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ASME DETC 1998
1
Robot Manipulators and Singularities
Vijay Kumar
ASME DETC 1998
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Outline
Jacobian matrix for a serial chain manipulator Singularities Parallel manipulator
ASME DETC 1998
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Serial Chain Linkages
Velocity Equations
Let the end effector twist be T.
Consider two joints, 1 & 2.
The effect of twists about two joints connected in series is to produce a composite twist that is obtained by adding the two twists (in the same coordinate system).
Axis 1
Axis 2
y
z
O
u2
u1
x
Axis n
21
1211
11
211
1
12121
1
1
1
21
TT
ATAT
AAAAAA
AAAA
AAT
dt
d
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Serial Chain Linkages
Velocity Equations for a n-joint serial chain
The effect of twists about n joints connected in series is to produce a composite twist that is obtained by adding the n joint twists (in the same coordinate system).
Axis 1
Axis 2
y
z
O
u2
u1
x
Axis n
n
nnn
nnn
nn
n
dt
d
TTT
AAATAAAATAT
AAAAAAAAAAAAAA
AAAAAA
AAT
21
11211211211
1121
11211
121
11
12121
1
1
1
21
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Serial chain linkages
Assume Single degree-of-freedom, axial joints ith joint twist
Ti = Si i
revolute joints: prismatic joints:
Velocity equationsT = T1 + T2 + … + Tn
“Standard form”
i i
i id
T S S S
1 2
1
2n
n
v
J
1
2
n
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Serial chain equations
Jacobian matrix
Geometric significance of the columns of the matrix Matrix can be constructed by inspection Physical insight into the kinematic performance
v
J
1
2
n
End effectortwist Joint ratesJacobian matrix Axis 1
Axis 2
y
z
O
u2
u1
x
Axis n
J S S S
1 2 n
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Jacobian matrix
Axis 1
Axis 2Axis 3
Axis 4
Axis 5
Axis 6
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Singularities C3 = 0
S5 = 0
Jacobian matrix
Axis 1
Axis 2Axis 3
Axis 4
Axis 5
Axis 6
ml
n
zy
J
s s c s
c s s
c c
l c ms nc n s s
mc n s nc s
l s mc
23 4 4 5
4 4 5
23 5
23 3 4 4 5
2 4 4 5
23 3
0 0 0
0 1 1 0
0 0 1 0
0 0
0 0 0
0 0 0 0
Axis1
Axis2
Axis3
Axis4
Axis5
Axis 6
det J m C n S m C n S3 23 2 5
n S m C23 2 0
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Example
J
s s c s
c s s
c c
l c ms nc n s s
mc n s nc s
l s mc
23 4 4 5
4 4 5
23 5
23 3 4 4 5
2 4 4 5
23 3
0 0 0
0 1 1 0
0 0 1 0
0 0
0 0 0
0 0 0 0
Axis1
Axis2
Axis3
Axis4
Axis5
Axis 6
Singularities C3 = 0
S5 = 0
det J m C n S m C n S3 23 2 5
n S m C23 2 0
1
2
3
4
56
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Example
Revolute Joints
Prismatic Joints
4
1
2
3
5
6
1
2
3
5
6
x
z
00010
00000
0000
0100
0010
000
2
23
32
52
454
4542
lS
Sd
dlC
CC
CSC
CSSS
J
u
u
vt
O
uvt
0d
O
ii
ii u
uS
ii u
0S
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Singularities
Algebra
Jacobian matrix becomes singular
Geometry
The joint screws (lines) are linearly dependent
Kinematics
The manipulator (instantaneously) loses one or more degrees of freedom
Statics
There exists one or more wrenches that can be resisted without turning on the actuators
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Case 1 C3 = 0
Zero pitch wrench reciprocal to all joint screws
Line intersects all six joint axes Rows 1, 5, and 6 are dependent It is not possible to effect the twist
[n l S2 , 0, 0, 0, - l S2 , n S2+mC2]T
Singularities
Axis 1
Axis 2Axis 3
Axis 4
Axis 5
Axis 6
ml
zy
J
s s c s
c s s
c c
l c m nc n s s
mc n s nc s
l s
2 4 4 5
4 4 5
2 5
2 4 4 5
2 4 4 5
2
0 0 0
0 1 1 0
0 0 1 0
0 0
0 0 0
0 0 0 0 0
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Case 2 S5 = 0
Axes 4 and 6 are dependent Joints 4 and 6 have the same
instantaneous motions The end effector loses a degree of
freedom
