Optimal Planner for Lawn Mowers
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Optimal Planner for Lawn Mowers
Ping-Min Hsu and Chun-Liang Lin
Abstract —The task of planning trajectories for mobile robots navigation of autonomous mobile robots in a completely
has received considerable attention in the research literature. unknown environment. In [9], a geometric algorithm was
However, rare research addresses the issue relate to lawn mower proposed for generating paths via a modified sweeping line
designs. An optimal and efficient path planner for lawn strategy that tries to minimize extra relocation moves. In [10],
mowers is proposed here. There are two issues concerned in a method for cooperation of multiple robots for complete
the design of the planner: (i) low working time or energy coverage path planning was proposed on the basis of chaos
consumption, and (ii) human safety. The optimal mowing path synchronization. In [11], the optimal path was designed based
for time or energy consumption with safety concern is
on genetic algorithms. However, it didn’t take energy
achieved via combining these factors in the mowing path
consumption into consideration. In [12, 13], a cooperative
planning. Experimental results show that the proposed
approach yields excellent performance with a variety of sweeping strategy of path planning for multiple cleaning
scenarios1. robots was developed and a biologically inspired neural
network was adopted. In [14], the authors proposed an
Index Terms—Lawn mower, path planning, obstacle autonomous mower along with a global positioning system
avoidance, optimization.
(GPS), it could thus navigate itself by the aid of a navigation
device. With the GPS, it can achieve related coordinates
I. INTRODUCTION
concerned as the temporary destinations in the mowing
Nowadays, consumers pay a considerable high attention to process; however, it doesn’t possess the functions of obstacle
fundamental service robots, such as autonomous lawn mowers avoidance and path planning for optimal efficiency. In [15], a
and vacuum cleaners. Due to increased demand to this kind of method of localization for robot mowers covering an area was
robots, research related to this issue has also been addressed in proposed, in which a neural network was used to enhance the
the literature. tracking accuracy. A path planner without considering
In [1], a neural network approach was proposed for a optimization of operating time and energy consumption for a
complete coverage path planning with obstacle avoidance of robotic vacuum cleaner has been proposed in [16]. In [17], a
cleaning robots in nonstationary environments. In [2], a service robot has been successfully implemented, however,
solution to vicinity problem of obstacles in complete coverage the path planning mechanism was not considered.
path planning was proposed. The path planner is designed on In this paper, an optimal mowing path planner including
the basis of a biologically inspired neural network. However, minimal time, minimal energy consumption, and minimal
the path planned was lacking of design optimization. In [3], time/energy modes as well as a particular solution of complete
the complete coverage motion planning and control system coverage path planning (CCPP) is developed. Especially,
were designed on the basis of visual sensor. In [4], a path incorporating the three modes enhances user convenience and
planning method based on a neural network, rolling path reduces mowing cost in time and energy. The complete task of
planning, and heuristic searching approach was proposed in lawn mowing plan consists of two stages. The first stage is for
which the neural network was used to model the environment. the rough mowing path planning which considers the factor of
In [5], a strategy of combined coverage path planning was working time or energy consumption. The second stage
proposed; it combined the random path planning with local considers avoidance of static and dynamic obstacles. It
complete coverage path planning. In [6], a path planning modifies the details of the mowing paths by introducing a
method integrating rolling windows and biologically inspired geography method, in which the idea of potential field is
neural networks was proposed in which the moving object had incorporated for obstacle avoidance. Applicability of the
been considered in the obstacle avoidance. In [7], the authors proposed design is verified via real-world experiments.
