We setthe mean and standard deviation values of the input parame-ters same as in the experimental results. The maximum andminimum values were determined by adding and subtractingtwice of the standard deviation to and from the mean value,respectively. Based on Eq. (17) with the probabilistic modelinput parameters in Table 6, we performed 1000 simulations.The frequency histogram of the movement duration is shownin Fig. 9, which has a mean of 8.85 s and a standard devia-tion of . s. Its range is [5 . , . s. The experimentalresults of movement duration [5 . , . s all fall withinthe simulated distribution. This indicates that our predictionmodel of movement duration can successfully replicate theexperimental data of movement duration.In order to reflect the effect of fatigue on travel time,we considered the fatigue coefficient 𝑓 𝑎 and the relationship J. Kwak et al .: Preprint submitted to Elsevier
Page 8 of 14atient Bed Horizontal Movement Performance Estimation F r e q u e n c y Movement duration (s)
Figure 9:
Frequency histogram of movement duration in acorner area estimated from numerical simulation. Red shadedarea indicates the range of movement duration measured fromthe experiment. with the distance traveled 𝑠 in the estimation of movementduration. Like Eq. (13), the movement duration in a straightcorridor 𝑡 𝑠 is given as 𝑡 𝑠 = 𝑁 𝑠 ∑ 𝑖 =1 Δ 𝑡 𝑗 , (18)where Δ 𝑡 𝑗 is the travel time in segment 𝑗 and 𝑁 𝑠 is the num-ber of segments. Note that Δ 𝑡 𝑗 is not constant here, meaningthat the travel speed is different for each segment. The nor-malized speed ̂𝑣 is obtained after we rearranged Eq. (9), ̂𝑣 = 𝑣𝑣 = 1 − 𝑓 𝑎 , (19)again, 𝑣 is entering speed. The average travel speed in seg-ment 𝑗 is given as Δ 𝑠 𝑗 Δ 𝑡 𝑗 = 𝑣 𝑗 −1 + 𝑣 𝑗 , (20)where Δ 𝑠 𝑗 is the distance traveled in segment 𝑗 and 𝑣 𝑗 −1 isthe speed at the beginning point of segment 𝑗 and 𝑣 𝑗 for theend point of segment 𝑗 . After combining Eqs. (19) and (20),we obtained the travel time of segment 𝑗 as Δ 𝑡 𝑗 = Δ 𝑠 𝑗 𝑣 ( ̂𝑣 𝑗 −1 + ̂𝑣 𝑗 ) . (21) In this case study, we demonstrate how the presentedapproach can be applied to predict the movement durationof a handler group transporting multiple bedridden patients.We selected the current Singapore General Hospital Emer-gency Department (SGH ED) for the case study. In Fig. 10,we present a schematic representation of the bedridden pa-tient evacuation route in the current SGH ED. The evacua-tion route starts from the critical care unit (CCU) at (0 , -15 0 15 30 45 60-15 0 15 30 45 60 y ( m ) x (m) Figure 10:
Schematic representation of bedridden patientevacuation route in current Singapore General Hospital Emer-gency Department (SGH ED). The evacuation route startsfrom the critical care unit (red square) which is placed at (0 , to the place of safety (green triangle) at (45 , . The evac-uation route includes three straight corridor sections and twocorner areas. The evacuation route is 2 m wide, same as theexperiment condition. Table 7
Bedridden patient evacuation route segments
Segment type from to Length (m)1 Straight corridor (0, 0) (0, 11) 112 Corner (0, 11) (4, 15) 7.65 ± ± to the place of safety at (45 , . The evacuation route is2 m wide (the same width as our experiment condition), andincludes three straight corridor sections and two corner ar-eas, as shown in Table 7. Note that we modeled CCU andthe place of safety as single points in order to focus on theevacuation of bedridden patients between those areas. Theactual geometry of the areas is much more complicated, thusthe movement duration within the areas is not considered forsimplicity. As in Section 3.3, the evacuation time in this casestudy was estimated for a patient bed moving with two han-dlers. Another important point to note is that evacuation ofa bedridden patient with a handler group cannot be simplyextended to the case of a population of bedridden patientswith multiple handler groups.This is mainly because the in-teractions among the handler groups are likely to affect theirmovement speed.We estimated the movement duration of a handler groupcarrying a bedridden patient from CCU to the place of safetyfive times. It is reasonable to suppose that the handler groupin this case study were expected to move the patient bed asfast as possible and their movement characteristics are thesame as in the experimental study. Equations (17) and (21) J. Kwak et al .: Preprint submitted to Elsevier
Page 9 of 14atient Bed Horizontal Movement Performance Estimation
Table 8
Probabilistic model input parameters. parameters mean sd min max 𝑠 𝑐 𝑣 𝑡 𝑝 𝑡 𝑞 𝑠 × 𝛽 .
