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ROBOTICS: ADVANCED CONCEPTS
&
ANALYSIS
MODULE 2 ELEMENTS OF ROBOTS: JOINTS, LINKS,
ACTUATORS & SENSORS
Ashitava Ghosal1
1Department of Mechanical Engineering&
Centre for Product Design and Manufacture
Indian Institute of ScienceBangalore 560 012, India
Email: [email protected]
NPTEL, 2010
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1 CONTENTS
2 LECTURE 1
Mathematical Preliminaries
Homogeneous Transformation3 LECTURE 2
Elements of a robot JointsElements of a robot Links
4 LECTURE 3
Examples of D-H Parameters & Link TransformationMatrices
5 LECTURE 4
Elements of a robot Actuators & Transmission
6 LECTURE 5
Elements of a robot Sensors
7 ADDITIONAL MATERIAL
Problems, References, and Suggested Reading
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OUTLINE1 CONTENTS
2 LECTURE 1
Mathematical PreliminariesHomogeneous Transformation
3 LECTURE 2
Elements of a robot JointsElements of a robot Links
4 LECTURE 3
Examples of D-H Parameters & Link TransformationMatrices
5 LECTURE 4
Elements of a robot Actuators & Transmission6 LECTURE 5
Elements of a robot Sensors
7 ADDITIONAL MATERIAL
Problems, References, and Suggested Reading
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ORIENTATION OF A RIGID BODY
Position of one point on the rigid body not enough todescribe it in 3D space.
Orientation of a rigid body B with respect to {A}
YA
YB
Bp
Rigid Body B
k
ZA
XB
{B}
{A}
ZB
OA,OB
XA
Figure 2: Orientation of a rigid body
Attach coordinatesystem,
{B
}, to rigid
body B.
Origin of{B}coincident with origin of{A} (see Figure 2).Obtain description of{B} with respect to{A}.
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ORIENTATION DIRECTION COSINES
Unit vectors XB, YB, and ZB, attached to B, can bedescribed in
{A
}AXB = r11XA + r21YA + r31ZAAYB = r12XA + r22YA + r32ZA (2)AZB = r13XA + r23YA + r33ZA
rij, i,j = 1,2,3 are called direction cosinesr11 =
A XB XAMagnitude of unit vectors are 1 r11 is cosine of anglebetween AXB and XA. All rijs are cosines of angles.
Define 3 3 rotation matrixA
B[R] with rij, i,j = 1,2,3 asits elements. Columns of AB[R] are
AXB,AYB, and
AZB.AB[R] completely describes all three coordinate axis of{B}with respect to {A}.A
B
[R] gives orientation of rigid body B in{
A}
.
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ORIENTATION PROPERTIES OF AB[R]
The vector Bp in rigid body B can be described in {A} byA
p =AB[R]
B
p (3)The columns of the rotation matrix are unit vectors,orthogonal to one another AXB A XB =A YB A YB =A ZB A ZB = 1, andAXB
A YB =
A YBA ZB =
A ZBA XB = 0
The determinant of the rotation matrix is +11.AB[R]
TAB[R] = [U], where [U] is a 3 3 identity matrix.
Inverse is same as transpose BA [R]= AB[R]
1= AB[R]
T.
Three eigenvalues of AB[R] are +1, e, where =
1,and = cos
1
(
r11+r22+r33
1
2 ).The eigenvector corresponding to +1 isk = (1/2sin)[r32 r23, r13 r31, r21 r12]T, = {0,n},where n is a natural number. For = {0,2n}, there is norotation and = 2(n
1) needs to handled specially!
1det(AB[R]) = +1 is due to the use of a right-handed coordinate system.ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 7 / 1 38
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ORIENTATION USING (k,)
Rotation axis k fixed in {A} and {B}: Ak = AB[R]Bk = 1Bk.First equality is transformation of a vector from
{B
}to
{A} and the second equality follows from equation (3) andthe definition of an eigenvector.Elements of AB[R] in terms of (kx,ky,kz)
T and angle
r11 = kx2(1
cos) + cos
r12 = kxky(1 cos) kzsinr13 = kzkx(1 cos) + ky sinr21 = kxky(1 cos) + kzsinr22 = ky
2(1
cos) + cos (4)
r23 = kykz(1 cos) kxsinr31 = kzkx(1 cos) kysinr32 = kykz(1 cos) + kxsinr33 = kz
2(1
cos) + cos
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ORIENTATION SIMPLE ROTATION
Rotation axis k is parallel to XA and hence to XB Rotation about X axis.
AB[R] = [R(X,)] =
1 0 00 cos sin
0 sin cos
(5)
ZA
Bp
Rigid Body BXA, XB
k
{B}
{A}
ZB
OA, OB YA
YB
Figure 3: Rotation about X by angle
Rotation about Xshown in Figure 3.
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SIMPLE ROTATIONS (CONTD.)
Rotation about Y and Z
[R(Y,)] =
cos 0 sin0 1 0
sin 0 cos
(6)
[R(Z,)] =
cos sin 0sin cos 0
0 0 1
(7)
Rotation matrices in equations (5) through (7) calledsimple rotations.
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SUCCESSIVE ROTATIONS
Two successive rotations:1 Initially B is coincident with {A}.2
First rotation relative to {A}. After first rotation{A} {B1}.3 Second rotation relative to {B1}. After second rotation
{B1} {B}.
{B}
{A}
ZB
ZA
YA
YB
Rigid Body BXA
XB1
XB
YB1ZB1
OA,OB1,OB
Rigid Body B1
Figure 4: Successive rotation
Resultant rotation:AB[R] =
AB1
[R] B1B [R] Note orderof matrix multiplication.
Resultant of n rotations AB[R] =
AB1
[R] B1B2 [R] ...Bn1B [R]
Matrix multiplication is noncommutative in general AB1
[R] B1B [R] =B1B [R]
AB1
[R] Order of rotation is important!
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ORIENTATION THREE ANGLES
Orientation described by 3 independent parameters Three rotations completely describe orientation of a rigidbody.
Three successive rotations about axes fixed to moving body
Rotations about three distinct axes: 6 combinations
X-Y-Z, X-Z-Y, Y-Z-X, Y-X-Z, Z-X-Y & Z-Y-XRotations about two distinct axes: 6 combinations X-Y-X, X-Z-X, Y-X-Y, Y-Z-Y, Z-X-Z, & Z-Y-Z
Rotations about axes fixed in space 12 possible
combinations for 3 and 2 distinct axes.Minimal representation of orientation of rigid body Onlythree parameters (angles) and no constraints.
Three angles also called Euler angles.
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XYZEULER ANGLES
Rotation about X AB1 [R] = [R(X,1)] =
1 0 00 cos1 sin10 sin1 cos1
Rotation about Y B1B2 [R] = [R(Y,2)] =
cos2 0 sin20 1 0
sin2 0 cos2
XA, XB1
YA
YB1
ZA
ZB1
XA, XB1
YA 2
ZA
ZB1
ZB2
XA, XB1
{B2}
{A}, {B1}
ZA
YA
YB1
YB
ZB, ZB2
XB2XB2
{B}
OA, OB OA, OB
3
XB
{B2}
YB1 , YB2
ZB1
{A}{B1}
1
OA, OB
{A}, {B1}
Figure 5: XYZ Euler angles
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XYZEULER ANGLES (CONTD.)
Rotation about Z
B2B [R] = [R(Z,3)] =
cos3 sin3 0sin3 cos3 0
0 0 1
Resultant rotation AB
[R] = AB1
[R] B1B2
[R] B2B
[R] = c2c3 c2s3 s2s1s2c3 + s3c1 s1s2s3 + c3c1 s1c2
c1s2c3 + s3s1 c1s2s3 + c3s1 c1c2
Note: ci, si denote cosi and sini, respectively.
AB[R] gives orientation of{B} given XYZ angles.AB[R] will be different if order of rotations is different.
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XYZEULER ANGLES (CONTD.)
Given AB[R], find XYZ Euler angles.
Algorithm rij i for XYZ rotationsIf r13 = 1, then
2 = atan2(r13,
(r211 + r212))
1 = atan2(r23/cos2, r33/cos2)3 = atan2(r12/cos2, r11/cos2)Else
If r13 = 1, 1 = atan2(r21, r22), 2 = /2, 3 = 0If r31 = 1, 1 = atan2(r21, r22), 2 = /2, 3 = 0
atan2(y,x): four-quadrant arc-tangent function (see
function atan2 in MATLAB R) 1, 2, 3 [,].Two sets of values of1, 2 and 3.
2 = /2 1, 2 not unique 1 3 can be found.Singularities in Euler angle representation.
