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Toward Human-like Motion Planning in Urban Environments
Tianyu Gu1and John M. Dolan2
Abstract— Prior autonomous navigation systems focused on
the demonstration of the technological feasibility. But as the
technology evolves, improving user experience through learning
expert’s or individual’s driving pattern emerges as a promising
research direction. As a first step toward this goal, we inves-
tigate methods to learn from human demonstrations in urban
scenarios without any environmental disturbances (traffic-free).
We propose a path model that generates a reference path with
smooth and peak-value-reduced curvature, and a parameterized
speed model to be fitted by human driving data. Model
parameters are then learned through regression methods, and
certain statistical human driving patterns are revealed. The
learned model is then evaluated by comparing the generated
plan with the collected data by the same human driver.
I. INTRODUCTION
The high cost of car crashes, the trillions of hours squan-
dered in traffic jams, and the lost urban space given over
to parking lots yield many difficult transportation problems.
Autonomous passenger vehicle technology is a promising
solution to those problems.
Work by Dickmanns [1] and NavLab [2] at Carnegie
Mellon University demonstrated semi-autonomous driving
with vision-based nagivation systems in the late 1980s.
Since then, many path generation schemes, like arc-line
[3], spiral[4] and polynomial spline[5], have been developed
for mobile robots. These path generation methods specify
only the geometric curve (path), without simultaneously
generating the speed plan (trajectory). Trajectory planning
was then studied in many race car projects. [6], [7] and
[8] used convex optimization routines to generate trajectories
that push the race cars to their limits, accounting for complex
vehicle dynamics. These offline methods are computationally
unaffordable for real-time application.
The 2005 Grand Challenge and the 2007 Urban Challenge
made further progress in creating fully autonomous vehicles.
For both contests, maps were given, on which the entries
used high-accuracy GPS to register the host vehicle. Grand
Challenge-winning entry Stanley [9] sampled trajectories in
the control space based on the waypoints corridor from the
map. Urban Challenge-winning entry Boss [10] spawned
short-horizon reactive trajectories by laterally shifting from
the lane centerline and picking the fastest collision-free one
for execution. The reactive nature of both short-horizon plan-
ning methods makes them only applicable for the simplified
contest environments.
1Tianyu Gu is with the Department of Electrical & Computer Engineer-
ing, Carnegie Mellon University tianyu@cmu.edu
2John M. Dolan is with the Department of Electrical & Computer Engi-
neering and the Robotics Institute, School of Computer Science, Carnegie
Mellon University
A deliberative planning quality was later introduced in
several lattice-based trajectory sampling methods [11], [12].
Reactiveness was maintained through exhaustive sampling
and the computation overhead was significant. [13], [14]
reduced the computational overhead by performing two-step
(coarse and fine) trajectory planning.
These prior works have demonstrated the technological
feasibility of achieving vehicle autonomy. They usually
had weighted cost terms as the measurement of optimality.
Costs were often manually tuned for maximum performance
(speed), rather than systematically adjusted for passenger
comfort based on human driving patterns. Only a few works
have concerned human driving patterns, for example: [15]
investigated the proper speed model that explains a driver’s
modulation of speed when entering or exiting a curve. [16]
attempted to identify abnormal on-road driving behavior in
freeway situations by learning a steering-speed mapping.
Both works assumes overly simplified models that were
not expressive in characterizing general curve negotiation,
especially for high-curvature turnings.
II. SCOPE OF STUDY
“Human-like” planning is defined as an effort to imitate
human driving patterns while staying within the bounds of
safe driving. Individuals are likely to have different driving
styles; for example, a smooth turning maneuver by an elderly
driver may be too slow for young drivers.
In the high-level view of our motion planner (Fig. 1),
traffic-free planning generates a reference trajectory consid-
ering only road geometry while other traffic and obstacles
are ignored. Traffic-based planning produces a spatially and
temporally varied reference based on the traffic-free plan to
account for other traffic and interfering objects. The tracking
trajectory module then generates a dynamically feasible tra-
jectory through model-based evaluation for tracking control.
