Singularity-free Region Research Articles

Kinematic and Dynamic Analysis of a 3-PRUS Spatial Parallel Manipulator

Parallel Kinematic Machines (PKMs) are being widely used for precise applications to achieve complex motions and variable poses for the end effector tool. PKMs are found in medical, assembly and manufacturing industries where accuracy is necessary. It is often desired to have a compact and simple architecture for the robotic mechanism. In this paper, the kinematic and dynamic analysis of a novel 3-PRUS (P: prismatic joint, R: revolute joint, U: universal joint, S: spherical joint) parallel manipulator with a mobile platform having 6 Degree of Freedom (DoF) is explained. The kinematic equations for the proposed spatial parallel mechanism are formulated using the Modified Denavit-Hartenberg (DH) technique considering both active and passive joints. The kinematic equations are used to derive the Jacobian matrix of the mechanism to identify the singular points within the workspace. A Jacobian based stiffness analysis is done to understand the variations in stiffness for different poses of the mobile platform and further, it is used to decide trajectories for the end effector within the singularity free region. The analytical model of the robot dynamics is presented using the Euler-Lagrangian approach with Lagrangian multipliers to include the system constraints. The gravity and inertial forces of all links are considered in the mathematical model. The analytical results of the dynamic model are compared with ADAMS simulation results for a pre-defined trajectory of the end effector.

Determination of the maximal singularity-free workspace of 3-DOF parallel mechanisms with a constructive geometric approach

This paper proposes a novel approach to obtain the maximal singularity-free regions of planar parallel mechanisms, which is based on a constructive geometric reasoning. The proposed approach consists of two algorithms. First, the borders of the singularity-free region associated with an arbitrary starting point of the moving platform are obtained. Then, the second algorithm finds the center of the maximal singularity-free circle, which is obtained using the so-called offset curve algorithm. The procedure is applied to a 3-PRR planar parallel mechanism as an example and the obtained results illustrate graphically the effectiveness of the proposed algorithm. The proposed approach can be directly applied to obtain the maximal singularity-free circle of similar parallel mechanisms, which is not the case for other approaches proposed in the literature that are limited to a given parallel mechanism, namely, the 3-RPR planar parallel mechanism. Moreover, as the main feature of the proposed approach, it can be implemented both in a CAD system and in a computer algebra system where non-convex and reentrant curves can be considered.

Forward displacement analysis of a quadratic 4-DOF 3T1R parallel manipulator

A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) and the Type II singularity analysis of a quadratic 4-DOF 3T1R (SCARA) parallel manipulator: the Quadrupteron. It will be proved that there exists a one-to-one correspondence between the two formulas, each producing one solution to the FDA, and the two singularity-free regions. 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. The Type II singularity analysis in the joint space will also be investigated in order to identify the conditions on the inputs to keep the Quadrupteron working in the same singularity-free region in its Cartesian workspace.

Forward Displacement Analysis of a Quadratic Planar Parallel Manipulator: 3-RP̱R Parallel Manipulator With Similar Triangular Platforms 1

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.

우주비행체의 랑데부 및 도킹을 위한 적응 제어기법

본 논문에서는 일반적인 타원 궤도상에서의 두 위성체간의 랑데부와 도킹을 수행하기 위한 적응 제어기법을 개발하였다. 직교좌표계를 이용해서 나타낸 두 비행체간의 상대운동방정식을 일반적인 해밀토니안 운동방정식의 형태로 변환한 후, 불확실한 시스템 파라미터를 가진 동적시스템을 위해 개발된 적응제어기법을 적용하여 제어 알고리즘을 유도하였다. 시스템 파라미터를 추정하는데 투사기법을 적용하여 파라미터 추정값의 변화에 의한 특이점을 회피할 수 있도록 하였으며, 수치해석을 통하여 추적비행체의 질량이 불확실한 경우에 대하여 제어 알고리즘의 성능을 검증하였다. An adaptive control algorithm for spacecraft rendezvous and docking in a Keplerian orbit is presented. The equations of relative motion of two spacecrafts expressed in a local-vertical-local-horizontal rectangular frame are converted to a general Hamiltonian form, then an adaptive control method developed for the uncertain Hamiltonian system is applied to the rendezvous and docking problem. A smooth projection algorithm is applied to keep the parameter estimates inside a singularity-free region, and a numerical example shows that the developed controller successfully deals with the unknown mass of the chaser spacecraft.

Traversing the singularity hypersurface by applying the input disturbances to 6-SPS parallel manipulator

The singular points of a 6-SPS Stewart platform are distributed on the multi-dimensional singularity hypersurface in the task-space, which divides the workspace of the manipulator into several singularity-free regions. Because of the motion uncertainty at singular points, while the manipulator traverses this kind of hypersurface from one singularity-free region to another, its motion cannot be predetermined. In this paper, a detailed approach for the manipulator to traverse the singularity hypersurface with its non-persistent configuration is presented. First, the singular point transfer disturbance and the pose disturbance, which make the perturbed singular point transfer horizontally and vertically, respectively, are constructed. Through applying these disturbances into the input parameters within the maximum loss control domain, the perturbed persistent configuration is transformed into its corresponding non-persistent one. Under the action of the disturbances, the manipulator can traverse the singularity hypersurface from one singularity-free region to another with a desired configuration.

