MECH572A Introduction To Robotics

flybittencobwebΤεχνίτη Νοημοσύνη και Ρομποτική

2 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

174 εμφανίσεις

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

Quadratic equ in

4
(

4
)


4


5


6

Radical


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



MSS/Canadarm2


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