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2008-01-1228

Stability and Control Considerations of

Vehicle-Trailer Combination
Aleksander Hac, Daniel Fulk and Hsien Chen
Delphi Corporation

Copyright © 2008 SAE International

ABSTRACT The influences of physical parameters of the vehicle and

trailer on both types of instability have been investigated in
In this paper, dynamics and stability of an articulated detail using both validated analytical models and vehicle
vehicle in the yaw plane are examined through analysis, testing [1-6]. It has been shown [2, 5] that the divergent
simulations, and vehicle testing. Control of a vehicle- type of instability depends only on the parameters of the
trailer combination using active braking of the towing towing vehicle and on the vertical hitch load. The latter is
vehicle is discussed. A linear analytical model describing affected by the trailer mass and the location of the trailer
lateral and yaw motions of a vehicle-trailer combination is center of gravity in the longitudinal direction. Reducing the
used to study the effects of parameter variations of the hitch load, or equivalently moving the trailer center of
trailer on the dynamic stability of the system and gravity rearward, improves the stability margin with respect
limitations of different control strategies. to divergent motion. In contrast, stability with respect to the
snaking motion depends on the parameters of both the
The results predicted by the analytical model are towing vehicle and the trailer. More specifically, the system
confirmed by testing using a vehicle with a trailer in becomes unstable beyond a certain speed. This speed
several configurations. Design of the trailer makes it decreases, thus rendering the system less stable, as: 1)
possible to vary several critical parameters of the trailer. the mass of the trailer (relative to the vehicle’s mass)
The test data for vehicle with trailer in different increases, 2) the center of gravity of the trailer moves
configurations is used to validate the detailed non-linear rearward, 3) the moment of inertia of the trailer increases,
simulation model of the vehicle-trailer combination. 4) cornering stiffness of trailer tires decreases, 5)
Analysis and the simulation model are used to examine cornering stiffness of the vehicle’s rear tires decreases 6)
the advantages and limitations of two active brake the distance from the vehicle rear axle to the hitch point
control methods: the uniform braking and the direct yaw increases, 7) vehicle wheelbase decreases. Even for a
moment (DYM) control of the towing vehicle. specific vehicle-trailer combination, most of these
parameters can be greatly altered by changing parameters
INTRODUCTION that are in control of the user, such as weight distribution
or tire type and tire pressure. Hence, stability of the
Handling behavior of articulated vehicles is more vehicle-trailer combination cannot be guaranteed by
complex and less predictable than that of non-articulated selecting the passive design parameters, thus
vehicles. This poses challenges to non-expert drivers of necessitating an active control approach.
light vehicles, who occasionally tow recreational trailers
and may not have good understanding of dynamic Several active control methods have been proposed to
complexities of a vehicle-trailer combination. The task of improve handling and stability of a vehicle-trailer
controlling the vehicle becomes the most difficult when combination. For example, active rear steering of the
the system becomes unstable. Apart from rollover, which towing vehicle was considered by Kageyame and Nagai
is not discussed in this paper, the articulated vehicle may [7] and active front and rear steering by Deng et al. [8].
experience two types of instability in the yaw plane. The Active control of trailer braking was investigated by Lugner
first one is a divergent instability such as jackknifing, in et al. [9] and by Fernandez and Sharp [10], while
which the hitch angle increases without experiencing Kimbrough et al. [11] considered active control of both
oscillations. This occurs when, at a particular operating vehicle and trailer braking. Aside from active vehicle brake
condition, the understeer gradient of a vehicle-trailer control, these types of systems are seldom available in
combination becomes negative and the speed is above production vehicles or trailers. Furthermore, the proposed
critical velocity. The second type is dynamic in nature control strategies typically required a prior knowledge of
and may lead to oscillatory response with increasing trailer parameters and the variables describing trailer
amplitude known as snaking or sway. states (e.g. hitch angle, rate, trailer yaw rate or lateral
acceleration of trailer center of gravity) which necessitated

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additional sensors. More recently, cost effective solutions center of mass, e1 is the distance of the rear axle of the
to improve stability and handling of vehicle-trailer vehicle to the hitch point and c1 = b1 + e1 is the distance of
combinations have been proposed and implemented by the hitch point to the vehicle center of mass. Symbols a2
utilizing only brake actuators and sensors from the towing and b2 denote the distances of the trailer center of mass to
vehicle’s electronic stability control (ESC) system. In the hitch point and trailer axle, respectively, and l2 = a2 + b2
essence, two control methods have been considered: one is the distance of the trailer axle to the hitch point. The yaw
is a symmetric braking of the towing vehicle with the rates of the vehicle and trailer are denoted by ω1 and ω2,
objective of bringing the vehicle speed below the critical respectively, and lateral velocities at their centers of mass
speed (e.g. [12]); the second is control of the yaw moment are vy1 and vy2, respectively. Under the small angle
of the towing vehicle via asymmetric braking [13]. Since no assumption, the longitudinal velocities of vehicle and trailer
direct feed-back information about the trailer motion or are the same vx1 = vx2 = vx and are denoted by vx. Fyf, Fyr
even knowledge of trailer parameters is available, and Fyt are the lateral tire forces per axle acting at front
maintaining robustness of the control method with respect and rear axles of the vehicle and the trailer axle,
to variation in trailer parameters is one of the most respectively. Yh represents the hitch force in the lateral
important issues for systems intended for production. direction. Additionally, the vehicle may be subjected to the
yaw moment ΔMz1 resulting from active asymmetric
In this paper, dynamics, stability, and stabilizing control braking of the towing vehicle, which is considered the
of the articulated vehicle is investigated using first an control input. Under the above conditions and
analytical model and then a combination of testing and assumptions, the equations of lateral and yaw motions of
full-vehicle simulations. the vehicle and trailer are:

