1. CS 326 A: Motion Planning http://robotics.stanford.edu/~latombe/cs326/2003Nonholonomic and Under-Actuated Robots

2. Fewer actuators / controls than dimensions of configuration space  Some “confusion” in literature about what a degree of freedom is: dimension of C-space or control?

3. How can we span a C-space of dimension n > m with only m actuators/controls?  Mechanics of the task, e.g.: - Rolling contact: car, bicycle, roller skate - Conservation of angular momentum: cat, satellite robot -Others: submarine, plane, object pushing Why? - Fewer actuators (less weight) - Design simplicity - Convenience (think about driving a holonomic car!)

5. Example: Car-Like Robot y x    dx/dt = v cos  dy/dt = v sin  d  dt = (v/L) tan    | <  dy/dx = tan  Configuration space is 3-dimensional: ( x, y,  ) But control space is 2-dimensional: ( v,  ) L A robot is nonholonomic if its motion is constrained by a non-integrable equation of the form f(q,q’) = 0 Lower-bounded turning radius

6. How Can This Work? Tangent Space/Velocity Space x y  (x,y,  ) 

7. Nonholonomic Path Planning Approaches  Two-phase planning: (Laumond’s paper)  Compute collision-free path ignoring nonholonomic constraints  Transform this path into a nonholonomic one  Efficient, but possible only if robot is “controllable”  Plus need to have “good” set of maneuvers  Direct planning: (Barraquand-Latombe’s paper)  Build a tree of milestones until one is close enough to the goal (deterministic or randomized)  Robot need not be controllable  In general, more time than 2-phase approach

8. Path Transform Holonomic path Nonholonomic path

9. Nonholonomic vs. Dynamic Constraints  Nonholonomic constraint: f(q,q’) = 0  Dynamic constraint: g(q,q’,q’’) =0 (1) Let s = (q,q’). Eq. (1) becomes: G(s,s’) = 0  Similar techniques to handle nonholonomic and dynamic constraints (kinodynamic planning)