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Greyhound racing ideal trajectory 
path generation for straight 
to bend based on jerk rate 
minimization
Md. imam Hossain✉, David eager & paul D. Walker
this paper presents methods for modelling and designing an ideal path trajectory between straight 
and bend track path segments for racing greyhounds. to do this, we numerically generate clothoid and 
algebraic curve segments for racing quadrupeds using a sequential vector transformation method as 
well as using a helper equation for approaching ideal clothoid segments that would respect greyhound 
kinematic parameters and boundary conditions of the track. further, we look into the limitations of 
using a clothoid curve for racing dog track path design and propose a smooth composite curve for 
track transition design which roughly maintains G3 curvature continuity for smooth jerk to overcome 
limitations of a clothoid transition. finally, we show results from race data modelling and past injury 
data, which provide a strong indication of clothoid curve segments improving the dynamics and safety 
of racing greyhounds while reducing injuries.
In the greyhound racing sports industry, injuries to dogs are highly prevalent1. The sport has grown exponentially 
in recent years due to live wagering accessibility and various revenue sharing programs2. As a result, it has become 
evident that better track design is required to reduce the likelihood of racing greyhound injuries at the tracks. 
Observations3 confirmed that in greyhound racing congestion occurs at the entrance to the first bend. Also, 
researchers theorized that a smooth-running path is required for curved track design without which quadrupeds 
are more likely to lose coordination at specific transitions4. Similarly, it was shown that various track shapes have 
considerable effects on greyhound injury rates indicating track curvature influences5. When it comes to track 
shapes and smooth paths, transition curves are an essential part of path design in many areas such as road design 
and train track designs6. Similarly, transition curves help reduce disturbances in quadruped gait symmetry4. This 
is because, quadrupeds are subject to a centrifugal force which induces an outward pull on the curved track path, 
forcing quadrupeds to deviate from navigating the track path4. Theoretically, a transition curve would also assist 
navigation of the body around the curved path even if it is not sufficiently banked7.
Clothoid transition curves are extensively found in road and rail track designs such as it was found from the 
analysis that the Tokaido Shinkansen high-speed rail uses a 600 m clothoid transition in one of the 2.1 km radius 
bends to achieve a maximum travelling speed of 270 km/hr with minimal track path camber. Clothoid curves 
are essential for generating continuous curvature paths with straight and perfect arc segments8. This is achieved 
by linking constant curvature segments with clothoid segments8. For example, a clothoid can join a line and a 
circle with G2 curvature continuity where both the tangent vector and curvature at the line-circle intersection 
are continuous9.
The performance of clothoid and other transition curves trajectories can be effectively analyzed by looking 
into their curvature profiles. Curvature is an import factor in trajectory designs as it affects the maximum speed 
a vehicle can travel without skidding or whether the pilot of an aeroplane suffers blackout as a result of g-forces10. 
Also, a valid curve is one which respects upper bound curvature constraints set by kinematics properties of mov-
ing bodies11.
In this paper, we illustrate numerical methods to approach clothoid curves and other transition curves to 
model and generate smooth running paths for greyhound racing. We also show galloping greyhound trajectory 
performance, relating it to injury rates and track shapes. The paper is organized as follows. Sections one and two 
Faculty of Engineering and Information Technology, University of Technology Sydney, Broadway, 2007, NSW, 
Australia. ✉e-mail: MDImam.Hossain@uts.edu.au
open
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describe greyhound trajectory and trajectory dynamics, respectively. In sections three and four, clothoid transi-
tion generation and approaching ideal clothoid transitions for racing greyhounds are presented. Ideal transition 
curves developed for galloping greyhounds are presented in section five. Finally, section six evaluates racing 
greyhound trajectory performance for existing tracks.
trajectory of a racing greyhound. 
In the greyhound racing industry, the trajectory of a racing greyhound 
is oftentimes overlooked for track designs and injury prevention measures despite its significance in dynamic out-
comes for the animal. One key parameter which determines the trajectory of a racing greyhound is the curvature 
of its running path. Curvature, κ(s), is the change of heading relative to distance travelled8. Also, the curvature 
can be thought of as the inverse of the radius of curvature, which denotes the turning radius at any point in 
the path12. Furthermore, a related variable, sharpness α, is the change of curvature for distance travelled which 
also forms the basis for constructing continuous curvature path trajectories8. While designing a path, curvature 
change must remain smooth throughout the trajectory of a moving object as the centrifugal acceleration expe-
rienced is directly proportional to the path curvature12. As a result, in trajectory generation for motion planning 
the smoothness of a trajectory is directly related to the smoothness of its curvature profile13. Likewise, for the path 
to be feasible, it must conform to continuous position, heading, as well as curvature at all points8. Now, if the path 
of the trajectory is defined by a function y = f(x) then the radius of curvature ρ at any given point can be found 
from the following equation:14
ρ =



