Simbody
3.6

A UnitInertia matrix is a unitmass inertia matrix; you can convert it to an Inertia by multiplying it by the actual body mass. More...
Public Member Functions  
UnitInertia_ ()  
Default is a NaNed out mess to avoid accidents, even in Release mode. More...  
UnitInertia_ (const RealP &moment)  
Create a principal unit inertia matrix with identical diagonal elements. More...  
UnitInertia_ (const Vec3P &moments, const Vec3P &products=Vec3P(0))  
Create a unit inertia matrix from a vector of the moments of inertia (the inertia matrix diagonal) and optionally a vector of the products of inertia (the offdiagonals). More...  
UnitInertia_ (const RealP &xx, const RealP &yy, const RealP &zz)  
Create a principal unit inertia matrix (only nonzero on diagonal). More...  
UnitInertia_ (const RealP &xx, const RealP &yy, const RealP &zz, const RealP &xy, const RealP &xz, const RealP &yz)  
This is a general unit inertia matrix. More...  
UnitInertia_ (const SymMat33P &m)  
Construct a UnitInertia from a symmetric 3x3 matrix. More...  
UnitInertia_ (const Mat33P &m)  
Construct a UnitInertia from a 3x3 symmetric matrix. More...  
UnitInertia_ (const Inertia_< P > &inertia)  
Construct a UnitInertia matrix from an Inertia matrix. More...  
UnitInertia_ &  setUnitInertia (const RealP &xx, const RealP &yy, const RealP &zz) 
Set a UnitInertia matrix to have only principal moments (that is, it will be diagonal). More...  
UnitInertia_ &  setUnitInertia (const Vec3P &moments, const Vec3P &products=Vec3P(0)) 
Set principal moments and optionally offdiagonal terms. More...  
UnitInertia_ &  setUnitInertia (const RealP &xx, const RealP &yy, const RealP &zz, const RealP &xy, const RealP &xz, const RealP &yz) 
Set this UnitInertia to a general matrix. More...  
UnitInertia_  shiftToCentroid (const Vec3P &CF) const 
Assuming that this unit inertia matrix is currently taken about some (implicit) frame F's origin OF, produce a new unit inertia matrix which is the same as this one except measured about the body's centroid CF. More...  
UnitInertia_ &  shiftToCentroidInPlace (const Vec3P &CF) 
Assuming that this unit inertia matrix is currently taken about some (implicit) frame F's origin OF, modify it so that it is instead taken about the body's centroid CF. More...  
UnitInertia_  shiftFromCentroid (const Vec3P &p) const 
Assuming that the current UnitInertia G is a central inertia (that is, it is inertia about the body centroid CF), create a new object that is the same as this one except shifted to some other point p measured from the centroid. More...  
UnitInertia_ &  shiftFromCentroidInPlace (const Vec3P &p) 
Assuming that the current UnitInertia G is a central inertia (that is, it is inertia about the body centroid CF), shift it in place to some other point p measured from the centroid. More...  
UnitInertia_  reexpress (const Rotation_< P > &R_FB) const 
Return a new unit inertia matrix like this one but reexpressed in another frame (leaving the origin point unchanged). More...  
UnitInertia_  reexpress (const InverseRotation_< P > &R_FB) const 
Rexpress using an inverse rotation to avoid having to convert it. More...  
UnitInertia_ &  reexpressInPlace (const Rotation_< P > &R_FB) 
Reexpress this unit inertia matrix in another frame, changing the object in place; see reexpress() if you want to leave this object unmolested and get a new one instead. More...  
UnitInertia_ &  reexpressInPlace (const InverseRotation_< P > &R_FB) 
Rexpress using an inverse rotation to avoid having to convert it. More...  
operator const SymMat33P & () const  
This is an implicit conversion to const SymMat33. More...  
const Inertia_< P > &  asUnitInertia () const 
Recast this UnitInertia matrix as a unit inertia matrix. More...  
UnitInertia_ &  setFromUnitInertia (const Inertia_< P > &inertia) 
Set from a unit inertia matrix. More...  
Public Member Functions inherited from SimTK::Inertia_< P >  
Inertia_ ()  
Default is a NaNed out mess to avoid accidents, even in Release mode. More...  
Inertia_ (const P &moment)  
Create a principal inertia matrix with identical diagonal elements, like a sphere where moment=2/5 m r^2, or a cube where moment=1/6 m s^2, with m the total mass, r the sphere's radius and s the length of a side of the cube. More...  
Inertia_ (const Vec< 3, P > &p, const P &mass)  
Create an Inertia matrix for a point mass at a given location, measured from the origin OF of the implicit frame F, and expressed in F. More...  
Inertia_ (const Vec< 3, P > &moments, const Vec< 3, P > &products=Vec< 3, P >(0))  
Create an inertia matrix from a vector of the moments of inertia (the inertia matrix diagonal) and optionally a vector of the products of inertia (the offdiagonals). More...  
Inertia_ (const P &xx, const P &yy, const P &zz)  
Create a principal inertia matrix (only nonzero on diagonal). More...  
Inertia_ (const P &xx, const P &yy, const P &zz, const P &xy, const P &xz, const P &yz)  
This is a general inertia matrix. More...  
Inertia_ (const SymMat< 3, P > &inertia)  
Construct an Inertia from a symmetric 3x3 matrix. More...  
Inertia_ (const Mat< 3, 3, P > &m)  
Construct an Inertia matrix from a 3x3 symmetric matrix. More...  
