Multiobjective trajectory optimization of intelligent electro-hydraulic shovel

doi:10.1007/s11465-022-0706-2

Front. Mech. Eng.

2022, Vol. 17

Issue (4) : 50 https://doi.org/10.1007/s11465-022-0706-2

RESEARCH ARTICLE

Multiobjective trajectory optimization of intelligent electro-hydraulic shovel

Rujun FAN, Yunhua LI(

), Liman YANG

School of Automation Science and Electrical Engineer, Beihang University, Beijing 100191, China

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Abstract

Multiobjective trajectory planning is still face challenges due to certain practical requirements and multiple contradicting objectives optimized simultaneously. In this paper, a multiobjective trajectory optimization approach that sets energy consumption, execution time, and excavation volume as the objective functions is presented for the electro-hydraulic shovel (EHS). The proposed cubic polynomial S-curve is employed to plan the crowd and hoist speed of EHS. Then, a novel hybrid constrained multiobjective evolutionary algorithm based on decomposition is proposed to deal with this constrained multiobjective optimization problem. The normalization of objectives is introduced to minimize the unfavorable effect of orders of magnitude. A novel hybrid constraint handling approach based on ε-constraint and the adaptive penalty function method is utilized to discover infeasible solution information and improve population diversity. Finally, the entropy weight technique for order preference by similarity to an ideal solution method is used to select the most satisfied solution from the Pareto optimal set. The performance of the proposed strategy is validated and analyzed by a series of simulation and experimental studies. Results show that the proposed approach can provide the high-quality Pareto optimal solutions and outperforms other trajectory optimization schemes investigated in this article.

Keywords trajectory planning electro-hydraulic shovel cubic polynomial S-curve multiobjective optimization entropy weight technique

Corresponding Author(s): Yunhua LI

Just Accepted Date: 31 May 2022 Issue Date: 06 January 2023

Cite this article:

Rujun FAN,Yunhua LI,Liman YANG. Multiobjective trajectory optimization of intelligent electro-hydraulic shovel[J]. Front. Mech. Eng., 2022, 17(4): 50.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0706-2
https://academic.hep.com.cn/fme/EN/Y2022/V17/I4/50

Fig.1 Scheme of novel electro-hydraulic shovel with dipper energy recovery system (DERS).

Fig.2 Profiles of acceleration and velocity based on cubic polynomial S-curve.

Fig.3 Coordinate system of electro-hydraulic shovel.

$θ i / (∘)$	$l i / m$	$ϑ i / (∘)$	$d i / m$
θ₁	l₁	0	0
0	0	90	0
0	0	0	d
0	$l A C 2 sin ? γ$	0	$l A C 2 sin ? γ$

Tab.1 D?H parameters in Eq. (4)

Fig.4 Velocity analysis for the dipper handle.

Fig.5 Distribution of forces in excavation.

Fig.6 Pareto fronts for (a) Case 1, (b) Case 2, and (c) Case 3.

Fig.7 Pareto fronts for Case 4 using different algorithms: (a) MOEA/D-CDP, (b) E-NSGA-III, (c) C-MOEA/D-FRRMAB, and (d) proposed algorithm.

Tab.2 Computational time of different algorithms

Tab.3 Best, average, and worst HV results for Cases 1–4

Tab.4 Best, medium, and worst HV results for different population sizes

Fig.8 Pareto front with uncertainty of pile angle.

Fig.9 Pareto front with uncertainty of density.

Tab.5 Average HV results with and without acceleration constraints

Fig.10 Distribution of the nondominated solutions.

