A SIMPLE THROW MODEL FOR FRONTAL VEHICLEPEDESTRIAN COLLISIONS
Milan Batista
University of Ljubljana
Faculty of Maritime Studies and Transportation
Pot pomorščakov 4, SI- 6320 Portorož, Slovenia, EU
milan.batista@fpp.edu
ABSTRACT
This paper discusses a simple theoretical throw model for frontal vehicle-pedestrian
collisions. The model is based on the simple assumption that pedestrian movement after
impact can be approximated by movement of a mass point. Two methods of
reconstruction of vehicle-pedestrian collision are discussed: one knowing only the
throw distance and the other when also impact to ground contact distance is known. The
model is verified by field data available in the literature and by comportment with full
scale numerical simulation.
1
INTRODUCTION
The investigation of vehicle-pedestrian collisions must have begun in the middle of the
sixties mainly for the purpose of accident reconstruction. From that time several models
describing the motion of pedestrians after impact with vehicles was developed
([1],[2],[3],[4],[5],[6],[8],[9],[10],[11],[13],[14],[15],[17],[18]). Basically there are two
types of models: theoretical, based on laws of mechanics, and empirical. Theoretical
models yield reliable results; however, considerable input data from real world
collisions is needed to solve the equations. On the other hand empirical models--usually
consisting of a single regression formula which connects the vehicle impact speed with
pedestrian throw distance ([5],[17])--need no particular data; however their application
is limited only to well defined scenarios and the accuracy of models is within, say,
±10 km/h ([7]). Typically the empirical models do not include road grade, which can be
an influence factor when one determines vehicle impact velocity from throw distance.
The hybrid models try to combine features of both basic models ([3]).
In the present paper the model of frontal vehicle-pedestrian collision closely following
the Han-Brach approach ([3],[6]) is developed. The details of derivation of equations
are included for comprehensiveness. In addition to Han-Brach equations the equations
for total flying time and total throw time and throw distance are also given. The basic
equation of reconstruction--i.e., the equation for calculation of pedestrian launch
velocity--is then obtained by inverting the equation for total throw distance. This
equation is, in the special case of a horizontal road, reduced to the so called Searle
equation ([15][16]). The four methods of reconstruction are then discussed: the method
when one knows the pedestrian launch angle and friction between pedestrian and road;
the Serale method ([15]) where the launch angle is estimated on the basis of extreme of
3/17/2008
1
launch velocity; and two new methods where in addition to throw distance the distance
from impact to ground contact is also known.
2
THE MODEL
2.1
Assumptions
Only the frontal impact of the vehicle with the pedestrian is considered. In the case
when the vehicle has enough speed or it is braking the pedestrian will, after impact, be
thrown from the vehicle hood, fly through the air, impact the ground and then
slide/roll/bounce on the ground to a rest. The possible impacts of pedestrian with the
road obstacles in the last phase are excluded from consideration. To describe these
events mathematically the following assumptions are made:
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
the car-pedestrian impact is symmetric so all events happens in a single plane
the initial velocity of the pedestrian is zero
after launch the pedestrian is considered as a mass point
the ground is flat
the pedestrian-ground friction is constant
all air resistance is neglected
y
g
vC0
θ
t0
x
t1
vP0
h
s2
s1
α
tP
μ
sP
s0
Figure 1. Vehicle-pedestrian collision variables and events
According to events description and the above assumptions the following basic
variables are included in the model (Figure 1):
ƒ
ƒ
ƒ
ƒ
gravity acceleration g = 9.8 m/s 2 ,
mass of the vehicle mC and mass of the pedestrian mP ,
initial pedestrian launch height h (not pedestrian COG),
total pedestrian throw distance sP ; i.e., the distance the pedestrian travels from
impact to his rest position on the ground,
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2
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
total pedestrian throw time t P ,
vehicle impact velocity vC 0 ,
pedestrian launch velocity vP 0 ,
road gradient angle α ,
pedestrian launch angle θ ,
coefficient of friction μ between the pedestrian and the ground.
It is further assumed that the total throw distance sP and total throw time t P can be
expressed as the sum of three phases--contact phase, flying phase and
sliding/rolling/bouncing phase ([4],[11]). The total throw distance is therefore
sP = s0 + s1 + s2
(1)
t P = t0 + t1 + t2
(2)
and the total throw time is
where indices 0, 1, 2 belong consecutively to impact, flying and sliding.
2.2
Contact phase
This phase roughly consists of ([4])
• vehicle-pedestrian contact
• impact--i.e., acceleration of the pedestrian's body
• movement on the vehicle's hood
In the scope of the present paper, the movement of the body onto the vehicle can be
roughly of two types:
• wrap trajectory - here the pedestrian is wrapped over the front of vehicle , usually
involving a decelerating vehicle
• forward projection - in this case COG of pedestrian is below the leading edge of
the vehicle at impact
The main goal in this phase is to connect vehicle impact velocity vC 0 with pedestrian
launch velocity vP 0 and also to determine the contact path length s0 and contact time t0 .
This last is beyond the scope of this paper and therefore will not be discussed. However
in the case of forward projection one can approximately take s0 = 0 and t0 = 0 . More
detailed analysis of impact and future references can be found in [4], [6] and [18].
Despite the fact that this phase of throw influences others, only a simple model will be
present: it is assumed that impact between vehicle and pedestrian is plastic. In this case
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3
from conservation of momentum--i.e., from mC vC 0 = ( mC + mP ) uC 0 --the initial contact
velocity of the vehicle vC 0 is
uC 0 =
vC 0
1 + mP mC
(3)
where uC 0 is vehicle/pedestrian post impact velocity. The case of non-plastic impact is
discussed in [8].
Because the velocity uC 0 and the pedestrian launch velocity vP 0 differ for the case of
wrap trajectory, a coefficient η called pedestrian impact factor is introduced to relate
them ([6][15][18]):
vP 0 = η uC 0 =
η vC 0
1 + mP mC
(4)
In general the coefficient η can not be constant and it is in general dependant on
various factors, including vehicle impact velocity, geometry of vehicle front, pedestrian
height, etc. ([18]).
2.3
Flying phase
Following Figure 1 and Newton’s 2nd Law the equations of motion of pedestrian COG
are the well known equation of a projectile in a vacuum:
dx
= vx
dt
mP
dvx
= −mP g sin α
dt
(5)
dy
= vy
dt
mP
dv y
dt
= − mP g cos α
where t is time, x, y are coordinates of COG of pedestrian, and vx , v y its velocity
components. The equation is completed with the following initial conditions
x ( 0) = 0
vx ( 0 ) = vP 0 cos θ
(6)
y (0) = h
v y ( 0 ) = vP 0 sin θ
Carrying out the integration and imposing initial conditions one finds velocity
vx = vP 0 cos θ − g sin α t
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v y = vP 0 sin θ − g cos α t
(7)
4
and position coordinates
x = vP 0 cos θ t − g sin α
t2
2
y = h + vP 0 sin θ t − g cos α
t2
2
(8)
At impact time t1 --i.e., the time when the pedestrian impacts the ground--the following
conditions are reached: y ( t1 ) = 0 and x ( t1 ) = s1 . From these, by using (7) and (8), one
obtains the flying time
vP 0 sin θ + vP2 0 sin 2 θ + 2 gh cos α
t1 =
g cos α
(9)
and the flying distance
s1 = vP 0 cos θ t1 − g sin α
2.4
t12
2
(10)
Impact with ground
At pedestrian impact with the ground the Newton dynamical equations take the
following impulse form
m ( v y+ − v y− ) = I y
m ( vx+ − vx− ) = − I x
(11)
where superscripts + and – denote velocities before and after impact, and I x and I y are
impulses in road horizontal and vertical direction, respectively. Here one needs further
assumptions about the nature of impact. The simplest are:
• the impact is plastic--i.e., v y+ = 0
• the Coulomb friction law is valid at impact--i.e., I x = μ I y
On the basis of these assumptions one can from the first of (11) find impulse in vertical
direction I y = − mv y− and from the second the horizontal velocity after impact
vx+ = vx− − I x m . From those, by using friction law, one obtains vx+ = vx− + μ v y− . By using
(7) this becomes
vx+ = vP 0 ( cos θ + μ sin θ ) − g ( sin α + μ cos α ) t1
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(12)
5
2.5
Sliding phase
After impact with ground the pedestrian slides to rest. The dynamical equations
governing this motion are
dx
= vx
dt
m
dvx
= − mg sin α − N x
dt
0 = − mg cos α + N y
(13)
where N x and N y are horizontal and normal reaction of the ground, respectively, and
the initial conditions are
x ( 0) = 0
vx ( 0 ) = vx+
(14)
By assuming that the Coulomb friction law is valid--i.e., N x = μ N y --one, after
integration and imposing the initial conditions obtain the velocity
vx = vx+ − g ( sin α + μ cos α ) t
(15)
t2
2
(16)
and the distance
x = vx+ t − g ( sin α + μ cos α )
At the end of pedestrian sliding one has vx ( t2 ) = 0 and x ( t2 ) = s2 . From these, by using
(15) and(16), the sliding time t2 is
t2 =
vx+
g ( sin α + μ cos α )
(17)
and the sliding distance s2 is
s2 =
2.6
(v )
+ 2
x
2 g ( sin α + μ cos α )
(18)
Total throw time and throw distance
The total time and distance can be, after calculation, obtained by summing the times and
distance of flying and sliding phase. However it turns out that a simple formula exists.
That is, by substituting (12) into (17) one obtains the total pedestrian throw time tP
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6
t P = t0 +
vP 0 ( cos θ + μ sin θ )
g ( sin α + μ cos α )
(19)
and by introducing (19) into (17) one finds the total throw distance
vP2 0 ( cos θ + μ sin θ )
sP = s0 + μ h +
2 g ( sin α + μ cos α )
2
(20)
Figure 2. Spreadsheet program for analyzing
vehicle-pedestrian collision
2.7
Analysis of vehicle-pedestrian collision
On the basis of the above formulas one can analyse the pedestrian motion from the
contact with vehicle to rest on the ground if the following ten data are available
mC , mP , μ , α , vC 0 , η , t0 , s0 , h , θ
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7
Among these the last five can in practice only be estimated. Nevertheless since all the
equations describing the collision are algebraic and explicit they can be easily
implemented into a spreadsheet program. An example of that is shown by Figure 2. The
data used in the example of Figure 2 will be used in all the following examples.
3
RECONSTRUCTION
In the reconstruction of pedestrian accidents one usually knows the throw distance and
asks for launch velocity on which vehicle impact speed can be estimated. Two cases
will be considered: the case of known throw distance and the case where the flying
distance is also known.
3.1
Known throw distance
If the pedestrian throw distance sP is known, then from (20) one can express pedestrian
launch velocity
vP 0 =
2 g ( sin α + μ cos α )( sP − s0 − μ h )
cos θ + μ sin θ
(21)
By denoting
p ≡ tan α
(22)
the formula (21) can be written in the following form
vP 0 =
2 g ( p + μ ) (1 + tan 2 θ ) ( sP − s0 − μ h )
(1 + μ tan θ ) 4 1 + p 2
(23)
The confidence analysis of the formula is given in Appendix A. For p = 0 and s0 = 0
one obtains the Searle formula ([15]). If the pedestrian launch velocity vP 0 is known,
the vehicle impact velocity is, from (4)
vC 0 = (1 + mP mC )
vP 0
η
(24)
The formulas (23) and (24) can be used for reconstruction of vehicle-pedestrian
collision if the following data are available
mC , mP , η , μ , p, sP , s0 , h , θ
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8
The spreadsheet program for reconstruction of vehicle-pedestrian collision on the basis
of (23) and (24) is shown in Figure 3. Note that by taking throw distance 16 m and data
from the previous example, the impact velocity of the vehicle is 50 km/h, as it should
be.
Figure 3. Spreadsheet program for reconstruction of a
vehicle-pedestrian collision
Before proceeding, the effect of grade on pedestrian launch velocity will be discussed in
some details. Let vP 0 = vP 0 (α , θ ) . In the special case of no grade--i.e., α = 0 --eq (21)
becomes
vP 0 ( 0,θ ) =
2 g μ ( sP − s0 − μ h )
cos θ + μ sin θ
(25)
v0 (α ,θ )
= sin α μ + cos α and is independent of θ
v0 ( 0, θ )
so one can define normalized launch velocity
The quotient of (21) and (25) is
uα ( μ , p ) ≡
3/17/2008
μ+ p
μ 1 + p2
(26)
9
The diagram of (26) for various values of μ is shown in Figure 4.
Now it follows that uα ( μ , 0 ) = 1 and lim uα ( μ , 0 ) =
1
μ
p →∞
. Thus if μ ≤ 1 then if the
pedestrian is launched uphill--i.e., if p > 0 --then the launch velocity for reaching the
distance sP is greater than that of horizontal ground; while if μ > 1 this is true only for
2μ
grades satisfying p ≤ 2
. But since the grades of the roads are limited to
μ −1
approximately p < 0.3 and the friction coefficient is limited approximately to μ < 2 this
limit can practically never be reached. In the case of downhill launch--i.e. for p < 0 -one should have μ > − p in order that the pedestrian can attain the rest position and in
this case the launch velocity for reaching the distance sP is lesser than that of horizontal
ground.
1.6
1.4
0.5
1.0
uα=v0(α,θ)/v0(0,θ)
1.2
1.0
0.8
1.5
0.6
0.4
0.2
0.0
-2
-1
0
1
2
3
4
p
Figure 4 Effect of grade on normalized launch velocity for
various values of μ
3.2
Known throw distance - Searle method
The difference between grade p of the road and the angle θ in Eq (23) is practical:
while the grade of the road can be measured, the launch angle θ can only be estimated.
Here it was Searle’s idea that the formula (23) can be used to determine lower and upper
boundaries, by considering the launch angle θ that will minimize and maximize the
expression ([15]). Therefore the effect of launch angle on pedestrian launch velocity
will now be considered.
