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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
Description of end-effector task
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
Three-link planar arm (Pp50 2-58)
?)( 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
A measure of the ability to return to a previously
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
relative to the task
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
Three-link Planar ArmThree-link Planar ArmThree-link Planar ArmThree-link Planar Arm
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
Three-link Planar ArmThree-link Planar ArmThree-link Planar ArmThree-link Planar Arm
Two-link planar arm
one-link planar arm
21 ,
3
Class problem
Three-link Planar ArmThree-link Planar ArmThree-link Planar ArmThree-link Planar Arm
Algebraic approach
Three-link Planar ArmThree-link Planar ArmThree-link Planar ArmThree-link Planar Arm
no admissible solution If c2 is out of this range
Elbow up and elbow down
Three-link Planar ArmThree-link Planar ArmThree-link Planar ArmThree-link Planar Arm
Geometric approach
Three-link Planar ArmThree-link Planar ArmThree-link Planar ArmThree-link Planar Arm
?
?
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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