Z. Naturforsch. 2022; aop
Mehmet Pakdemirli* and Volkan Yıldız
Nonlinear curve equations maintaining constant
normal accelerations with drag induced
tangential decelerations
https://doi.org/10.1515/zna-2022-0253
Received October 8, 2022; accepted November 21, 2022;
published online December 8, 2022
Abstract: A nonlinear curve equation is derived for a
vehicle exposed to drag force only. At each point on the
curve, the vehicle maintains constant normal acceleration
component. The resulting curve equation is a highly
nonlinear third order ordinary differential equation. By
defining dimensionless coordinates, the equation is cast
into a non-dimensional form and a special path parameter
is defined. Two different perturbation type solutions as well
as a series solution are constructed as approximations to
the curves. The three approximate solutions are contrasted
with the numerical solution of the problem. The validity
range of the approximate solutions is discussed. The curves
may be used in determining the routes of land, marine and
aerial vehicles.
Keywords: acceleration; numerical solutions; perturbation solutions; series solutions; vehicle navigation paths.
1 Introduction
It is inevitable for the vehicles to track curved paths.
Transition from a straight path to a curved path must be
smooth for the comfort of the travel. Abrupt changes in
the directions of motion or sharp curves may lead to side
slip or turn over type accidents. Excess forces are exerted
on the vehicles during tracking curved routes. Fixing the
normal acceleration component under a prescribed value
would add to the safety and comfort of the travel.
The technological problem posed in this study is
to seek for a specific curve for which the vehicle is
*Corresponding author: Mehmet Pakdemirli, Department of Mechanical Engineering, Celal Bayar University, Muradiye, Yunusemre,
Manisa, Türkiye, E-mail: pakdemirli@gmail.com.
https://orcid.org/0000-0003-1221-9387
Volkan Yıldız, Department of Mechanical Engineering, Celal Bayar
University, Muradiye, Yunusemre, Manisa, Türkiye,
E-mail: volkan.yildiz@cbu.edu.tr
exposed to drag force only but maintains a constant normal
acceleration during the travel. Therefore, the tangential
deceleration is assumed to be proportional to the square
of the velocity. This means that the vehicle does not apply
any type of brake system and reduces its speed under the
influence of drag force only. In previous studies [1, 2], the
case of constant tangential deceleration has been treated
already. The curve equations corresponding to constant
tangential and normal coordinates were derived and solved
approximately and numerically. Results were applied to the
curvature designs of highway exits.
In this work, the nonlinear equation determining
the curves for a drag induced deceleration with constant
normal acceleration component is derived using the principles of dynamics. The resulting third order nonlinear
ordinary differential equation is transformed into a nondimensional form for expressing the results in a compact
form with less number of parameters. In fact, the family
of curves depends only on one path parameter which
simplifies the expression of the results.
Since the equations are highly nonlinear, approximate analytical solutions are sought first. Two different
perturbation type solutions are constructed. In the first
perturbation solution, the path parameter is selected as
the perturbation parameter assuming this parameter to be
small and an approximate solution with one correction
term is developed. In the second perturbation solution,
no assumption has been made for the path parameter
but the dependent variable itself is assumed to be small
by artificially introducing a small parameter. A different
approximate solution was constructed under the new
assumption. A power series solution for the problem is
given also. The provided recursion relation enables to
compute the coefficients of the series up to any arbitrary
order desired. Finally, the ordinary differential system is
numerically solved by employing an adaptive step size
Runge–Kutta algorithm. It is shown that the first perturbation solution performs well for small path parameters.
For large path parameters, all approximate solutions yield
similar results with some discrepancy with the numerical
solution at the right hand side of the domain of interest.
