0

IN THIS ISSUE

### Editorial

J. Mechanisms Robotics. 2011;3(1):010201-010201-3. doi:10.1115/1.4003039.
FREE TO VIEW
Koetsier, T., 1986, “ From Kinematically Generated Curves to Instantaneous Invariants: Episodes in the History of Instantaneous Planar Kinematics,” Mech. Mach. TheoryMHMTAS 0094-114X, 21(6), pp. 489–498.[[XSLOpenURL/10.1016/0094-114X(86)90132-1]]Moon, F. C., 2009, “ History of the Dynamics of Machines and Mechanisms From Leonardo to Timoshenko,” International Symposium on History of Machines and Mechanisms , H.S.Yan and M.Ceccarelli, eds.Koetsier, T., 1983, “ A Contribution to the History of Kinematics—II,” Mech. Mach. TheoryMHMTAS 0094-114X, 18(1), pp. 43–48.[[XSLOpenURL/10.1016/0094-114X(83)90006-X]]Kempe, A. B., 1976, “ On a General Method of Describing Plane Curves of the Nth Degree by Linkwork,” Proc. London Math. Soc.PLMTAL 0024-6115, 7, pp. 213–216.[[XSLOpenURL/10.1112/plms/s1-7.1.213]]Jordan, D., and Steiner, M., 1999, “ Configuration Spaces of Mechanical Linkages,” Discrete Comput. Geom.DCGEER 0179-5376, 22, pp. 297–315.[[XSLOpenURL/10.1007/PL00009462]]Connelly, R., and Demaine, E. D., 2004, “ Geometry and Topology of Polygonal Linkages,” Handbook of Discrete and Computational Geometry, J.E.Goodman and J.O’Rourke, eds., CRC, Boca Raton, FL, Chap. 9.Denavit, J., and Hartenberg, R. S., 1955, “ A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices,” ASME J. Appl. Mech.JAMCAV 0021-8936, 22, pp. 215–221.Freudenstein, F., 1973, “ Kinematics: Past, Present and Future,” Mech. Mach. TheoryMHMTAS 0094-114X, 8, pp. 151–160.[[XSLOpenURL/10.1016/0094-114X(73)90049-9]]Duffy, J., 1980, The Analysis of Mechanisms and Robot Manipulators, Wiley, New York.Lee, H. -Y., and Liang, C. -G., 1988, “ Displacement Analysis of the General Spatial 7-Link 7R Mechanisms,” Mech. Mach. TheoryMHMTAS 0094-114X, 23(3), pp. 219–226.[[XSLOpenURL/10.1016/0094-114X(88)90107-3]]Canny, J., and Emiris, I., 1993, “ An Efficient Algorithm for the Sparse Matrix Resultant,” Lect. Notes Comput. Sci.LNCSD9 0302-9743, 673, pp. 89–104.Neilsen, J., and Roth, B., 1995, “ Computational Kinematics,” Solid Mechanics and Its Applications, J.P.Merlet and B.Ravani, eds., Kluwer, Dordrecht, Vol. 40, pp. 51–62.Husty, M. L., 1996, “ An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mech. Mach. TheoryMHMTAS 0094-114X, 31(4), pp. 365–379.[[XSLOpenURL/10.1016/0094-114X(95)00091-C]]Freudenstein, F., and Sandor, G. N., 1959, “ Synthesis of Path Generating Mechanisms by Means of a Programmed Digital Computer,” ASME J. Eng. Ind.JEFIA8 0022-0817, 81, pp. 159–168.Sheth, P. N., and Uicker, J. J., 1972, “ IMP (Integrated Mechanisms Program), A Computer-Aided Design Analysis System for Mechanisms and Linkages,” ASME J. Eng. Ind.JEFIA8 0022-0817, 94, pp. 454–464.[[XSLOpenURL/10.1115/1.3428176]]Suh, C. H., and Radcliffe, C. W., 1978, Kinematics and Mechanism Design, Wiley, New York, p. 458.Paul, R. P., 1981, Robot Manipulators: Mathematics, Programming and Control, MIT, Cambridge, MA.Kaufman, R. E., and Maurer, W. G., 1971, “ Interactive Linkage Synthesis on a Small Computer,” ACM National Conference , Aug. 3–5.Rubel, A. J., and Kaufman, R. E., 1977, “ KINSYN III: A New Human-Engineered System for Interactive Computer-Aided Design of Planar Linkages,” ASME J. Eng. Ind.JEFIA8 0022-0817, 99, pp. 440–448.