Description

PART 5: Trajectory generation and control

Consider the following 1-D quintic spline:
x(t) = a0 + a1t + a2t2 + a3t3 + a4t4 + a5t5.
Find the constants ai, i = 0,1,2,3,4,5, to respect the constraints x(0) = 0, x˙(0) = 0, x¨(0) = 0, x(1) = 1, x˙(1) = 0, and ¨x(1) = 0.
Exercise 5.2: B-splines

Given the Cox-de Boor recusion formula:
(
1 if ti ≤ t < ti+1
Ni,0(t) =
0 otherwise
,
Compute the two first order basis functions N0,1(t) and N1,1(t) with ti = i and combine them linearly to interpolate between the two points (1,2) and (2,3).
Exercise 5.3: computation of a DCM ?

In Mellinger and Kumar (2011),
a Direction Cosine Matrix (DCM) is found knowing the upward direction of the drone zB and the yaw angle ψ:
.
However, this is based on the Z-X-Y Euler angles convention, and the convention we are using in this course is Z-Y-X, i.e. RB = Rz(ψ)Ry(θ)Rx(φ) with the roll, pitch and yaw angles φ, θ and ψ (respectively), and
and .
Compute RB based on zB and ψ in the Z-Y-X Euler angles convention.
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