1. Motion Planning

2. Basic Topology Definitions  Open set / closed set  Boundary point / interior point / closure  Continuous function  Parametric curve  Trajectory  Connected set  Topological mapping (homeomorphism)

3. Basic Motion Planning Problem  A – a single rigid object ( robot ) moving in a Euclidean space W ( workspace ) with no kinematical constraints  B i – fixed rigid objects ( obstacles) distributed in W  Given an initial and a goal position and orientation of A in W, generate a path specifying a continuous sequence of positions and orientations of A avoiding contact with the B i ’s, starting at the initial position and orientation, and terminating at the goal position and orientation. Report failure if no such path exists.

4. Configuration Space  Configuration: 2D: q = (x,y,θ) ∈ R² 3D: q = (x,y,z, θ 1, θ 2, θ 3 ) ∈ R³  C-obstacle CB i = {q ∈ C : A(q) ⋂ B i ≠ ⊘ } a closed subset of C  Free C-space C free = C – ⋃ int( CB i ) an open subset of C

5. Example 1 : translation

7. Properties  CB i includes obstacle B i ( i=1..n )  A and B i ( i=1..n ) are both polygons => CB i polygon  A and B i ( i=1..n ) are both convex => CB i is convex  Vertices of CB i are ( for the example picture ): d kj = b j – r k, (k=1..4, j=1..3 )

8. Definitions:  Free space configuration – any configuration in free space C free.  Free path between two free configurations q init and q goal is a continuous map τ : [0,1]→ C free, with τ(0) = q init and τ(1) = q goal.  Two configurations belong to the same connected component of C free iff they are connected by a free path.

9.  Semi-free path between two configurations q init and q goal is a continuous map τ : [0,1]→ cl( C free), with τ(0) = q init and τ(1) = q goal.  Contact space is the subset of C made of configurations at which A touches one or several obstacles without overlapping any.  Valid space – the union of free space and contact space; the set of configurations achievable by A when we accept contacts between A and obstacles.

10.  A valid path between two configurations q init and q goal in C valid is a continuous map τ : [0,1]→ C valid, with τ(0) = q init and τ(1) = q goal  A contact path between two configurations q init and q goal in C contact is a continuous map τ : [0,1]→ C contact, with τ(0) = q init and τ(1) = q goal