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Citation: Liu, C.; Liu, A.; Wang, R.;
Zhao, H.; Lu, Z. Path Planning
Algorithm for Multi-Locomotion
Robot Based on Multi-Objective
Genetic Algorithm with Elitist
Strategy. Micromachines 2022,13, 616.
https://doi.org/10.3390/mi13040616
Academic Editors: Zhangguo Yu
and Marco Ceccarelli
Received: 8 March 2022
Accepted: 13 April 2022
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micromachines
Article
Path Planning Algorithm for Multi-Locomotion Robot Based on
Multi-Objective Genetic Algorithm with Elitist Strategy
Chong Liu , Aizun Liu * , Ruchao Wang, Haibin Zhao and Zhiguo Lu
School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China;
congliu@me.neu.edu.cn (C.L.); 2110099@stu.neu.edu.cn (R.W.); hbzhao@mail.neu.edu.cn (H.Z.);
zglu@me.neu.edu.cn (Z.L.)
*Correspondence: 2000371@stu.neu.edu.cn
Abstract:
The multi-locomotion robot (MLR), including bionic insect microrobot, bionic animal robot
and so on, should choose different locomotion modes according to the obstacles it faces. However,
under different locomotion modes, the power consumption, moving speed, and falling risk of MLR
are different, and in most cases, they are mutually exclusive. This paper proposes a path planning
algorithm for MLR based on a multi-objective genetic algorithm with elitist strategy (MLRMOEGA),
which has four optimization objectives: power consumption, time consumption, path falling risk,
and path smoothness. We propose two operators: a map analysis operator and a population diversity
expansion operator, to improve the global search ability of the algorithm and solve the problem so that
it is easy to fall into the local optimal solution. We conduct simulations on MATLAB, and the results
show that the proposed algorithm can effectively optimize the objective function value compared
with the traditional genetic algorithm under the equal weight of the four optimization objectives,
and, under alternative weights, the proposed algorithm can effectively generate the corresponding
path of the decision maker’s intention under the weight of preference. Compared with the traditional
genetic algorithm, the global search ability is improved effectively.
Keywords:
multi-locomotion robot; path planning; genetic algorithm; elitist strategy; multi-objective
optimization
1. Introduction
Inspired by bionics, many bionic robots have been developed, including bionic insect
microrobot and medical microrobot, bionic humanoid robot, and so on. These types of
robots usually have multiple modes of locomotion called multi-locomotion robot (MLR).
This kind of robot has a broad prospect in a special work environment, such as pipeline
maintenance, drugs transportation inside of human body, terrain exploration, and so
on. The authors of [
1
] designed a humanoid robot system which supports bipedal or
quadrupedal walking, climbing, brachiation, and even flying. The algorithm proposed
in this paper is mainly based on this kind of robot, but the idea of this algorithm is also
applicable to other kinds of MLRs. In the related studies, researchers realized bipedal
walking on the flat ground [
2
,
3
], quadruped walking on the slope [
4
], swinging on the
ladder [
5
–
7
], and climbing a vertical ladder [
8
]. In [
2
], the researchers proposed the 3-D
biped dynamic walking algorithm based on the PDAC, and validated the performance
and the energy efficiency of the proposed algorithm. In [
4
], the researchers determined
an optimal structure for a quadruped robot to minimize the robot’s joint torque sum.
In [
5
], the researchers presented a control method to realize smooth continuous brachiation.
The authors of [
6
] proposed an energy-based control method to improve the stability of
continuous brachiation. The authors of [
7
] designed a type of brachiation robot with three
links and two grippers, and designed a control method based on sliding-mode control,
which improved the robustness and swing efficiency of the robot. In [
8
], the researchers
Micromachines 2022,13, 616. https://doi.org/10.3390/mi13040616 https://www.mdpi.com/journal/micromachines
Micromachines 2022,13, 616 2 of 30
introduced a vertical ladder climbing algorithm of the MLR only by the posture control
without any external sensors.
With the diversification of robot locomotion mode, the application environment of
robots has also expanded from the laboratory to the field environment. Due to its high
adaptability to the environment, the MLR should choose different locomotion modes
according to the obstacles it faces, as shown in Figure 1.
Figure 1. Different locomotion modes for different obstacles.
So, multi-locomotion robots can pass obstacles that single locomotion robots could not
pass safely in the past by switching locomotion modes. For example, as shown in Figure 2,
there is a little bridge above a river, the MLR will choose to switch its locomotion mode
from bipedal walking to quadrupedal walking, to go on the slope, and go through the
bridge with minimal probability of falling down. For a normal biped robot, walking on the
slope of the small bridge in the bipedal way, there will be a high probability of falling down.
Figure 2. MLR goes over a bridge above a river.
Therefore, in the same field environment, the number of possible paths for MLR is
greatly increased compared with robots with single locomotion mode. How to choose
among many possible paths is a problem to be solved.
Different locomotion modes, such as bipedal and quadrupedal walking, have specific
and different capabilities because a robot’s mobility is constrained by the physical and
structural conditions for a motion [
9
]. For example, bipedal walking means low power
consumption but is prone to falling down according to [
2
,
10
,
11
]; so, it is suitable to go over
flat ground; quadrupedal walking is suitable to climb a slope because of its high stability,
but its speed is rather slow [
4
,
12
]. Thus, it is necessary to take power consumption, time
Micromachines 2022,13, 616 3 of 30
consumption, and falling risk all into consideration in path planning, as shown in Figure 3.
These three goals are normally mutually exclusive; so, the path planning problem is a
multi-objective optimization problem.
Figure 3. Multi-objective path planning.
Many researchers have proposed path planning algorithms for multi-objective opti-
mization, including multi-objective path planning based on the genetic
algorithm [13,14]
,
multi-objective path planning based on the particle-swarm optimization algorithm
(PSO) [15–17], and multi-objective ant colony algorithm path planning [18].
In [
19
], inspired by the frogs’ behavior, the researchers proposed a multi-objective
shuffled frog-leaping path planning algorithm. This algorithm has three objectives: path
safety, path length, and path smoothness. In this research, the optimization method of path
safety was realized by ensuring the distance between path and obstacle. When the path
generated by the algorithm passes through the obstacle, the path safety operator will look
for two candidate regions in the vertical direction of this section of the path crossing the
obstacle, and determines the new path node according to the vertical distance between the
candidate regions and this section of the path in order to generate two new path sections,
to avoid passing through the obstacle.
