Drake
Drake C++ Documentation ContactSurface< T > Class Template Reference

## Detailed Description

### template<typename T> class drake::geometry::ContactSurface< T >

The ContactSurface characterizes the intersection of two geometries M and N as a contact surface with a scalar field and a vector field, whose purpose is to support the hydroelastic pressure field contact model as described in:

R. Elandt, E. Drumwright, M. Sherman, and Andy Ruina. A pressure
field model for fast, robust approximation of net contact force
and moment between nominally rigid objects. IROS 2019: 8238-8245.


## Mathematical Concepts

In this section, we give motivation for the concept of contact surface from the hydroelastic pressure field contact model. Here the mathematical discussion is coordinate-free (treatment of the topic without reference to any particular coordinate system); however, our implementation heavily relies on coordinate frames. We borrow terminology from differential geometry.

In this section, the mathematical term compact set (a subset of Euclidean space that is closed and bounded) corresponds to the term geometry (or the space occupied by the geometry) in SceneGraph.

We describe the contact surface 𝕊ₘₙ between two intersecting compact subsets 𝕄 and ℕ of ℝ³ with the scalar fields eₘ and eₙ defined on 𝕄 ⊂ ℝ³ and ℕ ⊂ ℝ³ respectively:

           eₘ : 𝕄 → ℝ,
eₙ : ℕ → ℝ.


The contact surface 𝕊ₘₙ is the surface of equilibrium eₘ = eₙ. It is the locus of points Q where eₘ(Q) equals eₙ(Q):

         𝕊ₘₙ = { Q ∈ 𝕄 ∩ ℕ : eₘ(Q) = eₙ(Q) }.


We can define the scalar field eₘₙ on the surface 𝕊ₘₙ as a scalar function that assigns Q ∈ 𝕊ₘₙ the value of eₘ(Q), which is the same as eₙ(Q):

         eₘₙ : 𝕊ₘₙ → ℝ,
eₘₙ(Q) = eₘ(Q) = eₙ(Q).


We can also define the scalar field hₘₙ on 𝕄 ∩ ℕ as the difference between eₘ and eₙ:

         hₘₙ : 𝕄 ∩ ℕ → ℝ,
hₘₙ(Q) = eₘ(Q) - eₙ(Q).


It follows that the gradient vector field ∇hₘₙ on 𝕄 ∩ ℕ equals the difference between the gradient vector fields ∇eₘ and ∇eₙ:

         ∇hₘₙ : 𝕄 ∩ ℕ → ℝ³,
∇hₘₙ(Q) = ∇eₘ(Q) - ∇eₙ(Q).


By construction, Q ∈ 𝕊ₘₙ if and only if hₘₙ(Q) = 0. In other words, 𝕊ₘₙ is the zero level set of hₘₙ. It follows that, for Q ∈ 𝕊ₘₙ, ∇hₘₙ(Q) is orthogonal to the surface 𝕊ₘₙ at Q in the direction of increasing eₘ - eₙ.

Notice that the domain of eₘₙ is the two-dimensional surface 𝕊ₘₙ, while the domain of ∇hₘₙ is the three-dimensional compact set 𝕄 ∩ ℕ. Even though eₘₙ and ∇hₘₙ are defined on different domains (𝕊ₘₙ and 𝕄 ∩ ℕ), our implementation only represents them on their common domain, i.e., 𝕊ₘₙ.

## Discrete Representation

In practice, hydroelastic geometries themselves have a discrete representation: either a triangular surface mesh for rigid geometries or a tetrahedral volume mesh for compliant geometry. This discretization leads to contact surfaces that are likewise discrete.

Intersection between triangles and tetrahedra (or tetrahedra and tetrahedra) can produce polygons with up to eight sides. A ContactSurface can represent the resulting surface as a mesh of such polygons, or as a mesh of tesselated triangles. The domains of the two representations are identical. The triangular version admits for simple, high-order integration over the domain. Every element is a triangle, and triangles will only disappear and reappear as their areas go to zero. However, this increases the total number of faces in the mesh by more than a factor of three over the polygonal mesh. The polygonal representation produces fewer faces, but high order integration over polygons is problematic. We recommend choosing the cheapest representation that nevertheless supports your required fidelity (see QueryObject::ComputeContactSurfaces()).