3/17/2008
10
First, for horizontal launch--i.e., for θ = 0 --eq (21) reduces to
v0 (α , 0 ) = 2 g ( sin α + μ cos α )( sP − s0 − μ h )
(27)
vP 0 ( α , θ )
1
=
is independent of grade angle α so one can
vP 0 (α , 0 ) cos θ + μ sin θ
introduce the normalized launch velocity
The quotient
uθ ( μ ,θ ) ≡
1
cos θ + μ sin θ
(28)
The graph of this function for various values of μ is shown in Figure 5
2.0
1.8
μ = 0.0
uθ=v0(α,θ)/v0(α,0)
1.6
0.2
1.4
0.4
1.2
0.6
0.8
1.0
0.8
1.0
1.2
0.6
1.4
1.6
0.4
0
15
30
45
60
75
90
θ [deg]
Figure 5 Normalized launch velocity as function of launch angle
for various values of friction coefficients
Since the launch angle is limited to 0 ≤ θ <
π
2
the two limits of the functions are
⎛ π⎞ 1
uθ ( μ , 0 ) = 1 and uθ ⎜ μ , ⎟ = . Now, if μ > 1 then it follows from (28) that
⎝ 2⎠ μ
uθ ( μ , θ ) < 1 . The function in this case reaches its maximum at θ = 0 . Thus the maximal
velocity for reaching the distance sP is obtained at θ = 0 and its value is
v0max = 2 g
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(μ + p)
1 + p2
( sP − s0 − μ h )
(μ >1
∨ θ ≤ θ* )
(29)
11
For μ ≤ 1 one sees from Figure 6 that uθ ( μ , 0 ) = uθ ( μ , θ* ) = 1 where θ* is the critical
launch angle ([15]) given by
θ* = arctan
2μ
1− μ2
The expression (29) is thus valid only for θ ≤ θ* . If θ > θ* then the maximal velocity
obtained for the launch angle is θ →
v0max <
1
μ
2g
(μ + p)
1 + p2
π
2
with
( sP − s0 − μ h )
(μ ≤1
∧ θ > θ* )
(30)
Now as can be seen from the graph of Figure 3 the velocity given by (28) has a
du
minimum. The minimum is obtained from the condition θ = 0 , which gives ([15])
dθ
tan θ = μ
(31)
For this value of angle, the function has minimal value
uθ min =
1
1+ μ2
(32)
But since for μ > 0 one has uθ min < 1 it follows that v0 (α , θ ) < v0 (α , 0 ) . The minimal
velocity for reaching the given distance s p is thus
v0 min =
2 g ( μ + p ) ( s p − s0 − μ h )
(1 + μ )
2
1 + p2
(33)
The above formulas generalize the Searle formulas ([15]) in a way that they include
road grade and contact distance. They provide means to estimate velocity bounds if
throw distance is known. The spreadsheet program for reconstruction of vehiclepedestrian collision based on these formulas is shown by Figure 6. Note that if one
considers the example in Figure 2 as a reference, the Searle method estimates the impact
velocity between 48.3 km/h to 56.3 km/h, while the 'true' value is 50 km/h.
3/17/2008
12
Figure 6 Spreadsheet program for reconstruction of a
vehicle-pedestrian collision by the Searle method.
3.3
Known flying and throw distance
In a case when besides throw distance the flying distance of the pedestrian is also
known - for example from physical evidence - we can calculate both launch angle and
launch velocity. If s* is impact to ground contact distance then the flying distance is
s1 = s* − s0
(34)
From (10) and (9) one can express the flying time
t1 =
2 ( s1 sin θ + h cos θ )
g ( cos α cos θ − sin α sin θ )
(35)
and the pedestrian launch velocity
v0 = ( s1 + ph )
g (1 + tan 2 θ )
2 ( s1 tan θ + h )(1 − p tan θ ) 1 + p 2
(36)
Remark. The launch angle for minimal velocity of (36) is
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13
tan θ =
(s
2
1
+ h 2 )(1 + p 2 ) − ( h − ps1 )
s1 − ph
(37)
Since the pedestrian launch velocity is also given by (21) one can--by equating (36) and
(21)--find the compatibility equation
( s1 p + h ) (1 + μ tan θ )
2
2
= 4 ( p + μ )( s1 tan θ + h )(1 − p tan θ )
× (1 + p 2 ) ( sP − s0 − μ h )
(38)
From this either θ or μ can be calculated. In each of the cases the above relation
reduces to a quadratic equation.
Figure 7. Results of calculation of vehicle-pedestrian collision.
Known impact to ground contact distance with estimated μ .
Given μ . For a given μ one can arrange (38) to the following quadratic for unknown
launch angle θ
Aμ tan 2 θ + Bμ tan θ + Cμ = 0
(39)
where
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14
Aμ = −4 p 2 s1 ( s p − s0 ) − 4 μ ps1 ( sP − s0 − ph ) − μ 2 ( s1 − ph )
2
Bμ = 4 p ( s p − s0 ) ( s1 − ph ) + μ ⎡⎣ 4 ( s p − s0 ) ( s1 − ph ) − 2 ( s12 + 4 phs1 − p 2 h 2 ) ⎤⎦
−4 μ 2 h ( s1 − ph )
(40)
Cμ = 4 ph ( s p − s0 ) − ( s1 + ph ) + 4μ h ( sP − s0 − ph ) − 4μ 2 h 2
2
The spreadsheet program for reconstruction of vehicle-pedestrian collision based on
equation (39) is shown in Figure 7.