Repose angle/(° )	$k 1$	$k 2$	$t 1 /s$	$t 2 /s$	$t 4 /s$	$t 5 /s$	$θ 0$ /(° )	Energy/ $kJ$	Time/s	Volume/m³
35	0.3627	0.3547	2.2610	4.8802	2.6056	4.2697	26.08	9449.87	9.48	38.47
40	0.4055	0.4621	2.1754	5.3587	2.1767	5.3688	36.15	9367.98	9.72	38.33
45	0.4238	0.3901	2.1414	5.0035	2.2187	4.9748	44.09	9591.25	9.41	38.28

Tab.6 Most satisfied solutions at different repose angles

Fig.11 Evolution of kinematic characteristics and energy consumption at different repose angles: (a) vertical displacement, (b) stretching length, (c) retraction length of wire rope, (d) crowd velocity, (e) hoist velocity, (f) crowd acceleration, (g) hoist acceleration, (h) crowd force, (i) hoist force, (j) crowd power, (k) hoist power, and (l) total energy.

Tab.7 Comparisons of performance metrics between EHS with and without DERS

Fig.12 Comparisons of two trajectory planning methods at 35° repose angle: (a) crowd speed, (b) hoist speed, (c) vertical distance, (d) stretching length, (e) retraction length of wire rope, (f) crowd power, (g) hoist power, and (h) total energy consumption.

Tab.8 Comparisons between 4-degree polynomial and CPS-curve methods

Method	Energy/kJ	Time/s	Volume/m³	$f per$
Proposed	9188.53	9.93	38.27	62.30
PF-GA	10252.27	10.39	38.49	71.91
PF-ABC	10584.32	10.51	37.78	77.94
PF-PSO	9602.66	11.33	36.99	79.52
NSGA-II	9657.77	10.63	38.49	69.29

Tab.9 Excavating performance under different algorithms

Fig.13 Schematic diagram of real time simulation of excavation trajectory.

Fig.14 Real-time simulation device for excavating trajectory of electro hydraulic shovel.

Fig.15 Real-time and theoretical trajectories of EHS: excavating trajectory at (a) 35°, (b) 40°, and (c) 45°.

Repose angle/(° )	$e max$	$IAE$	$R 2$
35	0.0403	0.0792	0.9990
40	0.0458	0.0848	0.9979
45	0.0486	0.0924	0.9978

