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Research Papers

J. Mechanisms Robotics. 2018;10(5):051001-051001-10. doi:10.1115/1.4040433.

The Boerdijk–Coxeter helix (BC helix, or tetrahelix) is a face-to-face stack of regular tetrahedra forming a helical column. Treating the edges of these tetrahedra as structural members creates an attractive and inherently rigid space frame, and therefore is interesting to architects, mechanical engineers, and roboticists. A formula is developed that matches the visually apparent helices forming the outer rails of the BC helix. This formula is generalized to a formula convenient to designers. Formulae for computing the parameters that give proven edge-length minimax-optimal tetrahelices are given, allowing transformation through a continuum of optimum tetrahelices of varying curvature while maximizing regularity. The endpoints of this continuum are the BC helix and a structure of zero curvature, the equitetrabeam. Only one out of three members in the system change their length to transform the structure into any point in the continuum. Numerically finding the rail angle from the equation for pitch allows optimal tetrahelices of any pitch to be designed. An interactive tool for such design and experimentation is provided. A formula for the inradius of optimal tetrahelices is given. The continuum allows a regular Tetrobot supporting a length change of less than 16% in the BC configuration to untwist into a hexapodal or n-podal robot to use standard gaits.

Topics: Rails
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051002-051002-13. doi:10.1115/1.4040435.

This paper presents a method to minimize the base attitude disturbance of a space robot during target capture. First, a general dynamic model of a free-floating space robot capturing a target is established using spatial operator Algebra, and a simple analytical formula for the base angular velocity change during the impact phase is obtained. Compared with the former models proposed in the literature, this model has a simpler form, a wider range of applications, and O(n) computation complexity. Second, based on the orthogonal projection matrix lemma, we propose the generalized mass Jacobian matrix (GMJM) and find that the base angular velocity change is a constant multiple of the component which the impact impulse projects to the column space of the GMJM. Third, a new concept, the base attitude disturbance ellipsoid (BADE), is proposed to express the relationship between the base attitude disturbance and the impact direction. The impact direction satisfying the minimum base attitude disturbance can be straightforwardly obtained from the BADE. In particular, for a planar space robot, we draw the useful conclusion that the impact direction unchanged base attitude must exist. Furthermore, the average axial length of the BADE is used as a measurement to illustrate the average base attitude disturbance under impact impulses from different directions. With this measurement, the desired pre-impact configuration with minimum average base attitude disturbance can be easily determined. The validity and the efficiency of this method are verified using a three-link planar space robot and a 7DOF space robot.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051003-051003-13. doi:10.1115/1.4040439.

Rigid origami is a restrictive form of origami that permits continuous motion between folded and unfolded states along the predetermined creases without stretching or bending of the facets. It has great potential in engineering applications, such as foldable structures that consist of rigid materials. The rigid foldability is an important characteristic of an origami pattern, which is determined by both the geometrical parameters and the mountain-valley crease (M-V) assignments. In this paper, we present a systematic method to analyze the rigid foldability and motion of the generalized triangle twist origami pattern using the kinematic equivalence between the rigid origami and the spherical linkages. All schemes of M-V assignment are derived based on the flat-foldable conditions among which rigidly foldable ones are identified. Moreover, a new type of overconstrained 6R linkage and a variation of doubly collapsible octahedral Bricard are developed by applying kirigami technique to the rigidly foldable pattern without changing its degree-of-freedom. The proposed method opens up a new way to generate spatial overconstrained linkages from the network of spherical linkages. It can be readily extended to other types of origami patterns.

Topics: Linkages , Kinematics
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051004-051004-16. doi:10.1115/1.4040462.

