Rigid Body Configuration and Velocity, Lecture notes of Electrical and Electronics Engineering

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MAE263B: Dynamics of Robotic Systems

Discussion Section – Week

: Jacobian (SCARA)

Seungmin Jung

Contents

❑ Jacobian with SCARA example

• Velocity propagation

• Direct differentiation

❑ Frame of Representation

A minimal representation of orientation - Euler angles



 



A minimal representation of orientation - Euler angles

Kinematics Relations - Forward & Inverse

• The robot kinematic equations relate the two description of the robot tip location

 N





 

2

1

 

 

N

N

r

P

X

0

0

Tip Location in Joint Space

Tip Location in Cartesian Space

X = FK( )

 =IK(X)

Jacobian Matrix - Introduction

• The velocity relationship

: The relationship between

the joint angle rates ( )

and the translation and rotation velocities of the end

effector ( ).

• The relationship between

the robot joint torques ( )

and the forces and moments ( )

at the robot end effector ( Static Conditions ).

N

x

x^ =J( ) ^ 



F

F

J ( ) F

T  = 

Jacobian Matrix - Calculation Methods

Jacobian Matrix

Differentiation the

Forward Kinematics Eqs.

Iterative Propagation (Velocities or Forces / Torques)

Jacobian Matrix - Introduction

• This expression can be expanded to:
• Where:
  • is a 6x1 vector of the end effector linear and angular velocities
  • is a 6xN Jacobian matrix
  • is a Nx1 vector of the manipulator joint velocities
  • is the number of joints

( )

z N

y

x

z J

y

x

2

1

N

x

J( )

N

6x1 (^) 6xN Nx

Position Propagation

• The homogeneous transform matrix provides a complete description of the

linear and angular position relationship between adjacent links.

• These descriptions may be combined together to describe the position of a link

relative to the robot base frame {0}.

T TT T

i i

o o i

1 1 1 2

− = 

1 1 1 0 1

i i i (^) i i i

R P

T

− − −

Velocity Propagation – Intuitive Explanation

• Three Actions
  • The origin of frame B moves as a function of time with respect to the origin of frame A
  • Point Q moves with respect to frame B
  • Frame B rotates with respect to frame A along an axis defined by B

A 

B

A (^)  Q B (^) P

• Linear and Rotational Velocity
  • Vector Form
  • Matrix Form

Q

A B B B

A Q

A B BORG B

A Q

A V = V + R V +   R P

( (^) Q)

A B B

A Q B

A B BORG B

A Q

A V = V + R V + R R P

Velocity Propagation – Intuitive Explanation

• Three Actions
  • The origin of frame B moves with respect to the origin of frame A
  • Point Q moves with respect to frame B
  • Frame B rotates with respect to frame A about an axis defined by B

A 

B

A (^)  Q B (^) P

Frame - Velocity

• As with any vector, a velocity vector may be described in terms of any frame,

and this frame of reference is noted with a leading superscript.

• A velocity vector computed in frame {B} and represented in frame {A} would be

written

Q

B

A

Q

A B P dt

d ( V )=

Q

Computed (Measured)

Represented (Reference Frame)

BORG=^0

A V

R= 0

A B

Angular Velocity - Rigid Body

Q Q

B (^) P

• Given: Consider a frame {B} attached

to a rigid body whereas frame {A} is fixed. The vector is constant as view from frame {B}

• Problem: describe the velocity of the

vector representing the the point Q relative to frame {A}

• Solution: Even though the vector

is constant as view from frame {B} it is clear that point Q will have a velocity as seen from frame {A} due to the rotational velocity

Q

B P

Q =^0

B V

Q =^0

B V

Q

B P

Q

B P

B

A  BORG=^0

A V

Q

A B B B

A Q

A B BORG B

A Q

A V = V + R V +   R P (^ Q)

A B B

A Q B

A B BORG B

A Q

A V = V + R V + R R P