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Geometric algebra gives robotics a single algebraic language for geometry: points, lines, planes, circles and spheres are elements of one algebra, and the transformations that move them are elements of the same algebra. That uniformity is what makes it useful in practice — an objective written for a point works unchanged for a line, and a transformation applies to every primitive by the same operation.
These pages introduce the parts of the theory used throughout our research, starting from the products that define a geometric algebra and building up to the conformal model and its use in robot modeling.
- 1 Why Geometric Algebra? What geometric algebra unifies, what it buys over matrices and quaternions, and where it costs more.
- 2 Basics of Geometric Algebra The geometric product, how it splits into the inner and outer products, and how grades organise the algebra.
- 3 The Conformal Model Why CGA uses a 4,1 signature, the null basis of origin and infinity, and the conformal embedding of Euclidean points.
- 4 Geometric Primitives Constructing lines, circles, planes and spheres by spanning points with the outer product, and the projection and meet operators.
- 5 Transformations Rotors, translators, dilators and motors as versors, with their exponential and logarithmic maps.
- 6 Robot Modeling Forward kinematics as a product of motors, Jacobians as multivector matrices, and what geometric objectives buy in control.