roboticslab::PadenKahanThree Class Reference

Third Paden-Kahan subproblem. More...

#include <ScrewTheoryIkSubproblems.hpp>

Inheritance diagram for roboticslab::PadenKahanThree:

Public Member Functions
	PadenKahanThree (const MatrixExponential &exp, const KDL::Vector &p, const KDL::Vector &k)
	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)
Public Member Functions inherited from roboticslab::ScrewTheoryIkSubproblem
virtual	~ScrewTheoryIkSubproblem ()=default
	Destructor.
bool	solve (const KDL::Frame &rhs, const KDL::Frame &pointTransform, Solutions &_solutions)

Private Attributes
const MatrixExponential	exp
const KDL::Vector	p
const KDL::Vector	k
const KDL::Rotation	axisPow

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, single revolute joint geometric IK subproblem given by \( \left \| e\,^{\hat{\xi}\,{\theta}} \cdot p - k \right \| = \delta \) (rotation screw for moving \( p \) to a distance \( \delta \) from \( k \)).

Constructor & Destructor Documentation

PadenKahanThree()

PadenKahanThree::PadenKahanThree	(	const MatrixExponential &	exp,
		const KDL::Vector &	p,
		const KDL::Vector &	k
	)

Parameters

exp	POE term.
p	First characteristic point.
k	Second characteristic point.

Member Function Documentation

describe()

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

inlineoverridevirtual

Implements roboticslab::ScrewTheoryIkSubproblem.

solutions()

int roboticslab::PadenKahanThree::solutions ( ) const

inlineoverridevirtual

Implements roboticslab::ScrewTheoryIkSubproblem.

solve() [1/2]

bool PadenKahanThree::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