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  \longrightarrow PP^{N,n_c}(I, \mathbb{R}^n))
such that
 - f \|_{\infty} < \varepsilon)
Theorem For every there exists an and map  \longrightarrow \mathbb{R}^{n_c N n})
such that if
)
such that
_i(t)= \sum_{k=k_1}^{k_1%2Bn_c} a_k p_k(t))
Definitions
- We call
)
the polynomial approximation of 
- We call
)
the finite-dimensional approximation of
Remark Let  \longrightarrow C^0(I, \mathbb{R}^m))
. 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
 = \dot q )
Integral
 = \int_I q dt)
## Newton-Euler
 = M(q)\ddot q %2B S(q, \dot q) \dot q %2B g(q))
Here the finite dimensional approximation using the Lagrange interpolation at some points can be computed with
)\ddot q(t_i) %2B S(q(t_i), \dot q(t_i)) \dot q(t_i) %2B g(q(t_i)))
Direct kinematics
Here we need to represent a manifold (quaternions, euler angles. etc)
 = \prod A(q))
Arc lent of the direct kinematics
 = \int_I \Bigg\| \frac{d q}{d t}\Bigg\|)
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
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  \longrightarrow PP^{N,n_c}(I, \mathbb{R}^n))
such that
Theorem For every
there exists an 
and map  \longrightarrow \mathbb{R}^{n_c N n})
such that if
)
such thatDefinitions
Remark Let \longrightarrow C^0(I, \mathbb{R}^m))
. Then we can define
Requirement: We need a class that can represent arbitrary functions an its derivative
Derivative
Integral
 = \int_I q dt)
## Newton-Euler = M(q)\ddot q %2B S(q, \dot q) \dot q %2B g(q))
Here the finite dimensional approximation using the Lagrange interpolation at some points can be computed withDirect 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
Remarks
ReturnType_1which can behave as both, a vector and aGSpline.GSpline::get_coefficients.ReturnType_2which can behave as both, a matrix and aFunctional.ReturnType_2does not need explicitly the interval lenghts because we don't have to cast it into aGSpline