In [
20
], in order to solve the problem of multi-objective optimization in the global
path planning of autonomous intelligent patrol vehicle, which is the shortest path length
and the smallest total angle change in the path, the researchers proposed a path planning
method based on a multi-objective Cauchy mutation cat-swarm optimization algorithm.
The multi-objective problem proposed in this algorithm only considered the path length
and path smoothness; thus, the practicability is limited.
In [
21
], the researchers proposed an improved ant colony optimization-based algo-
rithm for user-centric multi-objective path planning for ubiquitous environments. This
algorithm uses the ant colony algorithm to plan a path for vehicle navigation in an urban
map considering length, traffic condition, and power consumption; the traffic condition in
this research is very similar to the path safety, which provides a reference for considering
the path safety in our research.
In [
22
], the researchers proposed an aging-based ant colony optimization algorithm.
The researchers introduced a modification based on the age of the ant into the standard ant
colony optimization. This algorithm can find the shortest and the most free-collision path
under static or dynamic environment, and, when compared with other traditional algo-
rithms, it proves its superiority. However, path safety is not considered in this algorithm.
In [
23
], the researchers proposed a multi-objective path planning algorithm based on
an adaptive genetic algorithm. In this algorithm, the self-defined genetic operator is used to
realize the optimization of the path length and smoothness, and the artificial potential field
theory is introduced to realize the planning of the path safety which inspired our research
in the optimization of path safety. However, the path safety mentioned in this algorithm
only considers the distance between the robot and the impassable obstacle, and does not
involve road conditions and the locomotion mode of the robot.
Micromachines 2022,13, 616 4 of 30
In [
24
], the researchers proposed a multi-objective genetic algorithm path planning
method for reconfigurable robots. This kind of robots can provide high dexterity and
complex actions by reconfiguring its shape. This method proposed four objective functions:
goal reachability, time consumption, path smoothness, and path safety. Even the reconfig-
urable robots can provide different shapes to move, but the path safety considered in this
method only considered the distance between robots and obstacles rather than considering
the falling risk of a robot in different shapes.
In [
25
], the researchers proposed a multi-objective path planning algorithm for mobile
robot, this algorithm has three objectives: length, smoothness and safety. The applicable
environment of this algorithm is relatively simple, the environment was only divided into
passable and impassable, and the safety optimization only considers the distance between
the robot and the obstacles.
In [
26
], the researchers proposed a path planning method based on interval multi-
objective PSO; this method concentrates on three objectives: path time, path safety, and
path length. In the optimization of path safety, this method takes the road condition into
consideration, which has enlightening values to our works, but the applied objects of these
methods are traditional wheeled robots; so, this method is not suitable for MLR.
Based on the path planning research described above, and taking the high adaptability
to the complex environment of MLR into consideration, a path planning algorithm for
multi-locomotion robot (MLR) based on multi-objective genetic algorithm with elitist
strategy (MLRMOEGA) is proposed in this paper. The algorithm considers the power
consumption, moving speed, and falling risk of the MLR in different locomotion modes
and different environment, and proposes four optimization objective functions: power
consumption, time consumption, path falling risk, and path smoothness. Compared with
previous studies, optimal safety mostly refers to the distance between the robot and the
obstacle. This paper proposes the concept of robot global path falling risk, that is, when
the MLR is moving through alternative locomotion modes, there will be a certain falling
risk, so there will be a falling risk of each possible path. We calculate the falling risk of each
path, and take it into the multi-objective problem to be considered. To solve the problem
of premature convergence of the Genetic Algorithm, we propose two operators: a map
analysis operator and a population diversity expansion operator, to improve the population
diversity in the algorithm process.
The rest of this paper is organized as follows: Section 2is the introduction of the
environment building method and the problem statement. Section 3is the introduction of
the Genetic Algorithm and the proposed algorithm. Section 4is the implementation of the
multi-objective path planning including building global environment and details of the
algorithm. Section 5is simulation experiment. Section 6is the conclusion.
2. Problem Statement and Preliminaries
2.1. Method of Building the Global Path Planning Environment
The path planning of a robot is divided into two steps: first, abstract out a global map
containing real environment information, then, execute the path planning algorithm and
the path will be generated.
In this paper, we map the environment on a grid of size n
×
n, where the grid size
nis decided depending on the accuracy required in the path planning problem [
27
]; the
bigger nis, the more accurate the map will be. Each grid of the map contains environmental
information of its location. There is an example of a 10
×
10 grid map, as shown in Figure 4,
in this example, black grids represent impassable obstacles, white grids are traversable.
The position of each grid can be obtained by index or coordinate values, which can be
converted to each other as Equation (1):
ind =A×y+x−(A+1)
(y=bind
Ac+1
x=ind%A+1
(1)
Micromachines 2022,13, 616 5 of 30
In this equation, ind is the index value of gird, Ais the size of the grid map,
b c
means
to round down the value to the closest integer, % means to take the remainder as the result.
Figure 4. Example of a 10 ×10 grid map: (a) index to the grid; (b) coordinate of the grid.
2.2. Optimization Objective Functions
This paper proposes four optimization objectives for MLR: power consumption, time
consumption, path falling risk, and path smoothness.
One path consists of many grid cells from the grid map and we call them nodes of
paths, and we divide one path into several sections. We call them path sections based on
the locomotion modes that the MLR takes. d
i
is the Euclidean distance of each path section
as shown in Equation (2) [23]:
di=∑J
j=1q(xj−xj−1)2+ (yj−yj−1)21≤x,y≤A(2)
In this equation, iis the number of this path section; jis the node number in this
section; Jis the total number of nodes in this section; Ais the size of the grid map.
1.
Power consumption: In order to pass different road conditions, the MLR will choose
alternative locomotion modes with different power consumption. In the real environ-
ment, the energy stored by the MLR itself is very limited, which greatly restricts the
activity duration of the MLR. Therefore, it is very necessary to include power con-
sumption into the optimization objective in path planning. The power consumption
is calculated by Equation (3):
f1→
p=∑N
i=1Ci•di(3)
In this equation,
→
p
is a vector that consists of the nodes’ coordinate of one path; C
i
is
the power consumption of different path sections with different locomotion modes; Nis
the total path section count of this path.
2.
Time consumption: Different locomotion modes bring different time consumption.
