A System for Automated Fixture Planning with Modular Fixtures

3.4.2. Computation of Clamping Force
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Given clamp locations and an estimation of the direnion and magnitude of cutting forces. our g d k to
estimate the least clamping force to mainrain the force and moment equilibrium of the part during
machining.
Here we have proposed a new method to estimate the clamping force. The problem is fonnulated as an
optimization problem with consnaints from static equations and a friction law m avoid any slippage at each
contact. The objective function is a clamping fme at a contact point between a clamp and the part. Since
our pal is to achieve force and moment equilibrium with the least clamping force, the problem is to
minimize the objective function. This appoach is similar m Keds approach in computing minimal
grasping force in each finger to maintain force and moment equilibrium oC the object [Ken 841.
In esrablishing a force model, only contacts between the part and clamping and Supporring components
are considered because side locating components are primarily used for locaring the part and can be ignored
for the conservative analysis.
The force analysis assumes the following:
hand tixturecompoaents are rigid.
Clamping force is the same at every active(or clamping) contax
Coloumb’s law is wed to model friction at each contact
Friction coefficient is the same at every contact
The friction coefficient between the wortpitoe and a fixture component m y vary drastically depending
on the contact surfaces’ conditions and the marerial The contact sorfaces between the part and fixture
compiments are wet due to a cutting fluid, and a wet friction coefficient should be used 6or the worst case.
Using a Coloumb fiiction model, the constraint equatmn for friction force at the ith active contact is given
as foflows Em 841:
Where Air Aiy are the X and Y component of frkrion force at the ith conract. p is the fiction coefficient,
and Ni is the clamping force a! the ith active contact point Quaiion (3.7) describes a cone that resiricts the
direction of the vector summation of two friction force components. The djreetion oC force at the contact
should remain within the cone. and-tbe magnitude of friction force at a contact is limited by a circle whose
radius is pi; (see Figure 3-9).
Since the friction constraints fmm Coloumb’s law are expressed as inequality equations. the optimization
can be done by a l i i propmning (LP) method as long as constraint equations are linear. For example,
the simplex method can be used when the optimization problem is farmdated with linear equasims Faha
821. ’Ihe standard format for the simplex memod is given in the previous section. Since the constmint
equation at each contact from Coulomb’s friction law is non-lioear, it is li- by a set of multiple
planes. For example, an invd pyramid within the friction cone can be used to approximate the cone.