Drake
SpatialMomentum< T > Class Template Reference

## Detailed Description

### template<typename T> class drake::multibody::SpatialMomentum< T >

This class is used to represent the spatial momentum of a particle, system of particles or body (whether rigid or soft.) The linear momentum l_NS of a system of particles S in a reference frame N is defined by:

  l_NS = ∑l_NQi = ∑mᵢv_NQi


where mᵢ and v_NQi are the mass and linear velocity (in frame N) of the i-th particle in the system, respectively. Their product l_NQi = mᵢv_NQi is the linear momentum of the i-th particle in the N reference frame. The angular momentum h_NSp of a system of particles S in a reference frame N about an arbitrary point P is defined by:

  h_NSp = ∑ p_PQi x l_NQi


where p_PQi is the position vector from point P to the i-th particle position Qi. The definitions above extend to a continuum of particles as:

  h_NSp = ∫p_PQ(r) x v_NQ(r) ρ(r)d³r
l_NS = ∫v_NQ(r) ρ(r)d³r


where ρ(r) is the density of the body at each material location r. In particular, the continuum version above also applies to rigid bodies.

Spatial momenta are elements of F⁶ (see [Featherstone 2008]) that combine both rotational (angular momentum) and translational (linear momentum) components. Spatial momenta are 6-element quantities that are pairs of ordinary 3-vectors. Elements 0-2 are the angular momentum component while elements 3-5 are the linear momentum component. As with any other spatial vector, both vector components must be expressed in the same frame.

Neither the expressed-in frame nor the about-point are stored with a SpatialMomentum object; they must be understood from context. It is the responsibility of the user to keep track of the about-point and the expressed-in frame. That is best accomplished through disciplined notation. In source code we use monogram notation where L is used to designate a spatial momentum quantity. We write a point P fixed to body (or frame) B as $$B_P$$ which appears in code and comments as Bp. Then we write as $$[^NL^{S/B_P}]_E$$, which appears in code as L_NBp_E, the spatial momentum of a body B in a reference frame N, about a point P and, expressed in frame E. Very often the about-point will be the body origin Bo; if no point is shown the origin is understood, thus L_NB_E means L_NBo_E. For a more detailed introduction on spatial vectors and the monogram notation please refer to section Spatial Vectors.

• [Featherstone 2008] Featherstone, R., 2008. Rigid body dynamics algorithms. Springer.
Template Parameters
 T The underlying scalar type. Must be a valid Eigen scalar.

#include <drake/multibody/math/spatial_velocity.h>

## Public Member Functions

SpatialMomentum ()
Default constructor. More...

SpatialMomentum (const Eigen::Ref< const Vector3< T >> &h, const Eigen::Ref< const Vector3< T >> &l)
SpatialMomentum constructor from an angular momentum h and a linear momentum l. More...

template<typename Derived >
SpatialMomentum (const Eigen::MatrixBase< Derived > &L)
SpatialMomentum constructor from an Eigen expression that represents a six-dimensional vector. More...

SpatialMomentum< T > & ShiftInPlace (const Vector3< T > &p_PQ_E)
In-place shift of a SpatialMomentum from one "about-point" to another. More...

SpatialMomentum< T > Shift (const Vector3< T > &p_PQ_E) const
Shift of a SpatialMomentum from one application point to another. More...

SpatialMomentum< T > & operator+= (const SpatialMomentum< T > &L_NSp_E)
Adds in a spatial momentum to this spatial momentum. More...

SpatialMomentum< T > & operator-= (const SpatialMomentum< T > &L_NSp_E)
Subtracts a spatial momentum from this spatial momentum. More...

dot (const SpatialVelocity< T > &V_NBp_E) const
Given this spatial momentum L_NBp_E of a rigid body B, about point P and, expressed in a frame E, this method computes the dot product with the spatial velocity V_NBp_E of body B frame shifted to point P, measured in an inertial (or Newtonian) frame N and expressed in the same frame E in which the spatial momentum is expressed. More...

Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
SpatialMomentum (const SpatialMomentum &)=default

SpatialMomentumoperator= (const SpatialMomentum &)=default

SpatialMomentum (SpatialMomentum &&)=default

SpatialMomentumoperator= (SpatialMomentum &&)=default

Public Member Functions inherited from SpatialVector< SpatialMomentum, T >
SpatialVector ()
Default constructor. More...

SpatialVector (const Eigen::Ref< const Vector3< T >> &w, const Eigen::Ref< const Vector3< T >> &v)
SpatialVector constructor from an rotational component w and a linear component v. More...