076 × 10 −4 .
780 × 10 −5 .
520 × 10 −4 .
476 × 10 −4 𝛽 −3 .
492 × 10 −4 .
580 × 10 −5 −4 .
714 × 10 −4 −1 .
976 × 10 −4 𝛽 + 𝛽 .
840 × 10 −5 .
895 × 10 −6 .
447 × 10 −5 .
619 × 10 −5 F r e q u e n c y Movement duration (s)
Figure 11:
Numerical simulation results of movement durationfor five round trips between CCU and the place of safety. Theprobability of finishing the movement within the reference time 𝑇 𝑟𝑒𝑓 = 999 . s is 23.6%. However, 76.4% of prediction resultsare longer than the reference time (indicated by the red shadedarea), indicating that the reference time is not likely to beachievable. Here, the reference time 𝑇 𝑟𝑒𝑓 is computed withthe traditional deterministic evaluation method using the meanvalue of input parameters. were applied to estimate the movement duration in the cor-ner areas and straight corridor sections, respectively. We as-sumed that the fatigue effect is in effect both in the outgoingand return trips to the CCU, thus we applied the same fatiguecoefficient function in Eq. (10) for the both trips.In addition to the distance traveled in the corner area 𝑠 𝑐 and entering speed 𝑣 shown in Table 6, we introduced addi-tional input parameters to the probabilistic model: prepara-tion time 𝑡 𝑝 , positioning time 𝑡 𝑞 , breakpoint 𝑠 × , and fatiguecoefficient curve slopes 𝛽 and 𝛽 . The preparation time islength of time required to prepare a patient bed for move-ment from CCU and the positioning time is the time takento place a patient bed at the place of safety after its arrival.We set the mean and standard deviation of 𝑡 𝑝 and 𝑡 𝑞 based onthe observation reported by Strating (2013) [10], and basedon Tables 4 and 5 for 𝑠 × , 𝛽 , 𝛽 , and 𝛽 + 𝛽 .Based on our movement duration prediction model withthe probabilistic model input parameters in Table 8, we per-formed 10000 simulations. In each simulation, the handler Table 9
The probability and the corresponding available safe egresstime (ASET) that a handler group can safely evacuate fivebedridden patients. Note that the handler group can transportone bedridden patient to the place of safety for each roundtrip.
Probability correspondingASET (s)10% 892.8920% 973.7230% 1041.1740% 1109.6550% 1181.3160% 1266.2770% 1368.3080% 1506.4890% 1773.08 group makes five round trips between CCU and the place ofsafety. The frequency of histogram of the movement dura-tion is shown in Fig. 11, which has a mean of 1272.5 s anda standard deviation of 375.46 s. The estimated movementduration ranges from 718.1 s to 2699.7 s. One can observea large spread of the numerical simulation results presentedin Fig. 11 due to the right-skewed distributions of 𝑡 𝑐 and 𝑡 𝑠 .We refer the readers to Appendix A for further details.By introducing a probabilistic model of movement dura-tion prediction, we can consider the effect of uncertain fac-tors in evacuation process, like entering speed, preparationtime, positioning time, and fatigue effect coefficients. Basedon the results shown in Fig. 11, one can estimate the proba-bility that an evacuation can be safely performed within cer-tain amount of time. Table 9 shows the probability and thecorresponding available safe egress time (ASET) that a han-dler group can safely evacuate five bedridden patients. Notethat the handler group can transport one bedridden patientto the place of safety for each round trip. For instance, theprobability of successful evacuation is estimated as 90% ifASET is 1773.08 s. In fire evacuation studies, ASET is the J. Kwak et al .: Preprint submitted to Elsevier
Page 10 of 14atient Bed Horizontal Movement Performance Estimation duration measured from the start of a fire to the onset of haz-ardous conditions in which people cannot be evacuated dueto the fire. ASET is influenced by various factors includingfire detection devices, location of fire origin, growth of fire,and ventilation paths [30].In order to quantify the effect of uncertainty in the in-put parameters, we compared the movement duration pre-diction results against the reference time which is based onthe traditional deterministic evaluation method. As stated inZhang et al. [29], the value of reference time was computedbased on the mean value of input parameters: evacuationroute length ( 𝑙 = 99 . m), preparation time ( 𝑡 𝑝 = 7 . s),positioning time ( 𝑡 𝑞 = 6 . s), and entering speed ( 𝑣 =1 . m/s). For a handler group making five round trips be-tween CCU and the place of safety, the reference time 𝑇 𝑟𝑒𝑓 is computed as 𝑇 𝑟𝑒𝑓 = 𝑀 ( 𝑡 𝑝 + 𝑡 𝑞 + 𝑡 𝑡 ) , (22)where 𝑀 is the number of round trips that the handler groupmakes and 𝑡 𝑡 = 2 𝑙 ∕ 𝑣 is the time required for the handlergroup to make a round trip between CCU and the place ofsafety excluding the time for preparation and positioning. Ascan be seen from Fig. 11, the probability that the handlersevacuate five bedridden patients within the reference time 𝑇 𝑟𝑒𝑓 = 999 . s is 23.6% while 76.4% of prediction resultsare longer than the reference time. It appears that the tra-ditional deterministic evaluation method based on the meanvalue of input parameters underestimates the movement du-ration.