Tab.10 Error analysis of different repose angles

Abbreviations
AdaW	Adaptive weight strategy
C-MOEA/D-FRRMAB	Constrained MOEA/D with fitness-rate-rank-based multiarmed bandit
CPS-curve	Cubic polynomial S-curve
DERS	Dipper energy recovery system
D?H	Denavit–Hartenberg
EHS	Electro-hydraulic shovel
E-NSGA-Ⅲ	Extended nondominated sorting genetic algorithm Ⅲ
GOF	Goodness-of-fit
H-MOEA/D-AdaW	Improved MOEA/D algorithm with hybrid constraint strategy
HV	Hypervolume
IAE	Integrated absolute error
MOEA	Multiobjective evolutionary algorithm
MOEA/D	Multiobjective evolutionary algorithm based on decomposition
MOEA/D-CDP	MOEA/D with constraint-domination principle
MOP	Multiobjective optimization problem
NSGA-Ⅱ	Nondominated sorting genetic algorithm Ⅱ
PBI	Penalty-based boundary intersection
PF	Penalty function
PF-ABC	PF-base artificial bee colony algorithm
PF-GA	Genetic algorithm based on PF
PF-PSO	PF-based particle swarm optimization algorithm
PSO	Particle swarm optimization
TOPSIS	Technique for order preference by similarity to an ideal solution
Variables
$a c$	Crowd acceleration
$a c m a x$	Maximum crowd acceleration
$a h$	Hoist acceleration
$a h m a x$	Maximum hoist acceleration
$b i$	Coefficient of the 4-degree polynomial, $i = 0, 1, . . ., 4$
$B i$	Calculation coefficient, $i = α, γ, c, q, N$
$c$	Material cohesion
$c a$	Rock?metal adhesion
$C C i$	Closeness
$d$	Distance between axis $x i$ and $x i − 1$
$d 0$	Value of $d$ at $t = 0$
$d f$	Final extension displacement of the dipper handle
$d i$	Distance between axis $x i$ and $x i − 1$ , $i = 1, 2, . . ., 4$
$d offset$	Distance between point $C 1$ and point $C 2$
$d p$	Vertical distance from point $C 2$ to the axis $x 1$
$d p b i 1$	Parallel distance from the reference point to the weight vector
$d p b i 2$	Vertical distance from the objective vector to the weight vector
$d ˙$	Derivative of $d$
$d ¨$	Second derivative of $d$
$e max$	Maximum absolute error
$e t$	Error between the real-time trajectory and theoretical trajectory
$e m i j$	$j t h$ objective function value of the $i t h$ solution
$E j$	Entropy of the $j t h$ evaluation criteria
$E t o t a l$	Total energy consumed by excavation, including crowd energy and hoist energy
$E D i +$	Positive Euclidean distance
$E D i −$	Negative Euclidean distance
$E M$	Evaluation criteria matrix
$f p e r$	Excavating performance index
$f p (X i)$	$p t h$ objective value of population $X i$
$F b$	Cutting resistance on the cutting blade
$F c$	Crowd force
$F h$	Hoist force
$F n$	Normal excavation force
$F s$	Side resistance
$F t$	Tangential excavation force
$F t 1$ , $F t 2$	First and second component of tangential excavation force, respectively
$F v$	Velocity-induced resistance
$F (X i)$	Objective value vector of the $i th$ subproblem
$F ′ (X i)$	Penalized objective vector
$F ~ (X i)$	Normalized objective function vector
$F ~ max$	Maximum value of each objective vector
$g$	Gravitational constant
$g j (X i)$	Value of the $j t h$ constraint, $j = 1, 2, . . ., 10$
$g j max$	Maximum violation degree for the $j th$ constraint
$g p b i$	PBI value
$h$	Excavation depth
$H u p$	User-predefined value
$I d$	Inertia of the dipper and dipper handle
$I p$	Inertia of the payload
$J 1$	Total consumption energy
$J 2$	Execution time
$J 3$	Excavation volume
$J 4$	Weighted optimization objective
$J$	Optimization objective vector
$k$	Current evolutionary generation
$k 1$ , $k 2$	Coefficients of the crowd speed and hoist speed, respectively
$k r$	Number of solutions replaced by a child
$K$	System kinetic energy
$l i$ (i = 1,2)	Length of common vertical line between axis $z i$ and $z i − 1$
$l A M 0$	Value of $l A M$ at $t = 0$
$l B H 0$	Value of $l B H$ at $t = 0$
$l O$	Length from point O to the ground
$l i j$	Length from point i to point j, i = A, B, O, O_s, and j = B, C₂, H, M, N, O
$L$	Lagrangian function
$m d$	Mass of the dipper handle and dipper
$m p$	Mass of payload
$N gen$	Control parameter of generation
$N gmax$	Maximum generation
$N nei$	Number of neighborhoods