Controller InterfacingReference Planning
Traffic-Free
Planning
Traffic-Based
Planning
Tracking Trajectory
Generation
Fig. 1: Planning Structure. On-road navigation uses two plan-
ning phases (traffic-free & traffic-based) and one trajectory
generation phase for tracking control. The focus of this paper
is on the traffic-free reference planning in bold font.
The focus of this paper is to investigate the appropriate
model and the parameter learning methods for urban traffic-
free planning, as illustrated by Fig. 2. The rest of this paper
is arranged as follows. Section III proposes the expressive
Traffic-free
Path Models
Traffic-free
Speed Models
Map Input Reference Trajectory
Parameter
Identification
Human Driving Datum
Learning Component
Planning Component
Fig. 2: Overview of human-like traffic-free model learning
and planning. It consists of a planning component that
generates the human-like traffic-free reference plan, and a
learning component that performs parameter identification
and statistical interpretation.
trajectory models to capture human driving patterns. Section
IV formulates the model identification problems. Sections
V and VI illustrate the results of learning, summarize the
contribution and propose future work.
III. MODEL DESIGN
In this section, models are proposed to fit human maneuver
data and to be efficiently evaluated for regeneration. To
efficiently serve these two purposes, path and speed models
are devised and explained independently.
A. Path Model
The path model makes use of a pre-stored digital map
to generate a smooth reference path. Road information is
represented by a sequence of map waypoints in a global
two-dimensional plane, augmented with lane width as well as
speed limits. The map waypoints are usually not distributed
uniformly, the interval sometimes spanning from 5m to
300m. To generate a smooth reference path in a sequence
of dense and uniformly sampled waypoints, the proposed
path model has three steps (Fig. 3).
Interpolation
Smoothing
Curvature Reduction
Map Waypoint
Reference Path
Fig. 3: Path Model. The path model takes three steps to
generate a smooth human-like reference path, including
interpolation, smoothing and curvature reduction.
The centerline is first generated by connecting 2D map
waypoints (X, Y ) with interpolation. Neither the xnor the y
coordinates of the map waypoints are necessarily monotone.
We create a monotonically increasing variable “station” (s)
by (linearly) estimating the longitudinal position of each
waypoint. Splines interpolate the map waypoints smoothly
by separately generating two spline functions x(s)and y(s),
which are then evaluated at uniformly sampled stations to
represent the centerline in a non-parametric form:
{si,[xi,y
i],✓i,i}
where si,[xi,y
i],✓iand irespectively represent the ith lon-
gitudinal station, global coordinates, global heading and path
curvature. Note that with the polynomial representations, ✓i
and ican be calculated analytically from their definitions.
However, this centerline can easily create high-curvature
bumps near waypoints, which causes un-humanlike jerky
steering with controllers sensitive to the jerky curvature. Note
that experienced drivers do not always stay precisely on
the centerline. We use a local smoothing routine to create
a smooth result. Two independent smoothings (quadratic
least-square fitting) are performed locally to get two smooth
polynomial functions x⇤(s)and y⇤(s). The size of the
smoothing window is chosen to constrain the maximum shift.
Polynomial evaluation is conducted similarly as before to
obtain a non-parametric form:
{si,[x⇤
i,y
⇤
i],✓⇤
i,⇤
i}
Experienced drivers also take advantage of lane width to
reduce the maximum curvature during tight turn negotiations.
The subsequent optimization for curvature reduction provides
visible benefits in such situations. It minimizes the following
cumulative term by laterally nudging the points {pi}:
{o⇤
i}= argmin
{oi}X|pipi1
|pipi1|pi+1 pi
|pi+1 pi||(1)
where
pi=x⇤
ioi·sin(✓⇤
i)
y⇤
i+oi·cos(✓⇤
i)
where oiis the lateral nudge distance. Note that this opti-
mization is applied over the region whose absolute value of
curvature is above ⇢·max, where max is the maximum
curvature corresponding to the steering angle limit of the
vehicle and ⇢is a percentage threshold.
The optimality criterion has a least-square formulation.
The Levenberg-Marquardt algorithm is used to numerically
solve this optimization problem quickly in a local sense. The
overall output of the path model is the reference path.
