Fractal branching ultra-dexterous robots (Bush robots) #
NASA ACRP Quarterly Report, August 1998, Hans Moravec

PR-Number 10-86888 Appropriation: 806/70110
CMU Cooperative Agreement NCC7-7

Quarterly Report
June 1, 1998 - August 31, 1998

Fractal branching ultra-dexterous robots
(Bush robots)

Hans Moravec

Carnegie Mellon University
Robotics Institute
5000 Forbes Avenue
Pittsburgh, PA 15213

August 30, 1998

Table of Contents


Section 1: Triangulating Bush Robot Skin

Section 2: Stewart Platform Branch Actuation

Section 3: Arbitrary Pose Triangulations

Appendix: “Rubber Tree” Bush Robot Configuration Program with Triangular Skin Tessellation


This is the seventh quarterly report for “Fractal branching ultra-dexterous robots.”

In this quarter we took steps toward producing a stereolithographic solid model of a bush robot with fingers in a planar fractal configuration. We devised a tessellation of the surface of a bush robot into triangles, as required by the software controlling stereolithography machines. The tessellation expresses each branch as six triangles. The edges of the tessellation are in the pattern of the actuators of a six-degree-of-freedom Stewart platform, suggesting a new, elegant way of implementing the motor functions of an animated bush robot. We had no trouble producing bushes with the triangulated skin in poses with constant branch angles. For general poses, we modified our “rubber tree” relaxation algorithm to make triangulated skin. Getting the smallest details right has proven to be a time-consuming trial-and-error job, and the program still generates some imperfections in the smallest, sharply angled, fingers that prevent stereolithographic implementation. We expect to resolve these in the next, final, quarter.

1: Triangulating Bush Robot Skin

This quarter’s efforts were directed at producing a 10 cm stereolithographic solid sculpture of a bush robot. The software controlling stereolithography machines depends on having descriptions of the objects to be fashioned, encoded in “STL” files, as enclosing skins composed solely of triangles joined only at vertices, with no gaps or overlaps. Until this quarter, our 3D models consisted largely of cylindrical sections somewhat sloppily penetrating spherical hubs. The geometric precision required in STL files make the job of constructing models much more demanding, as does the need to minimize the number of triangles in the enclosing skin.

It is possible to form a bush robot branch out of six triangles, connecting a three-sided base to a smaller three-sided apex, rotated about 60 degrees:

(each 3D image in this report will be labeled with a URL pointing to its generating Open Inventor file)

For a three-way branching robot, our preferred configuration, the bases of three small branches can be perfectly joined to the apex of a larger branch enclosing a tetrahedral-shaped void. The following illustration shows this done to two levels of branching:

Note that in the above construction, only the six triangles in the surface of each tapered branch are necessary to form the structure. All other triangles, for instance the faces of the tetrahedron internal to each joint, are only virtual.

There are six edges pairwise joining the six triangles in each branch. Interestingly, these edges suggest a plausible implementation for the mechanical actuation of actual working bush robots, using only length-changing linear actuators.

2: Stewart Platform Branch Actuation

A Stewart platform is an elegant configuration of six linear actuators in a circular triangulated truss that provides a full six degrees of 3D translational and rotational freedom. The edges of our bush robot skin triangulation are in the pattern of the actuators of a Stewart platform. A Stewart platform allows the position and angles of the apex plate (including the effective branch length) to be changed over significant ranges, though it only provides about + or – 30 degrees axial rotation. Since many conceivable bush robot motions involve multi-turn axial rotation oft one branch level or another, it may be desirable to add a redundant axial rotor at the base of each joint. Ignoring this refinement for the moment, here are two views of a Stewart platform bush branch. Each cylindrical strut is assumed to be a length-changing actuator like a piston or linear motor. The equilateral triangular end plates are rigid, and the pivots freely rotate:

An additional three equilateral triangles can be erected on the apex triangle, turning it into a tetrahedron, and providing support surfaces for the three smaller branches constituting the next level of the tree:

Adding scaled-down versions of the structure gives us a two-level actuated bush robot. Note that for this actuated structure, unlike the bounding skin depictions in the last section, the internal tetrahedrons must exist to give the structure essential rigidity. Of course, they need not be solid, as shown, but could consist of simply an open rigid framework. On the other hand, the enclosed tetrahedral volumes might be an ideal place to package the energy storage and computation machinery needed to control a real robot bush. Following are depictions of two-level and three-level Stewart platform robot bushes:

3: Arbitrary Pose Triangulations

It is quite easy to triangularly tessellate bushes with modest branch angles and little axial rotation at each branch. Here are views of a 7-level triangulated bush with equal-angle, non-twisting, branching:

It is much more difficult to find a triangulation for an arbitrary bush robot pose, which may contain sharp bend angles and significant twists at each joint. In particular, the only regular way for a B=3 D=2 bush to cover a surface is with a fractal pattern that involves a 45 degree twist from level to level. The pattern also tends to generate a pose where the smaller branchlets have increasingly wide splay angles, approaching 90 degrees for the smallest fingers. We have modified our old “rubber tree” bush posing program to generate triangular tessellated skin, but did not manage to achieve a perfect skin in this quarter - we still find overlaps in some of the triangles in the smallest fingers. We were thus unable to provide a suitable STL file for making a stereolithographic model in this period. At present debugging the configuration process is a time-consuming trial-and-error process. Some insights are emerging that may improve the process in future. Here are images of the results so far. We are trying to construct a model of a simple tree configuration where all the smallest fingers are distributed in a fractal pattern on a planar surface.


“Rubber Tree”
Bush Robot
Configuration Program
Triangular Skin Tessellation

/* Bush robot surface triangulation program tree.c */
/* */