We expect MLR to perform time-sensitive tasks, such as terrain detection, material
transport, rescue and so on; so, time consumption must also be considered in path
planning. It is calculated by Equation (4):
Micromachines 2022,13, 616 6 of 30
f2→
p=∑N
i=1
di
Si
(4)
S
i
represents the moving speed of the MLR in this path section with one locomotion
mode; i,N, and →
prepresent the same quantities as in Equation (3).
3.
Path falling risk: Different locomotion modes not only affect the power consumption
and time consumption of one path, but also the falling risk. Compared with ordinary
wheeled mobile robots, the locomotion modes of MLR have higher risk of falling, and
its design is aimed at a more complex field environment; so, it is also important to
ensure that the falling risk of MLR in the process of moving is within an acceptable
range. In addition to considering the falling risk of the MLR in different locomotion
modes, the condition when the MLR is too close to an impassable obstacle should also
be taken into consideration; so, we refer to the theory of artificial potential field [
28
]
and transform the diffusion of the potential field into the influence on the falling risk
of surrounding girds. When one grid in the map represents an impassable obstacle,
the falling risk of the eight surrounding grids will be increased, as shown in Figure 5.
Figure 5. The diffusion of artificial potential field.
We define the influence of impassable obstacles on the surrounding grid as shown in
Equation (5):
(Psg(f)=Praw(f)+0.05 D=1
Psg(f)=Praw(f)+0.03 D=√2
D=qxip −xsg2+yip −ysg 2
(5)
In this Equation,
Psg(f)
represents the falling risk of surrounding girds of the impass-
able obstacle;
Praw(f)
represents the falling risk of the grids when there is no impassable
obstacle around; Dis the Euclidean distance between the impassable obstacle and the
surrounding grids; x
ip
,y
ip
are the coordinates of the impassable obstacle; x
sg
,y
sg
are the
coordinates of the surrounding girds.
We assume that whether the MLR falls or not is independent of the path at any
different nodes and we calculate the falling risk with Equation (6):
f3→
p=1−∏n
k=1[1−Pk(f)] (6)
Micromachines 2022,13, 616 7 of 30
kis the count of node in an individual. nis the total node count in an individual. P
k
(f)
is the falling risk when the MLR passes one node.
4.
Path smoothness: We want the path generated by the algorithm to be as smooth as
possible, and we define the sum of all the angles in the path as the smoothness of that
path. The smoothness of a path is calculated by Equation (7) [23]:
f4→
p=∑T
t=1π−arccos(b2+c2−a2)
2bc (7)
In this Equation:
a=q(xk+1−xk−1)2+(yk+1−yk−1)2
,
b=q(xk−xk−1)2+(yk−yk−1)2
,
c=q(xk+1−xk)2+(yk+1−yk)2
,kis the count of
node in one path, Tis the total number of turns in one path.
5.
The synthetic objective function: We synthesize the above four objective functions and
linearly weighted them as the final objective function. As shown in Equation (8) [
23
]:
(f(→
p) = cw ×f1(→
p) + tw ×f2(→
p) + rw ×f3(→
p) + aw ×f4(→
p)
cw +tw +rw +aw =1cw ≥0, tw ≥0, rw ≥0, aw ≥0(8)
In this equation, cw is the power consumption weight, tw is the time consumption
weight, rw is the falling risk weight, and aw is the smoothness weight. The weight values are
determined by decision makers through experience or practical requirements; they repre-
sent the importance that decision-makers attach to different objective functions. The higher
a weight value is, the more attention it receives in multi-objective optimization problems.
3. Introduction of MLRMOEGA
3.1. Overview of Genetic Algorithm
Genetic Algorithm (GA) was proposed by Professor J. Holland in 1975. This algorithm
simulates the genetic processes in nature, that is, starting from an initial population, through
selection, crossover and mutation operation, a new population with higher adaptability
to the environment can be obtained. In theory, the population will keep approaching to a
better search space and, finally, will become a group of individuals which is most adaptable
to the environment.
In our research, we use grid map to describe the environment; with this method, the
path planning problem becomes a discrete problem and the GA has a good performance
in dealing with discrete problems. In the process of GA, each grid can play the role of
genes very well, and the path composed of grids is a chromosome. Combining the grid
number of each path and the environmental information contained in each grid, the power
consumption, time consumption, path falling risk, and path smoothness of each path can
be clearly obtained, and then, the fitness of each path can be obtained. We explain this in
Section 4.1.
There are other intelligent algorithms, such as ant colony algorithm (ACO) and particle
swarm optimization (PSO). ACO works efficiently in graph and network space, such as
the traveling salesman problem (TSP) and the scheduling problem [
21
], but is unsuitable
to solve path planning problems in a grid map. The original POS has been designed to
work in continuous space, and it needs some major changes to adapt to path planning
problems [
21
]. As we use a grid map to describe the environment in our research, it turns
out that POS is not very suitable.
The application of GA in path planning is shown in Figure 6.
Micromachines 2022,13, 616 8 of 30
Figure 6. GA used in path planning.
3.2. Elitist Strategy
When GA only includes selection, crossover, and mutation, it is called Simple Genetic
Algorithm (SGA). It is proved that the SGA does not converge almost surely to the set of
populations that contains the optimum point as one of its points [
29
]; so, we introduce
the Elitist Strategy into SGA. It is shown that Elitist Strategy helps in achieving better
convergence in SGA [
30
]. The Simple Genetic Algorithm with Elitist Strategy is called EGA,
the flowchart of EGA is shown in Figure 7.
Elitist strategy improves the global convergence of GA, but on the other hand, it also
makes a local optimal individual not easy to be eliminated and continues to be retained in
the genetic processes, thus affecting the global search ability of algorithm, and eventually
leads to premature convergence; so, it is usually used in conjunction with other operators.
In the following section, we propose two operators to work with EGA, in order to improve
the global search ability of the algorithm.
Micromachines 2022,13, 616 9 of 30
Figure 7. Flowchart of EGA.