B. Speed Model
With the smooth reference path generated, the speed model
specifies the speed of the vehicle’s motion. Together, they
form a reference trajectory. The trajectory determines many
dynamic indicators that are important to a passenger’s expe-
rience of comfort, like lateral & longitudinal acceleration.
Speed Generation
(Normal)
Speed Generation
(Tight-turn)
Smoothing
Reference Path
Reference Trajectory
Fig. 4: Speed Model. The speed model takes three steps
(normal & tight-turn speed generation and smoothing) to
generate a smooth human-like reference trajectory.
A three-step speed model is designed to reflect a human
driving pattern (Fig. 4). The first step uses the geometry
information of the reference path to generate a baseline
speed profile for general urban driving cases. For many
road segments, speed limits can be obtained from the map.
Driving at these speed limits might already yield the optimal
speed model, such as in most highway situations. For situ-
ations like driving on many curvy urban roads or traversing
intersections, however, a valid nominal speed might not exist
in the map, so a proper speed profile should be generated in
response to the path geometry. We use the normal speed
model Mnormal:
{vi}=Mnormal({i},v
max,P)(2)
such that
vivmax
i·vi2alat
˙vialon
˙vidlon
where {i}is the curvature sequence of the reference path,
vmax is the speed limit, and P=[alat,a
lon,d
lon]Tare the
tunable parameters that consist of preferred lateral accelera-
tion, longitudinal acceleration and deceleration, respectively.
An iterative numerical algorithm [14] is used to apply
several constraint parameters, including speed limit, lateral
acceleration and longitudinal acceleration and decelerations.
Instead of using the vehicle’s extreme values of these con-
straining terms, human-preferred values are identified and
applied. Note that by changing the maximum speed at one
particular reference point to zero, we can generate stopping
speed profiles.
Mnormal restricts the local curvature peak to correspond
to a local speed minimum. While sufficient for most urban
driving, driving behaviors for tight1turning maneuvers can-
not be modeled precisely, since human drivers tend to behave
in a cautious manner by decelerating earlier to perform
corner negotiation at a lower speed, as shown in Fig. 5. Both
drivers’ minimum speed is reached prior to the maximum
curvature. The absence of this feature could potentially cause
passengers’ anxiety both when entering and exiting the turns.
To model this driving pattern, several path features are
marked first to position the speed profile as shown by the
upper plot in Fig. 6, including local peak curvature peak, the
center point spand the length lprepresenting the “principal”
turning region of a tight turning maneuver, defined by the
segment whose curvatures are above ⌘·peak. A tight-turn
specific speed method Mtight is proposed to describe a three-
phase piecewise-linear process that consists of decelerate-to-
enter, maintain-low-speed, and accelerate-to-exit, as shown
by the lower plot in Fig. 6:
{vi}=Mtight({i},Q)(3)
1A “tight” turning manuver is defined as a turning process whose peak
curvature is above certain threshold, 0.07 in our setup.
Fig. 5: Tight Turning Maneuver by Human. Human driving
data of tight turning maneuvers are collected for two individ-
ual drivers on the same road segment. Driver 2 has a more
cautious driving style than driver 1 by slowing down earlier
and more significantly.
where {i}is the curvature sequence of the reference path,
and Q=[vmin,s
,l
v,˜alon,˜
dlon]Tdefines the shape of the
speed profile. Another parameter s=spsvdefines the
longitudinal distance from the centerpoint of the “principal”
turning region to the local speed minimum point.
Station
Longitudinal Spped
Decelerating
Enter
Low-Speed
Maintain
Accelerating
Exit
lv
sv
!
dlon
!
alon
vmin
Station
Curvature
κ
peak
η
⋅
κ
peak
lp
sp
sΔ
Fig. 6: Tight Turn Speed Model. Plot above illustrates the
path features on curvature plot. Plot below shows the three-
phase speed model for a tight-turning maneuver.
Note that Mtight, as well as Mnor mal, potentially yields
speed profiles with non-smooth (huge longitudinal jerk
value) transition points connecting linear segments. In order
to improve the speed smoothness, the third step constrains the
numerically interpreted jerk iteratively until the maximum
jerk value is below a certain threshold.
|¨v|jlon (4)
IV. MODEL FITTING
The proposed models have multiple parameters to be fitted
from the human driving data. The path model performs
smoothing and optimization with respect to two curvature-
related criterias, i.e. smoothness and peak value. We assume
that a natural human driving pattern is to achieve optimalities
in these two aspects, in which case, path model fitting (and
subsequent learning) becomes unnecessary. The focus of this
section is to identify the parameters of the speed models
Mnormal and Mtight .
The parameters Pof Mnormal are fitted by performing an
optimization that minimizes the least-square error for model
Mnormal and human driving data:
ˆ
P= argmin
P
k{vhuman
i}Mnormal({i},v
limit,P)k(5)
where {vhuman
i}represents the human driving speed data
for normal urban curves. Note that the parameter alat in P
can be determined by other methods, as will be shown in the
following section. Only the two remaining parameters in P
will be identified with equation (5).
The parameters Qof model Mtight are fitted with an
optimization via a similar least-square error formulation:
ˆ
Q= argmin
Q
k{vhuman
i}Mtight({i},Q)k(6)
where {vhuman
i}represents the human driving speed data
on tight turns. Note that the parameter vmin in Qcan be
obtained simply by scanning the human driving data to fit.
Only the four remaining parameters in Qwill be identified
with equation (6).
In Fig. 7, an example turning maneuver is fitted with the
appropriate optimization routine above, in which the speed
model shows good adaptability to the human driving pattern.
0 5 10 15 20 25 30 35
0
0.02
0.04
0.06
0.08
0.1
Station(m)
Curvature(1/m)
Human Data by Driver 2
0 5 10 15 20 25 30 35
4.5
5
5.5
6
6.5
7
7.5
Station(m)
Speed(m/s)
Human Data by Driver 2
Fitted Profile by Tight−turn Speed Generation
Fitted Profile by Tight−turn Speed Generation with Jerk Smoothing
Fig. 7: Speed Model Fitting Example. The black dashed star
curves represent the human driving. The blue solid circle
curve is the speed profile fitting speed model Mtight. The red
dashed square curve is after further applying jerk smoothing.
V. E XPERIMENTS AND RESULTS
Two experiments were designed to support the claims of
this paper. The first explains parameter learning results of
the proposed speed models. The second evaluates the learned
model by comparing the plan with real human driving data.
A. Parameter Learning
1) Data collection: In order to obtain human driving
data, we set up a differential GPS-based localization system
mounted at the center of the rear axle. An experienced human
driver was required to drive in a comfortable manner in three
distinct scenarios including highway (S1), urban roads (S2)
and parking lots with multiple tight turns (S3), as illustrated
in Fig. 8. The high-accuracy output data consisting of po-
(a) (b) (c)
Fig. 8: Three experiment scenarios. (a) S1: Highway I376
and I79 connecting S2and S3. (b) S2: Urban driveways near
Schenley Park, Overlook Dr, Pittsburgh, PA, USA. (c) S3:
parking lots near Ernie Mashuda Dr, Cranberry, PA, USA.
sition, heading and path curvature, as well as longitudinal
speed and acceleration were recorded:
{[xhuman
i,y
human
i,✓human
i,human
i,v
human
i,a
human
i]}
2) Learning: Learning was performed to distill and dis-
cover important characteristics of human driving pattern. Due
to the variations in human behavior, two types of statistical
regression approaches were performed after identifying the
model parameters at the multiple curve negotiation locations.
Three parameters of the speed model Mnormal must be
identified. We use an asterisk to represent a statistically
learned parameter, e.g. P⇤=[a⇤
lat,a
⇤
lon,d
⇤
lon]T. The pre-
ferred lateral acceleration a⇤
lat is first learned from the data
by creating a state-action mapping, where the “state” is
the curvature of the reference path and the “action” is the
value of alat. Scattering {[human
i,v
human
i]}data points
on the curvature-speed plane (scattered grey circles in Fig.
