lugre_model – FRICTION

type, public, extends(friction_model) :: lugre_model

Defines the Lu-Gre friction model.

The Lu-Gre model is a bristle-type model that attempts to describe friction using a bristle interpretation of the frictional surfaces.
The bristle-type models assume that the frictional behavior is represented by an average deflection of elastic springs. These springs have their own stiffness and damping properties and act as a typical spring-damper pair under small velocities; however, once sufficient velocity occurs, the bristles slip resulting in Coulomb-like sliding behavior.

The Lu-Gre model is defined as follows.

$F = \sigma_{0} z + \sigma_{1} \frac{dz}{dt} + \sigma_{2} v$ $\frac{dz}{dt} = v - \frac{\left| v \right| z}{g(v)}$ $g(v) = a_{1} + \frac{a_2}{1 + s^{\alpha}}$ $a_{1} = \frac{\mu_c N}{\sigma_{0}}$ $a_{2} = \frac{\mu_s N - \mu_c N}{\sigma_{0}}$ $s = \frac{\left| v \right|}{v_s}$

where:

$F =$ Friction Force

$N =$ Normal Force

$v =$ Velocity

$\mu_c =$ Coulomb Friction Coefficient

$\mu_s =$ Static Friction Coefficient

$\sigma_{0} =$ Bristle Stiffness

$\sigma_{1} =$ Bristle Damping Coefficient

$\sigma_{2} =$ Viscous Damping Coefficient

$\alpha =$ Stribeck Curve Shape Factor

$v_s =$ Stribeck Velocity Coefficient

Components

Type	Visibility		Name
real(kind=real64),	public	::	coulomb_coefficient	The Coulomb (dynamic) friction coefficient.
real(kind=real64),	public	::	damping	The bristle damping coefficient.
real(kind=real64),	public	::	shape_parameter	The Stribeck curve shape parameter.
real(kind=real64),	public	::	static_coefficient	The static friction coefficient.
real(kind=real64),	public	::	stiffness	The bristle stiffness.
real(kind=real64),	public	::	stribeck_velocity	The Stribeck velocity term.
real(kind=real64),	public	::	viscous_damping	The viscous damping coefficient.

Type-Bound Procedures

procedure, public :: constraint_equations => fmdl_constraints

private subroutine fmdl_constraints(this, t, x, dxdt, nrm, f, rst)

Overload this routine to establish constraings for the model to be enforced as part of the fitting operation.

Arguments

Type	Intent	Attributes		Name
class(friction_model),	intent(in)		::	this	The friction_model object.
real(kind=real64),	intent(in),	dimension(:)	::	t	An N-element array containing the time points at which the data to be fit was sampled.
real(kind=real64),	intent(in),	dimension(:)	::	x	An N-element array containing the relative motion data.
real(kind=real64),	intent(in),	dimension(:)	::	dxdt	An N-element array containing the relative velocity data.
real(kind=real64),	intent(in),	dimension(:)	::	nrm	An N-element array containing the normal force data.
real(kind=real64),	intent(in),	dimension(:)	::	f	An N-element array containing the friction force data.
real(kind=real64),	intent(out),	dimension(:)	::	rst	An M-element array where the results of the constraint equations will be written. M must be equal to the number of constraint equations for the model.

procedure, public :: evaluate => lg_eval

private function lg_eval(this, t, x, dxdt, nrm, svars) result(rst)

Evaluates the Lu-Gre friction model.

Arguments

Type	Intent	Optional	Attributes		Name
class(lugre_model),	intent(inout)			::	this	The lugre_model object.
real(kind=real64),	intent(in)			::	t	The current simulation time value.
real(kind=real64),	intent(in)			::	x	The current value of the relative position between the contacting bodies.
real(kind=real64),	intent(in)			::	dxdt	The current value of the relative velocity between the contacting bodies.
real(kind=real64),	intent(in)			::	nrm	The current normal force between the contacting bodies.
real(kind=real64),	intent(in),	optional,	dimension(:)	::	svars	An optional array containing any internal state variables the model may rely upon.

