Identify how to define complex functions of GSplines and its derivatives

[rafaelrojasmiliani]

Motivation, Approximation with GSplines

Let be an interval, be a function and let be the set of piecewise polynomials of degree with pieces.

Theorem For every there exists an and map such that

Theorem For every there exists an and map such that if

such that

Definitions

• We call the polynomial approximation of
• We call the finite-dimensional approximation of

Remark Let . Then we can define

• The polynomial approximation of the restriction of to the polynomials
• The finite-dimensional approximation of the restriction of to the polynomials

Requirement: We need a class that can represent arbitrary functions an its derivative

Derivative

Integral

## Newton-Euler

Here the finite dimensional approximation using the Lagrange interpolation at some points can be computed with

Direct kinematics

Here we need to represent a manifold (quaternions, euler angles. etc)

Arc lent of the direct kinematics

How we want to use this representation: We want to build new functionals from old ones

• Gspline to Gspline functional a simple function that we cal call as

GSpline torque = NewtonEuler::evaluate(q);
GSplineFunctional diff = NewtonEuler::evaluate(q);

GSpline pose = DirectKinematic::evaluate(q);

class ArcLen: public Functinal{
public:
ArcLen(GSplineSet);/*Domain, codom dim, number of intervals, interval width*/
ReturnType_1 value(const GSpline &_in) const{
return _in | gspline_set_.derivative | gspline_set_.norm | gspline_set_.integral;
}
ReturnType_2 diff(const GSpline &_in){
return gspline_set_.integral / gspline_set.norm(_in | derivative) * derivative
}
};

Remarks

• In the above example we desire to have a ReturnType_1 which can behave as both, a vector and a GSpline.
  • Cast from GSpline to Vector: This is not simple, because a gspline is constituted by two vectors. We can return the vector that represent a GSpline as an element of a vector space GSpline::get_coefficients.
  • Cast Vector to GSpline, from where we take the intervals?: This is impossible without adding more data structures.
• In the above example we desire to have a ReturnType_2 which can behave as both, a matrix and a Functional.
  • this is done here
  • This case is simpler, because ReturnType_2 does not need explicitly the interval lenghts because we don't have to cast it into a GSpline