[[XSLOpenURL/10.1115/1.3439257]]Erdman, A. G., and Gustafson, J., 1977, “ LINCAGES—A Linkage Interactive Computer Analysis and Graphically Enhanced Synthesis Package,” Chicago, IL, ASME Paper No. 77-DTC-5.Hunt, L., Erdman, A. G., and Riley, D. R., 1981, “ MicroLINCAGES: Microcomputer Synthesis and Analysis of Planar Linkages,” Proceedings of the Seventh OSU Applied Mechanisms Conference .Chuang, J. C., Strong, R. T., and Waldron, K. J., 1981, “ Implementation of Solution Rectification Techniques in an Interactive Linkage Synthesis Program,” ASME J. Mech. Des.JMDEDB 0161-8458, 103, pp. 657–664.[[XSLOpenURL/10.1115/1.3254967]]Ruth, D. A., and McCarthy, J. M., 1997, “ SphinxPC: An Implementation of Four Position Synthesis for Planar and Spherical Linkages,” Proceedings of the ASME Design Engineering Technical Conferences , Sacramento, CA, Sept. 14–17.Furlong, T. J., Vance, J. M., and Larochelle, P. M., 1999, “ Spherical Mechanism Synthesis in Virtual Reality,” ASME J. Mech. Des.JMDEDB 0161-8458, 121, pp. 515–520.[[XSLOpenURL/10.1115/1.2829491]]Liao, Q., and McCarthy, J. M., 2001, “ On the Seven Position Synthesis of a 5-SS Platform Linkage,” ASME J. Mech. Des.JMDEDB 0161-8458, 123, pp. 74–79.[[XSLOpenURL/10.1115/1.1330269]]Roth, B., and Freudenstein, F., 1963, “ Synthesis of Path-Generating Mechanisms by Numerical Methods,” ASME J. Eng. Ind.JEFIA8 0022-0817, 85B, pp. 298–306.Freudenstein, F., and Roth, B., 1963, “ Numerical Solution of Systems of Nonlinear Equations,” J. ACMZZZZZZ 1535-9921, 10(4), pp. 550–556.[[XSLOpenURL/10.1145/321186.321200]]Watson, L. T., 1979, “ A Globally Convergent Algorithm for Computing Fixed Points of $C2$ Maps,” Appl. Math. Comput.AMHCBQ 0096-3003, 18, pp. 297–311.[[XSLOpenURL/10.1016/0096-3003(79)90020-1]]Morgan, A. P., 1986, “ A Homotopy for Solving Polynomial Systems,” Appl. Math. Comput.AMHCBQ 0096-3003, 5, pp. 87–92.[[XSLOpenURL/10.1016/0096-3003(86)90030-5]]Tsai, L. W., and Morgan, A. P., 1985, “ Solving the Kinematics of the Most General Six- and Five-Degree-of-Freedom Manipulators by Continuation Methods,” ASME J. Mech., Transm., Autom. Des.JMTDDK 0738-0666, 107, pp. 189–200.[[XSLOpenURL/10.1115/1.3258708]]Wampler, C. W., and Morgan, A. P., 1992, “ Complete Solution for the Nine-Point Path Synthesis Problem for Four-Bar Linkages,” ASME J. Mech. Des.JMDEDB 0161-8458, 114(1), pp. 153–159.[[XSLOpenURL/10.1115/1.2916909]]Raghavan, M., and Roth, B., 1993, “ Inverse Kinematics of the General 6R Manipulator and Related Linkages,” ASME J. Mech. Des.JMDEDB 0161-8458, 115(3), pp. 502–508.[[XSLOpenURL/10.1115/1.2919218]]Raghavan, M., and Roth, B., 1995, “ Solving Polynomial Systems for Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators,” ASME J. Mech. Des.JMDEDB 0161-8458, 117, pp. 71–79.[[XSLOpenURL/10.1115/1.2836473]]Raghavan, M., 1993, “ The Stewart Platform of General Geometry Has 40 Configurations,” ASME J. Mech. Des.JMDEDB 0161-8458, 115(2), pp. 277–282.[[XSLOpenURL/10.1115/1.2919188]]
Commentary by Dr. Valentin Fuster