proposed a method of capability for the path planning to deal
with the priori mapped or expected obstacles in the working II. PATH PLANNING ALGORITHM
space. However, the situation of moving objects wasn’t
A method is first proposed to determine three kinds of
considered. In [8], a neural-dynamics-based approach was
mowing paths in minimum time, minimum energy, and mixed
proposed for real-time map building and complete coverage operation to achieve the best efficiency. The minimal time
mode means that the time consumed for cutting the whole
1
This work was sponsored in part by the Ministry of Education, Taiwan, lawn is minimal. The minimal energy mode means that the
R.O.C. under the ATU plan. whole energy consumed during the period of mowing is
C. L. Lin is with the Department of Electrical Engineering, National Chung minimal. The mixed mowing mode simultaneously considers
Hsing University, Taichung, 402 Taiwan, R.O.C. (e-mail:
chunlin@dragon.nchu.edu.tw)
time saving and energy consumption for lawn mowing to
P. M. Hsu is with the Department of Electrical Engineering, National achieve a best compromise between two modes.
Chung Hsing University, Taichung, 402 Taiwan, R.O.C.Before deriving the index of working time, assume that the
unit energy consumption is a constant E0 , no matter what
kind of paths the mower follows. The index of working
time T is then defined as
E0
T (3)
Fig. 1. Example for computing k P
To proceed, a variable k is introduced, which is defined as where P is defined as in (2). It can be easily shown that the
the number of the turning operation with the change of time T achieves its minimum when k is maximal. Therefore,
veering angle of the steering direction being ±180° . For the value of k of the optimal path in the minimal time mode is
example, the number of k in Fig. 1 is 9. Dw 1 .
Next, an index of cutting difficulty is defined in the follows
The index of energy consumption E is defined as
which represents virtually the moving distance:
d
difficulty Bw Dw 1 d S Dw 1 E PT0 (4)
2 (1)
ª§ S · §S · º
« ¨ 1 ¸ k ¨ 1 ¸ 2 Dw 2k 2 »d where T0 denotes the unit working time. It can be easily seen
¬ © 2 ¹ © 4 ¹ ¼
Ll that E is minimal when k is minimal, i.e. k 1 .
where Bw 1 with Ll being the length of the working area,
d Before developing the algorithm to the mixed mode, denote
Lw the maximal velocity of the mower as V , the maximal angular
Dw 1 with Lw being the width of the working area,
d velocity is We , the change of the motor’s input voltage to time
d 10800S Rearth m/mmin with Rearth (km) being the radius is m1 , the change of the mower’s velocity to time is m2 , the
of Earth, and slope of change of the mower’s angular velocity to time is m3 ,
S
1 the distance difference for a turning V We
2 and t .
operation with the turning angle being S m2 m3
S v m
1 the distance difference for a turning s
4
S
operation with the turning angle being
2 V
Bw Dw 1 d total distance for the straight navigation
d
S Dw 1 total distance for the turning navigation
2
ª§ S · §S · º t s
«¨ 2 1¸ k ¨ 4 1¸ 2 Dw 2k 2 » d total distance difference t1 T t1 T
¬© ¹ © ¹ ¼
induced in the period of turning operation Fig. 2. Change of velocity versus time during the straight section of
navigation
Technically, the cutting difficulty in the form of (1) means
the more distance the mower walks, the more difficulty the W rad
s
mowing job is. It is reasonable to assume that the mowing
power is proportional to the difficulty to mow the same area
We
with the same efficiency. Thus, one may let the ratio between
mowing difficulty and mowing power to be P0 , and the index
for power consumption of the mower be
ª d § S· º t s
P P0 « Bw Dw 1 d S Dw 1 ¨ 2 ¸ Dw d S d kd » (2) t2 T t2
¬ 2 © 2 ¹ ¼ T
Fig. 3. Change of angular velocity versus time during the turning
where k >1, Dw 1@ . section of navigation
A. Preliminary path planning Voltage (V )
A complete route planning task consists of two stages. The
first stage devises a preliminary mowing path which plans the Vd
optimal mowing route in the off-line manner and only focuses
on the conservation of working time and energy. The next
stage is a detailed path planning which worked on-line. It
t s
concerns safety of the planned mowing path and enhances the t1 T t1 T
human safety. Fig. 4. Change of input voltageL T
On the basis of Figs. 2-3, the working time for the straight where T0 4 Dw 3 t ,
V We
or turning section of navigation can be derived as
Lm m12 t 2 L m12 t 2T m2t 3
Tsm t (5) E0 4 Dw 3 1 , and A is a weighing
V RV RWe 3R
Tn factor. The optimal k , which minimizes (11), is given by
Tcn t (6)
We
tT0 m12 t 3 E0
where Tsm is the working time for the m th straight section,
k A 3R (12)
Tcn is the working time for the n th turning section, Lm is 2 m 4 6
t
2t 2 1 2
the distance of the m th straight path, and T n is the veering 9R
angle of the n th turning section.
