Describing the position of backhoe dredge buckets
C.F. Hofstra[1],A.J.M. van Hemmen[2],S.A. Miedema[3],J. van Hulsteyn[4]
Abstract
There is
growing interest in the automation of the production cycle of backhoe dredges.
In order to realise an effective control mechanism for a non-rigid body it is necessary
to acquire an adequate insight in not only the dynamic behaviour of the
hydraulic system but also of the mechanical systems. This second factor is also
influenced by the fact that bigger and bigger machines are being used.
As there is
no real reference material on this subject the first step has been the
description of the kinematics of a backhoe. This was followed by determining of
the Denavit-Hartenberg matrix to describe the mechanical motions of the system
with respect to the position and orientation of the bucket.
Using this
method a dynamical model has been developed. The paper will give a description
of the dynamical model (using Matlab and Adams) and show some of the simulation
results with respect to the influences of the flexibility of the hydraulic
fluid and the steel structure on the achievable accuracy.
Keywords: Denavit-Hartenberg, trajectory control,
flexible bodies, accuracy.
Introduction
Over the
last decades the backhoe has come to replace the bucket dredge as the primary
tool for the excavation of trenches and localised sites. Mainly because of the
noise production and the ability to perform environmental dredging operations.
This has been accompanied by an increase in the size of these machines in order
to work in deeper locations. The increase in size and capacity leads to a
reduction in achievable accuracy of the dredging process.
In order to
improve the accuracy of the dredged level one must first determine the prime
factors influencing the accuracy of the process. This is especially true if one
wishes to automate the process in the future. This paper describes a dynamical
model that was used to determine the effects of the flexibility of the boom and
the stick with respect to the position of the tip under various loading conditions
during the dredging cycle compared to the rigid body approach.
First the
conventional approach to bucket position is described which was used to
determine the points of interest. The next phase is the description of the
degrees of freedom as available independent of the control system. To round off
the system the influence of the (non-) rigidity of the steel structure in two
digging situations and the responses to external loads are described and
analysed.
The model
used in this paper is based on the Komatsu H245S with a 12 m boom and a 8.5 m
stick.
Trajectory
planning
The
trajectory described by the bucket tip is determined beforehand. This
trajectory is then converted to the machine co-ordinates (angles) after which the
necessary sequence of piston positions is determined.
The
relationship between the orientation of the machine and the position of the
piston is determined from figure 1.
Figure 1
Model with angle definitions
Boom
with: 
(1)
Stick
with: 
(2)
Bucket
(3)
with :
(4)
(5)
The next
step is to determine the relation between the machine orientation and the
desired trajectory. In order to do this effectively while describing the
position and orientation of the bucket the Denavit-Hartenberg (DH) approach
based on homogenous co-ordinates is utilised.
Derivation of
the position of the bucket (tip) using Denavit-Hartenberg
As stated the
position of the bucket, specifically the bucket tip, is derived from the
orientation of boom, stick and bucket with respect to each other. This is
measured by means of their respective angles. For the DH approach the modelling
of figure 1 is modified to give a "chain' as depicted in figure 2.
Figure 2 Backhoe “chain”
Using
homogenous co-ordinates the Denavit-Hartenberg matrices for boom, stick and
bucket based on an orthogonal Cartesian system are:
(6)
In these
matrices Cf=cos(f) and Sf=sin(f).
Multiplication
of these matrices yields the "hand" matrix for the position and orientation
of the bucket tip:
(7)
With 
, 
This is the
forward kinematics of the backhoe. Given a trajectory f(t) in time for the bucket we can determine the accompanying
angles by solving the following equation:
(8)
Control of
the machine is achieved using the hydraulic cylinders. If there is a small
change in the orientation of the machine, what is the corresponding change in
the piston position? The equations (1 to 5) clearly show the difficulty in
extracting the cylinder length, which in turn leads to the piston position.