### Research Papers

J. Mechanisms Robotics. 2010;3(1):011001-011001-9. doi:10.1115/1.4002513.

In this paper, we implement a characterization based on eigentwists and eigenwrenches for the synthesis of a compliant mechanism at a given point. For 2D mechanisms, this involves characterizing the compliance matrix at a unique point called the center of elasticity, where translational and rotational compliances are decoupled. Furthermore, the translational compliance may be represented graphically as an ellipse and the coupling between the translational and rotational components as vectors. These representations facilitate geometric insight into the operations of serial and parallel concatenations. Parametric trends are ascertained for the compliant dyad building block and are utilized in example problems involving serial concatenation of building blocks. The synthesis technique is also extended to combination of series and parallel concatenation to achieve any compliance requirements.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2010;3(1):011002-011002-6. doi:10.1115/1.4002696.

In this paper, the unique form of the screw based Jacobian is suggested for lower mobility parallel manipulators. Utilizing the concept of the reciprocal Jacobian, the forward statics relation for each of the serial kinematic chains of a parallel manipulator can be first obtained and then used to derive both the forward statics and the inverse velocity relations of the manipulator. The screw based Jacobian of a parallel manipulator can be formulated from the inverse velocity relation in such a way that it consists of the reciprocal Jacobians of the serial kinematic chains. Since any reciprocal Jacobian is unique to the corresponding serial chain, the suggested form of the screw based Jacobian is also determined uniquely to the lower mobility parallel manipulator. Two examples are given to illustrate the proposed method, one for the 3DOF parallel manipulator with three identical prismatic-revolute-spherical joints-serial chains and the other for the 4DOF parallel manipulator with nonidentical serial chains (two spherical-prismatic-spherical- and one revolute-revolute-prismatic-revolute joints-serial chains).

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2010;3(1):011003-011003-8. doi:10.1115/1.4002697.

With the introduction of generalized function sets ($GF$ set) to represent the characteristics of the end-effectors of parallel mechanisms, two classes of $GF$ sets are proposed. The type synthesis of parallel mechanisms having the second class $GF$ sets and two dimensional rotations, including 2-, 3-, and 4DOF parallel mechanisms, is investigated. First, the intersection algorithms for the $GF$ sets are established via the axiom of two dimensional rotations. Second, the kinematic limbs with specific characteristics are designed according to the axis movement theorem. Finally, several parallel mechanisms having the second class $GF$ sets and two dimensional rotations have been illustrated to show the effectiveness of the proposed methodology.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2010;3(1):011004-011004-9. doi:10.1115/1.4002815.

During the last 15 years, parallel mechanisms (robots) have become more and more popular among the robotics and mechanism community. Research done in this field revealed the significant advantage of these mechanisms for several specific tasks, such as those that require high rigidity, low inertia of the mechanism, and/or high accuracy. Consequently, parallel mechanisms have been widely investigated in the last few years. There are tens of proposed structures for parallel mechanisms, with some capable of six degrees of freedom and some less (normally three degrees of freedom). One of the major drawbacks of parallel mechanisms is their relatively limited workspace and their behavior near or at singular configurations. In this paper, we analyze the kinematics of a new architecture for a six degrees of freedom parallel mechanism composed of three identical kinematic limbs: revolute-revolute-revolute-spherical. We solve the inverse and show the forward kinematics of the mechanism and then use the screw theory to develop the Jacobian matrix of the manipulator. We demonstrate how to use screw and line geometry tools for the singularity analysis of the mechanism. Both Jacobian matrices developed by using screw theory and static equilibrium equations are similar. Forward and inverse kinematic solutions are given and solved, and the singularity map of the mechanism was generated. We then demonstrate and analyze three representative singular configurations of the mechanism. Finally, we generate the singularity-free workspace of the mechanism.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2010;3(1):011005-011005-8. doi:10.1115/1.4002694.

To solve the velocity and acceleration of legs with different structures is a fundamental and challenging issue for dynamics analysis of parallel manipulators (PMs). In this paper, the kinematics of linear legs with different structures for limited-degree of freedom (DOF) PMs is studied. First, based on kinematics/statics of general limited-DOF PM, the formulas are derived for solving the angular velocity/acceleration of some linear legs with different structures. Second, the velocity and acceleration of the piston/cylinder in the legs are represented by velocity and acceleration of platform in PM. Finally, the solving procedures are illustrated by applying this approach to a 4DOF PM.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2010;3(1):011006-011006-8. doi:10.1115/1.4002695.

It is commonly assumed that the singularities of kinematic mappings constitute generically smooth manifolds but this has not yet been proven. Moreover, before this assumption can be verified, the concept of genericity needs to be clarified. In this paper, two different notions of generic properties of kinematic mappings are discussed. One accounts for the stability of the manifold property with respect to small changes in the geometry of a mechanism while the other concerns the likelihood that a mechanism possesses smooth manifolds of singularities. Singularities forming smooth manifolds is the condition for singularity-free motion of overconstrained mechanisms but also has consequences for reliable control of serial manipulators. As basis for establishing genericity, a formulation of the kinematic mapping is presented that takes into account the type of joints and feasible link geometries. The continuous transition between link geometries defines a deformation of a kinematic mapping. All mappings obtained in this way constitute a class of kinematic mappings. A basic characteristic of a kinematic chain is its motion space. An explicit expression for the motion spaces of individual as well as classes of kinematic mappings is given. The actual conditions for genericity will be addressed in a forthcoming publication.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011007-011007-6. doi:10.1115/1.4003079.

A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) of a linearly actuated quadratic spherical parallel manipulator. An alternative formulation of the kinematic equations of the quadratic spherical parallel manipulator is proposed. The singularity analysis of the quadratic spherical parallel manipulator is then dealt with. A new type of singularity of parallel manipulators—leg actuation singularity—is identified. If a leg is in a leg actuation singular configuration, the actuated joints in this leg cannot be actuated even if the actuated joints in other legs are released. A formula is revealed that produces a unique current solution to the FDA for a given set of inputs. The input space is also revealed for the quadratic spherical parallel manipulator in order to guarantee that the robot works in the same assembly mode. This work may facilitate the control of the quadratic spherical parallel manipulator.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011008-011008-18. doi:10.1115/1.4003005.

A biped mountable robotic baby head was developed using a combination of Biometal fiber and Flexinol shape memory alloy actuators (SMAs). SMAs were embedded in the skull and connected to the elastomeric skin at control points. An engineered architecture of the skull was fabricated, which incorporates all the SMA wires with 35 routine pulleys, two firewire complementary metal-oxide semiconductor cameras that serve as eyes, and a battery powered microcontroller base driving circuit with a total dimension of $140×90×110 mm3$. The driving circuit was designed such that it can be easily integrated with a biped and allows programming in real-time. This 12DOF head was mounted on the body of a 21DOF miniature bipedal robot, resulting in a humanoid robot with a total of 33DOFs. Characterization results on the face and associated design issues are described, which provides a pathway for developing a humanlike facial anatomy using wire-based muscles. Numerical simulation based on SIMULINK was conducted to assess the performance of the prototypic robotic face, mainly focusing on the jaw movement. The nonlinear dynamics model along with governing equations for SMA actuators containing transcendental and switching functions was solved numerically and a generalized SIMULINK model was developed. Issues related to the integration of the robotic head with a biped are discussed using the kinematic model.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011009-011009-7. doi:10.1115/1.4003078.

