KINEMATICS OF MINI HYDRAULIC BACKHOE EXCAVATOR – PART: I · KINEMATICS OF MINI HYDRAULIC BACKHOE...

166

Int. J. Mech. Eng. & Rob. Res. 2013 Bhaveshkumar P Patel and J M Prajapati, 2013

KINEMATICS OF MINI HYDRAULIC BACKHOEEXCAVATOR – PART: I

Bhaveshkumar P Patel1* and J M Prajapati2

*Corresponding Author: Bhaveshkumar P Patel,[email protected]

Backhoe excavators are construction machines can be use for hazardous and even poisonousenvironmental conditions, worst working conditions, severe weather, and dirty areas where it isvery difficult to operate machine by human operator. The excavation task can be performingsuccessfully and effectively for above mentioned working conditions by semiautonomous orfully computer controlled of excavation machines. For this it is important to understand thekinematics of this machine. For autonomous excavation operation of a backhoe, it is desirableto place the bucket to a specified location. The bucket configuration can be determined by thedirect 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 backhoeexcavator. 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 ofan automated excavation control system for light duty construction work and can be applied forheavy duty or all types of backhoe excavators.

Keywords: Autonomous, Excavation, Bucket configuration, Direct kinematic, Backhoeexcavator

INTRODUCTIONRapidly growing rate of industry of earthmoving machines is assured through the highperformance construction machineries withcomplex mechanism and automation ofconstruction activity. Backhoe excavators arewidely used for most arduous earth moving

ISSN 2278 – 0149 www.ijmerr.comVol. 2, No. 1, January 2013

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Int. J. Mech. Eng. & Rob. Res. 2013

1 Mechanical Engineering Department, JJT University, Jhunjhunu 333001, Rajasthan, India.2 Mechanical Engineering Department, Faculty of Engineering & Technology, M S University of Baroda, Vadodara 390002, Gujarat,

India.

work in engineering construction to excavatebelow the natural surface of the ground onwhich the machine rests. Hydraulic system isused for operation of the machine while diggingor moving the material. An excavator iscomprised of three planar implementsconnected through revolute joints known as the

Research Paper

167


boom, arm, and bucket, and one verticalrevolute joint known as the swing joint (Howard,1999).

Autonomous excavation is the best solutionto carry out excavation task in hazardous andpoisonous environment conditions, worstworking conditions, severe weather, and dirtyareas where it is very difficult to operatemachine by human operator. Some times atconstruction site, the excavation machines arenot utilized in effective way as their capacitiesif they are operated manually or semiskilledoperator. The semiautonomous or automaticcomputer control is essential to improve theproductivity and the effective use of expensiveconstruction machines. Moreover,automatically operated machines can oftenperform a task faster and with better precisionthan manually operated machines (Koivo,1994). To automate the excavation task, it isnecessary to understand the kinematics of thismachine. Kinematics of backhoe linkmechanism is one of the steps in the directionto automate the excavation operation. Thepresented basics of kinematics and kinematicmodel can be applied to other loader andhydraulic backhoe excavators which are usedto excavate the surface materials of terrain.Kinematics is the science of motion whichtreats motion without regard to the forces thatcause it. Within the science of kinematics onestudies the position, velocity, acceleration, andall higher order derivatives of the positionvariables (with respect to time or any othervariables) (John, 1989).

A backhoe excavator is designed toperform a task in the 3-D space. The bucketof the backhoe is required to follow a plannedtrajectory to carry out the digging task in the

workspace. This requires control of positionof each link (swing link, boom, arm, and thebucket) and joints of the backhoe to controlboth the position and orientation of the bucket.To program the bucket motion and the joint linkmotions, a mathematical model of the backhoeis required to refer all geometrical and/or timebased properties of the motion. In other wordskinematic model purely encodes thegeometric relationship of the mechanismunder study. Moreover; the kinematic modelgives relation between the position andorientation of the bucket and spatial positionsof joint-links. A problem of describing the directkinematic model for an autonomous operationof the backhoe excavator is presented here.

RELATED WORKVaha and Skibniewski (1993) firstly developedkinematics of the excavator by appropriateframe assignments. Vaha developed thiskinematic model only as a prerequisite for thedynamic model. He developed the kinematicmodel for a hydraulic excavator in the generalform only, thus giving up to the generaltransformation matrix relating two consecutiveframes (relating frame {i – 1} to the frame {i}).His kinematic model was not thorough, andthus not giving a clear insight in terms of theforward kinematics, inverse kinematics, andvelocity equations.

