Single-parameter Lie group-valued function. More...

#include <spline.hpp>

Public Member Functions
	Spline (const G &ga=Identity< G >())
	Default constructor creates an empty Spline starting at a given point.

	Spline (double T, Eigen::Matrix< double, Dof< G >, K > &&V, G &&ga=Identity< G >())
	Create Spline with one segment and given velocity control points.

template<typename Derived >
	Spline (double T, const Eigen::MatrixBase< Derived > &V, const G &ga=Identity< G >())
	Create Spline with one segment and given velocity control points.

template<std::ranges::range Rv> requires (std::is_same_v<std::ranges::range_value_t<Rv>, Tangent<G>>)
	Spline (double T, const Rv &vs, const G &ga=Identity< G >())
	Create Spline with one segment and given velocities.

	Spline (const Spline &)=default
	Copy constructor.

	Spline (Spline &&)=default
	Move constructor.

Spline &	operator= (const Spline &)=default
	Copy assignment.

Spline &	operator= (Spline &&)=default
	Move assignment.

	~Spline ()=default
	Destructor.

std::size_t	size () const
	Number of Spline segments.

bool	empty () const
	True if Spline has zero size()

void	reserve (std::size_t capacity)
	Allocate space for capacity segments.

double	t_min () const
	Start time of Spline (always equal to zero).

double	t_max () const
	End time of Spline.

G	start () const
	Spline start value (always equal to identity).

G	end () const
	Spline end value.

void	make_local ()
	Move start of Spline to Identity()

Spline &	concat_global (const Spline &other)
	In-place concatenation with a global Spline.

Spline &	concat_local (const Spline &other)
	In-place local concatenation.

Spline &	operator+= (const Spline &other)
	Operator overload for local concatenation.

Spline	operator+ (const Spline &other)
	Local Spline concatenation.

template<typename S = double>
CastT< S, G >	operator() (const S &t, OptTangent< CastT< S, G > > vel={}, OptTangent< CastT< S, G > > acc={}) const
	Evaluate Spline at given time.

Tangent< G >	arclength (double t) const
	Get approximate arclength traversed at time T.

Spline	crop (double ta, double tb=std::numeric_limits< double >::infinity(), bool localize=true) const
	Crop Spline.

Static Public Member Functions
static Spline	ConstantVelocityGoal (const G &gb, double T=1, const G &ga=Identity< G >())
	Create constant-velocity Spline that reaches a given target state.

static Spline	ConstantVelocity (const Tangent< G > &v, double T=1, const G &ga=Identity< G >())
	Create constant-velocity Spline.

static Spline	FixedCubic (const G &gb, const Tangent< G > &va, const Tangent< G > &vb, double T=1, const G &ga=Identity< G >())
	Create Spline with given start and end position and velocities.

Detailed Description

template<int K, LieGroup G>
class Spline< K, G >

Single-parameter Lie group-valued function.

Template Parameters

K	Spline degree
G	Lie group

A Spline is a continuous function \( x : [0, T] \rightarrow \mathbb{G} \). Internally it is a piecewise collection of cumulative polynomial segments.

Definition at line 27 of file spline.hpp.

Constructor & Destructor Documentation

◆ Spline() [1/4]

template<int K, LieGroup G>

Spline< K, G >::Spline ( const G & ga = Identity< G >() )

Default constructor creates an empty Spline starting at a given point.

Parameters

ga	Spline starting point (defaults to identity)

◆ Spline() [2/4]

template<int K, LieGroup G>

Spline< K, G >::Spline	(	double	T,
		Eigen::Matrix< double, Dof< G >, K > &&	V,
		G &&	ga = `Identity< G >()`
	)

Create Spline with one segment and given velocity control points.

Parameters

T	duration (strictly positive)
V	velocities for segment
ga	Spline starting point (defaults to identity)

◆ Spline() [3/4]

template<int K, LieGroup G>

template<typename Derived >

Spline< K, G >::Spline	(	double	T,
		const Eigen::MatrixBase< Derived > &	V,
		const G &	ga = `Identity< G >()`
	)

Create Spline with one segment and given velocity control points.

Parameters

T	duration (strictly positive)
V	velocities for segment
ga	Spline starting point (defaults to identity)

◆ Spline() [4/4]

template<int K, LieGroup G>

template<std::ranges::range Rv>
requires (std::is_same_v<std::ranges::range_value_t<Rv>, Tangent<G>>)

Spline< K, G >::Spline	(	double	T,
		const Rv &	vs,
		const G &	ga = `Identity< G >()`
	)

Create Spline with one segment and given velocities.

Parameters

T	duration (strictly positive)
vs	velocity constants (size K)
ga	Spline starting point (defaults to identity)

Member Function Documentation

◆ arclength()

template<int K, LieGroup G>

Tangent< G > Spline< K, G >::arclength ( double t ) const

Get approximate arclength traversed at time T.

