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Robotica:
Investigating Reduced Path Planning
Strategy for Differential Wheeled Mobile
Robot
Raouf Fareh†∗, Mohammed Baziyad‡, Mohammad
H. Rahman¶, Tamer Rabie†and Maamar Bettayeb†,§
†Electrical and Computer Engineering Department, University of Sharjah, UAE
E-mails: trabie@sharjah.ac.ae,maamar@sharjah.ac.ae
‡Research Institute of Sciences and Engineering (RISE), University of Sharjah, UAE
E-mail: mbaziyad@sharjah.ac.ae
¶Mechanical/Biomedical Engineering Department, University of Wisconsin-Milwaukee,
Milwaukee, USA
E-mail: rahmanmh@uwm.edu
§MCEIES, King Abdulaziz University, Jeddah, KSA
(Accepted March 30, 2019)
SUMMARY
This paper presents a vision-based path planning strategy that aims to reduce the computational time
required by a robot to find a feasible path from a starting point to the goal point. The proposed algo-
rithm presents a novel strategy that can be implemented on any well-known path planning algorithm
such as A*, D* and probabilistic roadmap (PRM), to improve the swiftness of these algorithms.
This path planning algorithm is suitable for real-time scenarios since it reduces the computational
time compared to the basis and traditional algorithms. To test the proposed path planning strategy, a
tracking control strategy is implemented on a mobile platform. This control strategy consists of three
major stages. The first stage deals with gathering information about the surrounding environment
using vision techniques. In the second stage, a free-obstacle path is generated using the proposed
reduced scheme. In the final stage, a Lyapunov kinematic tracking controller and two Artificial
Neural Network (ANN) based-controllers are implemented to track the proposed path by adjust-
ing the rotational and linear velocity of the robot. The proposed path planning strategy is tested on
a Pioneer P3-DX differential wheeled mobile robot and an Xtion PRO depth camera. Experimental
results prove the efficiency of the proposed path planning scheme, which was able to reduce the
computational time by a large percentage which reached up to 88% of the time needed by the basis
and traditional scheme, without significant adverse effect on the workability of the basis algorithm.
Moreover, the proposed path planning algorithm has improved the path efficiency, in terms of the path
length and trackability, challenging the traditional trade-off between swiftness and path efficiency.
KEYWORDS: Path planning; Image processing; Depth sensor; Mobile robot; Kinematic control;
Probabilistic roadmap; Obstacle avoidance; Stability; Trajectory tracking.
1. Introduction
The path planning problem of an autonomous robotic system and its requirements have attracted
researchers to this research area in the recent years.1–10 An automated robot has the ability to accli-
mate to the surrounding environment and interact with it without direct human intervention. In
∗Corresponding author. E-mail: rfareh@sharjah.ac.ae
2Investigating reduced path planning strategy
order to build an effective autonomous system, fast and robust techniques are required to deal with
the real-time nature of such applications. An autonomous robotics system tries to imitate human
behaviors with an acceptable level of intelligence and precision in a real-time fashion. Avoiding
moving obstacles, generating a free-obstacle path to the destination, face detection and recognition
are all examples of such behaviors. However, real-time algorithms suffer from the enormous compu-
tational complexity and the massive resources required. As a result, a traditional trade-off between
simplicity and efficiency have formed a challenging problem for researchers in this field, as enhanc-
ing the efficiency increases the complexity of a system, which may end up with a system that fails
with real-time scenarios. One of the critical objectives for an autonomous robot is to compute a fea-
sible path to its destination without colliding into an obstacle. To find a free-obstacle path, the robot
must sense the environment instantaneously to find obstacles, calculate distances, and compute the
shortest path all in a real-time fashion. In robotics, various tools are utilized for sensing the environ-
ment such as sonar, laser, and cameras (vision). In recent years, with the massive improvements in
the processing power of modern computers, cameras (vision) became a widely used tool since it is
the most human-like sensor available.11–14
In ref. [13], a non-holonomic mobile robot tracks a pre-defined path generated with the help of a
vision system. Vision was also used as a sensor to estimate the location of the robot instantaneously
without the need of the traditional location measurements. Wallack et al. [11] proposed a system that
calibrates a vision system and a robot. The proposed system is equipped with multiple cameras in
a calibration mode to successfully detect and locate objects of interest with reference to the robot’s
initial position and orientation.
In ref. [14], a framework for implementing computer vision functions on cognitive robotics is
proposed. The proposed framework targets to combine visual perception and cognition processes in
one platform. The framework provides capabilities for locating 3D objects in the real world shown in
a 2D image plane. The framework was tested on an iCub robot and showed outstanding performance.
The path planning algorithm starts right after gathering the required information about the envi-
ronment such as the location of obstacles. The objective of a path planning algorithm is to find a
feasible path to the destination without colliding into an obstacle. Path planning algorithms can be
classified into two main categories: the grid-based algorithms and the sampling-based algorithms.
First, grid-based algorithms15,16 install a grid on a configuration space and represent each config-
uration with a point. The robot can freely navigate from a grid point to another if the region between
these points is free and does not include obstacles. There is a substantial parameter when dealing
with grid-based algorithms which is the grid resolution. The resolution parameter is simply the size
of the grid in terms of points or pixels. The importance of choosing this parameter becomes more
critical when dealing with real-time applications. Increasing this value may result in a slow system
that may fail dealing with the real-time nature of an autonomous robotic system. However, having
a low-resolution value can affect the robustness and the efficiency of the system, and the algorithm
may fail finding a feasible path.
