Quantitative inheritance properties for simultaneous approximation by tensor product operators II: Applications

Abstract

We summarize several general results concerning quantitative inheritance properties for simultaneous approximation by tensor product operators and apply these to various situations.
 All inequalities are given in terms of moduli of continuity of higher order.

On the application of the Schoenberg quasi-interpolant for complexity reduction in trajectory generation

“…The Schoenberg operator [10], [13], [14] approximates a function (call it f (•)) by interpolating it as weighted sum of B-splines basis functions and the function's values in the Greville points of the B-spline basis:…”

“…Problem (13) still explicitly considers time as a variable. To bring it to a discrete time variant we exploit B-spline properties and provide sufficient conditions involving the associated control points.…”

“…is a sufficient condition for (12b). With varying degrees of success, the same reasoning may be applied to any term from (13).…”

“…To handle the issues raised in Remarks 3 and 4 we make use of the Schoenberg operator introduced in Section II-B to approximate terms appearing in cost and/or constraints. Using again the illustrative example from (13) we apply (7) thrice, once to approximate the nonlinear cost (13a), once to approximate the velocity in (13b) and once to approximate the roll angle in (13c). Correspondingly, the approximations are…”

“…Hence, we underline that the interest of applying the Schoenberg operator to approximate the constraints and the cost function is to solve the approximated optimization problem and obtain a feasible solution. Due to the approximation error, some constraints may have to be tightened, that is why we need to confirm that the control points obtained in (20) are a solution of the initial problem (13). The numerical values for the constraints in the optimization problem (20) are taken from the experimental results proposed in [11].…”