(Broad) Outline

1. Motivation
2. A Roboticist's Intro to Differential Geometry
3. Motion Planning in Non-Euclidean Spaces

(Broad) Outline

1. Motivation
2. A Roboticist's Intro to Differential Geometry
3. Motion Planning in Non-Euclidean Spaces

Motion Planning with GCS

• Represent C-Free with convex sets
• Formulate shortest path problem as MICP
• Can solve to optimality (branch and bound)
• In practice, relax and round yields a good result

What Is a Non-Euclidean Configuration Space?

(No joint limits)

"Locally" Euclidean Spaces

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Manifolds

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Continuity (Intuition)

Continuous functions take nearby points to nearby points

What is a Topological Space?

A rule for which functions are continuous by defining what "nearby" means

Explicitly: a rule for constructing arbitrarily small* neighborhoods around points

Examples:

(Path) Connectedness

A topological space is path connected if there is a continuous curve joining any two points.

A continuous curve is the image of a function \(f:[0,1]\to X\)

Compactness

A set is compact if every open cover has a finite subcover

Heine-Borel Theorem: In Euclidean spaces, compact iff closed and bounded

Extreme Value Theorem: Functions over compact sets have max and min values (useful for optimization theory)

Topology Summary

• Continuity
  • Does a function map nearby points to nearby points?
• Path-Connectedness
  • Can one "move" between two points?
• Compactness
  • Do functions have extrema?

Definition of a Manifold

A manifold is a locally-Euclidean topological space

Definition of a Manifold

A manifold is a locally-Euclidean topological space

Definition of a Manifold

Locally-Euclidean: any \(x\in\mathcal{M}\) is contained in an open set \(U\subseteq\mathcal{M}\) such that \(U\) is homeomorphic to a subset of Euclidean space

A manifold is a locally-Euclidean topological space

Charts and Atlases

Coordinate chart: An open set (sufficiently small neighborhood) \(U\subseteq\mathcal{M}\) with a homeomorphism \(\psi\)

Atlas: A collection of charts that cover \(\mathcal{M}\)

Transition Map: How to move from one chart to another

Smooth Manifolds

A manifold is smooth if its atlas has smooth transition maps

Intuition: no corners or edges

Can have different levels of smoothness (\(n\)-times continuously-differentiable vs analytic)

Derivatives and Integrals

We can compute directional derivatives for functions

We can take integrals of functions, vector fields, etc.

Interesting, but not relevant for the current project

The Tangent Space

Given a point \(x\in M\), we can consider the velocity of curves that pass through \(x\)

This forms a vector space \(T_xM\), called the tangent space

Maps Between Manifolds

Let \(M\) and \(N\) be two manifolds, and \(\phi:M\to N\) a smooth function

We can use \(\phi\) to "move information" between \(M\) and \(N\)

Pushforward

The pushforward \(d\phi\) moves tangent vectors from \(M\) to \(N\)

Pullback

The pullback \(\phi^*\) moves functions from \(N\) to \(M\)

\(\phi^*f=f\circ\phi\)

Application: RMPs

Riemannian Motion Policies use pushforwards and pullbacks to unify controllers across different configuration spaces

Product Manifolds

The Cartesian product of two manifolds is itself a manifold

Lie Groups

What if a manifold is also a group?

Need the group operation and inversion to be smooth maps

Examples: SO(3), SE(3)

Riemannian Manifolds

Motivation: Distance

Arc length depends on the parameterization of the charts

Motivation: Curvature

Triangles and angles look different

Riemannian Metric

A Riemannian Metric assigns an inner product to each tangent space

The inner product must be a positive definite bilinear form, and it must vary smoothly along the manifold

Allows us to define curvature, and compute distances

Curvature

• Describes how the local geometry differs from Euclidean
• Explicitly, a fourth-order tensor (with complex symmetries)
• Intuition: what do angles and triangles look like?

