5 Robotic Systems Kinematics Control
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Transcript of 5 Robotic Systems Kinematics Control
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Robotic Systems(5)
Dr Richard Crowder
School of Electronics and Computer Science
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Workspace
Workspace is the volume that therobot can operate in.
Function of the joints
Which have limited values
Either Revolute or Prismatic
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Workspace Formal definition
W is the workspace
p(q), position for all joint values
k
= 1 for revolute and 0 for
prismatic
qmin and qmax are the joint limits
R6 is tool space (Cartesian space)
Rn is joint space (AKAconfiguration space)
Qq:Rp(q)
W 6
d)-(1+q kkkkk
)qCq:R(qQ maxminn
Q represents the set of vectors of the robots
joint coordinates with respect the joint limits
Means defined as
Means a set of
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Mapping
Tool SpaceJoint SpaceActuator Space
Forward
Reverse
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Inverse Kinematics
General properties
Tool tip must be within the workspace
Dexterous workspace is a subset ofReachableworkspace
A manipulator of less than 6 DoF cannot attain the generalgoal solution (i.e. has restriction on orientation or reach).
Note to completely control the end effectors position andorientation a minimum of 6 joints requires (6 DOF)
A clear understanding of the task requirements is requiredin all cases (particularly for less than 6DoF manipulators)
Inverse Kinematic Problem: given location and
orientation of EE, find joint variables
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Redundancy
When a manipulator can reach a point with more than oneconfiguration the manipulator is redundant.
The number of possible solutions is 2n where n is the
number of redundancies
Two solutions
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Infinitely Redundant Manipulators
Manipulator has more than 6 DoF
Increased flexibility: e.g. operating in pipes, through holes
Space Station
Manipulator system
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Degeneracy
Degeneracy occurs when control is lost over one or moredegrees of freedom.
Consider a wrist with co-linear (lying in the same straight
line)axis
Joint 3
Joint 2
Joint 1
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Degeneracy
In this position the move shown can be undertaken by bothjoint 1 and 3.
The possibility exist of one joint moving +180o and the
other -180o
The position of the end effector does not change, but themotion can be violent, leading to manipulator damage.
Solution to the problem Avoid these position.
Place software limits into the system to lock one ofthe joints when degeneracy is detected.
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Inverse Kinematic Solution
Geometric decompose the problem into a number ofsimple plane geometry problems-see Inverse KinematicsSolutions using Conformal Geometric Algebra.pdf
Algebraic equate the elements in the [A] and [T]matrices to give soluble equations.
Iterative approaches hill following
Neural Networks
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Algebraic Approach
The approach to reverse kinematics is to equate thevariables in the [T] matrix with those in the tool spacematrix. Hence, all the elements in the tool matrix mustequal those of the forward kinematic equations:
1000
Cd-Sa-Sa-dC-SSCS-
)Sd-Ca+Ca(SSS-CC-CS-CC-CS
)Sd-Ca+Ca(CSC-CS+SCC-SS+CC
pnml
pnml
pnml
T
234523322123452345234
23452332212341512341512341
234523322123415152341512341
zzzz
yyyy
xxxx
1000
][
Position of the end effector
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Atan2 Function
As the solutions contain sin-1 and cos-1, uncertainties canarise. These can be minimised by using the Atan2 function.
20and)tan(
0)sgn()(
0)sgn(2
0)sgn(
),(2Atan
x
y
xy
xy
xy
xy
Note that cos-1 (0) yields
two angles, -90 and +90,
which means two possible
solutions
Sign
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Uniqueness of the solution
Consider a three axis planar robot four possible solutions
Right handed, Elbow down
Left handed, Elbow up
Elbow up
Right handed
Elbow downLeft handed
3
360-3
360 2
1+180
2
1
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Robot's components
An industrial robot consists of "manipulator" which movesand performs tasks, "controller" which actuates andcontrols the manipulator, and "programming pendant"which teaches the manipulator movement