+

( ) 
1,
,
(1)
dy
dx
d y
dx
2 3/2
2
2
Then, the curvature is,
κ
ρ
= 1
(2)
However, if the path of the trajectory cannot be translated into a continuous function, then any three adjacent 
data points lying on the path can be used to calculate the radius of curvature at any given point using the circum-
radius of a triangle formula15. The circumradius formula (3) provides the radius of the circumcircle of a triangle 
which is inherently cyclic16,17. The triangle is defined by the adjacent data points lying on the path, as shown below 
in Fig. 1.
ρ =
R =
abc
4A
(3)
Where, a, b, and c denote the three sides of a triangle defined by three adjacent data points on the path, and A is 
the area of the triangle.
An ideal racing greyhound trajectory would involve looking into two major control factors, greyhound head-
ing which deals with curvature and sharpness of the running path and greyhound kinetics which deals with the 
acceleration/deceleration of a greyhound.
Racing greyhound trajectory dynamics. 
The trajectory of a racing greyhound induces dynamic grey-
hound conditions such as centrifugal acceleration, centrifugal jerk, and greyhound heading yaw rate. It also influ-
ences racing greyhound states such as leaning, braking forces as a result of ground reaction force, centripetal 
force, stride frequency, and stride length. A sharp discontinuity in any of the dynamic conditions would result in 
a significantly unpredictable dynamic imbalance for a racing greyhound. During racing, such a situation would 
put a greyhound in considerably uncontrollable situations where there are already racing situations such as con-
gestion and tight bends with variable track cross-falls along the width of the tracks. To design a trajectory for 
Figure 1. Calculating an arbitrary path’s instantaneous radius of curvature using data points lying on the path.
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racing greyhounds which would meet the specific track design goals, the trajectory performance can be evaluated 
by looking into two dynamic factors of racing greyhounds namely: centrifugal jerk and yaw rate. These two fac-
tors are highly sensitive to the trajectory performance of an object in motion as both are related to the radius of 
curvature of the trajectory.
Modelling centrifugal jerk. 
Jerk is the rate change of acceleration. Like centrifugal acceleration, the effect 
of jerk is also experienced in the body18. Essentially, jerk is the increasing or decreasing of the force in the body18. 
Eager, et al.18 explains the use of jerk as a measure of safety in various disciplines including mechanical engineer-
ing and civil engineering as well as its application in greyhound racing. Lower jerk values are essential as they 
indicate that the change in centrifugal acceleration is minimal for a greyhound while it is navigating its trajectory. 
For, humans, there are derived maximum acceleration change and corresponding time duration for this change 
for roller coaster rides18. No such derivations exist for racing quadrupeds yet. As a result, modelling of the centrif-
ugal jerk for racing animals becomes an essential part of optimum trajectory generation.
The first step to centrifugal jerk analysis is finding the instantaneous radius of the trajectory or calculating the 
radius of curvature at all points in the trajectory path. For cars and trains, calculation of the instantaneous radius 
of curvature is found by using geometric primitives and splines and approximated using continuous functions as 
their respective heading change is continuous. However, for greyhounds, the heading change by the greyhound 
is not continuous and expected to occur at every stride. Furthermore, greyhounds are known to have a stride 
frequency greater than 3 Hz19. This implies a greyhound would change its heading if required more than three 
times a second where the magnitude of each heading change could vary from stride to stride. Therefore, we can 
gather all the location coordinate data for strides of a single racing greyhound and calculate the instantaneous 
radius of curvature ρ of the racing greyhound using either the circumradius formula (3) or the perpendicular 
bisectors method. Then, we can calculate the racing greyhound’s instantaneous centrifugal acceleration from the 
instantaneous speed and radius of curvature. Finally, the instantaneous jerk is derived from the rate of change in 
the centrifugal acceleration.
Modelling yaw rate. 
The yaw rate is the rate change of heading or turning. It relates a racing greyhound’s 
angular displacement to its forward speed. It also provides an indication of the stability of the path a racing grey-
hound is taking. For example, it was shown from the race kinematic simulation and race data that racing grey-
hounds’ yaw rate is not smooth immediately after jumping out from the starting boxes20. For a constant radius 
curve path, the yaw rate is simply the radius of curvature over speed (4) which is used for calculating a vehicle’s 