Arc Length

Consider a piecewise-differentiable curve \(\gamma\in\mathcal{C}^{1}_{\operatorname{pw}}([a,b],\mathcal{M})\)

Its arc length is \(\displaystyle L(\gamma)=\int_a^b \sqrt{\left<\gamma'(s),\gamma'(s)\right>^{\mathcal{M}}_{\gamma(s)}}\,ds\)

Geodesics

A geodesic is a (locally) length-minimizing curve -- solution to

\[\operatorname{arg}\,\inf\{L(\gamma)\,|\,\gamma\in\mathcal{C}^{1}_{\operatorname{pw}}([0,1],\mathcal{M}),\gamma(0)=p,\gamma(1)=q\}\]

Variational calculus problem -- solution satisfies Euler-Lagrange system of differential equations

As a result, each geodesic is uniquely defined by its initial conditions

Computation: forward integrate the differential equations

Examples of Geodesics

Exponential Map

Recall: tangent vectors are velocities of curves through a point

Exponential map intuition:

• Can extend a tangent vector to a geodesic
• Length of the vector corresponds to distance along the geodesic
• Yields a coordinate chart, called the "normal" coordinates

Geodesic Convexity

• Bring the ideas of convexity to Riemannian manifolds
• Enables optimization with strong guarantees
• Young but growing area of study

Convex Sets

A set \(X\subseteq\mathbb{R}^n\) is convex if \(\forall p,q\in X\), the line connecting \(p\) and \(q\) is contained in \(X\)

Some Sets Look Convex

(But Aren't)

Sets can be convex in local coordinates, but not geodesically-convex on the manifold

Geodesics can go "the wrong way around"

Radius of convexity: how large can a set be and still be convex?

Convex Functions

A function \(f:X\to\mathbb{R}\) is convex if for any \(x,y\in X\) and \(\lambda\in[0,1]\), we have

\(f(\lambda x+(1-\lambda)y)\le\)

\(\lambda f(x)+(1-\lambda) f(y)\)

Convex Functions

A Graph of Geodesically Convex Sets (GGCS)

A directed graph \(G=(V,E)\)

• For each vertex \(v\in V\), there is a strongly geodesically convex manifold \(\mathcal{Y}_v\)
• For each edge \(e=(u,v)\in E\), there is a cost function \(\ell_e^\mathcal{Y}:\mathcal{Y}_u\times\mathcal{Y}_v\to\mathbb{R}\)
• \(\ell_e^\mathcal{Y}\) must be geodesically convex with respect to the product metric on \(\mathcal{Y}_u\times\mathcal{Y}_v\)

GCS

GGCS

Shortest Paths in GGCS

Mixed-Integer Riemannian Optimization? No existing literature.

Pullback of Sets and Edges

Shortest Paths in GGCS

Mixed-Integer Riemannian Optimization? No existing literature.

Problem Statement

Configuration space \(\mathcal{Q}\) as a Riemannian manifold

\(\mathcal{M}\subseteq\mathcal{Q}\) is the set of collision free configurations, \(\bar{\mathcal{M}}\) its closure

The shortest path between \(p,q\in\bar{\mathcal{M}}\) is the solution to the optimization problem

\[\begin{array}{rl}    \argmin & L(\gamma)\\    \textrm{subject to} & \gamma\in\mathcal{C}^{1}_{\operatorname{pw}}([0,1],\bar{\mathcal{M}})\\    & \gamma(0)=p\\    & \gamma(1)=q\end{array}\]

Additional Requirements

\(\mathcal{Q}\) is zero-curvature

• Negative-curvature manifolds not common in robotics
• Distance metric on positive curvatures is nonconvex 😡

Zero-Curvature is Still Useful

• All 1-dimensional manifolds
• Continuous revolute joints
• \(\operatorname{SO}(2)\), \(\operatorname{SE}(2)\)
• Products of flat manifolds

Fetch Robot Configuration

Space: \(\operatorname{SE}(2)\times \mathbb{T}^3\times \mathbb{R}^7\)