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ZYZEULER ANGLES
OA, OB OA, OB
ZA, ZB1 ZA, ZB1
YB1, YB2
XB1
XB2
XB
{A}{B1}
1
ZA, ZB1
XA
XB1
YA
YB1
OA, OB
{B2}{A}, {B1}
2
XA
XB1 XB2
YA
YB1, YB2
ZB2
3
ZB2 , ZB
XA
YA
YB
{A}{B}
Figure 6: ZYZ Euler angles
A
B[R] =
c1 s1 0s1 c1 00 0 1
c2 0 s20 1 0
s2 0 c2
c3 s3 0s3 c3 00 0 1
=
c1c2c3 s1s3 c1c2s3 s1c3 c1s2s1c2c3 + c1s3 s1c2s3 + c1c3 s1s2
s2c3 s2s3 c2
(8)
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ZYZEULER ANGLES (CONTD.)
Algorithm rij ZYZ Euler anglesIf r33
=
1, then
2 = atan2(
(r231 + r232), r33)
1 = atan2(r23/sin2, r13/sin2)3 = atan2(r32/sin2,r31/sin2)
Else
If r33 = 1, then1 = 2 = 0, 3 = atan2(r12, r11)
If r33 = 1, then1 = 0, 2 = , 3 = atan2(r12,r11)
Two possible sets of ZYZ Euler angles (1,2,3) for agiven AB[R].
r33 = 1 Singularity 1 3 can be found.For unique 1 and 3 when r33 = 1 Choose 1 = 0.
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OTHER REPRESENTATIONS
Euler parameters (see Kane et al., 1983, McCarthy, 1990):
4 parameters derived from k = (kx,ky,kz)T and angle 3 parameters = k sin/2fourth parameter 4 = cos/2One constraint 1
2 + 22 + 3
2 + 42 = 1
Quaternions (pp. 185 Arfken(1985), Wolfram website2): 4parameters
Sum of a scalar q0 and a vector (q1,q2,q3)T
Q= q0 + q1i + q2j + q3k.i, j, and k are the unit vectors in 3.Product of two quaternion also a quaternion.
Inverse of quaternion exists.q20
+ q21
+ q22
+ q23
= 1 Unit quaternion representorientation of a rigid body in 3.
2
See Known Bugs in Module 0, slide #10 for viewing links.ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 1 8 / 1 38
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ORIENTATION OF A RIGID BODY
SUMMARY
Orientation of a rigid body in 3 is specified by 3
independent parameters.Various representation of orientation with their ownadvantages and disadvantages
Rotation matrix AB[R] 9 rijs + 6 constraints Toomany variables and constraints but ideal for analysis.
Axis (kx,ky,kz)T
and angle 4 parameters + oneconstraint k2x + k
2y + k
2z = 1 Useful for insight and
extension to screws, twists and wrenches.Euler angles: 3 parameters and zero constraints Minimalrepresentation but suffer from problem of singularitiesand
sequence must be known.Euler parameters and quaternions: 4 parameters +1constraint Similar but not exactly same as axis-angleform, no singularities, used in motion planning.
Can convert from one representation to any other for
regular cases!ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 1 9 / 1 38
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COMBINED TRANSLATION AND
ORIENTATION OF A RIGID BODY
{A}
OA
OB
ApAOB
Bp
Rigid Body B
XA
{B}
YA
ZA
XB
YB
ZB
Figure 7: General transformation
{A} and {B}, OA andOB not coincident
Orientation of{B} withrespect to {A} AB[R]Ap =A OB +
AB[R]
BpAOB locates OB withrespect to OA.
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OUTLINE1 CONTENTS
2 LECTURE 1
Mathematical PreliminariesHomogeneous Transformation
3 LECTURE 2
Elements of a robot JointsElements of a robot Links
4 LECTURE 3Examples of D-H Parameters & Link TransformationMatrices
5 LECTURE 4
Elements of a robot Actuators & Transmission6 LECTURE 5
Elements of a robot Sensors
7 ADDITIONAL MATERIAL
Problems, References, and Suggested Reading
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4 4 T M
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44 TRANSFORMATION MATRIXCombined translation and orientation AP = AB[T]
BPA
P andB
P are 4 1 homogeneous coordinates (seeAdditional Material) constructed by concatenating a 1 toAp and BpAP = [Ap | 1]T and BP = [Bp | 1]T4
4 homogeneous transformation matrix AB[T] is formed as
AB[T] =
AB[R]
AOB0 0 0 1
(9)
In computer graphics and computer vision
Last row is
used for perspective and scaling and not [0 0 0 1].Upper left 3 3 matrix is identity matrix Puretranslation.
Top right 3 1 vector is zero Pure rotation.
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4 4 TRANSFORMATION MATRIX
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4 4 TRANSFORMATION MATRIX PROPERTIES
Inverse of AB[T] is given by
AB[T]
1=
AB[R]
T AB[R]TAOB
0 0 0 1
Two successive transformations {A} {B1} {B} givesAB[T] =
AB1
[T] B1B [T].
AB[T] =
AB1
[R]B1B [R]AB1
[R]B1OB +AOB1
0 0 0 1
n successive transformationsAB[T] =
AB1
[T] B1B2 [T] ...Bn1B [T]
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PROPERTIES OF A [T ] (CONTD )
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PROPERTIES OF AB[T] (CONTD.)
Four eigenvalues of AB[T] are +1, +1,e, same as for
AB[R].
Eigenvectors for +1 is k No other eigenvector!AB[T] represents the general motion of a rigid body in 3Dspace 6 parameters must be present.General motion of rigid body as a twist Rotation abouta line and translation along the line.
Direction of the line: (kx,ky,kz)T 2 independentparameters.Rotation about the line: angle 1 independentparameter.Location of the line in 3: (k,Y k) whereY = ([U]
AB[R]
T
)A
OB
2(1cos) 4 independent parameters in k and Y k.Translation along the line by an amount (AOB k).
More details about twists, eigenvalues and eigenvectors
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SUMMARY
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SUMMARY
A rigid body B in 3D space has 6 DOF with respect toanother rigid body A: 3 for position & 3 for orientation.
Definition of a right-handed coordinate system {A} X,Y, Z and origin OA.Rigid body B conceptually identical to a coordinate system{B}.Position of rigid body
Position of a point of interest on
rigid body with respect to coordinate system {A} 3Cartesian coordinates: Ap = (px,py,pz)
T.Orientation described in many ways: 1) by 3 3 rotationmatrix AB[R], 2) (, k) or angle-axis form, 3) 3 Eulerangles, 4) Euler parameters & quaternions.Algorithms available to convert one representation toanother.4 4 homogeneous transformation matrix, AB[T], representposition and orientation in a compact manner.
Properties ofA
B[T] can be related to a screw.ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 2 5 / 1 38
OUTLINE
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OUTLINE1 CONTENTS
2 LECTURE 1
Mathematical PreliminariesHomogeneous Transformation
3 LECTURE 2
Elements of a robot JointsElements of a robot Links
4 LECTURE 3Examples of D-H Parameters & Link TransformationMatrices
5 LECTURE 4
Elements of a robot Actuators & Transmission6 LECTURE 5
Elements of a robot Sensors
7 ADDITIONAL MATERIAL
Problems, References, and Suggested Reading
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JOINTS INTRODUCTION
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JOINTS INTRODUCTION
A joint connects two or more links.
A joint imposes constraints on the links it connects.2 free rigid bodies have 6 + 6 degrees of freedom.Hinge joint connecting two free rigid bodies 6 + 1degrees of freedom.Hinge joint imposes 5 constraints, i.e., hinge joint allows 1
relative (rotary) degree of freedom.
Degree of freedom of a joint in 3D space: 6 m where m isthe number of constraint imposed.
Serial manipulators
All joints actuated
One-degree-of-freedom joints used.Parallel and hybrid manipulators Some joints passive Multi-degree-of-freedom joints can be used.
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TYPES OF JOINTS
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TYPES OF JOINTS
D O F
Figure 8: Types of joints
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CONSTRAINTS IMPOSED BY A
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CONSTRAINTS IMPOSED BY A
ROTARY (R) JOINTRigid bodies, {i1} and
{i}, connected by a rotary(R) joint (Figure 9).
{i} can rotate about k,with respect to {i1}, byangle i0i [R] = 0i1[R] i1i [R(k,i)]Three independentequations in the matrixequation above; i is
unknown 2 constraintsin the 3 equations.
