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Velocity Kinematics -Examples
Dr.-Ing. John Nassour
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik
09.01.2017 J.Nassour 1
Elbow Manipulator
09.01.2017
Work out the linear velocity Jacobian at the end-effector then discuss the singular configurations.
J.Nassour 2
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Elbow Manipulator
09.01.2017 J.Nassour 3
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Find DH parameters for thisrobot. Identify the jointvariables.
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1
2
3
Elbow Manipulator
09.01.2017 J.Nassour 4
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1
2
3
Find the A matrices
Reminder: 𝑨𝒊
Elbow Manipulator
09.01.2017 J.Nassour 5
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Find the T matrices
Elbow Manipulator
09.01.2017 J.Nassour 6
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Find the linear velocity Jacobian.
For Prismatic joint:
𝒥𝑣𝑖= 𝑧𝑖−1
0
For Revolute joint:
𝒥𝑣𝑖= 𝑧𝑖−1
0 × 𝑜𝑛 − 𝑜𝑖−1
Elbow Manipulator
09.01.2017
Work out the linear velocity Jacobian at the end-effector then discuss the singular configurations.
J.Nassour 7
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Elbow Manipulator
09.01.2017
Work out the linear velocity Jacobian at the end-effector then discuss the singular configurations.
J.Nassour 8
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Elbow Manipulator
09.01.2017
Work out the linear velocity Jacobian at the end-effector then discuss the singular configurations.
J.Nassour 9
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Elbow Manipulator
09.01.2017
Singularities …
J.Nassour 10
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
The elbow is fully extended or
fully retracted
Elbow Manipulator
09.01.2017
Singularities …
J.Nassour 11
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
The wrist center intersects the axis of the base
rotation, Z0.
There are infinitely many singular configurations and
infinitely many solutions to the inverse position
kinematics when the wrist center is along this axis.
Elbow Manipulator
09.01.2017
Work out the angular velocity Jacobian at the end-effector.
J.Nassour 12
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Elbow Manipulator
09.01.2017
Work out the angular velocity Jacobian at the end-effector.
J.Nassour 13
Link 0 (fixed)
Joint 1
Link 1
𝜽1
Joint 2
Link 2
𝜽2
Link 3
𝒛𝟐
Joint 3
𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
𝒥𝜔 = 𝜌1𝑧0 𝜌2𝑧1 … 𝜌𝑛𝑧𝑛 − 1
𝜌1 =? , 𝜌2 =?, 𝜌3 =?
09.01.2017
The robot has three revolute joints that allow the endpoint to move in the three dimensional space. However, this robot mechanism has singular points inside the workspace. Analyze the singularity, following the procedure below.
J.Nassour 14
Link 0 (fixed)
Joint 1
Link 1
Joint variable 𝜽1
Joint 2
Link 2
Joint variable 𝜽2
Link 3
𝒛𝟐
Joint 3
Joint variable 𝜽3
𝒛𝟑
𝒙𝟎
𝒚𝟎
𝒛𝟎
𝒛𝟏 𝒙𝟏
𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
Link 1= 2 mLink 2= 2 mLink 3= 2 m
Step 3 Find the joint angles that make det J =0.Step 4 Show the arm posture that is singular. Show where in the workspace it becomes singular. For each singular configuration, also show in which direction the endpoint cannot have a non-zero velocity.
Step 1 Obtain each column vector of the Jacobian matrix by considering the endpoint velocity created by each of the joints while immobilizing the other joints.
Step 2 Construct the Jacobian by concatenating the column vectors, and set the determinant of the Jacobian to zero for singularity: det J =0.
Elbow Manipulator
Spherical Manipulator
09.01.2017 J.Nassour 15
Singularities …
𝒥𝑣 =
𝐿3𝑠1𝑠2 −𝐿3𝑐1𝑐2 −𝑐1𝑠2−𝐿3𝑐1𝑠2 −𝐿3𝑠1𝑐2 −𝑠1𝑠2
0 𝐿3 𝑠2 −𝑐2
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
Spherical Manipulator
09.01.2017 J.Nassour 16
Singularities …
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
The manipulator is in a singular configuration when the wrist center intersects z0, any rotation about the base leaves this point fixed.
09.01.2017 J.Nassour 17
Reminder
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