ROBOTICS An Introduction

1
ROBOT KINEMATICS
Purpose The purpose of this chapter is to
introduce you to robot kinematics, and the
concepts related to both open and closed
kinematics chains. Forward kinematics is
distinguished from inverse kinematics.
2

• In particular, you will
• Review the concept of a kinematics chain
• Distinguish serial from parallel robots
  structures.
• Relate a robots degrees-of-freedom (dof) to the
  joint structure constraints.
• Define robot redundancy.
• Define serial robot types based on primary dof.
• Introduce D-H coordinates.
• Examine forward and inverse kinematics for serial
  robots.
• Examine forward and inverse kinematics for
  parallel robots.

3
Kinematics chain Mechanisms can be configured as
kinematics chains. The chain is closed when the
ground link begins and ends the chain otherwise,
it is open.
4
Degrees-of-freedom (F) of a mechanism
5
Grublers criterion It is also true that l ci
fi which leads to Grubler's Criterion F l
(n - j - 1) Example - 6-axis revolute robot(ABB
IRB 4400) Referencing figure and using
previous equation F 6 (7 - 1) - 6 (5) 6
"as expected or by Grublers equation F
6(7 6 1) 6 (1) 6 "as expected
6
Redundant degrees-of-freedom .
A redundant joint is one that is unnecessary
because other joints provide the needed position
and/or orientation. Can you see the redundancy
among the last 3 joints on the IRB 4400?
7
Redundant degrees-of-freedom
A passive dof allows an intermediate link to
rotate freely about an axis defined by the two
joints. A passive dof cannot transfer torque.
8
Loop Mobility Criterion
The number of independent loops (L) is the total
number of loops excluding the external loop. For
multiple loop chains it is true that j n L -1
which gives Euler's equation L j - n 1 The
figure applies this equation for a 2 loop
mechanism.
9
Loop Mobility Criterion
Combining Eulers equation with Grubler's
Criterion, we get the Loop Mobility Criterion
å fi F l L
10
Parallel robots A parallel robot is a closed
loop chain, whereas a serial robot is an open
loop chain. A hybrid mechanism is one with both
closed and open chains.
The figure shows the Stewart-Gough platform.
Determine the dof. Each S-P-S combination
generates a passive degree-of-freedom. Thus,
apply l 6 n 14 j1 6 j3 12
fp 6 to get F 6(14 - 18 -1) (12 x 3 6)
- 6 6

....as expected!
11
Serial Robot Types
Serial robots can be classified as revolute,
spherical, cylindrical, or rectangular
(translational, prismatic, or Cartesian). These
classifications describe the primary DOF
(degrees-of-freedom) which accomplish the global
motion as opposed to the distal (final) joints
that accomplish the local orientation.
12
Open Chain Link Coordinates According to
Denavit-Hartenberg (D-H) notation (Denavit, J.
and Hartenberg, "A Kinematic Notation for
Lower-Pair Mechanisms Based on Matrices," J. of
Applied Mechanics, June, 1955, pp. 215-221.),
only four parameters (a, d, q, a) are necessary
to define a frame in space (or joint axis)
relative to a reference frame
z
a minimum distance between line L (the z axis
of next frame) and z axis (mutually orthogonal
line between line L and z axis) d distance
along z axis from z origin to minimum distance
intersection point q angle between x-z plane
and plane containing z axis and minimum distance
line a angle between z axis and L .
L
a
d
a
d
q
y
x
13
Conventional D-H notation for serial links/joints
14
D-H parameters applied to Puma robot
15
D-H parameters applied to Puma robot
 
16
D-H parameters and the HT
 
Given a revolute joint a point located on the
ith link can be located in i - 1 axes by the
following transformation set which consist of
four homogeneous transformations (2 rotations and
2 translations). The set that will accomplish
this is Ai H(d,zi-1) H(q,z i-1) H(a
,xi) H(a,x i) (i 1, ...n)
Why are the D-H parameters applied as shown to
generate the Ai transformation?
Ai
17
Other D-H Notation (used in CODE)
 
18
Other D-H Notation (used in CODE)
 
Four parameters must be specified ai minimum
distance between joint i axis (zi) and joint i-1
axis (zi-1) di distance from minimum distance
line (xi-1 axis) to origin of ith joint frame
measured along zi axis. ai angle between zi and
zi-1 measured about previous joint frame xi-1
axis. qi angle about zi joint axis which
rotates xi-1 to xi axis in right hand sense.
The xi axis is the minimum distance line defined
from zi to zi1 zi is defined as the joint
rotation or translation axis axis and yi by the
right hand rule (zi x xi). The origin of each
joint frame is defined by the minimum distance
line intersection on the joint axis.
19
Other D-H Notation (used in CODE)
 