4. Conclusion
We performed a series of controlled experiments to studythe dynamics of patient beds in horizontal movement. In ourexperiments, we examined the change of velocity in cornerturning movements and speed reductions in multiple tripsbetween both ends of a straight corridor. In the corner turn-ing movements, we observed that the start and end of turn-ing movement can be identified based on the trajectory de-viation. We also quantified common patterns in differentspeed curves by means of the normalized speed profile withrescaled time. In the straight corridor experiment, we dis-covered experimental relationship between the fatigue co-efficient and distance traveled 𝑠 ≤ m. Based on theexperiment results, we developed a movement duration pre-diction model and then applied the model for a patient bedhorizontally moving in a healthcare facility. In order to re-flect uncertainty in the horizontal movement, we introduceda probability distribution to the horizontal movement param-eters like entering speed, preparation and positioning time,and fatigue coefficients. According to our case study results,one can estimate the probability that an evacuation can besafely performed within certain amount of time. In addition,it is highly probable that the horizontal movement durationwould be longer than the prediction results from an exist-ing model which assumes constant movement speed. Thecase study results demonstrated that our model has potential in predicting emergency evacuation time of patient beds inhealthcare facilities.This study presents the experiment results from four han-dler groups in left and right turning movements, and thestraight corridor movements, respectively. The experimentdata was collected from a small number of handlers and theyare all young adults who are not professionally trained. Whenwe set up the experiment, we were mainly interested in ob-serving difference in male and female groups, so we did notconsider a group of mixed genders. In addition, the weightof the bedridden patient was fixed. The experiment resultsmight be different for various age groups, the mix of gen-ders in the handler groups, bedridden patient weight, andhandlersâĂŹ proficiency in moving patient beds. It is alsonoted that the handlers in the experiment were not necessar-ily in a rush as they would be in the real emergency evacu-ation. Thus, there exists the possibility that the movementspeed of handlers in the real emergency evacuation can behigher than the speed reported in this study. The presentedexperiment results can be generalized with larger number ofhandler groups having different conditions like age and pro-ficiency.As a first step to study the dynamics of patient bed move-ment in horizontal space, we focused on the case of a patientbed movement based on simple experiment setups. Furtherexperimental studies need to be carried out in order to extendthe presented model. In emergency evacuations in health-care facilities, there are several dozens of patient beds to betransported to the place of safety by multiple handler groups.During the evacuations, several patient beds are likely tomove in the same evacuation route at the same time and theinteractions among them might affect the movement speed.The examples of observable interactions include the distancebetween two moving patent beds and the acceleration anddeceleration in their speed profile. Future studies can bealso planned from the perspective of emergency evacuationsimulations. Hunt et al. [31] pointed out that the move-ment of evacuation devices has not been appropriately con-sidered in existing evacuation simulation models althoughthe movement of such devices is critical for vulnerable pa-tients. We are currently developing an evacuation simulationmodel that can incorporate our findings in patient bed move-ment dynamics to explicitly reflect their movement duringthe emergency evacuations in healthcare facilities. The evac-uation simulation models can check potential conflict withother evacuees and geometric elements such as doors andcorners, and predict the total evacuation time for the sce-narios in which patient beds are fleeing with other evacuees.Another interesting extension of the presented study is to op-timize the evacuation time of bedridden patients in health-care facilities. The optimization study can be performed todesign optimal evacuation routes and estimate the number ofhandlers for transporting patient beds to the place of safety. J. Kwak et al .: Preprint submitted to Elsevier
Page 11 of 14atient Bed Horizontal Movement Performance Estimation
Acknowledgements
This research is supported by National Research Founda-tion (NRF) Singapore, GOVTECH under its Virtual Singa-pore program Grant No. NRF2017VSG-AT3DCM001-031.The experiment was organized with the help of Ms. Yo-geswary Pasupathi (Research nurse at the Singapore GeneralHospital Emergency Department) and members of Complex-ity Institute, Nanyang Technological University, Singapore.We thank Mr. Joshua for his help in processing video records.