From Fig. 4, the consumed energies for the straight section where
or veering section of navigation is achieved as 4tT0 4 Dw 1 tT0
A 2 2
1 § m12 t 2 Lm m12 t 3 · § 2m t · 2 3
2 2 ª 2m12 t 3 º
Esm ¨ ¸ (7) 4t 2 ¨ E0 1
¸ 4 Dw 1 t « E0 Dw 1 »
R© V 3 ¹ © 3R ¹ ¬ 3R ¼
1 § m12 t 2T n m12 t 3 · The factor A is derived under the requirement to assure that
Ecn ¨ ¸ (8)
R © We 3 ¹ the values of J with k being the minimum or maximum will
where Esm is the consumed energy for the m th straight be the same.
section of the mowing path, Ecn is the consumed energy for
the n th turning section, and R is the equivalently inner B. Geography method in path planning and obstacles
resistance of the mower’s motor. avoidance
The total amount of the straight section for the entire path is After the preliminary mowing path planning, the mowing
N s 2 Dw k 1 . The total amount of the veering section for path will be specified by a series of critical points. However,
the whole mowing path is N c 2 Dw k 2 . After combining in practice, there might be expected situations which happen
in a sudden during the navigation of the mower. For example,
N c , N s with (5)-(8), the total working time and consumed some fixed obstacles or moving objects may appear on the
energy are derived, respectively, as path and disturb the pre-specified mowing process.
L T To solve the problem, a strategy based on the potential filed
T 4 Dw 2k 3 t (9)
V We with respect to obstacles is proposed. The strategy is named
the geography method. In which, two indices reflecting
m12 t 2 L m12 t 2T m2t 3 (10)
E 4 Dw 2k 3 1 dangerousness and difficulty of the mowing process are
RV RWe 3R proposed where the index of dangerousness is defined as
where
L n 2 2
total working time for the straight navigation xi x0 j yi y0 j
V
T
dangeri ¦h r
j 1
j (13)
total working time for the turning navigation
We
4 Dw 2k 3 t extra working time induced during the periods
where xi , yi denote the mower’s present coordinates,
of strating and stopping operations
m12 t 2 L
consumed energy druing the straight navigation
x0 j , y0 j are the coordinates of the j-th obstacle,
RV
2 2
m1 t T h j represents the value of the index of difficulty at the mass
consumed energy during the turning navigation
RWe x x
2
y y
2
m2t 3
center of the obstacle, r i 0 k i 0k
represents the
4 Dw 2k 3 1 consumed energy savfed during the decreasing rate of the obstacle strength, and n represents the
3R
processes of acceleration and deceleration number of obstacles. The index of difficulty representing the
and T is the total working time, L is the total distance of the objective field strength is defined as
mowing path, and T is the sum of all turning angles.
An integrated objective function can thus be defined as difficultyi xi xe
2
yi ye
2
(14)
2 2
§ T0 · § 2km12t 3 ·
J ¨ 2kt ¸ ¨ E0 ¸ (11) where xe , ye denote coordinates of the destination.
©A ¹ © 3R ¹
The overall objective function is the combination of the two
indices:Ji difficultyi dangeri (15)
The objective field is built according to the value of J i with
respect to the coordinates xi , yi in this filed. Fig. 5 Set xi , yi to be the
demonstrates an example of the objective field. next woking corrdi- Ti
nates
140
120
xi xi l cos Ti
100 yi yi l sin T i
80
J
60
40
20
0 0
0 20
20 40
40
60 60
80 80
x
y
Fig. 5. Example of the objective field
Fig. 6. Operational flowchart of the geography method
Once the objective field has been determined, the optimal
path can be found. The next mowing point is given by Figure 7 illustrates a flat lawn area with the optimal moving
xi , yi = xi l cos T i , yi l sin T i where l is a constant used path obtained in Fig. 5 and the path planned by the geometry
to specify the unit incremental working length. The optimal Ti method where the starting coordinates are at 5, 6 , the
with respect to the i th optimal angle, which achieves the destination at 50, 65 , and the obstacles located at 3, 20 and
minimal J i , is found by considering 60,35 , respectively.
