This article develops a general technique for differential analysis that can be applied to singularities of three related problems: path tracking for nonredundant robots, self-motion analysis for robots with one degree of redundancy, and displacement analysis of single-loop mechanisms. For each of these problems, the locus of displacement solutions generally forms a set of one-dimensional manifolds in the space of variable parameters. However, if singularities occur, the manifolds may degenerate into isolated points, or into curves that include bifurcations at the singular points. Higher-order equations, derived from Taylor series expansion of the matrix equation of closure, are solved to identify singularity type and, in the case of bifurcations, to determine the number of intersecting branches as well as a Taylor series expansion of each branch about the point of bifurcation. To avoid unbounded mathematics, branch expansions are derived in terms of an introduced curve parameter. The results are useful for identifying singularity type, for numerical curve tracking with continuation past bifurcations on any chosen branch, and for determining exact rate relations for each branch at a bifurcation. The noniterative solution procedure involves configuration-dependent systems of equations that are evaluated by recursive algorithm, then solved using singular value decomposition, polynomial equation solution, and linear system solution. Examples show applications to RCRCR mechanisms and the Puma manipulator. >

Differential analysis of bifurcations and isolated singularities for robots and mechanisms

Computation of kinematic paths and bifurcation points

The kinematic singularities of robot manipulators are studied from the point of view of the theory of singularities. The notion of a 'generic' kinematic map, whose singularities form smooth manifolds of prescribed dimension in the joint space of the manipulator, is examined. For three-joint robots, an equivalent algebraic condition for genericity using the Jacobian determinants is derived. This condition lends itself to symbolic computation and is sufficient for the study of decoupled manipulators. Orientation and translation singularities of manipulators are studied in detail. A complete characterization of orientation singularities of robots with any number of joints is given. The translation singularities of the eight possible topologies of three-joint robots are studied and the conditions on the link parameters for nongenericity are determined. >

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Genericity and singularities of robot manipulators

A new method, termed the offset modification method (OM method), for solving the manipulator inverse kinematics problem is presented. The OM method works by modifying the link offset values of a manipulator until it is possible to derive closed-form inverse kinematics equations for the resulting manipulator (termed the model manipulator). This procedure allows one to derive a set of three nonlinear equations in three unknowns that, when numerically solved, give an inverse kinematics solution for the original manipulator. The OM method can be applied to manipulators with any number of degrees of freedom, as long as the manipulator satisfies a given set of conditions (Theorem 1). The OM method is tested on a 6-degree-of-freedom manipulator that has no known closed-form inverse kinematics equations. It is shown that the OM method is applicable to real-time manipulator control, can be used to guarantee convergence to a desired endpoint position and orientation (if it exists), and allows one to directly choose which inverse kinematics solution the algorithm will converge to (as specified in the model manipulator closed-form inverse kinematics equations). Applications of the method to other 6-DOF manipulator geometries and to redundant manipulators (i.e. greater than 6 DOF geometries) are discussed. >

Iterative inverse kinematics with manipulator configuration control

In this paper a program of singularity theory is pro claimed to be of systematic use in robotics. Complete lists of normal forms are proposed and are regarded as candi date models of kinematics of robot manipulators. Argu ments for the applicability of candidate normal forms to manipulator kinematics are provided. Singularities and bifurcation diagrams of the normal forms are examined and consequences derived for the inverse kinematic prob lem in redundant kinematics with singularities.

Differential topology of the inverse kinematic problem for redundant robot manipulators

A general approach for the determination of singularities in motion and displacement functions of manipulators and linkages is proposed. Singularities in motion of a manipula tor occur at positions where the parameters in motions of the end-effector become dependent, or when the end-effector is at rest and some manipulator links become movable. Singular ities in motion of a linkage occur at positions where an over- constrained group of links becomes mobile while the driving link is held at rest. Both cases of singularities are determined with the condition that the rank r of the system matrix of six linear equations is less than 6. The system of linear equa tions relates the velocities in motion in joints with the veloci ties of the linkage driving link (manipulator end-effector). In the proposed approach, the elements of the system matrix can be derived directly by using the matrix representation of coordinate transformation. The requirement r < 6 provides an equation (the singularity equation) that rela...

Singularities in Motion and Displacement Functions of Constrained Mechanical Systems

Singularites du mouvement et fonctions de deplacement d'un mecanisme a trois articulations de rotation et deux articulations cylindriques

Singularities in Motion and Displacement Functions for the RCRCR Linkage

A helicopter tailrotor driveline is equipped with dampers to suppress the resonance of transmission shafts. A dry friction damper and a slender shaft form an assembly The comprehensive understanding of different responses of the assembly provides the basis for a more reasonable parameter configuration of the damper. To this end, the assembly is modeled as a dynamic system with rub impact between a rotor with continuous parameter distribution and a suspended elastic stator. Boundaries of rub impact and condition of Hopf bifurcation are derived. Pinned natural frequencies and backward whirl frequencies are solved by analytical deduction. The critical speed for triggering a backward whirl with consideration of eccentric excitation is obtained by the multiple scale perturbation method. Then, all-round responses and boundaries are solved by both analytical and numerical approaches based on the parameters of a helicopter. The effects of friction coefficient, internal damping, suspension stiffness, variable impact stiffness, etc. on the boundaries are presented in detail with the discussion of relationships between physical reasons and analytical conditions. Results show that smaller impact stiffness difference and friction coefficient, a greater mass damping ring, moderate suspension stiffness, greater internal and external damping are the key factors for the damper design. This study has reference significance not only for the assembly in a helicopter but also for similar mechanical systems. 

All-round responses and boundaries of a shaft and dry friction damper assembly

Compliant amplification mechanisms amplify input displacement in the desired output direction. However, owing to structural design, parasitic motion can easily be produced in an unexpected direction. The parasitic motion has a negative effect on the motion accuracy of the mechanism. To solve this problem, a topology optimization method for compliant amplification mechanisms with low parasitic displacement was proposed. Based on the variable density topology optimization method, the topology optimization model of the compliant amplification mechanism was established with the goal of increasing the output displacement and reducing the parasitic displacement. Volume ratio was set as constraint condition. The optimization criterion method were used to solve the problem and topology optimized amplification mechanisms (TOAMs) were obtained. Simultaneously, the configuration characteristics and displacement amplification ratios of the mechanism under different virtual spring stiffnesses were compared. To verify the validity of the method, the performance of the TOAM and the typical amplification mechanism (TAM) were compared using finite element simulation. The displacement amplification ratio is 5.95 and 3.17, and the relative parasitic displacement is 0.6% and 10.27%, respectively. Finally, the performance of the TOAM and the TAM was verified by experiments. The displacement amplification ratio is 5.72 and 3.06, and the relative parasitic displacement is 0.95% and 10.64%, respectively. Simulation and experimental results show that the TOAM has a larger displacement amplification ratio and a lower parasitic displacement, which verifies the validity of this method.

J. Tan

Papers

Singularities in Motion and Displacement Functions of Constrained Mechanical Systems

Singularities in Motion and Displacement Functions for the RCRCR Linkage

All-round responses and boundaries of a shaft and dry friction damper assembly

Topology optimization design of compliant amplification mechanisms with low parasitic displacement