Singularities (continued)
Axes 4 and 6 become colinear
Axis 4
Axis 5
Axis 6
P
Q
Link 3
J
s s
c
c
l c ms nc
mc n s
l s mc
23 4
4
23
23 3 4
2 4
23 3
0 0 0 0
0 1 1 0 0
0 0 1 0 1
0 0 0
0 0 0 0
0 0 0 0
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Case 3
Point of concurrence of axes 4, 5, and 6 lies on the plane defined by axes 1 and 2
Zero pitch wrench reciprocal to all the joint screws
Line intersects or is parallel to all joint axes
Rows 1 and 5 are dependent The end effector cannot move along the
twist: [-n, 0, 0, 0, 1, 0]T
Singularities (continued)n S m C23 2 0
J
mn
lmn
C s c s
c s s
c c
l c ms nc n s s
mc n s nc s
C mc
2 4 4 5
4 4 5
23 5
23 3 4 4 5
2 4 4 5
2 3
0 0 0
0 1 1 0
0 0 1 0
0 0
0 0 0
0 0 0 0
Axis 1
Axis 2Axis 3
Axis 4
Axis 5
Axis 6
ml
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Singularities: More Examples
Pa2
a3
Axes 4 and 6 become colinearManipulator is completely flexed/extended
Axis 4
Axis 5
Axis 6
P
QSpherical wrist
Link 3
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Singularities: More Examples
Case 1: the manipulator is completely extended or flexed
Case 2: the tool reference point lies on axis 1
Case 3: orientation singularity
Axes 4 and 6 are colinear
P
a2
a3
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Singular Structure
Six degree of freedom robot manipulator with an anthropomorphic shoulder and wrist1
2
34
56
Three axes intersecting at a point
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Special Third Order System: Type 2
System consists of zero pitch screws on all lines through a point There are no members with other pitches Screw system of spherical joint Self-reciprocal
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Manipulator Screw System
1
2
34
56l
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Parallel ManipulatorsStewart Platform
Each leg has five passive joints and one active (prismatic joint)
There is a zero pitch wrench reciprocal to all five passive joints.
Call it Si for Leg i.
The net effect of the prismatic joint must be to produce this zero pitch wrench.
Twists of freedom is a fifth order screw system defined by the five passive joints
Constraint wrench system is defined by the zero pitch reciprocal screw
The end effector wrench is the sum of the wrenches exerted by the six actuators (acting in parallel)
RR
P
S
BASE
END EFFECTOR
LEG 1
LEG 2
LEG 3LEG 4
LEG 5
LEG 6
LEG NO. iAxis of thereciprocal wrench
w S S S
1 2 6
1
2
3
4
5
6
f
f
f
f
f
f
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Parallel Manipulators
The columns of the transpose of the Jacobian matrix are the coordinates of the reciprocal screws.
The equations for force equilibrium (statics) for parallel manipulators are “isomorphic” to the equations for rate kinematics for serial manipulators.
A parallel manipulator is singular when Any of its serial chains becomes singular (kinematic singularity) The set of reciprocal screws (Si) becomes linearly dependent
w S S S
1 2 6
1
2
3
4
5
6
f
f
f
f
f
f
l
l
l
l
l
l
T
1
2
3
4
5
6
1 2 6
S S S t
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Parallel Manipulators: Example
Each serial chain consists of two revolute joints and 1 prismatic joint.
In the special planar three system, the joint screw reciprocal to the two revolute joints is the zero pitch screw in the plane whose axis intersects the two revolute joints.
Actuator i produces a pure force along the screw Si
The manipulator is singular when the axes of the reciprocal screws intersect at a point (or become parallel)
At this singularity, the actuators cannot resist a moment about the point of intersection (or a force perpendicular to the all the three axes)
END-EFFECTOR
ACTUATORS
1
2
3
S3
S2
S1
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