momentary radius of turn. For a racing greyhound trajectory, the yaw rate can be directly related to the sum of 
the lateral forces. A lower yaw rate would indicate lower lateral forces such as centrifugal force and frictional force 
acting on a greyhound. To maintain a smooth trajectory, a racing greyhound needs to maintain a smooth yaw 
rate. However, since the speed of the racing greyhound varies over time as well as the lateral frictional forces from 
the traction ground, maintaining a smooth yaw rate would also require careful balancing of these two factors 
while designing tracks to facilitate a smooth trajectory for a racing greyhound.
ψ
ρ
 =
s
(4)
clothoid track segments for deriving natural racing greyhound trajectory. 
The clothoid segment 
is a curve known for its curvature being proportional to its length21. This property of the clothoid is useful as it 
allows the gradual development of centrifugal acceleration or can act as centrifugal acceleration easement, which 
significantly reduces the risk of accidents occurring12. Recent research shows that there are different types of 
curves already developed, which can be used as centrifugal acceleration easement curves12. For example, Quintic 
polynomial and B-splines functions are computationally less expensive and also able to provide curvature con-
tinuity for curve design13. However, the drawbacks of these functions are complex curvature profiles which are 
hard to follow as they are not necessarily smooth13. This is where clothoids are useful as their curvature profile is 
a straight line making them easy to follow13. Furthermore, clothoids are characterized by a linear curvature, allow 
minimizing of curvature variation where piecewise clothoids exhibit excellent smoothness properties22. For these 
fundamental reasons, currently clothoids are extensively found in road design and robot path planning to achieve 
smooth transitions in the trajectories22.
We found that clothoids are essential at the race track not only for developing smooth path trajectories but also 
for reducing the likelihood of certain types of race dynamics hazards. From the race videos, it was noted that a 
greyhound is more likely to change lanes to a higher radius upon entering the first bend. This could be due to the 
track bend lacking adequate transition to accommodate for greyhound natural instantaneous yaw rate change and 
leaning rate change limits. As a result, the prospect of the greyhound bumping into another nearby greyhound 
increases significantly. This specific race dynamic outcome can be reduced or nearly eliminated if the track path 
has clothoid segments which match natural greyhound heading turning rate change limits.
Generating clothoid segments for track path design. 
There are many methods available for comput-
ing the clothoid. Most methods involve approximations to the clothoid21. For example, it can be approximated 
by high degree polynomial curves23, such as by an S-power series24 as well as by an arc spline9. Also, continued 
fractions and rational functions are commonly used for approximations9. A more recent development in the 
spline primitives found in much computer-aided design software makes it easy to approximate a clothoid while 
respecting boundary conditions such as curvature and tangent continuity. Also, spline primitives are known for 
good and fast controllability with positional and tangential constraints making them ideal for various applica-
tions22. Each of the methods available results in different degrees of accuracy and may not be suitable for efficient 
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greyhound track path design purposes. This is mainly due to less controllability in generating a clothoid accord-
ing to greyhound kinematics. Moreover, to accommodate the clothoid segment into the path design, a coordinate 
respecting system must be incorporated or derived from the existing clothoid methods which respects different 
design boundary conditions.
computing clothoid curves using existing methods. 
The most common method of computing a 
clothoid can be found in its definition in terms of Fresnel integrals24 where it is computed using the Fresnel sine 
and cosine functions as shown in Eqs. (5a) and (5b) and using some forms of Taylor series expansions on the 
functions which converge for an independent variable8. Series expansion functions are extensively used because 
the clothoid defining formulas are transcendental functions21. The parametric plot of Fresnel sine and cosine 
functions provides coordinate values of the clothoid curve. However, this does not respect any form of unit scal-
ing or boundary conditions as well as not allowing computing the clothoid for a specific rate of change of cur-
vature, sharpness or smoothing applications. Similarly, Eqs. (6a) and (6b) give an approximation of Fresnel sine 
and cosine functions which converge for all independent variables x. Another common method involves utilizing 
auxiliary functions8, as shown in Eqs. (7a) and (7b).
( ) = ∫
S x
t dt
sin( )
(5a)
x
0
2
( ) = ∫
C x
t dt
cos( )
(5b)
x
0
2
∫
∑





