1. from research .… to market
Self-locking Analysis in
Closed Kinematic Chains
Marco Leonesio, Giacomo Bianchi
marco.leonesio@itia.cnr.it
Milano, 13 Dicembre 2011

2. Content
• Introduction to self-locking
• Definition of self-locking borderline condition
• Self-locking equations
• Possible motions analysis
• Example on a 2DoFs parallel mechanism
• Conclusions and future works
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3. Introduction to self-locking
What is self-locking
In every mechanism where joints are affected by friction, the active forces applied to
independent DoFs - typically represented as actuators - are contrasted by corresponding
dissipative reaction forces. There exist particular structural conditions in which friction
forces systematically equilibrate active forces: it means that no actuator effort, even
arbitrarily great, is able to produce motion. This condition is commonly indicated as “self-
locking”.
Self-locking is a well-experienced phenomenon
fT
fM
fN
fA
x
rN
Irreversible mechanical transmissions
Cams for wrong when energy try to flow in the reverse Classical didactic examples
pressure angles direction
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4. Introduction to self-locking
If friction acts only in actuated joints, as it happens in mechanisms based on open
kinematic chains, the resulting effect is only an increased effort for the actuators. The
same happens if friction is a constant value: the actuators effort increase can be easily
computed:
If friction depends on joint reactions, the situation is more complicated: generally
actuator forces cannot completely cancel out the effects of friction forces on
constraints reactions.
i
qj
f M f Aj Eq. Tangent space
xi
f Aj h f Nj Friction model
f Nj e f M , f Aj
i
Eq. Normal space
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5. Definition of self-locking borderline condition
Borderline condition
friction
Not self-locking
Self-locking
friction
friction
Definition. A mechanical system, whose DoFs can be driven by input forces
applied in specific DoFs (the so called ‘‘actuators”) is in a self-locking borderline
condition if and only if at least one incipient motion exists with respect to which
actuator forces are balancedSystem in static equilibrium
by friction forces independently from the values of the
with the maximum friction forces
input forces.
System accelerates Given a motion, a kinematic System is iperstatic
model is sufficient to
A dynamic model is necessary in
calculate joint reactions and Joint reactions and friction forces
order to compute joint reactions
friction forces are undetermined (under rigid
and friction forces
bodies hypothesis)
Signs of joints motions must be considered as additional states!
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6. Objective
For a given mechanism pose (set of friction
coefficients) and an incipient motion, the objective is
to identify the friction coefficients (mechanism
poses) that lead the mechanism in a self-locking
borderline condition.
Self-locking borderline locus divides the friction
coefficients space (or the poses space) into regions
that can be associated to self-locking occurrence or
motion.
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7. Self-locking equations
fA: friction force/torque fM :actuator force/torque q: joints coordinates
x: actuator coordinates
Imposition of system equilibrium under
Friction model maximum friction forces (no external loads)
qj qj def
f Aj j
g j (max friction value) f i
M f A
j
0 f i
M f A
j
xi xi
def
f Aj j
g j sgn qj (1) h f Aj (2)
Which signs?
g j j u v
cuv f N f N All the feasible ones!
Constraints reactions
def
fN l fM , f A l h fA , fA v fA
Metric tensor to generalize
the norm g taking into account Being v(fA) linear:
possible anisotropic behaviour
k
fN v k f Aj (2)
j
Combining the (2) with the (1), one obtains:
fA w fA How does w look like?
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8. Self-locking equations
In order to simplify the notation, let us introduce the squared friction force module as it
follows:
j 2 j j 2
f A g
Substituting friction forces expression into constraint reactions expression, one obtains:
j 2
2 f
gj A
j
cuv vh f Ah vtv f At
j u
cuv bht f A h f At
j uv
uv
where bht vh vtv
u
Without losing generality, an isotropic friction forces behaviour can be assumed in order
to make tensor c unitary. Then,
j 2
fA Remember: there is an equations
j
bht f A h f At
j
0 set for each sign combination!
A system of NA homogeneus quadratic equations in NA variables fAj
NA: number of joints with friction Cones in RNA
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9. Self-locking equations
j 2
fA j h t
j
b fA fA
ht 0
Considering the whole system, in general there is only the solution f A 0 ,
j
but there exists also non-zero solutions if parameters µj or vh satisfy
t
particular conditions. In this case, a whole straight line solution exists, i.e.
at least one equation becomes dependent on the others. The identification
of such condition can be assimilated to the classical eigenvalues problem in
linear homogeneous systems, but, in nonlinear case, it requires specific
algebraic techniques (Dixon Polynomial). 
det(W( j | x)) 0 (locus in µj space) det(W(x | j )) 0 (locus in pose space)
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10. Possible motions analysis
Not all the sign combinations of joint relative velocities are feasible, as they are
constrained by the mechanism kinematics. In order to get a complete self-locking
borderline combination, the following set of signs must be identified:
qj
Q sk : x : sk sgn x
xi
The identification of the set Q is not in the scope of this work. It involves the
application of techniques borrowed by Linear Programming. In fact, the problem
can be traced back to the verification of consistency of an inequality set.
For simple 2 DoFs mechanisms, a graphical method is proposed.
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11. Example
2 DoFs parallel mechanism with friction in joints A and C
C D
B
BC=a +
l2
l1 CD=b
CF=c
EF=d
Y
+ FA=e α
A F E X
Geometry parameters [mm]
A=(0;0) B=(200;500) C=(300;600)
D=(500;800) E=(700;0) F=(400;0)
Mechanism pose identified by and β angles.
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12. Example
Identification of sign combinations
Both sign combinations
are feasible
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13. Example
FA , FC
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14. Example
det(W(x | j )) 0 (locus in pose space)
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15. Example
ZOOM A
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16. Example
Numerical verifications by ADAMS™
• Point A: the simulation fails for both combinations of velocity signs (always self-locking);
• Point B: the simulation fails only for concordant signs combination (partial self-locking);
• Point C,D,E: the simulation does not fail (motion is always possible, no self-locking).
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17. Conclusions
•A novel definition of self-locking condition for mechanisms
with more than 1DoF has been introduced;
•An approach for self-locking analysis of closed kinematic
chains has been developed: it is able to identify the friction
coefficients/the mechanism poses that provoke self-locking for
a particular mechanism pose;
•The approach has been exemplified on a 2 DoFs parallel
mechanism and the results validated via multibody simulation.
Future developments
•Extend the analysis considering also external loads and joints
preload;
•Try to apply the capability to predict self-locking occurrence in
the development metamorphic mechanisms.
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