Drake
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, the contact surface is approximated with a discrete triangle mesh. The triangle mesh's normals are defined per face. The normal of each face is guaranteed to point "out of" N and "into" M. They can be accessed via mesh_W().face_normal(face_index).

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

The normals of the mesh are discontinuous at triangle boundaries, but the pressure can be meaningfully evaluated over the entire domain of the mesh.

When available, the values of ∇eₘ and ∇eₙ are represented as a discontinuous, piecewise-constant function over the triangles – one vector per triangle. 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

ContactSurface (GeometryId id_M, GeometryId id_N, std::unique_ptr< SurfaceMesh< T >> mesh_W, std::unique_ptr< SurfaceMeshFieldLinear< T, T >> e_MN)
Constructs a ContactSurface. More...

ContactSurface (GeometryId id_M, GeometryId id_N, std::unique_ptr< SurfaceMesh< T >> mesh_W, std::unique_ptr< SurfaceMeshFieldLinear< T, T >> e_MN, std::unique_ptr< std::vector< Vector3< T >>> grad_eM_W, std::unique_ptr< std::vector< Vector3< T >>> grad_eN_W)
Constructs a ContactSurface with the optional gradients of the constituent scalar fields. More...

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

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

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

const SurfaceMeshFieldLinear< T, T > & e_MN () const
Returns a reference to the scalar field eₘₙ. More...

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

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

const Vector3< T > & EvaluateGradE_N_W (SurfaceFaceIndex index) const
Returns the value of ∇eₙ for the triangle 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< SurfaceMesh< T >> mesh_W, std::unique_ptr< SurfaceMeshFieldLinear< T, T >> e_MN )

Constructs a ContactSurface.

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.
Precondition
The face normals in mesh_W point out of geometry N and into M.
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() [4/4]

 ContactSurface ( GeometryId id_M, GeometryId id_N, std::unique_ptr< SurfaceMesh< T >> mesh_W, std::unique_ptr< SurfaceMeshFieldLinear< T, T >> e_MN, std::unique_ptr< std::vector< Vector3< T >>> grad_eM_W, std::unique_ptr< std::vector< Vector3< T >>> grad_eN_W )

Constructs a ContactSurface with the optional gradients of the constituent scalar 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.

## ◆ e_MN()

 const SurfaceMeshFieldLinear& e_MN ( ) const

Returns a reference to the scalar field eₘₙ.

## ◆ 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 ( SurfaceFaceIndex index ) const

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

Exceptions

 const Vector3& EvaluateGradE_N_W ( SurfaceFaceIndex index ) const

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

Exceptions

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.

## ◆ mesh_W()

 const SurfaceMesh& mesh_W ( ) const

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

## ◆ operator=() [1/2]

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

## ◆ operator=() [2/2]

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

## ◆ ContactSurfaceTester

 friend class ContactSurfaceTester
friend

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