Question

For 1b, I have the alphas for initial, midpoint, and final configuration. How do I calculate the spline from this point? The notes and hw solutions do not make sense to me.

Answer 1

From the notes,
\[P_{simp}*\left(\begin{matrix}\alpha_i \\ \alpha_v\\ \alpha_f\end{matrix}\right)=\left(\begin{matrix}a_0 \\ a_1\\a_2\\a_3\\b_0\\b_1\\b_2\\b_3\end{matrix}\right)\]

Answer 2

So Psimp = (alphai; alphav; alphaf)^-1 * (a0;a1;a2;a3;b0;b1;b2;b3)
Is that what we are solving for?

Answer 3

No we want the coefficients for our two splines: a0-b3. There is a formula for Psimp in the notes. It only depends on the quantity t which in this case is 2.5.

Answer 4

a0 = pi ; a1 = 0 ; a2 = 3/(tf)^2 * (pf -pi) ; a3 = 2/(tf)^3 * (pi-pf)
Is this correct? If so, what is pi and pf?

Answer 5

This would be correct if we were only trying to connect two points with a spline. Here we are trying to connect three points with 2 splines. For each joint pi = alpha(0), pf = alpha(5), and pv = alpha(2.5). You need one trajectory that goes from 0 to 2.5(accounted for by the as) and one that goes from 2.5 to 5(accounted for by the bs).

Answer 6

Ok, so using the equations I just posted, I can calculate a0-a3 for my two splines. (giving me p(t) for both splines).

For q(t): I don't see anything conclusive on how to calculate b0-b3

Answer 7

p(t) is the function for 0<t<tv and q(t-tv) is the function for 2.5<t<tf. The cannot be calculcated separately because we want their velocities and accelerations to match at t=2.5 All the a's and b's are calculated at once using the formula I gave you above.

Answer 8

All I see that is relevant in the notes is : "8 coefficients, 8 equations":
p(0) = alphai q(0) = alphav
p(t0) = a0 qdot(tf-tv) = 0
pdot(0) = 0 q(tf-tv) = alphaf
pdot(tv) = qdot(v) pdoubledot(tv) = qdoubledot(0)

Answer 9

I made two columns of four equations here

Answer 10

Yes that's part of the way we derived Psimp. I think if you look in your notes you will find an 8x3 matrix that depends on t. If you multiply that matrix by [alphai; alphav; alphaf], you will get all the 8 coefficients you need.

Answer 11

ok, i found that. What do we use as our t value for our two different splines?

Answer 12

There is only 1 t value, it's the time step and is 2.5 in this case.

Answer 13

So if I plug in 2.5 for in the 3x8 matrix and multiply it by my alpha matrix, I will get my final answer?

Answer 14

for t*

Answer 15

What do you mean by alpha matrix?

Answer 16

(alphai, alphav, alphaf)^T

Answer 17

Yes for each joint. So in total you will have three sets of a's and b's

Answer 18

So for my final answer, I will simply have three sets of a's and b's?

Answer 19

Yes. for each joint you need two splines: one that goes to 2.5s and the other goes 2.5-5s

Answer 20

So, a p(t) function and q(t) functin for each joint?

Answer 21

yep

Answer 22

how do we get our alphav values from our calculated g(2.5)?

Answer 23

inverse kinematics

Answer 24

the triangle method?

Answer 25

Yeah it uses triangle geometry. You did it 2 homework ago. You can check the solutions if you don't remember.