The Bennett plano-spherical hybrid 6R linkage is an overconstrained one-degree-of-freedom linkage whose motion-screw system degenerates at precisely characterized transitory singular configurations, allowing it to switch between an overconstrained 6R motion branch, a planar 4R branch, and a spherical 4R branch; assembled in three limbs, it yields a metamorphic parallel mechanism whose platform switches between helical, spherical, and vertical-translation modes. Zhang and Dai (2015) fully characterized this reconfiguration kinematically and proposed redundant actuation of joints R1 and R4 with branch-wise joint-angle domains, but explicitly left two implementation problems open: (1) an implemented actuation and control scheme able to supply the torques required at the degenerate configurations and to handle the dynamic effects there under load, with demonstrated dynamic feasibility and robustness; and (2) a concrete treatment of the singular configurations that do not permit branch switching (axes of R2 aligning with R6 and R3 with R5, where the linkage opens into an open kinematic chain). No complete actuation, dynamics, and control solution for this mechanism family is known, and both issues remain open.

Background

A closed kinematic chain built from revolute joints (R joints, hinge-like one-degree-of-freedom rotary joints) is called overconstrained when it moves with fewer joints locked than a generic mobility formula predicts, because special geometric relations among its joint axes make some constraints redundant. The Bennett linkage, a spatial four-revolute loop whose adjacent link lengths and twist angles satisfy a special ratio, is the classical example. A metamorphic mechanism is one that deliberately changes its effective topology or mobility during operation; such changes are typically possible only at singular configurations at which the mechanism's constraint structure degenerates. Zhang and Dai (2015) studied the Bennett plano-spherical hybrid linkage, a plane-symmetric spatial 6R closed loop that combines a spherical chain (axes of joints R1, R2, R6 meeting at one common point) with a planar chain (axes of R3, R4, R5 mutually parallel), with metric design parameters (r, h) and twist angles (α16\alpha_{16}, α25\alpha_{25}). In screw theory, each instantaneous joint motion is a screw (a line in Plücker coordinates with an associated pitch), and the span of the six joint screws forms the motion-screw system, whose reciprocal complement, the constraint-screw system, has dimension 6−d6-d when dd is the order of the motion-screw system, and represents the wrenches (force-and-moment combinations) the closed loop can transmit. Away from singularities the plano-spherical linkage's motion-screw system is a five-system and the linkage has one degree of freedom. The paper showed that in three parameter regimes, r>htan⁡α16r>h\tan\alpha_{16}, r<htan⁡α16r<h\tan\alpha_{16}, and the boundary r=htan⁡α16r=h\tan\alpha_{16}, the motion-screw system degenerates at explicit joint-angle values, and that at these transitory singular configurations the linkage can leave its original overconstrained 6R motion branch for a planar 4R branch (joints R2 and R6 becoming geometrically locked) or a spherical 4R branch (R3 and R5 locked), with a trifurcation point where all three branches meet. Zhang and Dai also assembled three of these hybrid limbs (each extended by a serial RR chain) into a metamorphic parallel mechanism whose platform correspondingly switches between a 1-DOF helical motion about the vertical axis, a 3-DOF spherical motion about the base centre, and a 1-DOF pure vertical translation. Reconfiguration thus requires physically driving the mechanism through configurations at which the constraint structure momentarily changes character. To effect this, the paper proposed redundant actuation of joints R1 and R4 of each plano-spherical limb, with tabulated joint-angle domains for each motion branch, but it did not develop the actuation hardware, control laws, or dynamic analysis. Subsequent work deepened the kinematics: Zhang and Dai (2016) reduced the branch-changing conditions of single-loop linkages to four Grassmann-type axis dispositions (coplanar, concurrent, parallel, and collinear axes) generating transitory positions; the same authors' companion analysis of the plane-symmetric double-spherical 6R linkage showed that the analogous configuration with coincident axis pairs reconfigures that linkage into a serial open chain rather than into another one-degree-of-freedom branch; Ma, Zhang, and Dai (2018) derived further metamorphic 6R linkages and, using motion-cycle and configuration-torus arguments, distinguished genuine bifurcation points from singular points that lie on a single branch. For actuation, Zhang and Liu (2021) presented a reconfigurable parallel manipulator derived from kirigami (the East Asian art of paper cutting, whose cut-and-folded sheets are kinematically equivalent to linkages) built from Bennett plano-spherical linkages driven by inflatable angular pouch motors (soft pneumatic actuators that contract angularly when pressurized), demonstrating branch reconfiguration in lightweight prototypes, while Nurahmi, Gan, and Setya (2023) developed unified dynamic models with consistency and indeterminate-form conditions for crossing the singularities of a different metamorphic parallel mechanism (the 3-(rR)PS with reconfigurable revolute joints), and Pagis, Bouton, Briot, and Martinet (2015) showed earlier that non-metamorphic parallel robots can cross Type-2 (Jacobian) singularities by exploiting dynamic effects. No existing work provides an implemented actuation and control design that reliably drives the Bennett plano-spherical linkage or its evolved parallel mechanism through the transitory singularities under load, together with a strategy for the singular configurations at which no branch switching is possible.

Problem Statement

Develop and validate an actuation and control scheme that makes motion-branch reconfiguration of the Bennett plano-spherical hybrid 6R linkage - and of the metamorphic parallel mechanism built from three such limb subchains - a deliberate, repeatable, load-bearing operation. The scheme must: (1) drive the mechanism through its transitory singular configurations, at which the motion-screw system degenerates and the constraint-screw system gains order, so that switching among the overconstrained 6R, planar 4R, and spherical 4R motion branches (and the corresponding helical, spherical, and vertical-translation platform modes of the evolved parallel mechanism) is effected on command, with the actuation sized for the torques required at and near the degeneracy, the dynamic effects at the configuration handled by the modeling and control design, and reliable reconfiguration demonstrated under a specified payload; and (2) treat the singular configurations that do not permit branch switching - those in which the axes of R2 align with R6 and R3 with R5 and the linkage opens into an open kinematic chain - through a concrete, mechanism-specific treatment, so that these configurations cannot trap, jam, or damage an implemented mechanism. The dynamic analysis must remain valid across the singularity and across the different-DOF branches, and repeated branch reconfiguration under load, together with the treatment of the non-switching singular configurations, must be demonstrated.

References

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