The representation of any ContactSurface instance can be reported by calling representation(). If it returns HydroelasticContactRepresentation::kTriangle, then the mesh and pressure field can be accessed via tri_mesh_W() and tri_e_MN(), respectively. If it returns HydroelasticContactRepresentation::kPolygon, then use poly_mesh_W() and poly_e_MN().

Regardless of representation (polygon or triangle), the normal for each mesh face is guaranteed to point "out of" N and "into" M. They can be accessed via the mesh, e.g., tri_mesh_W().face_normal(face_index). By definition, the normals of the mesh are discontinuous at triangle boundaries.

The pressure values on the contact surface are represented as a continuous, piecewise-linear function, accessed via tri_e_MN() or poly_e_MN().

When available, the values of ∇eₘ and ∇eₙ are represented as a discontinuous, piecewise-constant function over the faces – one gradient vector per face. These quantities are accessed via EvaluateGradE_M_W() and EvaluateGradE_N_W(), respectively.

## Barycentric Coordinates

For Point Q on the surface mesh of the contact surface between Geometry M and Geometry N, r_WQ = (x,y,z) is the displacement vector from the origin of the world frame to Q expressed in the coordinate frame of W. We also have the barycentric coordinates (b0, b1, b2) on a triangle of the surface mesh that contains Q. With vertices of the triangle labeled as v₀, v₁, v₂, we can map (b0, b1, b2) to r_WQ by:

         r_WQ = b0 * r_Wv₀ + b1 * r_Wv₁ + b2 * r_Wv₂,
b0 + b1 + b2 = 1, bᵢ ∈ [0,1],


where r_Wvᵢ is the displacement vector of the vertex labeled as vᵢ from the origin of the world frame, expressed in the world frame.

We use the barycentric coordinates to evaluate the field values.

Template Parameters
 T The scalar type, which must be one of the default nonsymbolic scalars.

#include <drake/geometry/query_results/contact_surface.h>

## Public Member Functions

ContactSurface (const ContactSurface &surface)

ContactSurfaceoperator= (const ContactSurface &surface)

ContactSurface (ContactSurface &&)=default

ContactSurfaceoperator= (ContactSurface &&)=default

GeometryId id_M () const
Returns the geometry id of Geometry M. More...

GeometryId id_N () const
Returns the geometry id of Geometry N. More...

bool Equal (const ContactSurface< T > &surface) const
Checks to see whether the given ContactSurface object is equal via deep exact comparison. More...

Constructors

The ContactSurface can be constructed with either a polygon or triangle mesh representation.

The constructor invoked determines the representation.

The general shape of each constructor is identical. They take the unique identifiers for the two geometries in contact, a mesh representation, a field representation, and (optional) gradients of the contacting geometries' pressure fields.

Parameters
 id_M The id of the first geometry M. id_N The id of the second geometry N. mesh_W The surface mesh of the contact surface 𝕊ₘₙ between M and N. The mesh vertices are defined in the world frame. e_MN Represents the scalar field eₘₙ on the surface mesh. grad_eM_W ∇eₘ sampled once per face, expressed in the world frame. grad_eN_W ∇eₙ sampled once per face, expressed in the world frame.
Precondition
The face normals in mesh_W point out of geometry N and into M.
If given, grad_eM_W and grad_eN_W must have as many entries as mesh_W has faces and the ith entry in each should correspond to the ith face in mesh_W.
Note
If id_M > id_N, the labels will be swapped and the normals of the mesh reversed (to maintain the documented invariants). Comparing the input parameters with the members of the resulting ContactSurface will reveal if such a swap has occurred.
ContactSurface (GeometryId id_M, GeometryId id_N, std::unique_ptr< TriangleSurfaceMesh< T >> mesh_W, std::unique_ptr< TriangleSurfaceMeshFieldLinear< T, T >> e_MN, std::unique_ptr< std::vector< Vector3< T >>> grad_eM_W=nullptr, std::unique_ptr< std::vector< Vector3< T >>> grad_eN_W=nullptr)
Constructs a ContactSurface with a triangle mesh representation. More...