A. Distribution of movement duration
In Sections 3.3 and 3.4, we presented the distribution ofmovement duration in a corner area and in the case study,respectively (see Figs. 9 and 11). This appendix providesevidence to suggest that the spread of movement durationcan be large.According to the setup of our numerical simulation studypresented in Section 3.4, the movement duration 𝑇 for around trip is given as a sum of preparation time 𝑡 𝑝 , position-ing time 𝑡 𝑞 , and total travel time in the corner areas 𝑡 𝑐 = ∑ 𝑡 𝑐,𝑖 , and total travel time in the straight corridors 𝑡 𝑠 = ∑ 𝑡 𝑠,𝑗 : 𝑇 = 𝑡 𝑝 + 𝑡 𝑞 + 𝑡 𝑐 + 𝑡 𝑠 = 𝑡 𝑝 + 𝑡 𝑞 + 𝑁 𝑐 ∑ 𝑖 =1 𝑡 𝑐,𝑖 + 𝑁 𝑠 ∑ 𝑖 =1 𝑡 𝑠,𝑗 , (23)where 𝑁 𝑐 = 4 is the number of corner areas in a round tripand 𝑁 𝑠 is the number of straight corridor segments in a roundtrip. Travel time in corner area 𝑖 is denoted by 𝑡 𝑐,𝑖 and 𝑡 𝑠,𝑗 for straight corridor section 𝑗 . Parameters 𝑡 𝑝 and 𝑡 𝑞 wereassumed to follow normal distribution. The distribution of 𝑡 𝑐 can be explained by means of the ratio distribution andthe distribution of 𝑡 𝑠 can be approximated by the reciprocalnormal distribution.As presented in Eq. (17), travel time in corner area 𝑖 isgiven as 𝑡 𝑐,𝑖 = 𝑠 𝑐 𝑣 ̂𝑣 𝑎𝑣𝑔 . (24)Here, the distance traveled in the corner area 𝑠 𝑐 and the en-tering speed 𝑣 are assumed to follow the normal distribu-tion while the average normalized speed profile ̂𝑣 𝑎𝑣𝑔 is setas a fixed value based on the experimental study result. Thatis, 𝑡 𝑐,𝑖 follows a distribution of a ratio of two independentnormally distributed variables 𝑠 𝑐 and 𝑣 . According to Cur-tiss [32] and Springer [33], the ratio distribution of 𝑡 𝑐,𝑖 canbe described by means of the joint distribution of 𝑠 𝑐 and 𝑣 : 𝑓 ( 𝑠 𝑐 , 𝑣 ) = 12 𝜋𝜎 𝑠,𝑐 𝜎 𝑣, ×exp { − 12 [( 𝑠 𝑐 − 𝜇 𝑠,𝑐 𝜎 𝑠,𝑐 ) + ( 𝑣 − 𝜇 𝑣, 𝜎 𝑣, ) ]} , F r e q u e n c y Movement duration (s)
Figure 12:
Frequency histogram of movement duration in acorner area estimated from numerical simulation (blue bars)and theoretically predicted by the ratio distribution (red solidline). We performed 1000 numerical simulations. (25)where 𝜇 𝑠,𝑐 = 7 . and 𝜎 𝑠,𝑐 = 0 . are the mean and standarddeviation of Gaussian distributed 𝑠 𝑐 > , and 𝜇 𝑣, = 1 . and 𝜎 𝑣, = 0 . for Gaussian distributed 𝑣 > . The prob-ability function of 𝑡 𝑐,𝑖 is given as 𝑃 ( 𝑡 𝑐,𝑖 )= ∫ +∞−∞ || 𝑣 || 𝑓 ( 𝑠 𝑐 , 𝑣 ) 𝑑𝑣 = ∫ +∞−∞ || 𝑣 || 𝑓 ( 𝑣 𝑡 𝑐,𝑖 ̂𝑣 𝑎𝑣𝑔 , 𝑣 ) 𝑑𝑣 = 12 𝜋𝜎 𝑠,𝑐 𝜎 𝑣, ∫ +∞0 𝑣 ×exp { − 12 [( 𝑣 𝑡 𝑐,𝑖 ̂𝑣 𝑎𝑣𝑔 − 𝜇 𝑠,𝑐 𝜎 𝑠,𝑐 ) + ( 𝑣 − 𝜇 𝑣, 𝜎 𝑣, ) ]} 𝑑𝑣 . (26)Based on Eq. (26), we theoretically predicted the frequencyhistogram of movement duration in a corner area and com-pared the result against one obtained from the numericalsimulation, see Fig. 12. As can be seen from Fig. 12, theratio distribution produced a right-skewed curve and the nu-merical