M. John D. Hayes

Kinematic Mapping Application to Approximate Type and Dimension Synthesis of Planar Mechanisms

Kinematic mapping is used for preliminary development of an algorithm for the approximate synthesis of planar four-bar mechanisms for rigid body guidance. Both dyad type and dimensions are determined. Planar mechanism coupler motions are represented as the curves of intersection of a pair of quadric constraint surfaces, one for each of two dyads. The problem reduces to identifying the two best constraint surfaces in the pencil of quadrics containing the curve. The overdetermined synthesis equations are linear in the unknown surface shape coefficients, and their products. Non-trivial solutions exist only in the nullspace of the coefficient matrix. While the algorithm remains incomplete, results presented herein are encouraging.

The kinematics of A-pair jointed serial linkages

A new kinematic pair called an algebraic screw pair, or A-pair, is introduced that utilizes the self-motions inherent to a specific configuration of Griffis-Duffy platform. Using the A-pair as a joint in a hybrid parallel-serial kinematic chain results in a sinusoidal coupling of rotation and translation between adjacent links. This motion affects both the direct and inverse kinematics of such chains. Presented in this paper are the direct kinematics of chains using A-pairs and an algorithm for the inverse kinematics of a 4A-pair chain.Copyright

Reconfigurable Planar Three-Legged Parallel Manipulators

A new class of reconfigurable planar three legged platforms is introduced. Kinematic mapping techniques are applied to solve the forward and inverse kinematic problems, and to visualize the reachable workspace of the platform. The results show that the new class of manipulator is viable from a kinematic modelling point of view.

Input-Output Equation for Planar Four-Bar Linkages

In this paper the generalised input-output (I-O) equation for planar 4R function generators is derived in a new way, leading to the algebraic form of the well known Freudenstein equation. The long term goal is to develop a generalised method to derive constraint based algebraic I-O equations that can be used for continuous approximate synthesis, where the synthesis equations are integrated between minimum and maximum input angle values resulting in a linkage whose objective function has been optimised over every output angle. In this paper we use a planar projection of Study’s soma and the Cartesian displacement constraints for the dyads. These are mapped to the image space leading to four constraint equations in terms of the image space coordinates and the sines and cosines of the input and output angles. Using the tangent of the half angle substitution the trigonometric equations are converted to algebraic ones. Algebraic methods are used to eliminate the image space coordinates, then the polynomial resultants are found to obtain common roots leading to the desired equations.

Integrated Type and Dimensional Synthesis of Planar Four-Bar Mechanisms

A novel approach to integrated type and approximate dimensional synthesis of planar four-bar mechanisms (i.e. linkages comprised of any two of RR, PR, RP, and PP dyads) for rigid-body guidance is proposed. The essence is to correlate coordinates of the coupler attachment points in two different coordinate frames, thereby reducing the number of independent variables defining a suitable dyad for the desired rigid-body motion from five to two. After applying these geometric constraints, numerical methods are used to size link lengths, locate joint axes, and decide between RR, PR, RP and PP dyads that, when combined, guide a rigid body through the best approximation, in a least-squares sense, of n specified positions and orientations, where n≥5. No initial guesses of type or dimension are required. An example is presented illustrating the effectiveness and robustness of this new approach.

Velocity Level Kinematic Analysis of Serial nA-Chains

The algebraic screw pair, or A-pair, represents a new class of kinematic constraint that exploits the self-motions inherent to a specific configuration of Griffis–Duffy platform. The A-pair causes a sinusoidal coupling of rotation and translation between adjacent links in the kinematic chain. The resulting linkage is termed an A-chain. This paper presents a derivation of the manipulator Jacobian of nA-chains in general, and a specific 4 degree-of-freedom hybrid serial-parallel 4A-chain.