What are the major differences between
conventional D-H and CODE D-H?
20
Class problem Derive the set of CODE D-H
parameters for the Puma robot being considered.
21
Forward Kinematics for Serial Robots
Given the A transformation matrices of one joint
frame relative to the preceding joint frame using
conventional D-H notation, we can relate any
point in the ith link to the global reference
frame by the following transformation. Let vi
be a point fixed to the ith link. Its
coordinates ui in the global frame are (n
dof) ui A1 A2....Ai vi ( i 1, 2, ...
n) Typically we represent the set of
transformations above by a single matrix called
the T matrix Ti A1 A2....Ai If i n, then
Tn locates the tool (or gripper) frame. We
usually drop the n subscript and simply use T.
22
Forward Kinematics using CODE D-H
 
The pose of a tool frame at the end of the robot
can be determined by the equation T
A1A2A3...AnG where T locates the tool relative
to the robot base frame and G locates the tool
relative to the last joint frame. Question Why
is G required in the CODE D-H notation, but not
in the conventional D-H notation?
23
Inverse kinematics for serial robots
 
Inverse kinematics raises the question Given
that I know the desired pose of the tool, what
are the joint values required to move the tool to
the desired pose? Forward kin T
A1A2A3...AnG unknown
known Inverse kin A1A2A3...An T G-1
unknown known
24
Inverse kinematics for Puma robot
 
The solution, calculated in two stages, first
uses a position vector from the robot base (place
at waist) to the wrist. This vector allows for
the solution of the first three primary DOF that
accomplish the global motion. The last 3 DOF
(secondary DOF) are found using the calculated
values of the first 3 DOF and the orientation
joint frames T4, T5, and T6 .
25
Inverse kinematics for Puma robot
 
sliding
y
6
approach
z
6
x
6
Gripper
26
Inverse kinematics for Puma robot
Let the gripper frame be defined by the unit
vector triad n, s, and a , unit vectors directed
along the tools x, y, and z axes respectively.
These are specified by the known pose of the tool
frame, i.e., the target frame. The origin of the
4th joint frame is determined by q and the known
offset of the tool frame, d6.
Zo
27
Inverse kinematics for Puma robot
Applying T A1 A2....A6 and using conventional
D-H notation, we multiply Ai together to get one
matrix with 6 unknowns q1, q2,..., q6. Setting
the unknown terms equal to the terms of the known
target frame we generate 12 equations with 6
unknown joint angles. The course notes have a
full description of these equations, but the
(1,1) term looks like
where C23 cos(q2 q3), etc.
28
Inverse kinematics for Puma robot
Since the last three joint axes intersect at a
common point (concurrent axes), which is the
origin of the 4th joint frame, we let q4 q5
q6 0 and also let d6 0 to reduce the invkin
(short for inverse kinematics) equations to those
of the 4th joint frame. At this point
29
Inverse kinematics for Puma robot
This gives the equations
qy S1 (d4 S23 a2 C2) d2 C1
Remember that qx, qy, and qz are known. Is there
any way that we can solve these equations for the
primary joint angles q1, q2 and q3. ?
30
Inverse kinematics for Puma robot
?What does this form imply?
31
Inverse kinematics for Puma robot
How many solutions exist for q1? Explain. Can you
relate these solutions to the physical
configuration of the Puma robot? The solution
for q3 can be found by squaring the q components
adding to get cos q3 sqrt 1 - sin2q3
q3 tan-1 The soln is for the elbow above
hand whereas the - soln is for the elbow below
hand.
32
Inverse kinematics for Puma robot
Now, given q3, we can expand and finally arrive
at (use atan2)
2
2
2
-
q


q
z
4
3
x
y
2
q
tan
-1
2
2
2
2
q

d

C
-


q
z
4
3
x
y
2
The - soln corresponds to the left arm
configuration, soln corresponds to right arm
configuration.
33
Inverse kinematics for Puma robot
Obviously knowing permits definition of 0T3. To
determine q4, q5, and q6 we assume that an
approach direction is known (a known) and that
hand orientation is specified (n, s). For the
PUMA robot we can arrange the joint axes such
that
(z4 axis direction cosines)
z5 a (z5 axis
direction cosines) y6 s
(y6 axis direction cosines)
34
Inverse kinematics for Puma robot
Now given the above criteria, we can solve for
q4 from
Determine x3 and y3 from 1st and 2nd columns of
0T3 to get (use atan2)
S
C

a
-

a
x
y
1
1
q
tan
-1

4
C
1
y
z
23
1
x
23
23
35
Inverse kinematics for Puma robot
Refer to the course notes to get similar
solutions for q5 and q6. How many total invkin
solutions are feasible for the Puma robot? How
many solutions exist in practical
applications? What limits the number of invkin
solutions that can be applied?
36
Geometric solutions for inverse kinematics
37
Kinematics summary serial robots

• Forward inverse kinematics are important to
  robotics.
• Robot teach-pendant uses direct joint control to
  place the robot tool at desired poses in space.
  
