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Optimization of a Multi-Axle Steered Heavy Vehicle Steering Mechanism by using the Bees Algorithm and the Hooke-Jeeves Algorithms Simultaneously

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In this study, an optimization scheme for a multi-axle heavy vehicle steering system is designed. The Bees Algorithm and the Hooke-Jeeves algorithm are used simultaneously in the optimization process in connection with each other in order to minimize the objective function which is obtained from the Ackerman error. The Ackerman error is computed by using the solid model of the multi-axle steering system achieved by a CAD application instead of using an analytical model. Considering some assumptions are performed in order to simplify the analytical model in the mathematical formulation of the steering system, steering angle computation by using an original solid model is expected to produce more precise results. Solidworks CAD application is used in Ackerman steering error computations. By using the solid model, the user can observe the current shape of the multi-axle steering system in every iteration of the optimization process. A software in the Visu-alBasic.Net programming language is generated in order to execute The Bees Algorithm in combination with the Hooke-Jeeves algorithm in the optimization of Ackerman steering error. Solidworks API is used in data shift from Solidworks to the generated software. The developed system is used in optimization of Ackerman steering error of a five-axle heavy vehicle steering system. Obtained numerical results are discussed and remarks on the use of the presented method in steering angle optimization are displayed. The participation of Hooke-Jeeves algorithm to the effectiveness of optimization method is also discussed.
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*Corresponding author, E-mail: aerdemir@mpg.com.tr
Optimization of a Multi-Axle Steered Heavy Vehicle Steering Mechanism
by using the Bees Algorithm and the Hooke-Jeeves Algorithms
Simultaneously
Abdullah Erdemir1* and Mete Kalyoncu2
1 MPG Machinery Production Group Co., 42250 Konya, Turkey
2 Department of Mechanical Engineering, Faculty of Engineering and Natural Sci., Konya Technical University, 42250 Konya, Turkey
Abstract
In this study, an optimization scheme for a multi-axle heavy vehicle steering system is designed. The Bees Algorithm and the
Hooke-Jeeves algorithm are used simultaneously in the optimization process in connection with each other in order to
minimize the objective function which is obtained from the Ackerman error. The Ackerman error is computed by using the
solid model of the multi-axle steering system achieved by a CAD application instead of using an analytical model. Considering
some assumptions are performed in order to simplify the analytical model in the mathematical formulation of the steering
system, steering angle computation by using an original solid model is expected to produce more precise results. Solidworks
CAD application is used in Ackerman steering error computations. By using the solid model, the user can observe the current
shape of the multi-axle steering system in every iteration of the optimization process. A software in the Visu-alBasic.Net
programming language is generated in order to execute The Bees Algorithm in combination with the Hooke-Jeeves algorithm
in the optimization of Ackerman steering error. Solidworks API is used in data shift from Solidworks to the generated software.
The developed system is used in optimization of Ackerman steering error of a five-axle heavy vehicle steering system.
Obtained numerical results are discussed and re-marks on the use of the presented method in steering angle optimization are
displayed. The participation of Hooke-Jeeves algo-rithm to the effectiveness of optimization method is also discussed.
Keywords: Hooke-Jeeves algorithm; Mechanism synthesis; Multi axle steering; Multivariable optimization, Steering optimization,
Solidworks API; The Bees algorithm.
1. Introduction
While computational sources developed, mechanism syntheses of complex multibody systems are a popular topic since 1970
[14]. While mechanisms are more complicated, it is hard to acquire an analytical solution.
Big sized and heavy load capacity vehicles need multi axles because of the requirement of reducing the pressure from vehicle
to the ground. Ackerman error in multi axle steering causes wearing on the tires. This also leads to undesired maneuver
trajectories and causes vibrations while maneuver process. In multivariable mechanisms, in order to track the desired trajectory,
it is important to determine the variable values which minimize the objective function. In order to do this, variable arrays must
be created and the objective function must be calculated according to the variable ar-ray until the variable array which gets the
minimum error is found within the desired tolerances. Let n is the number of the variables in an optimization process. The
variable array consists of n dimensional space. Moreover, the prediction of the minimum variable configuration in n
dimensional space heuristically is nearly impossible. Therefore, multivaria-ble optimization techniques must be used.
