Ch. 3. Manipulator Kinematics

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Ch. 3. Manipulator Kinematics
-- A video of Motoman -- A video of Adept
Lab. 1
Joints, Links, End-effector
Base frame, Tool frame
Joints
Rovelute joint Prismatic joint Helical
joint Cylindrical joint Spherical joint Planar
joints. ---lower pairs, all subgroups of SE(3)
Forward kinematics
Base(Link 0) Stationary Link 1 First movable
link Link n End-effector attached
Connecting link 0 to link 1
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Joint space
Revolute joint Prismatic joint
Joint Space
of R. joint
P-joint
Adept
Reference or nominal config
Final
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Classical approach
The product of exponentials formula
Consider Fig 3.2 again.
Step 1 Rotating about by
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Step 2 Rotating about by
What if another route is taken?
Step 1 Rotating about by
Step 2 Rotating about by
Let
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Independent of the route taken
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Procedure for forward kinematic map
Identify a nominal config.
Choose nominal config. So things become simple.
Revolute joint
Choose so is simple.
Prismatic joint
Write
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Example
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Example 2 (Elbow Manipulation)
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Simplify forward Kinematics Map
-- Choice of base frame or ref. Config.
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2.4 Manipulator Workspace

• Reachable Workspace

• Dextrous Workspace

E.q.
Reachable Workspace
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Dextrous Workspace
Note end-effector frame not aligned with the
center of the wrist
Manipulators maximum workspace (Paden).
Elbow manipulator and its dual
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3. Inverse Kinematics
Inverse Kinematics Given find
3.1 A planar example
Given , solve for
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Polar Coordinates
Law of cosines
filp solution
High Lights

• Subproblems
• Each has zero, one or two solutions!

3.2 Paden-Kahan Problem
Subproblem 1 Rotation about a single axis
Let be a zero-pitch twist, with unit magnitude
and Two points. Find s.t.
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Solution Let define
Also,
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The solution exists only if
If , then
If Infinite of solutions!
Subproblem 2
Let and be two zero-pitch, unit
magnitude twists, with intersecting axes, and
find and s.t.
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Solution If two axes of ,
coincide, Then Problem 1.
If the two axes not parallel, ,
then, let C be s.t.
As are linear
independent
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() has zero, one or two solution.

1. Two solutions when two circles intersect.
2. One solution when tangential
3. Zero solution when none

Subproblem 3
Let be a zero-pitch twist, with unit magnitude
and Two points. Find s.t.
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Define
Define
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Zero, one or two solutions !
3.3 Solving inverse kinematics using subproblems
Techniques 1
e.q.
Techniques 2 subtract a point
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E.q. 3.5 Elbow manipulator
Step 1 ( Solve for )
Let
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Subtract
Subproblem 1.
Step 2 Given
Step 3
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Step 4
Counting Maximum of 8 solutions.
E.q. 3.6 Inverse Kinematics of SCARA
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There are a maximum of two solutions!
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4. Manipulator Jacobian.
Given
What is the velocity of the tool frame
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is affected by only
The twist associated with joint i, at the present
configuration.
Body Jacobian
Joint twist written with respect to the body
frame at the current configuration!
If is invertible,
How to find , given
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E.q. Jacobian for a SCARA manipulator
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E.q. 2 Jacobian of Stanford arm
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T
4.2 End-effector force
S

• Given what is required to balance that
  force.

• If we apply a set of joint torques, what is the
  resulting end-effector wrench?

Structural force
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E.q. SCARA manipulator
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4.3 Singularities
A singularity config. Is a point at which
drops rank.

• Consequence n6
• Cant move in certain directions
• Large joint motion is required
• Structural force
• Cant apply end-effector force in certain
  direction force!

Singularities for 6R-manipulators
E.q.1. Two collinear revolute joints
is singular if there exists two joints
s.t. 1.The axes are parallel, 2. The axes are
collinear,
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Proof Elementary row or column operation do not
change rank of
Spherical wrist,
E.q.2. 3 parallel coplanar revolute joint axes
is singular if there exists two joints s.t.

1. The axes are parallel,
2. The axes are coplanar, I.e. there exists a plane
   with normal n s.t.

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Proof Changes of frame
Linear independent
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E.q. Elbow manipulator in its reference config.
E.q.3 Four intersecting revolute joints axes
is singular if there exists four revolute joints
s.t. With inter point q s.t.
Proof Choose the frame orgin at q,
4.4 Manipulability