2 | M. Pakdemirli and V. Yıldız: Nonlinear curve equations with drag induced decelerations
In transportation engineering, parts of the so called
“clothoid curves” are frequently used [3–7]. Clothoid
curves are also used for the path planning of unmanned
aerial vehicles (UAVs) [4]. For a clothoid curve, the
curvature and arc length has a linear relationship. The
curves are parametrically defined in terms of integrals
in the implicit form. In some of the studies, instead of
numerically calculating the integrals, the curves have
been approximated by s-power series [5], by polynomials,
power series, continued fractions and rational functions
[6]. Navigation at constant speed is possible at a constant
rate of angular acceleration in clothoids [7]. Instead of
the constant speed assumption of the clothoid curves,
reduced speed assumption which is more realistic when
tracking from a straight path to a curved path is considered
here. Side slip and turn over incidents are more associated
with the normal acceleration component rather that the
angular acceleration component. So, keeping the normal
acceleration component fixed and below a certain value
is safer and more comfortable during navigation. For land
transportation, the curves may be employed for design of
highway curves, highway exits and railways for example.
These designs can be made once and cannot be altered,
so the assumptions must be realistic and average values
should be used to calculate the path parameter. For marine
and aerial vehicles, the alternatives are widespread and the
route to be followed can be calculated simultaneously by
a computer and tracked by the vehicle according to the
calculated route.
2 Equation describing the curve
Consider a vehicle tracking a curved path as shown in
Figure 1. The curve diverges from a straight path at the
origin. The velocity of the vehicle at the origin is v0 and the
initial radius of curvature is 𝜌0 . The vehicle is mainly under
the influence of drag force and has a tangential deceleration a proportional to the square of its instantaneous
velocity. s is the curvilinear length coordinate starting
from the origin. For the curve equations, the cartesian
coordinates will be employed.
At an arbitrary distance s on the curve, the normal
acceleration component [8] is required to be fixed and
hence.
𝑣(s)2 𝑣20
= .
(1)
𝜌(s)
𝜌0
The total length at location s and the radius of
curvature are defined as [9]
Figure 1: Schematics of the curve.
x
s=
∫
√
1
=
1 + y′2 dx,
𝜌
0
y′′
(1 + y′2 )3∕2
(2)
where prime denotes differentiation with respect to x. The
reduced speed is mainly due to the drag force and the
deceleration equation
a = −k 𝑣 2
(3)
can be solved for the velocity at an arbitrary location
𝑣 = 𝑣0 e−ks
(4)
which is exponentially decaying. Differentiating (1) with
respect to x and employing (2) and (4) yields
(
)
)3∕2 ′′
(
y =0
(5)
1 + y′2 y′′′ − 3y′ y′′2 − 2k 1 + y′2
which is the differential equation determining the curves
under the given assumptions. The vehicle is initially at the
origin. For a smooth transition from the linear path to the
curved path, the slope is zero and the initial curvature is
given, so that the conditions are
y(0) = 0,
y′ (0) = 0,
y′′ (0) =
1
𝜌0
.
(6)
The curves depend on the initial curvature and the
coefficient k only and are independent of the initial
velocity v0 . To reduce further the number of parameters,
dimensionless coordinates are selected
x∗ =
x
𝜌0
,
y∗ =
y
𝜌0
leading to
(
)
(
)3∕2 ′′
1 + y′2 y′′′ − 3y′ y′′2 − 2𝜀 1 + y′2
y =0
(7)
(8)
M. Pakdemirli and V. Yıldız: Nonlinear curve equations with drag induced decelerations | 3
y(0) = 0,
y′ (0) = 0,
y′′ (0) = 1
(9)
where
𝜀 = k𝜌0
(10)
is the dimensionless path parameter. The asterisk symbol
is not shown in the dimensionless equations for simplicity.
Note that the dimensionless equations depend on only one
physical parameter. The equation is a highly nonlinear
third order equation. Numerical solution as well as the
approximate analytical solutions will be presented in the
following chapters.
3 Approximate analytical solutions
Two different perturbation type solutions and a power
series solution are presented in this section.