Koivo (1994) on the other hand developeda complete kinematic model for a hydraulicexcavator, considering the three degrees offreedom excavator mechanism or RRRconfiguration. He actually developed akinematic model for three degrees of freedomexcavator by neglecting the swing motion (inexcavator swing motion is applied by electric

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motor or actuator not by hydraulic actuator).He also incorporated the shortcomings ofVaha’s kinematic model as he proposed theforward kinematic model incorporatingequations relating joint shaft angles to thebucket pose (or bucket configuration), andequations relating lengths of hydraulicactuators to joint shaft angles. Thus firstlydetermining the bucket position andorientation in terms of joint shaft angles, andthen giving the relationship between the jointshaft angles and the lengths of the piston rodsin hydraulic actuators for boom, arm, andbucket cylinders. He also proposed acomplete inverse kinematic model by givingfirstly the joint shaft angles in terms of bucketpose, and then joint shaft angles in terms ofthe lengths of the hydraulic actuators. He alsoproposed a direct excavator Jacobian matrixto calculate the bucket velocity provided thatthe joint shaft velocities are known and alsoproposed velocity equations of actuatorpistons in terms of the speeds of the jointshafts. But his model can said to be incompletein some context as: he did not carry out thevelocities of each link (swing link, boom, arm,and bucket), he did not present the fullJacobian matrix of the size 6 4 and insteadpresented of the size 4 4 by neglecting thelast two rows of the matrix, as it was the factthat only one angular velocity of the bucket canbe controlled and that is in the Z direction or inthe direction of the bucket joint axis. He alsodid not carry out the inverse Jacobian on theexcavator mechanism which is necessarywhen the desired bucket velocity is to beachieved by controlling the joint velocities.Zygmunt (2003) developed few kinematicrelationships in terms of the transformationmatrices between the frames. He only

determined those kinematic equations that canbe helpful for him to develop a mathematicaldynamic model for an excavator. Apart fromthis, the transformation matrices establishedby him were only giving the rotationaltransformation, as it was the only need for hisstudy. Thus he also did not establish anysystematic kinematic relationships requiredencoding the geometrical relationships of theexcavator, and his kinematic model remainedincomplete. Shaban et al. (2007) developeda kinematic model for the excavator, but forthe arm, and boom only, thus representing thekinematic model in two degrees of freedomonly. He also established the inverse kinematicmodel for the two degrees of freedom only. Butthe shortcomings of his model are: he did notdetermine any systematic procedure to assignthe frames to the links, both the forward andinverse kinematic models presented by himwere incomplete in terms of the relationshipbetween the bucket position and orientation(collectively known as bucket configuration),and joint angles, and the relationship betweenthe actuator lengths and joint angles of theexcavator. So to be concluded, from these fourkinematic models, the kinematic model ofKoivo (1994) gives a complete kinematicrelationship for the geometry of a hydraulicexcavator assuming in three degrees offreedom. But a complete kinematicrelationship for the geometry of the backhoefor four degrees of freedom has not beenpresented so far, and this is one of the areasof research reported in this paper.

AFFIXING FRAMESTO THE LINKSFundamentally a backhoe excavator has fivelinks starting from the fixed link or base link,

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swing link, boom link, arm link (dipper link),and bucket link. These links are connected toeach other by joints, which allow revolutemotion between connected links each of whichexhibits just one degree of freedom. This leadsto the four degree of freedom R-RRRconfiguration of the backhoe, where R standsfor a revolute joint.

Figure 1 describes the schematic side viewof the backhoe excavator. To developkinematic relations for the geometry of thebackhoe; firstly the coordinate frames will beassigned to the backhoe excavator links.

To analyse the motion of the backhoeexcavator in Figure 1 for performing aspecific task, it becomes necessary todefine a world coordinate system todescribe the position and orientation of thebucket (collectively known as configurationof the bucket). A right-hand Cartesiancoordinate system X

wY

wZ

w is chosen, and

its origin is placed at an arbitrary point onthe ground level in the workspace of thebackhoe excavator. After assigning the worldcoordinate frame the local coordinateframes for all links are assigned by followingthe DH guideline for link frame assignmentalgorithm given in (Mittal and Nagrath, 2003).Figure 1 shows the schematic view of abackhoe excavator and frame assignments.

After assignment to the all coordinateframes is done, any point P in the ith coordinateframe can be written as iP. Where, point iP referto the point P with respect to frame {i}. The iP= [ip

xip

yip

z1]T, where the fourth component 1

is the scale factor. The scale factor has beentaken 1 to make the components of Cartesianrepresentation and homogeneousrepresentation identical. Where, ip

x, ip

y, and ip

z

indicate the x, y, and z coordinates of point Pwith respect to the ith coordinate frame,respectively.

Figure 1: Schematic View of a Backhoe Excavator and Frame Assignments

170


DIRECT KINEMATICS OFBACKHOE EXCAVATORFor the controlling and the automaticoperation of a backhoe, it is desirable toplace the bucket to a specified location. Thiscan be achieved by selecting the properlengths of the piston rods in the cylinder, andthus selecting the joint angles properly. Thekinematic equations are the mathematicalequations those relate the position andorientat ion of the bucket (bucketconfiguration) to the joint variables (jointangles in our case) or to the lengths of thepiston rods in the hydraulic actuators. If thelengths of the piston rods in the actuators orthe joint angles are given, the bucketconfiguration can be determined by thedirect or forward kinematic equations.Whereas; if the bucket configuration isspecified, then the corresponding jointangles or the lengths of the piston rods inactuators can be calculated from the inversekinematic equations, but it is covered in thepart-II of this paper (Bhaveshkumar andPrajapati, 2012). Here, firstly the directkinematic equations are presented.

