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IN THIS ISSUE

### In Memoriam

J. Mechanisms Robotics. 2009;1(2):020101-020101-2. doi:10.1115/1.3103563.
FREE TO VIEW

On January 11, 2009 my mentor, Dr. Jack Raymond Phillips, passed away quietly in the home in Birchgrove, Sydney, that he shared with his partner, Elayne Russell, after an extended illness. He is survived by his two daughters, Catherine and Odette.

Commentary by Dr. Valentin Fuster

### Research Papers

J. Mechanisms Robotics. 2009;1(2):021001-021001-9. doi:10.1115/1.3046125.

In this paper, a novel 3DOF fully decoupled translational parallel robot, called the Pantopteron, is presented. This manipulator is similar to the Tripteron Cartesian parallel manipulator, but due to the use of three pantograph linkages, an amplification of the actuator displacements is achieved. Therefore, equipped with the same actuators, the mobile platform of the Pantopteron moves many times faster than that of the Tripteron. This amplification is defined by the magnification factor of the pantograph linkages. The kinematics, workspace, and constraint singularities of the proposed parallel robot are studied in detail. Design considerations are also discussed, and a possible prototype is illustrated.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021002-021002-8. doi:10.1115/1.3046131.

Taking the 3DOF parallel mechanism within the Tricept robot as an example, this paper presents an analytical approach for the stiffness modeling of parallel kinematic machines having a properly constrained passive limb. The stiffness model is formulated using the $6×6$overall Jacobian. It takes particular interest in the precise formulation of the bending stiffness matrix of the properly constrained passive limb by considering the compatibility conditions of the system. Stiffness evaluation of a sample Tricept robot is carried out using two global indices obtained from singular value decomposition of the compliance matrix.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021003-021003-11. doi:10.1115/1.3046134.

The direct position analysis (DPA) of a manipulator is the computation of the end-effector poses (positions and orientations) compatible with assigned values of the actuated-joint variables. Assigning the actuated-joint variables corresponds to considering the actuated joints locked, which makes the manipulator a structure. The solutions of the DPA of a manipulator one to one correspond to the assembly modes of the structure that is generated by locking the actuated-joint variables of that manipulator. Determining the assembly modes of a structure means solving the DPA of a large family of manipulators since the same structure can be generated from different manipulators. This paper provides an algorithm that determines all the assembly modes of two structures with the same topology that are generated from two families of mechanisms: one planar and the other spherical. The topology of these structures is constituted of nine links (one quaternary link, four ternary links, and four binary links) connected through 12 revolute pairs to form four closed loops.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021004-021004-9. doi:10.1115/1.3046137.

The evaluation and representation of the orientation workspace of robotic manipulators is a challenging task. This work focuses on the determination of the theoretical orientation workspace of the Gough–Stewart platform with given leg length ranges $[ρimin,ρimax]$. By use of the roll-pitch-yaw angles $(ϕ,θ,ψ)$, the theoretical orientation workspace at a prescribed position $P0$ can be defined by up to 12 workspace surfaces. The defined orientation workspace is a closed region in the 3D orientation Cartesian space $Oϕθψ$. As all rotations $R(x,ϕ)$, $R(y,θ)$, and $R(z,ψ)$ take place with respect to the fixed frame, any point of the defined orientation workspace provides a clear measure for the platform to, respectively, rotate in order around the $(x,y,z)$ axes of the fixed frame. An algorithm is presented to compute the size (volume) of the theoretical orientation workspace and intersectional curves of the workspace surfaces. The defined theoretical orientation workspace can be applied to determine a singularity-free orientation workspace.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021005-021005-6. doi:10.1115/1.3046140.

