System and method for controlling an operation of a machine according to a task

A method for controlling an operation of a machine according to a task, the method comprising:

• collecting a feedback signal indicative of a current state of the operation of the machine;
• formulating an original quadratic program (QP) for optimizing an objective function subject to equality constraints and inequality constraints on one or a combination of state and control variables of the machine based on the task and the current state of the operation of the machine;
• lifting the equality constraints and the inequality constraints into a lifted space having a dimension higher than a dimension of an original space of the original QP by a lifting operation introducing an additional non-negative variable such that a subspace defined by the equality constraints in the lifted space intersects a subspace defined by the inequality constraints in the lifted space at least at a point of origin of the lifted space;
• transforming the objective function of the original QP into a quadratic objective function involving variables of the original QP and the additional non-negative variable, wherein the quadratic objective function subject to the lifted equality and inequality constraints forms a homogeneous QP in the lifted space such that first-order optimality conditions of the homogeneous QP correspond to first-order optimality conditions of the original QP lifted in the higher space by the lifting operation;
• solving the homogeneous QP to produce a solution in the lifted space;
• controlling the machine according to an infeasibility protocol when a value of the additional non-negative variable in the solution in the lifted space equals zero;
• and otherwise projecting the solution in the lifted space into the original space using a projection operation reversing the lifting operation to produce a solution of the original QP;
• and
• controlling the machine using a control command determined based on the solution of the original QP.

A non-transitory computer-readable storage medium embodied thereon a program executable by a processor for performing a method for controlling an operation of a machine according to a task, the method comprising:

• collecting a feedback signal indicative of a current state of the operation of the machine;
• formulating an original quadratic program (QP) for optimizing an objective function subject to equality constraints and inequality constraints on one or a combination of state and control variables of the machine based on the task and the current state of the operation of the machine;
• lifting the equality constraints and the inequality constraints into a lifted space having a dimension higher than a dimension of an original space of the original QP by a lifting operation introducing an additional non-negative variable such that a subspace defined by the equality constraints in the lifted space intersects a subspace defined by the inequality constraints in the lifted space at least at a point of origin of the lifted space;
• transforming the objective function of the original QP into a quadratic objective function involving variables of the original QP and the additional non-negative variable, wherein the quadratic objective function subject to the lifted equality and inequality constraints forms a homogeneous QP in the lifted space such that first-order optimality conditions of the homogeneous QP correspond to first-order optimality conditions of the original QP lifted in the higher space by the lifting operation;
• solving the homogeneous QP to produce a solution in the lifted space;
• controlling the machine according to an infeasibility protocol when a value of the additional non-negative variable in the solution in the lifted space equals zero;
• and otherwise
• projecting the solution in the lifted space into the original space using a projection operation reversing the lifting operation to produce a solution of the original QP;
• and
• controlling the machine using a control command determined based on the solution of the original QP.