Given θ . On the other hand, for given θ one can arrange (38) to the quadratic equation
for an unknown μ
Aθ μ 2 + Bθ μ + Cθ = 0
(41)
where
Aθ = 4h ( s1 tan θ + h )(1 − p tan θ ) (1 + p 2 ) + ( s1 p + h ) tan 2 θ
2
Bθ = −4 ( s1 tan θ + h )(1 − p tan θ ) (1 + p 2 ) ( sP − s0 )
+4 ph (1 − p tan θ )( s1 tan θ + h ) (1 + p 2 ) h + 2 ( s1 p + h ) tan θ
2
Cθ = −4 p (1 + p 2 ) (1 − p tan θ )( s1 tan θ + h )( sP − s0 ) + ( s1 p + h )
(42)
2
Since the launch angle θ can practically only be estimated the case is interesting for the
extremes, which give maximum and minimum value of launch velocity. The first
maximum value is forward projection. So, if θ = 0 then from (41)
μ=
sP − s0 − ph ±
( sP − s1 − s0 )( sP + s1 − s0 + 2 ph )
2h
(43)
This formula has h, which is usually small compared to sP , in denumerator, so one can
expect that it is very sensitive to its variation. The other extreme for maximal projection
velocity is θ =
π
2
. In this case (41) reduces to
2
( s1 − ph ) μ 2 + 4 ps1 ( sP − hp ) μ + 4 p 2 sP s1 = 0
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(44)
15
However this equation does not have any positive roots for p > 0 .
Figure 8. Results of calculation of vehicle-pedestrian collision. Searle method with
known flying distance.
Now by taking tan θ = μ for minimum launch velocity then the quadratic equation (41)
transforms to the quadratic equation of the form
a0 μ 4 + a1μ 3 + a2 μ 2 + a3 μ + a4 = 0
(45)
where
a0 = ( s1 − hp )
2
a1 = 4h ⎡⎣(1 − p 2 ) s1 − ph ⎤⎦ + 4 ps1 ( sP − s0 )
a2 = 4h ⎡⎣(1 − p 2 ) s1 + ph ⎤⎦ + 2s1 ( s1 + 4 ph ) + 2 ( 2 − p 2 ) h 2
(46)
a3 = −4h ⎡⎣ ps1 + (1 − p 2 ) h ⎤⎦ ( sP − s0 ) + 4 ph 2
a4 = ( s1 + hp ) − 4 ph ( sP − s0 )
2
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16
The roots of this equation can be calculated numerically by the Newton iteration
method. The example of the spreadsheet program implementing the above formulas is
shown in Figure 8. The estimated boundary for data from Figure 1 are 45 km/h to 49
km/h. The first maximal value of vehicle impact velocity is calculated by using (29) and
calculated μ for minimum value of vehicle impact velocity. The horizontal launch
based on (43) gives the unrealistic friction coefficient 1.99 which leads to the upper
velocity limit 93 km/h. Here one can recommend that in general the maximal value of
vehicle impact velocity based on (43) can be used in the cases when the values of μ
are, say, μ < 1.1 .
4
VERIFICATION
4.1
Comportment with field data
The first comportment is made for test data given in [4]. The test was performed by a
vehicle of mass 1542 kg. Since the mass of the pedestrian was not reported a value of
80 kg was taken into calculation. Results of the calculation using the Searle method are
shown in Table 1. Note the huge discrepancy of calculated and measured vehicle impact
velocity for Test no 69, performed by an non-braking vehicle. All other results are
within acceptable limits of 15%.
Tabela 1. Results of calculation on test data [4]. Searle method.
( s0 = 2 m , η = 0.95 , h = 1.1m , μ = 0.6 ) *Brake off
Input
Test
no.
65
67
68
69*
72
vC 0
[ km/h ]
32.2
48.3
48.3
64.4
48.3
sP [ m ]
7.9
13.4
17.4
61.9
20.7
Output
vC 0min
vC 0max
vC 0
Err
27.96
38.27
44.28
86.51
48.69
32.61
44.63
51.64
100.89
56.78
30.29
41.45
47.96
93.70
52.74
5.95
14.18
0.70
-45.50
-9.18
[ km/h ] [ km/h ] [ km/h ] [%]
For verification of the reconstruction method where besides the pedestrian throw
distance are known the impact to ground is also known are given in [12]. The lack of
data--the result of a real pedestrian accident--is that there is no mass of vehicle and
pedestrian included in the report and that the vehicle impact velocity is below 36 km/h.
The mass 1500 kg for vehicle and 80 kg for pedestrian was thus assumed in calculation.
Results of comportment are shown in Table 2. It can be seen from the table that in trail
case 1 the calculated velocity overestimates the impact velocity by about 25 %, and in
case 2 the calculated values underestimate the velocity by about 50 %. In the other four
cases the calculated impact velocity is between calculated limits within an error
maximum of 13 %.
3/17/2008
17
Tabela 2. Comportment with field data [12].
