ArticlePDF Available

Kinematics of mini hydraulic backhoe excavator - part II

Authors:
  • Ganpat University, U.V.Patel College of Engineering, Mehsana, Gujarat, India

Abstract and Figures

An excavator is a typical hydraulic heavy-duty construction machine used in hazardous, worst working conditions, severe weather, and dirty areas where it is very difficult to operate machine by human operator. The excavator operations require coordinated movement of boom, arm and bucket in order to control the bucket tip position to follow a desired trajectory. This can be achieved through the implementation of automatic control system for excavation task and it required understanding of kinematics of backhoe excavator. This paper focuses on the mathematical kinematic modelling of the backhoe excavator attachment to understand relations between the position and orientation of the bucket and spatial positions of joint-links, helpful for improving the operating performance of the backhoe excavation machine. This paper covers the inverse kinematics, differential motions, inverse Jacobian, and static model of backhoe excavator. The kinematic model developed for autonomous excavation of light duty construction work can be applied for heavy duty or all types of backhoe excavators.
Content may be subject to copyright.
Int. J. Mechanisms and Robotic Systems, Vol. 1, No. 4, 2013 261
Copyright © 2013 Inderscience Enterprises Ltd.
Kinematics of mini hydraulic backhoe
excavator – part II
Bhaveshkumar P. Patel*
Mechanical Engineering Department,
JJT University, Chudela,
Dist. Jhunjhunu-333001, Rajasthan, India
E-mail: bppmech@gmail.com
*Corresponding author
J.M. Prajapati
Faculty of Technology and Engineering,
Maharaja Sayajirao University of Baroda,
Vadodara – 390002, Gujarat, India
E-mail: drjmprajapati@gmail.com
Abstract: An excavator is a typical hydraulic heavy-duty construction machine
used in hazardous, worst working conditions, severe weather, and dirty areas
where it is very difficult to operate machine by human operator. The excavator
operations require coordinated movement of boom, arm and bucket in order to
control the bucket tip position to follow a desired trajectory. This can be
achieved through the implementation of automatic control system for
excavation task and it required understanding of kinematics of backhoe
excavator. This paper focuses on the mathematical kinematic modelling of the
backhoe excavator attachment to understand relations between the position and
orientation of the bucket and spatial positions of joint-links, helpful for
improving the operating performance of the backhoe excavation machine. This
paper covers the inverse kinematics, differential motions, inverse Jacobian, and
static model of backhoe excavator. The kinematic model developed for
autonomous excavation of light duty construction work can be applied for
heavy duty or all types of backhoe excavators.
Keywords: trajectory; bucket configuration; inverse kinematic; differential
motion; backhoe excavator.
Reference to this paper should be made as follows: Patel, B.P. and
Prajapati, J.M. (2013) ‘Kinematics of mini hydraulic backhoe excavator –
part II’, Int. J. Mechanisms and Robotic Systems, Vol. 1, No. 4, pp.261–282.
Biographical notes: Bhaveshkumar P. Patel completed his graduation in
Mechanical Engineering from S.V. National Institute of Technology, Surat,
Gujarat, India in the year 2002 and post-graduation (ME) with specialisation in
CAD/CAM from L.D. College of Engineering, Ahmedabad, Gujarat, India in
the year 2006 with good academic record. He has more than six years industrial
experience and seven years experience in the academe. He is currently working
as an Assistant Professor in Mechanical Engineering Department of U.V. Patel
College of Engineering, Ganpat University, Kherva, Dist. Mehsana, Gujarat,
India. He has guided 12 MTech students in their dissertation. There are 15
research papers published in various international journals in the field of design
262 B.P. Patel and J.M. Prajapati
to his credit. He has a total of 29 research papers presented in national and
international conferences and published in their proceedings.
J.M. Prajapati has completed his graduation in Mechanical Engineering and
PG (ME) with specialisation in Machine Design from S.P. University, V.V.
Nagar, Gujarat, India in the year 1995 and 2003 respectively. He has
successfully completed his PhD in Machine Design from HNG University,
Patan, Gujarat, India in the year of 2009. He has more than two years industrial
experience and 15 years experience in the academe. He is currently working as
an Associate Professor in the Faculty of Technology and Engineering, The
M.S. University of Baroda, Vadodara. Gujarat, India. He has guided three
MTech dissertations. He has 12 research papers published in various
international journals in the field of machine design and robotics. He has ten
research papers presented and published in national and international
conferences. He has many expert lectures delivered in the area of design
optimisation and robotics at various institutions in India.
1 Introduction
Excavation is of prime importance in mining, earth removal and general earthworks.
Backhoe excavators are widely used for most arduous earth moving work in engineering
construction to excavate below the natural surface of the ground on which the machine
rests (Patel and Prajapati, 2011). Hydraulic system is used for operation of the machine
while digging or moving the material (Mehta Gaurav, 2008). Excavation productivity
(amount of work done), efficiency (cost of work done in terms of labour and machinery)
and operator safety, particularly in underground mining or during the removal of
hazardous waste, are constantly under pressure from industry. Full or partial automation
may offer the possibility of improving each metric but has been only slowly accepted by
industry. After decades of increases in machine size and power, practical limits are now
being approached and automation is being sought for further improvements. Furthermore,
computing and sensing technologies have reached to a stage where they can affordably
and reliably be applied to automatic excavation. Beyond the industrial arena, which is
motivated mainly by economic considerations, automated excavators are needed in
workplaces that are hazardous to humans (Oza, 2006). An excavator is comprised of
three planar implements connected through revolute joints known as the boom, arm, and
bucket, and one vertical revolute joint known as the swing joint (Cannon, 1999).
Kinematics is the science of motion which treats motion without regard to the forces that
cause it. Within the science of kinematics one studies the position, velocity, acceleration,
and all higher order derivatives of the position variables (with respect to time or any other
variables) (Craig, 1989). The excavator linkage, however, is a complex link mechanism
whose motion is controlled by hydraulic cylinders and actuators. To programme the
bucket motion and joint-link motion, a mathematical model of the link mechanism is
required to refer to all geometrical and/or time-based properties of the motion. Kinematic
model describes the spatial position of joints and links, and position and orientation of the
bucket. The derivatives of kinematics deal with the mechanics of motion without
considering the forces that cause it. The basic problem in the study of mechanical link
mechanism is of computing the position and orientation of bucket of the backhoe
attachment when the joint angles are known, which is referred to as forward kinematics.
Kinematics of mini hydraulic backhoe excavator – part II 263
The inverse kinematics problem is, thus to calculate all possible sets of joint angles,
which could be used to attain a given position and orientation of the bucket tip of the
backhoe attachment. The problem of link mechanism control requires both the direct and
inverse kinematic models of the backhoe attachment of the excavator (Mittal and
Nagrath, 2003). The kinematic modelling helpful to follow the defined trajectory as well
as digging operation can be carried out successfully at required location of the terrain
using proper positioning and orientation of the bucket and ultimately digging task can be
automated.
The mathematical direct kinematic model and related work with their limitations are
described in the part I of this paper. As concluded from related work mentioned in part I
of the paper, the kinematic model of Koivo (1994) gives a complete kinematic
relationship for the geometry of a hydraulic excavator assuming in three degrees of
freedom. But a complete kinematic relationship for the geometry of the backhoe for four
degrees of freedom has not been presented so far, and this is one of the areas of research
reported in this paper. This paper includes the the inverse kinematics, differential motions
for velocity and acceleration, inverse Jacobian, and static model of backhoe excavator
and their MATLAB codes also developed. This can be helpful to automate the excavation
operation.
2 Affixing frames to the links
Fundamentally, a backhoe excavator has five links starting from the fixed link or base
link, swing link, boom link, arm link (dipper link), and bucket link. These links are
connected to each other by joints, which allow revolute motion between connected links
each of which exhibits just one degree of freedom. This leads to the four degrees of
freedom R-RRR configuration of the backhoe, where R stands for a revolute joint.
Figure 1 describes the schematic side view of the backhoe excavator and frame
assignments. To develop kinematic relations for the geometry of the backhoe; firstly the
coordinate frames will be assigned to the backhoe excavator links.
To analyse the motion of the backhoe excavator (Figure 1) for performing a specific
task, it becomes necessary to define a world coordinate system to describe the position
and orientation of the bucket (collectively known as configuration of the bucket). A
right-hand Cartesian coordinate system Xw Yw Zw is chosen, and its origin is placed at an
arbitrary point on the ground level in the workspace of the backhoe excavator. After
assigning the world coordinate frame the local coordinate frames for all links are
assigned by following the DH guideline for link frame assignment algorithm (Mittal and
Nagrath, 2003).
The kinematic equations are the mathematical equations those relate the position and
orientation of the bucket (bucket configuration) to the joint variables (joint angles in our
case) or to the lengths of the piston rods in the hydraulic actuators. If the lengths of the
piston rods in the actuators or the joint angles are given, the bucket configuration can be
determined by the direct or forward kinematic equations. The complete direct kinematic
model is described in the part I of the paper. The direct kinematics of backhoe excavator
includes the bucket frame transformation matrix and the relation between lengths of the
piston rods in actuators and joint angles. The next section covers the complete inverse
kinematic model for the backhoe excavator.