{0}
{i 1}
{i}
Oi1
Oi
ip
i1p
0Oi1
0Oi
0p
k
Li
iRigid Body i 1
Rigid Body i
Rotary Joint
Figure 9: A rotary joint
For a common point P on the rotation axis along line Li0p =0 Oi1 + 0i1[R]
i1p =0 Oi + 0i [R]ip 3 constraints
present.ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 2 9 / 1 38
CONSTRAINTS IMPOSED BY A
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CONSTRAINTS IMPOSED BY A
ROTARY (R) JOINT IN A LOOP
Rotary joint in aloop 2 ends {L}and
{R
}.
k
Li
i
Rotary Joint
Rigid Body i 1
Oi1
Oi
{L}
LOi1
i1p
i
p
{R}
Rigid Body i
ROi
LD
{i 1}
{i}
Figure 10: A rotary joint in a loop
Two constraints in Li [R] =Li1[R]
i1i [R(k,i)] =
LR[R]
Ri [R]
since i is an unknown variable.Three constraints inLp =L Oi
1 +
Li
1[R]i1p =L D +R Oi + Ri [R]
ip
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CONSTRAINTS IMPOSED BY A
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CONSTRAINTS IMPOSED BY A
PRISMATIC (P) JOINTTwo rigid bodies,
{i
1}
and{
i}
,connected by asliding/prismatic (P)joint
Orientation of
{i
}is
same as {i1} 3independent constraintsin 0i [R] =
0i1[R]
{i} can slide by di,
alongL
i, with respectto {i1}
{0}
k
Li
di
Rigid Body i 1
Rigid Body i
{i 1}
{i}
Oi1
Oi
0Oi1
i1p
ip
0Oi Prismatic Joint0p
Figure 11: A prismatic joint
2 constraints in 0Oi1 + 0i1[R]i1p + dik =0 Oi + 0i [R]
ip
since di is an unknown variable.
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CONSTRAINTS IMPOSED BY A
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CONSTRAINTS IMPOSED BY A
SPHERICAL (S) JOINT
Spherical(S) or ball andsocket joint allows threerotations.
S joint can berepresented as 3intersecting rotary(R)joints. {0}
Rigid Body i 1
Rigid Body i
{i}
{i 1}
0Oi10Oi
Spherical Joint
i1p
ip
0p
Figure 12: A spherical joint
3 constraints: 0p =0 Oi1 + 0i1[R]i1p =0 Oi + 0i [R]
ip
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CONSTRAINTS IMPOSED BY A
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CONSTRAINTS IMPOSED BY A
HOOKE (U) JOINT
Common in many parallel manipulators.
Equivalent to two intersecting rotary (R) joints Twodegrees of freedom
4 constraints.
For a common point P at the intersection of two rotationaxes 0p =0 Oi1 + 0i1[R]
i1p =0 Oi + 0i [R]ip 3
constraints.0i [R] =
0i1[R]
i1i [R(k1,1i)][R(k2,2i)] Only 1
constraint present as 1i and 2i are unknown rotationsabout k1 and k2
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CONSTRAINTS IMPOSED BY A
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CONS N S OS
SPHERICAL-SPHERICAL (SS) JOINT
PAIR
The SS pair appearin many parallelmanipulators.
Distance betweentwo spherical joint isconstant. {L}
Rigid Body i
Si
{i}
LSiLOi
iSi
{j}
Sjj
Sj
RORSj
lij
Rigid Body j
S-S Pair
LD
Figure 13: A SS pair in a loop
One constraint equation(LSi (LD +R Sj)) (LSi (LD +R Sj)) = l2ij, lij is aconstant.
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OUTLINE
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1 CONTENTS
2 LECTURE 1
Mathematical PreliminariesHomogeneous Transformation
3 LECTURE 2
Elements of a robot JointsElements of a robot Links
4 LECTURE 3Examples of D-H Parameters & Link TransformationMatrices
5 LECTURE 4
Elements of a robot Actuators & Transmission6 LECTURE 5
Elements of a robot Sensors
7 ADDITIONAL MATERIAL
Problems, References, and Suggested Reading
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LINKS INTRODUCTION
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A link A is a rigid body in 3D space Can be describedby a coordinate system {A}.A rigid body in 3 has 6 degrees of freedom 3 rotation+ 3 translation 6 parameters (see Lecture 1)For links connected by rotary (R) and prismatic (P),possible to use 4 parameters Denavit-Hartenberg(D-H)parameters (see Denavit & Hartenberg, 1955).
4 parameters since lines related to rotary(R) and prismatic(P) joint axis are used.
For multi-degree-of-freedom joints Use equivalentnumber of one-degree-of-freedom joints.
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LINKS DENAVIT-HARTENBERG
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(D-H) PARAMETERS
A word about conventions Several ways to derive D-Hparameters!
Convention used:
1 Coordinate system{
i}
is attached to the link i.2 Origin of{i} lies on the joint axis i Link i is after joint
i.3 after for serial manipulators Numbers increasing from
fixed {0} Link 1 {1} . . . Free end {n}.4 after for parallel manipulators Not so straight-forward
due to one or more loops.
Convention same as in Craig (1986) or Ghosal(2006).
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ASSIGNMENT OF COORDINATE AXES
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Three intermediate links {i 1}, {i} and {i+ 1}.Rotary joint axis Labeled Zi1, Zi and Zi+1 .
{0}
X0
Y0
Z0
Zi+1
Oi1
Oi{i 1}
{i}
Xi1
Link i 2
Link i 1
Link i
Link i + 1
Li1
Li
Li+1
ZiZi1
ai1
di
i
i1
Figure 14: Intermediate links and D-H parameters
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D-H PARAMETERS ASSIGNMENT
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OF COORDINATE AXIS (CONTD.)
For coordinate system {i 1}Zi1 is along joint axis i 1.Xi1 is chosen along the common perpendicular betweenlines Li1 and Li.
Yi1 = Zi1 Xi1 Right-handed coordinate system.The origin Oi1 is the point of intersection of the mutualperpendicular line and the line Li1.
For coordinate system {i}: Zi is along the joint axis i, Xiis along the common perpendicular between Zi and Zi+1,
and the origin of{i}, Oi, is the point of intersection of theline along Xi and line along Zi (see Figure 14).
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D-H PARAMETERS FOR LINK i
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Twist anglei1 Twist angle for link i has subscripti 1!
Angle between joints axis i
1 and i & measured aboutXi1 using right-hand rule (see Figure 14).Signed quantity between 0 and radians.
Link length ai1 Link length for link i is ai1!Distance between joints axis i 1 and i & measured alongXi
1 (see Figure 14).
Always a positive quantity.Link offset di
Measured along Zi from Xi1 to Xi Can be < 0.Joint i rotary di constant.Joint i prismatic
di is joint variable (see Figure 14).
Rotation angleiAngle between Xi1 and Xi measured about Zi usingright-hand rule Between 0 and radians.Joint i is prismatic i is constant.Joint i rotary
i is joint variable (see Figure 14).
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D-H PARAMETERS SPECIAL CASES
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Consecutive joints axis parallel commonperpendiculars
i1 = 0,, ai
1 along any common perpendicular.
Joint i is rotary (R) di is taken as zero.Joint i is prismatic (P) i is taken as zero.Consecutive P joints parallel ds not independent!
Consecutive joints axis intersecting ai1 = 0 and Xnormal to plane.
First link {1}: choice of Z0 and thereby X1 is arbitrary.R joint Choose {0} and {1} coincident &i1 = ai1 = 0, di = 0.P joint Choose {0} and {1} parallel &i
1 = ai
1 = i = 0.
Last link {n}: Zn+1 not definedR joint Origins of{n} and {n + 1} are chosencoincident & dn = 0 and n = 0 when Xn1 aligns with Xn.P joint Xn is chosen such that n = 0 & origin On ischosen at the intersection of Xn
1 and Zn when dn = 0.
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LINK TRANSFORMATION MATRICES
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Four D-H parameters describe link i with respect to linki 1
Orientation of of{
i}
with respect to{
i
1}
is given byi1i [R] = [R(X,i1)][R(Z,i)]Location of origin of{i} with respect to {i 1} is given byi1Oi = ai1 i1Xi1 + di i1Zi
Recall i1Xi1 = (1,0,0)T. The vector i1Zi is the last
column ofi
1
i [R] and is (0,si1 ,ci1)T3
.The 4 4 transformation matrix relating {i} with respectto {i1} is
i1i [T] =
ci si 0 ai1
sici1 cici1 si1 si1disisi1 cisi1 ci1 ci1di
0 0 0 1
(10)3The symbols si1 , ci1 denote sin(i1) and cos(i1),
respectively. Please seeNotations
in Module 0.ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 4 2 / 1 38
LINK TRANSFORMATION MATRICES
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i
1
i [T
] is a functiononly
one joint variable d
i or
i.i1Oi locates a point on the joint axis i at the beginning oflink i.
Position and orientation of link i determined by i1 andai
1. Note: subscript i
1 in the twist angle and length!
The mix of subscripts are a consequences of the D-Hconvention used!
Link i with respect to {0} 0i [T] = 01[T] 12[T] ... i1i [T]The positioin and orientation of any link can be obtained
with respect any other link by suitable use of 44 linktransformation matrices.
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SUMMARY
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Two elements of robots Links and joints.Joints allow relative motion between connected links Joints impose constraints.
Serial robots mainly use one-degree-of-freedom rotary (R)and prismatic (P) actuated joints.Parallel and hybrid robots use passivemulti-degree-of-freedom joints and actuatedone-degree-of-freedom joints.
One degree-o-freedom R and P joints represented by linesalong joint axis Z is along joint axis.
Formulation of constraints imposed by various kinds ofjoints.
Link is a rigid body in 3D space Represented by 4Denavit-Hartenberg (D-H) parameters.Convention to derive D-H parameters and for special cases.
4 4 link transformation matrix in terms of D-Hparameters.
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OUTLINEC
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1 CONTENTS
2 LECTURE 1
Mathematical PreliminariesHomogeneous Transformation
3 LECTURE 2
Elements of a robot JointsElements of a robot Links
4 LECTURE 3Examples of D-H Parameters & Link TransformationMatrices
5 LECTURE 4
Elements of a robot Actuators & Transmission
6 LECTURE 5
Elements of a robot Sensors
7 ADDITIONAL MATERIAL
Problems, References, and Suggested Reading
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SERIAL MANIPULATORS
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Steps to obtain D-H parameters for a n link serialmanipulator (see Lecture 2 for details)
1 Label joint axis from 1 (fixed) to n free end.2 Assign Zi to joint axis i, i = 1,2,...,n.3 Obtain mutual perpendiculars between lines along Zi1
and Zi Xi1.4 Origin Oi
1 on joint axis i
1.
5 Handle special cases a) consecutive joint axes paralleland perpendicular, b) first and last link
6 Obtain 4 D-H parameters for link i, i1, ai1, di and i.
For each link i obtain i1i [T] using equation (10).
Obtain link transform for link i with respect to fixedcoordinate system {0}, 0i [T], by matrix multiplication.Obtain, if required, ij[T] by appropriate matrix operations.
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PLANAR 3R MANIPULATOR
A
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ASSIGNMENT OF COORDINATE
SYSTEMS
All 3 rotary joint axisZi parallel and pointingout.
{0} Z0 is pointingout, X0 and Y0pointing to the rightand top, respectively.
Origin O0 is coincidentwith O1 shown infigure.
{2}
{3}
YTool
X0
Y0
Link 1
{0}
{1}
Y1
O2
l1
O1
2l2
l3
Link 3
O3
X2
Link 2
X3, XTool
{Tool}3
Y3
Y2 X1
1
Figure 15: The planar 3R manipulator
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PLANAR 3R MANIPULATOR
A
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ASSIGNMENT OF COORDINATE
SYSTEMS
For {1} origin O1 and Z1 are coincident with O0 and Z0.X1 and Y1 are coincident with X0 and Y0 when 1 is zero.
X1 is along the mutual perpendicular between Z1 and Z2.
X2 is along the mutual perpendicular between Z2 and Z3.
For {3}, X3 is aligned to X2 when 3 = 0.O2 is located at the intersection of the mutualperpendicular along X2 and Z2.
O3 is chosen such that d3 is zero.
The origins and the axes of {1}, {2}, and {3} are shown inFigure 15.
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PLANAR 3R MANIPULATOR D - H
TABLE
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TABLE
From the assigned axes and origins, the D-H parameters are as
follows:i i1 ai1 di i1 0 0 0 12 0 l1 0 23 0 l2 0 3
l1 and l2 are the link lengths and i, i = 1,2,3 are rotaryjoint variables (see Figure 15).
Length of the end-effector l3 does not appear in the table Recall: D-H parameters are till the origin which is at the
beginning of a link!For end-effector frame, {Tool}:
Axis of{Tool} parallel to {3} 3Tool[R] is identity matrix.Origin of{Tool} at the mid-point of the parallel jawgripper, at a distance of l3 from O3 along X3.
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PLANAR 3R MANIPULATOR LINK
TRANSFORMS
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TRANSFORMS
Substitute elements of first row of D-H table in
equation (10) to get 01[T] =
c1 s1 0 0s1 c1 0 00 0 1 00 0 0 1
The second row of the D-H table gives
12[T] =
c2 s2 0 l1s2 c2 0 00 0 1 00 0 0 1
Finally, third row of D-H table gives
23[T] =
c3 s3 0 l2s3 c3 0 00 0 1 00 0 0 1
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PLANAR 3R MANIPULATOR: LINK
TRANSFORMS (CONTD )
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TRANSFORMS (CONTD.)
The 3Tool[T] is obtained as 3Tool[T] =
1 0 0 l3
0 1 0 00 0 1 00 0 0 1
To obtain 03[T] multiply
01[T]
12[T]
23[T] and get
4
03[T] =
c123
s123 0 l1c1 + l2c12
s123 c123 0 l1s1 + l2s120 0 1 00 0 0 1
To obtain 0Tool[T], multiply
03[T]
3Tool[T]
0Tool[T] =
c123
s123 0 l1c1 + l2c12 + l3c123
s123 c123 0 l1s1 + l2s12 + l3s1230 0 1 00 0 0 1
4The symbols s12, c123 denote sin(1 +2) and cos(1 +2 +3),
respectively. Please see Notations in Module 0.ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 5 1 / 1 38
PUMA 560 MANIPULATOR
The PUMA 560 is a six degree of freedom manipulator with all
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The PUMA 560 is a six-degree-of-freedom manipulator with allrotary joints Figure 16 shows assigned coordinate systems.
X1
Y1
{1}
Z1X2
Z2
Y2
{2}
O1, O2
X3
Y3
Z3
{3}O3
X4Y4
Z4{4}
O4
a2
d3
(a) The PUMA 560 manipulator
d4
a3
Y3
X3 Z4
X4
Z6
X6
Y5
X5O3
{3}
{4}
{5}{6}
O4, O5, O6
(b) PUMA 560 - forearm and wrist
Figure 16: The PUMA 560 manipulator
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PUMA 560 MANIPULATOR D - H
PARAMETERS
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PARAMETERS
The D-H parameters for the PUMA 560 manipulator (seeFigure 16) are
i i1 ai1 di i
1 0 0 0 12 /2 0 0 23 0 a2 d3 34 /2 a3 d4 45 /2 0 0 56 /2 0 0 6
Note: i, i = 1,2,...,6 are the six joint variables.
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PUMA 560 MANIPULATOR LINK
TRANSFORMS
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TRANSFORMS
Substituting elements of row #1 of D-H table and using
equation (10), we get 01[T] =
c1 s1 0 0s1 c1 0 00 0 1 00 0 0 1
From row # 2, we get 12[T] =
c2 s2 0 00 0 1 0
s2 c2 0 00 0 0 1
From row # 3, we get 23[T] =
c3 s3 0 a2s3 c3 0 0
0 0 1 d30 0 0 1
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PUMA 560 MANIPULATOR: LINK
TRANSFORMS (CONTD )
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TRANSFORMS (CONTD.)
From row # 4 , we get 34[T] =
c4 s4 0 a30 0 1 d4
s4 c4 0 00 0 0 1
From row # 5 , we get 45[T] =
c5
s5 0 0
0 0 1 0s5 c5 0 00 0 0 1
From row # 6 , we get 56[T] =
c6
s6 0 0
0 0 1 0s6 c6 0 00 0 0 1
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PUMA 560 MANIPULATOR: LINK
TRANSFORMS (CONTD )
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TRANSFORMS (CONTD.)
{3}
with respect to{
0}
is given by03[T] =
01[T]
12[T]
23[T] =
c1c23 c1s23 s1 a2c1c2 d3s1s1c23 s1s23 c1 a2s1c2 + d3c1s23 c23 0 a2s2
0 0 0 1
{6} with respect to {3} is given by36[T] =
34[T]
45[T]
56[T] =
c4c5c6 s4s6 c4c5s6 s4c6 c4s5 a3
s5c6
s5s6 c5 d4
s4c5c6 c4s6 s4c5s6 c4c6 s4s5 00 0 0 1
Can obtain any required link transformation matrix onceD-H is table known!
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SCARA MANIPULATOR
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O0, O1 O2
1
2
d3
{ 1}
{ 2}
O3, O4
{ 3}
{ 4}
Z 1
X 1
Z 2
X 2
X
Z 3
4
Figure 17: A SCARA manipulator
SCARA Selective
Compliance AssemblyRobot Arm
Very popular for roboticassembly
Capability of desiredcompliance and rigidityin selected directions.
Three rotary(R) jointand one prismatic (P)joint.
A 4 DOF manipulator.
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SCARA MANIPULATOR: D-H
PARAMETERS
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PARAMETERS
{0} and {1} have same origin & origins of {3} and {4}chosen at the base of the parallel jaw gripper.