• Target for an end-effector requires invkin
  solutions to generate the necessary joint values.
  
• D-H parameters provide a simple way of relating
  joint frames to each other, although more than
  one D-H form proliferate the application methods.
  
• Invkin solutions can be complex depending on the
  robot structure . Both analytical and geometric
  methods can be applied.

38
Forward kinematics - parallel robots

• A parallel mechanism is symmetrical if the
• number of limbs is equal to the number
  ofdegrees-of-freedom of the moving platform
• joint type and joint sequence in each limb are
  the same
• number and location of the actuated joints are
  the same.

Otherwise, the mechanism is asymmetrical. We will
examine the kinematics for symmetrical
mechanisms.
39
Parallel robot definitions
limb a serial combination links and joints
between ground and the moving rigid
platform connectivity of a limb (Ck)
degrees-of-freedom associated with all joints in
a limb
40
Connectivity equation
Answer 7!
41
Forward kinematics example
The course notes present the forward kinematics
solution for the Maryland (or ABB picker) robot.
This solution is also found in Tsais text. It is
complex and will not be discussed here, but you
should review the solution to see how it is
done. The reason for not examining this solution
is found in the question How would you use a
teach pendant to program this robot?
42
Program picker robot?
In reality you would probably not command the
joints directly, but most likely command
translations in the u, v, and w directions. Thus,
you would not likely drive this robot using
forward kinematics but only apply inverse
kinematics.
43
Inverse kinematics for picker robot
Assume that a desired position vector p given.
Find the joint angles to place point P at p. In
reality, the gripper would not be located at P,
but be attached to the moving platform. This is
determined by gripper frame G relative to the
platform coordinate axes. A target frame is
specified as T. The frame for point P is
determined from the fourth column of Tp TG-1.
We designate this vector as p.
44
Inverse kinematics for picker robot
Given p we determine the location of point Ci.
This is simple because the moving platform cannot
rotate and thus the line between P and Ci
translates only. Thus, given P (as determined by
p) and the distance h, we can determine Ci as
displaced from P by a vector of length h that is
parallel to xi.
parallel to xi
Ci
h
P
p
P
zero state
xi
O
45
Inverse kinematics for picker robot
We can write a loop closure equation
AiBi BiCi OP PCi OAi and express this
equation in the limb i coordinate frame (xi, yi,
zi).
46
Inverse kinematics for picker robot
Expressing the previous limb vector loop
equation in the limb i coordinate frame, we get
the matrix equations in terms of the limb vector
c shown by the green vector in the figure
47
Inverse kinematics for picker robot
The locus of motion of link BiCi is a sphere with
center at Ci and radius b. From the previous
matrix equation we can determine a solution for
q3i as q3i cos-1(cyi/b)
P
h
Ci
b
Bi'
h
Bi
b
zi
a
O
r
xi
Ai
48
Inverse kinematics for picker robot
Given q3i we can determine an equation for q2i by
summing the squares of the matrix equation to
get 2ab sq3i cq2i a2 b2 cxi2 cyi2
czi2 which leads to a solution for q2i as
q2i cos-1(k) where k (cxi2 cyi2 czi2
- a2 - b2)/(2ab sq3i). Physically, we can
determine two solutions for q2i ("" angle and
"-" angle similar to elbow up/down case).
49
Inverse kinematics for picker robot
The two solutions for q1i can be determined from
the matrix equation by expanding the double angle
formulas, solving for the sine and cosine of q1i
and then using the atan2 function to get q1i.
50
Inverse kinematics for picker robot
It is possible that the target frame may fall
outside the robot's reach thus, we must examine
the special cases

• Generic solution - circle of link AB intersects
  the BC sphere at two points, giving two
  solutions.
• Singular solution - circle tangent to sphere
  resulting in one solution.
• Singular solution - circle lies on sphere --
  physically unrealistic case!
• No solution - circle and sphere do not intersect

51
Kinematics summary parallel robots

• Forward kinematics particularly useful.
• Parallel robots use invkin to program
  configurations
• Commercial parallel robots are symmetrical.
• Grubler's Criterion does not readily apply to
  this class of complex mechanisms, because of
  designed redundancies and special constraints.
• Parallel robots are inherently stiffer with less
  inertia thus, they can move much faster.