US 2025/0094668 A1 · ROBUST OPTIMIZATION DESIGN METHOD FOR MECHANICAL ARM BASED ON HYBRID INTERVAL AND BOUNDED PROBABILISTIC UNCERTAINTIES

Abstract

A robust optimization design method for a mechanical arm considering hybrid interval and bounded probabilistic uncertainties is provided. The method includes considering interval and bounded probabilistic uncertainties affecting a performance of a mechanical arm, describing a bounded probabilistic uncertainty by a generalized beta distributed random variable, and establishing a robust optimization design model of the mechanical arm; directly solving the optimization model based on a genetic algorithm, which includes analyzing, by the boundedness of the uncertainties, the robustness of a constraint performance function of an individual in a population, and determining whether the individual is feasible; calculating, a mean and a standard deviation of an objective function of a feasible individual by multi-layered refining Latin hypercube sampling (MRLHS); and ranking, according to a total feasibility robustness index and a distance to negative ideal solution (DNIS), individuals in a current population to obtain a robust optimal design of the mechanical arm.

A robust optimization design method for a mechanical arm based on hybrid interval and bounded probabilistic uncertainties, wherein the method is applied to a scenario of robust optimization design of the mechanical arm under an influence of uncertainties of multi-type distribution characteristics in practical engineering, in such a manner to ensure excellent and stable performance of the mechanical arm during an actual operation, and the method comprises:

1) considering uncertainties in a hydraulic cylinder pressure, manufacturing precision and a material property of the mechanical arm and classifying them into an interval uncertainty and a bounded probabilistic uncertainty, and describing each bounded probabilistic uncertain parameter as a random variable subjected to a generalized beta (GBeta) distribution:
1.1) obtaining, for a bounded probabilistic uncertain variable Xi, s samples through an experiment to construct a sample point set {Xi1, Xi2, . . . , Xis};
calculating, based on the sample point set, a value range of the variable Xi by Eq. 1, and calculating a mean and a variance of the variable Xi by Eq. 2:
{
a
i
=
min
⁢
{
X
i
1
,
X
i
2
,
…
,
x
i
s
}
b
i
=
max
⁢
{
X
i
1
,
X
i
2
,
…
,
X
i
s
}
,
and
Eq
.
1
{
μ
X
i
=
1
s
⁢
∑
k
=
1
s
⁢
X
i
k
S
X
i
2
=
1
s
⁢
∑
k
=
1
s
⁢
(
X
i
k
-
μ
X
i
)
2
;
Eq
.
2
1.2) describing, by the GBeta distribution, the variable Xi that is distributed within [ai, bi] and has a mean of μXi and a variance of SXi2;
firstly, normalizing the mean and the variance of the variable Xi by Eq. 3:
{
μ
^
X
i
=
μ
X
i
-
a
i
b
i
-
a
i
S
^
X
i
2
=
s
X
i
2
(
b
i
-
a
i
)
2
,
Eq
.
3
then, calculating distribution parameters αi and βi of the GBeta distribution of the variable Xi by Eq. 4:
{
α
i
=
1
-
μ
^
X
i
1
+
μ
^
X
i
·
1
S
^
X
i
2
β
i
=
(
1
-
μ
^
X
i
)
2
μ
^
X
i
(
1
+
μ
^
X
i
)
·
1
S
^
X
i
2
,
Eq
.
4
denoting the variable Xi subjected to the GBeta distribution within [ai, bi] with the distribution parameters αi and βi as Xi˜GBeta(ai, bi|αi, βi), wherein a probabilistic density function of the variable Xi is defined by Eq. 5:
f
X
i
(
X
i
;
α
i
,
β
i
❘
a
i
,
b
i
)
=
Γ
⁡
(
α
i
+
β
i
)
Γ
⁡
(
α
i
)
⁢
Γ
⁡
(
β
i
)
⁢
(
1
b
i
-
α
i
)
α
i
+
β
i
-
1
·
(
X
i
-
a
i
)
α
i
-
1
⁢
(
b
i
-
X
i
)
β
i
-
1
,
Eq
.
5
wherein in Eq. 5, ƒXi(·) is the probabilistic density function of the variable Xi, and Γ(·) is a gamma function;
2) establishing a robust optimization design model of the mechanical arm with the hybrid interval and bounded probabilistic uncertainties by taking a maximum loading capacity of the mechanical arm in operation under an influence of the hybrid interval and bounded probabilistic uncertainties as an optimization objective, and describing a performance index of the mechanical arm with a given maximum allowable value as a constraint performance function, the robust optimization design model being shown in Eq. 6:
min
d
{
μ
f
C
(
d
,
X
,
U
)
,
σ
f
C
(
d
,
X
,
U
)
,
μ
f
W
(
d
,
X
,
U
)
,
σ
f
W
(
d
,
X
,
U
)
}
Eq
.
6
s
.
t
.
[
g
i
L
*
(
d
,
X
,
U
)
,
g
i
R
*
(
d
,
X
,
U
)
]
≤
[
b
i
L
,
b
i
R
]
,
i
=
1
,
2
,
...
,
p
d
=
(
d
1
,
d
2
,
...
,
d
l
)
,
X
=
(
X
1
,
X
2
,
...
,
X
m
)
,
U
=
(
U
1
,
U
2
,
...
,
U
n
)
,
wherein in Eq. 6, d=(d1, d2, . . . , dl) is an I-dimensional design vector;
X=(X1, X2, . . . , Xm) is an m-dimensional bounded probabilistic uncertain vector;
U=(U1, U2, . . . , Un) is an n-dimensional interval uncertain vector;
Bi is an interval constant given based on a design requirement;
biL and biR are left and right bounds of Bi respectively, and when biL=biR, the interval constant Bi degenerates to a real number;
p is a number of constraint performance functions;
giL*(d, X, U) and giR*(d, X, U) are respectively left and right bounds of a performance variation interval of an i-th constraint performance function gi(d, X, U) under the influence of the hybrid interval and bounded probabilistic uncertainties, and giL*(d, X, U) and giR*(d, X, U) are calculated as follows:
a) rewriting the probabilistic uncertain vector X as an interval form XI=(X1I, X2I, . . . . XmI) utilizing boundedness of the probabilistic uncertain vector X, wherein XiI=[ai, bi](i=1,2, . . . , m) is an interval number corresponding to the bounded probabilistic uncertain variable Xi;
ai, bi are determined by Eq. 1;
I is a mark of an interval representation form corresponding to the bounded probabilistic uncertain variable;
b) merging the interval vector (and the interval form XI of the bounded probabilistic uncertain vector into a new interval uncertain vector HUXI=(XI,U), and calculating giL*(d,X,U) and giR*(d, X, U) with Eq. 7:
{
g
i
L
*
(
d
,
X
,
U
)
=
min
H
U
X
I
(
d
,
H
U
X
I
)
g
i
R
*
(
d
,
X
,
U
)
=
max
H
U
X
I
(
d
,
H
U
X
I
)
⁢
(
i
=
1
,
2
,
...
,
p
)
,
Eq
.
7
wherein in Eq. 6, μƒC(d,X,U), σƒC(d,X,U), μƒW(d,X,U), σƒW(d,X,U) are respectively a mean and a standard deviation of a center, and a mean and a standard deviation of a halfwidth of a variation interval of an objective performance function ƒ(d, X, U) under the influence of the bounded probabilistic uncertain vector X and the interval uncertain vector U, which are calculated as follows:
A) defining μX=(μX1, μX2, . . . , μXm) as a constant vector obtained by taking a mean of each probabilistic variable in the bounded probabilistic uncertain vector X, and denoting μX as a mean vector of the bounded probabilistic uncertain vector X;
substituting the bounded probabilistic uncertain vector X in the objective performance function ƒ(d, X, U) with the mean vector μX to transform the objective performance function into a function ƒ(d, μx, U), which comprises only the interval uncertain vector U and whose value is an interval number;
B) performing an interval analysis of ƒ(d, μX, U) through an interval analysis algorithm by Eq. 8 to obtain left and right bounds ƒL(d, μX) and ƒR(d, μX) of a variation interval of the objective performance function ƒ(d, μX, U) at the mean vector μX:
{
f
L
(
d
,
μ
X
)
=
f
L
(
d
,
μ
X
,
U
)
❘
U
=
U
min
*
=
min
U
f
⁡
(
d
,
μ
X
,
U
)
f
R
(
d
,
μ
X
)
=
f
R
(
d
,
μ
X
,
U
)
❘
U
=
U
max
*
=
max
U
f
⁡
(
d
,
μ
X
,
U
)
,
Eq
.
8
wherein in Eq. 8, Umin*and Umax* are interval uncertain vectors to minimize and maximize ƒ(d,μX, U), respectively;
C) calculating, by Eq. 9, a center ƒC(d, μX) and a halfwidth ƒW(d, μX) of the variation interval of the objective performance function ƒ(d, μX, U) at the mean vector μX:
{
f
C
(
d
,
μ
X
)
=
(
f
L
(
d
,
μ
X
)
+
f
R
(
d
,
μ
X
)
)
/
2
f
W
⁢
(
d
,
μ
X
)
=
(
f
R
⁢
(
d
,
μ
X
)
-
f
L
(
d
,
μ
X
)
)
/
2
,
Eq
.
9
wherein in Eq. 9, ƒL(d, μX), ƒR(d, μX), ƒC(d, μX) and ƒW(d, μX) have no uncertain variable and each has a real-number value;
D) restoring μX in ƒC(d, μX) and ƒW(d, μX) to the bounded probabilistic uncertain vector X;
performing multi-layered refining Latin hypercube sampling (MRLHS) within a probabilistic distribution range of the bounded probabilistic uncertain vector X;
calculating a value of the objective performance function corresponding to each sample point, wherein the objective performance function corresponding to each sample point has no uncertainty and has a real-number value;
calculating, by a Monte Carlo approach, the mean μƒC(d,X,U) and standard deviation σƒC(d,X,U) of the center and the mean μƒW(d,X,U) and standard deviation σƒW(d,X,U) of the halfwidth in the variation interval of the objective performance function ƒ(d, X, U) under the influence of the bounded probabilistic uncertain vector X and the interval uncertain vector U, specifically as follows:
D.1) determining an m-dimensional original sampling domain Dm=[a1, b1]×[a2, b2]×. . . ×[am, bm], where ai, bi (i=1,2, . . . , m) are boundary values of the bounded probabilistic uncertain variable Xi determined by Eq. 1, and × is a Cartesian product operator in a linear space;
D.2) constructing, by dividing and extracting the original sampling domain Dm, a mean neighborhood layer sampling domain δDμm and a transitional layer sampling domain Dtranm to form three layers of sampling domains, namely Dm, δDμm and Dtranm:
δ
⁢
D
μ
m
=
[
δ
⁢
X
1
L
,
δ
⁢
X
1
R
]
×
[
δ
⁢
X
2
L
,
δ
⁢
X
2
R
]
×
...
×
δ
⁢
X
m
L
,
δ
⁢
X
m
R
,
and
Eq
.
10
D
tran
m
=
[
X
1
⁢
t
L
,
X
1
⁢
t
R
]
×
[
X
2
⁢
t
L
,
X
2
⁢
t
R
]
×
...
×
[
X
mt
L
,
X
mt
R
]
,
Eq
.
11
wherein in Eq. 10 and Eq. 11, δXiL and δXiR (i=1,2, . . . , m) are left and right bounds of an i-th dimension in the m-dimensional mean neighborhood layer sampling domain δDμm respectively;
XitL and XitR (i=1,2, . . . , m) are left and right bounds of the i-th dimension in the m-dimensional transitional layer sampling domain Dtranm respectively;
the left and right bounds are determined by Eq. 12:
{
δ
⁢
X
i
L
=
F
X
i
-
1
(
0.3
,
α
i
,
β
i
)
δ
⁢
X
i
R
=
F
X
i
-
1
(
0.7
,
α
i
,
β
i
)
X
it
L
=
F
X
i
-
1
(
0.2
,
α
i
,
β
i
)
X
it
R
=
F
X
i
-
1
(
0.8
,
α
i
,
β
i
)
⁢
(
i
=
1
,
2
,
...
,
m
)
,
Eq
.
12
wherein in Eq. 12, FXi−1(·) is an inverse function of a probabilistic cumulative function FXi(·) of the bounded probabilistic uncertain variable Xi;
D.3) setting a total sample size to N, performing standard Latin hypercube sampling (LHS) with a size of N/3 in each of the three layers of sampling domains, and superimposing sample points of each layer to obtain a final sample point set;
D.4) calculating, by the Monte Carlo approach based on the obtained final sample point set, the mean μƒC(d,X,U) and standard deviation σƒC(d,X,U) of the center and the mean σƒW(d,X,U) and standard deviation σƒW(d,X,U) of the halfwidth in the variation interval of the objective performance function ƒ(d, X, U) under the influence of the bounded probabilistic uncertain vector X and the interval uncertain vector U;
and
3) directly solving the robust optimization design model of the mechanical arm based on a genetic algorithm (GA), a total feasibility robustness index and a distance to negative ideal solution (DNIS):
3.1) setting GA parameters, comprising population size, maximum number of iterations, mutation and crossover probabilities, and convergence criterion;
setting a current iteration number of the GA to 1, and generating an initial population of the GA;
3.2) performing robustness assessment for a constraint performance function of each individual in a current population, and calculating a total feasibility robustness index S corresponding to a design vector d;
3.3) classifying all the individuals in the current population according to the total feasibility robustness index S and marking an individual as (a) feasible if S=p, (b) semi-feasible if 0<S<p, and (c) infeasible if S=0;
3.4) calculating a mean and a standard deviation of an objective function corresponding to a feasible individual by an MRLHS-based Monte Carlo approach according to steps D.1) to D.4);
3.5) ranking, according to a classification result of the individuals in the current population in step 3.3) and a calculation result of the means and standard deviations of the objective function of the feasible individuals in step 3.4), all individuals in the population based on the total feasibility robustness indices and the DNISs to obtain a fitness of each individual in the current population;
3.6) determining whether the maximum number of iterations or the convergence criterion is satisfied;
if yes, outputting a design vector corresponding to an individual with a largest fitness as an optimal solution;
if not, performing crossover and mutation operations, increasing the iteration number by 1 to produce a new generation of population individuals, and returning to step 3.2);
4) manufacturing the mechanical arm according to lengths of guide linkages and joint positions corresponding to the optimal design vector obtained in step 3), in such a manner to ensure that the mechanical arm has excellent and stable performance under the influence of uncertainties of multi-type distribution characteristics in actual operation, wherein the optimal design vector is a design vector, output in step 3.6), corresponding to the individual with the largest fitness.

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Robust optimization design method for mechanical arm based on hybrid interval and bounded probabilistic uncertainties

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Robust optimization design method for mechanical arm based on hybrid interval and bounded probabilistic uncertainties

Life of this patent

Abstract

Claims4