Inertia_ &  setInertia (const P &xx, const P &yy, const P &zz) 
Set an inertia matrix to have only principal moments (that is, it will be diagonal). More...  
Inertia_ &  setInertia (const Vec< 3, P > &moments, const Vec< 3, P > &products=Vec< 3, P >(0)) 
Set principal moments and optionally offdiagonal terms. More...  
Inertia_ &  setInertia (const P &xx, const P &yy, const P &zz, const P &xy, const P &xz, const P &yz) 
Set this Inertia to a general matrix. More...  
Inertia_ &  operator+= (const Inertia_ &inertia) 
Add in another inertia matrix. More...  
Inertia_ &  operator= (const Inertia_ &inertia) 
Subtract off another inertia matrix. More...  
Inertia_ &  operator*= (const P &s) 
Multiply this inertia matrix by a scalar. Cost is 6 flops. More...  
Inertia_ &  operator/= (const P &s) 
Divide this inertia matrix by a scalar. More...  
Inertia_  shiftToMassCenter (const Vec< 3, P > &CF, const P &mass) const 
Assume that the current inertia is about the F frame's origin OF, and expressed in F. More...  
Inertia_ &  shiftToMassCenterInPlace (const Vec< 3, P > &CF, const P &mass) 
Assume that the current inertia is about the F frame's origin OF, and expressed in F. More...  
Inertia_  shiftFromMassCenter (const Vec< 3, P > &p, const P &mass) const 
Assuming that the current inertia I is a central inertia (that is, it is inertia about the body center of mass CF), shift it to some other point p measured from the center of mass. More...  
Inertia_ &  shiftFromMassCenterInPlace (const Vec< 3, P > &p, const P &mass) 
Assuming that the current inertia I is a central inertia (that is, it is inertia about the body center of mass CF), shift it to some other point p measured from the center of mass. More...  
Inertia_  reexpress (const Rotation_< P > &R_FB) const 
Return a new inertia matrix like this one but reexpressed in another frame (leaving the origin point unchanged). More...  
Inertia_  reexpress (const InverseRotation_< P > &R_FB) const 
Rexpress using an inverse rotation to avoid having to convert it. More...  
Inertia_ &  reexpressInPlace (const Rotation_< P > &R_FB) 
Reexpress this inertia matrix in another frame, changing the object in place; see reexpress() if you want to leave this object unmolested and get a new one instead. More...  
Inertia_ &  reexpressInPlace (const InverseRotation_< P > &R_FB) 
Rexpress in place using an inverse rotation to avoid having to convert it. More...  
P  trace () const 
operator const SymMat< 3, P > & () const  
This is an implicit conversion to a const SymMat33. More...  
const SymMat< 3, P > &  asSymMat33 () const 
Obtain a reference to the underlying symmetric matrix type. More...  
Mat< 3, 3, P >  toMat33 () const 
Expand the internal packed representation into a full 3x3 symmetric matrix with all elements set. More...  
const Vec< 3, P > &  getMoments () const 
Obtain the inertia moments (diagonal of the Inertia matrix) as a Vec3 ordered xx, yy, zz. More...  
const Vec< 3, P > &  getProducts () const 
Obtain the inertia products (offdiagonals of the Inertia matrix) as a Vec3 with elements ordered xy, xz, yz. More...  
bool  isNaN () const 
bool  isInf () const 
bool  isFinite () const 
bool  isNumericallyEqual (const Inertia_< P > &other) const 
Compare this inertia matrix with another one and return true if they are close to within a default numerical tolerance. More...  
bool  isNumericallyEqual (const Inertia_< P > &other, double tol) const 
Compare this inertia matrix with another one and return true if they are close to within a specified numerical tolerance. More...  
Static Public Member Functions  
static bool  isValidUnitInertiaMatrix (const SymMat33P &m) 
Test some conditions that must hold for a valid UnitInertia matrix. More...  
UnitInertia matrix factories  
These are UnitInertia matrix factories for some common 3D solids. Each defines its own frame aligned (when possible) with principal moments. Each has unit mass and its center of mass located at the origin (usually). Use this with shiftFromCentroid() to move it somewhere else, and with reexpress() to express the UnitInertia matrix in another frame.  
static UnitInertia_  pointMassAtOrigin () 
Create a UnitInertia matrix for a point located at the origin – that is, an allzero matrix. More...  
static UnitInertia_  pointMassAt (const Vec3P &p) 
Create a UnitInertia matrix for a point of unit mass located at a given location measured from origin OF and expressed in F (where F is the implicit frame of this UnitInertia matrix). More...  
static UnitInertia_  sphere (const RealP &r) 
Create a UnitInertia matrix for a unit mass sphere of radius r centered at the origin. More...  
static UnitInertia_  cylinderAlongZ (const RealP &r, const RealP &hz) 
Unitmass cylinder aligned along z axis; use radius and halflength. More...  
static UnitInertia_  cylinderAlongY (const RealP &r, const RealP &hy) 
Unitmass cylinder aligned along y axis; use radius and halflength. More...  
static UnitInertia_  cylinderAlongX (const RealP &r, const RealP &hx) 
Unitmass cylinder aligned along x axis; use radius and halflength. More...  
static UnitInertia_  brick (const RealP &hx, const RealP &hy, const RealP &hz) 
