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Ortiz et al. Intell Robot 2021;1(2):131-50 I http://dx.doi.org/10.20517/ir.2021.09 Page 139

Thus,
+1 − ≤ − +

where = max P −1 ( ¯ + ¯ ) + max [Λ] ¯ . If
+1
−1 2
min P k ( )k ≥
+2
then +1 − ≤ 0, k ( )k decreases. Thus, k ( )k converges to Equation (23).

3. GENETIC ALGORITHM AND SLAM FOR PATH PLANNING

Path planning is one key problem of autonomous robots. Here, the map is built by the sliding mode SLAM:
• The obstacle set is defined by ( ).
• The position is ( ), ( ) = \ ( ).
• The path planning is ( ( ), , ), = ( ).
The previous map is ( ), which requires the path ( ( ), , ).

We assume the previous map is obstacle-free, the initial point is , the target point is ∈ ,

= { ∈ | ( ) ∩ = ∅}

the obstacle is = , is the shape of the robot, and ( ) is the area of the robot. The objective of the
path planning is to find a path ( , , ) ∈ that allows the robot to navigate.

is defined as the search space. We use the GA to find an optimal trajectory ( , , ), such that

min ( , , ), where : → (36)
∈

Here, we use stochastic search for GA, and each iteration includes: reproduction or selection, crossing or

combination, and mutation. The population is ( ) = { , , .., } with being the size of the population

2
1
that represents the possible solutions:

(1) Every chromosome has a solution in

= [ , −1 , . . . , 2 , 1 ] with ∈ ∀ = 1, 2, . . . ,

(2) Crossing the chromosomes. An intersection in = [ , −1 , . . . , , ] and = [ , −1 , . . . , , ]

2
1
1
2
belongs to , such that ∩ ≠ ∅; then,

0 = [ , , . . . , . . . , , ]
−1 2 1

0 = [ , , . . . , . . . , , ]
−1 2 1

where 0 and 0 are the next generation from two compatible chromosomes by crossing.

(3) Mutation. It replace a number of chromosomes by chromosomes in .
The mutation operation is calculated by the fitness of each chromosome,
( ) = [ ( ), ( ), . . . , ( )], where is the number of mutations and uses the
1 2