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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
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Statics – 2
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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
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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
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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
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=
= ±
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
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Matrix review (from MSMS course) – 2
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Matrix review (from MSMS course) – 3
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Kineto-static duality – 2
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� 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
Kineto-static duality – 3
� When the robot is in a singular configuration:
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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)
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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)
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→∆F p
, , ,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 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
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