I'm not sure if Advanced Algebra is the correct forum for this, but if not maybe it can be moved elsewhere?
Anyways, I am taking this class where we want to solve coefficients for a cubic polynomial. This handout reads:
Suppose we want to find the motion trajectory for a path with only two knot points - the start and goal, which satisfier the time constraint that the tie duration of motion is T, and in addition, satisfier the velocity constraint that the omotion starts from zero velocity and ends with zero velocity. Then, for each variable qi, its trajectory as a function of time t should satisfy the following 4 constraints:
- qi(0) = si
- qi'(0) = 0
- qi(T) = gi
- qi(T) = 0
Now the question is if we try to interpolate the trajectory of qi, we should use a cubic polynomial of t
qi(t) = a0 + a1t + a2t2 + a3t3because we can determine the 4 coefficients ai, i=0,1,2,3 from the above 4 constraints. The following values for the coefficients can be easily derived:
- a0 = si
- a1 = 0
- a2 = (3/T2) * (gi - si)
- a3 = (2/T3) * (gi - si)
I don't understand how the a0,a1,a2,a3 are being derived. The things I have considered are:
For a0: Are we setting t=0? So then we have: si = a0 + a1(0) + a2(0) + a3(0).
For a1: We have si = si + a1t + a2(0) + a3(0) ? I don't think that is right, but it's the only way I see to set it up so that a1=0.
For a2: I'm pretty lost here. I see that qi - si is the displacement...but how are we getting 3/T2??
For a3: Same problem as with a2
Can anyone help me out here? I tried to summarize the handout text some, but I can try to copy more of it if the information is insufficient. Any help is appreciated.