SpatialVector (const Eigen::MatrixBase< OtherDerived > &V)
SpatialVector constructor from an Eigen expression that represents a six-dimensional vector. More...

int size () const
The total size of the concatenation of the angular and linear components. More...

const T & operator[] (int i) const
Const access to the i-th component of this spatial vector. More...

T & operator[] (int i)
Mutable access to the i-th component of this spatial vector. More...

const Vector3< T > & rotational () const
Const access to the rotational component of this spatial vector. More...

Vector3< T > & rotational ()
Mutable access to the rotational component of this spatial vector. More...

const Vector3< T > & translational () const
Const access to the translational component of this spatial vector. More...

Vector3< T > & translational ()
Mutable access to the translational component of this spatial vector. More...

const T * data () const
Returns a (const) bare pointer to the underlying data. More...

T * mutable_data ()
Returns a (mutable) bare pointer to the underlying data. More...

std::tuple< const T, const T > GetMaximumAbsoluteDifferences (const SpatialQuantity &other) const
Returns the maximum absolute values of the differences in the rotational and translational components of this and other (i.e., the infinity norms of the difference in rotational and translational components). More...

decltype(T()< T()) IsNearlyEqualWithinAbsoluteTolerance (const SpatialQuantity &other, double rotational_tolerance, double translational_tolerance) const
Compares the rotational and translational parts of this and other to check if they are the same to within specified absolute differences. More...

decltype(T()< T()) IsApprox (const SpatialQuantity &other, double tolerance=std::numeric_limits< double >::epsilon()) const
Compares this spatial vector to the provided spatial vector other within a specified tolerance. More...

void SetNaN ()
Sets all entries in this SpatialVector to NaN. More...

SpatialQuantitySetZero ()
Sets both rotational and translational components of this SpatialVector to zero. More...

CoeffsEigenTypeget_coeffs ()
Returns a reference to the underlying storage. More...

const CoeffsEigenTypeget_coeffs () const
Returns a constant reference to the underlying storage. More...

SpatialQuantity operator- () const
Unary minus operator. More...

SpatialVector (const SpatialVector &)=default

SpatialVector (SpatialVector &&)=default

SpatialVectoroperator= (const SpatialVector &)=default

SpatialVectoroperator= (SpatialVector &&)=default

## Additional Inherited Members

Public Types inherited from SpatialVector< SpatialMomentum, T >
enum
Sizes for spatial quantities and its components in three dimensions. More...

using SpatialQuantity = SpatialMomentum< T >
The more specialized spatial vector class templated on the scalar type T. More...

typedef Vector6< T > CoeffsEigenType
The type of the underlying in-memory representation using an Eigen vector. More...

typedef T ScalarType

Static Public Member Functions inherited from SpatialVector< SpatialMomentum, T >
static SpatialQuantity Zero ()
Factory to create a zero SpatialVector, i.e. More...

## ◆ SpatialMomentum() [1/5]

 SpatialMomentum ( const SpatialMomentum< T > & )
default

## ◆ SpatialMomentum() [2/5]

 SpatialMomentum ( SpatialMomentum< T > && )
default

## ◆ SpatialMomentum() [3/5]

 SpatialMomentum ( )

Default constructor.

In Release builds the elements of the newly constructed spatial momentum are left uninitialized resulting in a zero cost operation. However in Debug builds those entries are set to NaN so that operations using this uninitialized spatial momentum fail fast, allowing fast bug detection.

## ◆ SpatialMomentum() [4/5]

 SpatialMomentum ( const Eigen::Ref< const Vector3< T >> & h, const Eigen::Ref< const Vector3< T >> & l )

SpatialMomentum constructor from an angular momentum h and a linear momentum l.

## ◆ SpatialMomentum() [5/5]

 SpatialMomentum ( const Eigen::MatrixBase< Derived > & L )
explicit

SpatialMomentum constructor from an Eigen expression that represents a six-dimensional vector.

This constructor will assert the size of L is six (6) at compile-time for fixed sized Eigen expressions and at run-time for dynamic sized Eigen expressions.

## ◆ dot()

 T dot ( const SpatialVelocity< T > & V_NBp_E ) const

Given this spatial momentum L_NBp_E of a rigid body B, about point P and, expressed in a frame E, this method computes the dot product with the spatial velocity V_NBp_E of body B frame shifted to point P, measured in an inertial (or Newtonian) frame N and expressed in the same frame E in which the spatial momentum is expressed.

