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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