ROBOTICSROBOTICS

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Basilio BonaBasilio Bona

DAUIN DAUIN –– Politecnico di TorinoPolitecnico di Torino

StaticsStatics

Statics – 1

� We call GENERALIZED FORCESGENERALIZED 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

2τ

3τ

4τ

5τ

6τ

τ

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BASE

( )

( )

t

t

f⋯

Ν

1τ

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 iτ

− −= k NT

1 1,i i i iτ

− −= k fT

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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TCPWδ δ= F pT

gWδ δ= qτ 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 , ( )

gW W tδ δ δ δ= ∀ ⇔ =q q F pτ

T T

d , dδ δ= =q q p pdisplacements, i.e.,

� So …

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d , dδ 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 TThis is the relation between

TCP forces and joint forces.

It is an equivalenceequivalence relation

= J Fτ −T

If one needs to compute the joint

forces needed to equilibrateequilibrate

the TCP force, the relation isEquilibrate and Balance are synonymous

Kineto-static duality – 1

� Since

we speak of a kinetokineto--static dualitystatic 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 Fτ

ɺ ɺ

T

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g

g

=

= ±

p J q

J Fτ

ɺ ɺ

T

� The duality can be characterized considering the

mathematical concepts of range and kernel of the

transformations and g gJ 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

( )( )gJ qR � Null space

It contains the joint velocities that do

not produce any TCP velocities, for a

given pose

( )( )gJ qN

� Consider ( )g

= = p v J q qωɺ ɺ

T

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� Consider ( )g

= J q FτT

� Image space

It contains the joint generalized torques

that can balance TCP generalized forces,

for a given pose

( )( )gJ qTR � Null space

It contains the TCP generalized forces

that do not require balancing joint

generalized forces , for a given pose

( )( )gJ qTN

Kineto-static duality – 3

� When the robot is in a singularsingular 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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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 qτ

� We introduce an approximated description, considering the

elasticity due only to the joints (links are perfectly rigid)

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, , ,F p qτ

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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