264 B.P. Patel and J.M. Prajapati
Figure 1 Schematic view of a backhoe and frame assignments (see online version for colours)
3 Inverse kinematics of backhoe excavator
Inverse kinematic model determines the joint variable values, and the lengths of the
piston rods in the actuators corresponding to the specified position and orientation of the
bucket with respect to the base coordinate frame (Koivo, 1994). So, the inverse kinematic
model for the backhoe can be defined as “The determination of all possible and feasible
sets of joint variables, which would achieve the specified configuration of the bucket of
the backhoe with respect to the base frame”. Here assuming that, the coordinate of point
A3 in the base coordinate frame {0} are known, i.e., 3333
000
0[ 1],
x
yz T
AAAA
PPPP=, we
can find the joint variables (in our case these are joint angles) θ1, θ2, θ3, and θ4, and the
piston rod lengths of the actuators ST, UV, A5A6, A7A8, A9A10. The whole inverse
kinematic modelling task will now be divided into two sub-sections, one section
to find the joint angles in terms of bucket configuration, and other section to find
the piston rod lengths of the actuators in terms of the joint angles. As it is known that
the solution of the inverse kinematic problem in robotics is a difficult task as
compared to the direct kinematic problem (Mittal and Nagrath, 2003). But over
the period of time the solution techniques to solve the inverse kinematic problem have
been developed by the researchers, and the closed form solution is one of them. The
‘closed form’ in our context means a solution method based on analytical algebraic or
kinematic approach, giving expressions for solving unknown joint variables. We will
follow the closed form solutions for the solution of an inverse kinematic problem as
given in Mittal and Nagrath (2003). Equations (1) to (27) covers in part I of the paper,
whereas here the equations start from equation (28) provides continuity of the kinematic
mathematical modelling of backhoe excavator and for better understanding to the
readers.
Kinematics of mini hydraulic backhoe excavator – part II 265
3.1 Joint angles in terms of bucket configuration
Firstly, the joint angles θ1, θ2, θ3, and θ4, will be determined in terms of the bucket
configuration. It is assumed that the digging task is performed on the vertical plane
containing the line segment O0O1. The point A3 can be expressed in the first coordinate
frame as follows:
33 3
13 10
30
1
AA A
P== PT TP (28)
where 3
3
A
P = coordinates of point A3 in the coordinate frame {3}.
where 3333
0000
[1],
x
yz T
AAAA
PPP=P and 333
00 0
,,
xy z
A
AA
PPP are x, y and z-coordinates
of point A3 in the base coordinate frame respectively, and 3
3[0 0 0 1] ,
T
A=P and
1T0 = [0T1]–1, 1T3 = 1T2 2T3. Let us first find out 33
11
0 0.
A
A
=PTP This gives,
33
3
3
3
3
333
00
0111
11 1
0
0
1
000
11 11
0
0010
00
0001 11
xy
xAA
A
z
yA
A
Azxy
AAA
CPSPa
CS a P
P
P
SC PSP CP
++ −
⎡⎤
⎡⎤
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
==
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
⎣⎦
⎣⎦
P (29)
The equation (28) rewritten as:
[]
333
1
1123
2
AAA
⋅=⋅ TPTP (30)
where 1T2 and 2T3 are specified by equations (3), (4) and (5). The equation (30) yields:
()
()
33 3
33 3
33
00 0
1112 22
33
00 0 33
1112 2
00
11
0
1
1
xy z
AA A
xy z
AA A
xy
AA
CP SP aC PS a aC
aS
CP SP aS PC
PS PC
⎡⎤
+−+ −
⎡⎤
⎢⎥
⎢⎥
−++
⎢⎥
⎢⎥
∴=
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
⎣⎦
(31)
So, in the equation (31) LHS the first two equations are in terms of the unknown joint
variables θ1, and θ2, and in the RHS the first two equations are in terms of unknown joint
variable θ3. But by comparing the third equations of both the sides in equation (31) gives,
()
3
33
3
0
100
10
tan tan 2 ,
yAyy
AA
xA
P
θAPP
P
⎛⎞
∴= =
⎜⎟
⎝⎠
(32)
So, equation (32) gives the value of the joint 1 angle θ1 in terms of the bucket
configuration. Now, for solving the joint 2 angle θ2 in terms of bucket configuration, let
us first square the both sides of the first two equations of the equation (31), and this
yields,
()
()
()
()
()
33
222
002
22 22 2 3
22
zz
AA
Pa s xa c P x a a+=++ (33)
Now, in equation (33), let us assume, 3
02
(2 ) sin ,
zA
Pa r
=
⋅Φ and 2
(2 ) cos .xa r
⋅Φ So
this gives, 3
20 2 2
2
4( ) [( ) ]
zA
raPx=+ and 3
0
1
tan .
zA
P
x
Φ=
By substituting these
266 B.P. Patel and J.M. Prajapati
relations in equation (33), the resulting equation can be solved for (θ2Φ) to obtained
the expression for θ2.
()
()
{}
()
()
()
{}
()
()
()
3
33
3
0
1
2
1
22
22
22
2
0202
223
1
222
02
23
tan
4
tan
zA
zz
AA
zA
P
θx
aPx Pxaa
Pxa a
⎡⎤
=⎢⎥
⎣⎦
⎡⎤
⎡⎤
⎢⎥
+− ++ −
⎢⎥
⎣⎦
⎢⎥
+⎢⎥
++ +
⎣⎦
(34)
This equation (34) gives the value of the joint 2 angle θ2 in terms of bucket configuration,
because the rest of the values in equation (34) are known to us. Now, the joint 3 angle θ3
can be obtained by dividing the equation (2) by equation (1) of the equation (31), and this
yields,
()
()
33 3
33 3
00 0
1112 2
1
300 0
1112 22
tan
xy z
AA A
xy z
AA A
CP SP aS PC
θCP SP aC PS a
⎡⎤
−++
=⎢⎥
+−+ −
⎢⎥
⎣⎦
(35)
This equation (35) gives the value of joint 3 angle θ3 in terms of the bucket configuration.
So if the position of the point A3 is known, the joint angles θ1, θ2, and θ3 can be
determined by equations (32), (34), and (35) respectively. But to find the joint 4 angle θ4
this is not the case. One can determine the joint 4 angle θ4, if the orientation of the bucket
is known. When the coordinates of point A4 = O4 on the centre of the edge of the bucket
teeth and the orientation angle θ234 = θ2 + θ3 + θ4 of bucket relative to the X0-axis (or
equivalently the X1-axis ) are known, the solution to the inverse kinematic model is still
given by equations (32), (34), and (35). However, the following expressions are now
submitted into these equations for the components of 3
0:
A
P
33 34
34
00 00
1 4 234 1 4 234
004234
, ,
xx yy
AA AA
zz
AA
PPCaCPPSaC
PPaS
=− =−
=− (36)
These equations are obtained from the equations (9) and (10). Where, the bucket
orientation θ234 is given and 4444
0000
[1]
x
yz T
AAAA
PPP=P specifies the location of
point A4 in the base coordinate system.
The orientation of the bucket may, in some applications, be specified by an
alternative manner: Figure 2 shows the bottom plate of the bucket is defining as a plane,
and this plane contains the teeth of the bucket. So, the angle ρ is defined as the angle
made by this plane with the horizontal line, known as the digging angle of the bucket
(Koivo, 1994). The angle λ is the angle made by the plane defined by the bucket bottom
plate with the x4-axis as shown in Figure 2. From the geometry shown in Figure 2, it can
be written,
423
3θρλπθθ∴=++ −− (37)
Equation (37) gives the value of the joint 4 angle θ4 in terms of the bucket orientation.
Kinematics of mini hydraulic backhoe excavator – part II 267
Figure 2 Bucket orientation
3.2 The length of the piston rods of actuators in terms of joint angles
This part of the inverse kinematic problem is to determine the lengths of the piston rods
(the line segments between the attachment points of the actuators) when the values of the
joint angles are given. The piston rod length ST of the actuator 1 is determined by the
equation (12) when the joint 1 angle θ1 is known. The length of the piston rod of actuator
2 = UV is determined by the equation (15) when the joint 1 angle θ1 is known. The length
of the piston rod of actuator 3 = A5A6 is determined by the equation (17) when the joint 2
angle θ2 is known. The length of the piston rod of actuator 4 = A7A8 can be determined by
equation (19) when the joint 3 angle θ3 is known. But the length of the piston rod of
actuator 5 = A9A10 is tricky as compared to other actuator lengths to be determined.
Firstly, let us determine the length of the piston rod of actuator 5 = A9A10 in terms of the
unknown angle ζ1 as given in equation (21), and then determine the unknown angle ζ1 in
terms of the joint 4 angle θ4.