 =
=
−
+
+
=
∞
+
S x
t dt
x
n
n
sin( )
(
1)
(2
1)!(4
3)
(6a)
x
n
n
n
0
2
0
4
3
∫
∑





 =
=
−
+
=
∞
+
C x
t dt
x
n
n
cos( )
(
1)
(2 )!(4
1)
(6b)
x
n
n
n
0
2
0
4
1
Equations (5a) and (5b) then can be written in the auxiliary function form, as shown below:8
π
π
=
+ 






 − 







C x
f x
x
g x
x
( )
1
2
sin
2
cos
2
(7a)
2
2
π
π
=
− 






 − 







S x
f x
x
g x
x
( )
1
2
cos
2
sin
2
(7b)
2
2
Where auxiliary functions f and g are defined as:
π
π
= 


−












 − 


−













f x
S x
x
C x
x
( )
1
2
cos
2
1
2
sin
2
(8a)
2
2
π
= 


−












 + 


−










π



g x
C x
x
S x
n
x
( )
1
2
cos
2
1
2
si
2
(8b)
2
2
Likewise, for auxiliary function definition of the clothoid a good rational approximation to compute the 
clothoid is using the following auxiliary functions8.
=
+
.
+
.
+
.
f x
x
x
x
( )
1
0 926
2
1 792
3 104
(9a)
2
=
+
.
+
.
+
.
g x
x
x
x
( )
1
2
4 142
3 492
6 670
(9b)
2
3
Moreover, recently, researchers developed more efficient numerical methods where one such method is using 
arc length parameterisation12. While analytical methods lack parameterisation for different application case sce-
narios researchers are becoming more reliant on developing numerical techniques for computing the clothoids.
A numerical approach for generating the clothoid curve transitions for racing greyhounds 
and other quadrupeds. 
It is evident that existing methods lack greyhound kinematic parameterisation 
for racing greyhound transition design purposes. A numerical method is generally preferred as a first approach 
for incorporating different parametrisation into the clothoid curves. To develop a numerical technique for the 
clothoid which incorporates greyhound kinematics variables, we looked into the characteristics of the mathemat-
ical model of the clothoid curve. A clothoid curve transition accomplishes a gradual transition from the straight 
to the circular curve of the constant radius where the curvature changes from zero to a finite value. As a result, 
the tangent vector ti, which lies on the clothoid curve, also gradually rotates from zero to a finite angle Fig. 2. 
Furthermore, let us assume a greyhound changes its heading with every stride as noted from the race data and 
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galloping gait of a greyhound. With these two crucial pieces of information relating to the clothoid curve tangent 
vector and the greyhound heading step-change length, we can apply vector transformation to generate a clothoid 
curve positional vector Pi Fig. 2. Now, we define the clothoid tangent vector as a function of greyhound stride 
length constant as denoted by transition segment length and a variable denoted by transition deflection angle. 
The transition deflection angle ai defines the local rotation of the clothoid curve tangent vector at a specific tran-
sition segment location i relative to the horizontal axis. Moreover, as a clothoid curve transition would gradually 
increase its curvature with constant curvature acceleration, the transition deflection angle ai is a function of the 
transition deflection angle acceleration constant. The transition deflection angle acceleration d defines the rate 
change of curvature per transition segment length of the clothoid curve, which essentially tells us how quickly the 
clothoid tangent vector rotation is accelerating. Finally, once the transition deflection angle is calculated for local 
ith transition segment, the clothoid curve positional vector can be calculated as shown in Fig. 2 and Eq. (11). To 
generate the entire clothoid curve for the specified number of transition segments by the constant n the process 
of translating and then rotating the clothoid tangent vector is iterated to get the clothoid positional vectors for all 
the transition segments. For example, Fig. 3 shows a clothoid curve generated using this method when transition 
segment length s equals 1 m, the number of transition segments n equals 250 and transition deflection angle 
acceleration d is 0.02 degrees.
d = transition deflection angle acceleration
ai = transition deflection angle relative to horizontal axis
s = transition segment length
n = number of transition segments
ti = transition tangent vector
i = transition segment number
=






