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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.
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International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
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,
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
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
actuators for driving the two wheels, which are dependent on the variations in angular
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:
 
cccc θ,y,xP
where xc, yc, θc      direction angle respectively. TMR
kinematic model can be obtained as follows:
where r denotes the radius of a  D denotes the distance between its two
wheels, ωr denotes       angular velocity, and ωl denotes      
angular velocity.
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
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).
Figure 2. Bezier Curve-based path planning
0)()( iin,iuJAux
0)()()( uAu1u3Au1u3Au1A
0)()( iin,iuJBuy
0)()()( uBu1u3Bu1u3Bu1B 2
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.
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
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:
ty 00 0
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:
)t(h nn
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.
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
))()(()( uu uyΔuuyuxΔuuxuΔρ
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
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
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.
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
))(( ))((
Figure 4. Smooth trajectory, direction angle and angular velocity considering
maximum velocity limits
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
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):
rr rωv
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):
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].
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
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
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
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
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
Figure 11. Trajectories of TMR and trace of TMR driven by actuator velocity
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.
             
5. Conclusions
A velocity command trajectory generation method was proposed that enables a TMR to
                
smooth run and control.
International Journal of Control and Automation
Vol. 6, No. 2, April, 2013
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.
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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... Bézier curves are also popular in mobile robots, where a feasible path is also consistent with vehicle dynamics. Yang and Choi [14] used a cubic Bézier curve to make an S-curve path. Choi et al. [15] used Bézier curves as a seed function for the path planning algorithm as an alternative to cubic splines. ...
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This work is devoted to the second order rational Bézier curve coefficients estimation. We present the methodology of unique coefficients for each type of ship computation. In the presented formulas of ship’s length, a draft and angular path combined with a drift path are used. This approach leads to the simplest and most accurate Maritime Autonomous Surface Ships (MASS) path modeling. Three rational curve control points are waypoints (WPT). Using WPTs as curve control points allows integrating a trajectory intuitive for the navigator with a path predicting model used as a reference in the control system. Research was done based on real-time data originating from the MASS autonomous trajectory tracking system. The presented mathematical modeling tool may be treated as the best way of future trajectory prediction due to low computation power required.
... [Gordon 1974], [Böhm 1984], [Farin 2000]. In the robotics field, Bezier curves are used to plan trajectories, collision and obstacle avoidance for mobile robots [Jolly 2009], [Škrjanc 2010], [Yang 2013]. Further, shape estimation of a wire-driven flexible robot with multiple bending sections is done using Bezier curves [Song 2015b]. ...
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This work provides a new methodology to reconstruct the shape of continuum manipulators using a curve based approach. Pythagorean Hodograph (PH) curves are used to reconstruct the optimal shape of continuum manipulators using minimum potential energy (bending and twisting energy) criteria. This methodology allows us to obtain the optimal kinematics of continuum manipulators. The models are applied to a continuum manipulator, namely, the Compact Bionic Handling Assistant (CBHA) for experimental validation under free load manipulation. The calibration of the PH-based shape reconstruction methodology is performed to improve its accuracy to accommodate the uncertainties due to the structure of the manipulator. The proposed method is also tested under the loaded manipulation after combining it with a qualitative Neural Network approach. Furthermore, the PH-based methodology is extended to model multi-section heterogeneous bodies. This model is experimentally validated for a closed loop kinematic chain formed using two CBHAs manipulating jointly a rope.
To realize high-speed running of a warp knitting machine, the shogging motion should not only meet the requirement of high dynamic response but should also satisfy high positioning accuracy. Due to the large location disturbance and the dynamic response delay in the interpolation method or the single velocity planning curve method, an electronic shogging system for a warp knitting machine based on the mixed-velocity planning curve is proposed in the present study. Through the analysis of the shogging motion combined with the knitted structure, the optimal resolution of the instruction signal is calculated, which is 725 pulses for one needle step, and the velocity loop bandwidth of the servo driver is optimized. In addition, the motor with a load inertia ratio close to 1 is also selected. Analysis of the shogging motion vibration curve confirms that the shogging motion has advantages of high positioning accuracy and high dynamic response under the mixed-velocity planning curve. The response performance with the mixed curve is 12.5% higher than that with the quintic polynomial, and the positioning accuracy of the mixed curve is 26% higher than that with uniform acceleration–deceleration curve.
This paper presents a methodology for optimizing pre-calculated collision-free paths of differential-drive wheeled robots. The main advantage of this methodology is that optimization is done by considering the kinematics and mechanical constraints of the mobile robot. In accordance to this proposal, the optimized path is achieved by applying recursively a local smoothing on an initial path which is originally modeled as a one-dimensional piecewise linear function. By this recursive smoothing, it can be ensured that the original piecewise linear function can be transformed into a smooth one that fulfill the constraints established by the kinematic equations of the wheeled mobile in terms of a minimum radius of curvature. As a result of this, a trajectory which guarantees lower power consumption and lower mechanical wear, is obtained. To show the better performance of the proposed approach, numerical simulation results are contrasted to those obtained from other reported methods with regards to path length, minimum radius of curvature, cross track error, continuity and resulting acceleration.
