Proceedings of IMECE2003:

International Mechanical Engineering Congress and R&D Expo

16.-21. November 2003, Washington, D.C., USA
Proceedings of IMECE’03
2003 ASME International Mechanical Engineering Congress
Washington, D.C., November 15–21, 2003

IMECE2003-41457

MULTI CRITERIA DESIGN OPTIMIZATION OF BACKHOE LOADER FRONT

MECHANISM

Michael Rygaard Hansen Torben O. Andersen

Institute of Mechanical Engineering Institute of Energy Technology
Aalborg University Aalborg University
mrh@ime.auc.dk toa@iet.auc.de

ABSTRACT occupied space and avoidance of collision that typically appear

In this paper the design upgrade of a front shovel in practical mechanism optimization, see also [9] and [10].
mechanism of a backhoe loader is considered. The introduction
of computer aided optimization techniques for this task clearly NOMENCLATURE
illustrates the complexity of improving a mature and in many Rotation of
α w Weight factor
ways already optimal design. The design task is formulated as attachment
a weighted multi criteria optimization problem where the ϕi Rotation of i'th f Objective function
criteria comprise both kinematic and static performance as body
well as a substantial set of equality and inequality side L Link length h Equality constraint
constraints. A special purpose computer program has been x  Coordinates of i'th Inequality
developed to solve the optimization problem doing the ri =  i  g
y
 i point. constraint
minimization using the well known quasi-Newton method:
Davidon-Fletcher-Powell. The work has been carried out in ξ ( j ) Coordinates of i'th Penalty function
cooperation with the danish company: Hydrema ApS, during a s (i j)
=  i(  point in body j Φ contribution from
ηi
j)
design upgrade of their backhoe loader.  coord. system ineq. constraint
A Transformation
i O Penalty function
INTRODUCTION matrix of body i
In general, manufacturers of construction equipment are θ Cylinder rotation Y Design variables
challenged by increasing performance expectation from Index MC Main lift cylinder
customers, reduced time-to-marked and highly Index SC Shovel cylinder
multidisciplinary systems. This leads to complex design tasks
where it would be advantageous to employ some kind of CONSIDERED SYSTEM
computer based design optimization. This is especially true for The front shovel mechanism of the considered backhoe
problems where the toplogy is fixed and the design task is that loader is a two degree of freedom mechanism that is actuated
of sizing a set of parameters subject to criteria and constraints. by means of two hydraulic actuators referred to as the main lift
In this work the design of a front loader mechanism of a cylinder and the shovel cylinder, respectively, see Figure 1.
backhoe loader is considered using numerical optimization. The mechanism consists of 6 bodies: Four binary members
Numerous work on constrained optimization of mechanical (the pull rods, the vertical arm and the attachment), a ternary
mechanisms has been carried out since the original work by member (the lever) and a quintenary member (the main boom).
Fox and Willmert [1], typically with emphasis on the Structurally it should also be noted, that there is a double joint
kinematic criteria, see also [2..5]. With respect to front loader connecting the pull rods and the vertical arm.
mechansisms some work has already been done with empahsis The main functionality of the front loader is dual
on the purely kinematic criteria, [6], [7] and [8], however, in depending on the attached implement. Firstly, it may be used
this paper emphasis is on the overall optimization formulation for breaking and digging operations in which case a shovel is
and the handling of the numerous side constraints related to

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 mounted. Shovel operations include the capability of breaking The breaking force is simply defined as the vertical force
material from the ground as well as simply digging from a pile that may be held in static equilibrium by the shovel cylinder in
then hoist and subsequently dump the material in a truck. the position shown in Figure 3.
Secondly, it may also be used as a fork lift, replacing the Maximum shovel
shovel with a set of forks and now required to hoist and lower cylinder force
equipment positioned on pallettes.
Shovel cylinder

Lever
Pull rod Breaking force

Small pull rod

Figure 3. Definition of breaking force.

Frame
Main boom Beyond the main performance criteria a number of other
Main lift cylinder Vertical arm requirements and side constraints influence the design. These
Attachment design constraints will be dealt with in more detail when
Figure 1. Front loader mechanism of backhoe loader. discussing the optimization formulation.

