The class constructor function.

Parameters

model	the dynamical system model in which the virtual constraints are defined
ya	the symbolic expression of the actual outputs
name	the name of the virtual constraints
varargin	optional parameters. In details VirtualConstraint ( ..., DesiredType, PolyDegree, RelativeDegree, PhaseType, Holonomic, OutputLabel ) Required Parameters for varargin: DesiredType PolyDegree RelativeDegree PhaseType Holonomic OutputLabel

Required fields of model:

calculate the actual outputs

Note

: the last Jacobian output is the Jacobian of the dNya with respoect to the states, which is [x] for a first order system and [x;dx] for a second order system.

Parameters

x	the states
dx	the first order derivatives

Return values

ya	variable outputs in the following order: the actual output value
d1ya	the first order derivative ...
dNya	the N = RelativeDegree - 1 order derivative
JdNya	the Jacobian of dNya w.r.t. to the states

calculate the desired outputs

Note

: the last Jacobian output is the Jacobian of the dNyd with respoect to the states, which is [x] for a first order system and [x;dx] for a second order system.

Parameters

t	the time
x	the states
dx	the first order derivatives
a	the parameters of the desired output
p	the parameters of the phase variable

Return values

yd	variable outputs in the following order: the actual output value
d1yd	the first order derivative ...
dNyd	the N = RelativeDegree - 1 order derivative
JdNyd	the Jacobian of dNya w.r.t. to the states

calculate the phase variable

Note

: the last Jacobian output is the Jacobian of the dNya with respoect to the states, which is [x] for a first order system and [x;dx] for a second order system.

Parameters

t	the time
x	the states
dx	the first order derivatives
p	the parameter set

Return values

tau	variable outputs in the following order: the phase variable value
d1tau	the first order derivative ...
dNtau	the N = RelativeDegree - 1 order derivative
JdNtau	the Jacobian of dNya w.r.t. to the states

compiles the derivatives and Jacobian of the actual/desired outputs functions

Generated fields of obj:

export the symbolic expressions of the constraints matrices and vectors and compile as MEX files.

Parameters

export_path	the path to export the file
varargin	variable input parameters export ( ..., Vars, File, ForceExport, BuildMex, Namespace ) Required Parameters for varargin: Vars a list of symbolic variables File the (full) file name of exported file ForceExport force the export BuildMex flag whether to MEX the exported file Namespace the namespace of the function

impose virtual constraints as NLP constraints in the trajectory optimization problem nlp of the dynamical system

For each (vector) virtual constraints, we will enforce the following

constraints at the first node

{y, y', ..., y^N} % where N = RelativeDegree - 1

and this constraint

y^(N+1) + ep_N y^N + ... ep_1 y = 0 at all nodes.

Note

For example, ep = [kp, kd] for typical vector relative degree 2 holonomic virtual constraints.

For example, nzy = [0] for a relative degree 1 nonholonomic virtual constraint (velocity outputs) means we do not require y to be zero at the first node. Another example, nzy = [0,0] for a relative degree 2 virtual constraints, we do not require both y and y' to be zero at the first node, on the other hand, nzy = [1,0] for a relative degree 2 virtual constraints, we only enforce y being zero at the first node.

Parameters

nlp	the trajectory optimization NLP
ep	the coefficients of the derivatives
nzy	the derivatives that does not requires to be zero at the first node

Required fields of nlp:

save the symbolic expressions of the constraints matrices to MX binary files

Parameters

export_path	the path to export the file
varargin	variable input parameters saveExpression ( ..., ForceExport ) Required Parameters for varargin: ForceExport force the export

DesiredType sets the function type of the desired outputs.

Parameters

type	the function type
degree	the degree of polynomials

Generated fields of obj:

Holonomic sets the type of whether the virtual constraints are holonomic or nonholonomic.

Parameters

type	true for holonomic, false for nonholonomic

Generated fields of obj:

OutputLabel sets the naming labels of outputs.

Parameters

label

the cell array of labels

Generated fields of obj:

PhaseType sets the type of the phase variable.

Parameters

type	type of the phase variable

Generated fields of obj:

Phase variale sets the symbolic expression of the state-based timing phase variable.

Parameters

tau	the phase variable
p	the parameters of tau

Generated fields of obj:

RelativeDegree sets the relative degree of the virtual constraints.

Parameters

degree

relative degree of outputs

Generated fields of obj:

Name.

The actual outputs.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The actual outputs.

The actual outputs.

Note

This property has the MATLAB attribute Dependent set to true.

Matlab documentation of property attributes.

[readonly]

The desired outputs.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The desired outputs.

The desired outputs.

Note

This property has the MATLAB attribute Dependent set to true.

Matlab documentation of property attributes.

[readonly]

The type of the desired output function.

It could be one of the following functions

{Bezier,CWF,ECWF,MinJerk,Constant}

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The dimension of the virtual constraints.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

An indicator that shows there is an offset in the actual outputs.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.
Default: false

An indicator that shows there is a parameter variable for phase.

Default: false

Indicates whether the virtual constraint is holonomic or nonholonomic constraints.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The dynamical system model.

The name of the virtual constraints.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The name of the offset parameter variable.

Note

This property has the MATLAB attribute Dependent set to true.

Matlab documentation of property attributes.

[readonly]

The symbolic representation of offset of the actual outputs: ya = ya_orig + offset_params.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The virtual constraints ya - yd functions.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The virtual constraints ya - yd functions.

The label of the virtual constraint.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The name of the output parameter variable.

Note

This property has the MATLAB attribute Dependent set to true.

Matlab documentation of property attributes.

[readonly]

The symbolic representation of parameter sets of the desired outputs.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The phase variable.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The phase variable.

The name of the phase parameter variable.

Note

This property has the MATLAB attribute Dependent set to true.

Matlab documentation of property attributes.

[readonly]

The symbolic representation of parameters of the phase variable.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The type of the phase variable.

It could be either StateBased or TimeBased

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The phase variable.

Note

This property has the MATLAB attribute Dependent set to true.

Matlab documentation of property attributes.

[readonly]

The maximum degree of the polynomial function.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The relative degree of the output.

Note

This property has non-standard access specifiers: SetAccess = Protected, GetAccess = Public

Matlab documentation of property attributes.

The phase variables.

The actual outputs.

The desired outputs.