Directions of Z3 chosen pointing upward (see Figure 17).
Note: Actual SCARA manipulator has ball-screw at the
third joint We assume P joint.D-H Table for SCARA robot
i i1 ai1 di i1 0 0 0 12 0 a
10
23 0 a2 d3 04 0 0 0 4
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SCARA MANIPULATOR LINK
TRANSFORMS
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TRANSFORMS
Using equation (10) and the D-H table, link transforms can be
obtained as
01[T] =
c1 s1 0 0s1 c1 0 00 0 1 00 0 0 1
, 12[T] =
c2 s2 0 a1s2 c2 0 00 0 1 00 0 0 1
23[T] =
1 0 0 a20 1 0 00 0 1 d30 0 0 1
, 34[T] =
c4 s4 0 0s4 c4 0 00 0 1 00 0 0 1
The transformation matrix 04[T] is
04[T] =
01[T]
12[T]
23[T]
34[T] =
c124 s124 0 a1c1 + a2c12s124 c124 0 a1s1 + a2s12
0 0 1 d30 0 0 1
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PARALLEL MANIPULATORS
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Extend idea of D-H parameters to a closed-loopmechanism/parallel manipulator.
Key idea Break a parallel manipulator into serialmanipulators.
Obtain D-H parameters for serial manipulators.Several ways to breakChoose one that leads to simple serial manipulators
During analysis Combine serial manipulators using
constraints
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PARALLEL MANIPULATORS: 4-BA R
MECHANISM
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l0
{L}
2
1
YL
XL
YR
XR
{R}
OL
OR
Link 1l1
l3
Link 2
Link 3
l2
1
Figure 18: A planar four-bar mechanism
Rotary joints 1 and4 fixed to ground attwo places.
Break4-barmechanism into two
serial manipulators.
Break at joint 3 A 2R planar
manipulator + a 1Rmanipulator
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4-BA R MECHANISM D-H
PARAMETERS
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For 2R planar manipulator D-H parameter with respectto {L}
i i1 ai1 di i1 0 0 0 1
2 0 l1 0 2
For 1R planar manipulator D-H parameters with respectto {R}
i i1 ai1 di i
1 0 0 0 1The constant transform LR[T] is known.
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PARALLEL MANIPULATORS: THREE
DOF EXAMPLES i l 3 DOF ll l
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X
Y
ZMoving Platform
l3
l2
1
l1
2
3
p(x,y,z)
S1
S2S3
Axis ofR1
Base Platform
Axis ofR3
O
{Base}
Axis ofR2
Figure 19: A three DOF parallel
manipulator
Spatial 3- DOF parallelmanipulator.
Top (moving) platform &(fixed) bottom platformare equilateral triangles.
Figure 19 shows the home
position andZ need notpass through top platform
centre always.
Three legs Each leg isof R-P-S configuration.
Three P joints actuated.
First proposed by Lee andShah (1988) as a parallelwrist.
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THREE DOF EXAMPLE: D-H
PARAMETERS
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D-H parameter for first leg with respect to {L1}i i1 ai1 di i1 0 0 0 12 /2 0 l1 0
D-H parameter for all R-P-S leg same except the referencecoordinate system.
{L1}, {L2}, and {L3} are at the three rotary joints R1, R2,and R3, respectively.
{Base
}is located at the centre of the base platform &
BaseLi
[T], i = 1,2,3, are constant and known.
Note: The angle 1 shown in figure is same as /21.
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THREE DOF EXAMPLE: LINK
TRANSF ORMATION MATRICES
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L11
[T] =
c1
s1 0 0
s1 c1 0 00 0 1 00 0 0 1
, 12[T] = 1 0 0 0
0 0 1 l10 1 0 00 0 0 1
2S1
[T] is an identity matrix {S1} is located at the centreof the spherical joint and parallel to
{2
}.
BaseS1
[T] = BaseL1 [T]L11 [T]
12[T]2S1 [T]
Location of spherical joint S1 with respect to {Base} fromBaseS1
[T] BaseS1 = (b l1 cos1,0, l1 sin1)T, b is thedistance of R1 from the origin of{Base} (see figure 19).Location of S2 and S3BaseS2 = (b2 + 12 l2 cos2,
3b2
3l22 cos2, l2 sin2)
T
BaseS3 = (b2 + 12 l3 cos3,
3b2 +
3l32 cos3, l3 sin3)
T
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PARALLEL MANIPULATORS: SIX
DOF EXAMPLE
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l13
l12
l11
Z
ll l
l
ll
2122 23
31
32
33
1
1
1
2
2
2
3
3
b
b
b
1
2
3
pp
p
1
2
3
s
s
s
d
d
h
3
X
Y
Figure 20: A six DOF parallel (hybrid)manipulator
Moving platformconnected to fixed baseby three chains.
Each chain is R-R-R &S joint at top.
Model of a
three-fingered hand(Salisbury, 1982)gripping an object withpoint contact andno-slip.
Each finger modeledwith R-R-R joints andpoint of contactmodeled as S joint.
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SIX DOF EXAMPLE: D-H
PARAMETERS AND LINK TRANS-
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FORMS
D-H parameters for R-R-R chain.
i i1 ai1 di i1 0 0 0 1
2 /2 l11 0 13 0 l12 0 1
D-H parameter does not contain last link length l13.
D-H parameters for three fingers with respect to
{F
i}, i = 1,2,3 identical.
Can obtain transformation matrix Fipi [T] by matrixmultiplication.
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SIX DOF EXAMPLE: LINK
TRANSFORMS (CONTD.)
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Position vector of spherical joint i
Fipi =
cosi(li1 + li2 cosi + li3 cos(i +i))sini(li1 + li2 cosi + li3 cos(i +i))
li2 sini + li3 sin(i +i)
With respect to {Base}, the locations of{Fi}, i = 1,2,3,are known and constant (see Figure (20))Baseb1 = (0,d,h)T Baseb2 = (0,d,h)T Baseb3 = (0,0,0)TOrientation of{Fi}, i = 1,2,3, with respect to {Base} arealso known - {F1} and {F2} are parallel to {Base} and
{F3
}is rotated by about the Y (not shown in figure!).
The transformation matrices Basepi [T] isBaseF1
[T]01[T]12[T]
23[T]
3p1 [T] Last transformation includes
l13.
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SIX DOF EXAMPLE: LINK
TRANSFORMS (CONTD.)
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Extract the position vector Basep1
from the last column ofBaseF1
[T] Basep1 =Base b1 +
F1 p1 =
cos1(l11 + l12 cos1 + l13 cos(1 +1))
d+ sin1(l11 + l12 cos1 + l13 cos(1 +1))h + l12 sin1 + l13 sin(1 +1)
Similarly for second leg
Basep2 =
cos2(l21 + l22 cos2 + l23 cos(2 +2))d+ sin2(l21 + l22 cos2 + l23 cos(2 +2))
h + l22 sin2 + l23 sin(2 +2)
For third leg
Base
p3 =
[R(Y,)]
cos3(l31 + l32 cos3 + l33 cos(3 +3))sin3(l31 + l32 cos3 + l33 cos(3 +3))
l32 sin3 + l33 sin(3 +3)
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SUMMARY
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D-H parameters obtained for serial manipulators Planar3R, PUMA 560, SCARA
To obtain D-H parameters for parallel manipulators
Break parallel manipulator into serial manipulators.Obtain D-H parameters for each serial chain.
Examples of 4-bar mechanism, 3-degree-of-freedom and6-degree-of-freedom parallel manipulators.
Can extract position vectors of point of interest &orientation of links from link transforms.
Kinematic analysis, using the concepts presented here,discussed in Module 3 and Module 4.
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OUTLINE1 CONTENTS
2 LECTURE 1
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2 LECTURE 1
Mathematical Preliminaries
Homogeneous Transformation3 LECTURE 2
Elements of a robot JointsElements of a robot Links
4
LECTURE 3Examples of D-H Parameters & Link TransformationMatrices
5 LECTURE 4
Elements of a robot Actuators & Transmission
6 LECTURE 5Elements of a robot Sensors
7 ADDITIONAL MATERIAL
Problems, References, and Suggested Reading
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ACTUATORS FOR ROBOTS
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Actuators are required to move joints, provide power anddo work.
Serial robot actuators must be of low weight Actuators ofdistal links need to be moved by actuators near the base.
Parallel robots Often actuators are at the base.
Actuators drive a joint through a transmission device
Three commonly used types of actuators:
HydraulicPneumatic
Electric motors
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ACTUATORS FOR ROBOTS
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Source: http://www.hocdelam.org/vn/category/ho-tro/robotandcontrol/
Figure 21: Examples of actuators used in robots
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ACTUATORS FOR ROBOTS
HYDRAULIC
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Early industrial robots were driven by hydraulic actuators.
Pump supplies high-pressure fluid (typically oil) to a linearcylinders, rotary vane actuator or a hydraulic motor at thejoint!
Large force capabilities.
Large power-weight ratio The pump, electric motordriving the pump, accumulator etc. stationary and notconsidered in the weight calculation!
Control is by means of on/off solenoid valves or
servo-valves controlled electronically.The entire system consisting of Electric motor, pump,accumulator, cylinders etc. is bulky and often expensive Limited to big robots.
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ACTUATORS FOR ROBOTS
PNEUMATIC
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Similar to hydraulic actuators but working fluid is air.Similar to hydraulic actuators, air is supplied from acompressor to cylinders and flow of air is controlled bysolenoid or servo controlled valves.
Less force and power capabilities.Less expensive than hydraulic drives.
Chosen where electric drives are discouraged or for safetyor environmental reasons such as in pharmaceutical andfood packaging industries.
Closed-loop servo-controlled manipulators have beendeveloped for many applications.
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COMPARISON OF PNEUMATIC &
HYDRAULIC ACTUATORS
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Air used in pneumatic actuators is clean and safe.Oil in hydraulic actuator can be a health and fire hazardespecially if there is a leakage.
Pneumatic actuators are typically light-weight, portable
and faster.Air is compressible (oil is incompressible) and hencepneumatic actuators are harder to control.
Hydraulic actuators have the largest force/power densitycompared to any actuator.
With compressors, accumulators and other components,the space requirement is larger than electric actuators.
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ACTUATORS FOR ROBOTS
ELECTRIC MOTORS
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Electric or electromagnetic actuators are widely used inrobots.
Readily available in wide variety of shape, sizes, power andtorque range.
Very easily mounted and/or connected with transmissionelements such as gears, belts and timing chains.
Amenable to modern day digital control.
Main types of electric actuators:
Stepper motorsPermanent magnet DC servo-motorBrushless motors
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ELECTRIC ACTUATORS STEPPER
MOTORS
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Used in small robots with small payload and low speeds.
Stepper motors are of permanent magnet, hybrid orvariable reluctance type.
Actuated by a sequence of pulses For a single pulse,rotor rotates by a known step such that poles on stator and
rotor are aligned.Typical step size is 1.8 or 0.9.Speed and direction can be controlled by frequency ofpulses.
Can be used in open-loop as cumulative error andmaximum error is one step!
Micro-stepping possible with closed-loop feedback control.
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STEPPER MOTORS
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Parts of a Stepper Motor
http://www.engineersgarage.com/articles/stepper-motors
http://www.societyofrobots.com/member_tutorials/node/28
Variable Reluctance (VR) Stepper MotorsNumber of teeth in the inner rotor (permanent magnet) isdifferent than the number of teeth in stator.
1) Electro-magnet 1 is activated Rotor rotates up such that nearest teeth line up.2) Electro-magnet 1 is deactivated and 2 is turned on Rotor rotates such that nearest teeth line up
rotation is by astep (designed amount) of typically 1.8 or 0.9 degrees.
3) Electro-magnet 2 is deactivated and 3 is turned on Rotor rotates by another step.4) Electro-magnet 3 is deactivated and 4 is turned on and cycle repeated.Permanent Magnet (PM) Stepper Motors Similar to VR but rotor is radially magnetized.
Hybrid Stepper Motors Combines best features of VR and PM stepper motors.
Figure 22: Stepper motor components and working principle
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STEPPER MOTORS
Typically stepper motors have two phases.
Four stepping modes
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Four stepping modes
Wave drive Only one phase/winding is on/energized
Torque output is smaller.Full step drive Both phases are on at the same time Rated performance.Half step drive Combine wave and full-step drive Angular movement half of first two.
Micro-stepping Current is varying continuously Smaller than 1.8 or 0.9 degree step size, lower torque.
Choice of a stepper motor based on:Load, friction and inertia Higher load can cause slipping!Torque-speed curve and quantities such as holding torque,
pull-in and pull-out curve.Torque-speed characteristic determined by the drive Bipolar chopper drives for best performance.Maximum slew-rate: maximum operating frequency withno load (related to maximum speed).
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ELECTRIC ACTUATORS DC/AC
SERVO-MOTORS
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Rotor is a permanent magnet and stator is a coil.
Permanent magnets with rare earth materials(Samarium-Cobalt, Neodymium) can provide largemagnetic fields and hence high torques.
Commutation done using brushes or in brushless motor
using Hall-effect sensors and electronics.
Widely available in large range of shape, sizes, power andtorque range and low cost.
Easy to control with optical encoder/tacho-generators
mounted in-line with rotor.Brushless AC and DC servo-motors have low friction, lowmaintenance, low cost and are robust.
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DC SERVO MOTORS
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http://www.brushlessdcmotorparts.info http://www.rc-book.com/wiki Small RC Servo motors
/brushless-electric-motor http://www.drivecontrol-details.info
/rc-servo-motor.html
Slotted-brushless DC Servo motors Direct drive motor, Applimotion, Inc Brushless Hub motor for E-bike
http://www.motioncontroltips.com/ http://news.thomasnet.com/ http://visforvoltage.org/
company_detail.html?cid=20082162 system-voltage/49-60-volts
Figure 23: Examples of DC servo motors
See also Wikipedia entry on electric motors.ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 8 2 / 1 38
MODEL OF A DC PERMANENT
MAGNET MOTOR
LR
Rotor is apermanent magnet.
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ia
m
LaRa
Va
Motor
Figure 24: Model of a permanent magnet DCservo-motor
p g
Stator Armaturecoil with resistanceRa and inductanceLa.
Applied voltage Va,
ia current in coil.Rotation speed ofmotor m Mechanical part.
Torque developed Tm = Kt ia Kt is constant.
Back-emf V = Kg m Kg is constant.Motor dynamics can be modeled as first-order ODE
La ia + Raia + Kg m = Va
Mechatronic model Mechanical + Electrical/ElectronicsASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS N PTEL, 2 01 0 8 3 / 1 38
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TRANSMISSIONS USED IN ROBOTS
Purpose of transmission is to transfer power from source to
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Purpose of transmission is to transfer power from source toload.
The purpose of a transmission is also to transfer power atappropriate speed.
Typical rated speed of a DC motor is between 1800 & 3600rpm.3000 rpm = 60 rps
360 radians/sec.
For a (typical) 1 m link Tip speed is 36 m/sec Greater than speed of sound!Need for large reduction in speed.
Transmissions can (if needed) also convert rotary to linear
motion and vice-versa.Transmissions also transfer motion to different joints andto different directions.
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TRANSMISSIONS USED IN ROBOTS
Transmissions in robots are decided based on motion, loadand power requirements and by the placement of actuator
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and power requirements, and by the placement of actuatorrelative to the joint.
Transmissions for robots must be (a)stiff, (b) low weight,(c) backlash free, and (d) efficient.
Direct drives with motor directly connected to joint. It hasadvantages of low friction and low backlash but are
expensive.Typical transmissions
Gear boxes of various kinds Spur, worm and worm wheel,planetary etc..Belts and chain drives.
Harmonic drive for large reduction.Ball screws and rack-pinion drives To transform rotary tolinear motions.Kinematic linkages 4-bar linkage.
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TRANSMISSIONS USED IN ROBOTS
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Figure 26: Examples of transmissions used in robots
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OUTLINE1 CONTENTS
2 LECTURE 1
M h i l P li i i
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Mathematical Preliminaries
Homogeneous Transformation3 LECTURE 2
Elements of a robot JointsElements of a robot Links
4 LECTURE 3
Examples of D-H Parameters & Link TransformationMatrices
5 LECTURE 4
Elements of a robot Actuators & Transmission
6 LECTURE 5Elements of a robot Sensors
7 ADDITIONAL MATERIAL
Problems, References, and Suggested Reading
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INTRODUCTION
A robot without sensors is like a human being without eyes,ears, sense of touch, etc.
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Sensor-less robots require costly/time consuming
programming.Can perform only in playback mode.No change in their environment, tooling and work piececan be accounted for.
Sensors constitute the perceptual system of a robot,designed:
To make inferences about the physical environment,To navigate and localise itself,To respond more flexibly to the events occurring in itsenvironment, andTo enable learning, thereby endowing robots with
intelligence.
Sensors allow less accurate modeling and control.
Sensors enable robots to perform complex and increasedvariety of tasks reliably thereby reducing cost.
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INTRODUCTION (CONTD.)
Dynamical system
h h d b d ff l
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Changes with time Governed by differential equations.
For a given input there is a well defined output.Linear dynamical system Modeled by linear ordinarydifferential equations Can be analysed using Laplacetransforms and transfer function5.
