# MECH572A Introduction To Robotics

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2 Νοε 2013 (πριν από 4 χρόνια και 11 μέρες)

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MECH572A

Introduction To Robotics

Lecture 8

Dept. Of Mechanical Engineering

Review

Robot Kinematics

Geometric Analysis

Differential analysis

Forward (direct) vs. Inverse Kinematics problem

Inverse Kinematics Problem (IKP)

-

Problem description
:
Known
Q
EE

and
p
EE
, Seek

1
, …

n

-

Possibility of Analytical (closed form) solution depends on the
architecture of the manipulator

-

6
-
R Decoupled manipulator (e.g., PUMA)

Position problem

position of C (wrist centre)

Orientation problem

EE orientation

Review

IKP

6
-
R Decoupled Manipulator

Solution process overview:

Arm (position)

Wrist (orientation)

1
,

2
,

3

Equ's in

1
,

3

Quartic equ in

3
(

3
)

Eliminate

2

Eliminate

1

1

0

3

1

2

2

0

Max Number of Solution:
4

Elimination

Solution

4
,

5
,

6

4
(

4
)

4

5

6

0

Max Number of Solution:
2

Special geometry in
wrist axis

Manipulator Kinematics

Velocity (Differential) Analysis

Manipulator Kinematics

Velocity Analysis

Angular velocity of EE

Position of EE

Manipulator Kinematics

Velocity Analysis (cont'd)

Define

Let

Position vector from
O
i

to
P

Recall twist

Manipulator Kinematics

Velocity Analysis (cont'd)

Jacobian matrix:

i
th column (revolute joints)

Linear transformation between joint rates
and Cartesian rates (EE)

The Pl
ü
cker array of
i
th axis w.r.t
point P of EE

Manipulator Kinematics

Velocity Analysis (cont'd)

Prismatic joint:

The ith column of Jacobian matrix

Manipulator Kinematics

Velocity Analysis (cont'd)

For 6 joint manipulator, J is a 6

6 square matrix

Solve equations using Gauss
-
elimination (LU decomposition) algorithm

Compute y

(Forward substitution)

Compute

(Backward substitution)

Manipulator Kinematics

Velocity Analysis (cont'd)

Transformation of Jacobian matrix

In general, Jacobian can be defined wrt different points. For decoupled
manipulators:

Recall twist transformation

Property:

Wrist Centre

Point P at EE

Two point A
and B on EE

Manipulator Kinematics

Velocity Analysis (cont'd)

The Jacobian

matrix of decoupled Manipulator has special form

Partition arm and wrist rates:

Arm rate

Wrist rate

Manipulator Kinematics

Velocity Analysis (cont'd)

Decoupled Manipulator

solve 2 systems of three equations and three
unknowns

Manipulator Kinematics

Application Example

Operating and control overview

Robotic Work Station (RWS)

MSS

Commands

Telemetry

Hand Controllers

Display &
Control Panel

Manipulator Kinematics

Application Example (cont'd)

Kinematic aspects of Canadarm2 control modes

1
.
Human
-
in
-
the
-
loop modes

(commanding the arm via hand
controllers)

a) Manual Augmented Mode (MAM)

Description
: Control the manipulator by commanding EE rate

Kinematics
: IKP rate problem

b) Single Joint Rate Mode (SJRM)

Description
: Control the manipulator by commanding a single joint

Kinematics
: DKP rate problem

Manipulator Kinematics

Application Example (cont'd)

Kinematic aspects of Canadarm2 control modes (Cont'd)

2. Automatic Modes

a) Joint Modes

Operator Commanded/Pre
-
Stored Joint Auto Modes (OJAM & PJAM)

Description
: Execute joints movements to a pre
-
set joint positions

Kinematics
: DKP position/orientation problem

b) EE Modes (POR <Point Of Resolution> Mode)

Operator Commanded/Pre
-
Stored POR Auto Modes (OPAM & PPAM)

Description: Execute manipulator movements to a pre
-
set EE position/orientation

Kinematics: IKP position/orientation problem

Manipulator Kinematics

Singularity Analysis

Decoupled Manipulators

Observe the Jacobian Matrix

-

If neither
J
12
nor
J
21

is singular, IKP problem is solvable

-

Singularities of sub
-
Jacobian can be analyzed separately for decoupled
manipulators

a) Singularity of
J
21

-

Singularity of
J
21

depends on the relative orientation of the first three
column vectors

-

1

does not change relative orientation (viewpoint only)

Manipulator Kinematics

Singularity Analysis (cont'd)

General concept

L

L
3

L
2

L
1

Locus of
L

One sheet hyperboloid
surface

Manipulator Kinematics

Singularity Analysis (cont'd)

In summary: Let
L
1
,
L
2

and

L
3

represent
e
1
,
e
2

and
e
3
, respectively, if wrist
centre C fall on the surface of hyperboloid, singularity occurs.

Example

PUMA Robot

Case 1
:

C lies in the plane determined by

intersecting
e
1

and
e
2

e
1

r
1

and
e
2

r
2
are coplanar

Velocity of C along the direction

perpendicular to
e
3

r
3

and
n
12

(
L

direction) can not be produced

L

intersects with
e
1
at I, with

e
2

and
e
3

at

Manipulator Kinematics

Singularity Analysis (cont'd)

Case 2:

e
2

and
e
3
are parallel

r
2

and
r
3
lie in the same

plane

e
2

r
2

and
e
3

r
3

are

coplanar

Velocity of C in the plane

determined by
e
2

and
e
3

normal to
e
1

r
1

(
L

direction)

can not be produced

Manipulator Kinematics

Singularity Analysis (cont'd)

Geometric representation of singularity

Singularity : lies in the line that represents nullspace of

no mapping between and

The range of
J
21

is perpendicular to the nullspace of

Wrist singularity

J
12
is singular

e
4
,
e
5

and
e
6
are coplanar

e.g.,

Manipulator Kinematics

Acceleration Analysis

Rate relationship

Differentiate wrt time

Solving equation using LU decomposition

Compute
z

(forward substitution)

Compute (Backward substitution)

Manipulator Kinematics

Acceleration Analysis (cont'd)

Computing the Jacobian rate

Recall

Differentiate

Manipulator Kinematics

Acceleration Analysis (cont;d)

Computing the Jacobian rate (cont'd)

where

Static Analysis

Mapping between joint torques and EE wrench

Joint torques

Wrench acting at EE

Power at EE

Power at joints

X

Z

Y

1

n

3

2

f

n

Static Analysis

Mapping between joint torques and EE wrench (cont'd)

Power conservation condition:

Mapping EE wrench in Cartesian space to joint torques in joint space.

Recall

Static Analysis

Mapping between joint torques and EE wrench (cont'd)

6
-
R
Decoupled Manipulator

Solve the static problem for decoupled arm:

Arm torques

Wrist torques

Manipulator Kinematics

Interpretation Jacobian Matrix

Mapping from
n
-
D joint space to 6
-
D Cartesian space

The range of
J
(Column space) represents all possible EE twist that
can be produced by the manipulator

If
t

lies in the range of
J
, then there exist a that produces

t

at EE

The nullspace of
J

transpose represent all singularities

Assignment #3

Problems 4.4, 4.7, 4.19

Due in two weeks