Statics Statics Basilio Bona 1 ROBOTICA 03CFIOR Statics 1 - - PowerPoint PPT Presentation

Statics Statics

Basilio Bona ROBOTICA 03CFIOR 1

Statics – 1

Statics studies the relations between the task space forces/torques and the joint forces/torques in static equilibrium conditions The former derive from possible interactions with the environment (e.g., when the TCP pushes against a surface) environment (e.g., when the TCP pushes against a surface) The latter are due to the power provided by joint motors used to move the robot arm We call generalized forces generalized forces the whole set of forces/torques

Basilio Bona 2 ROBOTICA 03CFIOR

Statics – 2

TCP

τ

2

τ

3

τ

4

τ

5

τ

6

τ

τ  

Basilio Bona 3 ROBOTICA 03CFIOR

BASE

( ) ( ) t t                 f⋯ Ν

1

τ

def def 1 2 3 4 5 6

( ) ( ) ( ) ( ) t t t t τ τ τ τ τ τ                      = ⇔ =                           f F N τ ⋯

Cartesian (task space) generalized forces Joint generalized forces

Statics – 3

Prismatic joint Revolute joint To find the relation between we use the virtual work principle

1 1, i i i i

τ

− −

= k N

T 1 1, i i i i

τ

− −

= k f

T

and F τ we use the virtual work principle TCP generalized forces define a virtual work Joint generalized forces define another virtual work

Basilio Bona 4 ROBOTICA 03CFIOR

TCP

W δ δ = F p

T g

W δ δ = q τ T

Statics – 4

Virtual work principle states that a static equilibrium static equilibrium condition exists when Virtual displacements are equal to differential displacements, i.e.,

TCP

, ( )

g

W W t δ δ δ δ = ∀ ⇔ = q q F p τ T

T

d , d δ δ = = q q p p displacements, i.e., So …

Basilio Bona 5 ROBOTICA 03CFIOR

d , d δ δ = = q q p p d ( )d d ( )d = = = = p J q q q F J q q F J J F τ τ τ

T T T T T

This is the relation between TCP forces and joint forces. It is an equivalence equivalence relation

= J F τ −

T

If one needs to compute the joint forces needed to equilibrate equilibrate the TCP force, the relation is Equilibrate and Balance are synonymous

Kineto-static duality – 1

Since we speak of a kineto kineto-

• static duality

static duality between generalized (cartesian) forces and cartesian velocities. Considering the geometric Jacobian (that has is more geometrically meaningful than the analytical one) we have = = ± p Jq J F τ ɺ ɺ

T

Basilio Bona 6 ROBOTICA 03CFIOR

g g

= = ± p J q J F τ ɺ ɺ

T

The duality can be characterized considering the range and the kernel of the transformations and

g g

J J T

Matrix review – 1

Basilio Bona 7 ROBOTICA 03CFIOR

Matrix review – 2

Basilio Bona 8 ROBOTICA 03CFIOR

Matrix review – 3

Basilio Bona 9 ROBOTICA 03CFIOR

Kineto-static duality – 2

Image space

It contains the TCP velocities that can be generated by the joint velocities, for a given pose

( )

( )

g

J q R Null space

It contains the joint velocities that do not produce any TCP velocities, for a given pose

( )

( )

g

J q N Consider ( )

g

  = =     p v J q q ω ɺ ɺ

T

Basilio Bona 10 ROBOTICA 03CFIOR

Consider ( )

g

= J q F τ

T

Image space

It contains the joint generalized torques that can balance TCP generalized forces, for a given pose

( )

( )

g

J q

T

R Null space

It contains the TCP generalized forces that do not require balancing joint generalized forces , for a given pose

( )

( )

g

J q

T

N

Kineto-static duality – 3

When the robot is in a singular singular configuration:

There are non zero joint velocities that produce zero TCP velocities There are non zero joint generalized forces that cannot be balanced by TCP generalized forces

Basilio Bona 11 ROBOTICA 03CFIOR

There are TCP generalized forces that do not require any balancing joint generalized forces There are TCP velocities that cannot be obtained by any joint velocities

Conclusions

Statics is important since it allows to compute the equivalent effects on joints of TCP forces (and viceversa) Statics and velocity kinematics are linked by duality Remember that the product of a force by a velocity is the power For this reason they cannot be set at will. If you set a force you cannot set the corresponding velocity and viceversa, since the power is an external constraint Elastic forces were not considered

Basilio Bona 12 ROBOTICA 03CFIOR