Chapter 9. Trajectory Generation

Functions
double	mr::CubicTimeScaling (const double Tf, const double t)
	Computes s(t) for a cubic time scaling. More...
double	mr::QuinticTimeScaling (const double Tf, const double t)
	Computes s(t) for a quintic time scaling. More...
const std::vector< arma::vec >	mr::JointTrajectory (const arma::vec &thetastart, const arma::vec &thetaend, const double Tf, const size_t N, const Method &method)
	Computes a straight-line trajectory in joint space. More...
const std::vector< arma::mat44 >	mr::ScrewTrajectory (const arma::mat44 &Xstart, const arma::mat44 &Xend, const double Tf, const size_t N, const Method &method)
	Computes a trajectory as a list of N SE(3) matrices corresponding to the screw motion about a space screw axis. More...
const std::vector< arma::mat44 >	mr::CartesianTrajectory (const arma::mat44 &Xstart, const arma::mat44 &Xend, const double Tf, const size_t N, const Method &method)
	Computes a trajectory as a list of N SE(3) matrices corresponding to the origin of the end-effector frame following a straight line. More...

Detailed Description

Function Documentation

CartesianTrajectory()

const std::vector< arma::mat44 > mr::CartesianTrajectory	(	const arma::mat44 &	Xstart,
		const arma::mat44 &	Xend,
		const double	Tf,
		const size_t	N,
		const Method &	method
	)

Computes a trajectory as a list of N SE(3) matrices corresponding to the origin of the end-effector frame following a straight line.

Parameters

Xstart	The initial end-effector configuration
Xend	The final end-effector configuration
Tf	Total time of the motion in seconds from rest to rest
N	The number of points N > 1 (Start and stop) in the discrete representation of the trajectory
method	The time-scaling method, where 3 indicates cubic (third- order polynomial) time scaling and 5 indicates quintic (fifth-order polynomial) time scaling

Returns

The discretized trajectory as a list of N matrices in SE(3) separated in time by Tf/(N-1). The first in the list is Xstart and the Nth is Xend

This function is similar to ScrewTrajectory, except the origin of the end-effector frame follows a straight line, decoupled from the rotational motion.

CubicTimeScaling()

double mr::CubicTimeScaling	(	const double	Tf,
		const double	t
	)

Computes s(t) for a cubic time scaling.

Parameters

Tf	Total time of the motion in seconds from rest to rest
t	The current time t satisfying 0 < t < Tf

Returns

The path parameter s(t) corresponding to a third-order polynomial motion that begins and ends at zero velocity

JointTrajectory()

const std::vector< arma::vec > mr::JointTrajectory	(	const arma::vec &	thetastart,
		const arma::vec &	thetaend,
		const double	Tf,
		const size_t	N,
		const Method &	method
	)

Computes a straight-line trajectory in joint space.

Parameters

thetastart	The initial joint variables
thetaend	The final joint variables
Tf	Total time of the motion in seconds from rest to rest
N	The number of points N > 1 (Start and stop) in the discrete representation of the trajectory
method	The time-scaling method, where 3 indicates cubic (third- order polynomial) time scaling and 5 indicates quintic (fifth-order polynomial) time scaling

Returns

A trajectory as an N x n matrix, where each row is an n-vector of joint variables at an instant in time. The first row is thetastart and the Nth row is thetaend . The elapsed time between each row is Tf / (N - 1)

QuinticTimeScaling()

double mr::QuinticTimeScaling	(	const double	Tf,
		const double	t
	)

Computes s(t) for a quintic time scaling.

Parameters

Tf	Total time of the motion in seconds from rest to rest
t	The current time t satisfying 0 < t < Tf

Returns

The path parameter s(t) corresponding to a fifth-order polynomial motion that begins and ends at zero velocity and zero acceleration

ScrewTrajectory()

const std::vector< arma::mat44 > mr::ScrewTrajectory	(	const arma::mat44 &	Xstart,
		const arma::mat44 &	Xend,
		const double	Tf,
		const size_t	N,
		const Method &	method
	)

Computes a trajectory as a list of N SE(3) matrices corresponding to the screw motion about a space screw axis.

Parameters

Xstart	The initial end-effector configuration
Xend	The final end-effector configuration
Tf	Total time of the motion in seconds from rest to rest
N	The number of points N > 1 (Start and stop) in the discrete representation of the trajectory
method	The time-scaling method, where 3 indicates cubic (third- order polynomial) time scaling and 5 indicates quintic (fifth-order polynomial) time scaling

Returns

The discretized trajectory as a list of N matrices in SE(3) separated in time by Tf/(N-1). The first in the list is Xstart and the Nth is Xend