Smooth acceleration

A jolt at the start

Your cubic starts and ends at rest, but look at its acceleration:

$$\ddot q(t) = \frac{6\Delta}{T^2}\,(1 - 2\tau)$$

At $t = 0$ it leaps from nothing to $6\Delta/T^2$, which is 2.7 rad/s² for a 1 rad move over 1.5 s. A sudden change in acceleration means infinite jerk: the motors are asked for an instant step in torque. A real arm feels that as a knock. It shakes the payload, sets the links vibrating and wears the gearboxes.

Two more conditions

Ask for zero acceleration at both ends as well. Six conditions need six coefficients, which gives a quintic:

$$q(t) = q_0 + \Delta\,(10\tau^3 - 15\tau^4 + 6\tau^5)$$

Its acceleration, $\frac{\Delta}{T^2}(60\tau - 180\tau^2 + 120\tau^3)$, starts at zero, swells, and fades back to zero. This is the minimum-jerk curve that move_joints and move_to used in Pick & Place.

	cubic	quintic
boundary conditions	4	6
$\ddot q$ at the ends	$\pm 6\Delta/T^2$	$0$
peak speed	$1.5\,\lvert\Delta\rvert/T$	$1.875\,\lvert\Delta\rvert/T$

The smoothness has a price: for the same duration the quintic has to move faster in the middle.

Seeing it

The starter plots the base joint's acceleration for both curves, measured as a second difference of each profile:

$$\ddot q(t) \approx \frac{q(t + \delta) - 2q(t) + q(t - \delta)}{\delta^2}$$

It streams whatever quintic returns, which for now is still the cubic.

Your task

Implement quintic(q0, qf, T, t) (vectors in, vector out, $\tau$ clamped as before) and stream the same three 1.5 s moves with it.

The graders average each joint's commanded acceleration over the first and last 30 ms of every move. It must stay under 1 rad/s². The cubic's jolt is two to three times that.