Sub-optimal Solution of the Inverse Kinematic Task of Redundant Robots without Using Lagrange Multipliers

In the paper a novel approach is suggested for solving the inverse kinematic task of redundant open kinematic chains. Traditional approaches as the Moore-Penrose generalized inverse-based solutions minimize the sum of squares of the timederivative of the joint coordinates under the constraint that contains the task itself. In the vicinity of kinematic singularities where these solutions are possible the hard constraint terms produce high time-derivatives that can be reduced by the use of a deformation proposed by Levenberg and Marquardt. The novel approach uses the basic scheme of the Receding Horizon Controllers in which the Lagrange multipliers are eliminated by direct application of the kinematic model over the horizon in the role of the ”control force”, and no reduced gradient has to be computed. This fact considerably decreases the complexity of the solution. If the cost function contains penalty for high joint coordinate time-derivatives the kinematic singularities are ab ovo better handled. Simulation examples made for a 7 degree of freedom robot arm demonstrate the operation of the novel approach. The computational need of the method is still considerable but it can be further decreased by the application of complementary tricks.

Overcoming Kinematic Singularities for Motion Control in a Caster Wheeled Omnidirectional Robot

Omnidirectional planar robots are common these days due to their high mobility, for example in human–robot interactions. The motion of such mechanisms is based on specially designed wheels, which may vary when different terrains are considered. The usage of actuated caster wheels (ACW) may enable the usage of regular wheels. Yet, it is known that an ACW robot with three actuated wheels needs to overcome kinematic singularities. This paper introduces the kinematic model for an ACW omni robot. We present a novel method to overcome the kinematic singularities of the mechanism’s Jacobian matrix by performing the time propagation in the mechanism’s configuration space. We show how the implementation of this method enables the estimation of caster wheels’ swivel angles by tracking the plate’s velocity. We present the mechanism’s kinematics and trajectory tracking in real-world experimentation using a novel robot design.

Kinematics and Singularity Analysis of a 7-DOF Redundant Manipulator

A new method of kinematic analysis and singularity analysis is proposed for a 7-DOF redundant manipulator with three consecutive parallel axes. First, the redundancy angle is described according to the self-motion characteristics of the manipulator, the position and orientation of the end-effector are separated, and the inverse kinematics of this manipulator is analyzed by geometric methods with the redundancy angle as a constraint. Then, the Jacobian matrix is established to derive the conditions for the kinematic singularities of the robotic arm by using the primitive matrix method and the block matrix method. Then, the kinematic singularities conditions in the joint space are mapped to the Cartesian space, and the singular configuration is described using the end poses and redundancy angles of the robotic arm, and a singularity avoidance method based on the redundancy angles and end pose is proposed. Finally, the correctness and feasibility of the inverse kinematics algorithm and the singularity avoidance method are verified by simulation examples.

Singularity analysis of a 7-DOF spatial hybrid manipulator for medical surgery

This paper presents the singularity analysis of a 7-degrees of freedom (DOF) hybrid manipulator consisting of a closed-loop within it. From the past studies, it is well-known that the kinematic singularities play a significant role in the design and control of robotic manipulators. Kinematic singularities pose two-fold effects – first, they can induce the loss of one or more DOF of the manipulator and cause infinite joint rates at that particular joint, and second, they help to determine the trajectory or zone with high mechanical advantage. In current work, a 7-DOF hybrid manipulator is considered which is being developed at Council Of Scientific And Industrial Research–Central Scientific Instruments Organisation (CSIR–CSIO) Chandigarh to assist a surgeon during a medical-surgical task. To emulate the natural motion of a surgeon, the challenging configuration with redundant DOF is utilized. Jacobian has been computed analytically and analyzed at each instantaneous configuration with the evaluation of manipulability. Effect of a closed loop in the hybrid configurations is focused at, and utilizing the contour plots, good and worst working zones are identified in the workspace of the manipulator. The verification and validation of best and worst manipulability points (singularities) are done with the help of genetic algorithms, to determine locally and globally optimal configurations. Finally, on the basis of the singularity analysis, the present work concludes with few guidelines to the surgeon about the best and worst working zones for surgical tasks.

Generalized elastic positivity bounds on interacting massive spin-2 theories

We use generalized elastic positivity bounds to constrain the parameter space of multi-field spin-2 effective field theories. These generalized bounds involve inelastic scattering amplitudes between particles with different masses, which contain kinematic singularities even in the t = 0 limit. We apply these bounds to the pseudo-linear spin-2 theory, the cycle spin-2 theory and the line spin-2 theory respectively. For the pseudo-linear theory, we exclude the remaining operators that are unconstrained by the usual elastic positivity bounds, thus excluding all the leading (or highest cutoff) interacting operators in the theory. For the cycle and line theory, our approach also provides new bounds on the Wilson coefficients previously unconstrained, bounding the parameter space in both theories to be a finite region (i.e., every Wilson coefficient being constrained from both sides). To help visualize these finite regions, we sample various cross sections of them and estimate the total volumes.

Kinematic analysis of scissor-like elements using dual quaternions

This paper presents a standard approach, based on dual quaternions (DQ), to deal with the kinematic analysis of scissor-like elements (SLE). Kinematic relations of homogeneous SLE are simplified by using DQ exponentiation. The steps to identify kinematic singularities are detailed by an algorithm. A second algorithm is introduced to deal with the computation and plotting of reachable and dexterous workspace. In this approach, not just points and vectors, but lines and planes are represented by DQ. The motion of points, lines and planes can be appreciated in practical examples based on domotics.

An Application of Jacobian Matrix in Kinematic of Multi-loop Spatial Mechanism

Mechanism is an intended to turn input motion into a required set of output motion and forces. Now a days the spatial mechanism is widely used in various industrial applications like parallel kinematic machines as a robot manipulator. The main aim of the review paper is to use the Jacobian matrix into the kinematic analysis. From the review papers analysis, it has been concluded that the Kinematic analysiswith help of Jacobian matrix performed various Multi-loop Spatial Mechanism settings for Kinematic geometry, its topology and to carried out displacement modelling. It also used to Kinematic Design, synthesis and optimization. Finally, Comparison of existing methods for Kinematic Singularities with Jacobian in Parallel Manipulator.

Mechanism Singularities Revisited From an Algebraic Viewpoint

It has become obvious that certain singular phenomena cannot be explained by a mere investigation of the configuration space, defined as the solution set of the loop closure equations. For example, it was observed that a particular 6R linkage, constructed by combination of two Goldberg 5R linkages, exhibits kinematic singularities at a smooth point in its configuration space. Such problems are addressed in this paper. To this end, an algebraic framework is used in which the constraints are formulated as polynomial equations using Study parameters. The algebraic object of study is the ideal generated by the constraint equations (the constraint ideal). Using basic tools from commutative algebra and algebraic geometry (primary decomposition, Hilbert’s Nullstellensatz), the special phenomenon is related to the fact that the constraint ideal is not a radical ideal. With a primary decomposition of the constraint ideal, the associated prime ideal of one primary ideal contains strictly into the associated prime ideal of another primary ideal which also gives the smooth configuration curve. This analysis is extended to shaky and kinematotropic linkages, for which examples are presented.