roboticslab::PadenKahanTwo Class Reference

Second Paden-Kahan subproblem. More...

#include <ScrewTheoryIkSubproblems.hpp>

Inheritance diagram for roboticslab::PadenKahanTwo:

Public Member Functions
	PadenKahanTwo (const MatrixExponential &exp1, const MatrixExponential &exp2, const KDL::Vector &p, const KDL::Vector &r)
	Constructor.
bool	solve (const KDL::Frame &rhs, const KDL::Frame &pointTransform, const JointConfig &reference, Solutions &solutions) const override
	Finds a closed geometric solution for this IK subproblem.
int	solutions () const override
	Number of local IK solutions.
const char *	describe () const override
	Return a human-readable description of this IK subproblem.
virtual bool	solve (const KDL::Frame &rhs, const KDL::Frame &pointTransform, const JointConfig &reference, Solutions &solutions) const=0
	Finds a closed geometric solution for this IK subproblem.
bool	solve (const KDL::Frame &rhs, const KDL::Frame &pointTransform, Solutions &_solutions) const
Public Member Functions inherited from roboticslab::ScrewTheoryIkSubproblem
virtual	~ScrewTheoryIkSubproblem ()=default
	Destructor.
bool	solve (const KDL::Frame &rhs, const KDL::Frame &pointTransform, Solutions &_solutions) const

Private Attributes
const MatrixExponential	exp1
const MatrixExponential	exp2
const KDL::Vector	p
const KDL::Vector	r
const KDL::Vector	axesCross
const KDL::Rotation	axisPow1
const KDL::Rotation	axisPow2
const double	axesDot

Additional Inherited Members
Public Types inherited from roboticslab::ScrewTheoryIkSubproblem
using	JointConfig = std::vector< double >
	Joint configurations.
using	Solutions = std::vector< JointConfig >
	Collection of local IK solutions.

Detailed Description

Dual solution, double revolute joint geometric IK subproblem given by \( e\,^{\hat{\xi_1}\,{\theta_1}} \cdot e\,^{\hat{\xi_2}\,{\theta_2}} \cdot p = k \) (consecutive crossing rotation screws to a point).

Constructor & Destructor Documentation

PadenKahanTwo()

PadenKahanTwo::PadenKahanTwo	(	const MatrixExponential &	exp1,
		const MatrixExponential &	exp2,
		const KDL::Vector &	p,
		const KDL::Vector &	r
	)

Parameters

exp1	First POE term.
exp2	Second POE term.
p	Characteristic point.
r	Point of intersection between both screw axes.

Member Function Documentation

describe()

const char * roboticslab::PadenKahanTwo::describe ( ) const

inlineoverridevirtual

Implements roboticslab::ScrewTheoryIkSubproblem.

solutions()

int roboticslab::PadenKahanTwo::solutions ( ) const

inlineoverridevirtual

Implements roboticslab::ScrewTheoryIkSubproblem.

solve() [1/2]

bool PadenKahanTwo::solve ( const KDL::Frame &  rhs, const KDL::Frame &  pointTransform, const JointConfig &  reference, Solutions &  solutions  ) const

overridevirtual

Given the product of exponentials (POE) formula \( \prod_i e\,^{\hat{\xi}_i\,{\theta_i}} \cdot H_{ST}(0) = H_{ST}(\theta) \), invariant and known terms are rearranged to the right side (rhs) as follows:

\[ \prod_{i=j}^{j+k} e\,^{\hat{\xi}_i\,{\theta_i}} = \left [ \prod_{i=1}^{j-1} e\,^{\hat{\xi}_i\,{\theta_i}} \right ]^{-1} \cdot H_{ST}(\theta) \cdot \left [ H_{ST}(0) \right ]^{-1} \cdot \left [ \prod_{i=j+k+1}^{N} e\,^{\hat{\xi}_i\,{\theta_i}} \right ]^{-1} \]

where \( j = \{1, 2, ..., N\} \), \( k = \{1, 2, ..., N-1\} \), \( 1 <= j+k <= N \).

Given \( N \) terms in the POE formula, \( j \) of which are unknowns, any characteristic point \( p \) postmultiplying this expression could be rewritten as \( p' \) per:

\[ \prod_{i=1}^j e\,^{\hat{\xi}_i\,{\theta_i}} \cdot \prod_{i=j+1}^N e\,^{\hat{\xi}_i\,{\theta_i}} \cdot p = \prod_{i=1}^j e\,^{\hat{\xi}_i\,{\theta_i}} \cdot p' \]

where pointTransform is the transformation matrix that produces \( p' \) from \( p \).

Parameters

rhs	Right-hand side of the POE formula prior to being applied to the right-hand side of this subproblem.
pointTransform	Transformation frame applied to the first (and perhaps only) characteristic point of this subproblem.
reference	Known nearby solutions to be used as reference in case a singularity is found.
solutions	Output vector of local solutions.

Returns

True if all solutions are reachable, false otherwise.

Implements roboticslab::ScrewTheoryIkSubproblem.

solve() [2/2]

virtual bool roboticslab::ScrewTheoryIkSubproblem::solve ( const KDL::Frame &  rhs, const KDL::Frame &  pointTransform, const JointConfig &  reference, Solutions &  solutions  ) const

virtual

Given the product of exponentials (POE) formula \( \prod_i e\,^{\hat{\xi}_i\,{\theta_i}} \cdot H_{ST}(0) = H_{ST}(\theta) \), invariant and known terms are rearranged to the right side (rhs) as follows:

\[ \prod_{i=j}^{j+k} e\,^{\hat{\xi}_i\,{\theta_i}} = \left [ \prod_{i=1}^{j-1} e\,^{\hat{\xi}_i\,{\theta_i}} \right ]^{-1} \cdot H_{ST}(\theta) \cdot \left [ H_{ST}(0) \right ]^{-1} \cdot \left [ \prod_{i=j+k+1}^{N} e\,^{\hat{\xi}_i\,{\theta_i}} \right ]^{-1} \]

where \( j = \{1, 2, ..., N\} \), \( k = \{1, 2, ..., N-1\} \), \( 1 <= j+k <= N \).

Given \( N \) terms in the POE formula, \( j \) of which are unknowns, any characteristic point \( p \) postmultiplying this expression could be rewritten as \( p' \) per:

\[ \prod_{i=1}^j e\,^{\hat{\xi}_i\,{\theta_i}} \cdot \prod_{i=j+1}^N e\,^{\hat{\xi}_i\,{\theta_i}} \cdot p = \prod_{i=1}^j e\,^{\hat{\xi}_i\,{\theta_i}} \cdot p' \]

where pointTransform is the transformation matrix that produces \( p' \) from \( p \).

Parameters

rhs	Right-hand side of the POE formula prior to being applied to the right-hand side of this subproblem.
pointTransform	Transformation frame applied to the first (and perhaps only) characteristic point of this subproblem.
reference	Known nearby solutions to be used as reference in case a singularity is found.
solutions	Output vector of local solutions.

Returns

True if all solutions are reachable, false otherwise.

Implements roboticslab::ScrewTheoryIkSubproblem.

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

• libraries/ScrewTheoryLib/ScrewTheoryIkSubproblems.hpp
• libraries/ScrewTheoryLib/PadenKahanSubproblems.cpp