The leaf-type isosceles-trapezoidal flexural (LITF) pivot consists of two compliant beams and two rigid bodies. For a single LITF pivot, the range of motion is small while the center-shift is relatively large. The capability of performance can be improved greatly by the combination of two LITF pivots. Base on the pseudorigid-body (PRB) model of a LITF pivot, a method to construct the double-LITF pivots is presented by regarding a single LITF pivot as a the configurable flexure module. The trends of the center-shift are mainly considered by using this method with the combination of two LIFT pivots. Eight types of double-LITF pivots are synthesized. Compared with the single LIFT pivot, the stroke becomes larger, and stiffness becomes smaller. Four of them have the increased center-shift. The other four have the decreased center-shift. Two of the double-LITF pivots are selected as the examples to explain the proposed method. The comparison between PRB model and finite element analysis result shows the validity and effectiveness of the method.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021006-021006-8. doi:10.1115/1.3046142.

Congruent triangles establish that a class of intersecting-shaft couplings is constant velocity. These mechanisms employ a pair of linkages in parallel: a spherical joint at the intersection of the shafts and the intersection of straight-line tracks away from the shaft center to transmit rotation. A proof of constant velocity follows from the congruence of an initial pair of triangles with two matching sides and one excluded angle. This side-side-angle (SSA) condition is a pseudocongruence because it allows two different values for the included angle, indicating that such shaft couplings have symmetric and skewed assembly configurations. If the other excluded angle happens to be 90 deg, the SSA condition guarantees congruence because there is a single solution for the included angle. The 90 deg condition, however, occurs at a posture with a constraint singularity, where the shaft coupling is unable to transmit torque. Motion screw analysis establishes the same geometric condition for a coupling based on a revolute-spherical-revolute Clemens linkage. An upper bound on shaft deflection imposed by hyperextension of that linkage, along with a bound on deflection where constraint singularity occurs, identifies couplings where the constraint singularity can occur within the physical limits.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021007-021007-8. doi:10.1115/1.3046146.

It is possible to realize the desired compliance characteristics of a robot in a form of a passive compliance device, which demands the synthesis technique of a stiffness matrix by parallel connections of line and/or torsional springs. In this paper, the stiffness matrix is expressed in terms of the screw coordinates with respect to the basis consisting of its eigenvectors, thereby the synthesis equation is derived. Examination of the numbers of free design parameters involved in the synthesis suggests that a line or free vector for a spring can be freely selected from the induced wrench space depending on the rank of the stiffness matrix. The recursive synthesis method that allows one to select the positions or directions of the springs from the screw system spanned by the induced wrenches of the given stiffness matrix is proposed.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021008-021008-9. doi:10.1115/1.3046148.

In this paper, a pseudorigid-body (PRB) 3R model, which consists of four rigid links joined by three revolute joints and three torsion springs, is proposed for approximating the deflection of a cantilever beam subject to a general tip load. The large deflection beam equations are solved through numerical integration. A comprehensive atlas of the tip deflection for various load modes is obtained. A three-dimensional search routine has been developed to find the optimal set of characteristic radius factors and spring stiffness of the PRB 3R model. Detailed error analysis has been done by comparing with the precomputed tip deflection atlas. Our results show that the approximation error is much less than that of the conventional PBR 1R model. To demonstrate the use of the PRB 3R model, a compliant four-bar linkage is studied and verified by a numerical example. The result shows a maximum tip deflection error of 1.2% compared with the finite element analysis model. The benefits of the PRB 3R model include that (a) the model parameters are independent of external loads, (b) the approximation error is relatively small for even large deflection beams, and (c) the derived kinematic and static constraint equations are simpler to solve compared with the finite element model.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021009-021009-10. doi:10.1115/1.3046128.

