Robotic Systems (6) : DR Richard Crowder

Robotic Systems(6)
Dr Richard Crowder
School of Electronics and Computer Science

Environment
Overview of the problem

Controller

Joints
End Effector
Vision
Demand
Feedback
Robot Control
Control depends on the robots configuration and application
Conventional
Position Control
Speed Control
Force Control

Biologically inspired
Behaviour based
Artificial Intelligence
.

Joint Control
This relies on the real time solution of the reverse kinematic
equation (typically at 20-25Hz).
A number of problems are apparent,
Coupling of joint position and velocities with gravity
and inertia terms
Trajectory generation
Trajectory position, velocity and acceleration of each
degree of freedom
Path update rate is typically 60 to 2000 times per second
Typically we are aware of the initial and final points, need
to program in via points
Need to blend in the via points into a single fluid
movement, using polynomials path generations
Trajectory generation
t

Resolved motion control
Used in teleoperation, the input is normally either speed or
force
Lift (turn)
Reach (twist)
Sweep (tilt)
Z
X
Y
Linear (rotary)
Jacobian Matrix(1)
Jacobian matrix generalises the notion of the ordinary
derivative of a scalar function
We have defined the [T] matrix, hence we can state:
P(t) [p
x
(t) p
y
(t) p
z
(t)]
T
V(t) [v
x
(t) v
y
(t) v
z
(t)]
T
(t) q J =
(t)
V(t)
(
(
O
] q ..... q (t)] = [ q [
T
n 1

J is the Jacobian
Jacobian Matrix (2)

v
x
is the x component of the tool velocity as a function of an individual joint velocity

z
is the Z component of the tools angular velocity
J
11
is the partial derivative of the x component of the to0p position with respect to the variable J
1

(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
dq
dq
dq
dq
dq
dq

J
. . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . .
J
=
v
v
v
6
5
4
3
2
1
66
11
z
y
x
z
y
x
e
e
e
Jacobian Matrix (3)

p
i-1
is the position of the (i-1)
th
frame relative to the origin
p is the position of the tool relative to the origin
z
i-1
is the unit vector along the axis of rotation of the i
th
frame

int jo linear a f or
Z
int jo rotary a f or
Z
) p p ( Z
J
i
i
i i
i
0
1
1
1 1
Example
Consider a simple two link manipulator.
m
n
a d
1
1
90 0 n
2
2
0 m 0
(
(
(
(
=
1 0 0 0
mS n 0 C S
C mS C S S C S
C mC S S C C C
T
2 2 2
2 1 1 2 1 2 1
2 1 1 2 1 2 1
2
0
Jacobian
0
p
C mC
p
C mS
p
mS n p C mS p C mC p
1
y
2 1
1
y
2 1
1
x
2 z 2 1 y 2 1 x
=
u c
c
=
u c
c
=
u c
c
+ = = =
(
u
u
(
(
(

=
(
(
(
2
1
2
2 1 2 1
2 1 2 1
mC 0
S mS C mC
S mC C mS
z
x
y

Note Joint 1 has no impact on the velocity in the Z direction

The velocities are a function of
1
and
2
, hence J needs to recomputed as the
manipulator moves
Transformations of forces
If we consider a 61 representation of the velocity of any body [v ]
T
or
a force [F M]
T

As in previous cases a 66 transformation can be applied to map these
values from one frame to a second.
This can be achieved by considering an extension to the kinematics and
Jacobian analysis.
Considered the following example where the transformation a general
velocity vector in frame A to a second frame B is required. This
procedure is only valid if the two frames are rigidly connected
Transformations of forces
X
w
Z
w
Y
w
X
T
Z
T
Y
T
Sensor
Applied
force
Objective if a force is applied at the tip, what does the sensor measure
Solution(1)
We need to find
T
F (force at the sensor tip), knowing
S
F (the sensor output),
in addition we know the location of the tip with reference to the sensor, hence:
0
=
=
R R P
R
T
T T F T F
T
S
T
S SORG
T
T
S
T
S
T
S
S
T
S T
S
T
) computed be can known, is (as
X
w
Z
w
Y
w
X
T
Z
T
Y
T
Sensor Sensor:
S
F Applied
Force:
T
F

Solution(2)
(
(
(
(

=
(
(
(
(
=
1 0 0 0
5 . 2 5 . 0 86 . 0 0
3 . 4 86 . 0 5 . 0 0
0 0 0 1
T
1 0 0 0
5 5 . 0 87 . 0 0
0 87 . 0 5 . 0 0
0 0 0 1
T
T
S
S
T
5 0 86 0 0 7 3 23 1 5 2
86 0 5 0 0 1 2 14 2 3 4
0 0 1 0 0 0
0 0 0 5 0 86 0 0
0 0 0 86 0 5 0 0
0 0 0 0 0 1
=
. . . . .
. . . . .
. .
. .
T
T
S
Hence.
Solution(2)
(
(
(
(
(
(
(
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
5 . 2
3 . 4
2
0
0
1
0
0
2
0
0
1
5 . 0 86 . 0 0 7 . 3 23 . 1 5 . 2
86 . 0 5 . 0 0 1 . 2 14 . 2 3 . 4
0 0 1 0 0 0
0 0 0 5 . 0 86 . 0 0
0 0 0 86 . 0 5 . 0 0
0 0 0 0 0 1
Sensor readings Actual tip forces
Summary..
Considered the configuration of industrial manipulators
Determination of the DH matrix
Forward and inverse kinematics
Next step
Sensing Tactile, force and vision
Introduction to biologically inspired systems

Robotic Systems (6) : DR Richard Crowder

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Robotic Systems(6)

Note Joint 1 has no impact on the velocity in the Z direction

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