Review SessionA review of rigid motions, forward kinematics and velocity kinematics.

Big pictureI want to walk through the big picture a bit

Rigid Motions

Applying to Robots

Forward Kinematics

Velocity Kinematics

Next: Inverse Kinematics

Quiz

Quiz 1

How many degrees of freedom does a spatial body have?

How many values does it take to fully describe a spatial body’s pose (position and orientation)?

A homogeneous transform has 16 entries, why do we need so many?

Quiz 2

Is this Jacobian possible?

2 1 1 2

2 1 1 2

1 2

1 cos( )sin( ) cos( )

1 0 0

0 cos( )cos( ) sin( )

0 0 3cos( )

2 0 0

0 0 0

sJ

− − + − − +

= − +

−

Quiz 3

What is the difference between AdT(S) and [AdT]S?

Quiz 4

True or False?

A B A Be e e +=

At AtAe e A=

( )TT TB A AB=

AB BA=

( ) ( )A BC AB C=

Forward Kinematics

Forward Kinematics (Body Frame)

This is a PUMA manipulator. It is a very common arrangement.

1m

0.2m

0.5m

0.4m0.1m

z0

y0

x0

x1

y1

z1

Forward Kinematics (Body Frame)

4

5

6

Forward Kinematics (Body Frame)

Body Jacobian

z0

y0

x0

Body Jacobian

z0

y0

x0

Body Jacobian

z0

y0

x0

Body Jacobian

Singularities and Full Rank

2 3 3

2

2 3 3

cos( )sin( ) cos( ) 0

sin( ) 0 1

cos( )cos( ) sin( ) 0

0 0 0

0 0 0

0 0 0

bJ

− −

− −

=

22

= ?

Singularities and Full Rank

3

3

0 cos( ) 0

1 0 1

0 sin( ) 0

0 0 0

0 0 0

0 0 0

bJ

− − −

=

z0

y0

x0

z0

y0

x0

5

Space Jacobian

1 2 1

1 1 2

2

0 cos( ) cos( )sin( )

0 sin( ) cos( ) cos( )

1 0 sin( )

0 0 0

0 0 0

0 0 0

sJ

− −

− −

=

1

1

0 cos( ) 0

0 sin( ) 0

1 0 1

0 0 0

0 0 0

0 0 0

sJ

−

− −

=

22

= ?

Space Jacobian

Some problems worth looking at

Find the time derivative of a rotation matrix, given the angular velocity.

01R 0

0,1

0

0,1v

0u0u

We are given the following:

01T

Find the time derivative of u1:

1u

Frame 1 is moving with respect to Frame 0. These represent the instantaneous position, linear velocity and angular velocity. Linear velocity is wrt Frame 0 and angular velocity is wrt Frame 1.

1

0,1

0

0,1v 0u0u01T

1u1

0,1

0

0,1v 0

0,1

0u

We are given the following:

01T

Find the time derivative of u0:

1u1uCould you solve this similar problem?