dJ i xi l cos T i , yi l sin Ti
0 (16)
dT i
y
That is,
1
2 2
ª xi l cos Ti xe 2
yi l sin Ti ye º
¬ ¼
n
x
ª¬ xi xe l sin Ti yi ye l cos Ti º¼ ¦ h j ln r Fig. 7. Re-planned mowing path after appearance of an object
j 1
1
° 2 2
2 In this figure, the line constituted by the symbol “ ”
®lnr ª¬« xi l cos Ti xoj yi l sin Ti yoj º»
¼
(17) represents the original path in Fig. 5. An obstacle whose
°̄
position located at 55,50 appears unexpectedly during
1 2
½°
ª« xi l cos Ti xoj
2
yi l sin Ti yoj
2
º 2 ln r ¾
navigation. After re-planning, a revised mowing path is
¬ ¼» 2 °¿ generated, which is displayed in the figure with the line
constituted by “ o ”.
ª xi xoj l sin Ti yi yoj d cos Ti º 0 The decreasing rate of the density r should be redefined to
¬ ¼
avoid collision between the mower and the objects before
The optimal Ti can be found by solving the nonlinear applying the method to enhance the mowing efficiency. In
practical situation, the moving objects could be human being,
equation (17) and the coordinates of the next via point are
when the mower is in the invisible zone behind the people, the
obtained accordingly. The procedure proceeds until all
decreasing rate should be large to let the people be far away
working points have been determined. See Fig. 6 for the flow
from the mower. The decreasing rate is thus defined as
chart. The progress will be terminated once the absolute value
of error between xi , yi and xi 1 , yi 1 is less than a pre-
1 §1 ·
r cos ¨ T ¸ 0.1 (18)
specified threshold. An example of optimal mowing path is 2 ©2 ¹
shown in Fig. 5 where the pre-specified threshold was defined
as 0.1, l = 0.1 mmin and n 2 . JJJK JJJJK
where T is the angle between oi o and oi pi shown as in Fig.
8. In Fig. 8, o located by xo , yo represents the previouscoordinates of the moving object, oi located by xoi , yoi JJJJK JJJJK
represents the current coordinates of the moving object, and P2 P3 u P1 P3 ! 0 (19)
pi represents the current coordinates of the mower.
JJJJK JJJJK
oi where P2 P3 and P1 P3 are, respectively, the vectors between
a1 , a2 and b1 , b2 . This is equivalently to
T a1b2 a2 b1 ! 0 (20)
pi
If (19) is satisfied, then the point P2 should be a concave
o
Fig. 8. Definition of the decreasing rate corner and needed to be modified. To smooth the working
boundary, all concave section can be pushed outward. Fig. 11
III. INITIALIZATION AND MOWING MAP BUILDING shows an example of the original, extended and modified
working boundaries.
Initially, the mower should be pulled by an operator along
the working boundary and fixed obstacles to build up a
working map. At the meantime, the GPS receiver sends
positioning information to the embedded system which
decodes NEMA 0183 data and memorizes these data to
characterize the mowing boundary.
A. Modification of Mowing Boundary Coordinates
In practice, there might be via points missed during
initialization. Therefore, some intermediate via points should ( a ) Original boundary ( b ) Extended boundary
be interpolated between two adjacent points to smooth the
boundary of the working field. Fig. 9 shows the initial
coordinates and the smoothed mowing map of the example.
( c ) Modified boundary
Fig. 11. Example of original, extended, and modified boundaries
( a ) Original map: 5 GPS points ( b ) Extended map: 29 GPS points C. Coordinate Transformation and Inverse Coordinate
Transformation
Fig. 9. Initial and extended map
The modified map should be transformed to the one that the
B. Modification of Working Area coordination of this updated map follows the longitudinal
coordination.