ANALYTICAL MODEL OF VEHICLE-TRAILER m1 a y1 = F yf + F yr − Yh (1a)

COMBINATION I z1Z 1 F yf a1  F yr b1  Yh c1  'M z1 (1b)

For the purpose of studying the yaw plane dynamics of m 2 a y 2 = Yh + F yt (1c)

the vehicle-trailer combination, a single track 4-th order I z 2 Z 2 Yh a 2  F yt b2 (1d)
model illustrated in Figure 1 is selected. It is the simplest
model, which describes both types of instabilities and
Here, ay1 and ay2 are the lateral accelerations at the
permits studying of the effects of system parameters and
vehicle and trailer centers of mass. The following
different control strategies on the dynamic behavior of
kinematic relationships hold:
the system. Nearly identical models have been
considered in the past [1, 3, 5, 6] and have been shown
to represent the handling dynamics very well and to Z2 Z1  T (2a)
achieve excellent match between the values predicted by a y1 v y1  v xZ1 (2b)
the model and the test results.
a y2 v y1  v xZ1  c1Z1  a 2 Z1  T (2c)
x X
Fyf δ
The tire forces per axle are assumed to be proportional
vx1
ω1 to the tire slip angles:
m1, Iz1 a1
vy1
Fyr Fbx1 ⎛ v y1 + a1ω1 ⎞
F yf = −C yf α f = −C yf ⎜⎜ − δ ⎟⎟
Xh
ω2
- m2ax2 Yh b1 (3a)
m2, Iz2 y ⎝ vx ⎠
Fyt Yh c1
θ
a2 e1 v y1 − b1ω1
b2 F yr = −C yr α r = −C yr (3b)
l2 vx
Y § v y1  c1  l 2 Z1  l 2T ·
F yt C yt D t C yt ¨  T ¸ (3c)
¨ vx ¸
Figure 1: Simplified Model of Vehicle with Trailer in the Yaw Plane © ¹

In the model, the tires of each axle are represented as a Cyf, Cyr, and Cyt denote the cornering stiffness values for
single tire. The front wheel steering angle, δ, the hitch the tires of the vehicle’s front axle, rear axle and the
angle, θ, and the tire slip angles are assumed to be small. trailer axle, respectively. Combining equations (1)
In addition, the effects of aerodynamic drag forces and through (3) yields a system of linear equations, which is
deceleration on lateral dynamics are ignored. In the figure, written in a matrix form:
m1 and m2 are the masses of the vehicle and the trailer,
respectively, Iz1 and Iz2 are the yaw moments of inertia of Mx D v x x  Eu  F G (4)
the vehicle and the trailer about the respective yaw axes
passing through the centers of mass. The symbols a1 and M, D, E, and F are the system matrices which are given
b1 are the distances of the vehicle front and rear axle to the in the Appendix. While matrices M, E, and F are

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constant, matrix D is a function of vehicle speed, vx. The A good example of these points is the handling of a
state vector, x, and the control input, u, are given by vehicle without a trailer. In a steady-state turn, the yaw
>
x v y1 Z1 T T ,
T
@ u 'M z1 (5) rate of the vehicle, ω1, is

v xδ
By pre-multiplying equation (4) on both sides by M , the
-1 ω1 = (10)
l1 + K u v x2
state equation is obtained
Here, l1 is the vehicle wheel-base. The understeer
x A v x x  B u  GG (6)
gradient, Ku, is given by [14]
where
(
m1 C yr b1 − C yf a1 ) 1 ⎛Wf
⎜ W ⎞
⎟
A(v x ) = M −1 D(v x ) , B = M −1 E , G = M −1 F Ku = = − r (11a,b)
(7) C yf C yr l1 g ⎜ C yf C yr ⎟
⎝ ⎠
Matrix A is a function of vehicle speed, vx, while matrices
In the above, Wf/r denote the front and rear axle static
B and G are constant.
loads (due to gravity forces) and g is the gravity
acceleration. If the understeer gradient is negative,
It is known that the linear bicycle model of a vehicle
vehicle yaw response diverges when the vehicle speed
without a trailer is described by the state equation of the
same form as equation (6), specifically is above the critical speed − l1 / K u . Thus, increasing
the understeer gradient improves static (steady-state)
x v A v v x x v  B v uv  G vG (8) stability of the vehicle.

The matrices in equation (8) are given in the Appendix; At the same time, since the vehicle model is a 2nd order
the control input is the same as before and the state system, which is typically under-damped throughout
vector is most of the speed range, it is characterized by the
[
x v = v y1 ω1 ]T (9)
natural frequency and the damping ratio. The latter,
which is a good indicator of dynamic stability, is given by
the following function of speed [14]
The analytical model of a vehicle-trailer combination
described here is used primarily to study the effect of ζ0
ζ (v x ) = (12)
vehicle and trailer parameters on the stability of the 1 + v x2 (K u / l1 )
system at the onset of potential instability, when
amplitudes of motion are small and the linear
Here, ζ0 represents the damping ratio at zero speed.
assumptions are approximately satisfied. While
Hence, the larger the understeer gradient is, the faster the
extensions of the model are possible by employing
rate at which the damping coefficient decreases with
nonlinear tire models and avoiding the small angle
speed. At sufficiently high speed, the yaw response of the
assumption, in this paper a combination of vehicle testing
vehicle can become highly oscillatory when Ku is large.
and simulations based on a nonlinear full vehicle with
While increasing the understeer gradient improves static
trailer model are used to validate the analytical results.
stability, it can reduce dynamic stability at high speeds.
STABILITY ANALYSIS
STATIC STABILITY – In order to study the static
stability of a vehicle-trailer combination, the system
In this section, stability of the vehicle-trailer combination
described by equation (4) without the control and
is investigated using the model described in the previous
steering inputs is considered in steady-state condition
section. Although the concept of stability appears
(dx/dt = 0). That is
intuitively obvious, it is necessary to make two important
points. First, any system of at least the 2nd order may D(v x )x = 0 (13)
experience two types of instability: a static or divergent
type of instability, in which the variables describing the
Solving the above equation yields the steady-state
system increase exponentially in magnitude without
values of the state variables. In particular, vehicle yaw
oscillations and a “dynamic” instability in which the
rate, ω1, is given by
variables experience oscillations with increasing
amplitude. Changes in some parameters which increase v xδ
static stability may reduce the dynamic stability and vice ω1 = (14)
versa. Second, from an engineering point of view, it is l1 + (K u − ΔK u )v x2
not only important to maintain system stability, but a
certain degree of stability (stability margin) is necessary. Ku is the understeer gradient of the vehicle alone and ΔKu
For example, a system that is stable but highly oscillatory is the change in the understeer gradient due to the
may be difficult to control. presence of trailer. It is given by