Mechanism synthesis is consisting of three topics: Graphical, analytical and optimization methods [20]. For few points, the
graphical method finds fast solution. In the analytical method, mathematical model of mechanism is solved alge-braic methods
while tracking the precise points. In this method, the solution may be in complex numbers if too many desired points exist
[15]. Numerical techniques are generally used with variable optimization methods. Swarm optimi-zation algorithms are
commonly used in this field. Some of swarm optimization algorithms are listed as below:
1.Evolutionary Algorithms
2.Particle Swarm Optimization
3.Ant Colony Optimization
4.Bees Inspired Algorithms (the Bees System (BE), The Bees Colony Optimization (BCO), The Artificial Bees Colony
(ABC), The Bees Algorithm (BA)) [21]
The most commonly used optimization techniques are Evolutionary Algorithms (EA), Genetic Algorithms (GA) and Particle
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Swarm Optimization (PSO). These population based optimization techniques are usually reach the global min-imum. The
population based optimizations do not need initial conditions, either. They only need variable ranges [17].
There are two approaches in dimensional synthesis: Precise points and optimum synthesis [2, 5]. The precise point syn-thesis
aims to pass certain number of points. While the number of points increases, the problem becomes nonlinear and hard to solve.
In analytical methods, excessive number of precise points may result the solution to be in complex num-ber of dimensions.
This means the solution mechanism is physically not possible to build. On the other hand, the opti-mum synthesis uses
randomly generated mechanism dimensions to find the best solution which is the minimum error among the tried variable
configurations.
The objective function may also contain the constraint conditions. For example, the dimension length cannot be nega-tive; and
also one of the lengths of link may not be greater from a number or lower from a number. In order to do this, penalty functions
can be inserted into the objective function [1, 2].
Optimization is a process of finding the best solution among lots of alternatives. Optimization is based on numerical methods
and usually used for nonlinear problems. There are local search methods and global search methods in opti-mization. Most of
optimization techniques are based on the use of constraints [22].
Prasanna [9] has compared the Particle Swarm Optimization and its variants in a 5-linkage mechanism. The trajectory
synthesis was consisted of 11 to 25 points. Some of optimization algorithms search the space in neighbor of initial di-mension
array like Hooke-Jeeves. Hooke-Jeeves algorithm is good at finding the local minimum but is not good at find-ing the global
minimum. The resulting minimum which is found by the algorithms like Hooke-Jeeves, changes by the initial condition
because it converges according to the initial condition. Florbela [8] have started Hooke-Jeeves with different initial conditions
parallel and the algorithm has found different local minimums. Youxin [12] have used New-ton-Raphson method in a
mechanism synthesis. It has been seen that the Newton-Raphson method is also good at finding local minimum.
Some other optimization algorithms generate random arrays within the boundary conditions. These algorithms find the local
minimums and most likely find the global minimum [13].
D. T. Pham [16] have compared The Bees Algorithm, Particle Swarm Optimization, Evolutionary Algorithm and the Artificial
Bees Colony algorithms. In this benchmark the Bees algorithm solves most of benchmark functions and the Artificial Bees
Colony algorithm comes after it. Qiang [11] have combined the Hooke-Jeeves algorithm into the Genetic Algorithm. When
the both algorithm worked together, quite effective results were gained. The local search algorithm helped the global
optimization algorithm to search local regions. Coelho, Sacco and Henderson [19] have also com-bined the Hooke-Jeeves into
the Metropolis algorithm. And this increased the effective of the optimization process.
