Forward Kinematics

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Forward Kinematics
Introduction to ROBOTICS

• University of Bridgeport

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Kinematic

• Forward (direct) Kinematics
• Given The values of the joint variables.
• Required The position and the orientation of the
  end effector.
• Inverse Kinematics
• Given The position and the orientation of the
  end effector.
• Required The values of the joint variables.

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Why DH notation

• Find the homogeneous transformation H relating
  the tool frame to the fixed base frame

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Why DH notation

• A very simple way of modeling robot links and
  joints that can be used for any kind of robot
  configuration.
• This technique has became the standard way of
  representing robots and modeling their motions.

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DH Techniques

1. Assign a reference frame to each joint (x-axis
   and z-axis). The D-H representation does not use
   the y-axis at all.
2. Each homogeneous transformation Ai is represented
   as a product of four basic transformations

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DH Techniques

• Matrix Ai representing the four movements is
  found by four movements

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DH Techniques

• The link and joint parameters
• Link length ai the offset distance between the
  Zi-1 and Zi axes along the Xi axis.
• Link offset di the distance from the origin of
  frame i-1 to the Xi axis along the Zi-1 axis.

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DH Techniques

• Link twist ai the angle from the Zi-1 axis to
  the Zi axis about the Xi axis. The positive sense
  for a is determined from zi-1 and zi by the
  right-hand rule.
• Joint angle ?i the angle between the Xi-1 and Xi
  axes about the Zi-1 axis.

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DH Techniques

• The four parameters
• ai link length, ai Link twist , di Link
  offset and
• ?i joint angle.
• The matrix Ai is a function of only a single
  variable qi , it turns out that three of the
  above four quantities are constant for a given
  link, while the fourth parameter is the joint
  variable.

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DH Techniques

• With the ith joint, a joint variable is qi
  associated where
• All joints are represented by the z-axis.
• If the joint is revolute, the z-axis is in the
  direction of rotation as followed by the right
  hand rule.
• If the joint is prismatic, the z-axis for the
  joint is along the direction of the liner
  movement.

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DH Techniques

• 3. Combine all transformations, from the first
  joint (base) to the next until we get to the last
  joint, to get the robots total transformation
  matrix.
• 4. From , the position and orientation of the
  tool frame are calculated.

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DH Techniques
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DH Techniques
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DH Techniques
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DH Techniques
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Example I The two links arm

• Base frame O0
• All Z s are normal to the page

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Example I The two links arm

• Where (?1 ?2 ) denoted by ?12 and

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Example 2
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Example 2
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Example 3 The three links cylindrical
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Example 3 The three links cylindrical
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Example 3 The three links cylindrical
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Example 3 The three links cylindrical
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Example 4 Spherical wrist
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Example 4 Spherical wrist
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Example 4 Spherical wrist
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Example 4 Spherical wrist
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Example 5The three links cylindrical with
Spherical wrist
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Example 5The three links cylindrical with
Spherical wrist

• given by example 2, and given by
  example 3.

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Example 5The three links cylindrical with
Spherical wrist
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Example 5The three links cylindrical with
Spherical wrist
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Example 5The three links cylindrical with
Spherical wrist

• Forward kinematics
• 1. The position of the end-effector (dx ,dy ,dz
  )
• 2. The orientation Roll, Pitch, Yaw

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Roll Pitch Yaw

• The rotation matrix for the following operations
  

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Example 4The three links cylindrical with
Spherical wrist

• How to calculate
• Compare the matrix R with
• Of the matrix

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Module 1RRRRRR
Links a a ? d
1 90 0 10
2 0 10 0
3 -90 0 0
4 90 0 10
5 -90 0 0
6 0 0 0
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HW

• From Spong book page 112
• 3.2, 3.3, 3.4, 3.6 , 3.7, 3.8, 3.9, 3.11
• No Class on next Tuesday

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Module 1

• Where are
  Roll, Pitch, and Yaw

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Representing forward kinematics

• Forward kinematics

• Transformation Matrix

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Remember

• For n joints we have n1 links. Link 0 is the
  base
• Joints are numbered from 1 to n
• Joint i connects link i - 1 to link i.
• Frame i Xi Yi Zi is attached to joint i1.
• So, frame O0 X0 Y0 Z0, which is attached to the
  robot base (inertial frame) joint 1.