Minimal activation with maximal reach: Reachability clouds of bio-inspired slender manipulators

TL;DR

This work establishes an efficient and robust reduced-order method to generate reachability clouds of almost half a million points each to visualize the accessible workspace of a wide variety of manipulator designs and generates an atlas of 256 reachability clouds.

Abstract

In the field of soft robotics, flexibility, adaptability, and functionality define a new era of robotic systems that can safely deform, reach, and grasp. To optimize the design of soft robotic systems, it is critical to understand their configuration space and full range of motion across a wide variety of design parameters. Here we integrate extreme mechanics and soft robotics to provide quantitative insights into the design of bio-inspired soft slender manipulators using the concept of reachability clouds. For a minimal three-actuator design inspired by the elephant trunk, we establish an efficient and robust reduced-order method to generate reachability clouds of almost half a million points each to visualize the accessible workspace of a wide variety of manipulator designs. We generate an atlas of 256 reachability clouds by systematically varying the key design parameters including the fiber count, revolution, tapering angle, and activation magnitude. Our results demonstrate that reachability clouds not only offer an immediately clear perspective into the inverse problem of control, but also introduce powerful metrics to characterize reachable volumes, unreachable regions, and actuator redundancy to quantify the performance of soft slender robots. Our study provides new insights into the design of soft robotic systems with minimal activation and maximal reach with potential applications in medical robotics, flexible manufacturing, and the autonomous exploration of space.

Figures (4)

• Figure 1: Analytical model and minimal design of bio-inspired slender manipulator. (a) Summary of continuum mechanics quantities pertinent to the reduced-order active filament theory kaczmarskiActiveFilamentsCurvature2022c. A one-dimensional initial configuration deforms through a map $\bm{\chi}$ to an activated configuration with a centerline $\mathbf{r}$ and directors $\mathbf{d}_i$. (b) Helical fiber revolution around a tapered structure (left), with two cross sections (right) extracted at two different $Z$ values. (c) An example of three fiber architectures with $n_1=n_2=n_3=1$ activatable bundles in each architecture (top), and an example cross-sectional placement of the bundles (bottom). (d) Front and back view of the manipulator design resulting from fiber architectures and bundle placements in (c), which constructs a tapered minimal design kaczmarskiMinimalDesignElephant2024 with activations $\gamma_1$, $\gamma_2$, $\gamma_3$ in the three actuators.
• Figure 2: Reachability cloud atlas for varying fiber revolutions and tapering angles. (a) Atlas of $7\times 7$ reachability clouds with 400,000 activation samples each, generated using ranges of $\Omega\in[0\degree, 108\degree]$ for the helical fiber revolution and $\phi\in[0\degree, 3\degree]$ for the tapering angle. We use an RGB color scale to color-code the cloud points, where the activation magnitudes $|\gamma_{1,i}|\in[0, 1.67]$ in the red, green, and blue fiber bundles contribute to the R, G, and B components of the RGB point color. (b) Opposing views of the four designs of the four corner cases in the atlas: (1) no fiber helicity and no tapering, (2) no fiber helicity and maximal tapering, (3) maximal fiber helicity and no tapering, (4) maximal fiber helicity and tapering.
• Figure 3: Reachability cloud volumes for varying fiber revolutions and tapering angles. (a) Tight concave hull mesh boundaries for all $7\times7$ reachability clouds in the atlas. (b) Concave hull volume normalized by $L^3$ and plotted as a function of the fiber revolution and tapering angles (top). Volume fraction of the unreachable region inside the convex hull plotted as a function of the fiber revolution and tapering angles (bottom). We generated both contour plots for the full atlas of $16 \times 16$ clouds. (c) Comparison of configuration versatility between a non-tapered design, marked blue in (a), and a tapered design, marked orange in (a). We plot the distribution of curvatures $\kappa L$ at 6 uniformly-spaced points in the distal half of the structure, and show that the configuration space of the tapered design is richer than that of the non-tapered design.
• Figure 4: Reachability clouds of the minimal and redundant designs. (a) Arrangement of three and four independently actuated fibers in the minimal design, top, and redundant design, bottom. Both designs contain two helical fibers, red and green; the minimal design includes one longitudinal fiber, blue, the redundant design includes two longitudinal fibers, blue and yellow. (b) Reachability clouds of the two designs color-coded by the activation magnitude $|\gamma_{3}|$ in the blue fiber for the minimal design, and $|\gamma_{4}|$ in the yellow fiber in the redundant design. Each cloud consists of 2 million points. (c) Slices of the corresponding reachability clouds in (b). All slices pass through the origin and, for each cloud, the top slice is in the frontal plane, while the bottom slice is in the right-facing plane. (d) Mean activation-space distance visualized over a 10,000-point subset of each design's reachability cloud for $r_\mathcal{S} = L / 60$. Regions with large mean activation distances indicate a significant effect of redundancy caused by large activation changes within small sub-volumes of the cloud. Light gray points depict spheres that only bound the center point and no other cloud points.