Motivated by the problem of synthesizing a pattern of flexures that provide a desired constrained motion, this paper presents a new screw theory that deals with “line screws” and “line screw systems.” A line screw is a screw with a zero pitch. The set of all line screws within a screw system is called a line variety. A general screw system of rank $m$ is a line screw system if the rank of its line variety equals $m$. This paper answers two questions: (1) how to calculate the rank of a line variety for a given screw system and (2) how to algorithmically find a set of linearly independent lines from a given screw system. It has been previously found that a wire or beam flexure is considered a line screw, or more specifically a pure force wrench. By following the reciprocity and definitions of line screws, we have derived the necessary and sufficient conditions of line screw systems. When applied to flexure synthesis, we show that not all motion patterns can be realized with wire flexures connected in parallel. A computational algorithm based on this line screw theory is developed to find a set of admissible line screws or force wrenches for a given motion space. Two flexure synthesis case studies are provided to demonstrate the theory and the algorithm.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011010-011010-8. doi:10.1115/1.4003180.

This paper presents a new family of maximally regular $T2R1$-type parallel manipulators with bifurcated spatial motion. In each branch, the moving platform has two independent translations $(T2)$ and one rotation $(R1)$ driven by three actuators mounted on the fixed base. The rotation axis is situated in the plane of translation and can bifurcate in two orthogonal directions. This bifurcation occurs in a constraint singularity in which the connectivity between the moving and fixed platforms increases instantaneously, incurring no change in limb connectivity. The Jacobian matrix of the maximally regular solutions presented in this paper is a $3×3$ identity matrix in the entire workspace of each branch. This paper presents for the first time a family of maximally regular $T2R1$-type parallel manipulators with bifurcated spatial motion of the moving platform along with solutions using uncoupled motions.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011011-011011-6. doi:10.1115/1.4003269.

This paper presents the extension of the N-bar rotatability laws to N-bar chains containing prismatic joints. The extension is based on the principle that a prismatic joint may be regarded as a revolute joint located at infinity in the direction normal to the sliding path. The effects of long and short links, full rotatability, linkage classification, and formation of branches and sub-branches are discussed. The extension provides a consistent method to understand all aspects of linkage rotatability disregarding the existence of prismatic joints. The results are demonstrated by several examples.

Topics: Chain
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011012-011012-7. doi:10.1115/1.4003270.

This paper introduces mechanism state matrices as a novel way to represent the topological characteristics of planar reconfigurable mechanisms. As part of this new approach, these matrices will be used as an analysis tool to automatically determine the degrees of freedom (DOFs) of planar mechanisms that only contain one DOF joint. The DOF at each state can be combined with a mechanism state matrix to form an augmented mechanism state matrix. A series of examples will be used to illustrate the proposed concept.

Topics: Mechanisms
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011013-011013-8. doi:10.1115/1.4003271.

This paper presents a new approach to the velocity and acceleration analyses of lower mobility parallel manipulators. Building on the definition of the “acceleration motor,” the forward and inverse velocity and acceleration equations are formulated such that the relevant analyses can be integrated under a unified framework that is based on the generalized Jacobian. A new Hessian matrix of serial kinematic chains (or limbs) is developed in an explicit and compact form using Lie brackets. This idea is then extended to cover parallel manipulators by considering the loop closure constraints. A 3-P RS parallel manipulator with coupled translational and rotational motion capabilities is analyzed to illustrate the generality and effectiveness of this approach.