Return Value real(kind=real64)

The friction force.

procedure, public :: fit => fmdl_fit

private subroutine fmdl_fit(this, t, x, v, f, n, weights, maxp, minp, alpha, integrator, controls, settings, info, stats, fmod, resid, err)

Attempts to fit a friction model to the supplied data using a Levenberg-Marquardt solver.

Arguments

Type	Intent	Optional	Attributes		Name
class(friction_model),	intent(inout),		target	::	this	The friction model. On output, the model is updated with the final, fitted parameters.
real(kind=real64),	intent(in),		target, dimension(:)	::	t	An N-element array containing the time points at which the friction data was sampled. This array must contain monotonically increasing data.
real(kind=real64),	intent(in),		target, dimension(:)	::	x	An N-element array containing the relative position data.
real(kind=real64),	intent(in),		target, dimension(:)	::	v	An N-element array containing the relative velocity data.
real(kind=real64),	intent(in),		target, dimension(:)	::	f	An N-element array containing the friction force data.
real(kind=real64),	intent(in),		target, dimension(:)	::	n	An N-element array containing the normal force data.
real(kind=real64),	intent(in),	optional,	dimension(:)	::	weights	An optional N-element array that can be used to weight specific data points. The default is an array of all ones such that all points are weighted equally.
real(kind=real64),	intent(in),	optional,	dimension(:)	::	maxp	An M-element array (M = the number of model parameters) containing a maximum limit for each model parameter.
real(kind=real64),	intent(in),	optional,	dimension(:)	::	minp	An M-element array containing the minimum limit for each model parameter.
real(kind=real64),	intent(in),	optional		::	alpha	An optional input that defines the significance level at which to evaluate the confidence intervals. The default value is 0.05 such that a 95% confidence interval is calculated.
class(ode_integrator),	intent(inout),	optional,	target	::	integrator	An optional input, used in the event the model has internal state variables, that provides integration of the state equations. The defaults is a 4th order Rosenbrock method.
type(iteration_controls),	intent(in),	optional		::	controls	An optional input providing custom iteration controls.
type(lm_solver_options),	intent(in),	optional		::	settings	An optional input providing custom settings for the solver.
type(convergence_info),	intent(out),	optional		::	info	An optional output that can be used to gain information about the iterative solution and the nature of the convergence.
type(regression_statistics),	intent(out),	optional,	dimension(:)	::	stats	An optional output array of M-elements that can be used to retrieve statistical information regarding the fit of each of the M model parameters.
real(kind=real64),	intent(out),	optional,	target, dimension(:)	::	fmod	An optional N-element array used to provide the fitted model results.
real(kind=real64),	intent(out),	optional,	target, dimension(:)	::	resid	An optional N-element array containing the fitted residuals.
class(errors),	intent(inout),	optional,	target	::	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling.

procedure, public :: from_array => lg_from_array

private subroutine lg_from_array(this, x, err)

Converts an array into the parameters for the friction model.

Arguments

Type	Intent	Optional	Attributes		Name
class(lugre_model),	intent(inout)			::	this	The lugre_model object.
real(kind=real64),	intent(in),		dimension(:)	::	x	The array of parameters. See parameter_count to determine the size of this array. The parameter order is as follows: static_coefficient coulomb_coefficient stribeck_velocity stiffness damping viscous_damping shape_parameter
class(errors),	intent(inout),	optional,	target	::	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling.

procedure, public :: get_constraint_equation_count => fmdl_get_constraint_count

private pure function fmdl_get_constraint_count(this) result(rst)

Gets the number of constraint equations the model requires to be satisfied when fitting to data.

Arguments

Type	Intent	Optional	Attributes		Name
class(friction_model),	intent(in)			::	this	The friction_model object.

Return Value integer(kind=int32)

The number of constraint equations.

procedure, public :: get_state_variable_count => lg_get_state_var_count

private pure function lg_get_state_var_count(this) result(rst)

Gets the number of internal state variables used by the model.

Arguments

Type	Intent	Optional	Attributes		Name
class(lugre_model),	intent(in)			::	this	The lugre_model object.

Return Value integer(kind=int32)

The internal state variable count.

procedure, public :: has_internal_state => lg_has_state_vars

private pure function lg_has_state_vars(this) result(rst)

Returns a value stating if the model relies upon internal state variables.