In this study, in order to improve the local search of the Bees algorithm, Hooke-Jeeves method is simultaneously run with The
Bees Algorithm. In other words, the Bees algorithm will spread the space within the boundary conditions and the Hooke-
Jeeves will search the local regions learnt from The Bees Algorithm.
Kuang-Hua [3] have used CAD models in Solidworks and ProEngineer to optimize a kinematic and dynamic perfor-mance
of High Mobility Multi-Purpose Wheeled Vehicle. In the optimization process of 3D Ackermann steering mech-anism, usage
of a CAD model is more precise and preferred technique than simplified analytical model with assump-tions. An analytical
determination of the spatial mechanism kinematics is very complicated, so 3D modeling is more prevalent and affordable [6,
23].
In this study, the steering mechanism modeled in Solidworks CAD program, will be optimized with The Bees Algorithm and
the Hooke-Jeeves algorithm using Visualbasic.Net. The objective function will be calculated over Solidworks API calls and
the error results will be taken directly from the CAD model. By using The Bees Algorithm in this mechanism optimization
process, the global minimum and neighbor local minimum alternative solutions will be found. In other words, more than one
CAD configurations which minimizes the Ackerman error will be found.
2. Problem Definition
The steering error is the angle between the desired direction of the tire and the actual direction of the tire [7, 10, 18, 23].
Generally, the error is wanted to be less than 1°. The steering error is as shown in Fig. 1.
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Figure 1. Steering error
3. CAD Parameters and the Penalty Functions
In a five axles, heavy vehicle shown in Fig. 2, needs all-wheel steering mode and crab steering mode. So it is a requirement to
design the 3rd, the 4th and the 5th axles are mechanically independent from the 1st and 2nd axles. So there is only mechanical
connection between the 1st and the 2nd axles. But the 3rd, the 4th and the 5th axles are actuated with electro hydraulic actuators.
In this synthesis study, the mechanism between the first and the second axle is optimized. Moreover, all five axles steering
error will be checked when results are gained.
Figure 2. The 5 axle heavy truck CAD model
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The optimization parameters a1, a2, a3, L1, L2, L3, L4, θ1, θ2, θ3 and θ4 shown in Fig. 3 and Fig. 4. The initial conditions
(IC) of these parameters are shown in Table 1.
Figure 3. The links to be optimized
Figure 4. The complex parts to be optimized
Table 1. The parameters and the initial conditions.
Parameters
a1 [mm]
L1 [mm]
θ1 ]
L2 [mm]
θ2 ]
a2 [mm]
L3 [mm]
θ3 ]
θ4 ]
a3 [mm]
Initial
Conditions
787,80
250,60
89,00
294,38
89,62
1457,36
415,31
90,00
89,78
487,19
4. Penalty Functions
In order to prevent impossible mechanism configurations penalty functions are used. The penalty function checks the active
configuration and increases the error according to the condition.
For example, boundary conditions for a is shown in Eq. (1),
  (1)
Let the penalty for outside of this boundary is 1000. So the penalty function is
󰇛 󰇜 (2)
Link
length a1
Link
length a2
Link length a3
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The Eq. (2), if a1 < 623 than g1 is 1000, and if not a1 < 623 than g1 is 0. All penalty functions are shown in Table 2.
Table 2. Penalty functions.
g1 =
(a1<623) * 1000
g12 =
(a>823) * 1000
g2 =
(L1<170) * 1000
g13 =
(L1>370) * 1000
g3 =
(θ1<84°) * 1000
g14 =
(θ1>96°) * 1000
g4 =
(L2<100) * 1000
g15 =
(L2>300) * 1000
g5 =
(θ2<84°) * 1000
g16 =
(θ2>96°) * 1000
g6 =
(a2<1290) * 1000
g17 =
(a2>1490) * 1000
g7 =
(L3<200) * 1000
g18 =
(L3>450) * 1000
g8 =
(θ3<84°) * 1000
g19 =
(θ3>96°) * 1000
g9 =
(L4<91) * 1000
g20 =
(L4>291) * 1000
g10 =
(θ4<84°) * 1000
g21 =
(θ4>96°) * 1000
g11 =
(a3<390) * 1000
g22 =
(a3>590) * 1000
5. Objective Definition and Weights
The steering error is defined in the Fig. 1. In this mechanism, the steering error is wanted to be less than 1° at all times of
steering. So we will get the steering error at every nearly 5 degrees of steering. In addition, we will combine all the steering
errors with root-mean-square. Let e1 is the left tire of the 2nd axle steering error angle, and e2 is the right tire of the 2nd axle
steering error. The first objective function is shown in Eq. (3):
󰇛

󰇜


(3)
When some error regions are more important, it is good to use weight multipliers. For example, we want the steering error to
be less than 1° at all times. Changing the objective function like this, will make the objective function to adapt to reduce the
steering error less than 1°.
 󰇛󰇜
(4)
 󰇛󰇜
(5)
󰇛

󰇜


(6)
In the Eq. (4) and (5), we are multiplying 10 times weight to the error range from 0.80° to upper bound if the error is greater
than 1°. So the improved objective function Eq. (6) will return more error if the steering error is greater than 1°. The left tire
of the first axle is turning within range from 35° left to 28.5° right. Total range is 35 + 28.5 = 63.5°. There are 13 steps from
35° to -28.5°. Each step is 63.5 / 12 = 5.29°. The number n in the objective function is 13. m is the number of the penalty
functions. In this case m is 22.
6. Optimization Algorithm
The Bees Algorithm is a global search algorithm. The original The Bees Algorithm is shown schematically in Fig. 5. The
algorithm starts with n scout bees being placed randomly in the search space. The fitness of the population is calculated; this
means the error of all the scout bees are calculated. The array of the scout bees is reordered from minimum error to maximum
error. The best m sites are selected to be search for neighborhood search. The next bees will search those sites within the radius
of patch size which is ngh. But more scout bees will be sent to elite sites which are shown as number e. The remaining scout
bees are less than the number of elite bees. Each site is reordered from minimum error to maximum error. In addition, the
fittest bee is selected for that site. The remaining (n-m) bees are replaced with the new randomly created bees. The fitness of
the new population is recalculated and the loop continues until the stop condition occurs. The pseudo code for The Bees
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Algorithm in its simplest form is:
1. Initialize population with random solutions.
2. Evaluate fitness of the population.
3. While (stopping criterion not met)
4. Select sites for neighborhood search.
5. Recruit bees for selected sites (more bees for best e sites) and evaluate fitness.
6. Select the fittest bee from each patch.
7. Assign remaining bees to search randomly and evaluate their fitness.
8. End While
Figure 5. The original The Bees Algorithm
6.1. The Bees Algorithm combined with the Hooke-Jeeves Algorithm
The Hooke-Jeeves Algorithm is a local search algorithm. Now we will combine The Bees Algorithm and the Hooke-Jeeves
algorithm. When The Bees Algorithm has selected m best sites, it will send the bees into the pool. Simultaneously the Hooke-
Jeeves algorithm selecting one of the bees in the pool and deleting it from the pool and making a local search around it while
the Bees algorithm is working too.
Figure 6. The combination of The Bees Algorithm and the Hooke-Jeeves Algorithm
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The Bees algorithm is generating randomly new bees at 7th step. If the result of the Hooke Jeeves algorithm is fitter than the
first bee, than the result is replaced with the first randomly generated bee. The Bees Algorithm combined with the Hooke
Jeeves algorithm is shown Fig. 6.