3.3. The Proposed Algorithm
The premature convergence is generally due to the loss of diversity within the popu-
lation [
31
]. In order to solve this problem, the map analysis operator and the population
diversity expansion operator are proposed in this paper:
•
Map Analysis Operator: Once the algorithm starts, the map analysis operator will
analyze the grid map, divide various obstacles in the grid map into regions and store
the index values of each region, respectively. There are two methods to divide a
map into regions: the four-connected principle and the eight-connected principle. We
assume the current grid coordinate is (x,y), the four-connect principle considers the
four grids with (x+ 1, y), (x
−
1, y), (x,y+ 1) and (x,y
−
1) as the same region, and the
Micromachines 2022,13, 616 10 of 30
eight-connect principle considers the eight grids with (x+ 1, y), (x
−
1, y),
(x,y+ 1),
(x,y
−
1) (x+ 1, y+ 1), (x
−
1, y
−
1), (x
−
1, y+ 1) and (x+ 1, y
−
1) as the same
region. For example, as shown in Figure 8, the four-connected principle divides this
obstacle into three regions as shown in Figure 8a marked by circled numbers, while
the eight-connected principle into one region, as shown in Figure 8b. The MLR has
eight moving directions in the map; therefore, it is more reasonable to choose the
eight-connection principle.
Figure 8. (a) Four-connected principle; (b) eight-connected principle.
The pseudo-code for the Map Analysis Operator is as follows:
Map Analysis Operator:
Input: A matrix map of gray values
1: Store all slope grids’ index in the matrix map of gray values
2: Store all non-slope grids’ index in the matrix map of gray values
3: Set the gray values of all non-slope grids to 0 in the matrix map
4: Set the gray values of all slope grids to 1 in the matrix map, get a new matrix map
5: Divide the new matrix map into regions according to the eight-connected principle
6: The indexes of the slope grid are stored according to the divided regions and saved into a
cell array
//Other kinds of obstacles are divided into regions in the same way
7: Combine all grids’ index stored according to the divided regions into one cell array
Output: A cell array contains the grids’ index according to the divided regions
•
Population Diversity Expansion Operator: In the genetic process, in order to improve
the global search ability of GA, the most direct method is to improve population diver-
sity. For MLR, the diversity of the population is reflected in whether the locomotion
modes adopted by MLR are sufficiently diverse, that is, whether the path individuals
in the population have passed the obstacles that MLR needs to adopt alternative loco-
motion modes to pass in the global map. Therefore, we use the population diversity
operator to check whether all the regions obtained by the map analysis operator have
been gone through by the path in the population. If some regions have not passed
through, the paths going through those regions will be generated to improve the
population diversity.
Micromachines 2022,13, 616 11 of 30
The pseudo-code for the Population Diversity Expansion Operator is as follows:
Population Diversity Expansion Operator:
Input: The divided regions cell array from map analysis operator; the cell array composed of
populations of this generation
1: All path individuals in the population are intersected with the region in the divide regions
cell array
2: Take the region that has no intersection with any individual in the population, this is the
unpassed region
3: A random point in the unpassed region is selected to generate a new path individual
4: Put the new path individual into the population
Output: The population after diversity expansion
The flowchart of MLRMOEGA is shown in Figure 9.
Figure 9. Flowchart of MLRMOEGA.
Micromachines 2022,13, 616 12 of 30
4. Implementation
4.1. Encoding Method
One of the main factors in the implementation of SGA is the encoding method. In
conventional GA, solutions are encoded in binary strings, however, several different encod-
ings are also possible depending on the problems [
32
], and we use floating point encoding
in this paper. In this method, each gene value of the individual is represented by a real
number in a certain range, and the encoding length is equal to the number of decision
variables. The encoded individual is a set of grid index, representing a possible path, as
shown in Equation (9).
→
p={index1,index2, . . . , indexi, . . . , indexn}(9)
The composition of one individual in our research is shown in Figure 10.
Figure 10. Composition of one individual.
4.2. The Evaluation Function
The evaluation function is used to evaluate the quality of an individual in the algorithm
process. In the selection operation, the larger the value of the evaluation function is, the
more likely it is to be retained in the next generation.
1.
The power consumption evaluation function: To make the data dimensionless, we
take the natural logarithm of the data, and use the same method to calculate the other
data. According to Equation (3), the evaluation function of power consumption is
Equation (10):
F1(→
ppl ) = (ln(f1))−1(10)
2.
The time consumption evaluation function: According to Equation (4), the evaluation
function of time consumption is Equation (11):
F2(→
ppl ) = (ln(f2))−1(11)
3.
The path falling risk evaluation function: According to Equation (6) and to make
the data dimensionless, we have the evaluation function of falling risk as shown in
Equation (12):
F3(→
ppl ) = (ln(f3×1000 −BC))−1(12)
BC is a constant value related to the motion data of the robot, and its function is to
reduce the quantitative difference between the falling risk evaluation function and other
evaluation functions.
4.
The path smoothness evaluation function: The smoothness evaluation function is
quite different from Equation (7). First, we define a variable
αt
, it represents the value
of the angle formed by every three consecutive nodes as shown in Equation (13):
αt=π−arccos b2+c2−a2
2bc !(13)
Micromachines 2022,13, 616 13 of 30
The variables in this equation represent the same quantities as in Equation (7).
The evaluation function of the smoothness consumption is Equation (14):
F4→
ppl =
ln
5+∑T
t=1
10 90◦<αt<170◦
30 45◦<αt≤90◦
90 αt≤45◦
−1
(14)
Tis the total number of turns in an individual. In Equation (14), each individual has
an initial value which is 5, to prevent taking the natural logarithm of 0. We assign a penalty
value to every angle that occurs in each individual, when the angle is obtuse, the penalty
value is small; then, with the smaller αt, there will be a greater penalty value.
4.3. Execution Process of MLRMOEGA
(1)
Map analysis: According to the eight-connected principle, the obstacles in a grid map
are divided into regions.
(2) Initial population: Generate the required number of individuals for the initial population.
(3)
Calculate fitness value: Normally, a common method used to solve multi-objective
problems is by a linear combination of the objectives, in this way creating a single-
objective function to optimize [
33
]. In is step, we linearly weight the evaluation
functions as shown in Equation (15):
(F(→
ppl ) = cw ×F1(→
ppl ) + tw ×F2(→
ppl ) + rw ×F3(→
ppl ) + aw ×F4(→
ppl )
cw +tw +rw +aw =1cw ≥0, tw ≥0, rw ≥0, aw ≥0(15)
The variables in this equation represents the same quantities as in Equation (8).
(4)
Pick out the elitist individual: according to the previous step, we pick out the individ-
ual with highest
F(→
p)
called elitist individual, and retain it to step (10). This is the
first step of the Elitist strategy.
(5)
Population diversity expansion: Check population diversity and expand population
to improve population diversity when population diversity declines.
