0.4 Optimize for Gas, Time, and Comfort

0.2 optimized time. 0.3 optimized gas. Real deliveries care about both — and about whether the ride feels like a roller coaster. That third score is comfort, and it starts with jerk.

What is jerk?

Acceleration $a$ is how fast velocity changes. Jerk $j$ is how fast acceleration changes:

Units are m/s³. In the bang-bang plans of 0.2–0.3, force (and $a$) jumped instantly between levels. That is an infinite spike of jerk at each switch — fine for a math toy, rough on a pizza and a neck.

Here the vehicle still obeys $m\ddot{x} = F$ with $a = F/m$, but $a$ may only slew toward a commanded value at a limited rate $|j| \le j_{\max}$. Soft $j_{\max}$ means gentle ramps of $a$; large $j_{\max}$ recovers nearly bang-bang.

Comfort as peak |jerk|

Many comfort metrics exist ($\int j^{2}\,\mathrm{d}t$, ride indices, …). We use a simple one that matches the hardware knob:

If the plan never asks for more than $j_{\max}$, then $J_{\mathrm{peak}} \le j_{\max}$. Lower $j_{\max}$ → lower peak jerk → more comfortable (and usually slower).

A three-term cost with equal weights

Time $T$, gas $G = \int_0^{T}\lvert F\rvert\,\mathrm{d}t$, and comfort $J_{\mathrm{peak}}$ have different units. You cannot literally add seconds to newton-seconds to m/s³. So we normalize each term by a fixed reference, then sum with equal weights $w_T = w_G = w_C = 1$:

The references ($T_{\mathrm{ref}}$, $G_{\mathrm{ref}}$, $J_{\mathrm{ref}}$) are constants shown in the playground — yardsticks so a “1” means “about as bad as a harsh baseline on that axis.” Changing the references would change the trade-off; keeping weights equal and refs fixed makes comparisons fair across presets.

Still the same door constraints:

Try it

Compare Harsh, Balanced, Comfort, and Sip gas + soft. Read the three normalized terms and the total $\mathcal{J}$. Open Data for $a$, $j$, $v$, $x$, and the running cost.

Discussion

A single number $\mathcal{J}$ hides a fight among objectives. Slamming force and jerk wins the clock and loses comfort (and often gas). Feathering $j_{\max}$ wins comfort and loses time. Small peak force sips gas but stretches $T$. Equal weights after normalizing do not mean “equally important in the real world” — they mean “we chose to treat a reference amount of each as one unit of regret.”

Later, Section 4.1 (LQR) will put those tastes into matrices $Q$ and $R$. Here the pizza run is enough: move the sliders, watch $\mathcal{J}$, and feel that multi-objective control is about choosing what hurts.

A note about not reaching the door exactly

The math wants $x(T) = D$ on the nose. The playground usually lands a little short.

Braking starts when a quick preview says “a jerk-limited stop still fits in the remaining distance.” That preview is discrete and padded slightly so the car does not blow past the door. Starting a bit early, plus freezing $x$ while $a$ settles to zero in the rest phase, means $v \to 0$ with $x < D$ more often than not.

“At the door” here means within a small tolerance — about half a meter, or $2\%$ of $D$, whichever is larger — the same spirit as sections 0.2 and 0.3. Only if the trajectory overshoots does the sim snap $x$ back to $D$. Exact landing is a harder planning problem; for this chapter, a comfortable stop near the door is enough.

If you want to learn how to reach $x = D$ without error, keep reading — that comes in Section 4.2.