Control system is used to
Obtain desired response from dynamical system.Stabilize an unstable dynamical system.Improve performance.
Two kinds of system Open-loop and closed-loop orfeedback control systems.
Importance of sensors in modeling and control shown innext few slides.
5Other methods such as frequency domain analysis can also be used.
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OPEN-LOOP CONTROL SYSTEM
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Figure 27: An open-loop control system
Time domain linear function f(t) converted to s
domain
using Laplace transforms:F(s)
= L{f(t)} = 0 f(t)est dt
Time derivatives become polynomials in s.
Transfer function Ratio of output to input in sdomain.Open-loop system No feedback: Input R(s) SystemG(s) Output Y(s) (see Figure above)For error G(s) in the modeling of the plant Y(S)Y(s) =
G(s)G(s)
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CLOSED-LOOP CONTROL SYSTEM
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Figure 28: A closed-loop control system
Y(s) =D(s)G(s)
1 + D(s)G(s)R(s); Chosen D(s)G(s) >> 1
For error G(s) in the modeling of the plant
Y(S)
Y(s)=
G(s)
G(s)
1
1 + D(s)G(s)
Since 1 + D(s)G(s) >> 1
x% error in G(s) results in
much smaller error of(
11+D(s)G(s))
x% in Y(s).
With sensors and feedback, less complex and expensivecontrollers and/or models of system can be used for morerobust performance.
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CLOSED-LOOP CONTROL SYSTEM
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Figure 29: Closed-loop control with sensors
Sensors must have low noise and be good for effectiveness.
Consider a noisy sensor (see Figure above).
Output of system
Y(s) = D(s)G(s)1 + D(s)G(s) (R(s)N(s)); ChosenD(s)G(s)>> 1
Note that output is proportional to corrupted(R(s)N(s))input, hence output error can never be reduced!
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SENSORS IN ROBOTS
Sensor is a device to make a measurement of a physicali bl f i t t d t it i t l t i l i l
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variable of interest and convert it into electrical signal.
Desirable features in sensors areHigh accuracy.High precision.Linear response.Large operating range.
Low response time.Easy to calibrate.Reliable and rugged.Low costEase of operation
Broad classification of sensors in robotsInternal state sensors.External state sensors.
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SENSORS IN ROBOTS INTERNAL
Internal sensors measure variables for controlJoint position.Joint velocity
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Joint velocity.
Joint torque/force.Joint position sensors (angular or linear)
Incremental & absolute encoders Optical, magnetic orcapacitive.Potentiometers.
Linear analog resistive or digital encoders.Joint velocity sensors
DC tacho-generator & resolver.sOptical encoders.
Force/torque sensors.At joint actuators for control.At wrist to measure components of force/moment beingapplied on environment.At end-effector to measure applied force on gripped object.
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SENSORS IN ROBOTS EXTERNAL
Detection of environment variables for robot guidance,object identification and material handling.
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Two main types Contacting and non-contacting sensors.
Contacting sensors: Respond to a physical contactTouch: switches, Photo-diode/LED combination.Slip.Tactile: resistive/capacitive arrays.
Non-contacting sensors: Detect variations in optical,
acoustic or electromagnetic radiations or change inposition/orientation.
Proximity: Inductive, Capacitive, Optical and Ultrasoni.cRange: Capacitive and Magnetic, Camera, Sonar, Laserrange finder, Structured light.
Colour sensors.Speed/Motion: Doppler radar/sound, Camera,Accelerometer, Gyroscope.Identification: Camera, RFID, Laser ranging, Ultrasound.Localisation: Compass, Odometer, GPS.
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SENSORS IN ROBOTS
How to compute resolution of a sensor?Example: Optical encoder to measure joint rotation
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Figure 30: Resolution and required accuracy
Optical encoders and resolvers Desired accuracy e One must be able to measure a minimum angle e/L.Number of bits/revolution required: Compute 2/( eL ),express as a binary number and round-off to the next
higher power of 2.Example: L = 1m, e 0.5mm = eL = 5x104,2/( eL ) 10000 2
13 = 8192,214 = 16384 14 bitencoder needed.
IfL
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OPTICAL ENCODER
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Figure 31: Optical encoder
One of the most important and widely used internal sensor.Consists of an etched encoding disk with photo-diodes andLEDS. Disk made from
Glass, for high-resolution applications (11 to >16 bits).
Plastic (Mylar) or metal, for applications requiring morerugged construction (resolution of 8 to 10 bits).
As disk rotates, light is alternately allowed to reachphoto-diode, resulting in digital output similar to a squarewave.
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OPTICAL ENCODER
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Figure 32: Optical encoder outputs
Typically 3 signals available Channel A, B and I; A and Bare phase shifted by 90 degrees and I is called as the indexpulse obtained every full rotation of disk.
Signals read by a microprocessor/counter.
Output of counter includes rotation and direction.
Output can be absolute or relative joint rotation.
Can be used for estimating velocity.
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SENSORS FOR JOINT ROTATION
Potentiometers Voltage resistance and resistance rotation at joint
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Not very accurate but very inexpensive.
More suitable for slow rotations.Adds to joint friction.
Resolvers Rotary electrical transformer to measure jointrotation
Analog output, need ADC for digital control.Electromagnetic device Stator + Rotor (connected tomotor shaft).Voltage (at stator) sin(), = rotor angle.
Tachometers measures joint velocity
Similar to resolver.Voltage (at stator) , = angular velocity of the rotor.Analog output ADC required for digital control.
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FORCE/TORQUE SENSOR (CONTD.)
Performance specifications to ensure that the wrist motions
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Performance specifications to ensure that the wrist motions
generated by the force/torque sensors do not affect theposition accuracy of the manipulator:
High stiffness to ensure quick dampening of the disturbingforces which permits accurate readings during short timeintervals.
Compact design to ensure easy movement of themanipulator.Need to be placed close to end-effector/tool.Linear relation between applied force/torque and straingauge readings.
Made from single block of metal No hysteresis.
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FORCE/TORQUE WRIST SENSORF= [RF]W
wherea
RF =
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Figure 33: Six axisforce/torque sensors at wrist
RF =
0 0 r13 0 0 0 r17 0r12 0 0 0 r25 0 0 00 r32 0 r34 0 r36 0 r380 0 0 r44 0 0 0 r480 r52 0 0 0 r56 0 0
r61 0 r63 0 r65 0 r67 0
F= (Fx,Fy,Fz,Mx,My,Mz)T.
W = (w1,w2,w3,w4,w5,w6,w7,w8).wi are the 8 strain gauge readings.
aThe formula can be derived from statics,please see Module 0, Lecture 5.
Rij are found during calibration process using least-squaretechniques.
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EXTERNAL SENSORS TOUCH
Allows a robot or manipulator to interact with itsenvironment to touch and feel, see and locate.
Two classes of external sensors Contact and non contact
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Two classes of external sensors Contact and non-contact
Figure 34: Touch sensor
Simple LED-Photo-diode
pair used to detectpresence/absence of objectto be grasped
Micro-switch to detecttouch.
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EXTERNAL SENSORS SLIP
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Figure 35: Slip sensor
Slip sensor to detect if grasped object is slipping.
Free moving dimpled ball Deflects a thin rod on the axisof a conductive disk
Evenly spaced electrical contacts placed under diskObject slips past the ball, moving rod and disk Electricalsignal from contact to detect slip.
Direction of slip determined from sequence of contacts.
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EXTERNAL SENSORS TACTILE
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Figure 36: Robot hand with tactilearray
Skin like membrane tofeel the shape of thegrasped object
Also used to measureforce/torque required to
grasp object
Change inresistance/capacitance dueto local deformation from
applied force
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EXTERNAL SENSORS PROXIMITY
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Detect presence of an object near a robot or manipulatorWorks at very short ranges (
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Figure 38: A magnetic proximity sensor
Electronic proximity sensor based on change of inductance.
Detects metallic objects without touching them.
Consists of a wound coil, located next to a permanentmagnet, packaged in a simple housing.
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INDUCTIVE PROXIMITY SENSOR MAGNETIC
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Figure 39: Flux lines in a magnetic sensor
Ferromagnetic material enters or leaves the magnetic field
Flux lines of the permanent magnet change theirposition.
Change in flux Induces a current pulse with amplitudeand shape proportional to rate of change in flux.
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INDUCTIVE PROXIMITY SENSOR MAGNETIC
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Figure 40: Schematic of flux response in magnetic proximity sensor
Coils output voltage waveform For proximity sensing.Useful where access is a challenge.Applications Metal detectors, Traffic light changing,Automated industrial processes.
Limited to ferromagnetic materials.
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INDUCTIVE HALL EFFECTPROXIMITY SENSOR
Hall effect relates the voltage between two points in a
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conducting or semiconducting material subjected to astrong magnetic field across the material.