(6)
Selection: In the selection process, we use the proportion select method. The
F(→
p)
of
all the individuals in this generation has been calculated in the previous step, so we
sum them up, and the fitness of each individual divided by this sum is the probability
of being selected in the proportional selection method for one individual, that is,
Equation (16):
PApl =F(→
ppl )
∑PL
pl=1F(→
ppl )(16)
In this equation, event A
pl
represents the plth individual in the population which is
selected into the next generation, PL is the population quantity.
(7)
Crossover: We set a crossover probability called CP in advance, and then check
whether two individuals in the population share one gene, that is, whether they pass
through one same node, and if they do, we use the probability to determine whether
they do crossover. Once the crossover happened, all the genes that follow the same
gene in two individuals are swapped and generate two new individuals.
(8)
Mutation: We set a mutation probability called MP in advance, check each individual
in this population; if one individual needs to mutate, we delete randomly three
genes which are the nodes in this individual, then generate a new individual through
program operation.
(9)
Calculate fitness value: We calculate individuals’ fitness value again.
(10)
Delete the worst individual and insert the elitist individual: This is the second step of
the Elitist strategy. According to the previous step, we delete the individual with the
lowest F(→
p), and put in the elitist individual from step (4).
Micromachines 2022,13, 616 14 of 30
(11)
Check whether the iteration limit is reached: If the iteration limit is reached, output
the path information, otherwise, go back to step (3), and continue the iteration.
5. Simulation and Analysis
The simulation experiment is conducted by MATALB 2020b, the configuration of
computer is Core i5 CPU (3.8 GHz), 8 GB RAM, and Windows10 system. The idea of
our experiments in this section is to test the effectiveness of the proposed algorithm in
random 20
×
20 grid maps and compare the performance of the proposed algorithm with
other algorithms. Then, we test the proposed algorithm performance in a simulated field
environment which is abstracted into a 30 ×30 grid map.
5.1. The Software System
To conduct the simulation, we developed a user interface, as shown in Figure 11. In
this software, we can input and adjust the data of the MLR moving modes and edit the
global map by clicking the grid.
Figure 11. User interface (with 30 ×30 grid map).
There are 10 kinds of grids and 7 kinds of locomotion modes in total. In addition to
the mentioned flat ground, swing ladder, slope and vertical ladder and their corresponding
locomotion modes in Section 1, there are six other kinds of grids and other locomotion
modes: start and goal point seemed as flat ground, river, and high wall, which are impass-
able obstacles, stream for long jump mode, a small wall for which the MLR chooses the
high jump mode to go through, and a middle wall over which the MLR will climb.
To verify the validity of the algorithm, we assume a set of MLR data as shown in
Table 1:
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Table 1. The assumed MLR data.
Locomotion Mode Power Consumption Speed Falling Risk
Bipedal Walking 5 0.5 0.035
Quadruped Walking 10 0.3 0.02
Upward Climb 15 0.15 0.01
Swing 20 0.2 0.0015
High Jump 16 0.6 0.04
Climb Over 25 0.1 0.03
Long Jump 16 0.55 0.045
5.2. Verify the Validity of Artificial Potential Field Theory
Since the four objective functions will influence each other in the process of multi-
objective optimization, we first verify the effectiveness of the falling risk optimization
process with artificial potential field theory. The parameters of SGA and MLRMOEGA are
shown in Table 2. We use SGA and MLRMOEGA, respectively, to generate a path 10 times in
a random environment (20
×
20 grid map) and a simulate environment (
30 ×30 grid map
).
In the simple environment, there is only one kind of impassable obstacle, randomly placed
on a 20
×
20 grid map. The introduction of the simulated environment and the obstacles
in it has been given in Section 5.4.3. The results are shown in Tables 3and 4, the falling
risk of paths generated by MLRMOEGA is significantly lower than the falling risk of
paths generated by SGA (p-value of data in Table 3is 0.000395 and p-value of data in
Table 4is 0.00286). We pick out the paths with the minimum falling risk generated by
the two algorithms, as shown in Figures 12a and 13a, red lines mean the path generated
by algorithm, the path generated by MLRMOEGA effectively avoids walking along or
getting too close to the impassable obstacles that can generate artificial potential fields
which are black blocks shown in figures compared with the path generated by SGA in
Figures 12b and 13b.
Table 2. Parameters of SGA and MLRMOEGA.
Parameters SGA MLRMOEGA
Initial Population Quantity 300 300
Iteration Limit 150 150
Crossover Probability 0.95 0.95
Mutation Probability 0.2 0.2
Unit Grid Length 1 1
Vertical Ladder Height 2 2
cw - 0
tw - 0
rw - 1
aw - 0
Table 3. Results of Artificial Potential Field Theory Experiment (with 20 ×20 grid map).
Algorithm Falling Risk Average Value Algorithm Falling Risk Average Value
MLRMOEGA
0.7442
0.7498 ±0.0186 SGA
0.8194
0.7952 ±0.0248
0.7303 0.7826
0.7409 0.8053
0.7303 0.7326
0.7908 0.8133
0.7438 0.8053
0.7637 0.7788
0.7618 0.8194
0.7623 0.7901
0.7303 0.8053
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Table 4. Results of Artificial Potential Field Theory Experiment (with 30 ×30 grid map).
Algorithm Falling Risk Average Value Algorithm Falling Risk Average Value
MLRMOEGA
0.7303
0.7176 ±0.0073 SGA
0.7137
0.7760 ±0.0451
0.70868 0.71719
0.70868 0.75059
0.70868 0.76285
0.71458 0.78128
0.71458 0.78128
0.72168 0.78128
0.72168 0.79262
0.72168 0.79632
0.72604 0.88266
Figure 12.
Results of artificial potential field test (with 20
×
20 grid map): (
a
) MLRMOEGA path
(Falling risk: 0.7303); (b) SGA path (Falling risk: 0.7326).
5.3. Effect of Design Parameters on the Proposed Algorithm
In this section, we present the influence of the following design parameters: iteration
limit and initial population quantity. The MLR data are shown in Table 1, and other
parameters of the algorithm are shown in Table 5. The way to evaluate the influence of
design parameters on the algorithm is to calculate the synthetic objective function shown in
Equation (8). The smaller the synthetic objective function value is, the better the influence
of the design parameters.