3.1 Perturbation solution 1
The constant k is equal to dACD /2m and for a typical
automobile, the mass is m = 1050 kg, the vertical crosssectional area is A = 1.5 m2 , the drag coefficient is CD = 0.26,
the air density is d = 1.125 kg/m3 and hence the constant
is k = 2.09 × 10−4 . The path parameter is 𝜀 = k𝜌0 and for
an initial radius of curvature with 𝜌0 = 100 m, 𝜀 ≅ 0.02.
This rough calculation shows that the path parameter is
extremely small compared to 1. Even if the parameter value
becomes 10 times of this calculated value, i.e. 𝜀 = 0.2, the
small parameter assumption would not be violated. This
justifies a perturbation type of solution in terms of the small
parameter
y(x) = y0 (x) + 𝜀 y1 (x) + O(𝜀2 )
(11)
Substituting (11) into (8) and (9) and separating with
respect to orders lead to
)
(
2
(12)
O(1):
1 + y0′ y0′′′ − 3y0′ y0′′2 = 0
y0 (0) = 0,
O(𝜀):
y0′ (0) = 0,
y0′′ (0) = 1
)
1 + y0′2 y1′′′ + 2y0′ y1′ y0′′′ − 6y0′ y0′′ y1′′
)3∕2 ′′
(
− 3y1′ y0′′2 − 2 1 + y0′2
y0 = 0
(13)
(
y1 (0) = 0,
y1′ (0) = 0,
y1′′ (0) = 0
to maintain a constant normal acceleration component.
The first order solution
√
(16)
y0 (x) = 1 − 1 − x2
verifies this fact because the function expresses a circle
with radius 1 and center located at (0, 1). Inserting the
solution into (14) and solving subject to the conditions
given in (15) leads to
2 arccos h(x)
𝜋 − 2x
.
− √
2
1−x
x2 − 1
y1 (x) = √
Combining both solutions, the approximate solution
turns out to be
)
(
√
𝜋
−
2x
2
arccos
h(x)
− √
y(x) = 1 − 1 − x2 + 𝜀 √
1 − x2
x2 − 1
2
+ O(𝜀 ).
(18)
As is well known [10], the perturbation series are
usually asymptotic, such that, they get closer to the real
solution for the first few terms, but then addition of more
terms may result in a divergence from the real solution.
Because of this fact, and the complexity of the correction
term which will make the second correction term hardly
solvable in terms of the analytical functions, only the first
correction term has been calculated in the analysis.
3.2 Perturbation solution 2
Instead of selecting the path parameter as the perturbation
parameter, the dependent variable can be assumed to be
small
(19)
y(x) = 𝛼 u(x)
where 𝛼 is the perturbation parameter artificially introduced to express the smallness of y(x). Equations (8) and
(9) are transformed into
(
)
)
(
3
1 + 𝛼 2 u′2 u′′′ − 3𝛼 2 u′ u′′2 − 2𝜀 1 + 𝛼 2 u′2 u′′ = 0 (20)
2
u(0) = 0,
u′ (0) = 0,
u′′ (0) = 1∕𝛼
(15)
When 𝜀 = 0, k = 0 and hence the velocity is a
