Polynomial of degree at most N (N >= 0). More...

#include <roboptim/trajectory/polynomial.hh>

Inheritance diagram for roboptim::trajectory::Polynomial< N >:

Public Types
typedef Function::interval_t	interval_t

typedef std::vector< value_type >	roots_t
	Type of the vector of roots. More...

typedef Eigen::Matrix < value_type, N+1, 1 >	coefs_t
	Fixed-size coefficient vector (N+1 if N is the polynomial degree). More...

typedef std::vector< value_type >	values_t
	Type of the critical points. More...

typedef std::pair< value_type, value_type >	max_t
	Type of a maximum query: (t_max, P(t_max)) More...

typedef std::pair< value_type, value_type >	min_t
	Type of a minimum query: (t_min, P(t_min)) More...

typedef ::roboptim::Polynomial < Function::traits_t >	polynomialFunction_t
	Polynomial function. More...

Public Member Functions
	ROBOPTIM_FUNCTION_FWD_TYPEDEFS_ (Function)

	BOOST_STATIC_ASSERT (N >=0)
	Polynomial degree >= 0. More...

	Polynomial ()
	Default constructor: return a null polynomial. More...

	Polynomial (value_type t0, const_vector_ref coefs)
	Construct of a polynomial from its center and its coefficients. More...

	Polynomial (value_type t0,...)
	Variadic constructor. More...

template<int M>
	Polynomial (const Polynomial< M > &p)
	Copy constructor of polynomials of different orders. More...

Polynomial< N >	translate (value_type t1) const
	Return a new polynomial translated from (t-t₀) to (t-t₁). More...

void	translateInPlace (value_type t1)
	Translate the polynomial (in place) from (t-t₀) to (t-t₁). More...

template<int K>
Polynomial< N-K >	derivative () const
	Compute the derivative polynomial of a given order. More...

value_type	derivative (value_type t, size_type order=1) const
	Evaluate the derivative of a given order. More...

template<int M>
Polynomial< N+M >	operator* (const Polynomial< M > &poly) const
	Multiply polynomials of different orders. More...

Polynomial< N >	operator+ (const Polynomial< N > &poly) const
	Addition of polynomials. More...

Polynomial< N >	operator- (const Polynomial< N > &poly) const
	Subtraction of polynomials. More...

Polynomial< N >	operator* (value_type lambda) const
	Scalar multiplication of a polynomial. More...

void	operator+= (const Polynomial< N > &poly)
	Addition of polynomials. More...

value_type	operator() (value_type t) const
	Evaluate the polynomial with Horner's method. More...

const coefs_t &	coefs () const
	Const getter to coefs. More...

coefs_t &	coefs ()
	Getter to coefs. More...

value_type	t0 () const
	Const getter to t0. More...

value_type &	t0 ()
	Reference to t0. More...

value_type	operator[] (int i) const
	Get the i-th polynomial coefficient. More...

roots_t	realRoots (value_type epsilon=1e-6) const
	Return the real roots of the polynomial. More...

values_t	critPoints (const interval_t &interval) const
	Compute the critical values of the polynomial on an interval. More...

min_t	min (const interval_t &interval, bool acceptConstant=true) const
	Compute the minimum of the polynomial on an interval. More...

max_t	max (const interval_t &interval, bool acceptConstant=true) const
	Compute the maximum of the polynomial on an interval. More...

bool	isNull (value_type epsilon=Function::epsilon()) const
	Return whether the polynomial is null. More...

bool	isConstant (value_type epsilon=Function::epsilon()) const
	Return whether the polynomial is constant. More...

bool	isLinear (value_type epsilon=Function::epsilon()) const
	Return whether the polynomial is linear. More...

int	trueOrder (value_type epsilon=Function::epsilon()) const
	Return the "true" order of the polynomial. More...

polynomialFunction_t	asFunction () const
	Get the equivalent Polynomial function. More...

virtual std::ostream &	print (std::ostream &o) const
	Print the polynomial. More...

Static Public Member Functions
static int	order ()
	Return the order of such a polynomial. More...

Protected Types
enum	special_polynomials { all_zero_coefficients = 0, monomial_coefficients = 1 }
	Enum for special polynomials. More...

Protected Member Functions
value_type	impl_derivative (value_type t, size_type order, size_type start_coef=0) const

template<int K>
Polynomial< N-K >	impl_derivative () const
	Implementation of the compile-time derivative. More...

coefs_t	impl_translate (value_type t1) const

	Polynomial (value_type t0, special_polynomials key)
	Special constructor for Monomial<N> and some operators. More...

Static Protected Attributes
static const int	order_ = N
	order of the polynomial. More...

Detailed Description

template<int N>
class roboptim::trajectory::Polynomial< N >

Polynomial of degree at most N (N >= 0).

\[ P (t) = \sum_{i=0}^{N} a_i (t-t_0)^i \]

Member Typedef Documentation

template<int N>

typedef Eigen::Matrix<value_type, N+1, 1> roboptim::trajectory::Polynomial< N >::coefs_t

Fixed-size coefficient vector (N+1 if N is the polynomial degree).

template<int N>

typedef Function::interval_t roboptim::trajectory::Polynomial< N >::interval_t

template<int N>

typedef std::pair<value_type, value_type> roboptim::trajectory::Polynomial< N >::max_t

Type of a maximum query: (t_max, P(t_max))

template<int N>

typedef std::pair<value_type, value_type> roboptim::trajectory::Polynomial< N >::min_t

Type of a minimum query: (t_min, P(t_min))

template<int N>

typedef ::roboptim::Polynomial<Function::traits_t> roboptim::trajectory::Polynomial< N >::polynomialFunction_t

Polynomial function.

template<int N>

typedef std::vector<value_type> roboptim::trajectory::Polynomial< N >::roots_t

Type of the vector of roots.

template<int N>

typedef std::vector<value_type> roboptim::trajectory::Polynomial< N >::values_t

Type of the critical points.

Member Enumeration Documentation

template<int N>

enum roboptim::trajectory::Polynomial::special_polynomials

protected

Enum for special polynomials.

Enumerator
all_zero_coefficients	Null polynomial: 0.
monomial_coefficients	Monomial polynomial: (t-t₀)

Constructor & Destructor Documentation

template<int N>

roboptim::trajectory::Polynomial< N >::Polynomial ( )

Default constructor: return a null polynomial.

template<int N>

roboptim::trajectory::Polynomial< N >::Polynomial	(	value_type	t0,
		const_vector_ref	coefs
	)

Construct of a polynomial from its center and its coefficients.

Parameters

t0	polynomial of (t-t₀).
coefs	polynomial coefficients.

References roboptim::trajectory::Polynomial< N >::coefs().

template<int N>

roboptim::trajectory::Polynomial< N >::Polynomial	(	value_type	t0,
			...
	)

Variadic constructor.

Note: this is a legacy constructor used to keep Polynomial3's API.

Parameters

t0	polynomial of (t-t₀).
...	variadic arguments containing [a₀,a₁,...,a_N]

template<int N>

template<int M>

roboptim::trajectory::Polynomial< N >::Polynomial ( const Polynomial< M > & p )

Copy constructor of polynomials of different orders.

Warning: If N < M, coefficients of higher degrees will be discarded. This is only valid if you are considering |t-t₀|«1.

Template Parameters

M	degree of the copied polynomial.

Parameters

p	copied polynomial.

References roboptim::trajectory::Polynomial< N >::coefs().

template<int N>

roboptim::trajectory::Polynomial< N >::Polynomial	(	value_type	t0,
		special_polynomials	key
	)

protected

Special constructor for Monomial<N> and some operators.

Parameters

t0
key	one of all_zero_coefficients, monomial_coefficients.

References roboptim::trajectory::Polynomial< N >::all_zero_coefficients, and roboptim::trajectory::Polynomial< N >::monomial_coefficients.

Member Function Documentation

template<int N>

Polynomial< N >::polynomialFunction_t roboptim::trajectory::Polynomial< N >::asFunction ( ) const

Get the equivalent Polynomial function.

This can be used for all the methods that expect a RobOptim function (e.g. plotting).

Returns: Polynomial function with the appropriate coefficients set.

template<int N>

roboptim::trajectory::Polynomial< N >::BOOST_STATIC_ASSERT ( N >= 0 )

Polynomial degree >= 0.

template<int N>

const Polynomial< N >::coefs_t & roboptim::trajectory::Polynomial< N >::coefs ( ) const

Const getter to coefs.

Returns: const reference to polynomial coefficients.

Referenced by roboptim::trajectory::Polynomial< N >::operator*(), and roboptim::trajectory::Polynomial< N >::Polynomial().

template<int N>

Polynomial< N >::coefs_t & roboptim::trajectory::Polynomial< N >::coefs ( )

Getter to coefs.

Returns: reference to polynomial coefficients.

template<int N>

Polynomial< N >::values_t roboptim::trajectory::Polynomial< N >::critPoints ( const interval_t & interval ) const

Compute the critical values of the polynomial on an interval.

Parameters

interval time interval.

Returns: vector containing the critical values of the polynomial on the interval. Used by the min and max function.

Exceptions

std::runtime_error invalid polynomial. This is the case for constant polynomials since there is an infinity of critical points. This can be tested before calling min().

template<int N>

template<int K>

Polynomial< N-K > roboptim::trajectory::Polynomial< N >::derivative ( ) const

Compute the derivative polynomial of a given order.

Template Parameters

K	derivative order.

Returns: K-th derivative of the polynomial.

Referenced by roboptim::trajectory::BSpline< N >::basisFunctions(), roboptim::trajectory::BSpline< N >::impl_derivative(), roboptim::trajectory::CubicBSpline::impl_derivative(), roboptim::trajectory::ConstrainedBSpline< N >::variationDerivWrtParam(), and roboptim::trajectory::BSpline< N >::variationDerivWrtParam().

template<int N>

Polynomial< N >::value_type roboptim::trajectory::Polynomial< N >::derivative	(	value_type	t,
		size_type	order = `1`
	)		const

Evaluate the derivative of a given order.

Parameters

t	time of the evaluation.
order	order of the derivative.

Returns: derivative of a given order evaluated at t.

template<int N>

Polynomial< N >::value_type roboptim::trajectory::Polynomial< N >::impl_derivative	(	value_type	t,
		size_type	order,
		size_type	start_coef = `0`
	)		const

protected

template<int N>

template<int K>

Polynomial< N-K > roboptim::trajectory::Polynomial< N >::impl_derivative ( ) const

protected

Implementation of the compile-time derivative.

Template Parameters

K	order of the derivation.

Returns: K-th derivative.

template<int N>

Polynomial< N >::coefs_t roboptim::trajectory::Polynomial< N >::impl_translate ( value_type t1 ) const

protected

template<int N>

bool roboptim::trajectory::Polynomial< N >::isConstant ( value_type epsilon = Function::epsilon () ) const

Return whether the polynomial is constant.

Parameters

epsilon epsilon used.

Returns: true if the polynomial is constant, false otherwise.

template<int N>

bool roboptim::trajectory::Polynomial< N >::isLinear ( value_type epsilon = Function::epsilon () ) const

Return whether the polynomial is linear.

Parameters

epsilon epsilon used.

Returns: true if the polynomial is linear, false otherwise.

template<int N>

bool roboptim::trajectory::Polynomial< N >::isNull ( value_type epsilon = Function::epsilon () ) const

Return whether the polynomial is null.

Parameters

epsilon epsilon used.

Returns: true if the polynomial is null, false otherwise.

template<int N>

Polynomial< N >::max_t roboptim::trajectory::Polynomial< N >::max	(	const interval_t &	interval,
		bool	acceptConstant = `true`
	)		const

Compute the maximum of the polynomial on an interval.

Parameters

interval	time interval.
acceptConstant	boolean allowing constant polynomials.

Returns: pair containing t_max and the associated maximum of the polynomial on the interval. If the polynomial is constant and acceptConstant is not set to false, the call returns a couple of the middle of interval and the value of the polynomial.

Exceptions

std::runtime_error invalid polynomial. This is the case for constant polynomials if acceptConstant is set to false, since there is an infinity of critical points.

template<int N>

Polynomial< N >::min_t roboptim::trajectory::Polynomial< N >::min	(	const interval_t &	interval,
		bool	acceptConstant = `true`
	)		const

Compute the minimum of the polynomial on an interval.

Parameters

interval	time interval.
acceptConstant	boolean allowing constant polynomials.

Returns: pair containing t_min and the associated minimum of the polynomial on the interval. If the polynomial is constant and acceptConstant is not set to false, the call returns a couple of the middle of interval and the value of the polynomial.

Exceptions

std::runtime_error invalid polynomial. This is the case for constant polynomials if acceptConstant is set to false, since there is an infinity of critical points.

template<int N>

Polynomial< N >::value_type roboptim::trajectory::Polynomial< N >::operator() ( value_type t ) const

Evaluate the polynomial with Horner's method.

Parameters

t	point of evaluation.

Returns: P(t)

template<int N>

template<int M>

Polynomial< N+M > roboptim::trajectory::Polynomial< N >::operator* ( const Polynomial< M > & poly ) const

Multiply polynomials of different orders.

Template Parameters

M	order of the polynomial.

Parameters

poly	polynomial to multiply.

Returns: P₀P₁

References roboptim::trajectory::Polynomial< N >::coefs(), and roboptim::trajectory::Polynomial< N >::translate().

template<int N>

Polynomial< N > roboptim::trajectory::Polynomial< N >::operator* ( value_type lambda ) const

Scalar multiplication of a polynomial.

Parameters

lambda scalar.

Returns: λP

template<int N>

Polynomial< N > roboptim::trajectory::Polynomial< N >::operator+ ( const Polynomial< N > & poly ) const

Addition of polynomials.

Parameters

poly	polynomial to add.

Returns: P₀ + P₁

References roboptim::trajectory::Polynomial< N >::translate().

template<int N>

void roboptim::trajectory::Polynomial< N >::operator+= ( const Polynomial< N > & poly )

Addition of polynomials.

Parameters

poly	polynomial to add.

References roboptim::trajectory::Polynomial< N >::translate().

template<int N>

Polynomial< N > roboptim::trajectory::Polynomial< N >::operator- ( const Polynomial< N > & poly ) const

Subtraction of polynomials.

Parameters

poly	polynomial to subtract.

Returns: P₀ - P₁

References roboptim::trajectory::Polynomial< N >::translate().

template<int N>

Polynomial< N >::value_type roboptim::trajectory::Polynomial< N >::operator[] ( int i ) const

Get the i-th polynomial coefficient.

Parameters

i	number of the coefficient to get.

Returns: i-th polynomial coefficient αi, with: α_i (t-t₀)^i

template<int N>

static int roboptim::trajectory::Polynomial< N >::order ( )

inlinestatic

Return the order of such a polynomial.

Returns: order of such a polynomial.

References roboptim::trajectory::Polynomial< N >::order_.

template<int N>

std::ostream & roboptim::trajectory::Polynomial< N >::print ( std::ostream & o ) const

virtual

Print the polynomial.

Parameters

o	output stream.

Returns: output stream.

template<int N>

Polynomial< N >::roots_t roboptim::trajectory::Polynomial< N >::realRoots ( value_type epsilon = 1e-6 ) const

Return the real roots of the polynomial.

Warning: This function relies on Eigen's experimental polynomial solver. The polynomial should not be null or constant.

Parameters

epsilon epsilon used for testing constant polynomials.

Returns: vector of the real roots of the polynomial.

Exceptions

std::runtime_error invalid polynomial (e.g. null/constant).

template<int N>

roboptim::trajectory::Polynomial< N >::ROBOPTIM_FUNCTION_FWD_TYPEDEFS_ ( Function )

template<int N>

Polynomial< N >::value_type roboptim::trajectory::Polynomial< N >::t0 ( ) const

Const getter to t0.

Returns: t0.

template<int N>

Polynomial< N >::value_type & roboptim::trajectory::Polynomial< N >::t0 ( )

Reference to t0.

Returns: reference to t0.

template<int N>

Polynomial< N > roboptim::trajectory::Polynomial< N >::translate ( value_type t1 ) const

Return a new polynomial translated from (t-t₀) to (t-t₁).

Parameters

t1	new center.

Returns: polynomial translated to (t-t₁)

Referenced by roboptim::trajectory::Polynomial< N >::operator*(), roboptim::trajectory::Polynomial< N >::operator+(), roboptim::trajectory::Polynomial< N >::operator+=(), and roboptim::trajectory::Polynomial< N >::operator-().

template<int N>

void roboptim::trajectory::Polynomial< N >::translateInPlace ( value_type t1 )

Translate the polynomial (in place) from (t-t₀) to (t-t₁).

Parameters

t1	new center.

template<int N>

int roboptim::trajectory::Polynomial< N >::trueOrder ( value_type epsilon = Function::epsilon () ) const

Return the "true" order of the polynomial.

Leading coefficients may be null, which c

Returns: degree for the highest exponent with a nonzero coeffcient.

Member Data Documentation

template<int N>

const int roboptim::trajectory::Polynomial< N >::order_ = N

staticprotected

order of the polynomial.

Referenced by roboptim::trajectory::Polynomial< N >::order().

Public Types

Public Member Functions

Static Public Member Functions

Protected Types

Protected Member Functions

Static Protected Attributes

Detailed Description

template<int N> class roboptim::trajectory::Polynomial< N >

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<int N>
class roboptim::trajectory::Polynomial< N >