Multi-furcation in a derivative queer-square mechanism
Introduction
Most mechanisms maintain the same permanent mobility during their working processes [1], [2] while the degrees of freedom and motion types of the mechanisms generate no permanent change except the transitory infinitesimal mobility in singular positions [3]. Compared with a conventional mechanism with fixed mobility, mechanisms with variable mobility or motion types have been found and thereby have attracted much attention from many researchers in recent years [4], [5], [6], [7], [8], [9], [10], [11], [12].
The systematic study of mechanisms with variable mobility or motion types dates back to 1996 when Wohlhart [13] investigated the “Wunderlich mechanism” as well as the “Wren platform”. The “Queer Square” [13] paper folding was studied and its branching singularities were investigated by Gogu [4]. In the meantime in 1996, Dai and Rees Jones [14] investigated the origami folding and developed from them the kind of mechanisms that are foldable and erectable coined “metamorphic mechanisms” which are capable of changing topology [15] and mobility during motion following screw system analysis. More single-loop and multi-loop mechanisms with changing mobility or motion types were proposed by Galletti et al. [10], [11] in 2001 and 2002. In 2006, the characteristic analysis of variable kinematic joints which express the function of changing topology in a mechanism was studied and demonstrated by Yan and Kuo [12] with mechanisms that change topological structures in accordance with the varying mobility [16]. The study of reconfigurable mechanisms and their metamorphosis was taken on by Gan et al [19] by proposing the reconfigurable Hooke (rH) joint and by Zhang et al [20] by proposing the variable axis (vA) joint.
When it comes to reconfigurable mechanisms with variable mobility and motion types, there are two phenomena [13] depending whether there are changes in the joint directions.
The first phenomenon presents change of mobility or motion type irrelevant to the directions of the joints. The only reason causing mobility variation is the position parameters or configuration change across the constraint singularity. Up to now, fewer mechanisms belonging to phenomenon one are found in the related literature. Among the mechanisms belonging to type-one phenomenon, three kinematotropic linkages investigated by Wohlhart in 1996 [13] and two metamorphic mechanisms by Li and Dai with folded links [17] are typical examples.
The second phenomenon presents change of mobility as a result of variation of the directions of the joints. Type-two phenomenon has been studied with mechanisms of variable mobility. Specifically, investigation of this phenomenon includes several parallel mechanisms presented by Fanghella, Calletti and Giannotti [18] in 2006 and the metamorphic parallel mechanisms containing newly invented reconfigurable Hooke (rT) joints by Gan, Dai and Liao [19] and the metamorphic mechanisms containing newly invented variable axis (vA) joints by Zhang, Dai and Fang [20]. Mobility change of the metamorphic parallel mechanisms results from alignment of a limb with the axis of the reconfigurable Hooke (rT) joints or from the change of axis mutual alignment of the variable axis (vA) joints.
This paper presents a derivative mechanism from the “queer-square” and investigates the property of its multi-furcated motion. Based on the study of two phenomena in Section 2, two categories are investigated. This leads to the investigation of six states of the first category with only configuration change resulting in mobility change. Section 3 gives the description and Section 4 focuses on one pure translation and 5 on two pure translations. The second category with change of axis orientation is then investigated in Section 6.
Section snippets
The structure and screw system of the derivative queer-square mechanism
The derivative queer-square mechanism is composed of ten rigid links and twelve revolute joints. As illustrated in Fig. 1, Fig. 2, Fig. 3, the ten links are denoted from 1 to 10. The length of links 2, 4, 5, 6, 8, 9, the longer part of links 3 and 10 is defined as l3 which equals to l1 + l2. The length of the longer part of links 1 and 7 is equal to l1. The length of the shorter part of links 1 and 3 is l2 / 2, while the length of the shorter part of links 7 and 10 is l2. Link 3 is connected to
Phenomena, categories and states of the derivative mechanism
During the motion, the platform of the derivative queer-square mechanism permanently changes its mobility by crossing over the constraint singularity as illustrated in Fig. 2. With different values of angles α1, α2, β11, β12, β21 and β22, two phenomena, four categories and fourteen states are achieved by the derivative queer-square. The relations of the phenomena, categories and states are indicated in Fig. 4.
Phenomenon one and six states of configuration
As indicated before, the derivative queer-square mechanism has the same orientation of the joint axes in phenomenon one but changes its mobility when it comes across the constraint singularity while the specific geometrical constraint at different configurations presents. In phenomenon 1, angles β11 and β12 present the same values and angles β21 and β22 are also the same. According to the relation between α1 and α2, two categories are present. Section 4.1 reveals the first category with the
Phenomenon two and last eight states
In the first six states, the motion varies although the directions of the joints remain the same, however, the directions of joints change in the last eight states. Hence, when the mechanism moves from one of the first six states to one of the last eight states or between the last eight states, the motion change is generated from the variation of the direction, as phenomenon two presented.
In phenomenon 2, either angles β11 and β12 or angles β21 and β22 have the different value. Section 5.1
Conclusions
With regard to mechanisms with variable motion, there are two phenomena that exist. The first phenomenon is that the mechanisms change motion permanently with the same orientation of the joint axes. For the second phenomenon, the variation of the motion is accompanied by change of the orientation of the joint axes.
This paper presented a derivative queer-square mechanism and used it to illustrate both phenomena of the motion variation. The first six states of the derivative queer-square
References (22)
A family of overconstrained linkages
J. Mech.
(1967)Mobility of mechanisms: a critical review
Mech. Mach. Theory
(2005)- et al.
Single-loop kinematotropic mechanisms
Mech. Mach. Theory
(2001) - et al.
Constraint analysis on mobility change of a novel metamorphic parallel mechanism
Mech. Mach. Theory
(2010) Kinematic Geometry of Mechanisms
(1978)Branching singularities in kinematotropic parallel mechanisms, computational kinematics
- et al.
Constraint-based limb synthesis and mobility-change-aimed mechanism construction
J. Mech. Des. Trans. ASME
(2011) - et al.
Geometric constraint and mobility variation of two 3SvPSv metamorphic parallel mechanisms
J. Mech. Des. Trans. ASME
(2013) - et al.
Geometric analysis and synthesis of the metamorphic robotic hand
J. Mech. Des. ASME
(2007) - et al.
Simplified manufacturing through a metamorphic process for compliant ortho-planar mechanisms
Reconfiguration principles and strategies for reconfigurable mechanisms
Cited by (45)
Shape optimization of buckling-based deployable stiff structures
2024, Mechanism and Machine TheoryMultiple bifurcated reconfiguration of double-loop metamorphic mechanisms with prismatic joints
2022, Mechanism and Machine TheoryMotion/structure mode analysis and classification of n-RR planar parallelogram mechanisms
2022, Mechanism and Machine TheoryReconfigurability of the origami-inspired integrated 8R kinematotropic metamorphic mechanism and its evolved 6R and 4R mechanisms
2021, Mechanism and Machine TheoryHigh-order based revelation of bifurcation of novel Schatz-inspired metamorphic mechanisms using screw theory
2020, Mechanism and Machine TheoryCitation Excerpt :Gan et al. [13,14] used screw theory to study the mobility changes and bifurcated motion of novel metamorphic parallel mechanisms. Qin et al. [4] studied the multi-furcation phenomenon of the queer square mechanism, which can generate 14 bifurcation motion states through the constrained singular configuration. Gao et al. [15] presented a novel truss-shaped deployable grasping manipulator with mobility bifurcation, which changes its mobility from deployment to grasping motions.