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Page 136 Tong et al. Intell Robot 2024;4:125-45 I http://dx.doi.org/10.20517/ir.2024.08

Table 3. , , Δ affiliation table
Output
PB PB PM PM PS ZO ZO
PB PB PM PS PS ZO NS
PM PM PM PS ZO NS NS
PM PM PS ZO NS NM NM
PS PS ZO NS NS NM NM
PS ZO NS NM NM NM NB
ZO ZO NM NM NM NB NB

Table 4. , , Δ affiliation table
Output
PB PB PM PM PS ZO ZO
PB PB PM PS PS ZO NS
PM PM PM PS ZO NS NS
PM PM PS ZO NS NM NM
PS PS ZO NS NS NM NM
PS ZO NS NM NM NM NB
ZO ZO NM NM NM NB NB

Table 5. , , Δ affiliation table
Output
PB PB PM PM PS ZO ZO
PB PB PM PS PS ZO NS
PM PM PM PS ZO NS NS
PM PM PS ZO NS NM NM
PS PS ZO NS NS NM NM
PS ZO NS NM NM NM NB
ZO ZO NM NM NM NB NB

Its trigonometric affiliation function is defined as

 0, ≤ 
 
 
 
 − 
  ≤ ≤  
( ; , , ) = − (17)
−
 ≤ ≤ 
 − 
 
 
 0, ≤ 
 
Fuzzy inference involves deriving new conclusions based on existing fuzzy conditions or assumptions, with
the Mamdani inference method being a commonly used approach. The algorithm involves a direct product
operationforfuzzyimplications ( , , Δ ) asafuzzyset , , ,basedontheminimumconstraint
0
0
0

relation. Here, represents the error, is the error change rate, and is an adjustment value. The specific
algorithm involves direct product operations, where = 1 · · · 7, = 1 · · · 7 represent fuzzy set linguistic values,
expressed as
( , , Δ ) ∈ × × Δ 0 (18)
0
0

For a discrete domain fuzzy set , , Δ 0 represented as a matrix, it can be expressed as
0
0
p
0
( , ) = × = Λ 0 (19)
0
0

( , , Δ ) = ( , )Λ (20)
0
where ( , ) represents the transformation of the first-row elements into columns, and subsequent rows
follow suit. If the controller has fuzzy rules, the fuzzy relation is constructed from fuzzy implication
relations , defined as
=7, =7
Ø
= ( , , Δ ) (21)
=1, =1