The results below are from the 30 ×30 grid map. The more grid cells in the map, the
larger the number of the iteration limit and the initial population quantity required by the
algorithm; therefore, the design parameters of the algorithm based on the 30
×
30 grid map
test are also applicable to the 20 ×20 grid map.
Micromachines 2022,13, 616 17 of 30
Figure 13.
Results of artificial potential field test (with 30
×
30 grid map): (
a
) MLRMOEGA path
(Falling risk: 0.70868); (b) SGA path (Falling risk: 0.7137).
Table 5. Other parameters of the proposed algorithm.
Parameters Value
Crossover Probability 0.95
Mutation Probability 0.2
Unit Gird Length 1
Vertical Ladder Height 2
cw 0.25
tw 0.25
rw 0.25
aw 0.25
5.3.1. Iteration Limit
The initial population quantity is set to 100. By changing the iteration limit of the
proposed algorithm from 50 to 150, we execute the proposed algorithm 10 times at each
iteration limit, and then take the mean value of the synthetic objective function and exe-
cution time. As shown in Figure 14a, the mean synthetic objective function value of the
obtained path for testing the environment converged to a small range. In addition, with
the increase of the iteration limit, the mean execution time has an upward trend, as shown
in Figure 14b. Taking the mean synthetic objective function value and mean execution
time into consideration, the value 130 is selected to be the optimal iteration limit of the
environment with a 30 ×30 grid.
5.3.2. Initial Population Quantity
The iteration limit is set to 130. We execute the proposed algorithm 10 times at each
initial population quantity, and then take the mean value of the synthetic objective function
value and execution time. Through experiments, we come to the conclusion that the
optimal initial population quantity of environment with a 30
×
30 grid is 200. As shown
in Figure 15a, with the increase of the initial population quantity, the search space of
the proposed algorithm is expanded, search ability is improved; so, the mean synthetic
objective function value decreased, and, after reaching a certain initial population quantity,
the mean synthetic objective function value converged to a small range. In Figure 15b, with
the increase of initial population quantity, the amount of the data needed to be processed
increased. Considering the existence of genetic algorithm contingency, although there are
certain fluctuations, the mean execution time still shows a general upward trend.
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Figure 14.
Effect of the iteration limit on: (
a
) mean synthetic objective function value; (
b
) mean
execution time.
Figure 15.
Effect of the initial population quantity on: (
a
) mean synthetic objective function value;
(b) mean execution time.
5.4. Test Algorithm Performance
5.4.1. Compared with SGA
We compared the performance of the proposed algorithm with SGA in this section. In
the previous part, the optimal design parameters for a 30
×
30 grid map are obtained. With
the increase of grid map size, the initial population quantity and iteration limit required by
GA to achieve good results also increased; so, the design parameters of the 30
×
30 grid
map obtained in our test are also applicable to the 20 ×20 grid map.
The test environment is as shown in Figure 16. This 20
×
20 grid map was generated
randomly, we set the proportion of impassable obstacles to 20%, and the proportion of
other kinds of passable obstacles to 4%. We generate the map for 7 times; each time we
generate the map with flat ground and one kind of obstacle, then, we integrate the 7 maps
into one map. In the process of integration, there will inevitably be many obstacles in the
same grid; so, we set an obstacle in the coincidence grid on the premise of ensuring that
there will not be a large error in obstacle setting proportion.
Micromachines 2022,13, 616 19 of 30
Figure 16. The grid map of the testing environment (20 ×20 grid cells).
The design parameters of SGA and MLRMOEGA are shown in Table 6.
Table 6. The parameters of SGA and MLRMOEGA.
Parameters SGA MLRMOEGA
Crossover Probability 0.95 0.95
Mutation Probability 0.2 0.2
Iteration Limit 130 130
Initial Population Quantity 200 200
We first use SGA and MLRMOEGA, respectively, to generate a path 10 times in the
testing environment shown in Figure 16, and then take the mean value of the synthetic
objective function. The synthetic objective function value of SGA paths is calculated by the
same way as MLRMOEGA paths’, the results are shown in Table 7. To clarify, Figure 17
shows the paths generated by different algorithms.
Combined with the results in Table 7, it can be seen that the objective function values of
the path that passes through the region circled by red in Figure 17a are significantly smaller
than other paths. Due to the better global search ability and convergence of MLRMOEGA,
after 10 tests, 4 paths pass through the red-circled region and are finally retained. While
under the same design parameters, the objective function value of paths generated by SGA
are larger (the p-value of the runs in Table 7is 0.0159) and the global search performance of
SGA is worse than that of MLRMOEGA.
We also conducted experiments in five other random 20
×
20 grid maps, and the
results are shown in Table 8. It can be seen that in different random environments, the
paths’ objective function value generated by MLRMOEGA is significantly smaller than
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that generated by SGA in most cases (the p-values of these five tests are 0.05430, 0.002721,
0.000009037, 0.0008754, and 0.009397).
Table 7. The results of the test (20 ×20 grid map).
Algorithm MLRMOEGA SGA
Weight 0.25; 0.25; 0.25; 0.25 0.25; 0.25; 0.25; 0.25
1st 62.1595 83.4842
2nd 81.7038 114.4333
3rd 64.9642 85.7737
4th 62.7269 110.1055
5th 81.7038 83.2888
6th 63.0997 75.1382
7th 73.0977 118.8659
8th 65.7390 114.5052
9th 81.7038 86.3457
10th 104.9453 83.9620
Average Value 74.1844 ±12.9310 95.5902 ±15.8066
Figure 17. The paths of 20 ×20 grid map generated by: (a) MLRMOEGA; (b) SGA.
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Table 8. The results of 5 other tests (20 ×20 grid map).
