# Motion planning and control Misc.

Published:

• The notion of configuration space is the key insight to Lagrangian mechanics of rigid bodies, as it allows dynamics to be expressed using the precise degrees of freedom of a body. [1]

• Why are bidirectional RRTs better for getting out of “bug trap”? [1] Intuitively, extending the nearest node in the sampled nodes (most of them are within the trap) to $q_{rand}$, is very likely to hit the “bug trap” wall (convex), but extending from another side (concave) will be easier. Because the trap looks differently (opposite) from another side.
• Linear-quadratic regulator. A continuous-time linear system: $\dot{x}=Ax+Bu$ with a quadratic cost function defined as: $J=x^T(t_1)F(t_1)x(t_1)+\int_{t_0}^{t_1}(x^TQx+u^Ru+2x^TNu)dt$

• Pfaffian constraints: a set of k linearly independent constraints linear in velocity. It may come from rolling without slipping. [2] $A(q)\dot{q}=0$

• Differentially flat systems:

• Lie bracket: Given two vector fields, the Lie bracket of the vector fields tells if infinitely small motions along these vector fields can generate new motions that do not belong to these vector fields. For example, in the parallel parking of cars, although cars can not move sideways, by combining the turning and forward-backward motion, the car can indirectly move sideways.
$[g_1,g_2] = \frac {\partial{g_2}}{\partial{x}} g_1 - \frac {\partial{g_1}}{\partial{x}} g_2$

[1] Steven M. Lavalle. Motion Planning: The Essentials
[2] Choset, Howie M., et al. Principles of robot motion: theory, algorithms, and implementation. MIT press, 2005.

To be continued…