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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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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.
4 D.O.F. MANIPULATORS
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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.
5 D.O.F. MANIPULATORS
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
6 D.O.F. MANIPULATORS
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
Top view of configuration
Side view of configuration
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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
(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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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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Kinematics of Four Degree Of Freedom Parallel Manipulator
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)