ContactSurface (GeometryId id_M, GeometryId id_N, std::unique_ptr< PolygonSurfaceMesh< T >> mesh_W, std::unique_ptr< PolygonSurfaceMeshFieldLinear< T, T >> e_MN, std::unique_ptr< std::vector< Vector3< T >>> grad_eM_W=nullptr, std::unique_ptr< std::vector< Vector3< T >>> grad_eN_W=nullptr)
Constructs a ContactSurface with a polygonal mesh representation. More...

Representation-independent API

These methods represent sugar which masks the details of the mesh representation.

They facilitate querying various mesh quantities that are common to the two representations, so that code can access the properties without worrying about the representation. If necessary, the actual meshes and fields can be accessed directly via the representation-dependent APIs below.

int num_faces () const

int num_vertices () const

const T & area (int face_index) const

const T & total_area () const

const Vector3< T > & face_normal (int face_index) const

Vector3< T > centroid (int face_index) const

const Vector3< T > & centroid () const

Representation-dependent API

These functions provide insight into what representation the ContactSurface instance uses, and provide access to the representation-dependent quantities: mesh and field.

bool is_triangle () const
Simpley reports if this contact surface's mesh representation is triangle. More...

HydroelasticContactRepresentation representation () const
Reports the representation mode of this contact surface. More...

const TriangleSurfaceMesh< T > & tri_mesh_W () const
Returns a reference to the triangular surface mesh whose vertex positions are measured and expressed in the world frame. More...

const TriangleSurfaceMeshFieldLinear< T, T > & tri_e_MN () const
Returns a reference to the scalar field eₘₙ for the triangle mesh. More...

const PolygonSurfaceMesh< T > & poly_mesh_W () const
Returns a reference to the polygonal surface mesh whose vertex positions are measured and expressed in the world frame. More...

const PolygonSurfaceMeshFieldLinear< T, T > & poly_e_MN () const
Returns a reference to the scalar field eₘₙ for the polygonal mesh. More...

Evaluation of constituent pressure fields

The ContactSurface provisionally includes the gradients of the constituent pressure fields (∇eₘ and ∇eₙ) sampled on the contact surface.

In order for these values to be included in an instance, the gradient for the corresponding mesh must be well defined. For example a rigid mesh will not have a well-defined pressure gradient; as stiffness goes to infinity, the geometry becomes rigid and the gradient direction converges to the direction of the rigid mesh's surface normals, but the magnitude goes to infinity, producing a pressure gradient that would be some variant of <∞, ∞, ∞>.

Accessing the gradient values must be pre-conditioned on a test that the particular instance of ContactSurface actually contains the gradient data. The presence of gradient data for each geometry must be confirmed separately.

The values ∇eₘ and ∇eₘ are piecewise constant over the ContactSurface and can only be evaluate on a per-face basis.

const Vector3< T > & EvaluateGradE_M_W (int index) const
Returns the value of ∇eₘ for the face with index index. More...

const Vector3< T > & EvaluateGradE_N_W (int index) const
Returns the value of ∇eₙ for the face with index index. More...

## Friends

template<typename U >
class ContactSurfaceTester

## ◆ ContactSurface() [1/4]

 ContactSurface ( const ContactSurface< T > & surface )

## ◆ ContactSurface() [2/4]

 ContactSurface ( ContactSurface< T > && )
default

## ◆ ContactSurface() [3/4]

 ContactSurface ( GeometryId id_M, GeometryId id_N, std::unique_ptr< TriangleSurfaceMesh< T >> mesh_W, std::unique_ptr< TriangleSurfaceMeshFieldLinear< T, T >> e_MN, std::unique_ptr< std::vector< Vector3< T >>> grad_eM_W = nullptr, std::unique_ptr< std::vector< Vector3< T >>> grad_eN_W = nullptr )

Constructs a ContactSurface with a triangle mesh representation.

