6 robotic systems control
TRANSCRIPT
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Robotic Systems(6)
Dr Richard Crowder
School of Electronics and Computer Science
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Environment
Overview of the problem
Controller
Joints
End Effector
VisionDemand
Feedback
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Robot Control
Control depends on the robots configuration and application
Conventional
Position Control
Speed Control Force Control
Biologically inspired
Behaviour based
Artificial Intelligence
.
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Joint Control
This relies on the real time solution of the reverse kinematicequation (typically at 20-25Hz).
A number of problems are apparent,
Coupling of joint position and velocities with gravityand inertia terms
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Trajectory generation
Trajectoryposition, velocity and acceleration of eachdegree of freedom
Path update rate is typically 60 to 2000 times per second
Typically we are aware of the initial and final points, needto program invia points
Need to blend in the via points into a single fluid
movement, using polynomials path generations
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Trajectory generation
t
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Resolved motion control
Used in teleoperation, the input is normally either speed orforce
Lift (turn)
Reach (twist)
Sweep (tilt)
Z
X
Y
Linear (rotary)
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Jacobian Matrix(1)
Jacobian matrix generalises the notion of the ordinaryderivative of a scalar function
We have defined the [T] matrix, hence we can state:
P(t) [px(t) py(t) pz(t)]T V(t) [vx(t) vy(t) vz(t)]
T
(t)qJ=
(t)
V(t)
]q.....q(t)] = [q[T
n1 J is the Jacobian
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Jacobian Matrix (2)
vx is the x component of the tool velocity as a function of an individual joint velocity
zis the Z component of the tools angular velocity
J11 is the partial derivative of the x component of the to0p position with respect to the variable J1
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
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Jacobian Matrix (3)
pi-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
zi-1 is the unit vector along the axis of rotation of the ith frame
intjolinearafor
Z
intjorotaryaforZ
)pp(Z
Ji
i
ii
i
0
1
1
11
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Example
Consider a simple two link manipulator.
m
n
a d1 1 90 0 n
2 2
0 m 0
1000
mSn0CS
CmSCSSCS
CmCSSCCC
T222
2112121
2112121
2
0
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Jacobian
0p
CmCp
CmSp
mSnpCmSpCmCp
1
y
21
1
y
21
1
x
2z21y21x
2
1
2
2121
2121
mC0
SmSCmC
SmCCmS
z
x
y
Note Joint 1 has no impact on the velocity in the Z direction
The velocities are a function of1and2, hence J needs to recomputed as the
manipulator moves
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Transformations of forces
If we consider a 61 representation of the velocity of any body [v]T ora force [F M]T
As in previous cases a 66 transformation can be applied to map thesevalues from one frame to a second.
This can be achieved by considering an extension to the kinematics andJacobian analysis.
Considered the following example where the transformation a generalvelocity vector in frame A to a second frame B is required. Thisprocedure is only valid if the two frames are rigidly connected
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Transformations of forces
Xw
Zw
Yw
XT
ZTYT
SensorApplied
force
Objective if a force is applied at the tip, what does the sensor measure
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Solution(1)
We need to find TF (force at the sensor tip), knowing SF (the sensor output),
in addition we know the location of the tip with reference to the sensor, hence:
0=
=
RRP
RT
TTFTF
T
S
T
SSORG
T
T
ST
S
T
S
S
T
ST
S
T
)computedbecanknown,is(as
Xw
Zw
Yw
XT
ZTYTSensorSensor: SF Applied
Force: TF
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Solution(2)
1000
5.25.086.00
3.486.05.00
0001
T
1000
55.087.00
087.05.00
0001
T
T
S
S
T
5086007323152
8605001214234
001000000508600
000860500
000001
=
.....
.....
..
..
TTS
Hence.
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Solution(2)
5.2
3.4
2
0
0
1
0
0
2
0
0
1
5.086.007.323.15.2
86.05.001.214.23.4
001000
0005.086.00
00086.05.00
000001
Sensor readings Actual tip forces
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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