A First-Order Approach to Model Simultaneous Control of Multiple Microrobots

The paper derives the expected-value translational ODE mv̇ = F R_φ u(t) − 6π η r v and projects it onto R_φ u(t) to obtain v̇_φ + (6π η r/m) v_φ = F/m, with steady-state speed v_ss = F/(6π η r) and a microsecond-scale time constant; it similarly shows magnetic torque causes rapid rotational alignment, culminating in the first-order model ṗ(t) = v_ss R_φ u(t) (eqs. (9)–(14)) . The candidate solution reproduces the same steps, explicitly solves the scalar ODE, and justifies the kinematic reduction after O(max{τ_v, τ_ω}) transients. Aside from a small dimensional slip in one variation-of-constants formula and the paper’s minor notational typos (Tm vs. Tprop; J vs. I), the arguments agree in substance. The modeling assumptions (low-Re Stokes drag, negligible magnetic force in translation, zero-mean Brownian noise, rapid magnetic alignment) are the same as in the paper .