These different functionalities influence the criteria upon MODELING

which a front loader design is carried out. Clearly, it is not The type of analysis required to evaluate a front loader
only a case of multi criteria design but also of design based on mechanism design is of a purely static nature, i.e., inertia
multiple load cases. forces and dynamic variations in hydraulic cylinder forces may
The main performance criteria for the front loading be disregarded. Hence, a number of kinematically determined
mechanism are: analyses suffice which greatly reduced computational time.
Ø Parallel motion during hoisting In genereal, two different types of kinematically
Ø Breaking force. determined analyses are employed due to the many different
Parallel motion during hoisting is the ability to maintain the load cases.
attached implement substantially parallel during hoisting and
lowering. Perfect parallel motion can be obtained, simply by
actuating both cylinder continuously. However, in this context 8 L3
the parallel motion must be achieved while only acutating the y ϕ3
main lift cylinder. Also, this should be pursued for two x ϕ1 L5
0
different positions of the shovel cylinder. These two positions 3 9 ϕ5
are associated with the dual functionality, i.e., lifting with a 2 7
shovel or lifting with a set of forks, see also figure 2. 10
L4 ϕ4
1 4 L6
ϕ6
LSC
5
6 α
LMC
η8( 2 )
8

L1 ξ 8( 2 )
3
L2
7
Figure 2. Position of attachment and shovel cylinder in the two ϕ2
different situations where parallel motion is desired.
Figure 4. Variables associated with the kinematic analysis.

In the first case the rotation of the main boom, ϕ 1 , and

the attachment, α , are prescribed, see figure 4, and in the

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 second case the rotation of the main boom, ϕ 1 , and the length Note that the choice of configuration lies in the chosen
of the shovel cylinder, LSC , are prescribed. sign of ∆ y indicating that the intersection point with a positive

In the first case the procedure of analysis is as follows: y-coordinate in a local coordinate system pointing from point 5
Firstly the global coordinates, r of points 3 .. 6 are found as: to point 10 is chosen. The position of point 8 is computed
along exactly the same lines, only here the circles in question
have points 3 and 9 as center and the radii L3 and s(83 ) .
r i = A ⋅ s(i 1 ) i = 3..6 (1)
1
The position analysis is concluded with some straight
forward computation of the orientation of the lever, the
In equation (1) the standard formulation using the position of point 7 and the orientation and length of the
transformation matrix, A of the main boom together with the hydraulic actuators.
1
coordinates of the points of interest in the local coordinate
system of the main boom, s(i 1 ) are used. Next, the positions of ( )
ϕ3 = ∠ r8 − r3 − β + π r 7 = r 3 + A ⋅ s7( 3 )
3
(6)
points 10 and 9 are computed utilizing that a sub fourbar (
θ MC = ∠ r 4 − r 1 ) LMC = r 4 − r 1 (7)
linkage may be identified consisting of the attachment as the
driven link, the main boom as the frame and the small pull rod θ SC = ∠(r 7 − r 2 ) LSC = r 7 − r 2 (8)
and the vertical arm as the connecting rods. The position
analysis is basically a computation of the intersection points of
Hence, the entire kinematic analysis may be carried out
two circles, see figure 5, with center at point 10 and point 5
based on explicit expressions greatly reducing the
and radii L4 and L5 , respectively. The analysis also includes
computational time.
a unique routine that chooses the intersection point that The second type of kinematic analysis also employs
corresponds to the actual assembly configuration. equation (1). Next, point 7 is computed as the intersection
point between two circles centered at point 2 and point 3 and
cosθ − sin θ   ∆ x  with radii LSC and L2 , respectively. Next, point 8 may be
r9 = r5 +  ⋅  (2)
 sinθ cosθ  ∆ y  determined:

The different geometrical variables, see also figure 5, are given (

ϕ 3 = ∠ r7 − r 3 ) r 8 = r 3 + A ⋅ s (83 )
3
(9)
as:
(
θ = ∠ r 10 − r 5 ) d = r −r (3)
10 5 The next steps follow the usual circle-circle intersection
approach: Point 9 is computed as the intersection of the circles
L24 − L25 + d2
∆x = ∆ y = L24 − ∆2x (4) centered at point 8 and 5 with the radii L3 and L4 ,
2⋅d
respectively. Finally, point 10 is computed as the intersection
r 10 = r 6 + A ⋅ s (106 ) (5) of the circles centered at point 9 and 6 with the radii L5 and
6
L6 , respectively. This concludes the second type of kinematic
∆y analysis. Again, it is worth noting that the analysis is carried
9 out using explicit expressions.
Finally, static force analysis has also been required in
10 order to evaluate the breaking force. This has been carried
using static equilibrium for the lever, the pull rods, and the
β
shovel. The input has been the maximum available force from
θ the shovel cylinder applied at point 7 and the output has been
the corresponding breaking force as defined in figure 3.
5 ∆x
d OPTIMIZATION FORMULATION
Upgrading or improving a design as the current front
loader mechanism of Hydrema ApS is a challenging task due
Figure 5. Variables associated with the position analysis. to the mature level of this design. The design task was
formulated as a desired improvement in breaking force of 15%
within certain tolerances and as optimal parallel motion when

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 hoisting as possible. This naturally leads to an optimization contribution from any inequality constraint is computed as
formulation where the parallel motion is the objective function follows:
and the improvement in breaking force may be considered as a  0 g i ≥ 0.0
side criteria. Beyond these basic performance related criteria a g i < 0.0 : Φ i =  (13)
wg ⋅ g i g i < 0.0
2
large number of equality and inequality constraints had to be
imposed on the optimization formulation in order to achieve
an acceptable design. The development of the optimization In (13) wg is a weighing factor. The formulation in (13)
formulation itself was an iterative proces. Initially, only a few
does not penalize the design until it violates the inequality
side constraints, 4, were included, however, new side
constraint. This mean that the final design might very well
constraints had to be included continuously until the
violate certain inequality constraints and this should be taken
optimization routine returned a design taht was acceptable in
into consideration when defining the threshold or limit values
any respect.
which typically should be somewhat more restrictive than
The mathematical formulation is that of a weighted multi
necessary. This is typically a tuning task that is carried out
criteria optimization problem subjected to a set of equality and
during the work with the design optimization.
inequality constraints including lower and upper bounds on the
In the following the different contributions to the penalty
design variables, Y :
function is gone through in more detail. When a given design
min imize ω T f (Y ) is to be evaluated four extreme positions are examined as well
subject to : h(Y ) = 0 , g (Y ) ≤ 0
(10) as two motion intervals. The extreme positions are examined
with to check if the current design may be assembled and also
to evaluate certain inequality constraints. The motion intervals
The design variables comprise the following 12 variables: are used to evaluate the objective functions as well as a number
of inequality constraints. For future refefrence the inequality
[
Y = x2 y2 ξ 3( 1 ) η3( 1 ) ξ 5( 1 ) η5( 1 ) L2 .... constraints are referred to as ICx where x is the number of the
inequality constraint.
(11)
ξ 8( 3 ) η 8( 3 ) L3 L4 L5 ] Firstly, the inequality constraints that are independent of
the position of the front loader mechanism are considered: The
x-coordinate of point 2 must be positive to ensure the
Some of the potential design variables are fixed (equality
possibility of mounting the shovel cylinder (IC1). The distance
constraints). Hence, the main lift cylinder and its attachment
between the following points must all be at least 110 mm:
both to the vehicle and the main boom cannot be altered. The
Points 3 and 7, points 3 and 8, and points 5 and 6, (IC2..IC4).
total length of the main boom, L1 = 2735 mm , which secures
This value was determined from the experience of the
another major performance parameter namely the operational manufacturer with respect to the actual space occupied by a
height, is fixed together with the minimum, ϕ 1,min = −37.6° , point after adding physical dimensions. The design variable,
and maximum, ϕ 1,max = 47.2° , rotation of the main boom. η5( 1 ) should be less than -100 mm for manufacturing purposes.
The length of the attachment, L6 = 400 mm , is also fixed in
order to maintain compatibility with standard tools (shovel,
forks etc.).
The objective function and all the constraints are
combined in a penalty function that returns a single penalty
θ'
value/design index. This penalty value is computed as follows:

ng
O (Y ) = w1 ⋅ f 1 + w2 ⋅ f 2 + ∑Φ (12)
i =1 i

In equation (12) f 1 and f 2 are the objective functions 0

related to the deviation from parallel motion for the two
different hoisting situations, see also figure 2, and w1 and w2
are the corresponding weight factors. The number of inequality y'
constraints is ng and the contribution from the i'th inequality Figure 6. Extreme position 1.
constraint to the penalty function is: Φ i . In general, the

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 Extreme position 1, see figure 6: The main boom is in its
upper position, ϕ 1 = ϕ 1,max and the shovel is tilted forward in
upper dumping position, α = −45° . In this position the angle,
θ ' , must be large than than 15° to have a margin against the
cylinder and lever going over center (IC5). The distance, y' ,
between the shovel cylinder and point 0 must be larger than
120 mm (IC6). This gives a margin against collision between
shovel cylinder and main boom. 5
Extreme position 2, see figure 7: The main boom is in its min 30
lower position, ϕ 1 = ϕ 1,min and the shovel is tilted forward in 6
lower dumping position, α = −100° . In this position the tip of Figure 8. Extreme position 3.
the shovel should be at least 60 mm away from point 5 to give
some margin against collision between shovel and main boom The actual dead length of the shovel cylinder is given,
(IC7). The minimum length of the shovel cylinder is saved as: hence a penalty is imposed if d' deviates more than 20 mm
LSC ,min . from this value (IC10). The distance, y' , between the cylinder
LSC ,min centerlines must be at least 120 mm (IC11) to have a margin
against cylinder collision.

y'

5
min 60

Figure 9. Extreme position 4.

Figure 7. Extreme position 2. In all of the four extreme positions kinematic analysis of type 1
has been used. Next, the current design is evaluated in two
Extreme position 3, see figure 8: The main boom is in its motion intervals where kinematic analysis of type 2 is
lower position, ϕ 1 = ϕ 1,min and the shovel is plane, α = 0° . In employed.
Motion interval 1: The front loader mechanism is initially
this position point 5 must lie at least 30 mm above point 6 in a position corresponding to extreme position 3. With the
(IC8) to provide a margin against point 5 colliding with the shovel cylinder locked the main boom is moved to its upper
ground. Also, the breaking force is evaluated in this position. position. The first objective function is evaluated as:
The breaking force is penalized if it is either 2 kN below or
above the desired value, (IC9).
(α i − α 1 )2
m
Extreme position 4, see figure 9: The main boom is in its f1 = ∑ (15)
i =1
lower position, ϕ 1 = ϕ 1,min and the shovel is tilted upwards,
α = 45° . In this position the maximum length of the shovel In equation (15) m = 20 is the number of evaluated positions
cylinder is computed, LSC ,max . The necessary dead length of that should be viewed as a compromise between computational
the shovel cylinder in order to utilize the entire stroke is: cost and precision.
Motion interval 2: The front loader mechanism is initially
d' = 2 ⋅ LSC ,min − LSC ,max (14) in a position corresponding to extreme position 4. With the
shovel cylinder locked the main boom is moved to its upper
position.

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 The second objective function is evaluated as: The optimization is carried out by minimizing the penalty
function, O (Y ) as defined in equation (12). The minimization
(α i − α 1 )2
m
f2 = ∑ (16) is carried out by means of a combined gradient based approach
i =1 that alters between simple 1st order steepest descent and the
semi 2nd order method of Davidon-Fletcher-Powell. The
Specifically, for motion interval 2 the distance between the minimization routine uses Davidon-Fletcher-Powell as default,
cylinder centerlines should stay above 85 mm to supply margin but may jump to steepest descent if the direction suggested by
against collision between cylinder piston rods (IC12). Davidon-Fletcher-Powell is up-hill. The procedure for
A substantial number of inequality constraints, see also determining the step size was based on a typical golden section
figure 10, are evaluated based on the motion intervals by search but was augmented to take into account that the penalty
summation of each violation. function might not evaluate to anything (typically if the
x' mechanism could not assemble in all the extreme positions).

8 RESULTS
y' Applying the optimization formulation described
x' '
9 previously yields the design listed in table 1. As initial design
y' ' 3 in the optimization the best design developed "manually" by
0 10
7 Hydrema ApS was used. The non-zero contributions to the
2
minimized penalty function are listed in table 2.
4
5
6
1 Table 1. Design variables of optimized design. All dimensions
Figure 10. Different distances used to evaluate inequality in mm.
constraints associated with the motion intervals. x2 = 38.57 y2 = −170.71 ξ 3( 1 ) = 1000.53

The value x' should remain positive to give margin against η3( 1 ) = 184.75 ξ 5( 1 ) = 2432.65 η5( 1 ) = −50.76
collision between lever and steering house. The value y' L2 = 259.18 ξ 8( 3 ) = −565.24 η8( 3 ) = −125.18
should stay above 330 mm to allow room for a sufficient
bending stiffness of the main boom (IC13). L3 = 1489.17 L4 = 507.7 L5 = 371.28
Point 5 should lie at least 100 mm to the left of the line
connecting points 6 and 9 (IC14) and at least 110 mm below Table 2. Non-zero contributions to the penalty function.
the small pull rod (IC15). Points 7 and 3 should both lie at IC 5 = 9.21 IC6 = 10.62 IC 9 = 0.56
least 110 mm to the left of the vertical arm (IC16 and IC17). IC11 = 0.03 f1 = 411.70 f 2 = 237.46
The values x' ' and y' ' should remain less than 1250 mm and
more than -50 mm, respectively (IC18 and IC19). Inequality In figure 11 the shovel rotation are shown for the optimized
constraints (IC14..19) ensures margins against collision design corresponding to the two motion intervals.
between different members of the front loader mechanism.
Obviously, the vast number of side constraints and 50 α [deg]
corresponding threshold values have been decided in close
coorporation with Hydrema ApS. The weighing was carried 40
out as follows: All constraint violations related to dimensions Motion interval 2
was computed in mm 2
directly without any weighing. 30
Violation of the breaking force constraint was computed in
20
N 2 and weighed with a factor of 1 ⋅ 10 −5 . Violation of the
angle in IC5 was computed in deg 2 and weighed with a factor 10 Motion interval 1
of 3.0. The objective functions was directly computed in ϕ 1 [deg]
0
deg 2 without any weighing. The final optimization
-40 -30 -20 -10 0 10 20 30 40 50
formulation is the result of an iterative proces were weighing
-10
factors and side constraints continuously have been varied or
added/removed until the design returned by the procedure was Figure 11. Variation of shovel rotation as function of the
deemed acceptable. rotation of the main lift.

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 [6] Sandor, G. N. and Erdman, A. G., "Advanced Mechanism
The optimized design presented in the above was, with minor Design: Analysis and Synthesis,", Volume 2, Prentice-Hall,
justifications, used in the upgraded backhoe loader of Hydrema 1984.
ApS.
[7] Bruns, T. E. and Tortorelli, D. A., "A Coupled Approach to
CONCLUSIONS the Automated Analysis and Optimization of Flexible
A special purpose computer program has been developed Multibody Mechanisms," in Proceedings of the Twelfth
to solve the sizing task of designing a new backhoe loader by Conference of the Irish Manufacturing Committee, IMC 12
means of optimization. The nature of the optimization Competitive Manufacturing, S. M. De Almeida, ed., Gemini
formulation means that a design may be evaluated by means of International Limited, Dublin, Ireland, pp. 29-36, Sept. 1995.
a number of kinematically determined analyses which greatly
reduces the computational costs. The minimization is carried [8] Pesch, V. J., C. L. Hinkle, and D. A. Tortorelli, "Synthesis
out using a combination of the steepest descent and the quasi- and Optimization of Planar Mechanism Kinematics Using
newton method: Davidon-Fletcher-Powell. Symbolically Computed Design Sensitivities," Proceedings of
In general, the side constraints and weighing parameters the IUTAM Symposium on Optimization of Mechanical
has been decided in close coorporation with Hydrema ApS that Systems, Solid Mechanics and Its Applications, D. Bestle and
manufactures backhoe loaders, and a satisfying design, that W. Schiehlen, eds., Kluwer Academic Publishers, 43, pp. 221-
with minor justifications has been used as part of a general 230, 1996.
design upgrade, has been obtained.
[9] Hansen, M.R., "An Automated Procedure for Dimensional
ACKNOWLEDGMENTS Synthesis of Mechanisms", Structural Optimization, Vol. 5,
The contribution to this work from Poul H. Joensen, No. 3, pp. 145-152, 1993.
Hydrema ApS, is gratefully acknowledged.
[10] Hansen, M.R., "A Multi Level Approach to Synthesis of
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