This is certainly true for the bucket cylinder. Modifying the DH matrix to
include only boom and stick reduces this problem. Bucket position and
orientation are added to the equation as boundary conditions. Because rotation
of the machine influences neither the boom angle nor the stick angle it can
likewise be added as a boundary condition. The dredge cycle with respect to
boom, stick and bucket reduces to in plane motion. The reduced DH matrix now
reads:
(9)
With: 
Trajectory
control
To effectively
follow a prescribed path we have to know the relationship between small changes
in angles and the piston positions. Differentiating the DH matrix yields:
(10)
The
corresponding changes in the cylinder lengths can be determined from equations
1 and 2 in a similar fashion. Adding time to these equations gives us the
necessary tools to describe the motion of the bucket tip along a specified
trajectory.
The hydraulic
system
The use of
hydraulic cylinders introduces flexible elements into the system enabling it to
move independently of the controls. The magnitude of the flexibility is
determined by the bulk modulus and the volume of fluid between the control
block and the piston and the flexibility of the supply lines. This flexibility
is incorporated into the model by modelling the cylinders as springs. The
stiffness of the spring is determined by total change in the volume:
(11)
From which
the displacement of the piston is determined:
(12)
Resulting
in a spring stiffness:
(13)
This formula is used for both piston and rod
side incorporating the supply lines and hoses. This is necessary because in
contrast to normal practice the effects of both cannot be disregarded due to
their length. Damping in the cases studied was assumed to be limited to 1% of
the spring stiffness.
Additional
degrees of freedom - Flexible bodies
This study is
intended to determine the effects of flexibility of the steel structure. For
this purpose the Adams program was utilised. In the Adams program flexible
bodies are described using the eigen-frequencies and eigen-vector approach.
These are used to calculate the effective stiffness and damping matrices.
Frequencies and vectors are determined using the Ansys finite element program
and imported into the program. As these results vary with the orientation of
the machine they are reproduced here.
From past
experience it is known that the effective damping of steel structures can be
taken as equal or less than 1% of the effective stiffness matrix.
Forces on the
backhoe
Forces on
the backhoe can be subdivided into external and internal forces. Internal
forces are those caused by the friction in the hydraulic cylinders. Due to the
mismatch between the moment arms of the forces involved, friction forces in the
joints can be disregarded. External forces are digging force and those caused
by the movement of the parts through the water, to which are added the wave and
current forces.
External forces
The digging
forces on the bucket are determined using existing models for sand, Miedema
(1987), clay, Miedema (1997). In the absence of models for the cutting of loose
or sprung rock the following analogy is used:
The force
exerted by current, waves and moving through the water can be calculated by
combining the Morison equation with the formula for flow resistance. This results
in:
(15)
Except for
Cf, which is a function of the Reynolds number, the other
coefficients Ci have to be determined experimentally for the
submerged parts of the structure.
Internal forces
Friction in
the cylinders cannot be disregarded due to its necessity for determining the
equilibrium over the piston and its effects on the system damping. A linear
function for the friction is used according to:
(16)
Analysis of the
dredging cycle
Based on a
rigid body approach these aspects were used to model various dredging cycles
with the Matlab program. These show that during operations the largest
accelerations and therefore forces on the structure occur during the
positioning and the digging part of the cycle. During the other parts of the
cycle the available hydraulic power is not sufficient to induce significant
accelerations and the structure itself is not subjected to the rapid changes in
kinetic energy. As a whole these results were comparable to the results of
previous studies, for example Van Velzen (1999), Salcudean et al. (1999). The
main points of interest are therefore limited to the positioning of the bucket
and the digging part of the cycle.
Apart from
production the single most important factor of the dredging cycle is the
accuracy achieved during digging. The achievable accuracy decreases with
increasing machine size. This is ascribed in part to the flexibility of the
hydraulic and mechanical systems, the positioning of the pontoon and the
operator.
If the backhoe were automated the operator
influence would be taken up by the control system. For the design of such a
system we need to know in advance whether the basic assumptions for a rigid
system are applicable. To this end the effects of the flexibility of the steel
structure are studied first followed by flexibility of the hydraulic system.