When a semi-conductor magnet device is brought in closeproximity of a ferromagnetic material
the magnetic field at the sensor weakens due to bending ofthe field lines through the material,the Lorentz forces are reduced, andthe voltage across the semiconductor is reduced.
The drop in the voltage is used to sense the proximity.
Applications: Ignition timings in IC engines, tachometersand anti-lock braking systems, and brushless DC electricmotors.
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CAPACITIVE PROXIMITY SENSORS
Objects entry in electrostatic field of electrodes changes
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capacitance.Oscillations start once capacitance exceeds a predefinedthreshold.
Triggers output circuit to change between on and off.
When object moves away, oscillators amplitude decreases,changing output back to original state.
Larger size and dielectric constant of target, means largercapacitance and easier detection.
Useful in level detection through a barrier.
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ULTRASONIC PROXIMITY SENSORS
Electro-acoustic transducer to send and receive highfrequency sound waves.
Emitted sonic waves are reflected by an object back to the
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transducer which switches to receiver mode.Same transducer is used for both receiving and emittingthe signals Fast damping of acoustic energy is essentialto detect close proximity objects.
Achieved by using acoustic absorbers and by decoupling thetransducer from its housing.
Typically of low resolution.
Important specifications 1) Maximum operating distance,2) Repeatability, 3) Sonic cone angle, 4) Impulse frequency,
and 5) Transmitter frequency.Some well-known applications 1) Useful in difficultenvironments, 2) Liquid level detection and 3) Car parkingsensors.
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OPTICAL PROXIMITY SENSORS
Also known as light beam sensors Solid state LED actingas a transmitter by generating a light beam.
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A solid-state photo-diode acts as a receiver.
Field of operation of the sensor Long pencil like volume,formed due to intersection of cones of light from sourceand detector.
Any reflective surface that intersects the volume getsilluminated by the source and is seen by the receiver.
Generally a binary signal is generated when the receivedlight intensity exceeds a threshold value.
Applications: 1) Fluid level control, 2) Breakage and jamdetection, 3) Stack height control, box counting etc.
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RANGE SENSORS TRIANGULATIONA narrow beam of lightsweeps the plane definedby the detector, the objectand the source,
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Figure 43: Triangulation
illuminating the target.
Detector output is peakwhen illuminated patch isin front.
With B and known Obtain D = Btan.
Changing B and , onecan get D for all visibleportions of object.
Very little computation required.
Very slow as one point is done at a time.
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RANGE SENSORS STRUCTUREDLIGHTING
A sheet of light
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Figure 44: Structured lighting
generated through acylindrical lens or narrowslit is projected on a target.
Intersection of the sheet
with target yields a lightstripe.
A camera offset slightlyfrom the projector, viewsand analyze the shape ofthe line.
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RANGE SENSORS STRUCTUREDLIGHTING
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Distortion of the line is related to distance and can becalculated.
Horizontal displacement (in image) proportional to depthgradient.
Integration gives absolute range.
Calibration is required.
Advantages:
Fast, very little computation is required.Can scan multiple points or entire view at once.
Structured lighting should be permitted.
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RANGE SENSORS TIME-OF-FLIGHT
Utilizes pulsed lasers and ultrasonics to measure time takenby the pulse to coaxially return from a surface.
If D is the targets distance, c is the speed of radiation,
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and t is elapsed time taken for the pulse to return
D =ct
2
Light or electromagnetic radiation more useful for large
(kilometers) distances.Light not very suitable in robotic applications as c is large.
To measure range with 0.25 inch accuracy, one needs tomeasure very small time intervals 50 ps.
Suitable for acoustic (ultrasonic) radiation, since c
330m/s.
Can only detect distance of one point in its view Scanrequired for object.
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RANGE SENSORS TIME-OF-FLIGHT
Phase shift and continuous laser light for measuring distance
Laser beam ofwavelength split
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Figure 45: Range measurement using phaseshift
Two beams travel todetector through twodifferent paths.
Phase delay between
two beams is measured.Distance traveled byfirst beam L.
Total distance traveledby the second beamD = L + 2D D = L + 360, is thephase shift.
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RANGE SENSORS ULTRASONIC
Similar to the pulsed laser technique.
An ultrasonic chirp is transmitted over a short time period.
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From the time difference between the transmitted andreflected wave, D can be obtained.
Generally used for navigation and obstacle avoidance inrobots.
Much cheaper than laser range finder.
Shorter range as waves disperse.
Wavelength of ultrasonic radiation much larger Notreflected very well from small objects and corners.
Ultrasonic waves not reflected very well from plastics andsome other materials.
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VISION SENSORS
Most powerful and complex form of sensing, analogous tohuman eyes.
C i i f id i h i d
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Comprising of one or more video cameras with integratedsignal processing and imaging electronics.
Includes interfaces for programming and data output, and avariety of measurement and inspection functions.
Also referred to as machine or computer vision.Computations required are very large compared to anyother form of sensing.
Computer vision can be sub-divided into six main areas: 1)
Sensing, 2) Pre-processing, 3) Segmentation, 4)Description, 5) Recognition and 6) Interpretation.
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VISION SENSORS
Three levels of processing.
Low level vision
P i i i i i i lli h f
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Primitive in nature, requires no intelligence on the part ofthe vision functions.Sensing and pre-processing can be considered as low levelvision functions.
Medium level vision
Processes that extract, characterize and label componentsin an image resulting from a low level vision.Segmentation, description and recognition of the individualobjects refer to the medium level function.
High level vision: Processes that attempt to emulatecognition.
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VISION SENSORS
Smaller number of robotic applications Primarily due tocomputational complexity and low speed.
Vision system can
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Determine distances of objects.Determine geometrical shape and size of objects.Determine optical (color, brightness) properties of objectsin an environment.Can be used for navigation (map making), obstacle
avoidance, Cartesian position and velocity feedback,locating parts, and many other uses.Can learn about environment.Acquire knowledge and intelligence.
Vision systems extensively used in autonomous navigation
in mobile robots (Mars rovers).
Use of vision systems increasing rapidly as technologyimproves!
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NEW DEVELOPEMENTS IN SENSORTECHNOLOGY MEMS SENSORS
Consist of very small electrical, electronics and mechanical
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components integrated on a single chip.Provide an interface to sense, process and/or control thesurrounding environment.
Generally silicon based, allowing integration with
microelectronics.Sensing mechanisms employed are a) Capacitive, b)Piezoelectric, c) Piezoresistive, d) Ferroelectric, e)Electromagnetic and f) Optical.
Capacitive sensing most successful due to a) Norequirement of exotic materials, b) Low power consumptionand c) Stability over temperature ranges.
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SMART SENSORS
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Figure 47: Components of a smart sensor
Smart Sensor A single device combining data collectionand information output.
As per IEEE 1451.2, a smart sensor is:a transducer that provides functions beyond thosenecessary for generating a correct representation of a
sensed or control quantity.Integration of transducer + applications + networking.
Evolving concept Often sensors not complying with IEEEdefinition are also called smart sensors!
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INTELLIGENT SENSORS
For advanced robotic devices, large amount of sensory
information slows down the controllerll k l d ff h ll b
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information slows down the controller.Intelligent sensors take some load off the controller by
Taking some predefined action based on the input received.Efficiently and precisely measuring the parameters,enhancing or interrupting them.
Intelligent sensors: An adaptation of smart sensors withembedded algorithms for detection of noise,instrumentation anomalies and sensor anomalies.
Enabling technology for ISHM (Integrated Systems Health
Management) used in spaceships by NASA.
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SENSOR-BASED ROBOTICS ASMIOASIMO: Advanced Step in InnovativeMObility.
A humanoid robot developed byHONDA car company of Japan.
Sensors for vision speed balance
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Sensors for vision, speed, balance,force, angle, and foot area.34 degrees of freedom controlled byservo motors.
Capable of advanced movement
Walking and running.Maintaining posture and balance.Climbing stairs & avoiding obstacle.
Intelligence
Charting a shortest route.Recognizing moving objects.Distinguish sounds and recognizefaces and gestures.
Figure 48: ASIMOclimbing stairs
Figure 49: ASIMOplaying soccer
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SENSORS IN ROBOTS SUMMARY
Sensors enhance performance of robots/manipulators, andmake them more flexible. for different application.
Two main types of sensors
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Two main types of sensorsInternal sensors for measuring variables such as jointrotation, velocity, force, and torque.External sensors for measuring variables such as proximity,touch, range, and geometry.
Large varieties of sensors exist for large variety ofmeasurement tasks.
More than one sensor is typically used in a robot.
Vision system is the most powerful and complex sensor.
New and modern sensor technologies are becoming matureand are finding their way into robots.
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