Pop-up paper mechanisms use techniques similar to the well-studied paper folding techniques of origami. However, pop-ups differ in both the manner of construction and the target uses, warranting further study. This paper outlines the use of planar and spherical kinematics to model commonly used pop-up paper mechanisms. A survey of common joint types is given, including folds, interlocking slots, bends, pivots, sliders, and rotating sliders. Also included is an overview of common one-piece and layered mechanisms, including single-slit, double-slit, V-fold, tent, tube strap, and arch mechanisms. Each mechanism or joint is shown using both a paper representation and either a rigid-body or pseudo-rigid-body representation. In addition, this paper shows that more complex mechanisms may be created by combining simple mechanisms in various ways. The principles presented are applied to the creation of new pop-up joints and mechanisms. The new mechanisms employ both spherical and spatial kinematic chains. Understanding pop-up mechanism kinematics could lead to new applications in deployable structures, packaging, and instruments for minimally invasive surgery.

Topics: Mechanisms
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021010-021010-9. doi:10.1115/1.3046139.

This paper aims at providing a method to synthesize mechanical architectures of self-adaptive robotic fingers driven by linkages. Self-adaptive mechanisms are used in robotic fingers to provide the latter with the ability to adjust themselves to the shape of the object seized without any dedicated electronics, sensor, or control. This type of mechanisms has been known for centuries but the increased capabilities of digital systems have kept them in the shadows. Recently, because of the lack of commercial and industrial success of complex robotic hands, self-adaptive mechanisms have attracted much more interest from the research community and several prototypes have been built. Nevertheless, only a handful of prototypes are currently known. It is the aim of this paper to present a methodology that is able to generate thousands of self-adaptive robotic fingers driven by linkages with two and three phalanges. First, potential kinematic architectures are synthesized using a well-known technique. Second, the issue of proper actuation and passive element(s) selection and location is addressed.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021011-021011-10. doi:10.1115/1.3056476.

The current design algorithms for compliant mechanisms often generate solutions that imitate rigid-body linkages by means of point flexures or flexure pivots, by using the popular spring model formulation. This paper presents a kinetoelastic formulation for compliant mechanism optimization. With a state equation of the mechanism defined by the elasticity theory, the model incorporates not only the kinematic function requirements of the mechanism but, more importantly, the necessary conditions on the compliance characteristics of the mechanism’s structure. The kinematics of the compliant mechanism is defined on rigid bodies of input/output ports and is related to a set of kinetoelastic factors of the mechanism’s compliance matrix. The kinetoelastic formulation is applied to the problem of optimizing a compliant translational joint, producing compliant designs with compliance properties such as the leaf spring type sliding joint as opposed to the notch-type joint. This paper represents an initial development toward a more general methodology for compliant mechanism optimization.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2009;1(2):021012-021012-9. doi:10.1115/1.3046144.

In this paper, the mobility, the kinematic constraints, the pose of the end-effector, and the static constraints that lead to the kinematostatic model of a compliant parallel mechanism are introduced. A formulation is then provided for its instantaneous variation—the quasi-static model. This new model allows the calculation of the variation in the pose as a linear function of the motion of the actuators and the variation in the external loads through two new matrices: the compliant Jacobian matrix and the Cartesian compliance matrix that give a simple and meaningful formulation of the model of the mechanism. Finally, a simple application to a planar four-bar mechanism is presented to illustrate the use of this model and the new possibilities that it opens, notably the study of the kinematics for any range of applied load.

Commentary by Dr. Valentin Fuster

### Technical Briefs

J. Mechanisms Robotics. 2009;1(2):024501-024501-3. doi:10.1115/1.3046180.

A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) of a quadratic parallel manipulator: $3-RP̱R$ planar parallel manipulator with similar triangular platforms. Although it has been revealed numerically elsewhere that for this parallel manipulator, the four solutions to the FDA fall, respectively, into its four singularity-free regions (in its workspace), it is unclear if there exists a one-to-one correspondence between the four formulas, each producing one solution to the FDA, and the four singularity-free regions. Using an algebraic approach, this paper will prove that such a one-to-one correspondence exists. Therefore, a unique solution to the FDA can be obtained in a straightforward way for such a parallel manipulator if the singularity-free region in which it works is specified.

Commentary by Dr. Valentin Fuster