$N o b j$	Number of optimization objectives
$N p$	Population size
$N r max$	Maximal number of solutions replaced by a child
$N t$	Sampling number of excavating trajectory
$p c$	Crossover probability
$p m$	Mutation probability
$p m i j$	Normalized positive evaluation criteria matrix element
$P inf$	Penalty coefficient of infeasible solutions
$P p b i$	Penalty coefficient of PBI method
$P M$	Normalized positive evaluation criteria matrix
$q$	Surcharge
$r k$	Ratio of feasible solutions to total solutions in the kth generation
$r reg$	A parameter that determines the search preference in feasible and infeasible regions
$R 2$	Coefficient of determination
$R s$	Radius of sheave
$S i n d$	Index set of $N nei$ weights closest to the vector $λ i$
$S p$	Parent set
$s m i j$	Standardized evaluation matrix element
$S M$	Standardized evaluation matrix
$t$	Time
$t 1$ , $t 2$ , $t 3$	Acceleration time, cruise time, and deceleration of crowd speed, respectively
$t 4$ , $t 5$ , $t 6$	Acceleration time, cruise time, and deceleration of hoist speed, respectively
$t e$	Execution time
$T r$	Real-time trajectory
$T t$	Theoretical trajectory
$i i − 1 T$	Transformation matrix
$U$	System potential energy
$v (t)$	4-degree polynomial velocity
$v c$	Crowd speed
$v h$	Hoist speed
$v A$	Velocity of point A
$v A e$	Velocity of entrainment at point $A$
$v A r$	Relative velocity at point $A$
$v B e$	Velocity of entrainment at point $B$
$v B r$	Relative velocity at point $B$
$V$	Material volume
$V d$	Standard capacity of the EHS
$w 1$	Weight of $J 1$
$w 2$	Weight of $J 2$
$w 3$	Weight of $J 3$
$W d$	Dipper width
$x A$	Horizontal position of point $A$
$x f$	Final horizontal position of point A at the end of excavation
$X$	Design variables vector
$X i$	$i t h$ pareto optimal solution
$X i$	Decision vector of the $i th$ subproblem
$y A$	Vertical position of point $A$
$y f$	Final vertical position of point $A$
$y M$	Muck pile phase
$Y j$	Offspring
$z p ∗$	Minimum value of the $p t h$ objective value
$z ∗$	Reference point vector
$Z L$ , $Z S$	Larger and smaller value vectors of $F (X i)$ , respectively
$α 0$	Angle between $O O s →$ and $x$ axis, as shown in Fig.4
$α 1$	Angle between $O B →$ and $O H →$ , as shown in Fig.4
$α 2$	Angle between $O B →$ and $x$ axis, as shown in Fig.4
$α 3$	Angle between $B O →$ and $B O s →$ , as shown inFig.4
$α 4$	Angle between $B O s →$ and $B P →$ , as shown in Fig.4
$α 5$	Angle between $B P →$ and $v B e$ as shown in Fig.4
$α 6$	Angle between $P B →$ and $v B r$ as shown in Fig.4
$α 7$	Angle between $O A →$ and $O H →$ as shown in Fig.4
$β$	Excavation angle
$γ$	Angle between $C 1 C 2 →$ and $C 1 A →$ , as shown in Fig.3
$δ$	External friction angle
$ε$	User-defined $ε$ -comparison constant
$ε red$	A parameter that controls the reducing speed of $ε (k)$
$ε (k)$	Value of $ε$ in the kth generation
$ζ j$	Entropy weight of the $j t h$ evaluation criteria
$η c$	Distribution index for crossover
$η m$	Distribution index for mutation
$θ i$	Angle between axis $x i$ and $x i − 1$ , $i = 1, 2, . . ., 4$
$θ 0$	Initial value of the angle between $O A →$ and the vertical axis
$θ 10$	Value of $θ 1$ at $t = 0$
$θ ˙ 1$	Derivate of $θ 1$
$θ ¨ 1$	Second derivative of $θ 1$
$ϑ i$	Angle between axis $z i$ and $z i − 1$
$κ$	Shear plane angle
$λ p i$	$p t h$ weight of the $i th$ subproblem
$λ i$	Weight vector of the $i th$ subproblem
$ξ i j$	Weighted evaluation matrix element
$ξ +$	Positive ideal solution vector
$ξ −$	Negative ideal solution vector
$ϖ$	Internal friction angle
$ρ$	Material density
$τ i$	$i t h$ generalized force
$ϕ (X i)$	Constraint violation value of $X i$
$ϕ n o r (Y j)$	Normalized constraint violation value of $Y j$
$ϕ n o r (X i)$	Normalized constraint violation of $X i$
$ϕ n o r (X ς)$	Normalized constraint violation of the top $ς$ th individual in the initial population
$φ$	Repose angle
$ψ$	Ratio of normal and tangential resistance
$Γ i$	$i t h$ generalized coordinate

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