To establish the homogeneoustransformation matrix i–1T

i, which describes the

position and orientation of frame {i} relative toframe {i – 1}, and completely specifiesgeometric relationship between these links interms of four D-H parameters (

i, d

i,

i, a

i). i–1T

i

can be given as follow:

1000

01

iii

iiiiiii

iiiiiii

ii

dCS

SaSCCCS

CaSSCSC

T

...(1)

where, Ci = cos(

i), S

i = sin(

i), C

i =

cos(i), and S

i = sin(

i).

Here ai, and

i are two link parameters, and

di, and

i are two joint parameters. Out of these

four parameters, only one is a variable for linki, the joint displacement variable q

i (

i) and

other three are constant. These all parametersare defined in the Denavit-Hartenbergguidelines and notation given in (Mittal andNagrath, 2003).

Transformation Matrix of BucketFrame

The bucket configuration relative to the baseframe can be found by considering the 4consecutive link transformation matricesrelating frames fixed to adjacent links. Thus,

0T4 = 0T

1(

1) 1T

2(

2) 2T

3(

3) 3T

4(

4) ....(2)

So to determine the homogeneoustransformation matrix that relates the link 0 tolink 4 (0T

4), firstly other four transformation

matrices as given in Equation (2) must bedetermined. To determine i–1T

i we need to

determine the three parameters, i,

i, and a

i

(as given in Table 1). The values of ai,

i, and

i

are found out for our case based on guide linegiven in (Mittal and Nagrath, 2003). As for anexample link twist

1 determines the angle

between Z0– and Z

1– axes measured about the

X1– axis in the right hand side = +90. Similarly,

the values of other link twist angles aredetermined. The joint distance di for i = 1, 2, 3,4 is zero, because all joints are revolute joints.The link lengths a

1, a

2, a

3, a

4 are the lengths of

swing link, boom, arm, and bucket respectively,and taken as it is to keep the calculationprocedure more simplified, and later on thevalues of a

i for i = 1, 2, 3, 4 can be substituted

from the dimensions of the backhoe.

171


The substitution of the parameter values ofTable 1 into Equation (1) gives the followinghomogeneous transformation matrices:

1000

0010

0

0

1111

1111

10 SaCS

CaSC

T

...(3)

1000

0100

0

0

2222

2222

21 SaCS

CaSC

T...(4)

1000

0100

0

0

3333

3333

32 SaCS

CaSC

T...(5)

1000

0100

0

0

4444

4444

43 SaCS

CaSC

T...(6)

Now the location of the bucket from theFigure 1 can be specified by locating therotational axis of the bucket motion, i.e., todetermine the coordinates of point O

3 = A

3. The

coordinates of the point O3 with respect to base

frame 0, i.e., 0PA3

can be given as:

33

30

30

AAPTP ...(7)

But, 0T3 = 0T

11T

22T

3, therefore, 0P

A3 = 0T

11T

22T

33P

A3.

where, 3PA3

indicates the coordinates of thepoint A

3 with respect to the coordinate frame

{3}. So 3PA3

= [0 0 0 1]T. With the help of theEquations (3), (4) and (5) 0T

3 can be written

as follow:

1000

0 233222323

23322111231231

23322111231231

30

SaSaCS

CaCaaSCSSCS

CaCaaCSSCCC

T

...(8)

where, C23

= cos(2 +

3), and S

23 = sin(

2 +

3).

By neglecting the fourth raw and the fourthcolumn of the Equation (8) one yields 3 3rotational matrix 0R

3, which shows the rotation

of the frame {3} with respect to base frame{0}. By putting the values of 0T

3 from equation

(8), and 3PA3

= [0 0 0 1]T into Equation (7), oneyields;

0PA3

= [C1(a

1 + a

2C

2 + a

3C

23) S

1(a

1 + a

2C

2 +

a3C

23) a

2S

2 + a

3S

231]T ….(9)

Here superscript T refers to thetransposition of the raw matrix, i.e., a positionvector. If the values of the joint variables,

1,

2,

3 are known then the location of the point O

3 =

A3 in the base coordinate system can be

determined by Equation (9). Similarly thecoordinates of point A

4 with respect to the base

frame {0} can be found out as 0PA4

= 0T4

4PA4

,where 4P

A4 = [0 0 0 1]T. The transformation

matrix 0T4 can be found by multiplying 0T

3 with

3T4 and can be given as follows:

1000

0 234423322234234

23442332211123412341

23442332211123412341

40

SaSaSaCS

CaCaCaaSCSSCS

CaCaCaaCSSCCC

T

...(10)

Link i ai

i

di

i

qi

1 a1

90 0 1

1

2 a2

0 0 2

2

3 a3

0 0 3

3

4 a4

0 0 4

4

Table 1: Joint-Link Kinematic Parametersfor the Backhoe

172


where, C234

= cos(2 +

3 +

4), and S

234 =

sin(2 +

3 +

4). This will give the coordinates

of the point A4 as:

4PA4

= [C1(a

1 + a

2C

2 + a

3C

23 + a

4C

234) S

1(a

1 +

a2C

2 + a

3C

23 + a

4C

234) a

2S

2 + a

3S

23 + a

4S

234 1]T

This means that if the joint variable values

1,

2,

3,

4 are known, the position and

orientation of the point A4 or O

4 can be known

with respect to the base frame {0}.

