ROBOTICS ROBOTICS 01PEEQW 01PEEQW 01PEEQW 01PEEQW Basilio Bona - - PowerPoint PPT Presentation

ROBOTICS ROBOTICS 01PEEQW 01PEEQW 01PEEQW 01PEEQW

Basilio Bona Basilio Bona DAUIN DAUIN – – Politecnico di Torino Politecnico di Torino

Statics Statics

Statics – 1

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

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Statics – 2

TCP

τ

2

τ

3

τ

4

τ

5

τ

6

τ

τ  

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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

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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 “similar” 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 …

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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 geometrically more significant than the analytical one) we have = = ± p Jq J F τ ɺ ɺ

T

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g g

= = ± p J q J F τ ɺ ɺ

T

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

g g

J J T

Matrix review – 1

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Matrix review – 2

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Matrix review – 3

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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

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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

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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

See Example_2013_02

Elasticity of the structure

A perfectly rigid robot does not exist in practice Elastic effects can be localized in

• 1. Joints, due to the mechanical transmission elements: long

motor shafts, belts, chains, gearboxes, etc.

• 2. Links, due to distributed compliance of the mechanical

structure (flexion, torsion, compression)

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2 1

Elasticity – 1

When a generalized force is applied to the robot TCP a small deflection takes place We want to describe the relation in static conditions between the relevant variables → ∆ F p , , , F p q τ We introduce an approximated description, considering the elasticity due only to the joints (links are perfectly rigid)

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, , , F p q τ

Elasticity – 2

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Elasticity – 3

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Elasticity – 4

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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 a power For this reason forces and velocities cannot be set at will. If you set a force you cannot set at will the corresponding velocity and viceversa, since the power is an external constraint Elastic forces are usually not considered in the robot model, but they are very important in control

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