constant. The path for constant velocity should be circular
(21)
where the last term in the parenthesis is a Taylor approximation of the original term. The perturbation expansion
u(x) = u0 (x) + 𝛼 u1 (x) + 𝛼 2 u2 (x) + O(𝛼 3 )
(14)
(17)
(22)
separates with respect to orders after substitution into (20)
O(1):
u0 (0) = 0,
O(𝛼 ):
′′
u′′′
0 − 2𝜀u0 = 0
u′0 (0) = 0,
u′′
0 (0) = 1∕𝛼
′′
u′′′
1 − 2𝜀u1 = 0
(23)
(24)
(25)
4 | M. Pakdemirli and V. Yıldız: Nonlinear curve equations with drag induced decelerations
u1 (0) = 0,
u′1 (0) = 0,
u′′
1 (0) = 0
(26)
′′
′2 ′′′
′ ′′2
′2 ′′
u′′′
(27)
2 − 2𝜀u2 = −u0 u0 + 3u0 u0 + 3𝜀u0 u
O(𝛼 2 ):
u2 (0) = 0,
u′2 (0) = 0,
u′′
2 (0) = 0
(28)
The initial value problems at each level of approximation are solved
u0 (x) = −
1
4𝜀2 𝛼
−
1
2𝜀𝛼
x+
1
4𝜀2 𝛼
e2𝜀x
u1 (x) = 0
u2 (x) = −
1
1
where it is inevitable again to take the Taylor approximation of the last parenthesis. Substituting (33) into
(34), performing the necessary algebra, one reaches the
recursion relations between the coefficients
(i − 2)(i − 3)(i − 4)ai−2 +
j
i
∑
∑
× (i − j)(4k − 3 j + 1)ak a j−k ai− j
(29)
i−1 j
∑
∑
− 2𝜀(i − 3)(i − 4)ai−3 − 3𝜀
(30)
5
k(k − 1)( j − k)
j=0 k=0
k(k − 1)( j − k)
j=0 k=0
× (i − j − 1)ak a j−k ai− j−1 = 0
2𝜀x
−
x+
e
36𝜀4 𝛼 3 24𝜀3 𝛼 3
64𝜀4 𝛼 3
(31)
1
1
7
2𝜀x
4𝜀x
6𝜀x
+
xe −
e +
e .
16𝜀3 𝛼 3
16𝜀4 𝛼 3
576𝜀4 𝛼 3
Substituting the solutions into the perturbation
expansion and returning back to the original variable y(x)
leads finally to
1
1
1
1
1
− x + 2 e2𝜀x −
−
x
4𝜀2 2𝜀
4𝜀
36𝜀4
24𝜀3
5 2𝜀x
1
1 4𝜀x
7
+
e +
xe2𝜀x −
e +
e6𝜀x
64𝜀4
16𝜀3
16𝜀4
576𝜀4
y(x) = −
i = 5, 6, 7 …
(35)
The initial conditions dictate
a0 = 0,
a1 = 0,
a2 =
1
2
(36)
and from the recursion relation, the higher order coefficients can be calculated
1
1
1
𝜀, a4 = + 𝜀2 ,
3
8 6
19
1
1
41
1
a5 =
𝜀 + 𝜀3 , a6 =
+ 𝜀2 + 𝜀4
60
15
16 90
45
a3 =
(37)
(32)
where the artificially introduced perturbation parameter
𝛼 is automatically eliminated from the final solution.
This second perturbation solution is derived under the
assumption of y(x) being small and hence it is not expected
to give precise results when y(x) starts getting larger.
Contrary to the first perturbation approach, the path
parameter is not assumed to be small however.
Note that in this analysis, one of the simplest cases
of embedding an artificial parameter is employed. For a
more systematical and detailed analysis of the relevant
topic applied to some nonlinear oscillators, the reader is
referred to ref. [11].
