ROBOTICS ROBOTICS 01PEEQW 01PEEQW 01PEEQW 01PEEQW Basilio Bona Basilio Bona DAUIN DAUIN – Politecnico di Torino Politecnico di Torino
ROBOTICSROBOTICS
01PEEQW01PEEQW01PEEQW01PEEQW
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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