Values marked with * are not included in calculation of vC 0max . (η = 1 , s0 = h = 1m )
Input
Trail
No.
vC 0
[ km/h ]
1
3
4
10
11
14
4.2
26.99
25.13
20.19
20.28
27.85
32.13
Output
s* [ m ]
sP [ m ]
μmin
μmax
8.23
2.59
3.53
3.65
5.52
8.24
10.38
5.42
7.41
4.73
8.52
8.51
0.50
0.13
0.19
0.34
0.35
0.71
1.70*
0.15
0.26
0.55
0.76
2.76*
vC 0min
vC 0max
vC 0
Err
31.67
12.30
17.83
17.08
25.09
30.13
35.38
13.37
21.28
22.29
38.01
36.96
33.53
12.84
19.56
19.69
31.55
33.55
24.21
48.93
3.15
2.93
13.29
4.40
[ km/h ] [ km/h ] [ km/h ] [ %]
Comportment with full scale numerical simulation
The full scale numerical simulation of vehicle-pedestrian collision was done by the PCCrash 7.1 computer program. The following data was used:
•
•
•
•
•
•
•
•
vehicle bumper height: 0.5 m
vehicle front height: 0.8 m
distance from vehicle front to windshield: 1.02 m
vehicle mass: 1460 kg.
coefficient of tire-road friction: 0.8
coefficient of car-pedestrian friction: 0.2
coefficient of road-pedestrian friction: 0.6
coefficient of restitution for pedestrian impact: 0.1
The first numerical experiment was conducted for a pedestrian of mass 80 kg and height
1.83 m. The path and speed of COG of pedestrian torso for various impact speeds are
displayed in Figure 9 and numerical values are given in Table 2.
70
3.0
60
2.5
50
Vp [km/h]
y
2.0
1.5
40
70
30
70
1.0
20
0.5
30
40
50
60
50
20
60
40
30
10
20
0.0
0
0
5
10
15
20
x
25
30
35
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
t [s]
Figure 9. Path and time dependence of velocity for pedestrian torso COG
for various vehicle impact speed. Pedestrian height is 1.83 m and mass 80 kg.
(scales of x and y are different)
3/17/2008
18
For comportment of results of numerical simulation with PC-Crash and the present
model the calculated values given in Table 3 are taken as input to the present
spreadsheets programs. Results of calculations are shown n Tables 4, 5, 6, 7 and 8. The
results of analysis of throw distance are within 20%. Among reconstruction methods the
Searle method is dominant with maximum error in impact vehicle velocity of about 6%
(Table 6). When impact to ground distance is included in input data the error raises to
about 9 % (Table 7). When using the Searle method with known impact to ground
distance the error raises to about 20 % (Table 8).
Tabela 3. Results of simulation of vehicle-pedestrian collision with PC-Crash.
for pedestrian of height 1.83 m and mass 80 kg
vC 0
vP 0
20
30
40
50
60
70
17.30
24.96
34.99
44.77
55.46
69.08
[ km/h ] [ km/h ]
θ ⎡⎣ 0 ⎤⎦
t0 [ s ]
-44.16
-6.10
3.52
7.42
9.88
14.16
0.79
0.40
0.27
0.22
0.20
0.20
s0 [ m ] h [ m ]
2.72
2.31
2.07
2.14
2.37
2.88
0.39
1.01
1.08
1.15
1.20
1.35
t* [ m ]
1.00
1.05
1.16
1.26
1.51
s* [ m ] sP [ m ]
5.78
8.62
11.76
15.20
20.71
7.56
11.85
16.87
23.20
34.25
η
0.92
0.87
0.92
0.95
0.97
1.04
Tabela 4. Results of calculation of vehicle-pedestrian collision with present method
for pedestrian of height 1.83 m and mass 80 kg
Input
vC 0
[ km/h ]
20
30
40
50
60
70
η
θ ⎡⎣ 0 ⎤⎦
t0 [ s ]
0.92
0.87
0.92
0.95
0.97
1.04
-44.16
-6.10
3.52
7.42
9.88
14.16
0.79
0.40
0.27
0.22
0.20
0.20
Output
s0 [ m ] h [ m ]
2.72
2.31
2.07
2.14
2.37
2.88
0.39
1.01
1.08
1.15
1.20
1.35
vP 0
[ km/h ]
t* [ m ]
24.47
34.89
45.03
55.18
69.02
0.79
0.80
0.9
1.03
1.39
s* [ m ] sP [ m ]
4.94
7.23
10.53
14.91
24.96
6.39
11.26
18.02
26.72
42.60
Tabela 5. Comportment of results from Table 3 and 4.
Relative error in % for various variables
vP 0
1.96
0.29
0.58
0.50
0.09
3/17/2008
t*
s*
sP
21.00
23.81
22.41
18.25
7.95
14.53
16.13
10.46
1.91
20.52
15.48
4.98
6.82
15.17
24.38
19
Tabela 6. Results of simulation of vehicle-pedestrian collision
for pedestrian of height 1.83 m and mass 80 kg. Searle method. ( s0 = 2 m , η = 0.95 ,
h = 1.1m , μ = 0.6 )
Input
vC 0
[ km/h ]
sP [ m ]
30
40
50
60
70
7.56
11.85
16.87
23.20
34.25
Output
vC 0min
vC 0max
vC 0
Err
26.03
35.65
44.33
53.30
66.10
30.36
41.57
51.70
62.15
77.08
28.20
38.61
48.02
57.73
71.59
6.02
3.48
3.97
3.79
2.27
[ km/h ] [ km/h ] [ km/h ] [%]
Tabela 7. Results of simulation of vehicle-pedestrian collision.
for pedestrian of height 1.83 m and mass 80 kg. Known impact to ground distance.
