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# Homogeneous vector Homogeneous transformation matrix Review: Homogeneous Transformations.

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Homogeneous vector Homogeneous transformation matrix Review: Homogeneous Transformations
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Homogeneous vector

Homogeneous transformation matrix

Review: Homogeneous Transformations

Compute the position and orientation of the end effector as a function of the joint variables

Review: Aim of Direct Kinematics

The direct kinematics function is expressed by the homogeneous transformation matrix

Review: Direct Kinematics

Computation of direct kinematics function is recursive and systematic

Review: Open Chain

Review: Denavit-Hartenberg ConventionReview: Denavit-Hartenberg Convention

Review : D-H ConventionReview : D-H Convention

1. Fill in the table of D-H parameters for the spherical wrist.

Class Problem: Spherical Wrist

2. write the three D-H transformation matrices (one for each joint) for the spherical wrist

3. Find the overall transformation matrix which relates the final coordinates (x6y6z6) to the “base” coordinates (x3y3z3) for the spherical wrist

Review : D-H ConventionReview : D-H Convention

Joint Space and Operational SpaceJoint Space and Operational Space

position: coordinates (easy)

orientation: (n s a) (difficult)

w.r.t base frame

Function of time

Operational space

Independent variables

Joint space

Prismatic: d

Revolute: theta

Joint Space and Operational SpaceJoint Space and Operational Space

Direct kinematics equation

6,),( mnmRqRxqkx nm

?)( qk

Generally not easy to express

Joint Space and Operational SpaceJoint Space and Operational Space

Joint Space and Operational SpaceJoint Space and Operational Space

Workspace

reachable workspace

dexterous workspace

Factors determining workspace

Manipulator geometry

Mechanical joint limits

Mathematical description of workspace

Workspace is finite, closed, connected

Workspace ExampleWorkspace Example

Performance Indexes of ManipulatorPerformance Indexes of Manipulator

Accuracy of manipulator

Deviation between the reached position and the

position computed via direct kinematics.

repeatability of manipulator

reached position.

Kinematic RedundancyKinematic Redundancy

Definition

A manipulator is termed kinematically redundant

when it has a number of degrees of mobility whic

h is greater than the number of variables that are

necessary to describe a given task.

Kinematic RedundancyKinematic Redundancy

Intrinsic redundancy

m<n

functional redundancy

Why to intentionally utilize redundancy?

Kinematic CalibrationKinematic Calibration

Kinematic calibration techniques are devoted to

finding accurate estimates of D-H parameters fr

om a series of measurements on the manipulato

r’s end-effector location.

Direct measurement of D-H is not allowed.

Inverse Kinematics

Inverse KinematicsInverse Kinematics

we know the desired “world” or “base” coordinates for the end-effector or tool

we need to compute the set of joint coordinates that will give us this desired position (and orientation in the 6-link case).

the inverse kinematics problem is much more difficult than the forward problem!

Inverse KinematicsInverse KinematicsInverse KinematicsInverse Kinematics

there is no general purpose technique that will guarantee a closed-form solution to the inverse problem!

Multiple solutions may exist Infinite solutions may exist, e.g., in the case

of redundancy There might be no admissible solutions

(condition: x in (dexterous) workspace)

Inverse KinematicsInverse KinematicsInverse KinematicsInverse Kinematics

most solution techniques (particularly the one shown below) rely a great deal on geometric or algebraic insight and a few common “tricks” to generate a closed-form solution

Numerical solution techniques may be applied to all problems, but in general do not allow computation of all admissible solutions

x is known, compute q

W can be expressed W can be expressed both as a function of both as a function of end-effector p&o, anend-effector p&o, and as a function of a red as a function of a reduced number of joinduced number of joint variablest variables

21 ,

3

Class problem

Algebraic approach

no admissible solution If c2 is out of this range

Elbow up and elbow down

Geometric approach

?

?

i

＝？＝？l

l

Feasible condition: a1+a2>l and |a1-a2|<l

Class Problem

what are the forward and inverse kinematics equations for the two-link planar robot shown on the right?

2nd Joint: Prismatic

1st Joint: Revolute

X0

Y0

90 deg

Attention: m= ?

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