# Class LagrangianDS¶

class LagrangianDS : public SecondOrderDS

Lagrangian non linear dynamical systems - $$M(q,z) \dot v = F(v, q, t, z) + p$$.

This class defines and computes a generic ndof-dimensional Lagrangian Non Linear Dynamical System of the form :

$\begin{split} M(q,z) \dot v + F_{gyr}(v, q, z) + F_{int}(v , q , t, z) = F_{ext}(t, z) + p \\ \dot q = v \end{split}$

where

• $$q \in R^{ndof}$$ is the set of the generalized coordinates,

• $$\dot q =v \in R^{ndof}$$ the velocity, i. e. the time derivative of the generalized coordinates (Lagrangian systems).

• $$\ddot q =\dot v \in R^{ndof}$$ the acceleration, i. e. the second time derivative of the generalized coordinates.

• $$p \in R^{ndof}$$ the reaction forces due to the Non Smooth Interaction.

• $$M(q) \in R^{ndof \times ndof}$$ is the inertia term (access : mass() method).

• $$F_{gyr}(\dot q, q) \in R^{ndof}$$ is the non linear inertia term (access fGyr() method).

• $$F_{int}(\dot q , q , t) \in R^{ndof}$$ are the internal forces (access fInt() method).

• $$F_{ext}(t) \in R^{ndof}$$ are the external forces (access fExt() method).

• $$z \in R^{zSize}$$ is a vector of arbitrary algebraic variables, some sort of discrete state.

The equation of motion is also shortly denoted as $$M(q,z) \dot v = F(v, q, t, z) + p$$

where $$F(v, q, t, z) \in R^{ndof}$$ collects the total forces acting on the system, that is $$F(v, q, t, z) = F_{ext}(t, z) - F_{gyr}(v, q, z) + F_{int}(v, q , t, z)$$

This vector is saved and may be accessed using forces() method.

q[i] is the derivative number i of q. Thus: q[0]= $$q$$, global coordinates, q[1]= $$\dot q$$ , velocity, q[2]= $$\ddot q$$, acceleration.

The following operators (and their jacobians) can be plugged, in the usual way (see User Guide, ‘User-defined plugins’)

If required (e.g. for Event-Driven like simulation), formulation as a first-order system is also available, and writes:

• $$n= 2 ndof$$

• $$x = \left[\begin{array}{c}q \\ \dot q\end{array}\right]$$

• rhs given by:

$\begin{split} \dot x = \left[\begin{array}{c} \dot q\\ \ddot q = M^{-1}(q)\left[F(v, q , t, z) + p \right]\\ \end{array}\right] \end{split}$

• jacobian of the rhs, with respect to x

$\begin{split} \nabla_{x}rhs(x,t) = \left[\begin{array}{cc} 0 & I \\ \nabla_{q}(M^{-1}(q)F(v, q , t, z)) & \nabla_{\dot q}(M^{-1}(q)F(v, q , t, z)) \\ \end{array}\right] \end{split}$

with the input due to the non smooth law:

$\begin{split} \left[\begin{array}{c} 0 \\ p \end{array}\right] \end{split}$

In that case, use the following methods:

Subclassed by LagrangianLinearDiagonalDS, LagrangianLinearTIDS

Public Functions

LagrangianDS(SP::SiconosVector position, SP::SiconosVector velocity)

constructor from initial state only, $$dv = p$$

Parameters
LagrangianDS(SP::SiconosVector position, SP::SiconosVector velocity, SP::SiconosMatrix mass)

constructor from initial state and mass, $$Mdv = p$$

Parameters
LagrangianDS(SP::SiconosVector position, SP::SiconosVector velocity, const std::string &plugin)

constructor from initial state and mass (plugin) $$Mdv = p$$

Parameters
inline virtual ~LagrangianDS()

destructor

virtual void resetToInitialState() override

reset the state to the initial state

virtual void initRhs(double time) override

allocate (if needed) and compute rhs and its jacobian.

Parameters

time – of initialization

virtual void initializeNonSmoothInput(unsigned int level) override

set nonsmooth input to zero

Parameters

level – input-level to be initialized.

virtual void computeRhs(double time) override

update right-hand side for the current state

Parameters

time – of interest

virtual void computeJacobianRhsx(double time) override

update $$\nabla_x rhs$$ for the current state

Parameters

time – of interest

virtual void resetAllNonSmoothParts() override

reset non-smooth part of the rhs (i.e.

p), for all ‘levels’

virtual void resetNonSmoothPart(unsigned int level) override

set nonsmooth part of the rhs (i.e.

p) to zero for a given level

Parameters

level

inline virtual void setRhs(const SiconosVector &newValue) override

set the value of the right-hand side, $$\dot x$$

Parameters

newValueSiconosVector

inline virtual void setRhsPtr(SP::SiconosVector newPtr) override

set right-hand side, $$\dot x$$ (pointer link)

Parameters

newPtr – SP::SiconosVector

virtual void computeForces(double time, SP::SiconosVector q, SP::SiconosVector velocity) override

Compute $$F(v,q,t,z)$$.

Parameters
• time – the current time

• q – SP::SiconosVector: pointers on q

• velocity – SP::SiconosVector: pointers on velocity

virtual void computeJacobianqForces(double time) override

Compute $$\nabla_qF(v,q,t,z)$$ for current $$q,v$$ Default function to compute forces.

Parameters

time – the current time

inline virtual void computeJacobianqDotForces(double time)

Compute $$\nabla_{\dot q}F(v,q,t,z)$$ for current $$q,v$$.

Parameters

time – the current time

virtual void computeJacobianvForces(double time) override

Compute $$\nabla_{\dot q}F(v,q,t,z)$$ for current $$q,v$$.

Parameters

time – the current time

inline virtual SP::SiconosVector q() const override

generalized coordinates of the system (vector of size dimension())

Returns

pointer on a SiconosVector

virtual void setQ(const SiconosVector &newValue) override

set value of generalized coordinates vector (copy)

Parameters

newValue

virtual void setQPtr(SP::SiconosVector newPtr) override

set value of generalized coordinates vector (pointer link)

Parameters

newPtr

virtual void setQ0(const SiconosVector &newValue) override

set initial state (copy)

Parameters

newValue

virtual void setQ0Ptr(SP::SiconosVector newPtr) override

Parameters

newPtr

inline virtual SP::SiconosVector velocity() const override

Returns

pointer on a SiconosVector

virtual void setVelocity(const SiconosVector &newValue) override

set velocity vector (copy)

Parameters

newValue

virtual void setVelocityPtr(SP::SiconosVector newPtr) override

Parameters

newPtr

inline virtual SP::SiconosVector velocity0() const override

get initial velocity (pointer)

Returns

pointer on a SiconosVector

virtual void setVelocity0(const SiconosVector &newValue) override

set initial velocity (copy)

Parameters

newValue

virtual void setVelocity0Ptr(SP::SiconosVector newPtr) override

Parameters

newPtr

inline virtual SP::SiconosVector acceleration() const override

Returns

pointer on a SiconosVector

inline SP::SiconosVector fInt() const

get $F_{int}$ (pointer link)

Returns

pointer on a plugged vector

inline void setFIntPtr(SP::SiconosVector newPtr)

set $F_{int}$ (pointer link)

Parameters

newPtr – a SP to plugged vector

inline SP::SiconosVector fExt() const

get $$F_{ext}$$ , (pointer link)

Returns

pointer on a plugged vector

inline void setFExtPtr(SP::SiconosVector newPtr)

set $$F_{ext}$$ , (pointer link)

Parameters

newPtr – a SP to a Simple vector

inline SP::SiconosVector fGyr() const

get $$F_{gyr}$$ , (pointer link)

Returns

pointer on a plugged vector

inline void setFGyrPtr(SP::SiconosVector newPtr)

set $$F_{gyr}$$ , (pointer link)

Parameters

newPtr – a SP to plugged vector

inline SP::SiconosMatrix jacobianFIntq() const

get $$\nabla_qF_{int}$$ , (pointer link)

Returns

pointer on a SiconosMatrix

inline SP::SiconosMatrix jacobianFIntqDot() const

get $$\nabla_{\dot q}F_{int}$$ , (pointer link)

Returns

pointer on a SiconosMatrix

inline void setJacobianFIntqPtr(SP::SiconosMatrix newPtr)

set $$\nabla_{q}F_{int}$$ , (pointer link)

Parameters

newPtr – a pointer to a SiconosMatrix

inline void setJacobianFIntqDotPtr(SP::SiconosMatrix newPtr)

set $$\nabla_{\dot q}F_{int}$$ , (pointer link)

Parameters

newPtr – a pointer to a SiconosMatrix

inline SP::SiconosMatrix jacobianFGyrq() const

get $$\nabla_{q}F_{gyr}$$ , (pointer link)

Returns

pointer on a SiconosMatrix

inline SP::SiconosMatrix jacobianFGyrqDot() const

get $$\nabla_{\dot q}F_{gyr}$$ , (pointer link)

Returns

pointer on a SiconosMatrix

inline void setJacobianFGyrqPtr(SP::SiconosMatrix newPtr)

get $$\nabla_{q}F_{gyr}$$ , (pointer link)

Parameters

newPtr – a SP SiconosMatrix

inline void setJacobianFGyrqDotPtr(SP::SiconosMatrix newPtr)

get $$\nabla_{\dot q}F_{gyr}$$ , (pointer link)

Parameters

newPtr – a SP SiconosMatrix

inline virtual SP::SiconosVector forces() const override

get $$F(v,q,t,z)$$ (pointer link)

Returns

pointer on a SiconosVector

inline virtual SP::SiconosMatrix jacobianqForces() const override

get $$\nabla_qF(v,q,t,z)$$ (pointer link)

Returns

pointer on a SiconosMatrix

inline virtual SP::SiconosMatrix jacobianvForces() const override

get $$\nabla_{\dot q}F(v,q,t,z)$$ (pointer link)

Returns

pointer on a SiconosMatrix

inline virtual const SiconosMemory &qMemory() override

get all the values of the state vector q stored in memory.

note: not const due to SchatzmanPaoliOSI::initializeWorkVectorsForDS

Returns

a memory

inline virtual const SiconosMemory &velocityMemory() override

get all the values of the state vector velocity stored in memory.

note: not const due to SchatzmanPaoliOSI::initializeWorkVectorsForDS

Returns

a memory

inline const SiconosMemory &pMemory(unsigned int level)

get all the values of the state vector p stored in memory

Parameters

level

Returns

a memory

inline virtual const SiconosMemory &forcesMemory() override

get forces in memory buff

Returns

pointer on a SiconosMemory

virtual void initMemory(unsigned int size) override

initialize the SiconosMemory objects with a positive size.

Parameters

size – the size of the SiconosMemory. must be >= 0

virtual void swapInMemory() override

push the current values of x, q and r in the stored previous values xMemory, qMemory, rMemory,

Todo:

Modify the function swapIn Memory with the new Object Memory

inline void setComputeMassFunction(const std::string &pluginPath, const std::string &functionName)

allow to set a specified function to compute the mass

Parameters
• pluginPath – std::string : the complete path to the plugin

• functionName – std::string : the name of the function to use in this plugin

inline void setComputeMassFunction(FPtr7 fct)

set a specified function to compute Mass

Parameters

fct – a pointer on the plugin function

void setComputeFIntFunction(const std::string &pluginPath, const std::string &functionName)

allow to set a specified function to compute FInt

Parameters
• pluginPath – std::string : the complete path to the plugin

• functionName – std::string : the name of the function to use in this plugin

void setComputeFIntFunction(FPtr6 fct)

set a specified function to compute fInt

Parameters

fct – a pointer on the plugin function

inline void setComputeFExtFunction(const std::string &pluginPath, const std::string &functionName)

allow to set a specified function to compute Fext

Parameters
• pluginPath – std::string : the complete path to the plugin

• functionName – std::string : the name of the function to use in this plugin

inline void setComputeFExtFunction(VectorFunctionOfTime fct)

set a specified function to compute fExt

Parameters

fct – a pointer on the plugin function

void setComputeFGyrFunction(const std::string &pluginPath, const std::string &functionName)

allow to set a specified function to compute the inertia

Parameters
• pluginPath – std::string : the complete path to the plugin

• functionName – std::string : the name of the function to use in this plugin

void setComputeFGyrFunction(FPtr5 fct)

set a specified function to compute FGyr

Parameters

fct – a pointer on the plugin function

void setComputeJacobianFIntqFunction(const std::string &pluginPath, const std::string &functionName)

allow to set a specified function to compute the jacobian w.r.t q of the internal forces

Parameters
• pluginPath – std::string : the complete path to the plugin

• functionName – std::string : the name of the function to use in this plugin

void setComputeJacobianFIntqDotFunction(const std::string &pluginPath, const std::string &functionName)

allow to set a specified function to compute the jacobian of the internal forces w.r.t.

q

Parameters
• pluginPath – std::string : the complete path to the plugin

• functionName – std::string : the name of the function to use in this plugin

void setComputeJacobianFIntqFunction(FPtr6 fct)

set a specified function to compute jacobian following q of the FInt

Parameters

fct – a pointer on the plugin function

void setComputeJacobianFIntqDotFunction(FPtr6 fct)

set a specified function to compute jacobian following qDot of the FInt

Parameters

fct – a pointer on the plugin function

void setComputeJacobianFGyrqFunction(const std::string &pluginPath, const std::string &functionName)

allow to set a specified function to compute the jacobian w.r.t q of the the external forces

Parameters
• pluginPath – std::string : the complete path to the plugin

• functionName – std::string : the name of the function to use in this plugin

void setComputeJacobianFGyrqDotFunction(const std::string &pluginPath, const std::string &functionName)

allow to set a specified function to compute the jacobian w.r.t qDot of the the external strength

Parameters
• pluginPath – std::string : the complete path to the plugin

• functionName – std::string : the name of the function to use in this plugin

void setComputeJacobianFGyrqFunction(FPtr5 fct)

set a specified function to compute the jacobian following q of FGyr

Parameters

fct – a pointer on the plugin function

void setComputeJacobianFGyrqDotFunction(FPtr5 fct)

set a specified function to compute the jacobian following qDot of FGyr

Parameters

fct – a pointer on the plugin function

virtual void computeMass() override

default function to compute the mass

virtual void computeMass(SP::SiconosVector position) override

function to compute the mass

Parameters

position – value used to evaluate the mass matrix

virtual void computeFInt(double time)

default function to compute the internal strengths

Parameters

time – the current time

virtual void computeFInt(double time, SP::SiconosVector position, SP::SiconosVector velocity)

function to compute the internal strengths with some specific values for position and velocity (ie not those of the current state).

Parameters
• time – the current time,

• position – value used to evaluate the internal forces

• velocity – value used to evaluate the internal forces

virtual void computeFExt(double time)

default function to compute the external strengths

Parameters

time – the current time

virtual void computeFGyr()

default function to compute the inertia

virtual void computeFGyr(SP::SiconosVector position, SP::SiconosVector velocity)

function to compute the inertia with some specific values for q and velocity (ie not those of the current state).

Parameters
• position – value used to evaluate the inertia forces

• velocity – value used to evaluate the inertia forces

virtual void computeJacobianFIntq(double time)

To compute the jacobian w.r.t q of the internal forces.

Parameters

time – the current time

virtual void computeJacobianFIntqDot(double time)

To compute the jacobian w.r.t qDot of the internal forces.

Parameters

time – the current time

virtual void computeJacobianFIntq(double time, SP::SiconosVector position, SP::SiconosVector velocity)

To compute the jacobian w.r.t q of the internal forces.

Parameters
• time – the current time

• position – value used to evaluate the jacobian

• velocity – value used to evaluate the jacobian

virtual void computeJacobianFIntqDot(double time, SP::SiconosVector position, SP::SiconosVector velocity)

To compute the jacobian w.r.t.

qDot of the internal forces

Parameters
• time – the current time

• position – value used to evaluate the jacobian

• velocity – value used to evaluate the jacobian

virtual void computeJacobianFGyrq()

function to compute the jacobian w.r.t.

q of the inertia forces

virtual void computeJacobianFGyrqDot()

function to compute the jacobian w.r.t.

qDot of the inertia forces

virtual void computeJacobianFGyrq(SP::SiconosVector position, SP::SiconosVector velocity)

function to compute the jacobian w.r.t.

q of the inertia forces

Parameters
• position – value used to evaluate the jacobian

• velocity – value used to evaluate the jacobian

virtual void computeJacobianFGyrqDot(SP::SiconosVector position, SP::SiconosVector velocity)

function to compute the jacobian w.r.t.

qDot of the inertia forces

Parameters
• position – value used to evaluate the jacobian

• velocity – value used to evaluate the jacobian

inline virtual void updatePlugins(double time) override

default function to update the plugins functions using a new time:

Parameters

time – the current time

double computeKineticEnergy()

To compute the kinetic energy.

virtual void display(bool brief = true) const override

print the data of the dynamical system on the standard output

void computePostImpactVelocity()

Computes post-impact velocity, using pre-impact velocity and impulse (p) value.

void init_generalized_coordinates(unsigned int level)

Allocate memory for q[level], level > 1 Useful for some integrators that need q[2] or other coordinates vectors.

Parameters

level – the required level

virtual void init_inverse_mass() override

Allocate memory for the lu factorization of the mass of the system.

Useful for some integrators with system inversion involving the mass

virtual void update_inverse_mass() override

Update the content of the lu factorization of the mass of the system, if required.

virtual void init_forces() override

Allocate memory for forces and its jacobian.