Velocity Kinematics -Examples

Dr.-Ing. John Nassour

Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik

25.01.2018 J.Nassour 1

Elbow Manipulator

25.01.2018

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

25.01.2018 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

25.01.2018 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

25.01.2018 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

25.01.2018 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

25.01.2018

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

25.01.2018

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

25.01.2018

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

25.01.2018

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

25.01.2018

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

25.01.2018

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

25.01.2018

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 =?

25.01.2018

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

25.01.2018 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

25.01.2018 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.

25.01.2018 J.Nassour 17

Reminder