Map Algorithm MLRMOEGA SGA
Weight 0.25; 0.25; 0.25; 0.25 0.25; 0.25; 0.25; 0.25
Map.1
1st 52.7883 60.9763
2nd 52.7883 62.0930
3rd 52.7889 53.3597
4th 53.8005 63.6244
5th 60.3864 58.0548
6th 52.7889 52.9650
7th 52.7883 52.7883
8th 52.7883 60.9776
9th 60.1569 55.8255
10th 52.7883 61.7462
Average Value 54.3863 ±2.9583 58.2402 ±3.9792
Map.2
1st 60.5829 79.3548
2nd 56.2152 95.4599
3rd 58.3483 95.6548
4th 57.6904 68.3099
5th 54.1988 68.3099
6th 56.2152 85.2334
7th 55.6137 97.6310
8th 64.8534 65.0661
9th 64.8534 66.8496
10th 56.2152 65.0661
Average Value 58.4786 ±3.5810 78.6936 ±13.0306
Map.3
1st 63.309 73.35
2nd 63.5583 89.7417
3rd 62.688 75.3931
4th 63.7569 105.3384
5th 63.309 88.2386
6th 63.3095 89.5278
7th 63.7569 87.9132
8th 60.7829 86.7099
9th 64.3559 87.9132
10th 63.7569 87.9132
Average Value 63.2583 ±0.9220 87.2039 ±8.2155
Map.4
1st 73.1337 103.478
2nd 73.1353 106.5394
3rd 73.1353 111.6177
4th 73.1337 106.3428
5th 81.6376 75.3689
6th 73.1337 75.7625
7th 73.1337 92.1121
8th 73.1353 104.9162
9th 73.1353 93.1986
10th 69.2510 89.6151
Average Value 73.5965 ±2.9198 95.8951 ±12.2319
Map.5
1st 52.5312 55.3959
2nd 52.5312 65.2409
3rd 52.5312 60.7812
4th 52.5312 56.6374
5th 52.3362 52.9624
6th 52.3362 54.7224
7th 52.3362 52.9624
8th 52.3362 54.7224
9th 52.5655 61.0621
10th 52.7605 53.1563
Average Value 52.4796 ±0.1339 56.7643 ±3.9826
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5.4.2. Compared with Multi-Objective A* Algorithm
The A* algorithm is a deterministic search method based on traversal. The original A*
algorithm is mainly used to solve the shortest path in global path planning. Due to its simple
and intuitive search process, it is a typical algorithm for global path planning problems.
The A* algorithm compares the heuristic function values Fof the eight neighbor grids
of the current grid to determine the next path grid. However, when there are multiple
minimum values, the A* algorithm will randomly determine one grid as the next path gird,
so it cannot guarantee the optimal path.
We use the proposed MLRMOEGA and compare it with an improved A* algorithm
for multi-objective optimization. For this kind of multi-objective, the A* algorithm only
has three optimization objectives: power consumption, time consuming, and falling risk,
without path smoothness. Therefore, in this test, these two algorithms will not optimize
the path smoothness, to ensure even test conditions. The test is performed on random
20 ×20 grid maps
, which were generated in the same way as in Section 5.4.1. In order
to make the contrast more obvious, we reduced the proportion of various obstacles. We
set the proportion of impassable obstacles to 10%, and the proportion of other kinds of
passable obstacles to 2%.
One of the tests for the 20
×
20 grid map and paths generated by the two algorithms
are shown in Figure 18, and the value of the objective function is shown in Table 9.
Figure 18. The test 20 ×20 grid map and paths.
As can be seen from the results, under the equal weight of the three objective functions,
the path generated by the multi-objective A* algorithm selected a different grid from
MLRMOEGA path when the multi-objective A* algorithm was searching for the next
grid at one grid, resulting in the final data being inferior to the MLRMOEGA path. Due
to MLRMOEGA’s better global search ability, the objective function values of the path
generated by MLRMOEGA are better.
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The values of the synthetic objective function of other ten tests are shown in Table 10.
As can be seen from the data in this table, due to its better global search ability, the
objective function values of the paths generated by MLRMOEGA is less than that of the
path generated by the multi-objective A* algorithm in most cases, and the optimization
effect of MLRMOEGA is significantly better than that of the multi-objective A* algorithm
(the p-value of the data in Table 10 is 0.004886).
Table 9. Results of test compared with multi-objective A* algorithm.
Algorithm MLRMOEGA Multi-Objective A* Algorithm
Weight 0.3; 0.3; 0.3; 0 0.3; 0.3; 0.3
Power Consumption 142.4 155.6
Time Consumption 48.6284 54.8182
Path Falling Risk 0.7756 0.8356
The synthetic objective function 57.5412 63.3762
Table 10. The values of synthetic objective function compared with multi-objective A* algorithm.
Algorithm MLRMOEGA Multi-Objective A* Algorithm
Weight 0.3; 0.3; 0.3; 0 0.3; 0.3; 0.3
Map 1 59.1161 90.9409
Map 2 60.8907 66.9772
Map 3 60.3342 77.0933
Map 4 58.9753 49.1492
Map 5 58.9010 64.8626
Map 6 59.5881 76.4095
Map 7 57.6954 68.5558
Map 8 58.9312 91.6395
Map 9 59.9457 76.3552
Map 10 59.8486 79.3438
5.4.3. Comparison with Multi-Objective ACO
In this section, we compared the MLRMOEGA with a kind of a multi-objective ACO
algorithm. The ant colony optimization (ACO) algorithm is a stochastic-based optimization
technique that replicates the behavior of real ants when searching for food. The ants move
along the same path by following one another. This is because every ant leaves a chemical
substance called pheromone while moving on the path. The other ants sense the intensity
of the pheromone and follow the path with a higher concentration of pheromone. This is
their tactic to find an optimized path. On their back tour, the ants sense the pheromone
intensity and choose the path with a higher concentration of pheromone [23].
This multi-objective ACO algorithm has the same four optimization objectives as the
proposed algorithm, and we set these four objective function weights to 0.25 to consider
an even performance of path planning. The test is performed on random 20
×
20 grids
maps, which were generated in the same way as in Section 5.4.2. One of the tests for the
20 ×20 grid map
and paths generated by the two algorithms are shown in Figure 19, and
the value of the objective function is shown in Table 11.
As can be seen from the results, under the equal weight of the four objective functions,
the multi-objective ACO algorithm generated a different path from MLRMOEGA. Because
of the positive feedback of pheromone, the ant colony algorithm is easy to fall into local
optimal solution in the process of approaching the optimal solution. With better global
search ability, the objective function values of the path generated by MLRMOEGA are
significantly lower.
The values of synthetic objective function of the other ten tests are shown in Table 12.
As can be seen from the data in this table, due to its better global search ability, the
objective function values of the paths generated by MLRMOEGA is less than that of
Micromachines 2022,13, 616 24 of 30
the path generated by the multi-objective ACO algorithm, and the optimization effect of
MLRMOEGA is significantly better than that of multi-objective ACO algorithm (the p-value
of the data in Table 12 is 0.008118).
Figure 19. The test 20 ×20 grid map and paths.
Table 11. Results of test compared with multi-objective ACO algorithm.