A* is an example of a grid-based algorithm.17,18 The algorithm’s strategy in finding the shortest
path is by first finding a number of possible paths to the destination and then electing the path with
the lowest traveling cost. This creates a tree of paths starting from the starting point, until one of the
paths reaches the goal point.
Then, A* associates each path with a cost value that reflects the path length from the starting
point to the destination using a special heuristic function. Finally, A* elects the path with the lowest
cost value. D* is another example of a grid-based path planning algorithm.19 Unlike A*, D* starts
searching for the feasible path inversely from the goal point to the start point. In D*, each point has
a pointer that points to the adjacent point that leads to the destination. The cost value is calculated to
the goal for each point. The algorithm ends when the pointer is pointing to the starting point, and the
path is obtained by tracking the pointer.
In the second category, sampling-based algorithms, a number of samples are randomly distributed
in the free space. These points are then connected together, which results in having a pack of paths
from the starting point to the goal point. Finally, the shortest path is selected by using Dijkstra’s
shortest path query. The probabilistic roadmap (PRM) algorithm is an example of a sampling-based
algorithm.7In PRM, there are two critical parameters. The first parameter is the number of samples
(N), and the second is the distance (D). N is simply the number of samples to be distributed in the
Investigating reduced path planning strategy 3
free space. PRM connects all points separated by this distance or less in the map. In ref. [8], a new
path planning technique based on PRM is presented. The idea is to use a two-level system, or a PRM
of PRMs. A manipulation graph, whose nodes represent stable positions of the manipulated objects
while the edges represent transfer and transit actions. In the second level, PRM planners plan the
actual motion for the transfer and transit paths. Another modification of PRM named Lazy-PRM is
presented in ref. [9]. The algorithm tries to reduce the number of collision checks performed during
planning.
Another sampling-based technique is the Rapidly-exploring Random Trees (RRTs) techniques.2
These algorithms operate by incrementally building two RRTs rooted at the start and the goal config-
urations. The tree increases randomly in the search space and tends to grow towards large unsearched
areas of the map. RRT techniques are popular in nonconvex, high-dimensional spaces, and are used
widely in autonomous robotic motion planning.20,21 In ref. [21], a modified RRT path planning tech-
nique is used in a 3D path planning system. The proposed system uses a Kinect V2 sensor to obtain
a point cloud to emphasize the interest regions and the obstacles of the environment.
Once the path is generated, it is fed into a controller. The kinematic controller is a type of con-
trollers that adjusts the robot’s wheels’ velocity to track a certain trajectory. The authors presented
in ref. [12] a sliding mode control design to stabilize the mobile robot to a trajectory. A vision-based
control system in ref. [11] is developed to solve real-time experiences. In ref. [22], a kinematic con-
troller is used to track a PRM-based path obtained using a single RGBD sensor mounted on the
ceiling. The stability of the system was proved using Lyapunov theory.
Similar to the resolution attribute in A* and D*, raising the values of the PRM parameters (N and
D) increases the possibility of finding a path from starting point to goal point (if it exists), however,
a very high value will increase the complexity and computation time of the system, which is not
suitable for real-time applications. On the other hand, low values can decrease the computational
time, but the planner might fail to find a path, even if it exists.
To overcome the time-consuming problem, this paper proposes a path planning strategy that can
be implemented on any traditional path planning algorithm, which aims to reduce the computational
time required to find a feasible path to the desired destination. The proposed path planning strategy
consists of four main phases. First, a linear line is calculated from the starting point to the goal.
In the second phase, the planner detects obstacles that intersect with the linear line. Next, the basis
algorithm is performed around the obstacle to obtain a feasible path around each obstacle. Finally,
the planner connects these paths together to establish a free-obstacle path from the starting point to
destination.
The proposed path planning strategy is named “the reduced path planning strategy”. Moreover, a
vision-based control system is implemented to test the effectiveness of the proposed path planning
strategy. First, information about the environment is collected using vision algorithms. Next, a free-
obstacle path is calculated using the proposed path planning reduced scheme. Finally, a Lyapunov
kinematic controller and two ANN-based controllers are developed to ensure a good tracking of
the desired path. Experimental comparisons prove the efficiency and the effectiveness of the pro-
posed path planning strategy. The proposed path planning strategy is able to reduce 88% of the
computational time, without any effect on the workability of the basis algorithm.
The rest of this paper is organized as follows. In Section 2, the system description is illustrated.
Section 3explains the vision and the image processing algorithms. The proposed reduced path plan-
ning scheme is defined in Section 4. Section 5presents the kinematic and dynamic modeling of
the mobile robot. The control law for the mobile robot is calculated in Section 6. The experimental
results are discussed in Section 7, and Section 8presents the conclusion.
2. System Description
The system used is composed of a depth-RGB camera fixed on the ceiling with a height of Hand a
differential wheeled mobile robot that has two primary wheels fixed in the same axis in front. A small
free castor rotational wheel is placed at the back of the robot for stability purposes. Figures 1and 2
illustrate the general description of the system.
The position and orientation of the robot is represented by the generalized coordinate
q=x,y,∅T.