Relation Between Length of thePiston Rods in Actuators and JointAngles

The backhoe excavator is a constructionmachine that uses hydraulic actuators to carryout the digging operation. In the hydrauliccylinder, the pressure on the piston iscontrolled. So ultimately the lengths of thepiston rods determine the joint angles of therespective links. This means that there is adirect relation between the piston rod lengthand the joint angles, and the lengths of thepiston rods are determined here by the linesegment joining two actuator’s attachmentpoints, e.g., from Figure 1 it can be seen thatthe length of the actuator 3 is A

5A

6.

Firstly the relation between actuator 1,actuator 2 and joint angle

1 will be determined.

Figure 2 shows the top view of the assemblyof base (fixed) link and the swing link of thebackhoe.

From the Figure 2 it can be seen that pointsS and U are the centre of the holes wherecylinders actuator 1 and actuator 2 (swingcylinders) link are attached to the fixed linkrespectively. Points T and V are the pointswhere piston rods of actuator 1 and actuator 2are attached to the swing link respectively. Thismeans the distances SU, and TV will always

remain constant. Apart from this if the point Xis taken as the middle point of the line segmentSU, then from the geometry point X aligns withthe origin of frame {0} (i.e., O

0) horizontally.

This gives the distances XS, XU, and XO0

to be constant. Similarly if the midpoint of theline segment TV is Y then the angle betweenlines YO

0, O

0T and between the lines YO

0,

O0V is same and it is YO

0V = YO

0T = ,

and it is also constant once the geometry isdesigned, as shown in Figure 2 (Because, TY= YV). Also the distances O

0T, and O

0V are

constant, similarly O0S, O

0T, O

0U, O

0V are also

constant from the geometry. Apart from this lineO

0Y always coincides with the line O

0O

1 (i.e.,

X1 axis). The

1 is the angle between axis X

0

and X1 measured from the axis X

0 in the clock

wise direction. This gives the angle betweenlines YO

0 and the axis X

0 to be

1 as shown in

Figure 2. This means that to find the length ofthe actuators 1, and 2 one may find the ST,and UV and then the remaining distance of thehydraulic cylinders should be added to thevalues of ST, and UV respectively (see

Figure 2: Geometry of Top View of theBase to Swing Link

173


Figure 2). In our case we will call only ST to bethe length of actuator 1, and UV to be the lengthof the actuator 2. From the geometry as shownin the Figure 2

(ST)2 = (ST' )2 + (TT' )2 …(11)

(ST)2 = [O0X + O

0T cos( –

1)2]

+ [XS – O0T sin(–

1)]2 …(12)

Equation (12) gives the length of the pistonrod of the hydraulic actuator 1 in terms of jointangle

1. This means that if the joint angle

1 is

known then the length of the piston rod ofactuator 1, ST can be found out from Equation(12). On the other hand, if the length of thepiston rod of actuator 1, ST is known then thejoint 1 angle

1 can be determined as follows:

xO

XS

0

11 tan

220

20

2

1222

02

02

02

01

4tan

STTOSO

STTOSOTOSO

...(13)

In Equation (13) gives the value of the joint1 angle

1 if the length of the piston rod of

actuator 1 = ST is known. Also from thegeometry as shown in the Figure 2,

(UV)2 = (UV' )2 + (VV' )2 …(14)

(UV)2 = [O0V cos( +

1) + O

0X]2

+ [O0V sin( +

1) – XU]2 …(15)

Equation (15) gives the length of the pistonrod of hydraulic actuator 2 in terms of jointangle

1. This means that if the joint angle

1 is

known then the length of the piston rod ofactuator 2, UV can be found out from Equation(15). On the other hand, if the length of the

piston rod of actuator 2, UV is known then thejoint 1 angle

1 can be determined as follows:

xO

XU

0

11 tan

220

20

2

1222

02

02

02

01

4tan

UVVOUO

UVVOUOVOUO

...(16)

In Equation (16) gives the value of the joint1 angle


actuator 2 = UV is known. The length of thepiston rod of actuator 3 = A

5A

6 can be found

out from the Figure 1, which moves the joint 2of angle

2. The length of the piston rod of

actuator 3 = A5A

6 can be express as follows, if

the joint angle 2 is known.

(A5A

6)2 = (A

1A

5)2 + (A

1A

6)2 – 2(A

1A

5) (A

1A

6)

cos( – 1 –

2 –

2) …(17)

where, A1A

5, and A

1A

6 have specific constant

values, also A5A

1A

2 = Y

1 and the angle

between the line A1A

6 and the axis X

0 = Y

2 are

constant, thus known to us. The Equation (17)can be used to determine the joint angle

2. If

the length of the piston rod of actuator 3 = A5A

6

is known, then the trigonometric equation of(17) can be solved for joint 2 angle

2 by using

the standard method, and it can be expressas follows.

