Creating a MatrixLieGroup

This guide explains how to create a MatrixLieGroup, a specific type of LieGroup that has a matrix representation. Please read Creating a Manifold Class and Creating a LieGroup first. This document focuses only on the additional requirements for a matrix Lie group.

A MatrixLieGroup has all the properties of a LieGroup, but its structure is tied to an underlying N x N matrix. This allows for additional operations specific to matrix algebra.

1. From Lie Group to Matrix Lie Group¶

To implement a MatrixLieGroup, you inherit from gtsam::MatrixLieGroup<MyGroup, D, N>, which in turn inherits from gtsam::LieGroup. This inheritance provides even more functionality for free.

2. Minimal Additional Requirements¶

You must implement everything from the LieGroup.md guide, plus the following:

MyGroup.h - Header File Additions¶

1. Class Definition and Inheritance:

   • Change your inheritance to public gtsam::MatrixLieGroup<MyGroup, D, N>, where N is the side-length of your matrix.

2. Lie Algebra Typedef:

   • using LieAlgebra = gtsam::MatrixN;: Defines the type of a Lie algebra element (which is an N x N matrix).

3. Matrix Lie Group Primitives:

   • const gtsam::MatrixN& matrix() const;: An accessor for the underlying matrix data member.
   • static gtsam::MatrixN Hat(const gtsam::VectorD& xi);: Maps a D-dimensional tangent vector to its N x N matrix representation in the Lie algebra.
   • static gtsam::VectorD Vee(const gtsam::MatrixN& X);: The inverse of Hat, mapping an N x N Lie algebra matrix back to a D-dimensional vector.

3. What You Get for Free (Additionally)¶

By inheriting from gtsam::MatrixLieGroup, you get:

• A default AdjointMap() implementation: This generic version works by repeatedly calling your Hat and Vee methods. While it is correct, it is often slow. For performance-critical applications, you should still provide a faster, closed-form AdjointMap() implementation in your class, which will override the default.
• A vec() method: This vectorizes the N x N matrix representation of your group element into an (N*N) x 1 vector.

4. Traits and Concept Checking¶

Finally, the traits specialization in your header file must be updated to reflect that your class is now a MatrixLieGroup.

1. GTSAM Traits Specialization:

   • This is the final, crucial step. Specialize the traits struct using the internal::MatrixLieGroup helper.

   namespace gtsam {
   template <>
   struct traits<MyGroup> : public internal::MatrixLieGroup<MyGroup, N> {};
   
   template <>
   struct traits<const MyGroup> : public internal::MatrixLieGroup<MyGroup, N> {};
   }

2. Concept Checking:

   • Update the macro in your unit test file to check the MatrixLieGroup concept.

   #include <gtsam/base/TestableAssertions.h>
   #include <gtsam/base/MatrixLieGroup.h>
   
   // Your class include
   #include <gtsam/geometry/MyGroup.h>
   
   // This macro will fail to compile if MyGroup is not a valid MatrixLieGroup.
   GTSAM_CONCEPT_MATRIX_LIE_GROUP_INST(MyGroup)
   
   // ... your unit tests ...

This modular approach ensures that your class provides all the necessary components for full integration into the GTSAM framework.