4Investigating reduced path planning strategy
Fig. 1. General description of the system.
Fig. 2. The differential wheeled mobile robot.
qd=xd,yd,∅dTdenotes the desired trajectory calculated from the path planning algorithm.
The main objective of the control system is to track the path generated by the proposed path
planning scheme using a kinematic control. Figure 3presents the block diagram of the control design.
It shows the main parts of the system; the vision part, the proposed reduced scheme planner, and the
kinematic controller. Vision techniques assist the robot to understand the surrounding environment.
The locations of obstacles, the starting point, and the goal point are all captured in this phase. The
path planner takes this information along with the election of the basis algorithm and finds a feasible
path from the starting point to the goal point. Finally, the kinematic controller helps the robot to track
the desired path, by adjusting the linear and rotational velocities (vand w), based on the current
measurements of the location and orientation of the robot with respect to the origin frame.
3. Vision and Image Processing Algorithms
Vision and image processing algorithms are essential tools that are used as a sensor in the system to
detect obstacles and to find the coordinates of the staring and goal points. Since the proposed sys-
tem targets real-time scenarios, these algorithms must meet some characteristics such as simplicity,
robustness, and fast execution. To achieve that, a depth-RGB camera is mounted on the ceiling to
detect and track obstacles. A depth camera is a device that can return an image where each pixel
value represents the distance from the camera. First, the distance matrix Dis calculated mathemat-
ically for all pixels where each index in the Dmatrix corresponds to the trigonometric real-world
distance of the corresponding location from the camera (the trigonometric distance from the floor to
ceiling (camera) for each point in the map). After that, the actual distance depth matrix Pis obtained
from the camera, and each pixel in the new matrix Pis compared to each pixel in the matrix D.Ifa
certain index (x,y)in the Pxy matrix has a value less than the corresponding index value in the Dxy
matrix, then the location (x,y)can be considered as an obstacle index. Figure 4shows the obstacle
tracking method used in the proposed system.
To detect the starting point and the goal point, an RGB image is required since these points are
represented by two distinct colored labels placed in the corresponding points.
Investigating reduced path planning strategy 5
Fig. 3. Block diagram of the proposed control strategy.
No obstacle
Obstacle
Fig. 4. Obstacle detection.
Fig. 5. Overview demonstration of the vision phase.
Before the path planning phase begins, these raw images must first run into a processing phase.
The objective of this phase is to equip the planner with some essential parameters such as obstacle
locations, starting point and goal point. This phase is composed of four subparts, namely the noise
elimination, obstacle tracking, obstacle padding, and color detection. Figure 5shows an overview of
the vision and image processing phase.
Image acquisition: Two images are captured from the RGB-depth sensor, the RGB, and the
depth image. The RGB image is retrieved in the three-channel format namely the red, green, and
6Investigating reduced path planning strategy
Fig. 6. Noise elimination.
(a) (b)
HD
C
C
X
Y
N = The total number of columns (in
pixels)
L = The real length of the lab (e.g.: in m)
R = The total number of rows (in pixels)
W = The real width of the lab (e.g.: in m)
Fig. 7. Obstacle tracking technique. (a) Calculating the D distance (b) Top view sketch illustrates the
calculations of the distance C.
blue channels. On the other hand, the depth image has pixels with values that represent the distances
of objects from the camera.
Image processing: The raw images are processed to extract some fundamental information such
as the coordinates of the starting point, the goal point and obstacles. A pre-processing step, which is
noise elimination, is performed first since the depth camera is not an ideal sensor as it sometimes fails
to estimate the distance. In this case, the corresponding pixel will have a value of zero which indicates
the failure. To deal with this noise issue, the average value of the surrounding pixels is calculated
and replaces the zero pixel. This operation is similar to filtering an image with an averaging filter,
but replacing only zero pixels with the average value. Figure 6presents an example of the noise
elimination process. In this example, the zero pixel was replaced by the average value of 3.8.
Next, the obstacle tracking sub-part begins. The first step in the obstacle tracking phase is to
mathematically calculate the distance of each point from the mounted camera. Figure 7shows the
procedure used to calculate the D distance, which is the real distance from the camera to each
(x,y)location in the map. The D distance can be computed easily by forming a right triangle com-
posed of the three sides D, H and C as shown in Fig. 7. The side D (the hypotenuse) is the distance
from the camera to the location (x,y). The side H is the distance from the camera to the floor, while
the side C is the distance from the camera projection on the floor to the location (x,y). Figure 7b
shows a top view sketch to illustrate the procedure used to find the distance C. As shown in Fig. 7b,
another right triangle arises in the floor plane, with sides of C, X and Y. The distances X and Y are
simply the distance from the location T to the center of the camera projection on the floor in the
X-direction and Y-direction, respectively. Since the total number of rows and columns of the depth
Investigating reduced path planning strategy 7
image and the real length and width dimensions of the laboratory are all known quantities, the X and
Y distances can be easily computed using the following ratio relationship:
X=m×L
N,Y=r×W
R(1)
where mand rare the number of columns and rows from the projection of the center of the camera
on the floor to the location T, respectively (number of pixels); Lis the real length of the map (in
meters); Nis the total number of columns (in pixels); Ris the total number of rows (in pixels) and
Wis the real width of the map (in meters).
Now, the side C can be computed using a simple Pythagoras equation:
C=X2+Y2(2)
Finally, the distance D can be calculated using:
D=C2+H2(3)
Combining Eqs. (2) and (3), the distance D can be computed as:
D=X2+Y2+H2(4)
After finding the distance Dxy for all (x,y)locations in the map, the actual depth data Pxy is
obtained from the camera and compared to the Dxy matrix. If a certain index (x,y)in the Pxy matrix
has a value less than the corresponding index value in the Dxy matrix,then the location (x,y)can be
considered as an obstacle index. The output of the obstacle tracking phase is a binary image where
zero pixels (black pixels) represent obstacles, while white pixels represent the free space in the map.
Thus:
grid (x,y)=0if P (x,y)+h<D(x,y)
255 otherwise (5)
where his the threshold value, D(x,y)is the calculated distance of each (x,y)location from the
camera. P(x,y)is actual location returned from the depth camera, grid is the new binary image. The
threshold value his introduced to neglect very small obstacles and to not consider them as obstacles,
as the robot can easily drive over them. Empirically, the recommended h value is half of the height
of the robot’s wheel.
After detecting the obstacles, a safety margin is added around each obstacle. The purpose of this
process is that the path planning algorithm treats the robot as a point instead of its actual size. Thus,
the size of the added margin should be equal to the size of the robot. The last sub-part in the image
processing phase is detecting the starting and the goal points. These points are represented by two
labels with two different colors placed in the desired places. To find these points, a range search
of these colors is performed instead of a single value search since the value might change due to
compression and other impairments.
4. Path Planning
The proposed path planning scheme is described in this section. The proposed path planning tech-
nique aims to reduce the computational time and complexity. The idea is that instead of using some
well-known algorithms such as A*, D* and PRM on the whole map, the new path planning scheme
suggests performing these algorithms only around obstacles. The proposed reduced path planning
strategy is divided into four sub-parts. First, a linear line is calculated from the starting point to
the goal. In the second phase, the planner detects obstacles that intersect with the linear path. Next,
the basis algorithm is performed around each obstacle to find a feasible path around each obstacle.
Finally, the planner connects these paths together to establish a free-obstacle path from the start-
ing point to the destination. Figure 8shows the final path produced by the proposed reduced path
planning scheme.
Figure 9illustrates and simplifies the idea of the proposed reduced scheme. For simplicity and
illustration purposes, the map in Fig. 9is assumed to be 12 ×12 pixels only. First (Fig. 9a), the
coordinates of obstacles are obtained by searching for black pixels (zeros). Next, a linear path is
8Investigating reduced path planning strategy
Fig. 8. The reduced path planning scheme.
Fig. 9. The proposed reduced path planning strategy step-by-step illustration.
calculated from the starting point to the goal point (Fig. 9b). Then, each pixel is checked whether
it is located inside an obstacle or not, by comparing its index with all obstacle indices (Figs. 9c, 9d
and 9e). Once a path pixel is detected to be located inside an obstacle, a margin is added around
this obstacle. This margin will be the basis path planning algorithm (A*, D*, PRM) working area
(Fig. 9f). Next, the basis algorithm will generate a path around the obstacle (Fig. 9g). The basis
algorithm path will lead the robot back to the linear path. The planner continues checking whether the
linear path pixel is located inside an obstacle (Fig. 9h). Finally, all path parts are combined together
Investigating reduced path planning strategy 9
to establish a free-obstacle path from the starting point to the goal point (Fig. 9i). More details about
different parts of the proposed path planning strategy can be found in the next discussion.
4.1. Linear planner
In Fig. 9b, the proposed reduced scheme starts by calculating a linear path from the starting point to
the goal using the following equation:
y=m(x−x1)+y1(6)
where:
x1,y1are the coordinates of the starting point or the goal point.
xis the vector: xs,xs+1,xs+2,...,xg.
xsis the xcoordinate of the starting point.
xgis the xcoordinate of the goal point.
mis the slope of the linear line as follows:
m=y2−y1
x2−x1
(7)
where x1and y1are any two points in the slope. The only known points are the starting point and
the goal point. The points (x1,y1) and (x2,y2) can be either the starting point or the goal point.
The election is based on the y value of both points, y2must be greater than y1. If the y value of
the starting point is greater than the y value of the goal point, then (x2,y2) is the starting point and
(x1,y1) is the goal point. On the other hand, if the y value of the goal point is greater than the y value
of the starting point, then (x2,y2) is the goal point and (x1,y1) is the starting point.
4.2. Obstacle detector
The obstacle detector is responsible to find obstacles that intersect with the linear path. Since the
map is a binary image with obstacles with black pixels (zeros) and free-space in white pixels (white),
the locations of obstacles can be easily obtained by performing a simple search for the indices of all
zero pixels. The obstacle detector checks each point in the linear path, starting from the starting point
towards the goal point if it is located inside an obstacle. This is done by comparing the index of the
linear path point with every obstacle index. If there is any match between two indices, then the linear
path intersects with an obstacle in that index.
Figures (9c, 9d and 9e) illustrate the operation of obstacle detector. The obstacle detector starts
the checking process with the pixel after the starting point as in Fig. 9c which has an index of (3,3).
The index (3,3) is not an obstacle index. Thus, it is approved to be part of the final path. The obstacle
detector moves to the next linear path index (4,4), which is also not an obstacle index. The index
(4,4) is also accepted to be in the final path list.
The next index (5,5) is found to be an obstacle index. Once the obstacle detects that an obstacle
lies in the linear path, it creates a small margin around the obstacle, and sends the obstacle with the
margin to the basis path planner. The margin is added to allow the basis planner to find a path around
the obstacle as in Fig. 9g. The first pixel of the linear path located within the margin (index 4,4) is
now the starting point for the basis function. Similarly, the last point of the linear path located within
the margin (index 9,9) is the goal for the basis function.
The width of the margin is a tuning parameter, increasing the width increases the computational
time, but increases the possibility of having a shorter path length. On the other hand, decreasing the
width will definitely decrease the computational time, but will decrease the possibility of having a
shorter path length. The reason is that by expanding the working area of the basis algorithm, the
number of samples N (PRM) or the grid resolution (A*, D*) will increase. This increment will for
sure affect negatively on the computational time, but positively on the path efficiency.
These parameters, such as the map (obstacle and margin), new starting point and the new goal
point are sent to the basis planner (A*,D*, PRM, etc.) to find the shortest feasible path from the new
staring point to the goal point. The obstacle detector does that for each obstacle intersecting with the
linear path.
10 Investigating reduced path planning strategy
Fig. 10. Reduced-PRM path planner general diagram.
4.3. Basis planner
The basis planner receives the required information, and finds the shortest path around each obstacle.
As discussed before, there are some parameters that are critical, and must be chosen carefully. These
parameters are the grid size (for A*, D*) and the number of samples N (for PRM). Minimizing these
parameters can help reducing the computational time, but that may affect the performance of the
system. However, in the proposed scheme, these values will be automatically small since the working
area is a much smaller area than the whole map. Moreover, these small values suit the working space,
which means that there is no significant adverse effect on the performance.
Figure 9g shows the operation of the basis planner. It is to be noted that the path generated by the
basis path planning algorithm will vary based on the basis path planner used. Fig. 9g shows just the
illustration, and not the actual path to be generated by the basis algorithm.
4.4. Path combiner
Once all of the paths are generated around each obstacle, the path combiner combines the different
linear segments with the non-linear paths generated by the basis planner and produces the final free-
obstacle path from the starting point to the goal point. Figure 10 shows a general diagram of the
proposed system.
5. Dynamic and Kinematic Modeling
The mobile robot considered in this paper consists of a wheeled mobile robot shown in Fig. 2. The
general dynamic equation is described as follows:23
M(q)¨q+C(q,˙q)˙q+G(q)=B(q)τ −AT(q)λ (8)
Investigating reduced path planning strategy 11
where τ=[τr,τ
l]T∈Rris the input vector and consists of motors’ torques τrand τl, which act on
the right and left wheels, λ∈Rmis the vector of constraint forces, M(q)∈Rn×nis a symmetric
and positive-definite inertia matrix, C(q,˙q)∈Rn×nis the centripetal and Coriolis vector, G(q)∈Rn
is the gravitational vector, B(q)∈Rn×ris the input transformation matrix and A(q)∈Rm×nis the
matrix associated with the constraints. The kinematic constraints can be denoted as:
A(q)˙q=0(9)
˙x sin∅− ˙ycos∅=0 (10)
When selecting a full rank matrix S1(q)to be a basis of null space A(q), the constraint equation
will be:
A(q)S1(q)=0.(11)
where
S1=⎡
⎣
r
2L(Lcos∅+dsin∅)r
2L(Lcos∅−dsin∅)
r
2L(Lsin∅−dcos∅)r
2L(Lsin∅+dcos∅)
r
2L−r
2L
⎤
⎦.
Therefore, we have:
˙q=S1(q)W(12)
where W=ωlωrTωr;ωlrepresents the angular velocities of the right and left wheels and
˙q=˙x˙y˙
∅T. If we consider v;ωas the linear angular velocities of the mobile robot, the relation
between v;ωand ωr;ωlcan be explained as:
ωl
ωr=1
rL
r
1
r−L
rv
ω→W=HV (13)
where V=vω
T. Note that:
S1H=S=⎡
⎣
cos∅dsin∅
sin∅−dcos∅
01
⎤
⎦(14)
The derivative of Eq. (12) gives:
˙q=S1HW ⇒˙q=SV ⇒¨q=˙
SV +S˙
V(15)
6. Control Law Design
Note that the objective is to track a reference trajectory by the mobile platform. In this section,
the velocity control based on kinematic model is designed to develop the desired forward and rota-
tional velocities. The desired position is qd=xdydφdand the desired velocity is denoted by
Vd=vdωdT. Using the Kanayama transformation,24 the tracking errors are obtained as follows:
⎡
⎣
˜x
˜y
˜
φ⎤
⎦=⎡
⎣
cosφsinφ0
−sinφcosφ0
001
⎤
⎦⎡
⎣
xd−x
yd−y
φd−φ⎤
⎦(16)
and the related error dynamics are expressed as follows:
˙
˜x=ω˜y−v+vdcos∅
˙
˜y=−ω˜x+vdsin∅
˙
˜
φ=ωd−ω
(17)
12 Investigating reduced path planning strategy
Proposition: when selecting the velocity control law given in (18), the error dynamics (17)are
asymptotically stable.