=



×
×



t
f s a
cos a
s
sin a
s
,
( )
( )
(10)
i
i
i
i
=
=
+
−
−
P
f t P
P
t
( ,
)
(11)
i
i
i
i
i
1
1
Where,
= ∑
×
=
×
+
×
+
×
+ … +
×
=
a
d
i
d
d
d
i
d
1
2
3
i
k
i
1
And,
κ
×
∝
d
i
Figure 2. Racing greyhound clothoid path generation using numerical method parameterization.
Figure 3. A clothoid curve with curvature combs containing 250 single meter segments and with a turning 
acceleration of 0.02 degrees per segment.
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Where κ denotes the curvature of the clothoid curve.
Now, for instance, using the numerical method explained above to generate a clothoid curve transition for 
racing greyhounds with a transition exit radius of approximately 52 m and a total transition length of 45 m, we 
would have to consider the d constant to be 0.69 degrees per transition segment, the s constant to be 5 m (assum-
ing that average stride length of a greyhound is 5 m) and the n constant to be 9. The curvature and jerk results of 
this clothoid transition curve for racing greyhounds are shown in Fig. 4. The numerical calculation of ai and Pi is 
shown in Table 1.
Using this numerical method approach, we showed how an optimized clothoid curve transition could be 
determined numerically by tweaking curve generating factors. The controlling of initial values as set by d, s, and 
n allows generating any combination of clothoid curves as required for different kinematic path design goals.
Figure 4. A clothoid curve transition for racing greyhounds with a total 45 m transition length having a an 
approximately 52 m turning radius at the end of the transition.
i
ai (deg. per segment)
Pi X coordinate (m)
Pi Y coordinate (m)
1
0.69
5.00
0.00
2
2.07
10.00
0.06
3
4.14
15.00
0.24
4
6.9
19.98
0.60
5
10.35
24.95
1.20
6
14.49
29.87
2.10
7
19.32
34.71
3.35
8
24.84
39.43
5.01
9
31.05
43.96
7.11
Table 1. Numerically calculated values of ai and Pi variables for a clothoid curve.
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Designing ideal clothoid segments for racing greyhounds and other quadrupeds. 
When design-
ing clothoid segments, it is essential that greyhound heading is not changing at the maximum performable rate 
since such a heading would put a greyhound into a limit state turning while maintaining a high speed. An ideal 
clothoid segment would have continuous curvature to allow a greyhound to navigate the path with a minimal 
amount of veering effort. In the next section, we derive a helper equation which can be used for specifying ideal 
clothoid transitions as well as for modelling dynamics for racing greyhounds at the tracks.
Deriving an equation for exact clothoid requirements for racing greyhounds and other quadru-
peds. 
Equation (3) produced a relationship between greyhound kinematics such as heading turning angle accel-
eration and turning radius at the end of a natural clothoid transition. First, let’s assume for a clothoid transition a 
racing greyhound would pass ns number of strides with a constant s meter stride length. Now, if the total clothoid 
transition length is T meters, then the number of greyhound strides ns in a transition is given by Eq. (13). Again, 
since the length of the greyhound’s strides remains unchanged in the clothoid transition, the greyhound’s turning 
angle a in the last stride of the transition can be defined by Eq. (14) if the greyhound heading turning angle is 
accelerating with d degrees per stride. Now, to calculate a greyhound’s heading radius of turn R near the end of the 
clothoid transition using Eq. (3), we use Heron’s formula (17) to calculate the area of the triangle A (17) formed 
by last two greyhound strides s1 and s2. Furthermore, using the cosine rule we calculate the unknown side s of the 
triangle formed by the last two greyhound strides s1 and s2. Finally, by plugging in values for R and simplifying the 
equation, we reach a final equation form (18) which defines a racing greyhound’s turning radius R at the end of the 
clothoid transition in terms of transition length T, greyhound heading turning acceleration a and greyhound con-
stant stride length s. Consequently, as Eq. (18) relates greyhound heading turning parameters to clothoid transition 
parameters, which is useful for modelling and designing ideal clothoid transitions for racing greyhounds. In the next 
section, we show some of the design and modelling of the clothoid transitions using Eq. (18).
d = transition deflection angle acceleration (per stride)
a = deflection angle of greyhound heading for last greyhound stride
ns = total number of greyhound strides in the transition
s = length of a single stride
R = transition last stride turn radius
T = transition length
ns =
T
s
(13)
=
×
−
a
d
(ns
1)
(14)
=
+
−
−
s
s
s
2s s Cos a
(
180)
(15)
1
2
2
2
1 2
Where s1 and s2 are a racing greyhound’s last two strides in the transition.
=
+
+
p
s
s
s
2
(16)
1
2
Where p is semi-perimeter of the inscribed triangle (Fig. 1) in the circle formed by a racing greyhound’s last two 
strides s1 and s2.
=
−
−
−
A
p p
s
p
s
p
s
(
)(
)(
)
(17)
1
2
=






+
−









−



π
π
−
−
(
)
(
)
R
s
s
s
s
2
2
cos
2
2
cos
1
(18)
d
d
2
2
1
180
2
4
1
90
T
s
T
s
clothoid design for constant radius bend. 
Every track has a bend radius requirement as calculated from 
the physical infrastructure and design goals. If a track requires a 52 m radius bend at the end of the transition, 
then using Eq. (18), we find the following expected greyhound kinematics and transition design possibilities as 
shown in Table 2. It should be noted that there could be a large number of design outcomes for a single parameter 
design such as a design for a specific bend radius. The greyhound yaw rate at the entrance is simply the greyhound 
angular displacement rate change per stride times greyhound stride frequency. Also, in generating the folllowing 
results racing greyhound speed was assumed to be 19.5 m/s and stride frequency to be 3.5 Hz.
As can be seen from Table 2, each of the clothoid transition possibilities can be applied at different locations at 
the track based on the race requirements. For instance, the clothoid transition Design No. 3 can be applied at the 
home turn bend exit since the greyhound speed and stride length would be much lower making it possible for a 
greyhound to adopt to higher yaw rate and angular displacement acceleration path navigation.
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clothoid design for greyhound angular displacement rate change limits. 
For racing greyhounds 
known to have certain angular displacement rate change limits based on greyhound training and health back-
ground histories, using Eq. (18) we can enumerate possible clothoid designs options. For example, if the expected 
racing greyhounds have a maximum angular displacement rate change limit of 0.5 deg/stride2, then we can con-
sider the following clothoid transition design options as shown in Table 3.
As can be seen from Table 3, using greyhound angular displacement rate change as a design constraint exhibits 
more diverse clothoid transitions in terms of transition length and transition exit bend radius. Design No. 1 shows 
that it is possible to have a short transition for a larger radius bend. Likewise, design no. 4 portrays a long transi-
tion for smaller radius bend. As a result, the angular displacement rate based design approach provides excellent 
freedom in choosing clothoid transitions based on track requirements.
Modelling of racing greyhound jerk dynamics. 
It is possible to calculate jerk exhibited by clothoid 
transitions using Eq. (18). Since a clothoid has uniform curvature acceleration, the jerk produced by a clothoid 
remains the same for the entire length of the transition. So, we can find jerk value at any arbitrary location in a 
clothoid transition to find overall jerk for the transition. For example, if we are interested in the jerk at the end 
of a clothoid transition, first we would calculate radius value R for both T and T-s for the transition. Then, we 
would calculate corresponding centrifugal acceleration values. Finally, since the jerk is the change in centrifugal 
acceleration over time, we simply divide the difference of centrifugal accelerations by the time taken by one 
stride. Table 4 presents some example calculations of racing greyhound jerk values for various clothoid transition 
designs considering instantaneous greyhound speed to be 19.5 m/s:
An approach to designing ideal transitions for racing greyhounds. 
As can be seen from Fig. 4, it 
was found that racing greyhound clothoid transition curves have a significant flaw. Although the development of 
the curvature is gradual as can be seen from the curvature plot of Fig. 4, the jerk profile is not smooth and almost 
jumps instantaneously from zero to a higher value (Fig. 4). This is important, as such a dramatic change of jerk 
would impose a high energy release in a short time resulting in considerably unstable conditions for greyhounds 
navigating in and out of the transitions. Furthermore, the clothoid curve generation for racing greyhounds using 
the numerical method above showed that regardless of transition curve length jerk goes through a step change 
within one transition segment or one racing greyhound stride. Consequently, a clothoid curve transition was 
deemed not to be an ideal fit for racing greyhound track path designs.
The clothoid transition curve does not maintain a smooth jerk initiation for a racing greyhound. Hence the 
curve can only be considered G2 continuous with matching curvature at the entrance and exit of the transition 
curve. This imposes several disadvantages in racing greyhound race dynamics at the tracks. For example, we can 
Design No.
Clothoid transition 
length, T (m)
*Greyhound yaw rate at the 
transition entrance (deg/s)
Greyhound angular displacement rate 
change per stride, d (deg/stride2)
Greyhound expected 
constant stride length, s (m)
1
75
1.2969
0.393
5.0
2
60
1.6533
0.501
5.0
3
40
2.3825
0.722
4.8
4
60
1.7952
0.544
5.2
Table 2. Clothoid transition options for 52 m radius bend.
Design No.
Clothoid transition 
length, T (m)
Radius of constant bent 
at the transition end (m)
Greyhound expected 
constant stride length, s (m)
1
45
71.6
5.0
2
50
70.5
5.25
3
60
52.0
5.0
4
70
53.7
5.5
Table 3. Clothoid transition options for racing greyhound accelerating with a maximum angular heading 
turning of 0.5 degrees per stride2.
Design 
No.
Clothoid transition 
length, T (m)
Radius of constant bend at 
the transition end (m)
Greyhound expected 
constant stride length, s (m)
Greyhound angular displacement rate 
change per stride, d (deg/stride2)
Absolute 
jerk (m/s3)
1
45
71.6
5.0
0.5
2.59
2
50
70.5
5.3
0.5
2.35
3
60
52.0
5.0
0.5
2.59
4
70
53.7
5.5
0.5
2.14
Table 4. Clothoid transitions racing greyhound’s jerk modelling using Eq. (18).
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break down the disadvantages into two main categories, namely clustering related problems and path smoothing, 
where each is entangled with the other. The clustering of racing greyhounds is a common issue during races. This 
happens mainly due to single lure convergence as a result of the number of following galloping greyhounds. A 
tight convergence of the racing greyhound pack is noticeable at race tracks in the locations where track path cur-
vature change is sudden and abrupt. As clustering is a precursor to various dynamics unstable conditions such as 
bumping of one greyhound by another, maintaining a smooth path profile such as G3 curvature continuity where 
the clustering occurs, becomes vital. As greyhounds follow the racing lure, they occupy different lanes such that 
they have different path radii and tend to cut corners forming various individual transitions into the bend which 
are all unique. A G2 curvature continuity as found in the clothoid transitions where the rate of change of the jerk 
is not smooth would induce all the racing greyhounds following the lure to follow one unique transition into the 
bend to keep instantaneous jerk to the minimum. This is not feasible.
To overcome the limitations of clothoid transitions, we applied the numerical method of generating clothoid 
curves discussed in the previous section to develop moderate G3 curvature continuity transition curves for racing 
greyhounds. Also, two different transition curve configurations were selected for generating the curves as these 
configurations best match the many current tracks found in Australia in terms of real estate requirements. The 
configurations are a 45 m transition with transition end radius of 52 m and a 75 m transition with transition end 
radius of 70 m.
First, we assume ai = X and plot for different X expressions to derive different curves where the curvature 
results for the curves are shown in Figs. 5 and 6. The X expression defines the nature of curvature function as the 
curve length increases from the origin. As seen from the plots (Figs. 5 and 6), when the X expression is linear it is 
a clothoid transition where the jerk is initiated immediately within one transition segment for both 45 m and 75 m 
transition configurations. To get G3 curvature continuity curves, we tried X0.6, X1.5, X2, and ((1.2)X−1) expres-
sions. As can be seen from the plots, all the curves except the clothoid curve X and X0.6 curve maintain a moderate 
G3 curvature continuity with a smooth jerk profile. However, as X expressions are in power and logarithmic func-
tion form for X0.6, X1.5, X2, and ((1.2)X−1) these curves result in higher jerk in the second half of the transition. 
This suggested that X1.5, X2, and ((1.2)X−1) curves could be used to develop a G3 curvature continuity transition 
curve for racing greyhounds if the jerk could be maintained in the second half of the transition. Thus, we decided 
to use these curves as auxiliary curves which would provide smooth jerk initiation for the transition. However, 
compared to other curves, the overall jerk and smoothness performance of the X1.5 is optimum.
Here, we generate composite transition curves with various degrees of G3 curvature continuity for racing 
greyhound ideal path design. Each composite transition curve generated combines the X1.5 curve as an auxiliary 
curve and a clothoid curve as the main curve. So, the overall transition curve generating function can be consid-
ered as a piecewise function shown in Eq. (19) where the auxiliary curve function g is applicable until q transition 
segment is reached.






=



<
f x
g x
z x
x
q
( )
( )
if
otherwise
(19)
Where,
x =
y d i
( , )
Figure 7 shows curvature and jerk results for four different composite curves as ideal transitions for racing 
greyhounds, plotted using the numerical method explained in the earlier section, the configurations for these 
composite curves are given in Table 5.
As can be seen from Fig. 7, composite transition curves have strong advantages over pure clothoid transition 
in terms of curvature and jerk continuities and excellent moderate G3 continuity for the first half of the transition. 
The overall instantaneous jerk is significantly lower in the composite curve transitions compared to clothoid tran-
sitions. This is because the window of jerk initiation is much longer in composite curve transitions because of the 
gradual development of jerk and on average it is four transition segments or four greyhound strides compared to 
just one stride in the clothoid transitions.
Greyhound racing data results
A racing greyhound getting injured at the tracks provides an indication of its overall racing trajectory perfor-
mance. Also, we can analyze the trajectory of a racing greyhound at the tracks to measure track path performance. 
Below, we present two such case scenarios by analyzing racing greyhound track data and injury rates.
Race injury data results for track path renovation. 
In the greyhound racing track path design, it 
was found that only circular arcs (constant curvature) and lines (zero curvature) were used extensively despite 
non-continuous curvature resulting at the segment intersections25. A discontinuity in the curvature implies that a 
greyhound must change its heading instantaneously and abruptly resulting in a path which is not feasible8. Also, 
track survey data from Australia shows that a brief transition is applied, made of an arc spline consisting of one or 
more circular arcs joined with continuous tangent vectors. This particular design practice also leads to multiple 
discontinuities in track path curvature.
We looked into one particular greyhound racing track (Track A) located in Australia and its two years of 
racing history. In the first year, it had a track path design with G1 continuity constituting half-circle bends and 
straights (Fig. 8). In the second racing year, the track was renovated with clothoid curve transitions into and out 
of the constant radius bends (Fig. 9). A 40 m clothoid transition was adjoined between a straight and a constant 
bend section for four bend and straight intersections. The outcome of this clothoid transition incorporation into 
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the original track path design eases centrifugal acceleration effect on the greyhounds where the centrifugal force 
is raised gradually from zero to an approximate nominal 240 N (Fig. 9). The renovation at Track A definitely 
would have changed centrifugal jerk performance significantly as the clothoid curve joining straights and bends 
would maintain G2 curvature continuity for the track path. To see whether this resulted in a significant decrease 
in racing injury rates, injury data for a two-year period were analyzed containing one year injury data for before 
and after renovation. By assuming differences in other contributing factors to injury rates such as variations in 
weather, track maintaining conditions, different greyhound breeds and training patterns, race operating condi-
tions between the years were minimal the injury rates should show general trends due to track path renovation 
changes. We found that before the clothoid intervention at Track A the normalized catastrophic and major injury 
Figure 5. Different smooth curves curvature and jerk results as 45 m transition curves for greyhound racing.
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rate per 1000 race starts was about 4.58 whereas after the clothoid intervention it was reduced to 4.22, a 7.9% 
reduction in this category of injury rates. Typically, this category of injury results from significant damage to 
greyhound physics. However, when we took into account all types of injuries at Track A for before and after 
the renovation, the normalized injury rates per 1000 race starts reduced to 26.71 from 44.68 injuries, a 40.2% 
reduction in overall injuries due to clothoid implementation at the track. Furthermore, under all injury types the 
most commonly occurring injury is happening in the greyhound forelegs responsible for turning assist for dog’s 
navigation. Metacarpal fractures and tibial fractures due to torsional stress occurring in the forelegs indicate 
navigational work stress on the greyhounds.
Figure 6. Different smooth curves curvature and jerk results as 75 m transition curves for greyhound racing.
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curvature of racing greyhound trajectory. 
Like any path following object, a racing greyhound has lim-
itations of the radius of curvature or extrema of curvature for its running path. Also, when a racing greyhound 
runs following a track path which has curvature discontinuity or non-optimal transitions, a deviation in the 
greyhound’s position occurs from the projected track path trajectory. This phenomenon was observed in the 
greyhound location data in the races. Furthermore, numerical racing greyhound simulations confirmed that 
Figure 7. Four different straight to bend curvature graphs and jerk results for ideal racing greyhound transition 
curves.
Composite curve 
configuration No.
Transition deflection 
angle acceleration for 
auxiliary curve (deg.)
Transition deflection angle 
acceleration for main 
clothoid curve (deg.)
Total transition 
length (m)
Transition exit 
radius (m)
1
0.3900
0.50
45
71.6
2
0.3900
0.52
45
68.9
3
0.1825
0.27
75
75.8
4
0.2500
0.32
75
64.0
Table 5. Kinematic and shape properties for four straight to bend composite curve transitions.
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when a greyhound is following the line of sight of a lure, its yaw rate gradually builds up for the bend for a track 
shape which is less circular5. To see if there is any difference between racing greyhound path trajectory to track 
path, we analyzed racing greyhound location tracking data for a track which has non-optimal transition length to 
reduce the jerk magnitude. From the racing greyhound location tracking data and track survey data, we generated 
curvature results for both racing greyhound trajectory and track path (Fig. 10). The greyhound location data for 
all greyhounds starts were averaged to plot the results where ten races or eighty starts were considered to plot the 
results below. As can be seen from the curvature plot, there is a significant difference between racing greyhound 
trajectory and track path. This indicates racing greyhounds deviating from the track path to accommodate a more 
natural trajectory according to their physics. Also, it was observed from the analysis that transitions occurring in 
racing greyhound trajectory is relatively gradual and longer as indicated by the green dashed marker compared 
to the black dashed marker for track path.
conclusions
This paper presents a numerical method for generating racing greyhound clothoid transitions for track path designs 
along with an equation for modelling any kind of clothoid curves. The numerical technique is robust and can be 
algorithmically controlled to achieve defined goals compared to existing approaches for designing racing greyhound 
clothoid transitions. Moreover, it can be extended to function as a generator of other curves rather than just clothoid 
curves. By looking into jerk modelling data, an ideal transition curve is presented suitable for racing greyhound 
track path designs which overcomes limitations set by clothoid transitions. The effect of clothoid transitions in an 
existing track was verified by measuring injury rates over a two-year period. The trajectory of racing greyhounds in 
an existing track with inadequate transitions was analyzed to show non-optimum track path conditions.
Finally, this paper showed evidence through modelling and injury data that clothoid and other composite 
curves improve racing dynamics safety for racing greyhounds. Furthermore, the methods presented here can 
also be used in designing and modelling trajectories for other moving bodies, including but not limited to horses, 
vehicles and trains.
Figure 8. Track path curvature as shown by curvature combs for Track A with G1 curvature continuity for 
bends.
Figure 9. Track path curvature as shown by curvature combs for Track A with G2 curvature continuity for 
bends.
Figure 10. Track path and greyhounds trajectory curvature comparison.
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Received: 1 December 2019; Accepted: 2 April 2020;
Published: xx xx xxxx
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Acknowledgements
This work was sponsored by Greyhound Racing NSW, Australia and Faculty of Engineering and Information 
Technology at the University of Technology Sydney, Australia. The authors would also like to acknowledge the 
support of Greyhound Racing Victoria for providing the greyhound location tracking and track survey data. 
The research was funded by Greyhound Racing NSW research grant “Identifying optimal greyhound race track 
design for canine safety and welfare Phase II”.
Author contributions
MIH conceived of the presented idea, developed the theoretical framework, performed the analytic calculations 
and performed the numerical modelling and simulations to derive the results. DE conceived of the presented 
idea, supervised the project and reviewed the manuscript. PW reviewed the manuscript.
competing interests
The authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to M.I.H.
Reprints and permissions information is available at www.nature.com/reprints.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and 
institutional affiliations.
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© The Author(s) 2020

Greyhound racing ideal trajectory path generation for straight to bend based on jerk rate minimization.
Hossain, Md Imam,Eager, David,Walker, Paul D