Now substituting the value of the angle ζ2 as ζ5ζ1 in equation (27), the angle ζ1 can
be determined in terms of the angle ζ5, by using the standard method as used to determine
the joint 2 angle θ2, and thus in terms of the joint angle θ4 as follows if angle
ζ5 = –π + η1 + η2 + θ4ζ3 and ζ2 = ζ5ζ1.
(
)
(
)
(
)
()()()
{}
()()
{}
()()()
()()()()()
{}
10 11 3 11 5
1
1
10 11 3 11 5 10 12 3 12
1
222 2
10 11 3 11 5
22
10 11 3 11 5 10 12 3 12 2
1
2
sin
tan cos
4sin
4cos
tan
AA AA ζ
ζAA AA ζAA AA
AA AA ζ
AA AA ζAA AA x
x
⎡⎤
=⎢⎥
⎣⎦
⎡⎤
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
+−
⎢⎥
⎣⎦
⎢⎥
⎣⎦
(38)
268 B.P. Patel and J.M. Prajapati
where x2 = (A10A11)2 + (A3A11)2 – (A10A12)2 – (A3A12)2 is constant and thus equation (38)
determines the angle ζ1 in terms of the joint 4 angle θ4. Note that the angle ζ3 has been
assumed to be known from the encoder. Then, the length of the piston rod of actuator
5 = A9A10 can be calculated by (21). When the joint shaft angles are known, the lengths
ST, UV, A5A6, A7A8, and A9A10 of the piston rods in hydraulic actuators can be determined
by equations (12), (15), (17), (19), and (21) respectively. Thus, relationship between the
joint angles and the lengths of the actuators has been established by inverse kinematic
equation. The complete inverse kinematic relations of the backhoe excavator are
presented by equations (12), (15), (17), (19), (21), (32), (34), (35) and (37).
4 Differential motions (velocity) of the backhoe excavator
In this study the use of Jacobian and use of derivative of kinematic equations for
determining the bucket velocity in terms of the joint velocities are presented. But at
certain location in joint space the Jacobian may lose its rank and become ill, these
locations are collectively known as Jacobian singularities. In addition Jacobian is also
useful for describing mapping between forces applied to the bucket and resulting torques
at the joints known as the statics. Firstly, let us find out the linear and angular velocities
of each link by derivative of kinematic equations. The joint accelerations are not the part
of this paper but it is covered in the paper of dynamics of backhoe excavator.
4.1 Determination of bucket velocity in terms of joint angle velocities by taking
the time derivative of kinematic equations
Firstly, the linear velocity components are derived for each link of the backhoe: swing
link, boom link, arm link, and the bucket link, and then angular velocity components for
each of the four links are derived. For determining the linear velocity of any link i with
respect to the base frame {0} = 0vi following procedure should be used (Mittal and
Nagrath, 2003).
() ()
00 0i
ii ii
dd
dt dt
== vD TD (39)
From equation (39) linear velocity of link i can be determined. Where 0Di is a
displacement vector from frame {i} to base frame {0}, 0Ti is a homogeneous
transformation matrix. iDi is a displacement vector from frame {i} to frame {i} and can
be given as [0 0 0 1]T in homogeneous coordinates.
But,
0012 1
12 3
.
i
ii
=………TTTT T
By putting this equation into equation (39) leads to;
()
00
1
i
i
iijj
j
j
θ
θ
=
=
vTD (40)
where
j
θ
is the speed of rotation of link j. With the use of equation (40) the linear
velocity of link 1 or swing link of the backhoe can be given by:
Kinematics of mini hydraulic backhoe excavator – part II 269
11 11 1
11 11 2
011
3
4
000
000
0 0 000
0 0 000
as as θ
ac ac θ
θθ
θ
−− ⎧
⎧⎫⎡ ⎤
⎪⎪
⎪⎪⎢ ⎥
⎪⎪ ⎪
⎢⎥
==
⎨⎬ ⎨
⎢⎥
⎪⎪ ⎪
⎢⎥
⎪⎪ ⎪
⎩⎭⎣ ⎦
⎩⎭
v (41)
The linear velocity of the boom of a backhoe can be given by:
()
()
()
()
11 22 1 22 1 2
11 22 1 221 2
02
222
122 1 221 1
122 1 221 2
22 3
4
0
00
00
000
0000
sθac a ascθ
cθac a assθ
acθ
sac a asc θ
cac a ass θ
ac θ
θ
⎧⎫
−+
⎪⎪
+−
⎪⎪
=⎨⎬
⎪⎪
⎪⎪
⎩⎭
⎧⎫
⎡⎤
−+⎪⎪
⎢⎥
−+⎪⎪
⎢⎥
=⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎪⎪
⎣⎦
⎩⎭


v
(42)
The linear velocity of the arm of a backhoe can be given by:
()
()
(
)
()
(
)
()
1 323 22 1 1 323 22 1 323 1
1323 22 1 1323 22 1323 2
03
323 22 323 3
4
0
0
0
0
0
000
sac ac a cas as cas θ
cac ac a sas as sas θ
ac ac ac θ
θ
⎧⎫
⎧⎫−++−+⎪⎪
⎪⎪
++ − +
⎪⎪
=⎨⎬
+
⎪⎪
⎪⎪
⎩⎭
⎩⎭
v (43)
The linear velocity of the bucket of a backhoe can be given by:
(
)
()
(
)
()
114234
1 4 234 3 23 2 2 1 4 234 3 23 1
11 4 234
1 4 234 3 23 2 2 1 4 234 3 23 2
04
4 234
4 234 3 23 2 2 4 234 3 23 3
4
()
()
0
00
00
sb cas
cas as as cas as θ
cb sas
sas as as sas as θ
ac
ac ac ac ac ac θ
θ
−−
⎧⎫
−++−+ ⎪⎪
⎪⎪
−++−+
⎪⎪
=⎨⎬
++ +
⎪⎪
⎪⎪
⎩⎭
⎩⎭
v (44)
where b = a4c234 + a3c23 + a2c2 + a1.
Here, all four equations from (41) to (44) can be used to determine the linear
velocities of the four backhoe links (swing link, boom, arm, and the bucket respectively)
in homogeneous coordinates. The procedure to find the angular velocities is given as
follows (Mittal and Nagrath, 2003):
1
1i
ii i
ωω ω
=+ (45)
where ωi is the angular velocity of link i, i–1ωi is the angular velocity of link i due to the
link i – 1 and can be determined by:
10
11
iiiii
ωzθ
−−
=
R (46)
For the backhoe excavator, the angular velocity of the swing link does not contribute to
the angular velocities of other three links, because the angular velocities of boom, arm,
and bucket only come into the picture while digging or excavating the ground. In other
270 B.P. Patel and J.M. Prajapati
words when swing link rotates other three links do not rotate and vice-versa. So, while
calculating the angular velocities of boom, arm, and bucket (the axis of rotation for these
three links are always parallel and thus the angular velocity of the boom contribute to the
angular velocities of arm and the bucket, while the rotational axis of the swing link is not
parallel but it is perpendicular to the axes of rotation of other three links and thus do not
directly contribute to the angular velocities of the other three links) the equation (46) will
be modified as:
11
11
iiiii
ωzθ
−−
=R (47)
Using equations (45) and (46) and assuming the angular velocity of the base link as zero
(stationary) the angular velocity of the swing link can be given by:
1
2
1
3
4
0000
0000
1000
0000
θ
θ
ωθ
θ
⎧⎫
⎡⎤
⎪⎪
⎢⎥
⎪⎪
⎢⎥
=⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎪⎪
⎣⎦
⎩⎭
(48)
Using equations (45) and (47) the angular velocity of boom can be given by:
1
2
2
3
4
0000
0000
0100
0000
θ
θ
ωθ
θ
⎧⎫
⎡⎤
⎪⎪
⎢⎥
⎪⎪
⎢⎥
=⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎪⎪
⎣⎦
⎩⎭
(49)
Using equations (45) and (47) the angular velocity of arm can be given by:
1
2
3
3
4
0000
0000
0110
0000
θ
θ
ωθ
θ
⎧⎫
⎡⎤
⎪⎪
⎢⎥
⎪⎪
⎢⎥
=⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎪⎪
⎣⎦
⎩⎭
(50)
Using equations (45) and (47) the angular velocity of bucket can be given by:
1
2
4
3
4
0000
0000
0111
0000
θ
θ
ωθ
θ
⎧⎫
⎡⎤
⎪⎪
⎢⎥
⎪⎪
⎢⎥
=⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎪⎪
⎣⎦
⎩⎭
(51)
Here, all four equations from (48) to (51) can be used to determine the angular velocities
of the four backhoe links in homogeneous coordinates. So, the bucket Cartesian velocity
vector (combination of both linear and angular velocity components) can be given by (by
removing the last rows from bucket linear velocity vector and from bucket angular
velocity vector because it only represents the homogeneous coordinates):
Kinematics of mini hydraulic backhoe excavator – part II 271
()
()
4 234 3 23
1 1 1 4 234 3 23 1 4 234
22
4
44 234 3 23
1 1 1 4 234 3 23 1 4 234
422
44 234 3 23 2 2 4 234 3 23 4 234
4
4
()
()
0
00 00
00 00
01 11
x
y
z
x
y
z
as as
sb c c as as cas
as
v
vas as
cb s s as as sas
vas
ωac ac ac ac ac ac
ω
ω
⎡+
⎛⎞
−− + −
⎜⎟
+
⎡⎤ ⎝⎠
⎢⎥ +
⎛⎞
⎢⎥ −+
⎜⎟
⎢⎥ +
⎝⎠
=
⎢⎥ ++ +
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
1
2
3
4
θ
θ
θ
θ
⎢⎥
⎢⎥
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
⎢⎥
⎢⎥
⎢⎥
(52)
where b = a4c234 + a3c23 + a2c2 + a1.