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For a large variety of Industrial applications dc motors are being used. The common requirement of the drives in industries is speed control under varying operating conditions. In several research studies, the controllers have been considered without modeling of final control element (FCE). In practical applications, the effect of dynamics and nonlinearity of FCE affects the performance of the system, so it is necessary to consider these for a dependable simulation study of the drive performance. The overall system becomes non-linear due to the dynamics of converter used as FCE. In this study a new approach is being used, first the transfer function of the buck converter is obtained by considering a small linear region near the operating point, then using overall transfer function of buck converter and dc motor, the PID settings are obtained. Now fuzzy logic is used to update these settings on-line corresponding to the changes that may occur in system operating conditions. This configuration of controller is also observed to have robust performance against parameter variations and uncertainties. For more close to actual performance evaluation PWM controlled buck converter, used as FCE has been simulated using sim-power system library of MATLAB. The comparative results are presented with PID and fuzzy adaptive PID control strategies implemented, for both types of control situations i.e. tracking speed control and load disturbance rejection and the strength of approach is demonstrated.
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We investigate the problem of a heaving and pitching hydrofoil in an inflow that consists of a uniform velocity field and a staggered array of vortices. The foil can exploit the energy in such a Karman vortex street for efficient propulsion of animals or submarines. Through a two-dimensional inviscid analysis, me find that the phase between foil motion and the arrival of inflow vortices is a critical parameter. Everything else being equal, the highest efficiency is seen when this phase is such that the foil moves in close proximity to the oncoming vortices. Different modes of vortex interaction in the wake results from a variation in this phase, and we show that the how downstream of the foil is related to the foil input power.
To improve the robot's locomotion performance, it definitely needs the HRI (Human Robot Interaction) technology, the location awareness technology, and the real time location correction technology. In this paper, an HRI method that uses a circular coordinate system is proposed. The proposed method can implement robot locomotion with variable speed and directions instead of conventional method that controls speed and direction separately. Also, the nine characteristics of proposed circular coordinate system are to be described and the principles of user interfaces using the circular coordinate system are to be analyzed. For the performance evaluation, we have used a mobile robot that can be controlled through the wireless LAN and can perform differential drive. In addition, we have configured and tested a smart phone environment that can control the robot. The evaluation results show that the circular coordinate system reduces the locomotion time with accuracy.
For a real-time system, the system correctness depends not only on the correctness of the logical result of the computation but also on the result delivery time. Real-time Operating System (RTOS) is widely accepted in designing real-time systems. The real-time performance is achieved by using real-time mechanisms through data communication and synchronization of inter-task communication (ITC) between tasks. Therefore, benchmarking the response time of real-time mechanisms is a good measure to predict the performance of real-time systems. This paper aims to analyze the response characteristics of real-time mechanisms in kernel and user space for real-time embedded Linux: RTAI and Xenomai. The performance evaluations of real-time mechanisms depending on the changes of task periods and load are also conducted in kernel and user space. Test metrics are jitter of periodic tasks and response time of real-time mechanisms including semaphore, real-time FIFO, Mailbox and Message queue. The results are promising to estimate deterministic real-time task execution in implementing real-time systems using RTAI or Xenomai.
During the past several years, researchers have demonstrated that when new wireless sensors are placed in the home environment, data collected from them can be used by software to automatically infer context, such as the activities of daily living. This context-inference can then be exploited in novel applications for healthcare, communication, education, and entertainment. Prior work on automatic context-inference has cleared the way to a reduction in costs associated with manufacturing the sensor technologies and computing resources required by these systems. However, this prior work does not specifically address another major expense of wide-scale deployment of the proposed systems: the expense of expert installation of the sensor devices. To date, most of the context-detection algorithms proposed assume that an expert carefully installs the home sensors and that an expert is involved in acquiring the necessary training examples. End-user sensor installation may offer several advantages over professional sensor installations: 1.) It may greatly reduces the high cost of time required for an expert installation, especially if large numbers of sensors are required for an application, 2.) The process of installing the sensors may give the users a greater sense of control over the technology in their homes, and 3.) End-User Installations also may improve algorithmic performance by leveraging the end-user's domain expertise. An end-user installation method is proposed using "stick on" wireless object usage sensors. The method is then evaluated employing two in-situ, exploratory user studies, where volunteers live in a home fitted with an audio-visual monitoring system. Each participant was given a phone-based tool to help him or her self-install the object usage sensors. They each lived with the sensors for over a week. They were also asked to provide some training data on their everyday activities using multiple methods. Data collected from the two studies is used to qualitatively compare the end-user installation with two professional installation methods. Based on the two exploratory experiments, design guidelines for user self-installation of home sensors are proposed.
Conference Paper
This paper suggests a trajectory generation method using convolution operation. The proposed trajectory method always generates differentiable S-curve shape regardless of irregular time intervals, even when a new target is inputted before the trajectory does not reach to the previous target, or when a target is lost due to the communication problems. Also, the proposed method generates a trajectory considering physical system limits, for instance, maximum velocity, acceleration, and jerk, through the successive digital convolution that can reduce computational load, by using a recursive form of convolution operation. The effectiveness of the proposed method is shown through simulations.
Conference Paper
Recently, there is a rising interest on studying fish-like underwater robots because of real fish's great maneuverability and high energy efficiency. However, the researches about the fish-like underwater robots have not been done so much and there are still diverse problems in respect of using of the fish robot in the real environment such as in the river. For example, the fish robot has a short operating time and cannot move narrow passage such as swimming between aquatic plants. Therefore, this paper mainly describes a control method according to propulsion algorithm for improving energy efficiency and obstacle avoidance. The fish robot ‘Ichthus’ has a 3-DOF serial link-mechanism and is developed in KITECH. Also, we propose a dynamic equation of the fish robot to use the underwater environment. In the control portion, response characteristics of the fish robot were analyzed according to the input parameters of tail fin's amplitude and oscillation frequency. In consequence of this result, Control parameters of robot fish were found. These parameters are useful to increase energy efficiency and it can be used when the fish robot moves in the real environment.