Adaptive spacecraft attitude tracking control with actuator uncertainties

An adaptive control algorithm for the spacecraft attitude tracking problem when the spin axis directions and/or the gains of the flywheel actuators are uncertain is developed. A smooth projection algorithm is applied to keep the parameter estimates inside a singularity-free region and avoid parameter bursting. Numerical examples show that the controller successfully deals with unknown misalignments of the axis directions as well as the unknown gains of the flywheel actuators.

Modified Disturbance Function Method for a 6-6 Gough–Stewart Parallel Manipulator to Traverse the Singularity Hypersurface

Due to the motion uncertainty at the singular point, while the manipulator traverses the singularity hypersurface and moves from one singularity-free region to another, its motion is uncertain. To obtain a desired motion after it traverses the singularity hypersurface, the disturbance function approach was presented by the authors. Because the configuration transformation process takes place within the maximum loss control domain (MLCD), the motion uncertainty within the MLCD still exists. In order to eliminate this kind of motion uncertainty, a modified disturbance function method is presented in this paper. With the aid of the optimization method, the modified disturbance function is constructed through setting up the constraint equations to ensure each trace point locating beyond the MLCD corresponding to the perturbed singular point, and permitting some components of the configuration parameters with a little deviation. Under the action of the modified disturbances, the manipulator moves beyond the MLCD and traverses the singularity hypersurface with a controllable motion.

Repeatability of inverse kinematics algorithms for mobile manipulators

We introduce and examine the property of repeatability of inverse kinematics algorithms for mobile manipulators. Similarly to stationary manipulators, repeatability of mobile manipulators is defined by requiring that a closed path in the task space should be transformed by the inverse kinematics algorithm into a closed path in the configuration space. In a simply connected, singularity-free region of the task space, a necessary and sufficient condition for repeatability is derived as the integrability condition of a distribution associated with the inverse kinematics algorithm.

Classical Signature Change in the Black Hole Topology

Investigations of classical signature change have generally envisaged applications to cosmological models, usually a Friedmann–Lemaître–Robertson–Walker model. The purpose has been to avoid the inevitable singularity of models with purely Lorentzian signature, replacing the neighbourhood of the big bang with an initial, singularity free region of Euclidean signature, and a signature change. We here show that signature change can also avoid the singularity of gravitational collapse. We investigate the process of re-birth of Schwarzschild type black holes, modelling it as a double signature change, joining two universes of Lorentzian signature through a Euclidean region which provides a "bounce". We show that this process is viable both with and without matter present, but realistic models — which have the signature change surfaces hidden inside the horizons — require nonzero density. In fact the most realistic models are those that start as a finite cloud of collapsing matter, surrounded by vacuum. We consider how geodesics may be matched across a signature change surface, and conclude that the particle "masses" must jump in value. This scenario may be relevant to Smolin's recent proposal that a form of natural selection operates on the level of universes, which favours the type of universe we live in.

Fracture parameter for thermoinelasticity

A path domain independent integral, S, that equals the energy release rate for an extending crack in a thermoinelastic field is presented. This paper develops the new parameter from its theoretical foundations in continuum mechanics, demonstrates that S can be calculated from finite element results and describes how S can be obtained from experiments. The S-integral is developed for simple uncoupled and linearized fully coupled quasi static thermoinelastic cases. Invoking thermoinelastic continuum mechanics linearized for small strain and small temperature changes, S emanates from a discrete Lagrangian describing the thermoinelastic system and Noether's theorem from classical field theory. S defines the force acting on an extending crack and represents a conservation law for a crack free body analogous to the Budiansky and Rice interpretation of the J-integral. The conservation law nature of S for a singularity free region is demonstrated by both computational and physical experiments. S can be calculated from finite element results via a two step postprocessing algorithm. Furthermore, S can be obtained from physical experiments. The S-integral offers a parameter to improve the understanding of the strength and reliability of materials subjected to thermomechanical loadings.

No horn of singularities for the double well anharmonic oscillator

We consider the one-dimensional quantum mechanical potential V= g 2( x 2−1/ g 2) 2, which has pseudo-particle solutions. It has been suggested that the work of Bender and Wu, and Simon on the anharmonic oscillator shows that the energy levels for V, considered as an analytic function of g 2, have a horn of singularities asymptotic to the real g 2 axis. This implies that the eigenvalues of the hamiltonian could not be written as a Borel sum. It is shown here that there is no such proof, and indeed, this potential does not have a horn of singularities. Further, the known singularity-free region allows the possibility of Borel summability.