The physical interpretation of the equation (52) is: out of three angular velocities of
the bucket, the bucket angular velocity with respect to the joint 4 Z4-axis can only be
controlled, and it also depends on the angular velocities of arm, and boom links, and this
is the actual case in the backhoe excavator as the bucket can only be rotated with respect
to its joint axis.
4.2 Determination of bucket velocity in terms of joint angle velocities using
Jacobian matrix
There are certain methods are used to generate the Jacobian, which are vector product
method, differential transform method, and direct differential method. In vector product
method, to calculate the ith column of the Jacobian matrix, we need to find two vectors.
These vectors are position of origin and the joint axis unit vector of the frame attached to
link (i 1), both expressed in the base frame. Practically, we find the Jacobian matrix
column by column. Each column is a Jacobian generating vector (Jazar, 2010). The
differential transform is an analytic method for solving differential equations. It can be
apply for linear and non-linear problems. This method constructs an analytical solution in
the form of a polynomial. By using differential transform method, we can get a series
solution, in practice a truncated series solution. The concept of differential transformation
is derived from the Taylor series expansion and it is iterative process (Biazar and Eslami,
2010; El-Shahed, 2008). Whereas we have used direct differential method, in which the
Cartesian velocities (linear as well as angular) of the end effector (for our case it is bucket
tip) are linearly related to the joint velocities. The relationship between differential joint
motions with differential changes in end-effector position and orientation is investigated
here. In the presented method, the derivative of the coordinates and orientation of end
effector point (bucket tip) describe three linear and three angular velocities of the end
effector point. This method is very simple and computationally easy compares to other
methods.
For our case (i.e., backhoe excavator) the end-effector is the bucket, and if we write
Cartesian bucket velocity vector as Vb(t), where t is function of time, Jacobian matrix as
J(θ), and the joint speed vector as ,θ
then the bucket velocity can be written as:
() ( )
btθθ=VJ (53)
where Vb(t) is a 6 × 1 Cartesian bucket velocity vector (three linear and three angular
velocities), J(θ) is a 6 × 4 backhoe Jacobian matrix (because there are only four joint
272 B.P. Patel and J.M. Prajapati
angles or four degrees of freedom under consideration), and θ
is a 4 × 1 vector of four
joint velocities 1,θ
2,θ
3,θ
and 4.θ
. Equation (53) can be written in column vectors of
the Jacobian, that is,
(
)
(
)
(
)
(
)
11 2 2 3 3 4 4
()
bt⎡⎤
=⎣⎦
VJθJθJθJθθ (54)
In equation (54) Ji(θ) for i = 1 to 4 is the ith column of the Jacobian matrix. The ith
column of the Jacobian matrix at the bucket edge point or tool point A4, can be given by:
()()()()()()
()()() ()
444444
444 4
00
00
0
()
0
() 0 0
T
xyzxyz
AAAAAA
i
iiiiii
T
xyz z
AAA A
i
iii i
PPPPPP
θθθθθθ
PPP θ
θθθ θ
⎡⎤
∂∂∂
=⎢⎥
∂∂∂
⎣⎦
⎡⎤
∂∂∂ ∂
∴=
⎢⎥
∂∂∂ ∂
⎣⎦
Jθ
Jθ
(55)
In equation (55) the fourth row and fifth row of the matrix Ji(θ) is taken as zero because
bucket is restricted to be rotated about X4- and Y4-axes, and allowed to be rotated only
about Z4-axis. This means bucket of the backhoe can be rotated about its own joint axis
(Z4) only. Moreover; bucket orientation angle depends on the orientation angles of boom
and arm also, but does not depend on the orientation angle of the swing link because
while excavating the swing link remains steady and while swinging the whole
mechanism, all other links boom, arm, and the bucket remain steady. So, the bucket
orientation angle can be written as:
4234 2 3 4
zA
θθ θθθ==++ (56)
By using the equations (55), and (56) the Ji matrices for the backhoe are found as
follows:
()
()
(
)
()
11 4 234 3 23 2 2 1 4 234 3 23
11 4 234 3 23 2 2 1 4 234 3 23
4 234 3 23 2 2 4 234 3 2 3
12 3
4
()
()
0, , , and
000
000
011
sb cas as as cas as
cb sas as as sas as
ac ac ac ac ac
⎡⎤−++ −+
⎡⎤ ⎢⎥
⎢⎥ −++ −+
⎢⎥
⎢⎥ ⎢⎥
⎢⎥ ++ +
== =
⎢⎥
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
JJ J
J
1 4 234
1 4 234
4234
0
0
1
ca s
sa s
ac
⎡⎤
⎢⎥
⎢⎥
⎢⎥
=⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
By assembling all the Ji matrices, the final Jacobian 6 × 4 matrix is given as follows:
Kinematics of mini hydraulic backhoe excavator – part II 273
()
(
)
()()
1 1 4 234 3 23 2 2 1 4 234 3 23 1 4 234
1 1 4 234 3 23 2 2 1 4 234 3 23 1 4 234
4 234 3 23 2 2 4 234 3 23 4 234
()
()
0
() 00 00
00 00
01 11
sb c as as as c as as cas
cb s as as as s as as sas
ac ac ac ac ac ac
θ
⎡⎤−− + + +
⎢⎥
−++−+
⎢⎥
⎢⎥
++ +
=⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
J
where b = a4c234 + a3c23 + a2c2 + a1.
The bucket velocity Vb now can be determined in terms of the joint angle velocities
by Jacobian matrix as follows:
()
()
4 234 3 23
1 1 1 4 234 3 23 1 4 234
22
4
44 234 3 23
1 1 1 4 234 3 23 1 4 234
422
44 234 3 23 2 2 4 234 3 23 4 234
4
4
() ( )
()
()
0
00 0
b
x
y
z
x
y
z
t
as as
sb c c as as cas
as
v
vas as
cb s s as as sas
vas
ωac ac ac ac ac ac
ω
ω
=
+
⎛⎞
−− + −
⎜⎟
+
⎡⎤ ⎝⎠
⎢⎥ +
⎛⎞
⎢⎥ −+
⎜⎟
⎢⎥ +
⎝⎠
=
⎢⎥ ++ +
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
VJθθ
1
2
3
4
0
00 00
01 11
θ
θ
θ
θ
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
⎢⎥
⎢⎥
⎢⎥
⎣⎦
(57)
Thus, equation (57) determines the velocity of bucket in terms of the joint angle
velocities, and joint angles and it is same as equation (52). But in actual practice all three
linear velocities and only one angular velocity ω4z of the backhoe bucket are possible to
control. Thus, the equation (57) can be rewritten as:
()
()
4 234 3 23
11 1423432314234
22
41
44 234 3 23
1 1 1 4 234 3 23 1 4 234
422
44 234 3 23 2 2 4 234 3 23 4 234
()
()
0
01 11
x
y
z
z
as as
sb c c as as cas
as
vθ
vas as θ
cb s s as as sas
vas
ωac ac ac ac ac ac
⎡+ ⎤
⎛⎞
−− + −
⎢⎥
⎜⎟
+
⎡⎤ ⎝⎠
⎢⎥
⎢⎥
⎢⎥
+
⎛⎞
⎢⎥
⎢⎥
=−+
⎜⎟
⎢⎥ +
⎢⎥
⎝⎠
⎢⎥
⎢⎥
++ +
⎣⎦
⎢⎥
⎢⎥
⎣⎦
2
3
4
θ
θ
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
(58)
()
(
)
()()
1 1 4 234 3 23 2 2 1 4 234 3 23 1 4 234
1 1 4 234 3 23 2 2 1 4 234 3 23 1 4 234
4 234 3 23 2 2 4 234 3 23 4 234
()
()
() 0
01 11
sb c as as as c as as cas
cb s as as as s as as sas
ac ac ac ac ac ac
⎡⎤−− + + − +
⎢⎥
−++−+
⎢⎥
=⎢⎥
++ +
⎢⎥
⎢⎥
⎣⎦
Jθ (59)
The equation (58) gives the velocity of the bucket (three linear and one angular) in terms
of the joint angles, where b = a4c234 + a3c23 + a2c2 + a1, with new Jacobian (a square
matrix) as equation (59).