This paper introduces a family of statically balanced five-degree-of-freedom (5DOF) parallel mechanisms (PMs) with kinematic and actuation redundancy. Moving platforms of this family of PMs can provide 4DOF Schönflies motion. Three applications are considered in this work. The first and second applications use kinematic redundancy to avoid parallel singularities and perform an auxiliary grasping task in sequence. The third application incorporates actuation redundancy into a kinematically redundant manipulator to increase the load-carrying capacity. Screw theory was used to derive the Jacobian of the 5DOF PM with kinematic and actuation redundancy. Parallel singularities can be completely alleviated by controlling the orientation of the redundant link, thereby obtaining a large rotational workspace, and actuation redundancy increases the load-carrying capacity. Using a commercially available multibody dynamic simulator, an example of trajectory was performed to illustrate the large rotational workspace of the first and second applications and compare the Euclidean norm of the vector of actuation torque of nonredundant and redundant PMs. Three prototypes were also developed to demonstrate the output motion and static balancing property.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051005-051005-16. doi:10.1115/1.4040488.

Based on the general degree-of-freedom (DOF) formula for spatial mechanisms proposed by the author in 2012, the early single open chain (SOC)-based composition principle for planar mechanisms is extended to general spatial mechanisms in this paper. First, three types of existing mechanism composition principle and their characteristics are briefly discussed. Then, the SOC-based composition principle for general spatial mechanisms is introduced. According to this composition principle, a spatial mechanism is first decomposed into Assur kinematic chains (AKCs) and an AKC is then further decomposed into a group of ordered SOCs. Kinematic (dynamic) analysis of a spatial mechanism can then be reduced to kinematic (dynamic) analysis of AKCs and finally to kinematic (dynamic) analysis of ordered SOCs. The general procedure for decomposing the mechanism into ordered SOCs and the general method for determining AKC(s) contained in the mechanism are also given. Mechanism's kinematic (dynamic) analysis can be reduced to the lowest dimension (number of unknowns) directly at the topological structure level using the SOC-based composition principle. The SOC-based composition principle provides a theoretical basis for the establishment of a unified SOC-based method for structure synthesis and kinematic (dynamic) analysis of general spatial mechanisms.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051006-051006-10. doi:10.1115/1.4040357.

In a classical mobility-one single loop linkage, the motion begins from an original position determined by the assembled condition and runs in cycles. In normal circumstances, the linkage experiences a full cycle when the input-joint completes a full revolution. However, there are some linkages that accomplish a whole cycle with the input-joint having to go through multiple revolutions. Their motion cycle covers multiple revolutions of the input-joint. This paper investigates this typical phenomenon that the output angle is in a different motion cycle of the input angle that we coin this as the multiple input-joint revolution cycle. The paper then presents the configuration torus for presenting the motion cycle and reveals both bifurcation and double points of the linkage, using these mathematics-termed curve characteristics for the first time in mechanism analysis. The paper examines the motion cycle of the Bennett plano-spherical hybrid linkage that covers an 8π range of an input-joint revolution, reveals its four double points in the kinematic curve, and presents two motion branches in the configuration torus where double points give bifurcations of the linkage. The paper further examines the Myard plane-symmetric 5R linkage with its motion cycle covering a 4π range of the input-joint revolution. The paper, hence, presents a way of mechanism cycle and reconfiguration analysis based on the configuration torus.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051007-051007-15. doi:10.1115/1.4040437.

This paper presents the design and experimental validation of a passive large-displacement constant-force mechanism (CFM). Unlike previous studies, without using extra stiffness-compensation components and active control devices, the presented CFMs can utilize the interaction between the components of a cam and sliders to directly achieve the constant-force characteristic over the entire flexibly designed large displacement once the cam is advisably designed with the consideration of friction effect by using the profile curve identification method (PCIM). Corresponding to the different requirements of conventional and extreme engineering environments, two versions of the mechanism, the basic and ultra-large-displacement CFM models are proposed, respectively. The basic version is designed directly based on the PCIM, whereas the ultra-large-displacement CFM is proposed using the relay-mode action of the multistage sliders. According to the theoretical design method, we design and fabricate two corresponding CFM prototypes. Validation experiments are then conducted, and the results show that both of the prototypes can satisfy the design requirements and possess large-displacement constant-force characteristics owing to the consistency of experimental and design data. Therefore, the proposed design theory for the cam-based large-displacement CFMs is validated and the designed CFMs will have extensive applications in relevant fields for force regulation and overload protection.

Commentary by Dr. Valentin Fuster

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