This dot-product is twice the kinetic energy ke_NB of body B in reference frame N. The kinetic energy ke_NB is independent of the about-point P and so is this dot product. Therefore it is always true that:

  ke_NB = 1/2 (L_NBp⋅V_NBp) = 1/2 (L_NBcm⋅V_NBcm)


where L_NBcm is the spatial momentum about the center of mass of body B and V_NBcm is the spatial velocity of frame B shifted to its center of mass. The above is true due to how spatial momentum and velocity shift when changing point P, see SpatialMomentum::Shift() and SpatialVelocity::Shift().

## ◆ operator+=()

 SpatialMomentum& operator+= ( const SpatialMomentum< T > & L_NSp_E )

Adds in a spatial momentum to this spatial momentum.

Parameters
 [in] L_NSp_E A spatial momentum to be added to this spatial momentum. It must be about the same point P as this spatial momentum and expressed in the same frame E.
Returns
A reference to this spatial momentum, which has been updated to include the given spatial momentum L_NSp_E.
Warning
This operation is only valid if both spatial momenta are applied about same point P and expressed in the same frame E.

## ◆ operator-=()

 SpatialMomentum& operator-= ( const SpatialMomentum< T > & L_NSp_E )

Subtracts a spatial momentum from this spatial momentum.

Parameters
 [in] L_NSp_E A spatial momentum to be subtracted from this spatial momentum. It must be about the same point P as this spatial momentum and expressed in the same frame E.
Returns
A reference to this spatial momentum, which has been updated to exclude the given spatial momentum L_NSp_E.
Warning
This operation is only valid if both spatial momenta are applied about same point P and expressed in the same frame E.

## ◆ operator=() [1/2]

 SpatialMomentum& operator= ( SpatialMomentum< T > && )
default

## ◆ operator=() [2/2]

 SpatialMomentum& operator= ( const SpatialMomentum< T > & )
default

## ◆ Shift()

 SpatialMomentum Shift ( const Vector3< T > & p_PQ_E ) const

Shift of a SpatialMomentum from one application point to another.

This is an alternate signature for shifting a spatial momentum's about-point that does not change the original object. See ShiftInPlace() for more information.

Parameters
 [in] p_PQ_E Shift vector from point P to point Q.
Return values
 L_NSq_E The equivalent shifted spatial momentum, now applied at point Q rather than P.
See also
ShiftInPlace() to compute the shifted spatial momentum in-place modifying the original object.

## ◆ ShiftInPlace()

 SpatialMomentum& ShiftInPlace ( const Vector3< T > & p_PQ_E )

In-place shift of a SpatialMomentum from one "about-point" to another.

this spatial momentum L_NSp_E for a system S in a reference frame N about a point P, and expressed in frame E, is modified to become the equivalent spatial momentum L_NSq_E of the same system about another point Q.

We are given the vector from point P to point Q, as a position vector p_PQ_E expressed in the same frame E as the spatial momentum. The operation performed, in coordinate-free form, is:

  h_NSq  = h_NSp -  p_PQ x l_NSp
l_NSq = l_NSp,  i.e. the linear momentum about point Q is the
same as the linear momentum about point P.


where h and l represent the angular and linear momentum components respectively. Notice that spatial momenta shift in the same way as spatial forces (see SpatialForce.)

The operation is linear, which [Jain 2010], (§2.1, page 22) writes using the "rigid body transformation operator" as:

  L_NSq = Φ(p_QP)L_NSp = Φ(-p_PQ)L_NSp


where Φ(p_PQ) is the linear operator:

  Φ(p_PQ) = | I₃ p_PQx |
| 0     I₃ |


where p_PQx denotes the cross product, skew-symmetric, matrix such that p_PQx v = p_PQ x v. This same operator shifts spatial forces in analogous way (see SpatialForce::Shift()) while the transpose of this operator allow us to shift spatial velocities, see SpatialVelocity::Shift().

• [Jain 2010] Jain, A., 2010. Robot and multibody dynamics: analysis and algorithms. Springer Science & Business Media, pp. 123-130.

For computation, all quantities above must be expressed in a common frame E; we add an _E suffix to each symbol to indicate that.

This operation is performed in-place modifying the original object.

Parameters
 [in] p_PQ_E Shift vector from point P to point Q, expressed in frame E.
Returns
A reference to this spatial momentum which is now L_NSq_E, that is, the spatial momentum about point Q rather than P.
See also
Shift() to compute the shifted spatial momentum without modifying this original object.

The documentation for this class was generated from the following files: