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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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Jacobian matrix

Axis 1

Axis 2Axis 3

Axis 4

Axis 5

Axis 6

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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Manipulator Screw System

1

2

34

56l

ASME DETC 1998

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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

ASME DETC 1998

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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

ASME DETC 1998

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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