Example
Using
prismatic beam theory the deflection of the end of a beam with dimensions
according to figure 3 under a load F of 20 tons amounts to 25.2 mm.
Figure 3
Cantilever beam
If the same
applies for a backhoe structure of comparable length it would lead to a
sizeable error in the achievable accuracy. In the case of the studied machine
the force reaches approximately 45 tons (deflection@6cm).
Influence of the
mechanical subsystem
To study
the effect of the flexibility of the steel structure two digging situations are
modelled. In the first the bucket digs horizontally starting at an inclined the
digging front (figure 4 - left). In the second the bucket scoops up material
(figure 4 - right). In these simulations dynamic effects due to motion are not
taken into account.
Figure 4
Digging profiles
The
position of the machine is of course but one of a number of possible ones.
However, the resulting deflections will not vary substantially from on
situation to another.
Case 1 - Digging
horizontally
Using the previously described flexibility the
bucket is placed at the digging front and commences digging. The digging force
is assumed to be zero at the beginning and increases to 40tons after which it is
kept constant. Figure 6 shows the resulting paths for (continuous) the rigid
simulation and (dash) the flexible situation.
Figure 5 Digging along horizontal profile
The main
difference occurs during the application of the digging force. This results in
an error of less than 1.5cm with respect to the rigid model in the vertical
direction and a slight lag in the horizontal.
Case 2 Scooping
up material
Figure 6 Scooping up a load
As can be seen the deviation during the entry
part is comparable to the previous case. When the bucket is rotated to the
horizontal digging position a vertical difference of about 3 cm appears. The
lag in the horizontal direction is about the same.
The hydraulic
subsystem
The hydraulic system exerts force by means of
pressure in the cylinders, which is then transmitted via the structure to the
bucket tip. The build-up of pressure in a cylinder compresses the hydraulic oil
and allows the oil to absorb energy due to the compressibility of the hydraulic
fluid. In normal situations the volume of oil under compression and therefore
the amount of energy absorbed is small and its effects are neglected except for
control purposes. However in this application the size of the cylinders and the
length of the hoses and pipes result in a large volume of oil. Release of
pressure due to the removal of an external load can lead to a change in the
volume of the oil and thereby to a significant change in the bucket position.
Two instances are described, the effect of the stepped removal of a horizontal
ramped load in the plane of the machine and a similar load perpendicular to
this plane.
The
external forces acting on the structure are the first two terms of equation 14
for components below the waterline. The experimental factors in this equation
are assumed to be equal to 1 as there is no available data. The structure is
assumed to be free of the ground after the release of the load.
Case 3 - Ramped
load in the working plane
In this example
the boom and stick mentioned in the introduction (12 m and 8.5 m respectively)
are used.. Boom and stick angles are 30° and -120° respectively, the bucket
angle is -15° (following the definitions of figure 2). For the hydraulic
stiffness and damping values according to table 1 are used, these are based on
a hydraulic bulkmodulus of 15000 kPa.
Table 1: Hydraulic stiffness and damping values
|
|
Stiffness |
Damping |
|
Boom
cylinder |
1x108 N/m |
1x106 Ns/m |
|
Stick
cylinder |
5x107 N/m |
5x105 Ns/m |
|
Bucket
cylinder |
5x106 N/m |
5x105 Ns/m |
Figures 7
and 8 show the resulting displacements of the bucket tip. Starting from the
original equilibrium position the bucket tip oscillates with amplitude of about
10 cm in horizontal direction and 5 cm in the vertical about the new
equilibrium.
Figure 7: Displacement bucket
tip in the horizontal direction in the working plane
As can be seen the addition of flexibility for
the steel structure leads to a slight increase in the displacements. This is
due to the higher level of potential energy when using flexible bodies.
Figure 8: Displacement bucket
tip in vertical direction in the working plane
It is
obvious from both figures that the actual amount of damping, both internal as
well as external, is small because the displacements are accompanied by
relatively low velocities. This in turn leads to low values for the damping
forces. The structure will oscillate for a considerable period of time due to
the absence of control input. Figures showing the paths of the corresponding
forces in the cylinders can be found in the appendix. As is the case with the
displacements the cylinder forces show little difference between the rigid and
the flexible models.
Case 4 - Ramped
load perpendicular to the working plane
The swing
mechanism of the backhoe is modelled also as a hydraulic cylinder (torque). The
ramped load leads to a displacement perpendicular to the working plane as seen
in figure 9. The resulting displacement is much larger than the in previous
case.
Figure 9: Displacement bucket
tip perpendicular to the working plane
This is due
to the distance between the tip of the bucket and the turning point of the
machine itself (a relatively small
force causes a large displacement). What can also be seen is that there is no
discernible difference between the rigid and the flexible model. The magnitude
of the displacements in the other directions is comparable to those described
in the previous case. They and the forces in the cylinders the are depicted in
the appendix.
The forces in the x- and y-direction are not
equal to zero due to the calculation of the equilibrium. The applied force does
not follow the rotation of the model, thereby causing a small pre-load on the
bucket in the working plane. The resulting motion is not significantly effected
by this error.
Discussion and
conclusions
In the
first two cases, modelling the elements of the steel structure as rigid, does
not lead to deviations from the specified path of the magnitude that the
deflection of the prismatic beam implied. The primary cause of this phenomenon
is that although the steel structure does bend, this bending takes place after
the bucket tip has been placed in the digging position. When force is applied
the structure bends but this leads to a very small alteration of the boom and
stick angles. The deflection of the structure is only visible in a slight lag
in the horizontal plane. This lag is slightly larger than the error in the
vertical direction but does not exceed 2 cm. The resulting differences imply
that direct bending of the steel structure can be ignored in the design of
control systems for this type of machine.
The other
cases show that the structure is capable of significant displacements in the
absence of control inputs. As can be seen in the appendix this in turn leads to
large variations in the cylinder forces. This can lead to the pressures in the
cylinders approaching zero in cases where the digging forces approach the
maximum. Modern machines are fitted with relief systems that deliver extra
fluid to combat the cavitation that could then occur. As a consequence the
resulting displacement would be even larger as the equilibrium positions of the
cylinder pistons change.
As was said
the motion in cases 3 and 4 is assumed to be free of the ground. Digging in a
continuous soil will of course limit the possible motions, but in for example
loose rock there can be quite large spaces along the digging profile. The
digging forces can also vary considerably. It will be necessary in future to
determine what the effects of these situations are.
As a result
we can state that accurate control of a backhoe's motion can be achieved using
a rigid body approach as the flexibility of the steel structure can be
disregarded. This is certainly true when the effects of pontoon motion on the
achievable accuracy are taken into account.
Abbreviations
A = Area, plate cross section [m2] b = Width of the plate [m]
Ca
= Experimentally deter. mass
coefficient [-] Cf = Flow coefficient [-]
CD = Experimentally deter. drag
coefficient [-] CM = Experimentally deter. inertia
coefficient [-]
E = Specific energy [kPa] F = Force [N]
h = Height [m] K = Stiffness coefficient [N/m]
l = (plate) length [m] p = Pressure [kPa]
u = Flow speed [m/s] V = Volume [m3]
w = Viscous friction term [Ns/m] x = Relative speed of with respect
to the water [m/s]
y’ = Piston speed [m/s] r =
Density [kg/m3]
mS =
Static coefficient of friction [-] sgn() = Sign function [-]
Bibliography
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(1995). Analytical robotics and
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water,
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University Press.
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Proceedings XIIIth World
Dredging Congress 1992, EADA, Bombay.
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Appendix
Case 3
Boom cylinder force
Stick cylinder force
Bucket cylinder force
Case 4
Displacement bucket tip x-direction
Displacement bucket tip y-direction
Boom cylinder force
Stick cylinder force
Bucket cylinder force