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Tony's Intertwingled Memex • Reference Manual

Electric Vehicle Acceleration & Trajectory Recovery

Formulas & Definitions

In electric and autonomous vehicle engineering, onboard inertial measurement units (IMUs) log high-frequency 3-axis accelerometer data. When GPS satellite signals are occluded (such as inside tunnels or subterranean parking garages), vehicle stability control systems reconstruct the vehicle's instantaneous velocity and position trajectory by performing successive vector integrations over time (dead reckoning).

An autonomous electric delivery vehicle enters an underground facility at t=0 secondst = 0\text{ seconds}. The onboard IMU records a 3-axis acceleration vector (in m/s2\text{m/s}^2) modeled by:

a⃗(t)=⟨6t,12t2,4⟩\vec{a}(t) = \langle 6t, 12t^2, 4 \rangle
At the moment of entering the facility (t=0t = 0), the vehicle's initial velocity is v⃗(0)=⟨2,−1,5⟩ m/s\vec{v}(0) = \langle 2, -1, 5 \rangle\text{ m/s} and its initial position vector relative to the entrance gate is r⃗(0)=⟨0,10,−3⟩ meters\vec{r}(0) = \langle 0, 10, -3 \rangle\text{ meters}.

  1. Use indefinite vector integration and the initial velocity condition to derive the vehicle's 3D velocity vector function v⃗(t)\vec{v}(t).
  2. Use indefinite vector integration and the initial position condition to derive the vehicle's 3D position trajectory function r⃗(t)\vec{r}(t).
  3. Calculate the net displacement vector Δr⃗\Delta \vec{r} of the vehicle during its first 3 seconds inside the facility (0≤t≤30 \le t \le 3) using a definite integral.