7. Software Implementation
The software developed in VisualBasic.Net and Solidworks API is used to connect to the Solidworks shown in Fig. 7. When
the program started Solidworks must be opened with the cad model. The connection between Solidworks and the software is
established by the Solidworks API. The software will create dimension arrays and the array will be sent to Solidworks. The
steering mechanism is checked while steering from left to right, and if there is no mechanical error than the software reads the
Ackerman errors.
Figure 7. The software schematics
According to the objective function (6), The Bees Algorithm (BA) and The Bees Algorithm Combined with the Hooke Jeeves
Algorithm (BAHJ) results are shown in Table 3.
Table 3. Penalty functions.
Original
BA
BAHJ
The Result of the Objective Function
222,14
11,18
10,30
Average left tire error ]
1,75
0,62
0,56
Average right tire error [°]
1,65
0,59
0,56
Maximum left tire error ]
6,39
0,99
1,00
Maximum right tire error [°]
5,46
1,00°
0,99
Optimization Duration [min]
-
362
245
a1 [mm]
787,80
781,33
791,68
L1 [mm]
250,60
352,56
310,64
θ1 [°]
89,00
88,83
92,57
L2 [mm]
294,38
230,61
237,22
θ2 [°]
89,62
93.56
84,40
a2 [mm]
1457,36
1466,52
1332,20
L3 [mm]
415,31
274,60
404,28
θ3 [°]
90,00
94,92
92,90
L4 [mm]
200
189,08
234,09
θ4 [°]
89,78
92,10
91,59
a3 [mm]
487,19
529,43
445,55
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While steering, the original and the optimized Ackerman steering errors are shown in Fig. 8.
Figure 8. The initial condition and the optimization results chart
The results obtained by The Bees Algorithm and The Bees Algorithm combined with the Hooke-Jeeves algorithm is shown
in Fig. 9. After the optimizations, the Ackerman steering error has been dropped under 1° at all times of steering angles as
seen in Fig. 9.
Figure 9. The comparison between The Bees Algorithm and the combination of The Bees Algorithm + Hooke-Jeeves Algorithm
The obtained results were commercialized using the design and manufacture of the steering system of the vehicle shown in
Fig. 10.
0
1
2
3
4
5
6
7
-40 -30 -20 -10 0 10 20 30 40
Ackerman Steering Error [°]
Steering Angle [°]
After BAHJ Left
Tire Error
After BAHJ Right
Tire Error
Initial Left Tire Error
Initial Right Tire
Error
0
0,2
0,4
0,6
0,8
1
1,2
-40 -30 -20 -10 0 10 20 30 40
Ackerman Steering Error [°]
Steering Angle [°]
After BAHJ Left
Tire Error
After BAHJ Right
Tire Error
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Figure 10. The commercialized five axle heavy vehicle, T0120.
7. Conclusion
In dimensional synthesis, using directly a CAD model is making a faster setup of an optimization process. In order to guarantee
the steering error is to be less than 1° at all times of steering, the objective function has been written to give more weight for
more than 0.80°.
The use of global optimization techniques like The Bees Algorithm is quite important in optimization study. In addition, it has
been seen that a local search algorithm could be helpful for global optimization algorithm to minimize for regional spaces.
The combination of Hooke-Jeeves local search algorithm has increased the speed of local searching of The Bees Algorithm
which is a global optimization method.
In this study, the used computer specifications are shown as below.
CPU: Intel Xeon CPU E3-1270- v3 @ 3.50 GHz
RAM: 16 GB memory
Operating System: Windows 7 Professional
Acknowledgment
This study was supported by MPG Machinery Production Group Co. / KONYA / TURKEY, we thank the MPG Machinery
Production Group Co.
Nomenclature
The length of the ith link [mm]
The length parameter of the complex part shown in the Fig. 4.
The angle parameter of the complex part shown in the Fig. 4.
Ackerman error of the left tire [°]
Ackerman error of the right tire [°]

Weighted Ackerman error of the left tire [°]

Weighted Ackerman error of the right tire [°]
The error function to be optimized
ith penalty function
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