Method for fast computation of optimal contact forces

A method for solving a force feasibility problem associated with a force optimization problem comprising the steps of:

• (a) formulating the force feasibility problem as a convex optimization problem;
• (b) formulating a primal barrier subproblem associated with the convex optimization problem;
• (c) formulating a dual problem having a dual objective such that if the dual problem is satisfied, the force optimization problem is not feasible;
• and
• (d) solving the primal barrier subproblem, using the dual objective in a convergence test.

A method for solving a feasible force optimization problem, comprising the steps of:

• (a) formulating the feasible force optimization problem as a convex optimization problem;
• (b) formulating a primal barrier subproblem associated with the convex optimization problem where the primal barrier subproblem has an accuracy parameter;
• and
• (c) solving the primal barrier subproblem for a fixed value of the accuracy parameter.

A method for solving a feasible force optimization problem, comprising the steps of:

• (a) formulating the feasible force optimization problem as a convex optimization problem;
• (b) formulating a primal barrier subproblem associated with the convex optimization problem;
• (c) formulating a dual problem having a dual objective;
• and
• (d) solving the primal barrier subproblem, using the dual objective in a convergence test.

A method for solving a feasible force optimization problem, comprising the steps of:

• (a) formulating the feasible force optimization problem as a convex optimization problem;
• (b) formulating a primal barrier subproblem associated with the convex optimization problem where the primal barrier subproblem has an accuracy parameter;
• and
• (c) solving the primal barrier subproblem to a guaranteed relative tolerance.

A method for solving a force optimization problem, comprising the steps of:

• (a) formulating the force optimization problem as a convex optimization problem;
• (b) constructing a system of linear equations associated with the convex optimization problem;
• and
• (c) solving the system of linear equations using a custom block elimination method.

A method for solving a set of force optimization problems, comprising the steps of:

• (a) formulating the set of force optimization problems problem as a set of convex optimization problems;
• (b) solving one or more of the convex optimization problems in the set so that a solution state is obtained for each solved convex optimization problem;
• (c) retaining one or more of the solution states;
• and
• (d) using one of the retained solution states as a starting point in solving subsequent convex optimization problems in the set.

A method for determining the minimum force required to satisfy any of a set of force optimization problems, comprising the steps of:

• (a) formulating each force optimization problem as a convex optimization problem;
• (b) solving one of the convex optimization problems;
• (c) retaining a best-case minimum value over the set of problems solved;
• (d) solving other convex optimization problems;
• (e) updating the best-case minimum value if the solution of any convex optimization problem produces a force smaller than the prior best-case minimum value;
• and
• (f) terminating the solution of one or more of the convex optimization problems if the minimum force for a convex optimization problem can be demonstrated to be larger than the best-case minimum value.

A method for determining the maximum force required to satisfy any of a set of force optimization problems, comprising the steps of:

• (a) formulating each force optimization problem as a convex optimization problem;
• (b) solving one of the convex optimization problems;
• (c) retaining a worst-case maximum value over the set of problems solved;
• (d) solving other convex optimization problems;
• (e) updating the worst-case maximum value if a solution to a convex optimization problem produces a force larger than the prior worst-case maximum value;
• and
• (f) terminating the solution of one or more of the convex optimization problems if the minimum force for a convex optimization problem can be demonstrated to be smaller than the worst-case maximum value.

A method for solving a force closure problem, comprising the steps of

• (a) computing a set of wrenches having a convex hull with an interior, such that the origin is in the interior of the convex hull;
• (b) forming a force optimization problem for each wrench in the set of wrenches;
• and
• (c) determining whether each of the force optimization problems is feasible.

A method for solving a feasible force optimization problem with bidirectional forces, comprising the steps of:

• (a) formulating the feasible force optimization problem as a convex optimization problem involving two classes of forces, the first class of forces being friction-related forces and the second class of forces being forces directed outward from contact surfaces;
• (b) formulating a primal barrier subproblem associated with the convex optimization problem;
• and
• (c) solving the primal barrier subproblem.