Algorithm MLRMOEGA Multi-Objective ACO Algorithm
Weight 0.25; 0.25; 0.25; 0.25 0.25; 0.25; 0.25; 0.25
Power Consumption 142.9371 222.4558
Time Consumption 55.7548 82.0036
Path Falling Risk 0.8036 0.9371
Path Smoothness 5.4978 11.7810
The synthetic objective function 51.2483 79.2944
5.4.4. Performance Test in Simulated Environment
In order to test the performance of the proposed algorithm in the real world, we
design a simulation environment close to the real situation as shown in Figure 20, in which
the MLR had to choose various paths to cross the river, such as jumping over the stream,
swinging on a ladder, walking over a bridge, and then climbing up slopes or walls to reach
the destination. We abstract the simulated environment into a 30
×
30 grid map as shown
in Figure 21. Then, we run MLRMOEGA and SGA in this map to compare the performance
of the proposed algorithm.
Micromachines 2022,13, 616 25 of 30
Table 12.
The values of synthetic objective function compared with multi-objective ACO algorithm.
Algorithm MLRMOEGA Multi-Objective ACO Algorithm
Weight 0.25; 0.25; 0.25; 0.25 0.25; 0.25; 0.25; 0.25
Map 1 51.1325 108.3110
Map 2 50.0654 195.0888
Map 3 48.8652 81.6816
Map 4 49.5720 73.6517
Map 5 50.4595 107.5477
Map 6 50.4761 67.1134
Map 7 50.8855 67.2223
Map 8 48.8173 68.8862
Map 9 49.8645 79.6593
Map 10 48.0516 68.6640
Figure 20. The simulated environment.
The design parameters of MLRMOEGA and SGA are shown in Table 6. The results
of the 30
×
30 grid map are shown in. To clarify, Figure 22 shows the paths generated by
different algorithms, obviously, under the same design parameters. Compared with SGA,
the MLRMOEGA has better global search performance for better paths through narrow
regions, for example, the stream region in the testing environment. In the 10-times test, the
SGA did not generate a path passing through the stream region while the MLRMOEGA
generated 3 paths passing through the stream region as shown in Figure 22a circled in red,
and the paths passing through the stream region have a better synthetic objective function
value (the p-value of the data in Table 13 is 0.00009459).
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Figure 21. The abstracted 30 ×30 grid map of the simulated environment.
Figure 22. Cont.
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Figure 22. The paths of 30 ×30 grid map generated by: (a) MLRMOEGA; (b) SGA.
Table 13. The results of test (30 ×30 grid map).
Algorithm MLRMOEGA SGA
Weight 0.25; 0.25; 0.25; 0.25 0.25; 0.25; 0.25; 0.25
1st 71.2978 84.6640
2nd 80.0754 85.9012
3rd 81.254 83.6812
4th 69.9468 85.6848
5th 80.6644 86.3729
6th 81.2546 87.3921
7th 70.5163 85.2559
8th 71.3646 85.1268
9th 70.5464 85.7125
10th 71.7590 87.3930
Average Value 74.8679 ±4.8867 85.7184 ±1.0906
5.4.5. Multi-Objective Optimization Performance Test
We test the multi-objective optimization performance of the proposed algorithm by
changing the weight of the optimization target, and the test environment is the abstracted
simulated environment shown in Figure 20.
The test results are shown in Table 14, the red text indicates the value of the objective
function with the weight of one which is emphasized by the decision makers. From
the result, we can see that the proposed algorithm can realize the intention of decision
makers and effectively optimize the value of one certain objective function emphasized
by the decision makers, and when the four optimization objectives are considered equally,
the path generated by the proposed algorithm is relatively balanced, which proves the
multi-objective optimization effectiveness of the proposed algorithm.
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Table 14. Results of Multi-Objective Optimization Performance Test.
Algorithm MLRMOEGA
Weights 1; 0; 0; 0 0; 1; 0; 0 0; 0; 1; 0 0; 0; 0; 1 0.25; 0.25; 0.25; 0.25
Power Consumption 200.7351 201.9777 227.3082 201.8792 207.7371
Time Consumption 74.0966 73.6447 85.0833 73.9689 76.312
Path Falling Risk 0.9357 0.9374 0.7217 0.7975 0.8412
Path Smoothness 10.9956 10.2102 8.6394 1.5708 7.0686
6. Conclusions
This paper proposed a multi-objective path planning algorithm for a multi-locomotion
robot based on a genetic algorithm with elitist strategy. First, we determine four opti-
mization objectives: power consumption, time consumption, falling risk, and smoothness,
then set their objective functions and evaluation functions. Then, to solve the problem
of premature convergence of SGA, we propose two operators: a map analysis operator
and a population diversity expansion operator, to improve the population diversity in the
algorithm process. We run the proposed algorithm in 30
×
30 grids testing environment,
and the optimal design parameters are determined by balancing the execution time and
the synthetic objective function value. After obtaining the optimal design parameters, we
test its performance and compare the proposed algorithm in multiple environments with
SGA. The results show that the proposed algorithm can effectively improve the global
search ability and convergence of SGA. We also compare the proposed algorithm with a
multi-objective A* algorithm. We run these two algorithms in a random 20
×
20 grid map;
the results show that the synthetic objective function value of the MLRMOEGA path is
better than that of the multi-objective A* algorithm path under equal weight due to the
better global search ability of MLRMOEGA. Then, we test the performance of the proposed
algorithm in a simulation environment which is close to the real field environment; the
results show that the global search ability and optimization ability of MLRMOEGA is better
than that of SGA. In addition, we test the multi-objective optimization performance under
alternative weights, and we find that the output path results can effectively optimize the
value of objective functions that the decision maker emphasizes.
According to the results above, the MLRMOEGA proposed in this paper can be
effectively applied to MLR tasks, such as pipeline maintenance, medical care by micro-
robots, cargo transporting, and terrain exploration by humanoid robots.
Author Contributions:
Conceptualization, C.L. and A.L.; methodology, A.L.; software, C.L. and A.L;
validation, C.L. and A.L.; formal analysis, A.L.; investigation, R.W.; resources, R.W.; data curation,
A.L.; writing—original draft preparation, C.L.; writing—review and editing, A.L.; supervision, H.Z.;
funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by the National Key R&D Program of China, grant num-
ber 2018YFB1304504.
Conflicts of Interest: The authors declare no conflict of interest.
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