12) makes a clear capping curve visible to characterize
the preferred maximum speed at a given path curvature.
However, it is unknown what the analytical expression should
be. [17] proposed a computational search method (Fig.
9) that performs symbolic regression on the experimental
data, simultaneously fitting both the symbolic form2and its
parameters.
Allowed operators:
- Constant
- Addition
- Subtraction
- Multiplication
- Division
- Exponential
- Logarithm
Symbolic Regression
Software — “eureqa”
Experimental data:
- (capping) curvature-
speed pairs
Analytical
model
Fig. 9: Symbolic Regression. The symbolic regression rou-
tine takes the allowed operations to distill an analytical model
that best explains the experimental data.
In practice, we first find an analytical expression v⇤
max()
that describes the capping curves based on the data points
collected from all S1,S2and S3scenarios with the analytical
2By restricting the allowed operators, we obtain certain control over the
complexity of this symbolic form.
(a)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
(1/m)
alat
* (m/s2)
(b)
Fig. 10: Symbolic regression result from human driving data.
(a) shows the scattered {[human
i,v
human
i]}human driving
datum and the plot of analytical capping curve. (b) shows
the learned curvature!lateral acceleration mapping.
expression (Fig. 10a):
v⇤
max =0.0348
||+0.832
0.0515 + ||(7)
The mapping of alat⇤can further be calculated with the
following equation, as shown in Fig. 10b:
alat⇤()=·v⇤
max
2
=0.6922||
(||+0.0515)2+0.05791
||+0.0515 +0.0012
||
(8)
For a⇤
lon and d⇤
lon, multiple parameter values are obtained
by model fitting of different (traffic-free) curve negotiation
maneuvers from scenario S2. A relation can be discovered
by linear regression:
a⇤
lon(peak )=0.2453 + 6.7456 ·|peak|(9)
d⇤
lon(peak )=0.1366 + 10.5464 ·|peak|(10)
As for the speed model Mtight, five parameters must
be identified, i.e., Q⇤=[v⇤
min,s
⇤
,l
⇤
v,˜a⇤
lon,˜
d⇤
lon]T. Tight
turning maneuvers from scenario S3are used to fit the
multiple recorded turns. Similarly, linear regression learns
a statistical mapping of the parameters we are interested in.
v⇤
min(peak )=7.5534 28.4011 ·|peak|(11)
s⇤
(peak)=1.6591 + 50.0945 ·|peak |(12)
l⇤
v(lp)=1.1873 + 0.4517 ·lp(13)
˜a⇤
lon(peak )=1.3784 2.2145 ·|peak|(14)
˜
d⇤
lon(peak )=1.3746 + 1.8192 ·|peak|(15)
3) Discussion: The symbolic regression discovered a new
relation between a⇤
lat and the path geometry, which poten-
tially can be adapted to other human driviers with its three
coefficients. The other parameters showed both strong and
weak correlation. The closer their determination coefficients
(R2) are to 1, the stronger linearity can be inferred. Based
on Table I and the scattering plots of each parameter (Fig.
11), we discover some human driving patterns that can be
explained in a statistical sense:
1) v⇤
min demonstrates the strongest linearity. It shows that
the tighter the maximimum curvature of a negotiation,
TABLE I: Determination coefficients of linear models
R2
a⇤
lon
R2
d⇤
lon
R2
˜a⇤
lon
R2
˜
d⇤
lon
R2
v⇤
min
R2
s⇤
R2
l⇤
v
0.34 0.29 0.02 0.01 0.68 0.08 0.58
the lower the maintaining speed vmin will be. The
necessity of speed model Mtight is also demonstrated,
since the learned maintaining speed v⇤
min is signifi-
cantly lower than the preferred maximum speed v⇤
max
of speed model Mnormal.
2) s⇤
demonstrates a weak linear correlation with peak
overall. This relation can be interpreted as meaning that
as the maximum curvature of a maneuver increases,
the driver tends to behave more cautiously be slowing
down earlier. The linearity is stronger when peak is
smaller and vice versa, which may also imply that
the human driver tends to behave more unpredictably
during high-curvature negotations.
