Review: Homogeneous Transformations
 Homogeneous vector
 Homogeneous transformation matrix
Review: Aim of Direct Kinematics
Compute the position and orientation of the end
effector as a function of the joint variables
Review: Direct Kinematics
 The direct kinematics function is expressed
by the homogeneous transformation matrix
Review: Open Chain
 Computation of direct kinematics function is
recursive and systematic
artenberg Convention
-H Convention
Class Problem: Spherical Wrist
1. Fill in the table of D-H parameters for the 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
-H Convention
Operational Space


Description of end-effector task
 position: coordinates (easy)
w.r.t base frame
 orientation: (n s a) (difficult)
Function of time
Operational space
Independent
variables

Joint space
Prismatic: d
Revolute: theta
Operational Space

Direct kinematics equation
x  k ( q ),

x R ,q R
m
Three-link planar arm
n
(Pp50 2-58)
k (q)  ?
mn
m6
Operational Space
Generally not easy to express
Operational Space


Workspace

Factors determining workspace
 reachable workspace
Manipulator geometry
 dexterous workspace
Mechanical joint limits
Mathematical description of workspace
Workspace is finite, closed, connected
ce Example
exes 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.
c Redundancy
 Definition
A manipulator is termed kinematically redundant
when it has a number of degrees of mobility
which is greater than the number of variables that
are necessary to describe a given task.
c Redundancy
 Intrinsic redundancy
m<n
 functional redundancy
 relative to the task
 Why to intentionally utilize redundancy?
c Calibration
Kinematic calibration techniques are devoted to
finding accurate estimates of D-H parameters
from a series of measurements on the
manipulator’s end-effector location.
Direct measurement of D-H is not allowed.
e Kinematics
e Kinematics



we know the desired “world” or “base” coordinates for
we need to compute the set of joint coordinates that w
the inverse kinematics problem is much more difficult
se Kinematics




there is no general purpose technique that will guaran
Multiple solutions may exist
Infinite solutions may exist, e.g., in the case of redund
There might be no admissible solutions (condition: x i
se Kinematics


most solution techniques (particularly the one shown
Numerical solution techniques may be applied to all p
link Planar Arm
x is known, compute q
link Planar Arm
W can be expressed
both as a function of
end-effector p&o,
and as a function of a
reduced number of
joint variables
Class problem
Two-link planar arm
 1 , 2
one-link planar arm
3
link Planar Arm

Algebraic approach
link Planar Arm
no admissible solution If
c2 is out of this range
Elbow up and elbow down
link Planar Arm
link Planar Arm

Geometric approach
l ?
＝？
＝？
i  ?
Feasible condition:
a1+a2>l and |a1a2|<l
l
Class Problem
what are the forward
and inverse
kinematics equations
for the two-link
planar robot shown
on the right?
Attention: m= ?
Y0
90 deg
2nd Joint:
Prismatic
1st Joint:
Revolute
X0