CS 8803RMP
Robot Motion Planning    

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Reading assignments should be completed before the lecture for which they are assigned.
References are abbreviated as listed on the Resources Page for the course.

Date Topic Reading Assignment  
Jan. 8 General course overview [Choset] Chapter 1
Jan. 10 Configuration space for rigid bodies: Rigid motions, SO(n), SE(n)
  • The concept of configuration spaces is introduced in Chapter 3 of [Choset], Chapter 2 of [Latombe], Chapter 4 of [LaValle]. The level of mathematical treatment, and the number of examples, varies considerably among these references.
  • An introductory-level presentation of SO(n) and SE(n) is given in Chap. 2 of [SHV].
  • A treatment of SO(n) as a manifold is given in Chap. 2 of [Latombe].
  • A general treatment, including exponential coordinates for rotation, is given in Chap. 2 of [MLS].
  • An introduction to SO(2) and SO(3), along with some nice examples, is given in Chapter 3 of [LaValle].
Jan. 15 Concepts in topology and differentiable manifolds I: metrics, open sets, neighborhoods, metrics, open sets, induced topology, homeomorphism, diffeomorphism, manifolds, the implicit function theorem This material is covered in Chapter 3 of [Choset], Chapter 3 of [Latombe], and Sections 4.1 and 4.2 of [LaValle], which is thorough and accessible, with several general examples.
Jan. 17 Concepts in topology and differentiable manifolds II: SO(2) revisited, charts and atlases, differentiable manifolds, charts and atlases for SO(n). This material is covered in Chapter 3 of [Choset], Chapter 3 of [Latombe], and Sections 4.1 and 4.2 of [LaValle], which is thorough and accessible, wish several general examples.
Jan. 22 Parameterizations of SE(n): Coordinate transformations and composition of transformations; Euler angles; axis/angle parameterizations; stereographic parameterization of SO(2)
Jan. 24 Quaternions Quaternions are discussed in Appendix E [Choset], and Section 4.2 of [LaValle].
Jan. 29 Snow Day (really)
Jan. 31 Path planning for Euclidean configuration spaces: Visibility graphs, trapezoidal cell decomposition
  • Visibility graphs are introduced in Section 5.1 of [Choset], Section 4.1 of [Latombe], and Section 6.2.4 of [LaValle]. The general sweep line algorithm for finding intersections of line segments in the plane is described in [Latombe] Appendix D.
  • Cell decomposition for polygonal configuration spaces (i.e., trapezoidal decomposition) is introduced in Section 6.1 of [Choset], Section 5.1 of [Latombe], and Section 6.3 of [LaValle].
Feb. 5 No Class
Feb. 7 No Class
Feb. 12 The generalized Voronoi diagram The generalized Voronoi diagrams is introduced in Section 5.2 of [Choset], Section 4.2 of [Latombe], and Section 6.2.3 of [LaValle].
Feb. 14 Path planing using Artificial Potential Field: Euclidean spaces This material is covered in Secs. 4.1-4.4 of [Choset], Secs. 5.1-5.3 of [SHV], and Sec. 7.1 of [Latombe].
Feb. 19 Path planning using potential functions: Navigation Functions This material is covered in Sec. 4.6 of [Choset].
Feb. 21 Path planning using potential functions: Non-Euclidean spaces This material is covered in Sec. 4.7 of [Choset], Secs. 5.1-5.3 of [SHV], and Sec. 7.2 of [Latombe].
Feb. 26 The manipulator Jacobian: Robot arm kinematics, geometric formulation of manipulator Jacobian This material is covered many intro robotics texts, e.g., [SHV] chpt 3 for kinematics and chpt 4 for the manipulator Jacobian.
Feb. 28 Randomized methods Randomized Path Planner (RPP), Introduction to PRMs Section 5.4.3 of [LaValle] describes RPP.
Mar. 5 Probabilistic Roadmap Planner (PRM)
  • PRM is introduced d in Section 7.1 of [Choset].
  • Section 5.4 of [LaValle] gives a good overview of sampling-based planners and various implementation issues, and Section 5.6 describes the basic PRM algorithm.
Mar. 7 Probabilistic completeness of PRM
  • A proof that PRM is probabilistically complete is given in Section 7.4.1 of [Choset].
  • A discussion of various notions of completeness is given in the begining of Chap. 5 of [LaValle].
Mar. 12 Variations on sampling-based planners This material is covered in Section 7.1.3 [Choset] and Section 5.6 of [LaValle].
Mar. 14 Rapidly Exploring Random Trees (RRTs)
  • Section 5.5 of [LaValle] covers this material very well.
  • This material is also covered in Section 7.2 in [Choset].
Mar. 19 Spring Break
Mar. 21 Spring Break
Mar. 26 Optimality for sampling-based methods: strong and weak delta-clearance (with a short digression on homotopy), connection complexity as a function of connection radius The algorithms PRM* and RRT* are described in an IJRR article by Karaman and Frazzoli.
Mar. 28 No Class
Apr. 2 Asymptotic optimality of PRM* (Part 1): building a sequence of paths that converges to an optimal path, tiling these paths with spheres, probability of failing to find a polygonal path whose vertices lie in the tiling, the Borel-Cantelli lemma
Apr. 4 Asymptotic optimality of PRM* (Part 2): proof of probabilistic completeness, proof of asymptotic optimality
Apr. 9 Introduction to Optimal Control:
Formulation of the optimal control problem, running and terminals cost, the value function, the principle of optimality for continuous time systems, and a derivation of the Hamilton-Jacobi-Bellman (HJB) equation. 		The material in this lecture is the classical approach to deriving the HJB equation. As such, it is covered in many text books, tutorials, and handbooks. Of course one ninety-minute lecture is not sufficient for a full development of this material, so it's worth choosing an optimal control book for your own library. My favorite is [Liberzon]. It's a nice addition to any e-library.
Apr. 11 Tools for Optimization-Based Planning I:
LQR (infinite and finite horizon), approximate linearization of nonlinear systems, Lyapunov theorems, candidate Lyapunov functions, basin of attraction, semi-definite programming. 		LQR is a classical topic. You can read about in many textbooks, tutorial articles, and handbook chapters. I recommend the treatment given in [Liberzon].
Apr. 16 Tools for Optimization-Based Planning II:
Approximate linearization of nonlinear systems, Lyapunov theorems, candidate Lyapunov functions, basin of attraction, semi-definite programming.
  • Nonlinear control, linearization, Lyapunov material is all classical in the world of nonlinear control, and you can read about these topics in many textbooks, tutorial articles, and handbook chpaters. As a genearal reference, I prefer [Khalil].
  • My favorite treatment of convex programming is this book by Stephen Boyd (pdf available at this link), but even better than the book is his set of online lectures, which you can find here. (I have these lectures on my iPad, for long-distance flights that have boring in-flight movies!)
Apr. 18 Tools for Optimization-Based Planning III:
Continuation of semi-definite programming, and introduction to sum of squares optimization, sampling-based planning 		The presentation of SOS optimization is heavily influenced by the approach taken by Russ Tedrake in his EdX course on underactuated systems.
The sampling-based approaches are based on the LQR-Trees, which are described in [LQRtrees]. Tedrake's group has published a number of related papers that also exploit SOS to characterize basins of attraction for nonlinear systems.
Apr. 23