Systems and methods for determining reactions due to friction of objects with multiple contacts

A computer-implemented method, executed by one or more processors and memory, for determining for an object having a set of two or more contacts and having at time t an initial generalized position qt and an initial generalized velocity qt, the object's generalized acceleration qt, the method comprising:

• computing a generalized force f which is a summary of forces acting on the object other than forces due to the set of contacts;
• computing a normal generalized reaction force R acting on the object;
• computing a tangential generalized reaction force Rt acting on the object using the computed generalized force f to account for contacts associated with zero relative velocity at time t;
• and
• using the tangential generalized reaction force Rt to compute the generalized acceleration qt of the object;
• wherein the computing the tangential generalized reaction force Rt further comprises applying the following constraints to an objective function maximizing power dissipated by friction:
• (a) associating each contact in the set of contacts with a friction cone in a space of generalized impulses to define a set of friction cones;
• (b) summing one term from each friction cone in the set of friction cones, minus the normal generalized reaction force Rn;
• and
• (c) constraining the tangential generalized reaction force Rt to be orthogonal to the normal generalized reaction force Rt, and to have a non-positive inner product with the normal direction of each contact in the set of contacts.

A system for determining, for an object having a set of two or more contacts and having at time t an initial generalized position qt and an initial generalized velocity qt, the object's end-of-interval generalized position qt+h and end-of-interval generalized velocity qt+h after an interval h, the system comprising:

• one or more processors;
• and
• memory including code that, when executed by the one or more processors perform to:
• compute a generalized force f over the interval, which is a summary of forces acting on the system other than forces due to the set of contacts, and which is used to define energy over the interval h;
• compute a normal generalized reaction impulse rn acting on the object;
• compute a tangential generalized reaction impulse rt acting on the object;
• and
• based on the tangential generalized reaction impulse rt, compute the end-of-interval generalized position qt+h and the end-of-interval generalized velocity qt+h of the object,
• wherein the computing the tangential generalized reaction impulse rt further includes applying the following constraints to an objective function minimizing energy over the interval h:
• (a) associating each contact in the set of contacts with a friction cone in a space of generalized impulses to define a set of friction cones;
• (b) summing one term from each friction cone in the set of friction cones, minus the normal generalized reaction impulse rn;
• and
• (c) constraining the tangential generalized reaction impulse rt to be orthogonal to the normal generalized reaction impulse rn and to have a non-positive inner product with the normal direction of each contact in the set of contacts.

A system, for determining for an object having a set of two or more contacts and having at time t an initial generalized position qt and an initial generalized velocity qt, the object's generalized acceleration qt, the system comprising:

• one or more processors;
• memory including code that, when executed by the one or more processors perform to:
• compute a generalized force f which is a summary of forces acting on the system other than forces due to the set of contacts;
• compute a normal generalized reaction force Rn acting on the object;
• compute a tangential generalized reaction force Rt acting on the object using the computed generalized force f to account for contacts associated with zero relative velocity at time t;
• and
• based on the tangential generalized reaction force Rt, compute the generalized acceleration qt of the object,
• wherein the computing the tangential generalized reaction force Rt further includes applying the following constraints to an objective function maximizing power dissipated by friction:
• (a) associating each contact in the set of contacts with a friction cone in a space of generalized impulses to define a set of friction cones;
• (b) summing one term from each friction cone in the set of friction cones, minus the normal generalized reaction force Rn;
• and
• (c) constraining the tangential generalized reaction force Rt to be orthogonal to the normal generalized reaction force Rn and to have a non-positive inner product with the normal direction of each contact in the set of contacts.