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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 (θ2 – Φ) 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 ζ5 – ζ1 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 + θ4 – ζ3 and ζ2 = ζ5 – ζ1.

(

)

(

)

(

)

()()()

{}

()()

{}

()()()

()()()()()

{}

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=θ…J (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

PPP=×P 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.

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