# What are the applications of Differential Geometry in Robotics?

I am taking up a Grad level course on Differential Geometry. Can any one please tell me the immediate applications of Differential Geometry in Robotics ?

Thanks

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hi. I think you meant to ask this in Mathematics forum? This forum is Mathematica (With that extra a at the end). – Nasser M. Abbasi Nov 12 '12 at 6:04
I don't know, but the "state" of a robot is probably described by several parameters -- various angles and such, concatenated into a single vector -- and all possible states form a manifold. As the robot moves, its state vector follows a path along this manifold. – littleO Nov 12 '12 at 7:49

I have no specialized knowledge in robotics itself, but I believe that the main connection between differential geometry and robotics comes from the powerful applications of differential geometry to mechanics in general. In particular, in its most general form, the mechanical motion of a system in the Hamiltonian formalism can be cast as a flow along a vector field, called a Hamiltonian vector field, on a special type of manifold called a symplectic manifold.

The first thing you learn in differential geometry is that you can calculate things about your manifold by passing to local coordinates. In a symplectic manifold, the local coordinates used are precisely the position and momentum coordinates, $(p,q)$. If you recall from mechanics, the coordinates $(p,q)$ define the phase space for a mechanic system. If we just have a particle in open space moving around under the influence of some potential, then our phase space is $\mathbb{R}^{6}$, because we have 3 spatial and 3 momentum coordinates. However, if our system has some constraints, such as some rigid body, then instead of imposing a whole bunch of equations that represent the constraint, it is more natural to simply model the system as motion along some blob (the manifold) in phase space that represents the constraints.

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There is some discussion of (a special case of) robotics in the following notes by Bjørn Dundas: Differential topology.

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Generally, the "Configuration space", that is, the collection of all possible configurations of a robot, forms a relatively high dimensional manifold (perhaps with boundary, or corners, or something like that). For example, if you're robot consists solely of an arm and hand, the shoulder joint provides $2$ degrees of freedom, the elbow $1$, the wrist $2$, the thumb $2$, and each of the rest of the fingers $3$ (one for each joint).

In total, that's a $19$ dimensional manifold.

Beyond this, consider the following question: You start with the robot hand raised in the air as if to say "hi", and then have the arm come down to shake hands. What's the best path of each joint individually to get it from "hi" to handshaking?

The answer to this depends on what you mean by "best", and mathematically, "best" depends on the choice of a Riemannian metric on this manifold. Once you have this metric, you can compute geodesics (at least in principle), which will give you best paths between configurations.

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