V(t)=v
ω=k1˜x+vdcos ˜
φ
ωd+k2vd˜y+k3vdsin ˜
φ(18)
where k1>0; k2>0; k3>0 are the controller gains.
To prove Proposition 2, let us define the following positive Lyapunov function:
W=1
2
∼2
x+1
2
∼2
y+1−cos ˜
φ
k1
(19)
The time derivative of Wis given as follows
˙
W(t)=˜x˙
˜x+˜y˙
˜y+˙
˜
φsin ˜
φ
k1
=˜x(ω˜y−v+vdcos∅)+˜y(−ω˜x+vdsin∅)+(ωd−ω)sin ˜
φ
k1
=−v˜x−ωsi n ˜
φ
k1
+˜xvdcos∅+ ˜yvdsin∅+ωd
sin ˜
φ
k1
Using the control law (18), the time derivative of the Lyapunov function becomes:
˙
W(t)=−k1
∼2
x(20)
The time derivative of W(t)is negative because k1is positive gain. By (20), Wis monotonously
decreasing. Therefore,
∼
x∈L∞,
∼
y∈L∞and
∼
φ∈L∞,and
∼
x∈L2. According to (17), we know that
.
∼
x,
.
∼
yand
.
∼
φare bounded and therefore
∼
x,
∼
yand
∼
φare uniformly continuous. Then, using Barbalat’s
theorem,22 we can conclude that
∼
x→0,
∼
y→0 and
∼
φ→0as t →∞ and the error dynamics are
asymptotically stable.
7. Experimental Comparative Results
The experiments were done in the laboratory where an Asus Xion pro depth camera was mounted on
the ceiling. The robotic system used in these experiments was the Pioneer P3-DX shown in Fig. 11.
The experimental setup is shown in Fig. 12.
The parameters used in the following experiments are shown in Table I. To test the efficiency
of the proposed path planning strategy in terms of the computational time, the proposed scheme
was tested with three different scenarios. In the first scenario, there are no obstacles “intersecting”
with the linear path from the starting point to the goal point (even though obstacles are presented in
the scene). In the second scenario, one obstacle is intersecting with the linear path. Finally, in the
third scenario, two obstacles are intersecting with the linear path. Each scenario has two different
configurations by changing the starting and end points. Figure 13 shows the RGB and depth images
captured by the ASUS Xtion Pro camera, while Fig. 14 shows the three different scenarios with
different configurations.
In the following experiments, A*, D* and PRM are chosen to be the basis algorithm to work on.
For A* and D*, a grid of 31×41 (1271 total points) was chosen for a 480×640 image. PRM was
configured to have 1000 samples (N), and a value of 50 is chosen to be the connection distance (D).
Since the workspace in the reduced area is smaller than the whole map, the proposed reduced strategy
allows theses parameters to be minimized. As explained earlier, the basis algorithm workspace is
composed of the obstacle and the surrounding margin. Thus, this area varies based on the obstacle
size and the margin width. To ensure having a fair comparison, N, D and the grid values were scaled
down linearly so that they are related to their workspace size.
Investigating reduced path planning strategy 13
Table I. Experiment parameters.
Parameters values
Image Processing:
H [Threshold] (m) 3.8
Ls (R,G,B) 255,255,0 yellow)
Lg (R,G,B) 0,128,0 (green)
Robot:
mc(kg) 7.4
mω(kg) 0.8
L(m) 0.19
r(m) 0.0975
d(m) 0.1
Ic(kg m2) 0.020
Im(kg m2) 0.00475
Fig. 11. Pioneer P3-DX.
Office Connect
Access Point ASUS Xion Pro
(RGB/Depth Camera)
Java
SimpleOpenNi
library
Matlab
R2015a
Remote
Desktop
Connection
Matlab / Simulink Aria
interface
|
On-Board PC
Fig. 12. Experimental setup.
Figure 15 shows the path calculated from both methods for the third scenario, including the
traditional algorithm and the reduced and proposed strategies.
7.1. Computational time comparison
A comparison of the time execution between the reduced and original versions of A*, D* and PRM
is shown in Table II. The experiment was done under different lighting conditions.
14 Investigating reduced path planning strategy
Fig. 13. (a) RGB image captured by the ASUS Xtion Pro. (b) Raw depth image captured by the ASUS Xtion Pro.
Table II clearly proves the efficiency of the proposed path planning scheme in terms of the compu-
tational time required to calculate the free-obstacle path to the goal point. It is obvious that scenario
1, where no obstacles are intersecting with the linear path, will have the greatest result, since all that
is required is computing a simple linear path using Eq. (6). Thus, there is no need to define grids, dis-
tribute points or establish lines between points, as all these processes consume time and effort. Figure
16 illustrates how it is inefficient to use PRM in scenario 1. Another important note, when comparing
scenario 2 with scenario 3, is that the computational time increases by only 0.029 seconds for reduced
A*, 0.062 for reduced D* and PRM. This indicates that even if we increase the number of obstacles
that intersect with the linear path, the reduced scheme will still be much faster than the original path
planning algorithms. Another noticeable fact is that the proposed scheme was not affected under dif-
ferent environmental (lighting) conditions. The average execution time results were independent of
the lighting condition. That is because the proposed system relies on the depth information (infrared
signals) in locating obstacles, which is robust against different lighting conditions.
