MBT compile topological information (RobotLocomotion#6120)

* Compiles BodyNodeTopology

* Updates multibody_tree_creation_tests.cc

* Adds unit tests for a complex tree topology

* Adds unit tests doc

* Adds more doc

* Addresses reviews

* Fixes gcc warning

* Addresses reviews

* Makes MBTTopology into a class

* Refactors multibody_tree_creation_tests to stop accessing private members of topology

* Makes MBTT::bodies_ private

* Makes MBTT::body_nodes_ private

* Makes MBTT::mobilizers_ private

* Makes MBTT::frames_ private

* Cleansup lint.

* Addresses reviews

* Get rid of copy-pasta

drake/multibody/multibody_tree/body.h

@@ -137,7 +137,7 @@ class Body : public MultibodyTreeElement<Body<T>, BodyIndex> {
   // At MultibodyTree::Finalize() time, each body retrieves its topology
   // from the parent MultibodyTree.
   void DoSetTopology(const MultibodyTreeTopology& tree_topology) final {
-    topology_ = tree_topology.bodies[this->get_index()];
+    topology_ = tree_topology.get_body(this->get_index());
     body_frame_.SetTopology(tree_topology);
   }
 

drake/multibody/multibody_tree/mobilizer.h

@@ -183,7 +183,7 @@ class Mobilizer : public MultibodyTreeElement<Mobilizer<T>, MobilizerIndex> {
   // At MultibodyTree::Finalize() time, each mobilizer retrieves its topology
   // from the parent MultibodyTree.
   void DoSetTopology(const MultibodyTreeTopology& tree_topology) final {
-    topology_ = tree_topology.mobilizers[this->get_index()];
+    topology_ = tree_topology.get_mobilizer(this->get_index());
   }
 
   const Frame<T>& inboard_frame_;

drake/multibody/multibody_tree/multibody_tree.cc

@@ -28,9 +28,12 @@ void MultibodyTree<T>::Finalize() {
         "MultibodyTree.");
   }
 
+  // Before performing any setup that depends on the scalar type <T>, compile
+  // all the type-T independent topological information.
+  topology_.Finalize();
+
   // TODO(amcastro-tri): This is a brief list of operations to be added in
   // subsequent PR's:
-  //   - Finalize non-T dependent topological information.
   //   - Compute degrees of freedom, array sizes and any other information to
   //     allocate a context and request the required cache entries.
   //   - Setup computational structures (BodyNode based).
@@ -49,8 +52,6 @@ void MultibodyTree<T>::Finalize() {
   for (const auto& mobilizer : owned_mobilizers_) {
     mobilizer->SetTopology(topology_);
   }
-
-  set_valid_topology();
 }
 
 // Explicitly instantiates on the most common scalar types.

drake/multibody/multibody_tree/multibody_tree.h

@@ -79,7 +79,7 @@ class MultibodyTree {
   const BodyType<T>& AddBody(std::unique_ptr<BodyType<T>> body) {
     static_assert(std::is_convertible<BodyType<T>*, Body<T>*>::value,
                   "BodyType must be a sub-class of Body<T>.");
-    if (topology_.is_valid) {
+    if (topology_is_valid()) {
       throw std::logic_error("This MultibodyTree is finalized already. "
                              "Therefore adding more bodies is not allowed. "
                              "See documentation for Finalize() for details.");
@@ -177,7 +177,7 @@ class MultibodyTree {
   const FrameType<T>& AddFrame(std::unique_ptr<FrameType<T>> frame) {
     static_assert(std::is_convertible<FrameType<T>*, Frame<T>*>::value,
                   "FrameType must be a sub-class of Frame<T>.");
-    if (topology_.is_valid) {
+    if (topology_is_valid()) {
       throw std::logic_error("This MultibodyTree is finalized already. "
                              "Therefore adding more frames is not allowed. "
                              "See documentation for Finalize() for details.");
@@ -285,7 +285,7 @@ class MultibodyTree {
       std::unique_ptr<MobilizerType<T>> mobilizer) {
     static_assert(std::is_convertible<MobilizerType<T>*, Mobilizer<T>*>::value,
                   "MobilizerType must be a sub-class of mobilizer<T>.");
-    if (topology_.is_valid) {
+    if (topology_is_valid()) {
       throw std::logic_error("This MultibodyTree is finalized already. "
                              "Therefore adding more bodies is not allowed. "
                              "See documentation for Finalize() for details.");
@@ -409,7 +409,7 @@ class MultibodyTree {
   /// When a %MultibodyTree is instantiated, its topology remains invalid until
   /// Finalize() is called, which validates the topology.
   /// @see Finalize().
-  bool topology_is_valid() const { return topology_.is_valid; }
+  bool topology_is_valid() const { return topology_.is_valid(); }
 
   /// Returns the topology information for this multibody tree. Users should not
   /// need to call this method since MultibodyTreeTopology is an internal
@@ -440,9 +440,6 @@ class MultibodyTree {
   // a method that verifies the state of the topology with a signature similar
   // to RoadGeometry::CheckInvariants().
 
-  // Sets a flag indicate the topology is valid.
-  void set_valid_topology() { topology_.set_valid(); }
-
   std::vector<std::unique_ptr<Body<T>>> owned_bodies_;
   std::vector<std::unique_ptr<Frame<T>>> owned_frames_;
   std::vector<std::unique_ptr<Mobilizer<T>>> owned_mobilizers_;

drake/multibody/multibody_tree/multibody_tree_indexes.h

@@ -14,6 +14,9 @@ using BodyIndex = TypeSafeIndex<class BodyTag>;
 /// Type used to identify mobilizers by index in a multibody tree system.
 using MobilizerIndex = TypeSafeIndex<class MobilizerTag>;
 
+/// Type used to identify tree nodes by index within a multibody tree system.
+using BodyNodeIndex = TypeSafeIndex<class BodyNodeTag>;
+
 /// For every MultibodyTree the **world** body _always_ has this unique index
 /// and it is always zero.
 // Note: