Research Overview

Our recent research focuses on practical parametrization and efficient path planning of linkage systems involving loops.  Many practical multi-body systems involve one or more loops -- physical (in parallel platforms, ring-type molecules, . . . ) or virtual (in inverse kinematics of serial manipulators or molecular chains,. . . ). Studying the kinematics of such systems has been challenging, partly because of the requirement of maintaining loop closure constraints, which have conventionally been formulated as highly  nonlinear equations in joint parameters.

[Deformation Space] Since loop closure constraints are intrinsic properties to the linkage systems and independent of rigid motions, we will mainly consider the Deformation Space (briefly DSpace) of the system under study, which is mathematically equivalent to the quotient space of the system configuration space modulo rigid motions of ambient space respecting all  system specifications.

DSpace = CSpace(system) / RM(system).

When a planar linkage has a link fixed in the plane, its configuration space and deformation space are identical. For a free-flying planar linkage, its configuration space is roughly the product of the deformation space and the group of rigid motions (SE(2)).

[Simplex-Based Approach and Results] We have recently made breakthroughs to the kinematics of linkages that have what we call construction trees of simplices.  Not every linkage has a construction tree of simplices, but many do, including all planar loops with revolute joints, all spatial loops with spherical joints, all single-vertex rigid origami folds and some multi-vertex origami folds.

For such linkages, our main published results, with only loop closure constraints and link length ranges considered, are summarized as follows and  further briefly explained in individual pages. For more details, please consult our papers published since 2006.

[Example Systems] To highlight our results in simple yet representative manners, we will mainly use 5-bar planar loops with revolute joints (briefly, 5R loops) as our example systems in the webpages here. For a 5R loop with fixed, generic link lengths, its DSpace has two(2) dimensions and, when viewed in our new parameters, consists of 8 copies of a convex polyhedron of at most 7 sides.  These convex tiles are glued into one connected component or two (ignoring all constraints other than loop closure) along proper boundaries  (just as the surface of a cube can be glued from 6 squares). Below is the DSpace of a 5R loop with link lengths (2,3,4,2,3), which consists of 8 copies of a pentagon and has one connected component, with corresponding edges drawn in same line styles.

In contrast, the DSpace of the same 5bar loop has very complicated geometry in some angle parameters. And the generation of this figure was based on the parametrization of the DSpace in our simplicial parameters. In conventional joint angle parameters, the DSpace would be a 2-dimensional surface sitting in a 4-dimensional ambient space.   The ambient joint angle space is 4-dimensional since one link can be considered as fixed in the plane and has a fixed joint angle.

Our approach and results are directly applicable to linkages with arbitrary numbers of links.  While we are still investigating how to visually convey the structures of high-dimensional DSpaces for linkages with large numbers of links , we can already plan paths for these linkages very efficiently. Below is an example path of a 100-link loop with spherical joints, whose path planning in our new parameters is no more difficult than path planning for open chains.

[Recent Progress] We have recently made interesting generalizations to other constraints such as link self-intersection avoidance and systems without simplicial construction trees. More manuscripts are under review or in preparation.