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### Research Papers

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

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
J. Mechanisms Robotics. 2018;10(5):051008-051008-8. doi:10.1115/1.4040631.
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This document introduces the holonomic flying capabilities of the Hexapodopter, a six-legged walking machine capable of vertical take-off and landing. For ground locomotion, each limb has two degrees-of-freedom (2DoF); while the thrust required for flying is provided by six motors mounted close to every knee, so the thrust vector can be reoriented depending on the configuration of each limb. The capacity of reorienting the thrust forces makes the Hexapodopter a true holonomic vehicle, capable of individually controlling its six degrees-of-freedom (6DoF) on the air without reorienting any of the thrust motors nor the body. The main design criteria and validation will be discussed on this paper, as well as a control law for the vehicle.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051009-051009-10. doi:10.1115/1.4040632.

A core issue in collaborative robotics is that of impact mitigation, especially when collisions happen with operators. Passively compliant structures can be used as the frame of the cobot, although, usually, they are implemented by means of a single-degree-of-freedom (DoF). However, n-DoF preloaded structures offer a number of advantages in terms of flexibility in designing their behavior. In this work, we propose a comprehensive framework for classifying n-DoF preloaded structures, including one-, two-, and three-dimensional arrays. Furthermore, we investigate the implications of the peculiar behavior of these structures—which present sharp stiff-to-compliant transitions at design-determined load thresholds—on impact mitigation. To this regard, an analytical n-DoF dynamic model was developed and numerically implemented. A prototype of a 10DoF structure was tested under static and impact loads, showing a very good agreement with the model. Future developments will see the application of n-DoF preloaded structures to impact-mitigation on cobots and in the field of mobile robots, as well as to the field of novel architected materials.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051010-051010-11. doi:10.1115/1.4040633.

Modular manipulators gained popularity for their implicit feature of “reconfigurability”—that is, the ability to serve multiple applications by adopting different configurations. As reported in the literature, most of the robotic arms with modular architecture used specific values of twist angles, e.g., 0 deg or 90 deg. Further, the number of degrees-of-freedom (DoF) is also kept fixed. These constraints on the design parameters lead to a smaller solution space for the configuration synthesis problems and may result as no-feasible solution in a cluttered work-cell. To work in a realistic environment, the task-based customized design of a manipulator may need a larger solution space. This work deals with the extension of the modular architecture from conventional values to unconventional values of design parameters, keeping the degrees-of-freedom also as variable. This results into an effective utilization of modular designs for highly cluttered environments. A three-phase design strategy is proposed in the current work. The design strategy starts with the decision of optimal number of modules required for the given environment in the first phase, which is followed by task-based “configuration planning” and “optimal assembly” in the second and third phase, respectively. Three types of modules are proposed with same architecture and different sizes—heavy (H), medium (M), and light (L). The configuration planning includes detailed discussion on the type-selection of the modules and their possible combinations. Comparison of all possible n-link combinations is analyzed based upon the optimized results with respect to the minimum torque values. Case studies of a power plant with two different workspaces are included to illustrate the three-phase strategy representing the importance of modularity in nonrepetitive maintenance tasks.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051011-051011-10. doi:10.1115/1.4040700.

This paper presents a pseudo-static modeling methodology for dynamic analysis of distributed compliant mechanisms to provide accurate and efficient solutions. First, a dynamic stiffness matrix of the flexible beam is deduced, which has the same definition and a similar form as the traditional static compliance/stiffness matrix but is frequency dependent. Second, the pseudo-static modeling procedure for the dynamic analysis is implemented in a statics-similar way based on D'alembert's principle. Then, all the kinematic, static and dynamic performances of compliant mechanisms can be analyzed based on the pseudo-static model. The superiority of the proposed method is that when it is used for the dynamic modeling of compliant mechanisms, the traditional dynamic modeling procedures, such as calculation of the elastic and kinetic energies as well as using Lagrange's equation, are avoided and the dynamic modeling is converted to a statics-similar problem. Comparison of the proposed method with an elastic-beam-based model in previous literature and finite element analysis for an exemplary XY precision positioning stage reveals its high accuracy and easy operation.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051012-051012-9. doi:10.1115/1.4040702.

In this work, a new parallel manipulator with multiple operation modes is introduced. The proposed robot is based on a three-degrees-of-freedom (3DOF) parallel manipulator endowed with a three-dof central kinematic chain, where by blocking some specific kinematic pairs, the robot can modify its mobility. Hence, the robot manipulator is able to assume the role of a limited-dof or a nonredundant parallel manipulator. Without loss of generality, the instantaneous kinematics of one member of the family of parallel manipulators generated by the reconfigurable parallel manipulator, the three-RPRRC + RRPRU nonredundant parallel manipulator with decoupled motions, is approached by means of the theory of screws. For the sake of completeness, the finite kinematics of the robot is also investigated. Numerical examples are included with the purpose to clarify the method of kinematic analysis.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051013-051013-9. doi:10.1115/1.4040703.