## ◆ ContactSurface() [4/4]

 ContactSurface ( GeometryId id_M, GeometryId id_N, std::unique_ptr< PolygonSurfaceMesh< T >> mesh_W, std::unique_ptr< PolygonSurfaceMeshFieldLinear< T, T >> e_MN, std::unique_ptr< std::vector< Vector3< T >>> grad_eM_W = nullptr, std::unique_ptr< std::vector< Vector3< T >>> grad_eN_W = nullptr )

Constructs a ContactSurface with a polygonal mesh representation.

## ◆ area()

 const T& area ( int face_index ) const

## ◆ centroid() [1/2]

 Vector3 centroid ( int face_index ) const

## ◆ centroid() [2/2]

 const Vector3& centroid ( ) const

## ◆ Equal()

 bool Equal ( const ContactSurface< T > & surface ) const

Checks to see whether the given ContactSurface object is equal via deep exact comparison.

NaNs are treated as not equal as per the IEEE standard.

Parameters
 surface The contact surface for comparison.
Returns
true if the given contact surface is equal.

 const Vector3& EvaluateGradE_M_W ( int index ) const

Returns the value of ∇eₘ for the face with index index.

Exceptions
Precondition
index ∈ [0, mesh().num_faces()).

 const Vector3& EvaluateGradE_N_W ( int index ) const

Returns the value of ∇eₙ for the face with index index.

Exceptions
Precondition
index ∈ [0, mesh().num_faces()).

## ◆ face_normal()

 const Vector3& face_normal ( int face_index ) const

Returns
true if this contains values for ∇eₘ.

Returns
true if this contains values for ∇eₙ.

## ◆ id_M()

 GeometryId id_M ( ) const

Returns the geometry id of Geometry M.

## ◆ id_N()

 GeometryId id_N ( ) const

Returns the geometry id of Geometry N.

## ◆ is_triangle()

 bool is_triangle ( ) const

Simpley reports if this contact surface's mesh representation is triangle.

Equivalent to:

representation() == HydroelasticContactRepresentation::kTriangle


and offered as convenient sugar.

## ◆ num_faces()

 int num_faces ( ) const

## ◆ num_vertices()

 int num_vertices ( ) const

## ◆ operator=() [1/2]

 ContactSurface& operator= ( const ContactSurface< T > & surface )

## ◆ operator=() [2/2]

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

## ◆ poly_e_MN()

 const PolygonSurfaceMeshFieldLinear& poly_e_MN ( ) const

Returns a reference to the scalar field eₘₙ for the polygonal mesh.

Precondition
is_triangle() returns false.

## ◆ poly_mesh_W()

 const PolygonSurfaceMesh& poly_mesh_W ( ) const

Returns a reference to the polygonal surface mesh whose vertex positions are measured and expressed in the world frame.

Precondition
is_triangle() returns false.

## ◆ representation()

 HydroelasticContactRepresentation representation ( ) const

Reports the representation mode of this contact surface.

If accessing the mesh or field directly, the APIs that can be successfully exercised are related to this methods return value. See below.

## ◆ total_area()

 const T& total_area ( ) const

## ◆ tri_e_MN()

 const TriangleSurfaceMeshFieldLinear& tri_e_MN ( ) const

Returns a reference to the scalar field eₘₙ for the triangle mesh.

Precondition
is_triangle() returns true.

## ◆ tri_mesh_W()

 const TriangleSurfaceMesh& tri_mesh_W ( ) const

Returns a reference to the triangular surface mesh whose vertex positions are measured and expressed in the world frame.

Precondition
is_triangle() returns true.

## ◆ ContactSurfaceTester

 friend class ContactSurfaceTester
friend

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