Smooth trajectory planning along Bezier curve for mobile robots with velocity constraints

Abstract

This paper presents a smooth path planning method considering physical limits for two-wheeled mobile robots (TMRs). A Bezier curve is utilized to make an S-curve path. A convolution operator is used to generate the center velocity trajectory to travel the distance of the planned path while considering the physical limits. The trajectory gained through convolution does not consider the direction angle of the TMR, so a transformational method for a center velocity trajectory following the planned path as a function of time of parameter for the Bezier curve is presented. Finally, the joint space velocity is computed to drive the TMR from the center velocity. The effectiveness of the proposed method was performed through numerical simulations. This algorithm can be used for path planning to optimize travel time and energy consumption.

Full text

International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
225
Smooth Trajectory Planning Along Bezier Curve for Mobile Robots
with Velocity Constraints
Gil Jin Yang and Byoung Wook Choi
Department of Electrical and Information Engineering
Seoul National University of Science and Technology, Seoul, South Korea
yang6495@gmail.com, bwchoi@seoultech.ac.kr
Abstract
This paper presents a smooth path planning method considering physical limits for two-
wheeled mobile robots (TMRs). A Bezier curve is utilized to make an S-curve path. A
convolution operator is used to generate the center velocity trajectory to travel the distance of
the planned path while considering the physical limits. The trajectory gained through
convolution does not consider the direction angle of the TMR, so a transformational method
for a center velocity trajectory following the planned path as a function of time of parameter
for the Bezier curve is presented. Finally, the joint space velocity is computed to drive the
TMR from the center velocity. The effectiveness of the proposed method was performed
through numerical simulations. This algorithm can be used for path planning to optimize
travel time and energy consumption.
Keywords: TMR, Bezier curve, Smooth path planning, Convolution, Physical Limits
1 Introduction
Two-wheeled mobile robots (TMRs) are recently becoming widely used as cleaning robots
and intelligent service robots; thus, extensive research is underway on trajectory planning to
minimize energy and optimize traveling time as well as to resolve issues regarding smooth
traveling toward the desired destinations in workspaces [1-4].
A navigation system for a TMR largely consists of a path planner, a trajectory generator
and a tracking controller. Path planning is about generating smooth paths while maintaining
the desired position in workspaces. The trajectory generator aims to generate a velocity
profile for the planned paths as a function of time. The tracking controller and driving
controller are control systems that allow a TMR to travel along the predefined trajectory at a
desired time while staying within its physical limits.
If the physical limits of a TMR during path planning and trajectory generation are
considered, potential damage to a TMR can be reduced; trajectory tracking accuracy and
tracking velocity can be improved [3]. To this end, velocity trajectory planning methods using
a convolution operator have been suggested that consider the physical limits of a TMR in
workspaces [2, 5]. However, the suggested method did not consider the direction angle,
which is part of a TMR ics. The method only considered the translational
velocity and path as opposed to the center of a TMR in Cartesian coordinates. A smooth
path planning method that considers the initial and final direction angles is the basic goal in
path planning for a TMR.
Application to an actual TMR is also difficult since only the translational velocity limits at
the center point in Cartesian coordinates is considered, and not the physical limitations of
 Corresponding author

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226
actuators for driving the two wheels, which are dependent on the variations in angular
velocity.
When generating a trajectory for a TMR, modifying the directions of a TMR after it stops
during operation has been used because a discontinuity point may cause a slip or path
deviation. In order to overcome this issue, research on path planning with continuous
curvature for a TMR with kinematic limits has been conducted. Path planning methods have
been studied for a TMR arriving in a desired position based on a starting position and
direction angle using a Bezier curve [6].
In this study, a path based on a Bezier curve was generated in order to build a smooth path
while considering the direction angle. A convolution operator was used to generate the central
velocity to travel the planned path. In this process, the velocity trajectory can be generated
while considering the maximum velocity and acceleration according to the physical limits of
a TMR. The velocity trajectory gained through convolution is a trajectory which a robot
travels such that the given distance does not consider the direction angle of the TMR. In order
to consider the direction angle of the TMR, a transformation method for the trajectory is
presented that consists of segmented paths along the designed Bezier curve with the central
velocity generated through convolution. The trajectory obtained through the transformation
process can be used for the TMR to smoothly follow the planned path while staying within
the physical limits. Finally, a trajectory generation method in joint space that can be used as
an actuator command for the TMR driving is proposed. The joint space trajectory limits the
             
limitations that depend on the direction angle of the central velocity.
In order to determine the effectiveness of the proposed method, numerical simulations
were performed. The application of the planned trajectory to a simulator showed that the
robot carried out desired tasks well while staying within its physical limits. This trajectory
can be used for path planning to optimize time and energy consumption.
2. Bezier Curve based Path Planning
As shown in Figure 1, a TMR is represented in the coordinate system using the
    direction angle. The position consists of a world frame
coordinate system and robot frame coordinate system. A TMR    Pc is defined
on the coordinate system as follows:
 
T
cccc θ,y,xP 
,
(1)
where xc, yc, θc      direction angle respectively. TMR
kinematic model can be obtained as follows:
























l
r
c
c
c
c
cω
ω
D
r
sinθ
2
r
cosθ
2
r
D
r
sinθ
2
r
cosθ
2
r
P
,
(2)
where r denotes the radius of a  D denotes the distance between its two
wheels, ωr denotes       angular velocity, and ωl denotes      
angular velocity.

