robotics term paper

7/29/2019 Robotics Term Paper

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Kinematics of Four Degree Of Freedom Parallel Manipulator

Four Degree Of Freedom parallel manipulator

- Kinematics

Vinu.K.S

Mechanical

ME

6337-410-091-07147


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Contents

SlNo

Description PageNo

1 Background 32 Types of Parallel manipulator 4

3 Kinematic analysis of 4 dof parallel manipulator 4a Inverse Kinematics 5b Direct Kinematics 6

c Velocity Equation and Jacobian 7d Singularity analysis 8e Workspace analysis 9

f Conclusion 94 References 13


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1.Background

A generalized parallel manipulator is a closed-loop kinematic chain mechanism

whose end-effectors are linked to the base by several independent kinematic chains.The definition of parallel manipulator is open: it includes for instance redundant

mechanisms with more actuators than the number of controlled degrees of freedom of

the end-effectors, as well as manipulators working in cooperation.

1.1Characteristics of parallel manipulatora.) At least two chains support the end-effectors. Each chain contains at least

one simple actuator. There is an appropriate sensor to measure the value of the

variables associated with the actuation (rotation angle or linear motion).

b.)The number of actuators is the same as the number of degrees of freedom of

the end-effectors.

c.)The mobility of the manipulator is zero when the actuators are locked.

1.2 Interesting aspect

i ) minimum of two chains allows us to distribute the load on the chains

ii) the number of actuators is minimal.

iii) the number of sensors necessary for the closed-loop control of the mechanism

is minimal.

iv) when the actuators are locked, the manipulator remains in its position;this is an important safety aspect for certain applications, such as medical robotics.


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1.3 Classification based on degree of freedom

The commonly used types are three, four, five and six degrees of freedom.

3 D.O.F. MANIPULATORS

Manipulators with 3 degrees of freedom in translation are extremely useful for pick-and-

place and machining operations. The most famous robot with three translation degrees

of freedom is the Delta. Another member of this family is the Star robot with the

notable difference that the Star is over-constrained (each leg restricts two rotational

degree of freedom of the platform) while the Delta is not.

Another interesting member of the same family is the Orthoglide robot, developed for

machine-tool application .The main interest of this robot is that it presents relatively

homogeneous kinematic performances in its useful workspace.


It is not possible to design a 4 d.o.f. with identical legs. Hence such a design will have to

rely either on a passive constraint mechanism, a specific geometry of the legs, different,

legs, less than 4 legs, or a specific mechanical design. In 1975, Kovermans realised a

flight simulator mechanism based on a passive constraint system. The degrees of

freedom are the three rotations and one translation about the z axis. One way to have

the same chains is to use flexible legs, as proposed by Rebman. Using less than 4 legs

may also lead to a manipulator with 4 d.o.f., either with an appropriate actuation

scheme.


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Koevermans flight simulator H4 robot

which uses various clever configurations of the platform to get 4 d.o.f., 3 translations

and one rotation, with a design that allows for large rotation ability.


Robots with 5 d.o.f will also have to rely on passive constraint mechanisms, specificgeometries or design. Such a structure is of particular interest in the machine-tool fieldfor so-called five-axis machining. Indeed 6 d.o.f. are not strictly necessary for machining

as the rotation of the spindle adds a degree of freedom.


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There are mainly three kinds PUS, RUS and UPS kinematic chain. Motion simulators

are mainly six degree of freedom systems. Eg include Stewart Platform, Kappel platform

etc.

The realizations of 6 degrees of freedom fully parallel manipulators are based on the

use of 6 generators of the motion group. These work with chains of the RRPS,

RPRS,PRRS, RRRS types.

Hexa glide robot used as machine tool Flexibility in the motion of prismatic joint.

Advantages of parallel Manipulator

Parallel manipulators have high stiffness, larger load capacity, low inertia, high

accuracy, High velocity, high acceleration, no accumulation of positional error. Hence

they are suitable candidate for industrial purposes.

List of 4 dof mechanisms

1. 4 UPU 17. RURR

2. 4 PUU 18. RURP

3. RRRU 19. RUPR

4. RRPU 20. PURR

5. RPRU 21. RUPP

6. PRRU 22. PURP

7. RPPU 23. PUPR

8. PRPU 24. UPRR

9. PPRU 25. UPRP

10.RRUR 26. UPPR

11.RRUP

12.RPUR

13.PRUR

14.RPUP


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15.PRUP

Description of PUU configuration

The figure shows PUU configuration with A, B, C, D, a, b, c and d represent the centres

of the Hooke joints. The manipulator consists of a

universal (UPU) in movable platform (rectangle ABCD), a base and four fixed length

limbs which connect the movable platform at point A,a Hooke joint and connect the

base at point a, b, c and d with a Hooke joint and a prismatic joint, The lengths of the

limbs are li (i=1,2,3,4). The four prismatic joints are active joints and are located on

three parallel rails where the distances are k and n, respectively. So, the manipulator

can have large workspace along rails. A reference frame (O-XYZ) is established with

start point O of the first rail being taken as origin . The x-axis is coincident with the rail,

the z axis is perpendicular to the base and Y-axis satisfies right hand rule.

A body fixed coordinate system P-uvw is created with geometry centre P of the movable

platform being taken as origin, u axis is parallel to side AB, the v axis is parallel to side

AD and the w axis is perpendicular to movable platform.


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Top view of configuration

Side view of configuration


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Front view of configuration

Inverse Position analysis

Inverse position analysis of parallel manipulator is concerned with the

determination of the displacements of the four active prismatic joints when the position

and orientation of the movable platform are given.