Additional Requirements

\(\mathcal{Q}\) is zero-curvature

• Negative-curvature manifolds not common in robotics
• Distance metric on positive curvatures is nonconvex

\(\mathcal{A}=\{(X_v,\psi_v)\}_{v\in V}\) is a finite atlas of \(\mathcal{M}\), with each \(X_v\) strongly geodesically convex, and each \(\psi_v\) an isometry

• Necessary for computation
• Leads to nice theoretical results

Reminder: Transition Maps

How to move from one chart to another

Reformulating as GCS

Decision variables:

\(\forall v\in V\), we have \(x_v=(x_{v,0},x_{v,1})\in\psi_v(X_v)^2\)

Edge costs:

\[\ell_e(x_u,x_v)=||x_{u,0}-x_{u,1}||_2,\quad\forall e=(u,v)\in E\]

Edge constraints:

\[\tau_{u,v}(x_{u,1})=x_{v,0},\quad\forall e=(u,v)\in E\]

(\(\tau_{u,v}\) is the transition map from \(X_u\) to \(X_v\))

Reformulating as GCS

Decision variables:

\(\forall v\in V\), we have \(x_v=(x_{v,0},x_{v,1})\in\psi_v(X_v)^2\)

Edge costs:

\[\ell_e(x_u,x_v)=||x_{u,0}-x_{u,1}||_2,\quad\forall e=(u,v)\in E\]

Edge constraints:

\[\tau_{u,v}(x_{u,1})=x_{v,0},\quad\forall e=(u,v)\in E\]

(\(\tau_{u,v}\) is the transition map from \(X_u\) to \(X_v\))

Nice Theoretical Results?

• There is a shortest path \(\gamma\) connecting \(p\) to \(q\) through \(\bar{\mathcal{M}}\) such that
  • \(\gamma\) is piecewise geodesic
  • Each geodesic segment is contained in some \(\bar{X}_u\)
  • \(\gamma\) passes through each \(\bar{X}_u\) at most once
• Ensures that a shortest path is feasible for GGCS
   
• Transition maps are affine functions (in particular, they are an orthogonal transformation plus a translation)
   
• Formal equivalence of GGCS and transformed GCS problem

Results

"Asteroids" World

Square with opposite edges identified

Configuration Space: \(\mathbb{T}^2\)

Planar Arm

A robot arm with 5 continuous revolute joints in the plane

Configuration Space: \(\mathbb{T}^5\)

(No self collisions)

(Modified) Kuka iiwa Arm

PR2 Whole Body Plans

Comparison: PRM + TrajOpt

• Build an initial plan with a PRM
• Post-process with kinematic trajectory optimization

Practicum

Generating Atlases

Key idea: generate within a chart in the first place

Use IRIS in configuration space, add bounds on the diameter of the set to preserve strong convexity

Further Motion Planning

All Bézier curve constructions also hold on the manifold

For higher-order continuity constraints, want tangent vectors to be equal. Use the pushforward!

Higher Order Continuity

Order \(0\) continuity, order \(k\) splines:

\[\tau_{u,v}(x_{u,k})=x_{v,0}\quad\forall e=(u,v)\in E\]

Order \(m\) continuity, order \(k\) splines:

\[(\tau_{u,v})_*(x^{(m)}_{u,k})=x^{(m)}_{v,0}\quad\forall e=(u,v)\in E\]

If \(\tau_{u,v}\) is an isometry, then \(\tau_{u,v}(x)=R(x)+T(x)\) for \(R\in\operatorname{O}(n)\) and \(T\in\operatorname{E}(n)\). Thus, \((\tau_{u,v})_*=R\).

Globally-Aligned Coordinates

In some (most?) cases, we can align the coordinate systems

If so, we have \(\tau_{u,v}\in \operatorname{T}(n)\) and \((\tau_{u,v})_*=I_n\), \(\forall e=(u,v)\in E\)