274 B.P. Patel and J.M. Prajapati
4.3 Actuator piston velocities in terms of joint speeds
The velocities of the actuator pistons are derived by using time derivatives of both sides
of the equations relating the piston rod length of the actuator and joint angles: equations
(12), (15), (17), (19) and (21) as follows:
By differentiating the equation (12) with respect to time yields the velocity of the
actuator 1 piston in terms of joint 1 angle speed 1
θ
or vice versa:
()
()()
()
{}
()
()
()
{}
()
00 1 1
0
011
1
cos sin
() sin cos
()
ST
OX OT αθ αθ
OT
XS O T αθ αθ
Vθ
ST
⎧⎫
⎡⎤
+−
⎪⎪
⎢⎥
⎨⎬
+− − −
⎢⎥
⎣⎦
=⎪⎪
⎩⎭
(60)
By differentiating the equation (15) with respect to time yields the velocity of the actuator
2 piston in terms of joint 1 angle velocity 1
θ
or vice versa:
()
()
()
{}
()
()
()
{}
()
01 1
0
010 1
1
sin cos
()cos sin
()
UV
OV αθ XU αθ
OV
OV αθ OX αθ
Vθ
UV
⎧⎫
⎡⎤
+− +
⎪⎪
⎢⎥
⎨⎬
−++ +
⎢⎥
⎣⎦
=⎪⎪
⎩⎭
(61)
By differentiating the equation (17) with respect to time yields the velocity of the actuator
3 piston in terms of joint 2 angle velocity 2
θ
or vice versa:
()()
(
)
56
15 16 1 2 2
2
56
sin
()
AA
AA AA πγ γ θ
Vθ
AA
⎧⎫−−
=⎨⎬
⎩⎭
(62)
By differentiating the equation (19) with respect to time yields the velocity of the actuator
4 piston in terms of joint 3 angle velocity 3
θ
or vice versa:
()()
(
)
78
27 28 1 2 3
3
78
sin 3
()
AA
AA AA πδ δ θ
Vθ
AA
⎧⎫
−−
=⎨⎬
⎩⎭
(63)
By differentiating the equation (21) with respect to time yields the velocity of the actuator
5 piston in terms of the angle velocity ζ1 or vice versa, and then ζ1 will be determined in
terms of the joint 4 angle velocity 4:θ
()( )
(
)
910
912 1012 1 1
1
910
sin 2
()
AA
AA A A πε ζ
Vζ
AA
⎧⎫
−−
=⎨⎬
⎩⎭
(64)
Equation (64) gives the bucket actuator 5 piston velocity in terms of the unknown angle
velocity ζ1. Now, let’s establish the relation between this unknown angle velocity ζ1 in
terms of the joint 4 angle velocity 4.θ
We know from the equation (26a) that:
12 1243
ζζ πηηθζ+=+++−
By differentiating this equation with respect to time gives:
12 43
ζζ θζ
+=− (65)
Because the angles η1, η2, and π are constants for the geometry.
Kinematics of mini hydraulic backhoe excavator – part II 275
Also, by differentiating the equation (27) with respect to time leads to:
()( )
()( )
312 1012 1
21
311 1011 2
sin
sin
AA A A ζ
ζζ
AA A A ζ
⎡⎤
=⎢⎥
⎣⎦
(66)
By substituting equation (66) into the equation (65) yields:
()( )
()( )
43
1
312 1012 1
311 1011 2
sin
1sin
θζ
ζAA A A ζ
AA A A ζ
⎡⎤
=⎢⎥
⎧⎫
⎢⎥
+⎨⎬
⎢⎥
⎩⎭
⎣⎦
(67)
Thus, equation determines the angle velocity ζ1 in terms of the joint 4 velocity 4.θ
If the
piston velocities are known then the joint speeds can be determined by the equations (60)
to (64).
4.4 Inverse Jacobian
In actual operation of a backhoe to make the bucket track a specified trajectory with a
given velocity profile, it is required to coordinate individual joint motions. In other
words, for a given bucket velocity Vb the corresponding joint velocities θ
must be found
that will cause the bucket to move at desired velocity. Now it is known from equation
(53) that,
() ( )
bt=
VJθθ
This leads to,
1() ()
b
θt
=
JθV (68)
For the automatic operation of the backhoe excavator, the piston velocities are required
that will cause the bucket to move at a desired trajectory and with a desired velocity.
These required piston velocities VST, VUV, 56
,
A
A
V 78
,
A
A
V and 910
A
A
V can be determined by
the equations from (60) to (67), once their respective joint angle speeds (Joint speed
matrix θ
) are known from equation (68). It is known that for an inverse of a matrix to
exist, it must be a square matrix. The earlier Jacobian matrix was of the size 6 × 4, so it
was not a square matrix. Thus, it was having four controllable joint rates, and six
controllable Cartesian velocities. But we reduced that matrix from 6 × 4 to 4 × 4 as given
by equation (59). This reduced matrix is a square matrix provided that there are only four
controllable joint rates and four controllable Cartesian velocity components, three
Cartesian linear velocities (v4x, v4y, v4z), and one Cartesian angular velocity (ω4z), and this
is acceptable for the jobs to be performed by the backhoe, so now the Jacobian is a 4 × 4
square matrix.
Apart from this J–1 exists only if Jacobian matrix (J) is non-singular at the current
configuration. The backhoe Jacobian J becomes rank deficient or singular at certain
configuration in Cartesian space. In such cases the inverse Jacobian does not exist and
equation (68) becomes an invalid equation. Those backhoe configurations at which J
becomes non-invertible are termed as Jacobian singularities (Mittal and Nagrath, 2003).
276 B.P. Patel and J.M. Prajapati
The computation of the singularities can be carried out by analysing the rank of the
Jacobian matrix. The Jacobian matrix loses its rank and becomes ill conditioned at values
of the joint angles θi at which the determinant of the Jacobian matrix vanishes, that is,
|J(θ) = 0|. In other words the values of joint angles at which the determinant of the
Jacobian matrix attains the value zero, are known as Jacobian singularities. Another way
to find the Jacobian inverse after putting the value of link lengths ai, and joint angles θi
into equation (59), the Gauss Jordan elimination method should be applied to the matrix
to find its inverse.
5 Backhoe static model
The work cycle of the backhoe includes the following operations or motions: Digging the
ground, loading the material into the bucket, and dumping the excavated material into the
dump truck or trolley, and then back to the digging operation again. From all the
operations the most important task is digging as it requires forceful interaction of the
bucket with the ground. While digging bucket is required to exert a force and/or moment
on the ground. The bucket makes the contact with the ground and all joints remain static
for that time. The contact between the bucket and ground results in interactive forces and
moments at the bucket ground interface. So, the static problem of a backhoe is to
determine the relationship between the joint torques and forces exerted by the ground on
the bucket teeth under static equilibrium conditions. So, the joint torques that must be
acting to keep the system in static equilibrium will now be considered. The relationship
between the joint torques and the bucket teeth torque vector can be derived using the
Jacobian in static equilibrium conditions (Mittal and Nagrath, 2003)
T
()
ii
τF=θJ (69)
where τi is a generalised drive torque applied by actuator i to driving joint i. Fi is the
endpoint force and moment vector known as bucket force vector. It is the vector of the
reaction forces and moments from the ground to the bucket. These reaction forces
known as resistive forces can be find by using different soil-tool interaction models based
on principle of soil mechanics. We have used the McKyes and D. Zeng models of
soil-tool interaction to determine the resistive forces exerted during the digging task
but it is not a part of this paper. After finding the reaction force it can be resolved in
three directions x, y and z and can be thought to be acting at the origin of the coordinate
frame {4}.
Assumptions in static modelling:
it is assumed that the joints of the backhoe are frictionless and joint torques are the
net torques that balance the end point force Fi
the reaction force is acting at the origin of frame {4}, and it is resolved in three
directions.
Now, by using equation (57) (the old Jacobian of size 6 × 4), in equation (69) one may
get:
Kinematics of mini hydraulic backhoe excavator – part II 277
()
()
()
()
()
11 1
2 1 4 234 3 23 2 2 1 4 234 3 23 2 2
3 1 4 234 3 23 1 4 234 3 23
4 1 4 234 1 4 234
4234 323 22
4 234 3 23
4234
() ()
0 000
000
001
001
x
y
z
x
y
z
τsb cb
τcas as as sas as as
τcas as sas as
τca s sa s
F
F
ac ac ac F
ac ac η
ac η
η
⎧⎫
⎪⎪
−++−++
⎪⎪
=
⎨⎬−+ −+
⎪⎪
⎪⎪ −−
⎩⎭
++
+
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩⎭
where b = a4c234 + a3c23 + a2c2 + a1. Matrix multiplication leads to:
()
()
()
()
()
()
()
()
4234 323
11
22 1
1
4 234 3 23 4 234 3 23
11
2
22 323 22
3
4 234 3 23
41 1 4 234 3 23
22
4 234 1 1 4 23
xy
xy zz
xy zz
xy
ac ac sF cF
ac a
τas as ac ac
cF sF F η
τas ac ac
τas as
τcF sF a s as F η
as
as cF sF as
+
⎛⎞
−+
⎜⎟
++
⎝⎠
⎧⎫ ++
⎛⎞ ⎛⎞
⎪⎪ −− + +
⎪⎪ ⎜⎟ ⎜⎟
+++
=
⎨⎬ ⎝⎠ ⎝⎠
⎪⎪ +
⎛⎞
⎪⎪ −− + + +
⎩⎭ ⎜⎟
+
⎝⎠
−− +
()
()
4zz
Fη
⎧⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
+
⎪⎪
⎩⎭
(70)
Equation (70) shows the joint torques in terms of the joint angles, when the system is in
equilibrium. The MATLAB codes developed for inverse kinematic model, bucket and
piston velocities model, inverse Jacobian model, and static joint-torque model.
Figure 3 The proposed 3D model of mini hydraulic backhoe excavator
(see online version for colours)
278 B.P. Patel and J.M. Prajapati
6 Results and discussion
The results of the complete kinematic model including inverse kinematics, bucket
velocity, piston rod velocities of actuators, inverse Jacobian and static model of backhoe
are discussed and the results are obtained using the developed MATLAB code. The input
values of the parameters are taken as per Table 1. The data given in the table captured for
maximum breakout force condition. Figure 3 shows the maximum breakout force
condition for the proposed 3D model of the mini hydraulic backhoe excavator.
Table 1 Values of the link parameters and geometry constants used in kinematics of backhoe
excavator’s MATLAB codes
Description Symbol Value Unit
a1 0.430
a2 1.347
a3 0.723
Swing link length, boom link length, arm link
length and bucket link length respectively
a4 0.547
m
α
52.72
γ1 46.23
γ2 28.53
δ1 33.23
δ2 139.54
ε1 197.79
η1 3.32
η2 80.14
Geometry constant angles
ζa 67.43
Degree
XS 0.092
OX 0.21507
OS 0.23386
O
T
0.11556
S
T
0.285
X
U0.092
OU 0.23386
O
V
0.11556
U
V
0.285
A1A5 0.67461
A1A6 0.21783
A5A6 0.70978
A2A7 0.91102
A2A8 0.28480
A7A8 0.86524
A9A12 0.74341
A10A12 0.220
A9A10 0.65907
A3A12 0.13254
A3A11 0.18103
Geometry constant distances, and the piston rod
lengths of the actuators for maximum breakout
force condition
A10A11 0.205
m
Kinematics of mini hydraulic backhoe excavator – part II 279
6.1 Results of inverse kinematic model
If the link lengths, backhoe geometry constants, and the coordinates of point A3 in the
base coordinate frame are known, i.e., 333
000
3
[0 1]
yz T
AAAA
PPPP then the lengths of
piston rods of all actuators, and joint angles to make the bucket hinge point reach the
required position and orientation can be determined from the following MATLAB code
of inverse kinematics. The values used in the MATLAB code are same as used in direct
kinematic model (Table 1) except the values of ST, UV, A5A6, A7A8, and A9A10
because these are the values to be found from the inverse kinematic model, and the values
of the angles ρ, and λ used are: 75.82º and 54.63º.
While using these all values into the MATLAB code of the inverse kinematic model
it will return the values of joint angles and the lengths of the actuators as: θ1 = 0º, θ2 = 15º,
θ3 = 295.462º, θ4 = 359.99º, ST = 0.28507 m, UV = 0.28507 m, A5A6 = 0.70977,
A7A8 = 0.86529, and A9A10 = 0.65916. These values can be compared with the direct
kinematic model as: θ1 = 0º, θ2 = 15º, θ3 = 295.468º, θ4 = 360º, ST = 0.285 m, UV = 0.285
m, A5A6 = 0.70978, A7A8 = 0.86524 and A9A10 = 0.65907.
As can be seen from the results, the differences in the values of joint angles θ1, θ2, θ3
and θ4 are: 0º, 0º, –0.006º, and –0.01º respectively, and the differences in the lengths of the
piston rods in the hydraulic actuators ST, UV, A5A6, A7A8, and A9A10 are: 0.07 mm,
0.07 mm, –0.01 mm, 0.05 mm, 0.09 mm. These differences are very small and thus the
proposed inverse kinematic model gives the accurate piston rod lengths to accurately
control the required joint angles to make the bucket traverse (from its hinge point) at a
desired trajectory, and thus can be used in an autonomous operation of the backhoe
excavator.
6.2 Results of bucket velocity and piston rod velocities
Three linear and one angular bucket velocities, and the velocities of the piston rods can
be determined from developed the MATLAB code, if the link lengths, backhoe geometry
constants, joint angles, and joint speeds are known. The values of link lengths and
geometry constants used in that MALAB code are same as used in direct kinematic
model (Table 1). The joint angle values of θ1, θ2, θ3, and θ4 used are: θ1 = 0º, θ2 = 15º,
θ3 = 295.47º, θ4 = 360º, and the joint speed values obtained from MATLAB code of
quadratic equation of motion to follow the trajectory, which are: 10θ=
deg/sec,
218.08θ=
deg/sec, 313.76θ=
deg/sec, and 419.72θ=
deg/sec.
The velocities found from the MATLAB code are:
444 4
[ ] [0.5702 0 1.1265 51.56] ,
TT
bxyzz
vvvω==V where; only ω4z is in deg/sec and
rest of the terms are in m/s, VST = 0 m/s, VUV = 0 m/s, 56 0.06534
AA
V
=
m/s,
78 0.06841
AA
V=− m/s, and 910 0.03457
AA
V
=
m/s. Thus, the model gives the required
piston velocities, and the bucket velocity to move the backhoe links at the desired joint
speeds.
6.3 Results of inverse Jacobian
From equation (64) it can be written that,
280 B.P. Patel and J.M. Prajapati
1() ()
b
θt
=
θJV
where J–1 is the inverse of a Jacobian matrix J of the size 4 × 4. This inverse Jacobian
problem determines the required joint speeds to move the bucket of the backhoe
excavator at a desired trajectory and velocity. And then so found joint speeds can be
utilised to find the required piston velocities of actuators to achieve the desired joint
speeds. A MATLAB code developed to determine the required piston velocities
to make the bucket move at a desired velocity. When the values of the link
lengths, geometry constants are used as given in Table 1 along with joint angles θ1 = 0º,
θ2 = 15º, θ3 = 295.47º, θ4 = 360º, and bucket velocity vector
444 4
[ ] [0.5702 0 1.1265 51.56] ,
TT
bxyzz
vvvω==V are used as inputs in the
MATLAB code, it will return the required joint speed matrix, and the required piston
velocities of the hydraulic actuators. The found joint speed matrix is
1234
[ ] [0 18.0768 13.7624 19.7212] .
TT
θθθθθ==
 Unit of the matrix is deg/sec,
and the required piston velocities are VST = 0 m/s, VUV = 0 m/s, 56 0.0653
AA
V
=
m/s,
78 0.0684
AA
V=− m/s, and 910 0.0346
AA
V
=
m/s. When these values are compared with
the input values of the bucket velocity model, the differences in joint speed are 0,
0.00482, 0, and 0 degree/sec. The differences in results are near to zero, so this model is
also validated and can be used directly to find the required piston velocities of the
hydraulic actuators of the backhoe excavator to cause the bucket of the backhoe to move
at a desired velocity for the autonomous operation of the backhoe excavator.
6.4 Results of backhoe static model
Static model of the backhoe excavator determines the resulting joint torques due to the
bucket ground interaction under the static equilibrium conditions (because when the
bucket makes the contact with the ground and at that time all joints remain static for that
time). If the link lengths, joint angles, and the resulting resistive forces, and moments due
to the ground-bucket interaction in the static equilibrium condition are known then the
resulting joint torques due to the ground-bucket interaction can be determined from the
developed MATLAB code.
The values of the link lengths, geometry constants are used in the MATAB code as
given in table 1 along with joint angles θ1 = 0º, θ2 = 15º, θ3 = 295.47º, and θ4 = 360º, and
the resistive force due to the soil tool interaction if taken as 7,626 N (considering the
maximum breakout force condition), and when it resolved in x and the y directions will
lead to two resistive force components Fx = 5,933 N, and Fy = 4,716 N, and moment
about the z direction ηz = 4,171.42 N · m (total resistive force × perpendicular distance or
bucket tip radius) when bucket cylinder is active and ηz = 9,684.71 N · m when arm
cylinder is active.
When these values are used for bucket cylinder active, the resulting joint
torque vector will be: 1234
[ ] [12, 049 7,836 9,903 6,640] .