In order to simplify the analysis we suppose that l1 is equal to l2. Because point

P(x,y,z)T

is the geometrical centre of the movable platform and side AB is parallel to the

X-axis,

xA=xD = x-L.

xB=xC = x+L. (1)

yA=yB = y-H*Cos (2)

yC=yD = y+H*Cos

zA=zB = z-H*Sin (3)

zc=zD = z+H*Sin

Let Sin be represented asS and Cos : C

abBA is a quadrangle - isosceles trapezoid.

xa + xb =2x.


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Constraint Equation

Fixed length of kinematic chain

aB=bA=l1 , cD=l3, dC=l4

(xB xa)2 + yB2 + zB2 =l12

(xA xb)2 + yA

2 + zA2 =l1

2

(xC xd)2 + (yC k)

2 + zC2 =l3

2

(xD xc)2 + (yD n)

2 + zD2 =l4

2

Substituting (1), (2), (3) in (5)

(x+L xa)2 + (y-H*C)2 + (z-H*S)2 =l1

2

(x+L xb)2 + (y-H*C)2 + (z-H*S)2 =l1

2

(x+L xd)2 + (y-H*C-k)2 + (z+H*S)2 =l3

2

(x+L xd)2

+ (y-H*C-n)2

+ (z+H*S)2

=l42

Solving for xa,xb,xc,xd

xa = x + L [l12(y-H*C)

2(z-H*S)

2]

xb = x - L [l12(y-H*C)

2(z-H*S)

2]

xc = x - L [l42(y-n+H*C)

2(z+H*S)

2]

xd= x - L [l32(y-k+H*C)

2(z+H*S)

2]

Forward Position Kinematics

With known set of actuated inputs to find position and orientation of platform. Used for

control, motion planning and calibration purposes.


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Given xa, xb, xc, xd

Define f1= (xB xa) = L + (xbxa)/2

f2 = (xC xd) = L -xd+ (xb +xa)/2

f3 = (xD xc) = L -xc+ (xb +xa)/2

Substituting (2),(3) and (11) into (5)

f12 + yB

2 + zB2 = l1

2 (12)

f22 + (yc-k)

2 + zc2 = l3

2 (13)

f32+ (yc-n)

2+ zc

2= l4

2(14)

Subtracting (13) from (14)

yc = (l32 - l4

2 f22 + f3

2 + n2 -k2)/2*(n-k) = f4

zc = (l32- f2

2- (yc-k)

2)

= f5

positive value of zc is considered

Now yB = yc2*H* C and zB = zc2*H* S

yB = f42*H* C (16)

zB = f52*H* S

Velocity Equations

Closed form solution of inverse kinematics is differentiated

Eqn (7),(8),(9) and (10)

Rearranging these equations give


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Parallel manipulator Jacobian is given as

Singularity Analysis

- Loss of Controllability

- Degradation of natural stiffness

Det JA = 0

xa-2*L-xb =0 or z = y*tan

First condition implies aB parallel to limb bA.


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Second condition implies

plane abBA is coincident with moving platform

Large tilting angle about X-axis

Det JB = 0 implies

x+L-xa =0

x+L-xb =0

x-L-xc =0

x+L-xd =0

which means one of the four limb is perpendicular to rail,any pair of limbs are mounted

crossly

Workspace analysis

For a given manipulator analysis of volume and shape of workspace. This part is used

for industrial application .

From inverse kinematics we get

(x-x1)2 + (y-y1)

2 + (z-z1)2 = l1

2


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(x-x2)2

+ (y-y2)2

+ (z-z2)2

= l12

(x-x3)2 + (y-y3)

2 + (z-z3)2 = l3

2

(x-x4)2 + (y-y4)

2 + (z-z4)2 = l4

2

x1=xa-L , y1=H* C, z1=H* S,

x2=xb+L ,y2=H* C, z2=H* S,

x3=xd-L , y3=k- H* C, z3=-H* S,

x4=xc+L , y4= n-H* C, z4=-H* S,

Specifying (xa,xb,xc,xd) and orientation

Constant Orientation workspace

Region that can be reached by reference point (P) on movable platform

Enveloping regions generated by four spheres.

centre of spheres roll on four lines parallel to three tracks

i denotes acute angle between ith limb and rail track

constraint equation is for limbs min i max

1 = tan-1[(yB

2+zB

2)]1/2/(xB - xa)

2 = tan-1[(yA

2+zA

2)]1/2/(xb - xA)

3 = tan-1[(yD n)

2+zA

2)]1/2/(xc - xD)

4 = tan-1[(yc k)

2+zc

2)]1/2/(xC - xd)

Reachable Work space

Region that can be reached by reference point with at least one orientation.

x1=xa-L, y12+ z1

2 = H2

x1=xb+L, y22+ z2

2 = H2

x3=xd-L, (y3k)2+ z3

2 = H2

x1=xc+L, (y4n)2 + z4

2 = H2

Envelope of 4 cylinders


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Conclusion

4 PUU configurations are suitable for industrial application because of the following

reason which has been obtained through kinematic analysis.

large tilting angle.

Larger workspace along the rail.

Closed form solution for inverse and direct kinematics problem.


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References

1. Kinematic Analysis Of a Novel 4-DOFs Parallel Manipulator- Jianfeng Yuan and

Xianmin Zhang.(Proceedings Of IEEE International Conference on information

Acquistion August 2006)

2 Mechanism Analysis Of a Novel four degree of freedom Parallel Manipulator

Based on larger Workspace Hairong Fang, Jianghong Chen.(Proceedings Of

IEEE International Conference on automation and Logistics August 2009).

3 Singularities of Parallel Manipulators a geometrical treatment-Guanfeng Liu,

Yunjiang Lou (IEEE transactions on Robotics and Automation)

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