212

265

261

251

2

122

652

612

512

612

511

4tan

AAAAAA

AAAAAAAAAA

...(18)

174


axis that is Z4. The task to find the length of the

actuator 5 is tricky, because unlike otheractuators, one end of the actuator 5 (point A

10)

is not directly attached to the bucket, and thismakes the geometry more complicated. Thelength of the piston rod of actuator 5 = A

9A

10 is

determined by the following Equation (21), ifthe value of

1 is known,

(A9A

10)2 = (A

9A

12)2 + (A

10A

12)2 – 2(A

9A

12)

(A10

A12

) cos(2 – 1 –

1) …(21)

where, the lengths A9A

12, and A

10A

12 are

constant, also 1 = the major A

9A

12A

3 is

constant and known to us. 1 = the A

3A

12A

10

as shown in Figure 1, but the 1 is not constant.

Here, we want to determine the length of thepiston rod of actuator 5 = A

9A

10 in terms of the

joint angle 4. Firstly, let us find

1 in terms of

the length of the piston rod of actuator 5 = A9A

10,

and that is given by,

11 2

2109

21210

2129

21

220`9

21210

2129

21210

2129

14

tan

AAAAAA

AAAAAAAAAA

...(22)

Now, let us find 1 in terms of

4. From the

Figure 1 it can be seen that,

2 = A4A

3A

11 + A

11A

3A

9 + A

9A

3A

12 +

A12

A3A

2 + A

2A

3A

4…(23)

Now, let’s concentrate on the quadrilateralA

10A

12A

3A

11. We already found the value of the

A10

A12

A3 =

1 in Equation (23). Let’s assume

the A10

A11

A3 =

2, the A

12A

10A

11 =

3, the

A12

A3A

11 =

4 and A

2A

3A

4 = (

4 – ). From

the Figure 1 it can be seen that,

4 = 2 – A

12A

3A

2 – (

4 – ) – A

4A

3A

11

…(24)

Thus the Equation (18) gives the value ofjoint 2 angle


actuator 3 = A5A

6 is known. Actuator 4 connects

the boom link to the arm link, known as armcylinder, and it controls the motion of the arm,at joint 3. The length of the piston rod ofactuator 4 = A

7A

8 moves the joint 3 angle

3.

From the Figure 1 it can be seen that theA

1A

2A

7 =

1, and A

8A

2A

3 =

2 are constant

for the geometry of boom and arm respectively,and thus are known to us, also it follows thatA

8A

2A

7 = 2–

1 –

2 – (

3 – ) = 3 –

1 –

2

– 3. Then for the A

8A

2A

7, if the cosine rule

for A8A

2A

7 is applied, one yields;

(A7A

8)2 = (A

2A

7)2 + (A

2A

8)2 – 2(A

2A

7) (A

2A

8)

cos (3 – 1 –

2 –

3) …(19)

Equation (19) gives the length of the pistonrod of actuator 4, if the joint 3 angle

3 is known.

The reverse is also possible that if the lengthof the piston rod of actuator 4 = A

7A

8 is known,

then the joint 3 angle 3 can be found out by

solving the Equation (19) in terms of tanfunction instead of sine or cosine functions for

3, as follows:

213 3

287

282

272

21

2287

282

272

282

272

14

tanAAAAAA

AAAAAAAAAA

...(20)

Now let’s move on to find the length of thepiston rod of actuator 5 (bucket actuator) =A

9A

10 (Figure 1) in terms of the joint 4 angle

4,

and later on find the angle 4 in terms of the

piston rod length of actuator 5 = A9A

10. It can

be seen from the Figure 1 that actuator 5causes the bucket to rotate about the joint 4

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Here, the A12

A3A

2 =

1 and A

4A

3A

11 =

2

are constant, and thus known for us. This leadsto,

4 = 3 –

1 –

2 –

4…(25)

Also for the quadrilateral A10

A12

A3A

11,

1 +

2 +

3 +

4 = 2 …(26)

This gives,

1 +

2 = – +

1 +

2 +

4 –

3…(26a)

If the angle 1 is to be found out in terms

of the joint 4 angle 4, then the values of the

angles 2, and

3 must be known in Equation

(26a). For this we will assume that the angle

3 is known to us with the help of encoder.

So still the angle 2 is to be determined in

terms of the angle 1. For this to be done,

let’s divide the quadrilateral A10

A12

A3A

11 into

two triangles as A10

A12

A3 and A

10A

11A

3. By

applying cosine rule for both the triangles, itgives,

(A3A

12)2 + (A

10A

12)2 – 2(A

3A

12) (A

10A

12)

cos(1) = (A

3A

11)2 + (A

10A

11)2 – 2(A

3A

11) (A

10A

11)

cos(2) …(27)

Equation (27) gives the value of 2 in terms

of 1, or vice versa. So from the Equation (29a)

it can be written as,

4 =

1 +

2 + –

1 –

2 +

3…(26b)

Thus, the Equations (13), (16), (18), (20),(22) and (26b) shows the relation betweenthe length of the actuators and joint angles

1,

2,

3 and

4. This means that if the joint

variable values 1,

2,

3,

4 are substituted

in Equations (9) and (10), the position andorientation of the point A

4 or O

4 can be find

with respect to the base frame {0}. Theseequations show the direct kinematics of thebackhoe excavator.

