robotics 2017 04 statics - polito.it · statics –2 robotics 01peeqw -2016/2017 4 ... robotics...

ROBOTICS

01PEEQW

Basilio Bona

DAUIN – Politecnico di Torino

Statics

Statics – 1

� We call Generalized Forces the vector of forces and torques

‒ It is not a vector in strict terms because the elements have different units

(forces are expressed in N, torques in N·m

� Statics studies the relations between the task space generalized

forces (TSGF) and the joint generalized forces (JGF) in static

equilibrium conditions

� The TSGF are generated from interactions with the environment

(e.g., when the TCP pushes against a surface)

� The JGF are generated by the power supplied by the joint motors

used to move the robot arms

3ROBOTICS 01PEEQW - 2016/2017

Statics – 2


BASE

TCP

( )

( )

t

t

f⋯

Ν

1τ

2τ

3τ

4τ

5τ

6τ

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 torques

� Revolute joint torques

� To find the relation between we apply the virtual

work principle

� TCP generalized forces define a virtual work

� Joint generalized forces define another virtual work


1 1,i i i iτ

− −= k NT

1 1,i i i iτ

− −= k fT

TCPWδ δ= F pT

gWδ δ= qτ T

and F τ

Statics – 4

� Virtual work principle says that a static equilibrium

condition exists when

� Virtual displacements are “similar” to differential

displacements, i.e.,

� Recalling that


TCP , ( )

gW W tδ δ δ δ= ∀ ⇔ =q q F pτ

T T

d , dδ δq 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 equivalence relation

= J Fτ −T

If one needs to compute the joint

forces needed to equilibrate

the TCP force, the relation isEquilibrate and Balance are synonymous

Kineto-static duality – 1

� Since

we speak of a kineto-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 Fτ

ɺ ɺ

T

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 (from MSMS course) – 1




� 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 the joint torques ( )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

� Consider the Cartesian velocity ( )g

= = p v J q qωɺ ɺ

T


� When the robot is in a 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

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_2014_02

Elasticity of the structure

� In real conditions a perfectly rigid robot does not exist

� Elastic effects are always present and 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)


21

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

� We introduce an approximated description, considering the

elasticity due only to the joints (links are assumed perfectly

rigid)


→∆F p

, , ,F p qτ

Elasticity – 2


Elasticity – 3


Elasticity – 4


Conclusions

� Statics is important since it allows to compute the equivalent effects

on joints of TCP forces when the TCP interacts with a surface (and

viceversa)

� Statics and velocity kinematics are linked by kinetostatic duality

� Remember that the product of a force by a velocity is a power

� For this reason forces and velocities cannot be set independently

when the power is an external constraint

� If you set a force you cannot set also the corresponding velocity and

viceversa

� Elastic forces are usually not considered in the robot model, but they

are very important for the control design of real systems


robotics 2017 04 statics - polito.it · statics –2 robotics 01peeqw -2016/2017 4 ... robotics...

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