Topics: Manipulators
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011014-011014-8. doi:10.1115/1.4003415.

The parallel connection of a revolute-spherical-revolute Clemens linkage with a revolute-universal joint (RU) spherical linkage realizes a two-axis spherical pointing device suitable for use in a robot wrist. The design is derived from a Clemens linkage constant-velocity coupling having a spherical constraint. A novel transverse orientation of the RU linkage facilitates control of the pointing direction through actuation of a pair of revolute joints at the base of the device, a capability not otherwise required when this type of mechanism is employed as a constant-velocity coupling. As the mechanism is confined to spherical space, spherical trigonometry and related geometric reasoning may be employed in the analysis of mobility and actuation. The analysis derives the forward and inverse kinematic relations, showing that the mechanism allows a full 180 deg of deflection of the pointer from the center before encountering a singularity.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):011015-011015-9. doi:10.1115/1.4003444.

New stiffness performance indices using the collinear stiffness value (CSV) associated with a given configuration of the machine are proposed. The minimal CSV (MinCSV) is applied to stiffness evaluation for all types of configurations. Similar to the determinant, the MinCSV equals zero in singular configurations. In regular configurations, the MinCSV is applied to evaluation of local stiffness for a given configuration and global stiffness in the workspace, wherein stiffness limitations are satisfied. A screw stiffness value, i.e., the CSV during a screw displacement, presents the general case of the CSV. There are two important special cases: rotational and translational stiffness values. Procedures for evaluation of the MinCSV are developed in natural and dimensionless forms. The CSV of the hexapod are simulated and compared with those of serial-type mechanisms. The proposed approach presents an effective design tool for evaluation and limitation of stiffness of machines and robots.

Topics: Stiffness , Machinery
Commentary by Dr. Valentin Fuster

### Technical Briefs

J. Mechanisms Robotics. 2010;3(1):014501-014501-7. doi:10.1115/1.4002693.

A time-varying instantaneous screw characterizes the motion of a rigid body. The kinematic differential equation expresses the path taken by any point on that rigid body in terms of this screw. Therefore, when a revolute joint is attached to a moving link in a planar kinematic chain, the path taken by the center of that revolute joint is the solution to such an equation. The instantaneous screw of a link in that chain is in turn determined by the action of the joints connecting that link to ground, where the contribution of each joint to that instantaneous screw is determined by its actuation rate and center point. Substituting power series expansions for joint rates into the kinematic differential equations for joint centers, and expressing loop closure as a linear constraint on the instantaneous screws of the links, a recurrence relation is established that solves for the coefficients in those power series. The resulting solution is applied to determine the equilibrium pendulum tilt of the United Aircraft TurboTrain. Comparing that power series approximation with an exact kinematic analysis shows convergence properties of the series.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):014502-014502-4. doi:10.1115/1.4002943.

This paper derives the expressions of an equivalent finite screw of two successive screw motions in a simplified form using purely vectorial analysis. This is achieved by tracing the trajectories of specific points on the moving body, which together with the known axis and angle of combined rotation, yield the expressions of the screw triangle. This paper also gives a short overview of different known expressions of the screw triangle and shows that the one given in this paper reduces the number of arithmetic operations by about a third compared with the most efficient algorithm in the literature.

Topics: Screws
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2011;3(1):014503-014503-5. doi:10.1115/1.4003077.

Physical parameters of the constituent modules and gait parameters affect the overall performance of snake-inspired robots. Hence, a system-level optimization model needs to concurrently optimize the module parameters and the gait. Incorporating a physics-based model of rectilinear gaits in the system-level optimization model is a computationally challenging problem. This paper presents a case study to illustrate how metamodels of the precomputed optimal rectilinear gaits can be utilized to reduce the complexity of the system-level optimization model. An example is presented to illustrate the importance of concurrently optimizing the module parameters and the gait to obtain the optimal performance for a given mission.

Topics: Robots , Optimization
Commentary by Dr. Valentin Fuster