RESULTS AND DISCUSSIONThe results of the direct kinematic model ofbackhoe are discussed and the results are

Description Symbol Value Unit

Swing link length, boom link a1

0.430

length, arm link length and a2

1.347 m

bucket link length a3

0.723

respectively a4

0.547

Geometry constant angles 52.72

Y1

46.23

Y2

28.53

1

33.23

2

139.54 Degree

1

197.79

1

3.32

2

80.14

3

67.43

Geometry constant XS 0.092

distances, and the piston OX 0.21507

rod lengths of the actuators OS 0.23386

for maximum breakout OT 0.11556

force condition ST 0.285

XU 0.092

OU 0.23386

OV 0.11556

UV 0.285

A1A

50.67461 m

A1A

60.21783

A5A

60.70978

A2A

70.91102

A2A

80.28480

A7A

80.86524

A9A

120.74341

A10

A12

0.220

A9A

100.65907

A3A

120.13254

A3A

110.18103

A10

A11

0.205

Table 2: Values of the Link Parametersand Geometry Constants Used

in Direct Kinematic MATLAB Code

176


obtained using the developed MATLAB code.The input values of the parameters are takenas per Table 2 for our case. The data given inthe Table 2 captured from Figure 3 forvalidation of direct kinematic mathematicalmodeling of backhoe excavator based onmaximum breakout force configuration asshown in Figure 3. The piston rod lengths ofthe actuators can be obtained from sensorsduring actual excavation operation but here forvalidation of work the piston rod lengthsconsidered for maximum breakout forcecondition. If the lengths of the piston rods inhydraulic actuators and the A

12A

10A

11 =

3 can

be determined by the sensors, and thebackhoe geometrical constants are knownthen the position of the bucket hinge point (A

3)

and point (A4) can be determined by the

MATLAB code developed. The values of theparameters used in the MATLAB code aregiven in Table 2.

While using the values as listed in Table 2for MATLAB code, the joint angle values comeout to be:

1 = 0°,

2 = 15°,

3 = 295.468°,

4 =

360.0° and the Cartesian coordinates for thepoint 0P

A3 = [2.19990 – 0.20151]T, and the

Cartesian coordinates of the point 0PA4

=[2.55500 – 0.61761]T.

While using the same values as listed inTable 2 in the proposed 3-D model of thebackhoe excavator attachment model, thevalues of joint angles and Cartesiancoordinates of points 0P

A3, 0P

A4 are:

1 = 0°,

2

= 15°, 3 = 295.47°,

4 = 360°, 0P

A3 = [2.199850

– 0.201541]T, and 0PA4

= [2.555820 –0.618711]T, as seen in Figure 3.

As can be seen from the results that thedifferences in the mathematical kinematicmodel and the proposed 3-D model for thejoint angles

1,

2,

3, and

4 are: 0°, 0°, –

0.002°, and 0.029 respectively.

Figure 3: Direct Kinematic Results of the Proposed 3D Model

177


It can be seen that the difference betweenthe joint 1 angle

1 is zero, and thus the

mathematical model gives accurate controllingof the swing link from actuators 1 and 2,difference between the joint 2 angle

2 is also

zero (0°), and thus while rotating the boom fromthe boom cylinder (actuator 3), themathematical model holds accurate controllingof the boom. The difference between the joint3 angle

3 is also very small (0.002°), and thus

while rotating the arm from the arm-cylinder(actuator 4), the mathematical model holdsaccurate controlling of the arm as well, and thedifference between the joint 4 angle

4 is also

very small (0.029°), and thus the mathematicalmodel holds accurate controlling of the bucketas well.

It can be seen that the bigger is the angle,the bigger is the difference. This may be dueto the accuracy of the CAD software as wellas of the Math tool, but these differences arevery small and near to zero, so the resultsare identical and thus acceptable.

Results also suggest the differences inthe coordinates of points 0P

A3, 0P

A4: [0.00005

0 0.00004 1]T, and [0.00082 0 0.00111 1]T

respectively. These show that the differencein the x direction for point A

3 is 0.05 mm and

for point A4 is 0.82 mm, the difference in the

y direction for both the points is zero and thedifference in the z direction for point A

3 is

0.04 mm and for point A4 is 1.11 mm. The

difference in the z direction is also small butcomparatively higher than other coordinatepoints because the origin of the frame {0}can be chosen to be anywhere on the Z

0 axis,

and as the position of the origin changesvertically on the Z

0 axis the z coordinates

also change.

Results of the direct kinematicmathematical model suggest that theproposed direct kinematic model is validateto be used to determine the coordinates of thebucket hinge point A

3, and the tool point A

4 for

an autonomous operation of the backhoeexcavator.

Above discussion shows the validation ofdirect kinematic model for maximum breakoutforce configuration. The direct kinematicmodel of backhoe excavator also validatedusing the MATLAB code and simulation resultsobtained from the modeling software ofAutodesk Inventor 2011. All the readings takenfor the position of backhoe assembly at whichthe link axes are aligned to each other for the

Figure 4: Results of the Coordinates ofBucket Tip Due to Variation of Angle 2, 3,

and 4 Obtained from MATLAB Code

178


purpose of validation of the direct kinematicmodel. Figure 4 shows the results ofcoordinates of bucket tip due to variation ofangle

2 (red curve),

3 (pink curve), and

4 (blue

curve) respectively obtained from MATLABcode.