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227
vl
vr
vc
xc
yc
θc
ωc
x
y
0
Figure 1. Kinematics of TMR
When planning a path for TMR, the position and direction angle at its starting point
and destination should be considered and a curved trajectory is commonly generated
using Bezier curves [6]. As shown in Figure 2, a trajectory is generated using a Bezier
curve consisting of an initial point Pi(A0, B0), end point Pf(A3, B3), and control points
C1(A1, B1) and C2(A2, B2). An equation for the Bezier curve is calculated using C1 and
C2. The equation of Bezier curve is given below in equation (3).
θi
C1
C2
Pi
Pf
θf
d1
d2
yc
xc
Figure 2. Bezier Curve-based path planning


3
0)()( iin,iuJAux
,
3
3
2
2
2
1
3
0)()()( uAu1u3Au1u3Au1A 
(3-a)


3
0)()( iin,iuJBuy
,
3
3
2
21
3
0)()()( uBu1u3Bu1u3Bu1B 2
(3-b)
In equation (3), u is an arbitrary value where 0 ≤ u ≤ 1 and can be used to generate a
smooth curve from a starting point to a target point: a more precise Bezier curve with a
smaller increase. The path given by equation (3) does not consider velocity and is only
parameterized by u.

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228
3. Convolution based Trajectory Planning Following Bezier Curve
There has been research that the path generation method may use a convolution operator to
create a central velocity trajectory of a TMR for smooth path generation while satisfying
physical limits [2, 3].
In order to use convolution, a square-wave function y0(t) is defined as follows:



otherwise
tt
,
,v
ty 00 0
0
)(
0
,
(4)
where the nth-applying convolution function hn(t) is defined as a square-wave function with
the unit area in 0 ≤ t ≤ tn as follows:



otherwise
tt0
,0
,t
)t(h nn
n1
.
(5)
If function yn(t) is a resulting function to which the nth convolution is applied, the result of
convolution y0(t) and h1(t) can be represented as y1(t) and y2(t) denotes the result of y1(t) and
h2(t) convolution. The velocity function vc(t) generates the velocity command of the
differentiable S-curve that considers the maximum velocity vmax for the robot to travel the
distance S, as shown in Figure 3.
vi
vf
vmax
t0t0+t1+t2
vc(t)
S
Figure 3. Convolution-based velocity command trajectory
Let a Bezier-curve-based path as shown in Figure 2 that considers the direction angle using
a constant value u be ρ(u). The distance traveled is calculated using formula (6) to generate
the central velocity trajectory for the robot to travel along the distance S, as shown in Figure 3.
The curved distance Bd along the path ρ(u) from Pi to Pf as in Figure 2, is calculated as
follows:




1
0
1
0
2
))()((
2
))()(()( uu uyΔuuyuxΔuuxuΔρ
d
B
(6)
The calculated distance Bd is the actual distance traveled along the path designed with
Bezier curve which has a smooth curve. To generate the center velocity trajectory of a TMR
using convolution, the distance S is thus used as an input value. Therefore, if the center
velocity trajectory vc(t) is generated to have the traveling distance as S = Bd, then the
trajectory using the advantages of convolution while considering velocity limits can make a
smooth path. Here, vi, vf, vmax and the sampling time can be arbitrarily set according to the
specifications of the TMR [2-3].
The generated central velocity trajectory of vc(t), as shown in Figure 3, travels along the

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229
distance S . However, the central velocity trajectory of TMR does not consider the direction
of the robot. In other words, for any position (x(ui), y(ui)), the robot travels with velocity vc(ti),
as shown in Figure 2. In equation (2), the subsequent position can be moved to an entirely
different position depending on the angle θi. In order to consider the positions in task space
that depend on velocities in paths with direction angles, the parameter u(t) of Bezier curve for
the distance during the sampling time should be determined and calculated using equation (7).
The trajectory ρ(u(t)) with the direction angle can be obtained by inputting the determined
u(t) into equation (3). In ρ(u(t)) , if the sampling time is shorter, the path can more accurately
follow ρ(u) as generated by constant parameter value u.
d
ttt
tc
B
tv
tu 



210
0)(
)(
(7)
Here, u(t) is defined as 0 ≤ u(t) ≤ 1 and represents the parameter of the Bezier curve that
depends on the central velocity. The trajectory generated by using u(t) satisfies the maximum
velocity allowed by the physical limits of a TMR while following the curved path with
respect to the direction angles.
4. Simulation Results
Figure 4 illustrates the central velocity trajectory that satisfies the physical limits
from the starting point Pi    ) to the target point Pf   and a Bezier curve
trajectory tracking it. In this figure, the distance between positions of the trajectory is
the distance driven by the central velocity function during sampling time. The results
show that the synthesized Bezier curved trajectory was generated depending on the
    Direction angles ∆θ at each section and the
angular velocity ωc at the center were calculated as follows and are illustrated in Figure
4.
))(( ))((
1tuΔxtuΔy
tanΔθ 

,
(8)
tΔ
Δ



c
(9)
Figure 4. Smooth trajectory, direction angle and angular velocity considering
maximum velocity limits

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The trajectory for the TMR was generated as shown in Figure 4. The generated
trajectory satisfies the physical limits described above and allows the TMR to travel
along the curved path using its central velocity. The actual command for actuating the
TMR is the angular velocity for both wheels. It can generate wheel velocity commands
in joint space using equations (10) and (11):
)2(
)2(
ccl
ccr
ωD/v
r
1
ω
ωD/v
r
1
ω


,
ll
rr rωv
rωv


.
(10)
(11)
The velocity command trajectory for two wheels obtained by formula (11) becomes
the actual velocity command for the TMR     

c is
   translational velocities as shown
in equation (12):
Dv-v
ωlr
c
.
(12)
Figure 5 shows that when physical limits are vmax=0.5m/s, amax=0.2m/s2 and
jmax=0.2m/s3, then the central velocity trajectory satisfies the physical limits moving
from the starting position  to the target position . When the velocity
commands for the two wheels are generated, angular velocity shown in Figure 4 is used.
The joint velocity commands for two wheels is used to drive two wheels to follow the
Bezier curve based trajectory
Figure 5. Velocity commands of two-wheels following smooth trajectory
Figure 6 shows the TMR  
velocity commands for two wheels as shown in Figure 5. The results show that the
robot successfully followed the Bezier curve along the planned path. Figure 7 shows the
simulation results driven by actuator commands on the anyKode, Marilou Robotics
Studio [10].

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231
Figure 6. S-Curve and Trajectory of TMR
Figure 7. Trace of TMR driven by Actuator Velocity Commands
Figure 8 shows the tracking error between the Bezier curve and the trajectory
generated according to sampling time of 1ms, 50ms, 10ms in equation (7). The error
increases as angular velocity and sampling time increases. The effect of sampling time
should be considered to control mobile robot.
Figure 8. Error Gap While Travelling Along an S-Curve

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The previous curve is S-curve so the tracking error could be compensated in whole
path. We applied the proposed algorithm to another path which has C-curve. Another
path is shown as a C-curve in Figure 9. The figure shows that when physical limits are
vmax=0.5m/s, amax=0.2m/s2 and jmax=0.2m/s3, the central velocity trajectory satisfies the
physical l                    

Figure 10 shows the velocity commands of the two wheels following the path shown
in Figure 9. In C-curve path, heading angle of TMR is changing monotonically. In this
case, Figure 11 shows trajectory of TMR driven by the joint velocity commands
described in Figure 10, which follows C-curve. Compared to the S-curve shown in
Figure 6, the error is greater in the C-curve.
Figure 9. Smooth trajectory, direction angle and angular velocity considering
maximum velocity limits
Figure 10. Velocity commands of two-wheels following smooth trajectory

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Figure 11. Trajectories of TMR and trace of TMR driven by actuator velocity
commands
Figure 12. Error Gap While Travelling Along the C-Curve
Figure 12 shows the error of the path between the ideal parametric Bezier curve by using
equation (3) and the generated path considering physical limits with various sampling time by
using equation (7). The error was computed by finding the difference between the ideal
smooth trajectory (as shown by the solid-line graph in Figure 11) and the trajectory of the
Bezier curve (as shown by the dotted-line graph in Figure 11). The tracking error resulted
from sampling time is also shown Cartesian trajectory simulated using the joint velocity
commands. Compared to Figure 8, the errors in x coordinates become larger as tracking C-
curve path. The shape of the error gap is different in depending on the target path shape.
             
decreased.
5. Conclusions
A velocity command trajectory generation method was proposed that enables a TMR to
                
smooth run and control.

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The proposed velocity trajectory generation method generates a trajectory to satisfy the
maximum velocity as opposed to a central velocity of TMR using the characteristics of
convolution, and the central velocity trajectory follows the Bezier curve based path to travel
smoothly. In the future, this trajectory generation method can be applied to obstacle
avoidance algorithms that satisfy velocity limits at the any points. Research on continuous
path generation at the any points to satisfy physical limits is currently underway.
Tracking errors according to the sampling time during convolution and
transformation process was examined. For smooth control, the effect of sampling time
should be considered. If the velocity trajectory performs at a real-time operating system,
then tracking error can be reduced. The performance evaluations of real-time
mechanisms can predict the tracking error according to the system, performance
evaluations of real-time mechanisms [7]. It can also analyze tracking error and apply
the results to the controller so that tracking error can be reduced [8-9].
The path planning method proposed in this paper can be utilized for a path planning with
optimized travelling time and an energy-efficient path planning that considers the limited
battery power of a running robot.
Acknowledgements
This research was supported by Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and
Technology (No. 2012-006057).
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