roboptim::trajectory::Polynomial< N > Class Template Reference

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

#include <roboptim/trajectory/polynomial.hh>

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

List of all members.

Public Types
typedef Function::interval_t	interval_t
typedef std::vector< value_type >	roots_t
	Type of the vector of roots.
typedef Eigen::Matrix < value_type, N+1, 1 >	coefs_t
	Fixed-size coefficient vector (N+1 if N is the polynomial degree).
typedef std::vector< value_type >	values_t
	Type of the critical points.
typedef std::pair< value_type, value_type >	max_t
	Type of a maximum query: (t_max, P(t_max))
typedef std::pair< value_type, value_type >	min_t
	Type of a minimum query: (t_min, P(t_min))
typedef ::roboptim::Polynomial < Function::traits_t >	polynomialFunction_t
	Polynomial function.
Public Member Functions
	ROBOPTIM_FUNCTION_FWD_TYPEDEFS_ (Function)
	BOOST_STATIC_ASSERT (N >=0)
	Polynomial degree >= 0.
	Polynomial ()
	Default constructor: return a null polynomial.
	Polynomial (value_type t0, const_vector_ref coefs)
	Construct of a polynomial from its center and its coefficients.
	Polynomial (value_type t0,...)
	Variadic constructor.
template<int M>
	Polynomial (const Polynomial< M > &p)
	Copy constructor of polynomials of different orders.
Polynomial< N >	translate (value_type t1) const
	Return a new polynomial translated from (t-t₀) to (t-t₁).
void	translateInPlace (value_type t1)
	Translate the polynomial (in place) from (t-t₀) to (t-t₁).
template<int K>
Polynomial< N-K >	derivative () const
	Compute the derivative polynomial of a given order.
value_type	derivative (value_type t, size_type order=1) const
	Evaluate the derivative of a given order.
template<int M>
Polynomial< N+M >	operator* (const Polynomial< M > &poly) const
	Multiply polynomials of different orders.
Polynomial< N >	operator+ (const Polynomial< N > &poly) const
	Addition of polynomials.
Polynomial< N >	operator- (const Polynomial< N > &poly) const
	Subtraction of polynomials.
Polynomial< N >	operator* (value_type lambda) const
	Scalar multiplication of a polynomial.
void	operator+= (const Polynomial< N > &poly)
	Addition of polynomials.
value_type	operator() (value_type t) const
	Evaluate the polynomial with Horner's method.
const coefs_t &	coefs () const
	Const getter to coefs.
coefs_t &	coefs ()
	Getter to coefs.
value_type	t0 () const
	Const getter to t0.
value_type &	t0 ()
	Reference to t0.
value_type	operator[] (int i) const
	Get the i-th polynomial coefficient.
roots_t	realRoots (value_type epsilon=1e-6) const
	Return the real roots of the polynomial.
values_t	critPoints (const interval_t &interval) const
	Compute the critical values of the polynomial on an interval.
min_t	min (const interval_t &interval, bool acceptConstant=true) const
	Compute the minimum of the polynomial on an interval.
max_t	max (const interval_t &interval, bool acceptConstant=true) const
	Compute the maximum of the polynomial on an interval.
bool	isNull (value_type epsilon=Function::epsilon()) const
	Return whether the polynomial is null.
bool	isConstant (value_type epsilon=Function::epsilon()) const
	Return whether the polynomial is constant.
bool	isLinear (value_type epsilon=Function::epsilon()) const
	Return whether the polynomial is linear.
int	trueOrder (value_type epsilon=Function::epsilon()) const
	Return the "true" order of the polynomial.
polynomialFunction_t	asFunction () const
	Get the equivalent Polynomial function.
virtual std::ostream &	print (std::ostream &o) const
	Print the polynomial.
Static Public Member Functions
static int	order ()
	Return the order of such a polynomial.
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.
coefs_t	impl_translate (value_type t1) const
	Polynomial (value_type t0, special_polynomials key)
	Special constructor for Monomial<N> and some operators.
Static Protected Attributes
static const int	order_ = N
	order of the polynomial.

---

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::BSpline< N >::cox_de_boor(), 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

(

)

[inline, static]

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 [static, protected]

order of the polynomial.

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