( s0 = 2 m , η = 0.95 , h = 1.1m , μ = 0.6 )
Input
vC 0
[ km/h ]
30
40
50
60
70
Output
s* [ m ] sP [ m ]
7.56
11.85
16.87
23.20
34.25
5.78
8.62
11.76
15.20
20.71
vC 0
Err
29.14
37.51
45.94
55.16
68.53
2.95
6.64
8.84
8.77
7.15
[ km/h ] [ %]
Tabela 8. Results of simulation of vehicle-pedestrian collision for pedestrian of height
1.83 m and mass 80 kg. Searle method. ( s0 = 2 m , η = 0.95 , h = 1.1m )
Input
vC 0
[ km/h ]
30
40
50
60
70
s* [ m ] sP [ m ]
5.78
8.62
11.76
15.20
20.71
7.56
11.85
16.87
23.20
34.25
Output
μ
0.37
0.42
0.44
0.43
0.42
vC 0min
vC 0max
vC 0
Err
22.95
32.32
40.70
48.73
59.85
24.48
35.02
44.42
53.13
65.01
23.715
33.67
42.56
50.93
62.43
20.95
15.83
14.88
15.12
10.81
[ km/h ] [ km/h ] [ km/h ] [%]
The second numerical experiment was conducted for vehicle impact velocity 60 km/h
with various pedestrians. The following pedestrian height/mass pairs were used for
calculation: 1.2/25, 1.4/40, 1.6/60, 1.83/80. The results of the calculations are shown in
Figure 10 and results of calculation are given in Table 9. Note that the pedestrian of
height 1.25 m is subject to forward projection while all others follow wrap trajectory.
3/17/2008
20
The comportment with results of calculation shows similar discrepancies as in the
previous case (Table 10 and 11).
3.0
80
70
2.5
1.2
60
2.0
Vp [km/h]
y
1.4
1.5
1.6
50
40
1.2
30
1.0
1.4
1.83
20
0.5
1.6
10
1.83
0.0
0
0
5
10
15
20
25
30
35
40
0.0
0.5
1.0
1.5
2.0
x
2.5
3.0
3.5
4.0
t [s]
Figure 10. Path and speed of COG of pedestrian torso for various height of pedestrian.
Vehicle impact speed 60 km/h.
Tabela 9. Results of simulation of vehicle-pedestrian collision with PC-Crash.
for vehicle collision velocity 60 km/h for various pedestrians
hP [ m ]
1.25
1.40
1.60
1.83
vP 0
[ km/h ]
70.39
63.38
57.81
55.46
θ ⎡⎣ 0 ⎤⎦
t0 [ s ]
12.94
8.74
12.15
9.88
0.22
0.21
0.20
0.20
s0 [ m ] h [ m ]
3.74
3.18
2.60
2.37
1.49
1.24
1.27
1.20
t* [ m ] s* [ m ] sP [ m ]
1.34
1.18
1.33
1.26
21.96
17.24
16.76
15.20
36.30
30.55
25.61
23.20
η
1.19
1.09
1.00
0.97
Tabela 10. Results of simulation of vehicle-pedestrian collision with present method.
for various pedestrian at 60 km/h
Input
hP [ m ]
η
θ ⎡⎣ 0 ⎤⎦
t0 [ s ]
1.25
1.40
1.60
1.83
1.19
1.09
1.00
0.97
12.94
8.74
12.15
9.88
0.22
0.21
0.20
0.20
Output
s0 [ m ] h [ m ]
3.74
3.18
2.60
2.37
1.49
1.24
1.27
1.20
vP 0
[ km/h ]
t* [ m ]
70.2
63.66
57.63
55.90
1.37
0.85
1.16
1.04
s* [ m ] sP [ m ]
25.66
17.97
17.58
15.16
44.36
34.88
29.89
27.34
Tabela 11. Comportmant of results between Table 9 and 10.
Relative error in % on various variables
3/17/2008
vP 0
t*
s*
sP
0.27
0.44
0.31
0.79
2.24
27.97
12.78
17.46
16.85
4.23
4.89
0.26
22.20
14.17
16.71
17.84
21
5
CONCLUSION
The vehicle-pedestrian collision is a complicated event which can not be exactly
modeled. The simple model present in this paper is comparable with the empirical
models and tested full scale model within an error of, say, 20%. The model has the
advantage over empirical models when the road has a grade or when the pedestrian
impact to ground contact distance is available as data. The model can therefore be used
for reconstruction purposes; however, one should be aware of its limitations and
accuracy.
All
present
spreadsheet
programs
are
available
from
www.fpp.edu/~milanb/pedestrian
REFERENCES
[1]R.Aronberg. Airborne Trajectory Analysis Derivation for Use in Accident
Reconstruction. SAE Technical Paper Series 900367.
[2]J. C. Collins. Accident Reconstruction. Charles C. Thomas Publisher, Springfield,
Illinois, USA, 1979, pp. 240-242.
[3]R.M.Brach, R.M.Brach. Vehicle Accident and Reconstruction Methods. SAE
International, 2005
[4]J. J. Eubanks, W. R. Haight. Pedestrian Involved Traffic Collision Reconstruction
Methodology. SAE Technical Paper Series 921591.
[5]T.F.Fugger Jr., B.C.Randles, J.L.Wobrock, J.L.Eubanks. Pedestrian Throw
Kinematics In Forward Projection Collisions. SAE Technical Paper 2002-010019.
[6]I. Han, R. M. Brach. Throw Model For Frontal Pedestrian Collisions. SAE
Technical Paper Series 2001-01-0898.
[7]A.Happer, M.Araszewski, A.Toor, R.Overgaard, R.Johal. Comprehensive
Analysis Method for Vehicle/Pedestrian Collisions. In: Accident
Reconstruction, SAE SP-1491, 2000
[8]R.Limpert. Motor Vehicle Accident Reconstruction and Cause Analysis. 5th
Edition. Lexis Publishing, Charlottesville, Virginia, USA, 1999. pp 539-554
[9]R.R.McHenry. B.G.McHenry. Occupant Trajectory. In: McHenry Accident
Reconstruction 2000. McHenry Software, Cary, NC, USA, 200, pp.95-102.
[10]A.Moser, H.Steffan, G.Kasanicky. The Pedestrian Model in PC-Crash – The
Introduction of Multi Body System and its Validation. SAE Technical Paper
Series 1999-01-0445
[11]A.Moser, H.Hoschopf, H.Steffan, G.Kasanicky. Validation of the PC-Crash
Pedestrian Model. SAE Technical Paper Series, 2000-01-0847
3/17/2008
22
[12]B.C.Randles, T.F.Fugger, J.J.Eubanks, E.Pasanen. Investigation and Analysis of
Real-Life Pedestrian Collisions. SAE Technical Paper Series, 2001-01-0171
[13]C.G.Russell. Equations & Formulas for the Traffic Accident Investigators and
Reconstructionist. Lawyers & Judges Publishing, Tuscon, AZ, USA, 1999
[14]H.Schnider. G.Beier. Experiment and Accident: Comparison of Dummy Tests
Results and Real Pedestrian Accidents. SAE Tecnhical Paper Series 741177.
[15]J.A.Searle, A.Searle. The Trajectory of Pedestrians, Motorcycles, Motorcyclists,
etc. Following a Road Accident. SAE Technical Paper Series 831622
[16]J.A.Searle. The Physics Of Throw Distance In Accident Reconstruction. SAE
Technical Paper Series 930659.
[17]A.Toor, M.Araszewski. Theoretical Vs. Empirical Solutions For
Vehicle/Pedestrian Collisions. 2003-01-0883, Accident Reconstruction 2003,
Society of Automotive Engineers, Inc., Warrendale, PA, 2003, pp. 117-126.
[18]D.P.Wood. Application Of A Pedestrian Impact Model To The Determination Of
Impact Speed. SAE Technical Paper Series 910814.
3/17/2008
23
APPENDIX A - Confidence analysis for Eq (21)
To estimate the errors in parameters that influence the pedestrian launch velocity Eq
(21) is viewed as a function of six variables
vP 0 = f ( h, sP , s0 , p, μ , θ )
(47)
The excepted value of this function is
vP 0 = f ( h , sP , s0 , α , μ ,θ
)
(48)
where h , sP , s0 , α , μ , θ is the given or measured values of variables. The coefficient of
variation in the result vP 0 is estimated as
cv =
( f h ch )
2
(
+ f sP csP
) +( f
2
c
s0 s0
) +( f c ) +( f c ) +( f c )
2
2
p p
2
θ θ
2
(49)
μ μ
where ck are the estimated coefficients of variation in k-th variable and f k =
ξ k ∂f
are
f ∂ξ k
dimensionless sensitivity coefficients whose explicit expressions are
fh =
h ⎛ ∂f ⎞
μh
⎜ ⎟=−
f ⎝ ∂h ⎠
2 ( sP − s0 − μ h )
(50)
f sP =
sP
f
⎛ ∂f ⎞
sP
⎜
⎟=
⎝ ∂sP ⎠ 2 ( sP − s0 − μ h )
(51)
f s0 =
s0 ⎛ ∂f ⎞
s0
⎜
⎟=−
f ⎝ ∂s0 ⎠
2 ( sP − s0 − μ h )
(52)
fp =
p ⎛ ∂f ⎞ p
1− μ p
⎜ ⎟=
f ⎝ ∂p ⎠ 2 (1 + p 2 ) ( p + μ )
(53)
θ ⎛ ∂f ⎞
tan θ − μ
⎜
⎟ =θ
f ⎝ ∂θ ⎠
1 + μ tan θ
(54)
μ ⎛ ∂f ⎞ μ ⎣⎡ sP − s0 − ( 2μ + p ) h ⎤⎦ − ⎡⎣( 2μ + p )( sP − s0 ) − pμ h ⎤⎦ tan θ
⎜
⎟=
f ⎝ ∂μ ⎠ 2
( μ + p )( sP − s0 − μ h )(1 + μ tan θ )
(55)
fθ =
fμ =
3/17/2008
24