L7 - Differential Motion and Jacobians 2 V1

8/9/2019 L7 - Differential Motion and Jacobians 2 V1

1/12

Continuing with Jacobian andits uses

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niversitiKualaLumpurMalaysiaFranceInstitute

Originally prepared by: Prof Engr Dr Ishkandar Baharin

Head of Campus & Dean

UniKL MFI


2/12

Connecting the Operator to theJacobian

Examination of the Velocity Vector:

If we consider motion to be made in

UNIT TIME: dt =t = 1

Then xdot (which is dx/dt) is dx

Similarly for ydot, zdot, and the s.

They are: dy, dz and x, y, and zrespectively

x

y

z

x

y

D z

=

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3/12

These data then can build the

operator:

0

0

0

0 0 0 0

z y x

z x y

y x z

d

d

d

=

Populate it with the outtakes from the DdotVector which was found from: J*Dqdot

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Institute


4/12

Using these two ideas: Forward Motion in Kinematics:

Given Joint Velocities and Positions

Find Jacobian (a function of Joint positions) & T0n

Compute Ddot, finding dis and is in unit time

Use the dis and is to build With and T0

n compute new T0n

Apply IKS to new T0n which gets new Joint Positions

Which builds new Jacobian and new Ddot and so on

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Institute


5/12

Most Common use of Jacobian is to MapMotion Singularities

Singularities are defined as: Configurations from which certain directions of motion are

unattainable

Locations where bounded (finite) TCP velocities may correspondto unbounded (infinite) joint velocities

Locations where bounded gripper forces & torques maycorrespond to unbounded joint torques

Points on the boundary of manipulator workspaces

Points in the manipulator workspace that may be unreachable

under small perturbations of the link parameters Places where a unique solutionto the inverse kinematic problemdoes not exist (No solutions or multiple solutions)

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aysiaFrance

Institute


6/12

Finding Singularities:

They exist wherever the Determinate of

the Jacobian vanishes: Det(J) = 0

As we remember, J is a function of the Joint

positions so we wish to know if there are anycombinations of these that will make thedeterminate equal zero

And then try to avoid them!

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Institute


7/12

Finding the JacobiansDeterminate

We will decompose theJacobian by Function:

J11 is the Arm Jointscontribution to Linear velocity

J22 is the Wrist Jointscontribution to AngularVelocity

J21 is the (secondary)contribution of the ARM jointson angular velocity

J12 is the (secondary)contribution of the WRIST

joints on the linear velocity Note: Each of these is a 3X3

matrix in a full function robot

11 12

21 22

J JJ J J

=

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Institute


8/12

Finding the Jacobians Determinate

Considering the case of the SphericalWrist:

J12:

Of course O3, O4, O5 are a single point soif we choose to solve the Jacobian

(temporally) at this (wrist center) pointthen J12 = 0! This really states that On= O3= O4= O5 (which

is a computation convenience but not a real

Jacobian)

( ) ( ) ( )3 3 4 4 5 5n n nZ O O Z O O Z O O

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9/12

Finding the Jacobians Determinate

With this simplification: Det(J) = Det(J11)Det(J22)

The device will be singular then whenevereither Det(J11) or Det(J22) equals 0

These separated Singularities would beconsidered ARM Singularities or WRISTSingularities, respectively

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10/12

Lets Compute the ARM Singularities for aSpherical Device

From Earlier efforts we found that:

To solve lets Expand by Minors along3rd row

3 3

3 3

3

1 2 1 2 1 2

11 1 2 1 2 1 2

0 2 2

d S C d C S C C

J d C C d S S S C

d C S

=

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11/12

Lets Compute the ARM Singularities for aSpherical Device

( ) ( ){ }

( ) ( ){ }( ) ( ){ }

3 3

3 3 3

3 3 3 3

11 0 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2

J d C S S C d S S C C

d C d S C S C d C C C C

S d S C d S S d C S d C C

=

+

After simplification: the 1st term is zero;

The second term is d32S2

2C2;

The 3rd term is d32C2

2C2U


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12/12

Lets Compute the ARM Singularities for a

Spherical Device, cont.

( )

2 2 2 2

3 2 2 3 2 2

2 2 2

3 2 2 2

23 2

11 0

11

11

J d C C d S C

J d C C S

J d C

= + +

= +

= This is the ARM determinate,it would be zero whenever

Cos(2) = 0 (90 or 270)U


Institute

L7 - Differential Motion and Jacobians 2 V1

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