Figure 5 shows the results of coordinatesof bucket tip due to variation of angle

2 (61.64°

to –56° = 117.64°) obtained from simulationat which the angles

3 and

4 are taken as zero

means axis of arm and bucket are aligned toaxis of boom.

Table 3 shows the comparison of bucket tipcoordinates obtained from MATLAB code andsimulation, due to variation in angle

2.

Figure 5: Results of the Coordinates ofBucket Tip Due to Variation of Angle 2

Obtained from Simulation

1. 61.6400 1673.1 2802.9 61.64 1672.98 2802.34

2. 48.5689 2161.7 2462.1 48.57 2164.21 2459.13

3. 35.4978 2560.6 2019.6 35.50 2561.48 2017.41

4. 22.4267 2849.1 1498.4 22.43 2848.58 1498.12

5. 9.3556 3012.2 925.4 9.36 3011.57 925.70

6. –3.7156 3041.5 330.4 –3.72 3041.44 338.58

7. –16.7867 2935.5 –255.8 –16.79 2936.63 249.96

8. –29.8578 2699.6 –802.9 –29.86 2699.02 802.81

9. –42.9289 2346.2 –1282.4 –42.93 2345.52 1282.28

10. –56.0000 1893.4 –1669.6 –56.00 1894.19 1668.38

Avg. 32.6778 2517.29 1404.95 32.68 2517.562 1404.471

Table 3: Comparison of Bucket Tip Coordinate 0PA4 Due to Variation in Joint Angle 2

S. No.

Results of MATLAB Code Results of SimulationInput Angle

2

(degree)Input Angle

2

(degree)X-Coordinate0xP

A4 (mm)

Z-Coordinate0zP

A4 (mm)

X-Coordinate0xP

A4 (mm)

Z-Coordinate0zP

A4 (mm)


3 (0° to

–120.45° = 120.45°) obtained from simulationat which the angles

2 is considered at an

angle of 61.64° and 4 is taken as zero means

axis of bucket is aligned to axis of arm.Table 4 shows the comparison of bucket tip

coordinates obtained from MATLAB code andsimulation, due to variation in angle

3.


4

(–145.41° to 30.04° = 175.45°) obtained fromsimulation at which the angles

2 is considered

at an angle of 61.64° and 3 is taken as zero

179


Figure 6: Results of the Coordinatesof Bucket Tip Due to Variation of Angle 3


1. 00.0000 1673.1 2802.9 00.00 1672.98 2802.34

2. –13.3833 1915.5 2632.9 –13.38 1915.18 2632.40

3. –26.7667 2111.8 2411.5 –26.77 2111.62 2410.82

4. –40.1500 2251.5 2150.6 –40.15 2251.30 2150.05

5. –53.5333 2327.1 1864.4 –53.53 2326.87 1863.90

6. –66.9167 2334.5 1568.5 –66.91 2334.17 1568.26

7. –80.3000 2273.1 1279.0 –80.30 2272.77 1278.59

8. –93.6833 2146.4 1011.5 –93.68 2146.07 1011.20

9. –107.0667 1961.2 780.6 –107.07 1960.72 780.18

10. –120.4500 1727.5 598.9 –120.45 1727.18 598.56

Avg. 60.225 2072.16 1710.08 60.224 2071.886 1709.63


S. No.


3

(degree)Input Angle

3


A4 (mm)

Z-Coordinate0zP

A4 (mm)

X-Coordinate0xP

A4 (mm)

Z-Coordinate0zP

A4 (mm)

means axis of arm is aligned to axis of boom.Table 5 shows the comparison of bucket tip

coordinates obtained from MATLAB code andsimulation, due to variation in angle

4.

Figure 7: Results of the Coordinatesof Bucket Tip Due to Variation of Angle 4


Ground Level

180


The comparison of results of MATLAB codeand simulation presented in Table 3 indicatethat the % variation in average value ofcoordinate X of bucket tip point 0P

A4 is

0.0108%, and % variation in average value ofcoordinate Y of bucket tip point 0P

A4 is

0.0341% due to % variation in average valueof angle

2 is 0.006732%. These results

indicate that the variations in results are veryless and the direct kinematic model isvalidated for angle

2.

Table 4 shows the comparison of results ofMATLAB code and simulation which presentthat the % variation in average value ofcoordinate X of bucket tip point 0P

A4 is

0.01322%, and % variation in average valueof coordinate Y of bucket tip point 0P

A4 is

0.02631% due to % variation in average valueof angle

3 is 0.00166%. These results indicate

that the variations in results are very less andthe direct kinematic model is validated forangle

3.

Table 5 shows the comparison of results ofMATLAB code and simulation which presentthat the % variation in average value ofcoordinate X of bucket tip point 0P

A4 is 0.113%,

and % variation in average value of coordinateY of bucket tip point 0P

A4 is 0.113% due to %

variation in average value of angle 4 is

0.04845%. These results indicate that thevariations in results are very less and the directkinematic model is validated for angle

4.