3.3 Power series solution
Another classical approach may be to seek for polynomial
type series solutions
y(x) =
∞
∑
i=0
ai xi
(33)
for the problem
(
)
)
(
3
1 + y′2 y′′′ − 3y′ y′′2 − 2𝜀 1 + y′2 y′′ = 0
2
(34)
2
151 3
2 5
𝜀+
𝜀 +
𝜀,
7
315
315
5
2357 2 205 4
1 6
a8 =
+
𝜀 +
𝜀 +
𝜀
128 3360
504
630
a7 =
a9 =
a10 =
25
18401 3
41 5
1 7
𝜀+
𝜀 +
𝜀 +
𝜀
96
15120
140
2835
7
2041 2 7858 4
872 6
1
+
𝜀 +
𝜀 +
𝜀 +
𝜀8
256 2240
4725
4725
14175
a11 =
(38)
(39)
(40)
1015
245683 3 71693 5
16202 7
𝜀+
𝜀 +
𝜀 +
𝜀
4224
110880
37800
155925
2
+
𝜀9
(41)
155925
and so on. The power series solution is
(
)
(
)
1
19
1
1
1
1
+ 𝜀2 x4 +
𝜀 + 𝜀3 x5
y(x) = x2 + 𝜀x3 +
2
3
8 6
60
15
(
)
1
41 2
1 4 6
+
+ 𝜀 + 𝜀 x
16 90
45
(
)
2
151 3
2 5 7
+ 𝜀+
𝜀 +
𝜀 x
7
315
315
(
)
5
2357 2 205 4
1 6 8
+
+
𝜀 +
𝜀 +
𝜀 x
128 3360
504
630
M. Pakdemirli and V. Yıldız: Nonlinear curve equations with drag induced decelerations | 5
(
0.8
0.7
Numerical
0.6
11 terms
8 terms
0.5
5 terms
y
)
25
18401 3
41 5
1 7 9
𝜀+
𝜀 +
𝜀 +
𝜀 x
96
15120
140
2835
(
2041 2 7858 4
7
+
+
𝜀 +
𝜀
256 2240
4725
)
872 6
1
+
𝜀 +
𝜀8 x10
4725
14175
(
1015
245683 3 71693 5
+
𝜀+
𝜀 +
𝜀
4224
110880
37800
)
16202 7
2
+
𝜀 +
𝜀9 x11 + O(x12 )
155925
155925
+
0.4
(42)
0.3
where the solution is truncated at O(x12 ) for brevity. There
are two errors introduced in this solution, the first is the
Taylor approximation of the last term parenthesis in the
original equation which requires the slope y′ (x) to be small
and the second is the truncation error. When 𝜀 = 0, the
infinite series reduces to the Taylor series expansion of the
circular arc given in Equation (16) as expected. The general
formula for an cannot be presented due to the complexity of
the recursion relations. However, for 𝜀 << 1 it can be shown
numerically that a2n+2 ∕a2n and a2n+3 ∕a2n+1 are always less
than 1, ensuring the convergence of the series for small
parameters.
Note that a new direct and simple series solution
method has been proposed for third order nonlinear
boundary value problems in [12]. It employs successive
differentiation of the original equation and determining the
coefficients in a Taylor expansion by using the conditions.
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Figure 2: Convergence of the series solution to the numerical
solution for 𝜀 = 0.1.
0.9
0.8
Numerical
Perturbation 1
0.7
Series
0.6
Perturbation 2
y
0.5
0.4
0.3
4 Numerical solutions and
comparisons
0.2
0.1
The original dimensionless initial value problem is cast
into a first order differential equation system
y1′ = y2
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 3: Comparisons of the numerical and analytical solutions for
𝜀 = 0.02.
y2′ = y3
(
)3∕2
3y2 y32 + 2𝜀 1 + y22
y3
y3 =
1 + y22
′
y1 (0) = 0,
0
y2 (0) = 0,
(43)
y3 (0) = 1
and the variable step size Runge–Kutta algorithm is
employed for seeking the numerical solutions. For this
purpose, the Matlab Ode45 subroutine is employed in
the calculations. Comparisons of the numerical solutions
and the approximate analytical solutions are given in
Figures 2–5.
First, the convergence of the series solution is depicted
in Figure 2. As the number of terms in the series increase,
solutions get closer to the numerical solution. The gain
is marginal in adding the last three terms to the 8-term
expansion. Despite this fact, in the calculations given in
the subsequent figures, the 11-term series is used.
In Figure 3, the path parameter is selected to be
small in accordance with the preliminary calculations
given for a real application in the previous section. Since
the path parameter is small, perturbation solution 1 is
the best approximation to the numerical solution with
an excellent compliance as expected, since the solution has been derived under the small path parameter
6 | M. Pakdemirli and V. Yıldız: Nonlinear curve equations with drag induced decelerations
0.8
0.7
Numerical
0.6
Perturbation 1
Series
y
0.5
Perturbation 2
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Figure 4: Comparisons of the numerical and analytical solutions for
𝜀 = 0.1.