simulation result is in good agreement with the the-oretical prediction.As shown in Eq. (21), the travel time in the straight cor-ridor section 𝑗 is given as 𝑡 𝑠,𝑗 = Δ 𝑠 𝑗 𝑣 ( ̂𝑣 𝑗 −1 + ̂𝑣 𝑗 ) = 2Δ 𝑠 𝑗 𝑣 𝑗 −1 + 𝑣 𝑗 . (27)Here, Δ 𝑠 𝑗 is a fixed length of straight corridor section 𝑗 , andthe speed at the beginning and end points of the straight cor-ridor section are denoted by 𝑣 𝑗 −1 and 𝑣 𝑗 , respectively. It is J. Kwak et al .: Preprint submitted to Elsevier
Page 12 of 14atient Bed Horizontal Movement Performance Estimation F r e q u e n c y Movement duration (s)
Figure 13:
Frequency histogram of movement duration in thestraight corridor sections (total length 𝑠 𝑗 = 168 m) estimatedfrom numerical simulation (blue bars) and theoretically pre-dicted by the ratio distribution (red solid line). We performed1000 numerical simulations. assumed that the value of 𝑣 𝑗 −1 follows the normal distribu-tion and the value of 𝑣 𝑗 is determined based on 𝑣 𝑗 −1 and thefatigue effect. It can be inferred from Section 3.2 that the fa-tigue effect becomes considerable when the handler grouptravels several hundred meters. In a round trip, the totallength of straight corridor sections is 𝑠 𝑗 = ∑ Δ 𝑠 𝑗 = 168 m,thus it may be reasonable to suppose that the difference be-tween 𝑣 𝑗 −1 and 𝑣 𝑗 is not significant. For simplicity, we ap-proximate 𝑣 𝑗 to 𝑣 𝑗 −1 and then simplify the equation of 𝑡 𝑠,𝑗 asa function of the total length of straight corridor sections 𝑠 𝑗 and the speed at the beginning point 𝑣 𝑗 −1 : 𝑡 𝑠,𝑗 ≈ 𝑠 𝑗 𝑣 𝑗 −1 , (28)indicating that the distribution of 𝑡 𝑠,𝑗 is can be described by areciprocal normal distribution. According to Gurarie et al. [34],the probability distribution function of 𝑡 𝑠,𝑗 is given as 𝑃 ( 𝑡 𝑠,𝑗 ) = 𝑠 𝑗 √ 𝜋𝜎 𝑣,𝑗 −1 𝑡 𝑠,𝑗 exp [ − 12 ( 𝑠 𝑗 − 𝜇 𝑣, 𝑗 −1 𝑡 𝑠,𝑗 𝜎 𝑣,𝑗 −1 𝑡 𝑠,𝑗 ) ] , (29)where 𝜇 𝑣, 𝑗 −1 = 1 . and 𝜎 𝑣,𝑗 −1 = 0 . are mean and stan-dard deviation of 𝑣 𝑗 −1 , respectively. Based on Eq. (29), wetheoretically predicted the frequency histogram of movementduration in the straight corridor and compared the result againstone obtained from the numerical simulation, see Fig. 13. Ascan be seen from Fig. 13, the reciprocal normal distributiongenerated a right-skewed curve that is consistent with thenumerical simulation result.We computed an estimate of the movement duration fora round trip by summing the mode values of 𝑡 𝑝 , 𝑡 𝑞 , 𝑡 𝑐 , and 𝑡 𝑠 shown in Table 10, i.e., 𝑇 ,𝑒𝑠𝑡 = 7 .
96 + 6 .
25 + 29 . . . s. One can notice that the total travel time in the cor-ner and straight corridor areas (i.e., 𝑡 𝑐 and 𝑡 𝑠 ) take a signif-icant portion of movement duration 𝑇 ,𝑒𝑠𝑡 . Due to the right-skewed distributions of 𝑡 𝑐 and 𝑡 𝑠 , the spread of the simulated Table 10
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