The arclength is defined as

\[ A(t) = \int_{0}^t \left| \mathrm{d}^r x_s \right| \mathrm{d} s, \]

where the absolute value is component-wise.

Note: This function is approximate for Lie groups with curvature.

◆ concat_global()

template<int K, LieGroup G>

Spline & Spline< K, G >::concat_global ( const Spline< K, G > & other )

In-place concatenation with a global Spline.

Parameters

other Spline to append at the end of this Spline.

The resulting Spline \( y(t) \) is s.t.

\[ y(t) = \begin{cases} x_1(t) & 0 \leq t < t_1 \\ x_2(t) & t_1 \leq t \leq t_1 + t_2 \end{cases} \]

◆ concat_local()

template<int K, LieGroup G>

Spline & Spline< K, G >::concat_local ( const Spline< K, G > & other )

In-place local concatenation.

Parameters

other Spline to append at the end of this Spline.

The resulting Spline \( y(t) \) is s.t.

\[ y(t) = \begin{cases} x_1(t) & 0 \leq t < t_1 \\ x_1(t_1) \circ x_2(t) & t_1 \leq t \leq t_1 + t_2 \end{cases} \]

That is, other is considered a Spline in the local frame of end(). For global concatenation see concat_global.

◆ ConstantVelocity()

template<int K, LieGroup G>

static Spline Spline< K, G >::ConstantVelocity	(	const Tangent< G > &	v,
		double	T = `1`,
		const G &	ga = `Identity< G >()`
	)

static

Create constant-velocity Spline.

The resulting Spline is

\[ x(t) = g_a \exp(t v), \quad t \in [0, T]. \]

Parameters

v	body velocity
T	duration
ga	Spline starting point

◆ ConstantVelocityGoal()

template<int K, LieGroup G>

static Spline Spline< K, G >::ConstantVelocityGoal	(	const G &	gb,
		double	T = `1`,
		const G &	ga = `Identity< G >()`
	)

static

Create constant-velocity Spline that reaches a given target state.

The resulting Spline is

\[ x(t) = g_a \circ \exp\left( \frac{t}{T} (g_b \ominus g_a) \right), \quad t \in [0, T]. \]

Parameters

gb	Spline target point
T	duration (must be positive)
ga	Spline starting point (defaults to Identity)

◆ crop()

template<int K, LieGroup G>

Spline Spline< K, G >::crop	(	double	ta,
		double	tb = `std::numeric_limits< double >::infinity()`,
		bool	localize = `true`
	)		const

Crop Spline.

Parameters

ta,tb	interval for cropped Spline
localize	cropped Spline start at identity

The resulting Spline \( y(t) \) defined on \( [0, t_b - t_a] \) is s.t.

If localize = true:
\[ y(t) = x(t_a)^{-1} \circ x(t - t_a) \]
If localize = false:
\[ y(t) = x(t - t_a) \]

◆ FixedCubic()

template<int K, LieGroup G>

static Spline Spline< K, G >::FixedCubic	(	const G &	gb,
		const Tangent< G > &	va,
		const Tangent< G > &	vb,
		double	T = `1`,
		const G &	ga = `Identity< G >()`
	)

static

Create Spline with given start and end position and velocities.

Parameters

gb	Spline target point
va,vb	start and end velocities
T	duration
ga	Spline starting point (default Identity)

◆ operator()()

template<int K, LieGroup G>

template<typename S = double>

CastT< S, G > Spline< K, G >::operator()	(	const S &	t,
		OptTangent< CastT< S, G > >	vel = `{}`,
		OptTangent< CastT< S, G > >	acc = `{}`
	)		const

Evaluate Spline at given time.

Template Parameters

S time type

Parameters

[in]	t	time
[out]	vel	output body velocity at evaluation time
[out]	acc	output body acceleration at evaluation time

Returns: value at time t

Note: Outside the support [t_min(), t_max()] the result is clamped to the end points, and the acceleration and velocity is zero.

◆ operator+()

template<int K, LieGroup G>

Spline Spline< K, G >::operator+ ( const Spline< K, G > & other )

Local Spline concatenation.

See also: concat_local()

◆ operator+=()

template<int K, LieGroup G>

Spline & Spline< K, G >::operator+= ( const Spline< K, G > & other )

Operator overload for local concatenation.

Parameters

other Spline to append at the end of this Spline.

See also: concat_local()

The documentation for this class was generated from the following file:

include/smooth/spline/spline.hpp

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ Spline() [1/4]

◆ Spline() [2/4]

◆ Spline() [3/4]

◆ Spline() [4/4]

Member Function Documentation

◆ arclength()

◆ concat_global()

◆ concat_local()

◆ ConstantVelocity()

◆ ConstantVelocityGoal()

◆ crop()

◆ FixedCubic()

◆ operator()()

◆ operator+()

◆ operator+=()