0.4
0.35
Numerical
0.3
Series
y
0.25
0.2
Perturbation 2
Perturbation 1
0.15
first perturbation solution and the second perturbation
solution still gives the worst approximation.
When the path parameter is further increased
(Figure 5), the series solution becomes the best approximation to the numerical solution and both perturbation
solutions perform similarly.
Although the calculation for a typical vehicle and
road example justifies for a very small value of the
perturbation parameter, i.e. 𝜀 = 0.02, for completeness
of the problem, large parameter values unlikely to occur
are also considered numerically. These large values are
taken to test the limitations of the analytical solutions.
As a general rule, for small path parameters, perturbation
solution 1 is recommended and if the path parameter is
larger, the series solution is better in approximating the
numerical solution.
Note that many more advanced semi analytical iterative methods have been developed in the past which may
be applied to this problem also. To name some of them, the
variational iteration method [13, 14], the homotopy perturbation method [15–17] and the perturbation-iteration
method [18–22] have already been tested over a wide
range of problems with success. A review of the vast
literature is beyond the scope of this study. Although, the
analytical solutions and their validity with respect to small
and large parameters have been already discussed in this
study together with the numerical solution and reasonable
agreements are presented, a future work may be to solve
the equations by the more advanced mentioned techniques
and test their validity with the numerical solutions.
0.1
5 Concluding remarks
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
Figure 5: Comparisons of the numerical and analytical solutions for
𝜀 = 0.5.
expansion. The series solution is the next best alternative
and the perturbation solution 2 which is derived under
small dependent parameter assumption diverges from the
numerical approximation much earlier than the two other
analytical solutions.
For an increased path parameter of 𝜀 = 0.1 (Figure 4),
the first perturbation solution is still the best approximation but the compliance with the numerical solution is
not as good as in the case of 𝜀 = 0.02 in the far end
of the domain. The series solution is closer now to the
New path equations maintaining constant normal accelerations with tangential decelerations proportional to the
square of the velocity has been derived. If the curves
are tracked from the reverse side, they represent tangential acceleration motions. The ordinary differential
equation describing the paths is derived and solved by
various methods. Three different analytical solutions are
presented, two of them being perturbation type solutions
and one is the series solution. The numerical solutions
are contrasted with the three analytical solutions. It is
concluded that, when the path parameter is small, as
in practical applications, the numerical solution is best
approximated by the first perturbation solution. As the
path parameter is increased, the series solution performs
better than the first perturbation solution.
Results of this work cannot be compared directly
with the existing methods using clothoid curves. The
M. Pakdemirli and V. Yıldız: Nonlinear curve equations with drag induced decelerations | 7
basic assumptions in deriving the curves are not the
same with each other. While the basic assumption of a
clothoid curve is the constant velocity assumption, the
present analysis assumes a drag induced deceleration. The
second difference is that; while the clothoid curves take
the angular acceleration as the main criteria, the normal
acceleration component is taken as the main criteria here.
The work can be extended in a number of ways.
Different velocity functions can be used which alters
the equation of motion. Another extension would be to
employ semi analytic iterative methods such as variational iteration method, homotopy perturbation method,
perturbation iteration method in search of approximate
analytical solutions.
The curves derived in this work can be applied to
the design of highway curvatures and their exits, to the
design of railways of high-speed trains. The paths of marine
and aerial vehicles can be determined using the curves
given in this work. When the thrust force of the vehicle
is annihilated, the vehicle is then under the influence of
drag forces only and the principles of this study may be
applied to track a path with constant normal acceleration
for comfort and safety of the travel. This work constitutes
a theoretical basis for possible applications in designing
highway curvatures, railways and routes of aerial and
marine vehicles.
Author contributions: All the authors have accepted
responsibility for the entire content of this submitted
manuscript and approved submission.
Research funding: One of the authors (M. Pakdemirli)
received funding from the Turkish Academy of Sciences
(TÜBA). The support is greatly appreciated.
Conflict of interest statement: The authors declare no
conflicts of interest regarding this article.
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