Support force vectors for a fixed rigid body

Hello all!

I'm a programmer looking to put together a system modeling the force a character's limbs exert on the environment as that character moves around. Characters can either grasp (with a hand) or press against (with a foot or something) viable contact points in the environment in order to support themselves. Were it not for grasping, I'd be able to do a simple check as to whether the center of gravity lies within the convex hull of the contact points as projected onto the plane perpendicular to the gravity vector. But this case is much more complicated.

I know by now that torque plays an important role in why, exactly, a person's hand and foot exert support forces in diagonal directions as they hold onto a ladder. But with no formal education in statics I'm having some difficulty locating the concepts I need to understand to compute those vectors. Any leads anyone can offer would be immensely helpful.

The specific problem I'm trying to solve (for now) is:

Given the gravity vector g, the set of contact points S, and the mass m and center of mass p of a rigid body (possibly also its rotational moment but I'm not sure that's needed), find the support-force vectors for each contact point.

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