To obtain the closed-form forward kinematics of parallel robots, researchers use algebra-based method to transform and simplify the constraint equations. However, this method requires a complicated derivation that leads to high-order univariate variable equations. In fact, some particular mechanisms, such as Delta, or H4 possess many invariant geometric properties during movement. This suggests that one might be able to transform and reduce the problem using geometric approaches. Therefore, a simpler and more efficient solution might be found. Based on this idea, we developed a new geometric approach called geometric forward kinematics (GFK) to obtain the closed-form solutions of H4 forward kinematics in this paper. The result shows that the forward kinematics of H4 yields an eighth degree univariate polynomial, compared with earlier reported 16th degree. Thanks to its clear physical meaning, an intensive discussion about the solutions is presented. Results indicate that a general H4 robot can have up to eight nonrepeated real solutions for its forward kinematics. For a specific configuration of H4, the nonrepeated number of real roots could be restricted to only two, four, or six. Two traveling plate configurations are discussed in this paper as two typical categories of H4. A numerical analysis was also performed for this new method.

Topics: Kinematics , Robots
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051014-051014-10. doi:10.1115/1.4040353.

This paper presents a uniform method of evaluating both transmission quality and singularity applicable for a class of parallel Schönflies-motion generators (SMGs) with four $RRΠRR$ limbs. It turns out that the determinant of the forward Jacobian matrices for this class of parallel robots can be expressed as the scalar product of two vectors, the first vector being the cross product of the four unit vectors along the parallelograms, and the second one being related to the rotation of the mobile platform (MP). The pressure angles, derived from the determinants of forward and inverse Jacobians, respectively, are used for the evaluation of the transmission quality and the detection of robot singularities. Four robots are compared based on the proposed indices as illustrative examples.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051015-051015-7. doi:10.1115/1.4040886.

In this paper, we present a thorough kinematics analysis of a humanoid two degrees-of-freedom (DoF) ankle module based on a parallel kinematics mechanism. Compared with the conventional serial configuration, the parallel kinematics ankle permits the distribution of the torque/power of the actuators to the two DoF of the ankle taking full advantage of available power/torque capacity of the two actuators. However, it complicates the kinematics study in return. In this work, a complete study of a parallel ankle mechanism is performed that permits the full characterization of the ankle module for the purpose of its design study, control, and performance evaluation. Screw theory is employed for mobility analysis to first determine the number and properties of the mechanism's DoFs. Then the inverse kinematics is solved analytically and the Jacobian matrix for describing the velocity relation between the ankle joints and motors is found. Based on these results, the forward kinematics of the parallel mechanism can be numerically computed using the Newton–Raphson method. The workspace of the ankle is also analyzed and the motor limits are decided accordingly. Finally, an experimental demonstration consisting of four tests is carried out to evaluate the proposed methods and ankle module.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051016-051016-18. doi:10.1115/1.4040885.

A kind of kinematic chain with parallel linear motion elements (PLMEs) is proposed and studied in this paper. Based on screw theory, the kinematic screw equations of these linkages are established. The two special categories of PLMEs, with pure translational motion and with pure rotational motion respectively, are identified. The mobilities and the singularities of these kinematic chains are also investigated. By the utilization of these PLMEs, three types of the compound limbs are invented and analyzed. Through assembling these compound limbs in different ways, a class of lower mobility symmetrical 3T, 3T-1R, and 3R mechanisms is synthesized and presented for the first time. The simplified kinematic equations for this class of mechanisms driven by the linear actuators are derived. And the workspaces, singularities, and kinematic performance are addressed. Finally, three typical prototypes with regard to 3T, 3T-1R, and 3R mechanisms are manufactured and experimented to validate the mobility and motion feasibility of these mechanisms.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2018;10(5):051017-051017-10. doi:10.1115/1.4040701.
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The purpose of this work is to introduce a new parallel actuated exoskeleton architecture that can be used for multiple degree-of-freedom (DoF) biological joints. This is done in an effort to provide a better alternative for the augmentation of these joints than serial actuation. The new design can be described as a type of spherical parallel manipulator (SPM) that utilizes three 4 bar substructures to decouple and control three rotational DoFs. Four variations of the 4 bar spherical parallel manipulator (4B-SPM) are presented in this work. These include a shoulder, hip, wrist, and ankle exoskeleton. Also discussed are three different methods of actuation for the 4B-SPM, which can be implemented depending on dynamic performance requirements. This work could assist in the advancement of a future generation of parallel actuated exoskeletons that are more effective than their contemporary serial actuated counterparts.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Mechanisms Robotics. 2018;10(5):054501-054501-5. doi:10.1115/1.4040630.

In this paper, nonprehensile throwing, catching, and balancing of a disk-shaped object by a two-link planar manipulator mounted with a disk-shaped end effector are presented. Given a goal position, which is out of the robot's reachable space, the required release position and velocity for throwing are first determined. The throwing manipulation is proposed in a way that the arm follows a planned trajectory between the ready and the release position to achieve the required velocity at the release position. Catching is performed in a way that it reduces impact when making contact. Balancing control is then applied to the disk-shaped end effector to prevent the object from falling after catching. The proposed approach was implemented on an experimental setup built for verification and the results are provided.

Commentary by Dr. Valentin Fuster