Ch. 5: Jacobian

5.1 Introduction

• relationship between the end effector velocity and the joint rates• differentiate the kinematic relationships to obtain the velocity relationship• Jacobian matrix• closely related to the static force/torque transformation by duality• an indication of singularity configuration• the reverse; inverse problem of determining joint rates for specified end effector velocity

Ch. 5: Jacobian

5.2 Angular Velocity

• it is a property of the frame or the body in contrast to the linear velocity, which is a property of the individual point• for the fixed axis of rotation, the motion is really a planar problem• develop the relationship between the derivative of the rotation matrix and the angular velocity use of skew symmetric matrix

Ch. 5: Jacobian

Skew symmetric matrix

• S is skew symmetric

•

••••

TS S↔ + = 0

( )3 2

3 1

2 1

00

0

a aS a a

a a

−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

a

( ) ( ) ( )S S Sα β α β+ = +a b a b( )S = ×a p a p( ) ( )TRS R S R=a a

T S 0x x =

Ch. 5: Jacobian

Derivative of Rotation Matrix and the Angular Velocity

• with and , it can be shown

• in particular, when is the angular velocity vector of the rotating frame at time t• but

( ) ( )TR R Iθ θ = TS S+ = 0

( ), ,d R S Rd θ θθ

=k kk

( ) ( )( ) ( )R t S t R t= ω ( )tω

( )0 1 0 0 11 2 2/1 1 2R R S R R⋅ = ⋅ω ( )0 0 0

2 2/ 0 2R S R= ω

Ch. 5: Jacobian

Resultant Angular Velocity

• determine the resultant angular velocity due to relative rotation of several frames• angular velocities (but not the rotation) can be added once they are expressed in the same frame• 0 0 0 0

/ 0 1/ 0 2/1 / 1

0 0 1 0 11/ 0 1 2/1 1 / 1

n n n

nn n nR R−

−− −

= + + +

= + ⋅ + + ⋅

ω ω ω ω

ω ω ω

Ch. 5: Jacobian

5.3 Linear Velocity

• two frames {0} and {1} are related by

• point P is rigidly attached to {1}, then

same as the relative velocity formula

( )0 0

0 1 11 0 1

R oT t

⎡ ⎤= ⎢ ⎥⎣ ⎦

( )

0 0 1 01 / 1

0 0 1 0 0 0 1 01 / 1 1/ 0 1 / 1

0 0 01/ 0 /

p o

p o p o

p o o

R o

p R o S R o

= ⋅ +

= ⋅ + = ⋅ +

= × +

p r

r r

r v

ω

ω

Ch. 5: Jacobian

5.4 Derivation of Jacobian

• Jacobian governs the relationship between the linear/angular velocity of the end effector (a point) to the vector of joint velocities

•

• note that the velocity vector is not the derivative of the position and orientation variables since the angular velocity vector is not the derivative of any particular orientation variables, such as Euler or angle/axis representative parameters

0

0vn

n

JJ

Jω

⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦ξ

ωv

q q

Ch. 5: Jacobian

Angular Velocity

• angular velocity of the end effector relative to the base is the sum of the angular velocity contributed by each joint relative to the base frame

• for revolute joint i• for prismatic joint i

•

•

[ ]/ 1 0 0 1 Tii i i iθ θ− = =ω k/ 1

ii i− = 0ω

0 0 0/ 0

1 1

n n

n i i i i i ii i

Rρθ ρθ= =

= ⋅ ⋅ ⋅∑ ∑ω k = z

0 01 1 n nJω ρ ρ⎡ ⎤= ⋅ ⋅⎣ ⎦z z

Ch. 5: Jacobian

Linear Velocity

• , therefore analytically0

0

1

n

ii i

qq=

∂=

∂∑ pp0

vii

Jq

∂=∂

p

Ch. 5: Jacobian

• for prismatic joint i,

•

0 0 0i i i id R d= ⋅ ⋅ = ⋅p k z

0vi iJ = z

Ch. 5: Jacobian

• for revolute joint i,

•

0 0 0 0 0, i i i i iRθ θ= ⋅ ⋅ = ⋅ = −k z r p oω

( )0 0 0vi i iJ = × −z p o

Ch. 5: Jacobian

Ex.

Ch. 5: Jacobian

5.5 Spatial Velocity Transformation

•

• transformation between two rigidly attached moving frame

TT Tvξ ω⎡ ⎤= ⎣ ⎦

( )0

A A AB B BA B

A BAB

R S d R

Rξ ξ

⎡ ⎤⋅= ⎢ ⎥⎢ ⎥⎣ ⎦

Ch. 5: Jacobian

5.6 Analytical Jacobian

• depend on the minimal representation for the orientation

•( )

( ) ( ) ( )

1

1 a

I vB

IJ J

B

α ω

α

−

−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦⎣ ⎦⎡ ⎤

= ⎢ ⎥⎣ ⎦

00

00

X

q q = q q

d

α

Ch. 5: Jacobian

•

• representational singularity at

( ) ( ) ( )

( )

[ ]

1

/ / 0

0/ / 1

when be the z-y-z Euler angle representation

a

T

IJ J

B

Ic s s s

Js c

c c s s c s

α

ψ θ ψ θψ ψ

ψ θ θ ψ θ θ

α φ θ ψ

−

⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥−⎢ ⎥− −⎣ ⎦

=

00

0

0

q q

q

0, 0 or sθ θ π= =

Ch. 5: Jacobian

5.7 Singularities

•

• spatial velocity is the linear combination of the columns of the Jacobian matrix need at least 6 independent columns to achieve arbitrary velocity• rank of the matrix depends on the configuration •• if rank is less than the max. value, the robot is at singular configuration• for be nxn matrix, it will be singular when

1 1 2 2 n nJ q J q J qξ = + + +

q( ) ( )rank min 6,J n≤

J ( )det 0J =

Ch. 5: Jacobian

At singularity,

• certain directions of motion may be unattainable• bounded end effector velocity may correspond to unbounded joint rates• bounded joint torque may correspond to unbounded spatial force• often, they are points on the boundary of the robot workspace

Ch. 5: Jacobian

Decoupling of Singularities

• for robots with spherical wrist, decouple the singularity determination into arm and wrist singularities

• partition by choosing

the reference frame so that ,i.e. the origin is located at the wrist center• because robot configuration is independent of the frames used to describe, the singularity happens at where • set of singular configurations is the union of arm configs. satisfying and wrist configs. satisfying

[ ] 11

21 22P O

JJ J J

J J⎡ ⎤

= = ⎢ ⎥⎣ ⎦

0

2 1n n no o o− −= = = 0

11 22det det det 0J J J= =

11det 0J =22det 0J =

Ch. 5: Jacobian

• wrist config singularities at where the last and the second to last joint axes line up

Ch. 5: Jacobian

• arm config singularities• for elbow arm,

11det 0J =

Ch. 5: Jacobian

• for spherical arm,

Ch. 5: Jacobian

• for SCARA arm,

Ch. 5: Jacobian

5.8 Static Force/Torque Relationship

• by principle of the virtual work• the equation relates the end effector forces to the joint torques required to equilibrate the robot when no gravity force acts upon it

TJ Fτ =