### Hugo Hadfield

Cambridge University PhD student, Signal Processing and Communications Laboratory

- This chapter introduces the concept of mirror objects, projectors and finding the rotor that takes one object to another
- Here is the first time we give an expression for the mirror object in terms of the
**addition**of two object blades

- Here we explore in detail the properties of the blade reprojection and linear interpolant objects
- This is the first time we touch on screw theory, although in a very limited way, in the context of the interpolation of lines
- Finally we look at some basic examples of applications

- In this chapter we describe
**higher order interpolations between objects**and how surfaces made of these objects would interact with rays of light - We then build a basic ray-tracer that can image a range of different objects and show how we can do texture mapping for these objects
- https://hugohadfield.github.io/tube_vis/

- In this chapter we make use of the rotor between objects that we described in Chapter 2 to construct cost functions between objects that we can use as an optimisation target for finding unknown transformations between noisy objects in an
**ICP+Ransac**like scheme - We then also leverage the rotor between objects to design
**fast heuristic algorithms**to solve the same problem

Constrained Rigid Body Dynamics

- This chapter is about statics, dynamics, and screw theory
- First we describe forces and moments as lines and compare and contrast this force and moment representation with other examples from the literature
- Then we work through how the operators of
**algebraic Screw Theory**embed into GA - We then do dynamics, showing how the
**inertia tensor**maps into CGA and PGA and how to represent non-axis-aligned inertia tensors - We show how to impose constraints on our dynamics via traditional
**virtual power**techniques and then via novel**multivector pinning** - Finally we show how integration could be done in the
**bivector domain**with a range of lie algebra mappings

Geometric Algebra

- This chapter is about multi-body systems: joints, articulated robots etc.
- We describe how to use the richness of CGA to model the various kinematic joints commonly used in robotic mechanisms.
**shared geometry**and**invariant bilinear operations**. - We show how to form a collection of joint constraints into a matrix and use this matrix to solve articulated body problems in robotics
- We compare and contrast this screw theory based constraint matrix approach vs a direct calculation for the forward and inverse kinematics of the delta robot

**We added blades together and it was, in general, a really good idea!**- We showed how to use interpolated objects in a range of applications in graphics, vision and robotics
- We described how Screw Theory embeds into CGA and PGA
- We showed how to do statics, kinematics and constrained dynamics in CGA and PGA
- We derived the kinematic equations for several mappings from se3 to SE3
- We showed how to model the kinematics of jointed robots in CGA and PGA using a screw theory inspired method

By Hugo Hadfield

- 131

Cambridge University PhD student, Signal Processing and Communications Laboratory