Respond to reviewer comments.

geometry/test/bad_geometryA.obj

@@ -1,6 +1,7 @@
 # File: bad_geometryA.obj
 # The geometry in this file produces invalid mass and inertia properties
 # due to the winding (order of the vertices) in some of the faces.
+# In Drake, the calculated volume of this mesh is zero (volume = 0).
 # Related files: bad_geometryB.obj and bad_geometry_corrected.obj
 # ----------------------------------------------------------------------
 

geometry/test/bad_geometryB.obj

@@ -1,6 +1,7 @@
 # File: bad_geometryB.obj
 # The geometry in this file produces invalid mass and inertia properties
 # due to the winding (order of the vertices) in some of the faces.
+# In Drake, the calculated volume of this mesh is negative (volume = -0.5).
 # Related files: bad_geometryA.obj and bad_geometry_corrected.obj
 # -----------------------------------------------------------------------
 

multibody/tree/geometry_spatial_inertia.cc

@@ -130,17 +130,25 @@ SpatialInertia<double> CalcSpatialInertia(
 
   // Volume should be inherently positive.
   const double volume = vol_times_six / 6.0;
-  constexpr double kEpsilon = std::numeric_limits<double>::epsilon();
+  // Throw an exception if volume is negative or nearly zero. This test of
+  // "reasonably positive" volume is more stringent than the mass ≥ 0 test in
+  // SpatialInertia::IsPhysicallyValid(). Reminder: The volume of a mesh can be
+  // calculated (and should be positive) whereas spatial inertia does not deal
+  // with volume. Instead, spatial inertia deals with mass, including idealized
+  // zero volume massive objects such as particles, rods, and plates.
+  constexpr double kEpsilon = 8 * std::numeric_limits<double>::epsilon();
   if (volume <= kEpsilon) {
     const std::string error_message = fmt::format(
         "{}(): The calculated volume of a triangle surface mesh is {} whereas "
         "a reasonable positive value was expected. The mesh may have bad "
-        "geometry, e.g., the winding (order of the vertices) of some faces do "
-        "not produce outward normals.", __func__, volume);
+        "geometry, e.g., it is an open mesh or the winding (order of vertices) "
+        "of one or more faces do not produce outward normals.",
+        __func__, volume);
     throw std::logic_error(error_message);
     // Note: In a Wavefront .obj file, each face's vertices should be stored
     // in a counter-clockwise order by default.
   }
+
   const double mass = density * volume;
   const Vector3d p_GoGcm = accum_com / (vol_times_six * 4);
   // We can compute I = C.trace * 1₃ - C. Two key points:

multibody/tree/geometry_spatial_inertia.h

@@ -48,6 +48,7 @@ SpatialInertia<double> CalcSpatialInertia(const geometry::Shape& shape,
 
  If these requirements are not met, a value *will* be returned, but its value
  is meaningless.
+ @throws std::exception if the volume of `mesh` is negative or nearly zero.
  @pydrake_mkdoc_identifier{mesh} */
 SpatialInertia<double> CalcSpatialInertia(
     const geometry::TriangleSurfaceMesh<double>& mesh, double density);

multibody/tree/test/geometry_spatial_inertia_test.cc

@@ -159,25 +159,35 @@ GTEST_TEST(GeometrySpatialInertiaTest, Convex) {
       /* tolerance = */ 1e-1));
 }
 
-// Check exception messages when CalcSpatialInertia(Shape) is called on a file
-// with bad geometry (e.g. not watertight or bad outward normals, or ...).
+// Throw an exception message when CalcSpatialInertia(Shape) calculates an
+// invalid volume for an associated geometry file.
 GTEST_TEST(GeometrySpatialInertiaTest, ExceptionOnBadGeometry) {
-  std::string geometry_file_path = FindResourceOrThrow(
-      "drake/geometry/test/bad_geometryA.obj");
+  // Throw an exception for the mesh in bad_geometryA.obj since
+  // its calculated volume is negative (volume = -0.5).
+  std::string geometry_file_path =
+      FindResourceOrThrow("drake/geometry/test/bad_geometryA.obj");
   const geometry::Mesh bad_geometryA_obj(geometry_file_path, 1.0);
-  DRAKE_EXPECT_THROWS_MESSAGE(CalcSpatialInertia(bad_geometryA_obj, kDensity),
+  DRAKE_EXPECT_THROWS_MESSAGE(
+      CalcSpatialInertia(bad_geometryA_obj, kDensity),
       ".*volume of a triangle surface mesh is.* whereas a reasonable "
       "positive value was expected. The mesh may have bad geometry.*");
 
-  geometry_file_path = FindResourceOrThrow(
-      "drake/geometry/test/bad_geometryB.obj");
+  // Throw an exception for the mesh in bad_geometryB.obj since
+  // its calculated volume is negative (volume = -0.5).
+  geometry_file_path =
+      FindResourceOrThrow("drake/geometry/test/bad_geometryB.obj");
   const geometry::Mesh bad_geometryB_obj(geometry_file_path, 1.0);
-  DRAKE_EXPECT_THROWS_MESSAGE(CalcSpatialInertia(bad_geometryB_obj, kDensity),
+  DRAKE_EXPECT_THROWS_MESSAGE(
+      CalcSpatialInertia(bad_geometryB_obj, kDensity),
       ".*volume of a triangle surface mesh is.* whereas a reasonable "
       "positive value was expected. The mesh may have bad geometry.*");
 
-  geometry_file_path = FindResourceOrThrow(
-      "drake/geometry/test/bad_geometry_corrected.obj");
+  // Ensure no exception is thrown for the mesh in bad_geometry_corrected.obj.
+  // This file has the same vertices as bad_geometryA_obj and bad_geometryB_obj,
+  // but its faces has vertices in an appropriate order.
+  // This mesh has an appropriate spatial inertia and positive volume.
+  geometry_file_path =
+      FindResourceOrThrow("drake/geometry/test/bad_geometry_corrected.obj");
   const geometry::Mesh ok_geometry_obj(geometry_file_path, 1.0);
   EXPECT_NO_THROW(CalcSpatialInertia(ok_geometry_obj, kDensity));
 }