Arguments

Type	Intent	Optional	Attributes		Name
class(lugre_model),	intent(in)			::	this	The lugre_model object.

Return Value logical

Returns true if the model utilizes internal state variables; else, returns false.

procedure, public :: parameter_count => lg_parameter_count

private pure function lg_parameter_count(this) result(rst)

Gets the number of model parameters.

Arguments

Type	Intent	Optional	Attributes		Name
class(lugre_model),	intent(in)			::	this	The lugre_model object.

Return Value integer(kind=int32)

The number of model parameters.

procedure, public :: reset => fmdl_reset

private subroutine fmdl_reset(this)

Resets the friction model to it's original state.

Arguments

Type	Intent	Optional	Attributes		Name
class(friction_model),	intent(inout)			::	this	The friction_model object.

procedure, public :: state => lg_state_model

private subroutine lg_state_model(this, t, x, dxdt, nrm, svars, dsdt)

Evaluates the time derivatives of the internal friction state model.

Arguments

Type	Intent	Attributes		Name
class(lugre_model),	intent(inout)		::	this	The lugre_model object.
real(kind=real64),	intent(in)		::	t	The current simulation time value.
real(kind=real64),	intent(in)		::	x	The current value of the relative position between the contacting bodies.
real(kind=real64),	intent(in)		::	dxdt	The current value of the relative velocity between the contacting bodies.
real(kind=real64),	intent(in)		::	nrm	The current normal force between the contacting bodies.
real(kind=real64),	intent(in),	dimension(:)	::	svars	An N-element array containing any internal state variables the model may rely upon.
real(kind=real64),	intent(out),	dimension(:)	::	dsdt	An N-element array where the state variable derivatives are to be written.

procedure, public :: to_array => lg_to_array

private subroutine lg_to_array(this, x, err)

Converts the parameters of the friction model into an array.

Arguments

Type	Intent	Optional	Attributes		Name
class(lugre_model),	intent(in)			::	this	The lugre_model object.
real(kind=real64),	intent(out),		dimension(:)	::	x	The array used to store the parameters. See @ref parameter_count to determine the size of this array. The parameter order is as follows: static_coefficient coulomb_coefficient stribeck_velocity stiffness damping viscous_damping shape_parameter
class(errors),	intent(inout),	optional,	target	::	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling.

lugre_model Derived Type

Contents

Variables

Type-Bound Procedures

type, public, extends(friction_model) :: lugre_model

Components

Type-Bound Procedures

procedure, public :: constraint_equations => fmdl_constraints

private subroutine fmdl_constraints(this, t, x, dxdt, nrm, f, rst)

Arguments

procedure, public :: evaluate => lg_eval

private function lg_eval(this, t, x, dxdt, nrm, svars) result(rst)

Arguments

Return Value real(kind=real64)

procedure, public :: fit => fmdl_fit

private subroutine fmdl_fit(this, t, x, v, f, n, weights, maxp, minp, alpha, integrator, controls, settings, info, stats, fmod, resid, err)

Arguments

procedure, public :: from_array => lg_from_array

private subroutine lg_from_array(this, x, err)

Arguments

procedure, public :: get_constraint_equation_count => fmdl_get_constraint_count

private pure function fmdl_get_constraint_count(this) result(rst)

Arguments

Return Value integer(kind=int32)

procedure, public :: get_state_variable_count => lg_get_state_var_count

private pure function lg_get_state_var_count(this) result(rst)

Arguments

Return Value integer(kind=int32)

procedure, public :: has_internal_state => lg_has_state_vars

private pure function lg_has_state_vars(this) result(rst)

Arguments

Return Value logical

procedure, public :: parameter_count => lg_parameter_count

private pure function lg_parameter_count(this) result(rst)

Arguments

Return Value integer(kind=int32)

procedure, public :: reset => fmdl_reset

private subroutine fmdl_reset(this)

Arguments

procedure, public :: state => lg_state_model

private subroutine lg_state_model(this, t, x, dxdt, nrm, svars, dsdt)

Arguments

procedure, public :: to_array => lg_to_array

private subroutine lg_to_array(this, x, err)

Arguments