What are the Kinematics of an Irregular Tripod? It is a common maxim (at least within the Scouting community) that a triangle is the most stable shape. In practice this means structures should have three legs whenever possible, and have cross-bars installed to support any more complex shape by making it into a lattice of triangles.
Suppose we have an irregular tripod (think of something like this but more irregular). Let's pretend it's perfectly rigid, even though the one in the picture will have a little flexibility at the joint.

Will the tripod always 'find' its balance? By this I mean will the weight borne by each leg be equal. This is obvious if the tripod is symmetric (meaning the base is an equilateral triangle) but not in general. 
In order for this to be true the torques exerted by each leg on the tripod would have to cancel out. And the torque should be equal to the downwards/upwards force exerted by that leg (or maybe some component of it) times the distance to the CoG. And a longer leg will pull the centre of gravity towards itself and away from the other legs, making them exert a greater torque and itself exert less.
What are the kinematics? Could anyone point me towards an analysis of free-body diagram?
 A: The shortest leg gets the most weight and the longest leg gets the least weight.
With a triangle $ABC$ with the center of gravity positioned over point $O$ the weight's that each leg will hold would be as follows:
$$W_a=W\frac{BO\times BC}{BA\times BC}$$
$$W_b=W\frac{AO\times AC}{AB\times AC}$$
$$W_c=W\frac{AO\times AB}{AC\times AB}$$
Where W is the total weight, and $\times$ denotes the cross product. This is saying the the weight that each leg will hold will be proportional to how close the center of gravity is to that leg, compared to the line between the other two legs.
If the center of gravity is outside the triangle, then the object will tip, thus one measure of stability is how close the center of gravity is to the edge of the supporting polygon.
If you compare to a four legged chair. If all four of the legs are on the ground then the center of gravity should be further from an edge then with a triangle of similar area, edge length, circumcircle diameter etc. However, if the ground is uneven then shifting the chair slightly will cause one of the legs to lift. Now instead of being supported by all four legs forming a support square, it's supported by three legs making a support right triangle. In this case the center of gravity is likely very close to the center of the hypotenuses, making the chair unstable. Just a slight shift will cause the chair to tip. Fortunately, it will tip towards the leg that was lifted, so that leg will then touch the ground, but the opposite leg will have lifted still giving you a support triangle with the center of gravity very near the edge. This is what causes tables, chairs, etc. to rock back and forth when the weight shifts slightly.
The reason tripods are considered stable, is that this shifting is impossible with the tripod as it is impossible for one or more legs to be off the ground and have the support area shift, no matter how uneven the terrain.
A: More fundamentally:
Given a body in contact with a planar surface, consider 1,2,3,4 and N number of legs supporting the body within a gravitational field that pulls the body towards the surface.
With one leg there is one point of contact within the planar surface - an unstable situation in which the body can freely rotate in any direction towards the planar surface.
With two legs there are 2 points of contact that determine a line within the planar surface - an unstable situation in which the body can rotate along the axis of the line in two possible directions towards the planar surface.
With three legs there are 3 points that determine a plane that uniquely coincides with the planar surface regardless of the lengths of the legs, and as long as the vertical projection of the center of gravity of the body lies within the three points of contact, the body remains stable. Furthermore the supporting surface need not be a plane - the three points of contact will still determine a plane and provide stability on a curved or irregular surface.
With 4 legs we have the possibility of having all 4 points of contact being coplanar and coincident with the planar surface if the lengths of the legs are chosen just right, but more generally the plane is overspecified. A special situation of bistable behavior can exist where the vertical projection of the center of gravity may lay within one set of three points or within another set of three points.
For N legs a similar situation exists for the 4 leg case, however with multistable equilibria.
