Optimising cam motion using piecewise polynomials

Abstract

The

Optimal Splines for Rigid Motion Systems: A Convex Programming Framework

“…If the τ i -set includes all analytical knots, ε p is numerically zero and the exact solution has been found to numerical precision (Table 2: m = {1, 2} and odd g). In the other case, the error can be made arbitrarily small by increasing g. The latter claim is not formally proved but numerically verified for m = 3 by solving (24) for 50 logarithmically spaced (even) g values between 10 and 1000. Figure 4 shows that the relative error ε p globally (but not monotonically) decreases for increasing g and drops just below 0.0001% for g = 1000.…”

“…(ii) for m = 3 the knots 2π{z, 1 − z} can never be part of the τ i -set for z (defined in Table 1) is an irrational number 10 . If the τ i -set includes all analytical knots, ε p is numerically zero and the exact solution has been found to numerical precision (Table 2: m = {1, 2} and odd g).…”

“…(iii) While nonlinear (but convex) extensions are required for the framework to reproduce the results [10,13], the original linear programming framework suffices to reproduce the results [11] (kinematic optimization example) and [7].…”

“…For this equality-constrained, convex quadratic program, a complicated solution procedure was developed of which the implementation "requires a symbolic mathematics software application, as the calculations involved are lengthy" [10]. This solution procedure relies on the fact that the solution of an equalityconstrained, convex quadratic program can be found by solving a set of linear equations.…”

“…Mermelstein and Acar [10] developed, as an improvement of [8], a method to minimize the mean-square jerk (that is, an objective of type (34), with all W j = 0, except W 3 = 1) of a spline subject to a number of prescribed positions, velocities and/or higher derivatives at specific time instants. For this equality-constrained, convex quadratic program, a complicated solution procedure was developed of which the implementation "requires a symbolic mathematics software application, as the calculations involved are lengthy" [10].…”