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ΔK u =
[
m2 b2 C yf (a1 + c1 ) + C yr e1 ] (15)
be asymptotically stable, all the roots must have negative
real parts. If any of the roots has a positive real part, the
C yf C yr l1l 2
system becomes unstable. Since matrix A is speed
dependent, the positions of the eigenvalues also change
By observing that the changes in the static vertical loads with speed. A convenient way of visualizing the changes
of the vehicle front and rear axles due to the vertical in system eigenvalues and the degree of stability with
hitch load are speed is to plot the damping coefficients and the damped
m 2 b2 e1 g m b (e + l )g natural frequencies as functions of vehicle speed. If the
ΔW f = − ; ΔW r = 2 2 1 1 (16) system has a complex pair of poles at
l1l 2 l1l 2
s1,2 = −d ± jω d (20)
it can be shown that the understeer gradient of the
vehicle trailer combination is given by where j is the imaginary unit, d is the damping coefficient,
and ωd is the damped natural frequency, then the
1 ⎛ W ft W rt ⎞ damping ratio, ζ, and the un-damped natural frequency,
K u − ΔK u = ⎜ − ⎟ (17)
g ⎜ C yf C yr ⎟ ωn, are
⎝ ⎠
−d
W ft = W f + ΔW f and W rt = W r + ΔW r are the static loads ς= , ω n = d 2 + ω d2 (21)
2
of the front and rear vehicle axle, respectively, when a d + ω d2
trailer is present. Thus, equations (14) and (17) are
identical to the corresponding equations for the vehicle A negative damping ratio ζ indicates an unstable system.
without a trailer. It follows from equations (15) or (17) that As a 4-th order system, the vehicle with trailer model has
the understeer gradient is reduced, as compared to that 2 pairs of complex conjugate poles with one damping
of the vehicle alone, when b2 > 0 (i.e. when center of ratio for each pair for the total of two. In contrast, the
mass of the trailer is ahead of the trailer axle), it vehicle without trailer possesses only one pair of
increases when b2 < 0 and remains the same when b2 = complex conjugate poles and one damping ratio. The
0. Thus in a typical case of b2 > 0, the presence of a damping ratios for the vehicle alone and the vehicle with
trailer reduces the static stability of the articulated vehicle trailer are plotted as functions of speed in Figure 2. The
and increases steady state yaw response. It should be numerical values of parameters correspond to a large
noted, however, that contrary to what is often assumed, pick-up truck with a 2000 kg (~ 4400 lbs) trailer.
the cornering stiffness values do not remain constant,
but change with the changes in axle normal load. If the Damping Ratios
1.2
change in cornering stiffness were proportional to the vehicle only
normal load, there would be no change in the understeer vehicle with trailer
1
gradient regardless of the dimension b2. In reality, the
cornering stiffness values change less than
0.8
proportionally with normal load [14], resulting in changes
in understeer gradient, which are directionally the same,
Damping Ratio

0.6
but significantly smaller than those predicted under the
assumption of constant cornering stiffness values. 0.4

DYNAMIC STABILITY – In order to study the inherent 0.2

dynamic stability of a vehicle-trailer combination,
consider a state equation of the free system (i.e. 0
equation (6) with the control and steering inputs both
equal to 0) under the assumption of constant speed: -0.2
20 40 60 80 100 120 140 160
x Ax (18) Velocity [kph]

Figure 2: Damping Ratios of Vehicle Alone and Vehicle with Trailer as

The system eigenvalues (poles) are the eigenvalues of Functions of Speed
matrix A, that is, the roots of the characteristic equation

det (sI − A ) = 0 (19) The vehicle without a trailer has the damping ratio which
decreases with speed from 1 to about 0.5 at 160 kph (~
Here, I denotes an identity matrix and s is the Laplace
100 mph), implying well damped response in the entire
operand which is the unknown variable in equation (19).
range of speeds. For the vehicle with trailer, one pair of
Since A is a matrix of dimension 4x4 with all real
eigenvalues has a slightly higher damping ratio than for
coefficients, equation (19) is a 4-th order equation with
the vehicle alone. The other pair of eigenvalues, in
real coefficients. The roots are therefore either real or
contrast, has a low damping ratio (below 0.3) at speeds
complex occurring in conjugate pairs. For the system to
above 60 kph (~ 37 mph); it becomes negative at about