The improvement percentage was calculated using the following equation:
Ip =1−tr
tc×100 (21)
where:
tris the computational time by the reduced scheme.
tcis the computational time by the classical scheme.
7.2. Path length comparison
As stated in the Introduction section, there is a trade-off between computational time and the path
efficiency. Path efficiency is studied in this paper from two aspects, path length and trackability. Since
Investigating reduced path planning strategy 15
(b) (e)
(c) (f)
(a) (b)
Fig. 14. Binary map after processing – configurations (a) and (b) represent scenario 1 (no obstacles intersecting
with the linear path). (c) and (d) configurations represent scenario 2 (1 obstacle intersecting with the linear
path). Configurations (e) and (f) represent scenario 3 (2 obstacles intersecting with the linear path).
the proposed reduced scheme has shown a huge improvement in the time execution calculations, it
is expected that the proposed reduced path will be less efficient in terms of length and trackabil-
ity. Table III shows a comparison between classical and reduced scheme in terms of path length
calculations for scenario 2.
Table III shows that the proposed reduced paths perform better than classical methods for sce-
nario 2. Even though the proposed reduced scheme focused on improving the computational time,
it could in some cases break the trade-off between path efficiency and computational time. Another
advantage of the reduced paths is that the differences between the lengths of experimental paths and
the lengths of the simulated paths of the reduced scheme are less than the equivalent differences in
the original scheme. For example, the path length of the reduced PRM is 4.3397 m in the real-life
experiment while it is 4.2010 m in simulations, with a difference percentage of 3.30%. On the other
hand, the path length of the original PRM is 4.2772 m in the real-life experiment while it is 3.4011
m in simulations, with a difference percentage of 25.76%. That is an indication that the reduced
scheme can be controlled easier with minimal errors between the real path and the desired simu-
lated path. To be noted here that the kinematic controller described in Section 6was used to obtain
the results in Table III. More details about the path trackability tests will be presented in the next
sub-section.
16 Investigating reduced path planning strategy
Fig. 15. Path planning with the basis and proposed algorithms. (a) Path calculated by original A*. (b) Path
calculated by reduced A*. (c) Path calculated by original D*. (d) Path calculated by reduced D*. (e) Path
calculated by original PRM. (f) Path calculated by reduced PRM.
7.3. Path trackability test
Next, the reduced PRM path – scenario 2 – is chosen to be tested with our kinematic controller and
two intelligent controllers based on Artificial Neural Networks (ANN), which are the Multi-Layer
Perceptron (MLP) and the Non-linear Auto-Regressive with Exogenous inputs (NARX). A popular
approach for function approximations is by using ANN. An ANN is trained to represent the forward
dynamics of a system. Training is done by minimizing the prediction error between the system output
and the neural network output. Using a two-layered network with a sigmoid activation function in the
hidden layer, and a linear activation function in the output layer is a good universal approximator.25
Thus, this MLP structure is going to be used as the first NN controller.
Another approach being popular in the system identification field is the usage of Recurrent Neural
Networks (RNN). An RNN is a category of ANN where a neuron can receive a previous predic-
tion and include it along with the input in the calculations. This is done by having some feedback
connections between neurons. Thus, RNN can capture temporal plant behavior for a time sequence.
Table II. Comparison between Reduced-PRM and the standard PRM.
Average Execution time (sec) Improvement percentage
Lighting condition Normal Dark Bright
Scenario Method Reduced Original Reduced Original Reduced Original Normal Dark Bright
Scenario 1 A* 0.243 0.589 0.223 0.569 0.233 0.609 58.7 60.8 61.7
(No obstacles intersecting with the linear path). D* 0.243 1.296 0.233 1.286 0.233 1.286 58.7 81.9 81.9
Configuration a Figure 14-a PRM 0.243 2.074 0.223 2.054 0.233 2.054 81.3 89.1 88.7
Scenario 1 A* 0.280 0.556 0.270 0.546 0.280 0.556 88.3 50.5 49.6
(No obstacles intersecting with the linear path). D* 0.280 1.37 0.270 1.360 0.280 1.370 49.6 80.1 79.6
Configuration b Figure 14-b PRM 0.280 3.75 0.280 3.750 0.270 3.740 79.6 92.5 92.8
Scenario 2 A* 0.304 0.649 0.314 0.659 0.284 0.629 92.5 52.4 54.8
(1 obstacle intersecting with the linear path). D* 0.458 1.262 0.448 1.252 0.448 1.252 53.2 64.2 64.2
Configuration c Figure 14-c PRM 0.558 2.149 0.558 2.149 0.578 2.169 63.7 74.0 73.4
Scenario 2 A* 0.331 0.672 0.321 0.662 0.331 0.672 74.0 51.5 50.7
(1 obstacle intersecting with the linear path). D* 0.480 1.30 0.460 1.280 0.490 1.310 50.7 64.1 62.6
Configuration d Figure 14-d PRM 0.80 3.77 0.820 3.790 0.810 3.780 63.1 78.4 78.6
Scenario 3 A* 0.333 0.601 0.333 0.601 0.353 0.621 78.8 44.6 43.2
(2 obstacles intersecting with the linear path). D* 0.52 1.26 0.540 1.280 0.530 1.270 44.6 57.8 58.3
Configuration e Figure 14-e PRM 0.62 2.27 0.620 2.270 0.610 2.260 58.7 72.7 73.0
Scenario 3 A* 0.354 0.685 0.344 0.675 0.354 0.685 72.7 49.0 48.3
(2 obstacles intersecting with the linear path). D* 0.54 1.316 0.520 1.296 0.550 1.326 48.3 59.9 58.5
Configuration f Figure 14-f PRM 0.90 3.71 0.920 3.730 0.890 3.700 59.0 75.3 75.9
18 Investigating reduced path planning strategy
Table III. Comparison between proposed reduced scheme and the classical schemes in the path length.
Path length – Path length –
experiment Improvement simulation Simulation-Experiment
Scenario Method (meters) percentage (meters) error (percentage)
2 A* Reduced 4.1611 20.07% 4.4343 6.16
(1 obstacle Classical 5.2061 4.5473 14.49
intersecting D* Reduced 4.1811 3.725% 3.9951 4.66
with the linear Classical 4.3429 4.2937 1.15
path). PRM Reduced 4.3397 −1.461% 4.2010 3.30
Figure 14 Classical 4.2772 3.4011 25.76
Fig. 16. Using PRM with scenario 1.
The NARX network is an RNN. The architecture contains some feedback connections that surround
several layers of the network. The NARX model is being widely used in time series modeling.26 The
NARX network can be defined as:
y(t)=f(y(t−1),y(t−2),...,yt−ny,u(t−1),u(t−2),...,u(t−nu)(22)
It is clear from Eq. (22) that the prediction is highly dependent on the previous predictions using
the feedback connections. The fact that distinguishes NARX networks in the RNN family is that the
final output is fed to be the input of the network.
The reference trajectory is chosen as qrd =xryr∅rTthen ˙xr=Vrcos∅r;˙yr=Vrsin∅r,
˙
∅r=ωr. where the linear velocity and the reference angular velocity are chosen as Vr=0.2m/s
and ωr=0.1rad/s, respectively.
The tracking experimental results obtained are shown in Figs. 17,18 and 19 for the kinematic
controller, the MLP controller and the NARX controller, respectively.
Regarding the tracking results, the three controllers were able to perfectly track the path with
minimal errors. Figures 17,18 and 19 (a, b, and c) show the tracking in the x, y, and phi, respec-
tively. This good tracking is confirmed by Figs. 17,18 and 19 (d, e, and f), which illustrate the
tracking error. The kinematic controller has achieved the best tracking performance. The errors
have small values that do not exceed 0.14 m for x-position, 0.05 for y-position and phi. Moreover,
Fig. 17 shows that in each sudden change in the path, the kinematic controller successfully deals
with that sudden change and makes the error approach zero. For the NN-based controllers, the
NARX controller has achieved slightly a better tracking performance than MLP. That is because
NARX networks are more suitable for time series signals. The feedback connections between
neurons have allowed this NN model to mimic the temporal dynamic behavior of a kinematic
controller.
Investigating reduced path planning strategy 19
(a)
0510 15 20
x-position (m)
–5
0
5
(d)
(b)
(c)
(e)
(f)
0510 15 20
error-x (m)
–0.2
0
0.2
0510 15 20
y-position (m)
–1
–0.5
0
0510 15 20
error-y (m)
–0.05
0
0.05
510 15
theta (rad)
–1
0
1
510 15
error phi (rad)
–0.1
0
0.1
Fig. 17. Kinematic control of the platform subsystem: (a) Tracking trajectory of x-position. (b) Tracking trajec-
tory of y-position. (c) Tracking trajectory of ϕ-direction. (d) Tracking error of x-position. (e) Tracking error of
y-position. (f) Tracking error of ϕ-direction.
Fig. 18. Intelligent control of the platform subsystem using MLP: (a) Tracking trajectory of x-position.
(b) Tracking trajectory of y-position. (c) Tracking trajectory of ϕ-direction. (d) Tracking error of x-position.
(e) Tracking error of y-position. (f) Tracking error of ϕ-direction.
8. Conclusion
This paper introduced a path planning strategy that is efficient for real-time scenarios. The novel
strategy can be applied on any path planning algorithm to reduce the computational time required
to calculate the path. The path planning strategy was tested on three control systems for differential
wheeled mobile robot, namely a kinematic controller, which was proved by the Lyapunov approach,
an MLP-based controller and an NARX-based controller. First, vision and image processing algo-
rithms were used to collect and analyze information. Then, a free-obstacle path was calculated by
using the proposed reduced-scheme path planner. This planner was the key feature of the proposed
system due to its simplicity and fast execution. Finally, three different controllers were developed to
20 Investigating reduced path planning strategy
Fig. 19. Intelligent control of the platform subsystem using NARX: (a) Tracking trajectory of x-position.
(b) Tracking trajectory of y-position. (c) Tracking trajectory of ϕ-direction. (d) Tracking error of x-position.
(e) Tracking error of y-position. (f) Tracking error of ϕ-direction.
follow the path. Experimental comparisons have proved that the path planning technique was able to
deduct the computational time needed by the corresponding classical path planning algorithm with-
out significant effects on the path quality. In some cases, it actually improved the quality as well
represented by the path length and the trackability. As a future work, the kinematic controller will be
upgraded to a dynamic intelligent controller that can directly send torque commands to the robot’s
motors.
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