Transcript Introduction to ROBOTICS
Introduction to ROBOTICS
Kinematics of Robot Manipulator
PROF.UJWAL HARODE
UJWALHARODE 1
Outline
Review Robot Manipulators -Robot Configuration -Robot Specification
•
Number of Axes, DOF
•
Precision, Repeatability Kinematics -Preliminary
•
World frame, joint frame, end-effector frame
•
Rotation Matrix, composite rotation matrix
•
Homogeneous Matrix -Direct kinematics
•
Denavit-Hartenberg Representation
•
Examples -Inverse kinematics
UJWALHARODE 2
Review
What is a robot?
By general agreement a robot is: A programmable machine that imitates the actions or appearance of an intelligent creature–usually a human.
To qualify as a robot, a machine must be able to: 1) Sensing and perception: get information from its surroundings 2) Carry out different tasks: Locomotion or manipulation, do something physical–such as move or manipulate objects 3) Re-programmable: can do different things 4) Function autonomously and/or interact with human beings Why use robots?
–Perform 4A tasks in 4D environments 4A: Automation, Augmentation, Assistance, Autonomous 4D: Dangerous, Dirty, Dull, Difficult UJWALHARODE 3
Manipulators
• • • • •
Robot arms, industrial robot Rigid bodies (links) connected by joints Joints: revolute or prismatic Drive: electric or hydraulic End-effector (tool) mounted on a flange or plate secured to the wrist joint of robot
UJWALHARODE 4
Manipulators
Robot Configuration
Cartesian: PPP Cylindrical: RPP Spherical: RRP Articulated: RRR
SCARA: RRP (Selective Compliance Assembly Robot Arm)
Hand coordinate: n: normal vector; s: sliding vector; a: approach vector, normal to the tool mounting plate 5
•
Manipulators
• •
Motion Control Methods
• •
Point to point control a sequence of discrete points spot welding, pick-and-place, loading & unloading
• •
Continuous path control follow a prescribed path, controlled path motion Spray painting, Arc welding, Gluing
UJWALHARODE 6
•
Manipulators
• • • • •
Robot Specifications Number of Axes
• • •
Major axes, (1-3) => Position the wrist Minor axes, (4-6) => Orient the tool Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration Degree of Freedom (DOF) Workspace Payload (load capacity) Precision v.s. Repeatability
UJWALHARODE 7
What is Kinematics Forward kinematics
z
Given joint variables
q
(
q
1 ,
q
2 ,
q
3 ,
q
4 ,
q
5 ,
q
6 ,
q n
)
Y
(
x
,
y
,
z
,
O
,
A
,
T
) End-effector position and orientation, Formula?
UJWALHARODE
y x
8
What is Kinematics Inverse kinematics End effector position and orientation (
x
,
y
,
z
,
O
,
A
,
T
)
q
(
q
1 ,
q
2 ,
q
3 ,
q
4 ,
q
5 ,
q
6 ,
q n
) Joint variables -Formula?
UJWALHARODE 9
Forward kinematics
x
0
l
cos
y
0
l
sin Inverse kinematics cos 1 (
x
0 /
l
) Example 1 UJWALHARODE
y
0
l y
1
x
1
x
0 10
Preliminary
•
Robot Reference Frames 1. World frame 2. Joint frame 3. Tool frame
z
T
z x y y
T
x
W R UJWALHARODE 11
Coordinate Transformation
z
Reference coordinate frame –XYZ Body-attached frame-ijk Point represented in XYZ
P xyz
P p xyz x
i x [
p p x y
, j y
p y
,
p z
]
T p z
k z Point represented in uvw:
P uvw
p u
i u
p v
j v
p w
k w
w
O, O’
v
UJWALHARODE Two frames coincide
P u y
12
x
Properties: Dot Product
Let and be arbitrary vectors in and be the angle from to , then
x
y
x y
cos
Properties of orthonormal co-ordinate frame
Unit vectors are mutually perpendicular
i
i k
j
k j
0 0 0 UJWALHARODE | | |
i
k j
| | | 1 1 1 13
Coordinate Transformation Reference coordinate frame OXYZ Body-attached frame O’uvw
z P P xyz
RP uvw w v u
O, O’ How to relate the coordinate in these two frames? UJWALHARODE
y
14
x
•
Basic Rotation
–
p x p p
OX, OY, OZ axes, respectively – Since
P
p u
i u
p v
j v
p w
k w
p x
i x
P
i x i u
p u
i x j v
p v
i x k w
p w p y
j y
P
j y i u
p u
j y j v
p v
j y k w
p w p z
k z
P
k z i u
p u
k z j v
p v
k z k w
p w
UJWALHARODE 15
Basic Rotation Matrix
p p y p x z
i k j x y z i u i i u u i j x k z y j v j v j v Rotation about x-axis with
Rot
(
x
, ) 1 0 0 0
C
S
C
0
S
j i x k z y k w k w k w
p p p w u v
w z
x u
UJWALHARODE
P v y
16
UJWALHARODE
Rot
(
y
, )
C
0
S
0 1 0
S
0
C
Rot
(
z
, )
C S
0
S
C
0 0 0 1 17
Example 2 • frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.
a
xyz
0
Rot
(
z
, 60 )
a uvw
0 0 .
5 .
866 0 .
866 0 0 .
5 0 0 1 4 3 2 4 0 .
2 598 .
964 UJWALHARODE 18
Composite Rotation Matrix
•
A sequence of finite rotations
– matrix multiplications do not commute – rules: • if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix • if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix UJWALHARODE 19
•
Find the rotation matrix for the following operations:
Rotation about OY axis Rotation about OW axis Rotation about OU axis
Answer
...
R
Rot
(
y
, )
I
3
Rot
(
w
, )
Rot
(
u
, ) C 0 S 0 1 0
C
C
S S
C
S 0 C
C S
0
S
S
C S
0
C
S
C
S
S C
C
C
C
S
0 0 1 1 0 0
C S
0
C
0
S
C C
S
C
S
C
S S
C
S
S
S
Pre-multiply if rotate about the OXYZ axes Post-multiply if rotate about the OUVW axes UJWALHARODE 20
•
Homogeneous Transformation
Special cases 1. Translation
A T B
I
3 3 0 1 3
A r o
' 1 2. Rotation
A T B
A R B
0 1 3 Homogeneous transformation matrix
F
R
3 3 0 0 3 1 1
P
3 1 1 UJWALHARODE 21
Homogeneous Transformation
• •
Composite Homogeneous Transformation Matrix Rules:
– Transformation (rotation/translation) w.r.t (X,Y,Z) (OLD FRAME), using pre-multiplication – Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post-multiplication UJWALHARODE 22
Orientation Representation
•
Description of Roll Pitch Yaw
Z
ROLL PITCH
Y X
YAW UJWALHARODE 23