Unitmass brick given by halflengths in each direction. More...  
static UnitInertia_  brick (const Vec3P &halfLengths) 
Alternate interface to brick() that takes a Vec3 for the half lengths. More...  
static UnitInertia_  ellipsoid (const RealP &hx, const RealP &hy, const RealP &hz) 
Unitmass ellipsoid given by halflengths in each direction. More...  
static UnitInertia_  ellipsoid (const Vec3P &halfLengths) 
Alternate interface to ellipsoid() that takes a Vec3 for the half lengths. More...  
Static Public Member Functions inherited from SimTK::Inertia_< P >  
static bool  isValidInertiaMatrix (const SymMat< 3, P > &m) 
Test some conditions that must hold for a valid Inertia matrix. More...  
static Inertia_  pointMassAtOrigin () 
Create an Inertia matrix for a point located at the origin – that is, an allzero matrix. More...  
static Inertia_  pointMassAt (const Vec< 3, P > &p, const P &m) 
Create an Inertia matrix for a point of a given mass, located at a given location measured from the origin of the implicit F frame. More...  
static Inertia_  sphere (const P &r) 
Create a UnitInertia matrix for a unit mass sphere of radius r centered at the origin. More...  
static Inertia_  cylinderAlongZ (const P &r, const P &hz) 
Unitmass cylinder aligned along z axis; use radius and halflength. More...  
static Inertia_  cylinderAlongY (const P &r, const P &hy) 
Unitmass cylinder aligned along y axis; use radius and halflength. More...  
static Inertia_  cylinderAlongX (const P &r, const P &hx) 
Unitmass cylinder aligned along x axis; use radius and halflength. More...  
static Inertia_  brick (const P &hx, const P &hy, const P &hz) 
Unitmass brick given by halflengths in each direction. More...  
static Inertia_  brick (const Vec< 3, P > &halfLengths) 
Alternate interface to brick() that takes a Vec3 for the half lengths. More...  
static Inertia_  ellipsoid (const P &hx, const P &hy, const P &hz) 
Unitmass ellipsoid given by halflengths in each direction. More...  
static Inertia_  ellipsoid (const Vec< 3, P > &halfLengths) 
Alternate interface to ellipsoid() that takes a Vec3 for the half lengths. More...  
Additional Inherited Members  
Protected Member Functions inherited from SimTK::Inertia_< P >  
const UnitInertia_< P > &  getAsUnitInertia () const 
UnitInertia_< P > &  updAsUnitInertia () 
void  errChk (const char *methodName) const 
Protected Attributes inherited from SimTK::Inertia_< P >  
SymMat< 3, P >  I_OF_F 
Related Functions inherited from SimTK::Inertia_< P >  
template<class P >  
Inertia_< P >  operator+ (const Inertia_< P > &l, const Inertia_< P > &r) 
Add two compatible inertia matrices, meaning they must be taken about the same point and expressed in the same frame. More...  
template<class P >  
Inertia_< P >  operator (const Inertia_< P > &l, const Inertia_< P > &r) 
Subtract from one inertia matrix another one which is compatible, meaning that both must be taken about the same point and expressed in the same frame. More...  
template<class P >  
Inertia_< P >  operator* (const Inertia_< P > &i, const P &r) 
Multiply an inertia matrix by a scalar. More...  
template<class P >  
Inertia_< P >  operator* (const P &r, const Inertia_< P > &i) 
Multiply an inertia matrix by a scalar. More...  
template<class P >  
Inertia_< P >  operator* (const Inertia_< P > &i, int r) 
Multiply an inertia matrix by a scalar given as an int. More...  
template<class P >  
Inertia_< P >  operator* (int r, const Inertia_< P > &i) 
Multiply an inertia matrix by a scalar given as an int. More...  
template<class P >  
Inertia_< P >  operator/ (const Inertia_< P > &i, const P &r) 
Divide an inertia matrix by a scalar. More...  
template<class P >  
Inertia_< P >  operator/ (const Inertia_< P > &i, int r) 
Divide an inertia matrix by a scalar provided as an int. More...  
template<class P >  
Vec< 3, P >  operator* (const Inertia_< P > &I, const Vec< 3, P > &w) 
Multiply an inertia matrix I on the right by a vector w giving the vector result I*w. More...  
template<class P >  
bool  operator== (const Inertia_< P > &i1, const Inertia_< P > &i2) 
Compare two inertia matrices for exact (bitwise) equality. More...  
template<class P >  
std::ostream &  operator<< (std::ostream &o, const Inertia_< P > &inertia) 
Output a humanreadable representation of an inertia matrix to the indicated stream. More...  
A UnitInertia matrix is a unitmass inertia matrix; you can convert it to an Inertia by multiplying it by the actual body mass.
Functionality is limited here to those few operations which ensure unit mass; most operations on a UnitInertia matrix result in a general Inertia instead. You can use a UnitInertia object wherever an Inertia is expected but not vice versa.
When constructing a UnitInertia matrix, note that we cannot verify that it actually has unit mass because every legal Inertia matrix can be viewed as the UnitInertia matrix for some differentlyscaled object.
Unit inertia matrices are sometimes called "gyration" matrices; we will often represent them with the symbol "G" to avoid confusion with general inertia matrices for which the symbol "I" (or sometimes "J") is used.
Typedefs exist for the most common invocations of UnitInertia_<P>:

inline 
Default is a NaNed out mess to avoid accidents, even in Release mode.
Other than this value, a UnitInertia_ should always be valid.

inlineexplicit 
Create a principal unit inertia matrix with identical diagonal elements.
This is the unit inertia matrix of a unit mass sphere of radius r = sqrt(5/2 * moment) centered on the origin.

inlineexplicit 
Create a unit inertia matrix from a vector of the moments of inertia (the inertia matrix diagonal) and optionally a vector of the products of inertia (the offdiagonals).
Moments are in the order xx,yy,zz; products are xy,xz,yz.

inline 
Create a principal unit inertia matrix (only nonzero on diagonal).

inline 
This is a general unit inertia matrix.
Note the order of these arguments: moments of inertia first, then products of inertia.

inlineexplicit 
Construct a UnitInertia from a symmetric 3x3 matrix.
The diagonals must be nonnegative and satisfy the triangle inequality.

inlineexplicit 
Construct a UnitInertia from a 3x3 symmetric matrix.
In Debug mode we'll test that the supplied matrix is numerically close to symmetric, and that it satisfies other requirements of an inertia matrix.

inlineexplicit 
Construct a UnitInertia matrix from an Inertia matrix.
Note that there is no way to check whether this is really a unit inertia – any inertia matrix may be interpreted as a unit inertia for some shape. So be sure you know what you're doing before you use this constructor!

inline 
Set a UnitInertia matrix to have only principal moments (that is, it will be diagonal).
Returns a reference to "this" like an assignment operator.

inline 
Set principal moments and optionally offdiagonal terms.
Returns a reference to "this" like an assignment operator.

inline 
Set this UnitInertia to a general matrix.
Note the order of these arguments: moments of inertia first, then products of inertia. Behaves like an assignment statement. Will throw an error message in Debug mode if the supplied elements do not constitute a valid inertia matrix.

