Lefèvre et al. ROBOMECH Journal
2014, 1:1
Page 3 of 14
http://www.robomechjournal.com/content/1/1/1
review is provided of the different methods for predicting
trajectories using these evolution models. Finally, the lim-
itations of Physics-based motion models are addressed in
Se ction ‘Limitations’.
Evolution models
Dynamic models
Dynamic models describe motion based on Lagrange’s
equations, taking into account the different forces that
affect the motion of a vehicle, such as the longitudinal and
lateral tire forces, or the road banking angle [1]. Car-like
vehicles are governed by complex physics (effect of driver
actions on the engine, transmission, wheels etc.), therefore
dynamic models can get extremely large and involve many
internal parameters of the vehicle. Such complex mod-
els are relevan t for control-oriented applications, but for
applications such as trajectory predic tion simpler models
are preferred. They are of ten based on a “bicycle” repre-
sentation, which represents a car as a two-wheeled vehicle
with front-wheel drive moving on a 2-D plane. Examples
of such simple dynamic models are found in several works
[2-7].
Kinematic models
Kinematic models describe a vehicle’s motion based on
the mathematical relationship between the parameters of
the movement (e.g . position, velocity, acceleration), with-
out considering the forces that affect the motion. The
friction force is neglected, and it is assumed that the
velocity at each wheel is in the direction of the wheel
[1]. Kinematic models are far more popular than dynamic
models for trajectory prediction because they are much
simpler and usually sufficient for this type of applica-
tions (i.e. applications which are not vehicle control). In
addition the internal parameters of a vehicle needed by
dynamic models are not observable by exteroceptive sen-
sors, which rules out the use of dynamical models for
ITS applications involving other vehicles than the ego-
vehicle. A survey of ki nematic models for car-like vehicles
wasdonebySchubertetal.[8].Thesimplestofthese
are the Constant Velocity (CV) and Constant Acceleration
(CA) models , which both assume straight motion for vehi-
cles [9-13]. The Constant Turn Rate and Velocity (CTRV)
and Constant Turn Rate and Acceleration (CTRA) models
take into account the variation around the z-axis by intro-
ducingtheyawangleandyawratevariablesinthevehicle
state vector [10,12,14-17]. The complexity remains low as
the velocity and yaw rate are decoupled. By considering
the steering angle instead of the yaw rate in the state vari-
ables, one obtains a “bicycle” representation, which takes
into account the correlation between the velocity and the
yaw rate. From this representation, the Constant Steer-
ing Angle and Velocity (CSAV) and the Constant Steering
Angle and Acceleration (CSAA) can b e derived.
Trajectory prediction
The evolution models described above can be used for
trajectory prediction in various ways, the main difference
being in the handling of uncertainties.
Single trajectory simulation
A straightforward manner to predict the future trajec-
tory of a vehicle is to apply an evolution model (see
Se ction ‘Evolution models)’ to the current state of a vehi-
cle, assuming that the current state is perfectly known
and that the evolution model is a perfect representation
of the motion of the vehicle. This strategy can be used
with dynamic models [2] or kinematic models [11,13,17],
and is illustrated in Figure 3. The advantage of this sin-
gle forward simulation is it s computational efficiency,
which makes it suitable for applications with strong real-
time constraints. However the predictions do not take
into account the uncertainties on the current state nor
the shortcomings of the evolution model, and as a result
the predicted tra jectories are not reliable for long term
prediction (more than one s econd).
Gaussian noise simulation
Uncertainty on the current vehicle state and on its evolu-
tion can be modeled by a normal distribution [9,10,12,16].
The popularity of the “Gaussian noise” representation of
uncertainty is due to its us e in the Kalman Filter (KF).
Kalman filtering is a standard technique for recursively
estimating a vehicle’s state from noisy sensor measure-
ments. It is a special case of Bayesian filtering where the
evolution model and the sensor model are linear
a
,and
uncertainty is represented using a normal distribution. In
a first step (prediction step) the estimated state at time t is
fed to the evolution model, resulting in a predicted state
for time t + 1 which takes the form of a Gaussian distribu-
tion. In a second step (update step) the sensor measure-
ments at time t + 1 are combined with the predicted state
into an estimated state for time t + 1, which is also a Gaus-
sian distribution. Looping on the prediction and update
Figure 3 Trajectory prediction with a constant velocity motion
model (based on [13]).