CONCLUDING REMARKSThis paper presents the complete fundamentalfoundation for the kinematics of the backhoeexcavators. Here, theoretical relations aredeveloped for the direct kinematics of thehydraulic backhoe excavator, which have notpreviously been presented in the literature for4-DOF. The developed relations can beutilized for autonomous digging operation.Here presented relations are developed withconsidering the links and joints are rigid.During the digging operation interactive forces

1. 30.0400 1397.2 2868.3 30.04 1401.01 2866.70

2. 10.5456 1580.6 2842.3 10.55 1584.26 2840.10

3. –8.9489 1744.8 2756.6 –8.95 1748.14 2753.87

4. –28.4433 1871.0 2621.0 –28.44 1873.85 2617.89

5. –47.9378 1944.7 2451.1 –47.94 1946.98 2447.72

6. –67.4322 1957.5 2266.4 –67.43 1959.13 2262.97

7. –86.9267 1907.9 2087.9 –86.93 1908.91 2084.71

8. –106.4211 1801.5 1936.3 –106.42 1802.09 1933.45

9. –125.9156 1650.7 1828.8 –125.92 1650.94 1826.51

10. –145.4100 1472.6 1777.8 –145.41 1472.76 1776.13

Avg. 65.77112 1732.85 2343.65 65.803 1734.807 2341.005


S. No.


4

(degree)Input Angle

4


A4 (mm)

Z-Coordinate0zP

A4 (mm)

X-Coordinate0xP

A4 (mm)

Z-Coordinate0zP

A4 (mm)

181


developed between soil and tool (for our caseit is bucket), also the digging forces and torquedeveloped by the actuators causes thebending effect in the link mechanism and theirjoints. Due to this the developed kinematicrelations will contain inaccuracies in the result.Here, for kinematic relations the resistiveforces offered by the soil-tool interactions anddigging forces developed by the actuators arenot considered. But the results obtained forvirtual movement of the backhoe excavatormechanism shows the excellent outcome interms of result. The proposed inversekinematic model with consideration ofdifferential motion of backhoe for velocity andacceleration, Inverse Jacobian and backhoestatic backhoe model are developed andcovered in the part-II of this paper(Bhaveshkumar and Prajapati, 2012).

A complete generalized mathematicaldirect kinematic model for four degrees offreedom backhoe excavator is developed andcan be applied to any backhoe excavator forautomated movements of the backhoeexcavator. The MATLAB codes are developedfor direct kinematics of the backhoe excavatorand the 3-D model of the backhoe excavatordeveloped using Autodesk InverterProfessional 2011. The parameters taken asinput of the MATLAB code of the proposeddirect kinematic model are obtained formaximum breakout force configuration of thebackhoe excavator, and the MATLAB codegives the identical results as compared to theproposed 3-D model of the backhoeattachment developed in Autodesk InventorProfessional 2011 for same configuration andthus the proposed kinematic model gives theexcellent accurate results. The proposed direct

kinematic model is validated through thecomparison of results of MATLAB code andsimulation and obtained results are identical.The developed direct kinematic relations canbe applied for autonomous operation for alltypes heavy duty backhoe excavators.

REFERENCES1. Bhaveshkumar P Patel and Prajapati J M

(2012), “Kinematics of Mini HydraulicBackhoe Excavator – Part: II”,International Journal of Mechanisms andRobotic Systems, September 10,Inderscience, Accepted for Publication.

2. Howard N Cannon (1999), “ExtendedEarthmoving with an AutonomousExcavator”, Unpublished Thesis of Masterof Science, The Robotics InstituteCarnegie Mellon University, Pittsburgh.

3. John J Craig (1989), Introduction toRobotics Machines and Control, 2nd

Edition, Addison-Wesley PublishingCompany, New York.

4. Koivo A J (1994), “Kinematics ofExcavators (Backhoes) for TransferringSurface Material”, Journal of AerospaceEngineering, Vol. 7, No. 1, pp. 17-32,American Society of Civil Engineering(ASCE).

5. Mittal R K and Nagrath I J (2003),Robotics and Control, 9th Reprint, TataMcGraw Hill, New Delhi.

6. Shaban E M, Ako S, Taylor C J andSeward D W (2008), “Development of anAutomated Vertically Alignment Systemfor a Vibro-Lance”, Journal of Automationin Construction, Vol. 17, pp. 645-655,Elsevier.

182


7. Vaha P K and Skibniewski M J (1993),

“Dynamic Model of Excavator”, Journal of

Aerospace Engineering, Vol. 6, No. 2,

pp. 148-158, American Society of Civil

Engineering (ASCE).

8. Zygmunt Towarek (2003), “Dynamics of aSingle-Bucket Excavator on a DeformableSoil Foundation During the Digging of theGround”, International Journal ofMechanical Sciences, Vol. 45, pp. 2053-1076, Elsevier.

KINEMATICS OF MINI HYDRAULIC BACKHOE EXCAVATOR – PART: I · KINEMATICS OF MINI HYDRAULIC BACKHOE...

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