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115 kph (~ 71 mph), indicating that at or above this capable of increasing damping in the system at any
speed, the model of a vehicle with trailer becomes speed.
unstable. In the speed range between 60 kph (~ 37 mph)
and 115 kph (~ 71 mph), the vehicle with trailer will SYMMETRIC BRAKING – In this approach, the natural
exhibit oscillatory response, but with oscillations slowly damping of the system is utilized, which changes
decreasing with time. Above 115 kph (~ 71 mph), the gradually with speed, as indicated in Figure 2. Hence,
oscillations initiated by any driver input or external vehicle speed has to be significantly reduced from the
disturbance will grow exponentially. critical speed, before the damping in the system is
sufficiently large to effectively damp the oscillations. In
The model of vehicle with trailer was used to study the addition, braking of the towing vehicle without the trailer
influence of various parameters on stability of a vehicle- braking has the following second-order effects on the
trailer combination. For example, the distance b2 of the yaw dynamics of the system:
trailer center of gravity to the trailer axle depends on the
1) It introduces longitudinal inertial force at the trailer
payload distribution. The effects of this distance on the
center of mass, which creates a destabilizing moment
damping ratio and the damped natural frequency of the
acting on the trailer (that is, a moment that tends to
least damped mode of the vehicle-trailer combination at
increase the hitch angle, θ);
the speed of 80 kph (~ 50 mph), are shown in Figure 3.
2) It changes the normal loads of the vehicle and trailer
The negative distance indicates that the center of gravity
axles, which in turn affect the cornering stiffness values
is behind the axle. The natural frequency does not
for each axle;
change dramatically, but the damping ratio of the least
3) It produces tire slips of front and rear axles of the
stable mode decreases significantly as the center of
towing vehicle which tend to reduce the lateral cornering
gravity moves to the rear of the trailer, making the
stiffness of these axles.
system less stable in the dynamic sense. Since moving
the center of gravity rearward improves static stability,
The last effect depends on brake proportioning, but is
generally an optimal range of distance b2 (and
generally small during light and moderate braking. The
corresponding hitch load) is recommended.
first two effects can be analyzed using the 4-th order
model (equation 6) with the following modifications. First,
Damping Ratio of the Least Damped Mode at 80 kph
0.4 the inertial force of the trailer, -m2ax2, causes a
corresponding longitudinal reaction force in the hitch
Damping Ratio

0.3
joint, which affects equation (1d) describing the yaw
0.2 motion of the trailer. This equation becomes
0.1
I z 2 Z 2 Yh a 2  F yt b2  m 2 a 2 a x1T (22)
0
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Distance b2 [m]
Damped Natural Frequency of the Least Damped Mode at 80 kph
Here, ax1 is longitudinal acceleration of the vehicle, which
5 within the linear approximation is the same as that of the
Natural Frequency [rad/s]

4 trailer. In addition, the changes in tire cornering stiffness

3 (per axle) resulting from the normal load transfer from
2
braking can be included in the model using quasi-static
1
equations. Consequently, the system equation (4) takes
the form
0
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Distance b2 [m]
Mx D v x1 , a x1 x  Eu  FG (23)

Figure 3: Damping Ratios and Damped Natural Frequency of the Least in which the matrix D is now a function of not only
Damped Mode of Vehicle as Functions of Distance b2 at Vehicle
Speed of 80 kph longitudinal speed, which is now time varying, but also
longitudinal acceleration.
STABILIZATION THROUGH ACTIVE CONTROL
Even though the method of determining stability of a
In this section, the effects of two control strategies using vehicle by investigating the location of eigenvalues
braking of the towing vehicle on stability of the vehicle- applies strictly only to time invariant systems, this
trailer combination are analyzed. The first one is slowing approach can be used when the system parameters vary
down the vehicle by symmetric braking until the vehicle slowly in time. Under this assumption, the critical speed
speed drops significantly below the critical speed where of the vehicle (i.e. the speed at which the system
natural damping of the system is sufficiently large. The comprising the vehicle and trailer becomes unstable)
second is the direct yaw moment control of the towing was determined at different levels of vehicle
vehicle via asymmetric braking, which influences deceleration. The results are shown in Figure 4.
eigenvalues of the closed loop system; it is therefore

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Critical Speed as Function of Deceleration vehicle and trailer dynamics including nonlinear
140
no effect of load transfer characteristics, thus removing most limitations of the
135 with effect of load transfer simpler analytical model. The trailer was designed to
130 facilitate variation of stability properties in a wide range
through changes in critical parameters.
125
Critical Speed [kph]

120 TEST RIG - Trailer testing was conducted with a custom

115 designed test trailer. The trailer, shown in Figure 5, can
be altered for a variety of loading conditions. This
110
flexibility offered by the trailer allows trailer sway motion
105 to be induced at a large range of speeds and therefore
100
promote safe testing scenarios and assist in stability and
control robustness studies.
95

90
-6 -4 -2 0 2 4 6
Deceleration [m/s 2]

Figure 4: Critical Speed of Vehicle-Trailer Combination as a Function

of Deceleration

In the figure, deceleration is positive and acceleration is

negative. The dotted line represents the results obtained
from the model (23), in which the effect of deceleration
on cornering stiffness is not included, while the results
depicted by the solid line include this effect. In both
cases the critical speed is decreasing as deceleration
increases. Thus the secondary factors decrease
dynamic stability of the system during uniform braking.

DIFFERENTIAL BRAKING – In contrast, differential Figure 5: Test Trailer

braking can directly affect the system eigenvalues at any
speed. Within the model considered here, differential The key feature which offers loading flexibility is an
braking is modeled as a direct yaw control input applied to adjustable axle carriage. It features two axles, providing
the towing vehicle. Let the output feedback control law be flexibility to mount wheels on one or both axles. More
importantly, the carriage can be moved in one of five
u = −K (v x )y = −K (v x )Cx (24) locations along the length of the trailer. Moving the axle
carriage affects the trailer’s Center of Gravity (CG)
Here, y = Cx is the output vector and K is the control location, thus affecting its dynamic response.
gain matrix, which may be speed dependent. Then the Furthermore, if additional ballast is required, an open
closed loop system is described by the state equation frame design with mounting holes allows that ballast to
x A  KC x (25) be secured to the trailer at any location.

The eigenvalues of the closed loop system are the Three primary safety features are incorporated in the
eigenvalues of matrix A-KC, which in general are trailer design. The first feature is an electro-hydraulic
different from those of the matrix A, corresponding to the braking system. A Tekonsha Prodigy brake controller
original system. Using equation (25), the closed loop installed in the towing vehicle controls a hydraulic pump
system eigenvalues and stability robustness with respect mounted to the axle carriage. Based on voltage input
to parameter variations can be investigated for various from the brake controller, the pump supplies hydraulic
control laws. Consequently, a control law can be pressure to vented, 4-piston caliper, disc brakes at each
selected, which uses only available signals while hub. In addition to the trailer brakes, there are devices
providing good robustness properties. designed to reduce the risks of roll-over and jack-knifing.
To prevent roll-overs during extreme test maneuvers,
TRAILER TESTING AND SIMULATION outriggers are attached to mounting plates which are
integrated into the trailer construction. In order to reduce
the risk of jack-knifing, tow straps linking the trailer
Using a custom built trailer, a series of experiments were
tongue to the outer, rear corners of the towing vehicle
performed to investigate sway characteristics at varying
are used to limit the hitch angle to ~ 30°.
speed and loading conditions. To perform this stability
investigation and to validate the earlier analysis, both
Instrumentation is mounted directly to the trailer to sense
real-world testing and simulation tools based were
and process its dynamic motions. An inertial
employed. The simulation model comprehends full

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measurement unit and optical Datron sensor are used to dimensional characteristics of the vehicle-trailer
measure yaw rate, lateral acceleration, and lateral and combination for a given configuration. From this
longitudinal velocities. The sensors are routed to a information, the locations of the resultant axle load and
dSPACE MicroAutobox which processes and transmits trailer center of gravity are determined and also displayed.
the signals through CAN to a second MicroAutobox Additionally, data is presented to explain the lateral force
mounted in the towing vehicle, which is also equipped capability, which is affected by changes in static loading.
with similar dynamic motion sensors. Therefore, both
vehicle and trailer data can be acquired simultaneously Static loading conditions for the three configurations are
with the same data file. detailed in figures 6, 7, and 8. In Figure 6, the static
loading conditions for Position A, Tandem Axle are
RESULTS – In this section, the results from testing and shown. The tongue load is 283 lbs (~ 1260 N), which
simulation are reported. Experiments were performed follows general towing guidelines that the tongue weight
with the axle carriage in one of two locations. Position A should be ~ 10 – 15% of the total trailer weight. With this
is with the carriage in its “home” position, that location configuration, the trailer CG is located 148 inches (~ 376
which is recommended by the trailer manufacturer for cm) from the hitch point, and it is ~ 17 inches (~ 43 cm)
normal towing operation. Position B is with the carriage in front of the resultant tire load which acts at 164.5
in its most forward location. The following list provides a inches (~ 418 cm) from the hitch point. Next, the static
brief description of configurations considered in this loading conditions for Position B, Single Axle are shown
paper. For each configuration, a 4 inch (~ 10 cm) drop in Figure 7. Compared to Position A, Tandem Axle, the
ball mount was used for securing the trailer to the towing tongue load is significantly reduced at 85 lbs (~ 378 N),
vehicle’s receiver. and there is a slight increase in the normal load acting at
the trailer axle. The trailer center of gravity location is
• Position A, Tandem Axle: Axle carriage is at its home
shifted forward on the trailer, but the axle load is shifted
position, Position A, with tires mounted to both axles.
even more, resulting in the distance between the two (b2)
• Position B, Single Axle: Axle carriage is at its most of only 4 inches (~ 10 cm). Finally, static loading for
forward position, Position B, with tires removed from Position B, Tandem Axle is shown in Figure 8. Simply
the front axle. re-mounting the tires to the front axle has significant
effects on the b2 distance and tongue loading. The
• Position B, Tandem Axle: Axle carriage is at its most
weight transfer from the vehicle onto the trailer results in
forward position, Position B, with tires mounted to
a negative tongue load of -280 lbs (~ - 1247 N). While
both axles.
the CG location changes by only one inch, the resultant
Static Loading and Tire Cornering Stiffness – To support tire normal load moves forward by 17 inches (~ 43 cm).
the results documented for each configuration in the As a result, the trailer CG is now located 12 inches (~ 30
following sections, static loading conditions and tire cm) behind the resultant axle load. As predicted by the
cornering stiffness data is provided. The figures for static analytical model and as will be shown in test results, this
loading illustrate tire normal loads, tongue loading, and has an adverse effect on trailer dynamic stability.

148 in
CG

36 in 283 lbs
39 in 63 in L = 119 in
L = 157.5 in
527 lbs 2079 lbs
164.5 in 3492 lbs 2870 lbs
2606 lbs
3060 lbs (w/out trailer) 3019 lbs (w/out trailer)
Combined Axle Load

Figure 6: Static Loading – Position A, Tandem Axle

134 in
CG

85 lbs
L = 119 in
94.5 in L = 138 in 63 in

3184 lbs 2980 lbs

2664 lbs 3060 lbs (w/out trailer) 3019 lbs (w/out trailer)

Figure 7: Static Loading – Position B, Single Axle

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133 in
CG

36 in -280 lbs
94.5 in 63 in L = 119 in
L = 102 in
1663 lbs 1497 lbs
2624 lbs 3175 lbs
3160 lbs 121 in 3060 lbs (w/out trailer)
Combined Axle Load 3019 lbs (w/out trailer)

Figure 8: Static Loading – Position B, Tandem Axle

It is necessary to characterize tire lateral force of the

trailer and cornering stiffness to provide modeling
accuracy for simulations and to comprehend the effect of
loading on lateral capability. Determining the tire lateral
force characteristics requires a plot of tire lateral force
vs. tire slip angle (α). For this paper, all such plots
combine the lateral tire forces for all trailer tires into a
single total force vector for the trailer, as was done in the
analytical model. Lateral force and slip angle data for the
trailer tires was obtained by subjecting the vehicle-trailer
system to a steady state cornering maneuver until the
point of maximum achievable lateral acceleration was
reached. To achieve this, a slowly increasing steer
maneuver was performed on dry asphalt at 30 mph (~ 48
kph). During the maneuver, lateral acceleration and yaw
rate were measured at the trailer’s center of gravity (CG)
using the inertial measurement unit. With this
information, the combined lateral tire force can be
Figure 9: Lateral Tire Force vs. Slip Angle – Comparison of Three
calculated by solving equations (1c) and (1d) for the Trailer Configurations
unknown lateral hitch and tire forces, Yh and Fyt. Since
under steady-state conditions ωh2 = 0 , this yields For characterization purposes, a critical parameter is the
m a cornering stiffness, which is the slope of the curve at 0°
F yt = 2 2 a y2 (26)
a 2 + b2 of slip angle (α). It is important to note that this value is a
function of many parameters which include, but are not
limited to, loading condition, tire pressure, tire-road
The Datron sensor measurement of trailer lateral velocity
surface friction, and suspension stiffness and damping.
(at CG), vy2, and longitudinal velocity, vx, can be
The cornering stiffness is 2337 N/deg for Position A,
combined with the yaw rate data to estimate tire slip
Tandem Axle; it is 2525 N/deg for Position B, Single
angle, , as follows
Axle, and 3093 N/deg for Position B, Tandem Axle. Since
v yt
v yt = v y 2 − b2 ω 2 , α = tan −1 ( ) (27) the normal tire loads are 2606 lbs (~ 11,590 N), 2664 lbs
vx (~ 11,850 N), and 3160 lbs (~ 14,060 N), it is apparent
that the cornering stiffness increases with normal load,
where vyt is the lateral velocity translated to the resultant but is not significantly affected by the number of axles.
tire load location.
SIMULATION VALIDATION Simulations of the nonlinear
Combined tire cornering stiffness values were full vehicle with trailer model were performed using the
determined by applying the above series of equations to CarSim® environment. The accuracy of the model was
data captured during the slowly increasing steer verified by comparing simulation results with test data
maneuver. Lateral force characteristics for each of the collected on a base truck (no active systems) towing a
three configurations are plotted in Figure 9. They were trailer. For each trailer loading configuration (Position A
obtained by a least square curve fitting method, in which or B), the axle and hitch loads in the vertical direction
a parabolic curve was forced to pass through the origin. were first validated against test data collected at the
static equilibrium condition by adjusting the center of
gravity location of the tractor and trailer. The tire data
collected based on the single axle trailer loading
configuration was used to represent the lateral force
dynamics of the trailer.

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As an example, comparisons between the test data instability for this condition, road testing for trailer sway
(solid line) and simulation (dashed line) for the steering was not conducted for safety reasons.
wheel input, vehicle speed, yaw rate, and lateral
acceleration of both the vehicle and trailer are shown in
Figure 10 for trailer configuration in Position B, Single
Axle. The figure demonstrates that the simulation model
matches well with the test data. The model was also
validated for other trailer configurations demonstrating
very good agreement with experiments in predicting the
onset of instability.

Figure 11: Simulation Results – Position A, Tandem Axle

To facilitate road testing of trailer sway at relatively low

speeds, the axle carriage was moved to its most forward
position (Position B). The front tires were removed to
create a single axle configuration. A series of straight-
line tests were conducted on a vehicle dynamics pad at
varying speeds. Once a steady state condition was
reached in which the vehicle-trailer combination was
traveling straight at the desired speed, an impulse
Figure 10: Model Validation for Position B, Single Axle steering input of ~ 50° from the towing vehicle was
applied to disturb the system and induce trailer
TRAILER RESPONSE – First, the trailer response was oscillation. The results obtained for the minimum and
characterized for Position A, Tandem Axle (configuration maximum speeds tested for this configuration are shown
used for normal towing activity). Both analysis and in Figure 12. The longitudinal speed, yaw rate, and
simulation of a validated vehicle-trailer model estimated hitch angle for the trailer are displayed. In
representing this configuration revealed stable response addition, a plot of the towing vehicle’s steering wheel
across a wide band of speeds. The selected results of angle is included to provide a frame of reference on
simulations performed at several speeds are displayed in when the disturbance was introduced. Whereas the
Figure 11. In the figure, the top two plots show yaw rate amplitude of each signal provides detail on the severity
and lateral acceleration data for the vehicle, and the of the maneuver, noting the damping effects is of greater
bottom plots show the same variables for the trailer. The importance for characterizing the stability margin. At
simulation used an impulse steering input to disturb the 72kph (45mph), shown by the solid line, the oscillations
system and induce dynamic response. At speeds less are fully damped within ~ 10 seconds. For 96kph
than 100 kph (~ 62 mph), the response dampens out (60mph), shown by the dashed line, the output signal
within a couple of cycles. As speed increases, greater amplitudes are ~ 50% greater. Although the system is
amplitudes of oscillations are present in the trailer data still damped, it takes almost twice as long for the sway
as well as decreased damping. Even at 160 kph (~ 100 motion to cease. The frequency of oscillations for both
mph), though, the system is still fairly well damped. In speeds, though, is similar. Recall that the analytical
part, the stability of this particular system can be model predicts that the damping ratio decreases as
attributed to increased lateral force capability at the distance b2 (measured from the trailer CG to the resultant
towing vehicle’s rear axle. The positive tongue load of axle load) decreases. Although Position B, Single Axle
283 lbs (~ 1260 N) increases the combined rear axle generates a higher cornering stiffness value than
normal load by ~ 400 lbs (~ 1780 N). Position A offers Position A, b2 is much smaller (4 inches (~ 10 cm) vs. 17
the widest dynamic stability margin of the three inches (~ 43 cm)). As a result, oscillations with higher
configurations tested and documented in this paper. amplitudes and less damping are observed at lower
Furthermore, due to the high speeds required to induce speeds.

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Using the same axle carriage position, Position B, tests

were performed with a tandem axle set-up to compare
against the single axle configuration. The trailer dynamic
responses for speeds of 48kph (30mph) and 72kph
(45mph) are shown in Figure 14. For the tandem axle,
72kph (45mph) appears to be the critical speed at which
the system becomes marginally stable. Shown with the
dashed line, the amplitude for yaw rate remains almost
constant as time continues beyond the initial disturbance.
Thus, the static loading conditions have a substantial
impact on the critical speed at which the sway motion
becomes unstable.

Figure 12: Trailer Sway Test Results – Position B, Single Axle

To augment the road testing, simulation was employed to

determine the critical speed at which this vehicle-trailer
combination becomes unstable. Simulation results for
three speeds: 96kph (60mph), 100kph (62mph), and
104kph (65mph) are shown in Figure 13. The simulations
used the same steering input disturbance that was
applied during testing. As with the test data, the
simulation shows the vehicle-trailer combination to have
oscillatory response at 96kph (60 mph), but it eventually
leads to a stable equilibrium condition. However, as
speed is increased to 104kph (65mph), the system
becomes unstable due to the onset of a negative
damping condition. The critical speed predicted by the
Figure 14: Trailer Sway Test Results – Position B, Tandem Axle
linear analytical model is virtually identical.
The lack of hitch angle data for the 72kph (45 mph) trace
is the result of severity of oscillations. Since the
calculation of this angle relies on integration of yaw rate
data from the IMU sensors, it is necessary to account for
yaw rate sensor biasing to remove errors that build over
time. To determine the bias factors, the vehicle-trailer
combination must be driven in a straight line without any
actual yaw rate at the beginning and end of the run. Due
to test surface size limitations, it was not possible to
obtain a zero yaw condition at the end of the run for
Position B, Single Axle tested at 72kph (45 mph)
because of persistent oscillations. Hence, an accurate
estimation of hitch angle could not be determined.

SUMMARY – The key parameters and results of the

three trailer configurations analyzed here are
summarized in Table 1. Those items are the resultant
axle normal load, combined trailer tire cornering stiffness,
center of gravity location with respect to resultant axle
load (distance b2), and the critical speed at which the
sway motion becomes unstable. Cornering stiffness
increases as axle normal load increases. However,
Figure 13: Simulation Results – Position B, Single Axle
there is not a direct correlation between cornering

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stiffness and critical speed. The primary indicator of No throttle input is applied. The base vehicle-trailer
achievable critical speed is the b2 distance. As that configuration (designated as No Braking) exhibits highly
distance decreases or even becomes negative when the oscillatory yaw rate and lateral acceleration responses of
CG is located behind the axle load, trailer stability is both the vehicle and the trailer. The direct yaw moment
adversely affected and instability occurs at lower speeds, control based on towing vehicle asymmetric braking
as predicted by the analytical model. significantly enhances the directional stability of the
vehicle-trailer combination. The uniform braking control
Resultant on the towing vehicle effectively reduces the vehicle
Tire Normal Cornering b2 Critical speed. However, it takes a longer time (10 seconds after
Load Stiffness Distance Speed brake is applied) to damp out the vehicle and trailer yaw
N N/deg m kph rate and acceleration responses. As shown in Figure 16,
Configuration (lbs) (lb/deg) (inches) (mph)
when the throttle input is applied to maintain a constant
Position A 11592 2337 0.419 >160 vehicle speed of 73 kph (45 mph) in the maneuver
Tandem Axle (2606) (525) (16.5) (>100) without braking, the trailer’s yaw rate of the “No Braking”
Position B 11850 2525 0.102 104 configuration leads to a dynamic instability (i.e., negative
Single Axle (2664) (568) (4) (65) damping). Once again, the direct yaw moment control
Position B 14056 3093 -0.305 72 (coarse dashed line) significantly enhances the vehicle
Tandem Axle (3160) (695) (-12) (45) and trailer yaw stability as compared to the uniform
braking control (dashed line).
Table 1: Data Summary Table

SWAY CONTROL METHODS

The validated simulation model (Position B, Tandem

Axle) is used to study advantages and limitations of two
active brake control methods: the uniform braking and
the direct yaw moment (DYM) control of the towing
vehicle. In both methods, only signals available from the
towing vehicle ESC system are utilized. The driver starts
with initial speed of 73 kph (45 mph), and the system is
subject to a sine wave steer input with magnitude of 50°
and frequency of 1Hz as shown in Figure 15.

Figure 16: Effects of Two Active Brake Control Methods on Trailer

Stability – Position B, Tandem Axle (With Throttle)

CONCLUSIONS

In this paper, the investigation of snaking oscillations of a

vehicle-trailer combination is documented. Analytical
modeling, simulation, and road testing were used to
study dynamic response as it is affected by various
speed and loading conditions. Furthermore, a brief
analysis of control methods for suppressing the
oscillation using only brake actuators and sensors from
the towing vehicle ESC system is presented. Results of
the experimentation performed give rise to the following
Figure 15: Effects of Two Active Brake Control Methods on Trailer
Stability – Position B, Tandem Axle (No Throttle)
conclusions:

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1. The analytical model provides good description of 4. Fratila, D. and Darling, J., 1995, “Improvements to
dynamic behavior of the vehicle-trailer combination; Car/Caravan Handling and High Speed Stability
it can be used to accurately predict the onset of Through Computer Simulation”, ASME Paper
snaking oscillations and to study the effects of WA/MET-9.
system parameters on stability. 5. Bevan, B. G., Smith, N. P., and Ashley, C., 1983,
“Some Factors Influencing the stability of
2. The test trailer proved to be an effective tool at
Car/Caravan Combinations”, I. Mech. Eng. Conf.
creating the desired testing scenarios.
Publ. 1983-5, MEP, London, pp. 221-227.
a. Dynamic behavior of the trailer can be 6. Anderson, R. J., Kurtz, E. F., 1980, “Handling
significantly altered without adding more weight. Characteristics Simulations of Car-Trailer Systems”,
SAE paper No. 800545.
b. Instability could be induced at low speeds which
7. Kageyama, I. And Nagai, R., 1995, “Stabilization of
promotes a safe testing environment.
Passenger Car-Caravan Combination using Four
3. Results of the experiments performed on the road Wheel Steering Control”, Vehicle System Dynamics,
conform to those learned through analytical modeling 24, pp. 313-327.
and simulation. 8. Deng, W., Lee, Y. H., and Tian, M., 2004, “An
Integrated Chassis Control for Vehicle-Trailer
4. The trailer’s center of gravity location relative to its
Stability and Handling Performance”, SAE paper No.
resultant axle load location is a critical factor in trailer
2004-01-2046.
stability.
9. Lugner, P., Plochl, M., and Riepl, A., 1996,
a. Small variations in CG location fore or aft of the “Investigation of Passenger Car_Trailer Dynamics
resultant tire normal load have significant effects Controlled by Additional Braking of Trailer”,
on achievable speed and stability levels. Proceedings from AVEC’96. International
Symposium on Advanced Vehicle Control, pp. 763-
b. In the cases shown here, trailer stability appears
778.
to be more dependent on CG location than on
10. Alonso Fernandez, M., A. and Sharp, R. S., 2001,
trailer tire cornering stiffness.
“Caravan Active Braking System-Effective
5. Control strategy involving direct yaw moment via Stabilization of Snaking of Combination Vehicles”,
asymmetric braking of towing vehicle is more SAE paper No. 2001-01-3188.
effective in stabilizing the snaking oscillations than 11. Kimbrough, S., Elwell, M. and Chiu, Ch., “Braking
symmetric braking. Controllers and Steering Controllers for Combination
Vehicles”, International Journal of Vehicle Design,
a. Direct yaw moment control can damp oscillations
1(2), pp. 195-222.
without significant reduction in speed
12. Waldbauer, D. and Krober, J., 2006, “Method and
b. Symmetric braking can also stabilize the system, System for Stabilizing a Car-Trailer Combination”,
but it has negative second-order effects on US Patent Application Publication No. US
system stability and requires large reduction in 2006/0033308 A1.
vehicle speed, significantly below the critical 13. Williams Jr., J. M. and Mohn, F.-W., 2004, “Trailer
speed, to utilize natural damping in the system. Stabilization through Active Braking of the Towing
Vehicle”, SAE paper No. 2004-01-1069.
14. Dixon, J. C., 1996, “Tires, Suspension and
Handling”, SAE, Inc., Warrendale, PA.
REFERENCES

1. Bundorf, R. T., 1967, “Directional Control Dynamics

APPENDIX
of Automobile-Travel Trailer Combination”, SAE
paper No. 670099, pp. 667-680.
Matrices in equation (6).
2. Kurtz, E. F. and Anderson, R. J., 1977, Handling
Characteristics of Car-Trailer Systems; A State-of-
the-Art Survey”, Vehicle System Dynamics, 6, pp.
217-243. ⎡m1 + m2 − m2 (c1 + a2 ) − m2 a2 0⎤
⎢ mc I z1 0 0⎥⎥
M=⎢ 1 1
3. Deng, W. and Kang, X., 2003, “Parametric Study on
(A1)
Vehicle-Trailer Dynamics for Stability Control”, SAE ⎢ − m2 a 2 I z 2 + m2 a2 (c1 + a 2 ) I z 2 + m2 a 22 0⎥
paper No. 2003-01-1321. ⎢ ⎥
⎣ 0 0 0 1⎦

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⎡ C + C yr + C yt − C yf a1 + C yr b1 + C yt (c1 + l 2 ) − (m1 + m 2 )v x2 C yt l 2 ⎤
⎢ − yf C yt ⎥
⎢ vx vx vx ⎥
⎢ ⎥
⎢ C yf (a1 + c1 ) + C yr e1 − C yf a1 (a1 + c1 ) + C yr b1e1 − m1c1v x2
0 ⎥
D = ⎢− vx vx
0
⎥ (A2)
⎢ ⎥
⎢ C yt l 2 − C yt l 2 (c1 + l 2 ) + m 2 a 2 v x2 C yt l 22 ⎥
⎢ − − C yt l 2 ⎥
⎢ vx vx vx ⎥
⎣⎢ 0 0 1 0 ⎦⎥

⎡ C yf 0 0⎤ ⎡1 1 ⎤
⎢C (a + c ) 1 0⎥⎥ ⎢c 0 ⎥⎥
E = ⎢ yf 1 1 F=⎢ 1 (A3, A4)
⎢ 0 0 1⎥ ⎢0 − a2 ⎥
⎢ ⎥ ⎢ ⎥
⎣ 0 0 0⎦ ⎣0 0 ⎦

Matrices in equation (8)

⎡ C yf + C yr − C yf a1 + C yr b1 ⎤ ⎡ C yf ⎤
⎢ − − vx ⎥ ⎡ 0 ⎤ ⎢ ⎥
m1v x m1v x ⎥ , B = ⎢ 1 ⎥ , G = ⎢ m1 ⎥
Av = ⎢ (A5)
⎢ − C yf a + C yr b1 C yf a12 + C yr b12 ⎥ v
⎢I ⎥
v
⎢ C yf a1 ⎥
⎢ 1
− ⎥ ⎣ z1 ⎦ ⎢ ⎥
⎢⎣ I z1v x I z1v x ⎥⎦ ⎣ I z1 ⎦

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