3) The mapping between lpand lvshows the second
strongest linearity. This has an intuitive explanation:
the longer the “principal” turning process lasts, the
longer the low-speed-maintaining process will be.
4) a⇤
lon and d⇤
lon from model Mnormal clearly show
stronger linearity than ˜a⇤
lon and ˜
d⇤
lon from model
Mtight, which also reflects that the human driver tends
to behave more unpredictably during high-curvature
negotations. But overall, they all have rather low de-
termination coefficient values, confirming the argument
made in [16] that the longitudinal accelerations have
a great variance depending on hard-to-access variables
like driver mood.
B. Model Evaluation
1) Evaluation: The learned models Mnormal ⇤and
Mtight⇤are fully specified by parameters P⇤and Q⇤,
respectively. To evaluate the quality of the models, speed
plans are generated and compared with two human driving
processes.
0 5 10 15 20 25 30
0.01
0.02
0.03
0.04
0.05
Station (m)
Curvature (1/m)
Curvature Log by Driver 1
0 5 10 15 20 25 30
9
9.5
10
10.5
Station (m)
Speed (m/s)
Speed Log by Driver 1
Speed Plan generated with Mnormal
(a)
0 5 10 15 20 25 30 35 40
0
0.05
0.1
0.15
0.2
Station (m)
Curvature (1/m)
Curvature Log by Driver 1
0 5 10 15 20 25 30 35 40
3
4
5
6
7
8
Station (m)
Speed (m/s)
Speed Log by Driver 1
Speed Plan generated with Mtight
(b)
Fig. 12: Learned speed model evaluation result. (a) evaluate
the planning output of Mnormal⇤on a mild turn. (b) evaluate
the planning output of Mtight⇤on a tight turn.
2) Discussion: From the evaluation results of both
Mnormal⇤and Mtight⇤speed models, it is illustrated that the
plan generated can rarely match perfectly with any one-time
human data. But we are able to generate traffic-free reference
0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
peak (1/m)
alon (m/s2)
(a)
0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
peak (1/m)
dlon (m/s2)
(b)
0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
peak (1/m)
alon (m/s2)
(c)
0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
peak (1/m)
dlon (m/s2)
(d)
0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16
3
3.5
4
4.5
5
5.5
6
6.5
7
peak (1/m)
vmin (m/s)
(e)
0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16
−5
0
5
10
15
20
25
peak (1/m)
s (m)
(f)
8 10 12 14 16 18 20 22 24 26 28
2
4
6
8
10
12
14
16
18
20
22
lp (m)
lv (m)
(g)
Fig. 11: Scattering plot of the speed model Mnormal and Mtight parameters. From subfigure (a) to (g), blue circles are the
scattered parameter values after model fitting. Black lines are the results after linear regression.
plans that capture the key human features (in the statistical
sense) of both mild and tight turning maneuvers. Moreover,
using these learned models guarantees the smoothness and
predictability of planning results.
VI. CONCLUSION
The general on-road planning is decoupled into traffic-
free and traffic-based planning procedures. In this paper,
efforts are made toward human-like traffic-free planning in
order to improve the user experience of autonomous driving.
Expressive and tunable trajectory models suitable for mim-
icking human traffic-free driving are proposed. Parameter
identifications are formulated and solved as least-square
optimization problems. The parameters are then generalized
statistically with regression methods in order to characterize
the driving patterns. Meanwhile, the regression character-
istics also reflect the sensibility of each parameter to the
unpredictability of human driving. Finally, the learned model
is evaluated on a new data set and is able to generate smooth
planning result that captures the user’s driving pattern.
One limitation of this work is that the reference speed
model is completely built on the geometric nature of the
reference path, while other factors affecting traffic-free hu-
man driving are ignored, like weather/road condition. An
immediate next step is to apply the learned model to the
planning system of a real autonomous car. Also, more
human driving data by different people must be collected and
analyzed in order to generate user-specific model parameters.
Moreover, human-like traffic-based planning models and
their parameter learning methods should be studied to handle
dynamic environments.
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