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ZJU-FAST-Lab / GCOPTER

A General-Purpose Trajectory Optimizer for Multicopters
MIT License
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MINCO vs B-splines #25

Open JzHuai0108 opened 9 months ago

JzHuai0108 commented 9 months ago

Dear Dr Wang,

GCopter is an amazing traj opt framework that allows efficient opt of control points and knot placements simultaneously. Your wonderful thesis gives detailed explanation of gcopter and its applications. I'd like to learn more about the minco traj representation, esp. regarding its comparison to the traditional b-splines. The below pics are from your thesis,

1708431291092

1708430813845 There are two points that are baffling to me.

  1. Why do the spatial temporal parameters of B-splines highly coupled compared to Bezier curves and MINCO? for instance, for the 4th order B spline, p(t) = f(c_i-3, c_i-2, c_i-1, c_i, t_i-3, t_i-2, t_i-1, t_i, t_i+1, t_i+2, t_i+3, t_i+4, t), where t is in [t_i, t_i+1), ie, p(t) = f(c_i-3, c_i-2, c_i-1, c_i, T_i-3, T_i-2, T_i-1, T_i, T_i+1, T_i+2, T_i+3, T_i+4, t - t_i), where T_i = t_i - t_i-1. Therefore, for a constraint W(c, T), we can efficiently compute its derivatives relative to the control points c_i and the time segments T_i by taking advantage of the B splines' local support property. The derivative computation for the B splines apparently is less involved than the derivations given in your thesis, eg, eq 3-84 and 3-93. Let's look at eq 3-93. The main complexity comes from G_i which is from solving M G = dK/dc. I think solving this equation causes G_i to be dependent on time segments other than T_i, may be all the way from T_1 to T_M. In contrast, the derivative of a B spline relative to time is only dependent on finite time segments, 8 for the above 4th order case, T_i-3, up to T_i+4. In this sense, the spatial temporal coupling of MINCO is higher than B-splines. Please correct me if I am wrong. I know that often times I am wrong.

  2. What is exactly the 凸包保守度？I know that B splines enjoy the convex hull property as explained in the ego planner paper, but the conservativeness of the convex hull sounds hard to understand to me.

JzHuai0108 commented 3 months ago

Hi, old self, regarding question 2, please refer to fig 1 of this paper.