Fig. 2. Path planning of microparticles in nonlinearly viscoelastic fluids.

a Open-loop control of a microparticle performing a square trajectory in 4% mucin. b Open-loop control of a microparticle in 4% mucin, which encounters a region of relatively high fluid resistance, disrupting its initial downward motion. Selected trajectories of a microparticle under closed-loop feedback control in 0.25% polyacrylamide (c, d) and 4% mucin (e–g), where magenta dots represent the intended targets and dashed lines represent shortest path. The times associated with each trajectory are (a) 80 s, (b) 95 s, (c) 156 s, (d) 145 s, (e) 94 s, (f) 60 s, and (g) 43 s. h Left: a simple 3D trajectory that was achieved using open-loop control. At the points indicated by the arrows, the microparticle translated downwards in the negative z-direction (black arrow) and then upwards in the positive z-direction (magenta arrow); the circular purple arrow signifies rotation direction from behind the heading vector, while the black circle and × represent out of the page and into the page, respectively. Right: the estimated 3D trajectory of the microparticles path and projections along the different planes; the total time of the trajectory was 114 s. Microparticles were rotated at (a–c, e–g) 19 Hz, (d) 40 Hz, and (h) 15 Hz. Source data are provided as a source data file. These experiments were repeatable for >90% of all particles examined in 4% mucin and >60% of all particles examined in 0.25% polyacrylamide.