TT
i
τττττ==
When the arm cylinder is active, the resulting joint torque vector will be:
1234
[ ] [12, 049 13,349 15,417 12,154] .
TT
i
τττττ== The so found joint
torques is in N · m. It is known that the digging operation is done with the help of the
bucket and arm cylinder respectively, so the torque acting at joint 3 = the resistive force ×
perpendicular distance from the point of action of the resistive force to the joint 3, and
Kinematics of mini hydraulic backhoe excavator – part II 281
this distance will be more than the perpendicular distance to the joint 4, so τ3 > τ4 for both
the cases.
7 Concluding remarks
This paper presents the complete fundamental foundation for the kinematics of the
backhoe excavators. Here, theoretical relations are developed for the kinematics of the
hydraulic backhoe excavator, which have not previously been presented in the literature
for 4-DOF. The presented inverse Jacobian can be used to move the bucket in specified
trajectory by finding the joint velocities with a desired known velocity of the bucket for
autonomous operation of backhoe excavator. This presented inverse Jacobian and the
static backhoe model have not previously been presented in the literature for backhoe
excavator application.
The developed relations can be utilised for autonomous digging operation. Here
presented relations are developed with considering the links and joints are rigid. During
the digging operation interactive forces developed between soil and tool (for our case it is
bucket), also the digging forces and torque developed by the actuators causes the bending
effect in the link mechanism and their joints. Due to this the developed kinematic
relations will contain inaccuracies in the result. Here, for kinematic relations the resistive
forces offered by the soil-tool interactions and digging forces developed by the actuators
are not considered. But the results obtained for virtual movement of the backhoe
excavator mechanism shows the excellent outcome in terms of result. The proposed
dynamic model developed with consideration of soil-tool interaction forces, gravitational
forces, moments developed by actuators and inertial forces, and also validated the results
obtained, but it is part of next paper.
8 Conclusions
A complete generalised mathematical kinematic model for four degrees of freedom
backhoe excavator is developed and can be applied to any backhoe excavator for
automated movements of the backhoe excavator. The complete kinematics of backhoe
excavator covers the direct kinematics (part I of the paper), inverse kinematics,
expressions differential motion of the backhoe for velocity, inverse Jacobian, and static
model of the backhoe. The results obtained from the developed MATLAB codes are
compared with the results captured from the proposed 3D model of the backhoe
excavator for same configuration. The outcomes of the results are identical, which shows
the good conformity of the results and pertinent accuracy of the model.
References
Biazar, J. and Eslami, M. (2010) ‘Differential transform method for quadratic Riccati differential
equation’, International Journal of Nonlinear Science, Vol. 9, No. 4, pp.444–447.
Cannon, H.N. (1999) Extended Earthmoving with an Autonomous Excavator, Unpublished thesis of
Master of Science, The Robotics Institute Carnegie Mellon University, Pittsburgh.
Craig, J.J. (1989) Introduction to Robotics Machines and Control, 2nd ed., Addison-Wesley
Publishing Company, New York.
282 B.P. Patel and J.M. Prajapati
El-Shahed, M. (2008) ‘Application of differential transform method to non-linear oscillatory
systems’, Communications in Nonlinear Science and Numerical Simulation, Elsevier, Vol. 13,
pp.1714–1720.
Jazar, R.N. (2010) Theory of Applied Robotics – Kinematics, Dynamics, and Control, 2nd ed.,
p.452, Springler. B
Koivo, A.J. (1994) ‘Kinematics of excavators (backhoes) for transferring surface material’, Journal
of Aerospace Engineering, January, Vol. 7, No. 1, pp.17–32, ASCE.
Mehta Gaurav, K. (2008) Design and Development of an Excavator Attachment, Unpublished
MTech thesis, Institute of Technology, Nirma University of Science and Technology,
Ahmedabad.
Mittal, R.K. and Nagrath, I.J. (2003) Robotics and Control, 9th Reprint, Tata McGraw-Hill, New
Delhi.
Oza, N.N. (2006) Finite Element Analysis and Optimization of an Earthmoving Equipment
Attachment – Backhoe, Unpublished MTech thesis, Institute of Technology, Nirma University
of Science and Technology, Ahmedabad.
Patel, B.P. and Prajapati, J.M. (2011) ‘A review on kinematics of hydraulic excavator’s backhoe
attachment’, International Journal of Engineering Science and Technology, Vol. 3, No. 3,
pp.1990–1997.
... Gu et al. 8 investigated various control systems for a robotic excavator, whose results demonstrated that state dependent parameter þproportional integral plus (SDP-PIP) controller can provide improved performance. However, current research mainly focus on the common excavator, [9][10][11][12] there are few reports about the research on the telescopic excavator, which has the greater flexible operation mode and the bigger operation range. So the research on spatial kinematics [13][14][15] of telescopic robotic excavator has significant value based on virtual prototype [16][17][18][19] and experiment research. ...
... Hence, the mapping model between l 1 and y 2 can be expressed as follows Mapping model of arm. According to the geometrical relationship of arm structure in Figure 4 ffDFE ¼ 2p À ffDFC À ffCFQ À ffQFE (12) where ffCFQ ¼ 3p=2 þ y 3 (y 3 is negative). Equation (12) may be written as ...
... According to the geometrical relationship of arm structure in Figure 4 ffDFE ¼ 2p À ffDFC À ffCFQ À ffQFE (12) where ffCFQ ¼ 3p=2 þ y 3 (y 3 is negative). Equation (12) may be written as ...
Article
Full-text available
In order to achieve autonomous operation, the kinematics model and experiment system of telescopic excavator are studied. Firstly, the operation space of the robotic excavator is classified and mapping models between different spaces are established based on the D-H method and geometrical relationship method. Then, the virtual prototype of the telescopic robotic excavator is built based on the SimMechanics software, leveling operation and digging operation are simulated; kinematics model and virtual prototype are tested and verified. Finally, experimental system is developed and trajectory tracking experiment is carried out. By comparing with the simulation results, the validity of the proposed control scheme is verified. The research of this article laid foundations for further study and application.
... If the lengths of the piston rods in the actuators or the joint angles are given, the bucket configuration can be determined by the direct or forward kinematic equations. Whereas; if the bucket configuration is specified, then the corresponding joint angles or the lengths of the piston rods in actuators can be calculated from the inverse kinematic equations, but it is covered in the part-II of this paper (Bhaveshkumar and Prajapati, 2012). Here, firstly the direct kinematic equations are presented. ...
... But the results obtained for virtual movement of the backhoe excavator mechanism shows the excellent outcome in terms of result. The proposed inverse kinematic model with consideration of differential motion of backhoe for velocity and acceleration, Inverse Jacobian and backhoe static backhoe model are developed and covered in the part-II of this paper (Bhaveshkumar and Prajapati, 2012). ...
Article
Full-text available
Backhoe excavators are construction machines can be use for hazardous and even poisonous environmental conditions, worst working conditions, severe weather, and dirty areas where it is very difficult to operate machine by human operator. The excavation task can be performing successfully and effectively for above mentioned working conditions by semiautonomous or fully computer controlled of excavation machines. For this it is important to understand the kinematics of this machine. For autonomous excavation operation of a backhoe, it is desirable to place the bucket to a specified location. The bucket configuration can be determined by the direct kinematic equations for given lengths of the piston rods in the actuators or the joint angles. This paper emphasize on the explicit expressions for the direct (forward) kinematics of backhoe excavator. It covers systematic procedure to assign Cartesian coordinate frames for the links (joints) of an excavator. The developed forward kinematic model is intended for development of an automated excavation control system for light duty construction work and can be applied for heavy duty or all types of backhoe excavators.
... The velocity of each actuator piston rod in terms of the velocity of the joint angle is calculated as Eqs. 26-28 [23]: Stick length 12 (m) a 3 Boom start to stick end (initial) d 1 Origin to boom start x-direction 1.4 (m) d 2 Origin to boom start y-direction 2.309 (m) θ 1 Absolute angle boom link (rad) ...
Article
This paper presents the development and implementation of an advanced control system for a hydraulic manipulator. Designing a suitable coordinate control system for materials handling machines is vital for desired machine performance. For this purpose, targets of developing a proper coordinate control system were defined and mathematical equations were formulated. The two common control methods, i.e. open-loop control and closed-loop control were specifically designed for the handler in question, and the performance of the machine was studied applying each control method. The simulation environment and the operation principle of the created models are discussed in detail in this paper. The two control strategies introduced were tested through real-time simulation, and the accuracy and the performance of the manipulator were investigated utilizing each control method. The limitations and problems of each control strategy are addressed, and suggestions for further research directions to improve the accuracy of tip control are recommended. The case study in this paper is a mobile harbor crane material handler provided by the Mantsinen Group.