inline 
Assuming that this unit inertia matrix is currently taken about some (implicit) frame F's origin OF, produce a new unit inertia matrix which is the same as this one except measured about the body's centroid CF.
We are given the vector from OF to the centroid CF, expressed in F. This produces a new UnitInertia matrix G' whose (implicit) frame F' is aligned with F but has origin CF (an inertia matrix like that is called "central" or "centroidal"). From the parallel axis theorem for inertias, G' = G  Gcom where Gcom is the inertia matrix of a fictitious, unitmass point located at CF (measured in F) taken about OF. (17 flops)

inline 
Assuming that this unit inertia matrix is currently taken about some (implicit) frame F's origin OF, modify it so that it is instead taken about the body's centroid CF.
We are given the vector from OF to the centroid CF, expressed in F. This produces a new UnitInertia G' whose (implicit) frame F' is aligned with F but has origin CF (an inertia matrix like that is called "central" or "centroidal"). From the parallel axis theorem for inertias, G' = G  Gcom where Gcom is the inertia matrix of a fictitious, unitmass point located at CF (measured in F) taken about OF. A reference to the modified object is returned so that you can chain this method in the manner of assignment operators. Cost is 17 flops.

inline 
Assuming that the current UnitInertia G is a central inertia (that is, it is inertia about the body centroid CF), create a new object that is the same as this one except shifted to some other point p measured from the centroid.
This produces a new inertia G' about the point p given by G' = G + Gp where Gp is the inertia of a fictitious point located at p, taken about CF. Cost is 17 flops.

inline 
Assuming that the current UnitInertia G is a central inertia (that is, it is inertia about the body centroid CF), shift it in place to some other point p measured from the centroid.
This changes G to a modified inertia G' taken about the point p, with the parallel axis theorem for inertia giving G' = G + Gp where Gp is the inertia of a fictitious, unitmass point located at p, taken about CF. Cost is 17 flops.

inline 
Return a new unit inertia matrix like this one but reexpressed in another frame (leaving the origin point unchanged).
Call this inertia matrix G_OF_F, that is, it is taken about the origin of some frame F, and expressed in F. We want to return G_OF_B, the same unit inertia matrix, still taken about the origin of F, but expressed in the B frame, given by G_OF_B=R_BF*G_OF_F*R_FB where R_FB is the rotation matrix giving the orientation of frame B in F. This is handled here by a special method of the Rotation class which rotates a symmetric tensor at a cost of 57 flops.

inline 
Rexpress using an inverse rotation to avoid having to convert it.

inline 
Reexpress this unit inertia matrix in another frame, changing the object in place; see reexpress() if you want to leave this object unmolested and get a new one instead.
Cost is 57 flops.

inline 
Rexpress using an inverse rotation to avoid having to convert it.

inline 
This is an implicit conversion to const SymMat33.

inline 
Recast this UnitInertia matrix as a unit inertia matrix.
This is just for emphasis; a UnitInertia matrix is already a kind of Inertia matrix by inheritance.

inline 
Set from a unit inertia matrix.
Note that we can't check; every Inertia matrix can be interpreted as a unit inertia for some shape.

inlinestatic 
Test some conditions that must hold for a valid UnitInertia matrix.
Cost is about 9 flops. TODO: this may not be comprehensive.

inlinestatic 
Create a UnitInertia matrix for a point located at the origin – that is, an allzero matrix.

inlinestatic 
Create a UnitInertia matrix for a point of unit mass located at a given location measured from origin OF and expressed in F (where F is the implicit frame of this UnitInertia matrix).
Cost is 11 flops.

inlinestatic 
Create a UnitInertia matrix for a unit mass sphere of radius r centered at the origin.

inlinestatic 
Unitmass cylinder aligned along z axis; use radius and halflength.
If r==0 this is a thin rod; hz=0 it is a thin disk.

inlinestatic 
Unitmass cylinder aligned along y axis; use radius and halflength.
If r==0 this is a thin rod; hy=0 it is a thin disk.

inlinestatic 
Unitmass cylinder aligned along x axis; use radius and halflength.
If r==0 this is a thin rod; hx=0 it is a thin disk.

inlinestatic 
Unitmass brick given by halflengths in each direction.
One dimension zero gives inertia of a thin rectangular sheet; two zero gives inertia of a thin rod in the remaining direction.

inlinestatic 
Alternate interface to brick() that takes a Vec3 for the half lengths.

inlinestatic 
Unitmass ellipsoid given by halflengths in each direction.

inlinestatic 
Alternate interface to ellipsoid() that takes a Vec3 for the half lengths.