It is quite common that there exist concave corners in the
initially built working map which usually cause mowing ª1 º ª0 º
Firstly, we change the basis from « » ʿ « »
difficulty. Therefore, the map should be modified prior to ¬ 0 ¼ ¬1 ¼
further process. ª x x º ª y2 y1 º
Firstly, an index is defined for identifying concavity within to « 2 1 » ʿ « » where pi { xi , yi , i 1, 2 .
the mowing map. In Fig. 10, P1 , P2 , and P3 are three adjacent ¬ y2 y1 ¼ ¬ x2 x1 ¼
points in the extended map. If the extended map possesses n critical points, relation of
P3 transformation is defined as follows
ª xik º ª axk cyk º
« » « bx dy » x2 x1 y2 y1 (21)
j
¬« yk ¼» ¬ k k¼
P2
where xik , j
yk denotes the transformed coordinates,
P1
1 d k d n and
Fig. 10. Three GPS coordinates of the extended result
Judgment on a concave corner is defined asx2 x1 y2 y1
a 2 2
,b 2 2
, TABLE II
x2 x1 y2 y1 x2 x1 y2 y1 RESULTS FOR THE MINIMAL ENERGY MODE
y2 y1 x2 x1 Scenario Consumed power (W)
c ,d
x2 x1
2
y2 y1
2
x2 x1
2
y2 y1
2
No obstacles 0.45
Fixed obstacles 0.542
The term x2 x1 y2 y1 represents a weighting factor used Moving objects (three fixed, 0.5
to achieve integer transformed coordinates. The inverse one moving)
coordinate transformation is correspondingly given by
TABLE III
RESULTS FOR THE MIXED MODE
ª xk º 1 ª x2 x1 xik y2 y1 jyk º Scenario Consumed power (W)
«y » « » (22)
¬ k¼ x2 x1 y2 y1 ¬« y2 y1 xik x2 x1 j
yk »¼ No obstacles 0.51
Fixed obstacles 1.0354
Moving objects (three fixed, 0.89
IV. DEMONSTRATION AND VERIFICATION one moving )
A. Results Note: The power consumed by the mowing blade wasn’t taken into account.
The experimental autonomous wheeled mower is displayed
in Fig. 12. The variable Rearth is 6378 (m), d is
1.85 m/mmin , l is 0.1, and r in (13) is 0.9. The ending
condition means that the tracking error on the x-y plane
converge to a reasonable level within the feasible range.
(a) Minimal time mode (b) Minimal energy mode
Fig. 12. The experimental autonomous wheeled mower
For comparison, three scenarios “no obstacles”, “fixed
obstacles”, and “moving objects” were considered. The (c) Mixed mode
Fig. 13. Optimal paths without obstacles existing in the field
experimental results are listed in Tables I-III. The ideal paths
under different conditions are shown as in Figs. 13-15. There
were three obstacles assumed in Fig. 14.
To test applicability of the geography method in dealing
with the moving objects avoidance, assumed an obstacle
located at 40471.8,7298 (mmin) and moved to
40476,7305.2 (mmin) with the speed of 1.11 m/s ; the
diagonal line illustrated represents the path of the moving
object in Fig. 15. Three fixed obstacles in Fig. 15 denoted by (a) Minimal time mode (b) Minimal energy mode
o3 , o1 and o2 are located, respectively,
at 40472.8,7300.5 , 40474.6,7304.3 , and
40471.4,7303 (mmin,mmin) . The resulting power
consumptions for three operational modes are listed in Tables
I to III.
TABLE I
RESULTS FOR THE MINIMAL TIME MODE (c) Mixed mode
Fig. 14. Optimal paths with fixed obstacles in the field
Scenario Consumed power (W)
No obstacles 0.76
Fixed obstacles 1.4
Moving objects (three fixed, 1.37
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V. CONCLUSION
The optimal path planning algorithm for the autonomous
lawnmowers with the capability to achieve minimum working
time, minimum energy consumption and mixed operation
mode as well as the solution to CCPP is developed. An
autonomous lawnmower equipped with a GPS enabling real-
time positioning and self-navigation has been built. The
theory proposed in path planning has been verified with
experiments. It is shown that the preliminary path planning
algorithm improves the operational efficiency either for
working time or energy consumption.
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