Article
Full-text available
Remote and descriptive visualization of spatio-temporal information of excavator activities may increase awareness about jobsite hazards and operational performance in earthwork operations. One of the emerging approaches to collect this information is to extract the 3D pose of an excavator from the video frames using a convolutional neural network (CNN). However, this method requires labeling the training datasets, which are difficult to prepare because of conditions unsuitable for installing the motion capture sensors. This study investigates the performance of a CNN for estimating the 3D pose when trained on a synthetic dataset. In particular, a kinematic constraint is proposed to update the model parameters efficiently during training. The results show that the proposed method estimated the 3D poses of a real excavator with an average pose error of 9.63°. Hence, the proposed data augmentation method could help address the training data issues and improves the learning of real data complexity.
Article
Full-text available
When choosing equipment for working out a coal-bearing zone of a quarry field, one of the most important factors is the use of excavator equipment available at the enterprise. From previously published works, the main advantages and disadvantages of the most common types of excavators (rope shovel and backhoe) are known. In this article, the authors propose an organizational scheme for the development of a coal-bearing zone using both types of excavators. A technological scheme of the development of a coal-bearing and a coalless zone of a quarry field by the complexes of these excavators is given. The advantages of the given scheme of mining are determined.
Conference Paper
A common wheeled excavator is used as a prototype and its monoblock boom is transformed into 2-piece boom., which form a new type of four-linkage excavator composed of arm and bucket. Kinematic analysis of the working device of the four-linkage excavator is carried out to grasp the position change of bucket tip and the relationship between the length of hydraulic cylinder and the joint angle. The model of four-linkage excavator is established and its kinematic simulation is carried out based on the robotic toolbox in MATLAB. Finally, actual measurements are made and the four-linkage excavator's parameters are compared with parameters of the prototype. The results show that the working ranges of the four-linkage excavator are much larger than that of the prototype excavator, which provide theoretical supports for the design of the working device of the excavator and the trajectory planning of the excavator.
Article
Full-text available
As is known from the classic works on open pit mining, the bench is a separately developed part of the rock layer, having the form of a step. It should also be noted that it is necessary to clearly differentiate the concepts of “bench height” and “height of the layer to be removed.” The benches are often divided into subbenches, developed by different excavation equipment or by the same equipment both sequentially and simultaneously, but having transport routes that are uniform for the bench. As an example, an excavator stripping of the upper and lower subbenches with loading, respectively, at the level of the excavator and above this level, is usually given, that is, the transport route (road) passes through an intermediate platform bench located in the middle of its height. Therefore, the excavation layer of any height, which is, in fact, a part of the working bench, can be considered as an independent bench with all its attributes, but in order to avoid duplication of definitions, this paper suggests the name “extraction layer”. When developing this element various digging modes can be applied. In this paper, we studied the main modes and selected the one that provides the highest performance.
Research
Full-text available
Hydraulic excavators are widely used in construction for multiple purposes and is considered to be one of the most commonly used construction tools. The arm of a hydraulic excavator is comprised of two hydraulic cylinders, a bucket and a boom, which is on the upper part of the arm. The arm moves much like a human arm, at the wrist and the elbow. A rod and piston inside the hydraulic cylinder enables the arm to move using oil. Oil is pumped through the end of the piston and in turn pushes the rod through the cylinder, thus creating movement of one or both parts of the arm. By controlling the amount of oil is pumped through the valve, the accuracy of the arm can be easily manipulated. The digging motion of an excavator can be divided into 3 mechanisms ; penetration, separation, and secondary separation, which constantly put different contributions to the generation of soil resistance in the digging process (Park, 2002). The digging motion and force applied by the end bucket is determined by the mixture of these 3 main mechanisms at different stages of digging process. The hydraulic cylinders in the excavator arms are limited to linear actuation. Hence, their mode of operation is fundamentally different from cable-operated excavators which use winches and steel ropes. The boom (upper arm) and bucket attached to the body work in conjunction to move materials, but are controlled independently. Hence, the optimal position for maximum efficiency regarding digging needs to be assessed. The movement of the excavators arms can be modelled as a ma-nipulator and the movement/position of the end bucket can be computed using the body and space Jacobians associated with the excavator. Using these techniques, our aim is to model the excavator motion and behavior with MATLAB and to estimate the optimum orientation and lengths of the excavator arms that maximizes the efficiency (power usage) of force applied during excavation. In this paper, we modelled a generic excavator arm as a 3 DOF manipulator and calculated its DH and screw parameters. The Jacobian calculated from the created model of the excavator in MATLAB was used to conducted paramet-ric studies regarding the joint angles and link length ratios. The geomechanical properties of the excavated soils were considered to provide a semi-optimal solution to the input joint angles for effective excavation.
Article
There are many efforts to mechanize the process for underwater port construction due to the severe and adverse working environment. This paper presents an underwater construction robot to level rubbles on the seabed for port construction. The robot is composed of a blade and a multi-functional arm to flatten the rubble mound with respect to the reference level at uneven terrain and to dig and dump the rubbles. This research analyzes the kinematics of the blade and the multi-functional arm including track and swing motions with respect to a world coordinate assigned to a reference depth sensor. This analysis is conducted interfacing with the position and orientation sensors installed at the robot. A hydraulic control system is developed to control a track, a blade and a multi-functional arm for rubble leveling work. The experimental results of rubble leveling work conducted by the robot are presented in land and subsea. The working speed of the robot is eight times faster than that of a human diver, and the working quality is acceptable. The robot is expected to have much higher efficiency in deep water where a human diver is unable to work.
Article
Full-text available
In this article differential transform method (DTM) is considered to solve quadratic Riccati dif-ferential equation. The results derived by differential transform method will be compared with the results of homotopy analysis method and Adomian decomposition method. It would be shown that this method used for quadratic Riccati differential equation is more effective and promising than homotopy analysis method and Adomain decomposition method. An efficient recurrence relation for solving these equations will be obtained.
Article
Full-text available
In this paper, the differential transform method is proposed for solving non-linear oscillatory systems. These solutions do not exhibit periodicity, which is the characteristic of oscillatory systems. A modification of the differential transform method, based on the use of Padé approximants, is proposed. We use alternative technique by which the solution obtained by the differential transform method is made periodic. The method is described and illustrated with examples. The results reveal that the method is very effective and convenient.
Article
Full-text available
An excavator is a typical hydraulic heavy-duty human-operated machine used in general versatile construction operations, such as digging, ground leveling, carrying loads, dumping loads and straight traction. These operations require coordinated movement of boom, arm and bucket in order to control the bucket tip position to follow a desired trajectory. This paper focuses on review of a work carried out by researchers in the field of kinematic modeling of the backhoe attachment to understand relations between the position and rientation of the bucket and spatial positions of joint-links. Kinematic modeling is helpful for understanding and improving the operating performance of the backhoe excavation machine. There are many research work done by researchers in the same field but still there is a scope to develop kinematic modeling of backhoe attachment to predict the digging trajectory as well as better controlling ofbackhoe attachment to carry out required digging task at desired location.
Article
Full-text available
The automation of earthmoving equipment is an endeavor that has the potential...
Article
To use construction machines effectively in the dark, severe weather, or hazardous and/or unhealthy environments, their operations should be controlled automatically. It can be realized if the kinematics and dynamics of the machine are understood. To help achieve this goal, the kinematics of specific construction machines -- excavators (backhoes and loaders) -- are investigated here. A systematic procedure is presented to assign Cartesian coordinate frames for the links (joints) of an excavator. Then, the homogeneous transformation matrices that relate two adjacent coordinate frames are given. The kinematic relations of the pose (position and orientation) of the bucket, the joint shaft angles, and the lengths of the cylinder rods in the hydraulic actuators for an excavator are studied. Explicit expressions for the forward and backward (inverse) kinematic relations are presented. Then, the corresponding kinematic velocity relations for the excavators are developed. The kinematic relations presented provide the foundation for engineers to realize the automatic computer-controlled operations of the machine.
Robotics and Control, 9th Reprint
  • R K Mittal
  • I J Nagrath
Mittal, R.K. and Nagrath, I.J. (2003) Robotics and Control, 9th Reprint, Tata McGraw-Hill, New Delhi.
Design and Development of an Excavator Attachment
  • K Mehta Gaurav
Mehta Gaurav, K. (2008) Design and Development of an Excavator Attachment, Unpublished MTech thesis, Institute of Technology, Nirma University of Science and Technology, Ahmedabad.
Finite Element Analysis and Optimization of an Earthmoving Equipment Attachment – Backhoe, Unpublished MTech thesis
  • N N Oza
Oza, N.N. (2006) Finite Element Analysis and Optimization of an Earthmoving Equipment Attachment – Backhoe, Unpublished MTech thesis, Institute of Technology, Nirma University of Science and Technology, Ahmedabad.
Finite Element Analysis and Optimization of an Earthmoving Equipment Attachment -Backhoe
  • N N Oza
Oza, N.N. (2006) Finite Element Analysis and Optimization of an Earthmoving Equipment Attachment -Backhoe, Unpublished MTech thesis, Institute of Technology, Nirma University of Science and Technology, Ahmedabad.