Singularity-free path planning for the Stewart platform manipulator

This paper addresses the problem of singularity-free path planning for the six-degree-of-freedom parallel manipulator known as the Stewart platform manipulator. Unlike serial manipulators, the Stewart platform possesses singular configurations within the workspace where the manipulator is uncontrollable. An algorithm has been developed to construct continuous paths within the workspace of the manipulator by avoiding singularities and ill-conditioning. Given two end-poses of the manipulator, the algorithm finds out safe (well-conditioned) via points and plans a continuous path from the initial pose to the final one. When the two end-poses belong to different branches and no singularity-free path is possible, the algorithm indicates the impossibility of a valid path. A numerical example has also been presented as illustration of the path planning strategy.

A Newton-Euler formulation for the inverse dynamics of the Stewart platform manipulator

This paper presents an inverse dynamic formulation by the Newton–Euler approach for the Stewart platform manipulator of the most general architecture and models all the dynamic and gravity effects as well as the viscous friction at the joints. It is shown that a proper elimination procedure results in a remarkably economical and fast algorithm for the solution of actuator forces, which makes the method quite suitable for on-line control purposes. In addition, the parallelism inherent in the manipulator and in the modelling makes the algorithm quite efficient in a parallel computing environment, where it can be made as fast as the corresponding formulation for the 6-dof serial manipulator. The formulation has been implemented in a program and has been used for a few trajectories planned for a test manipulator. Results of simulation presented in the paper reveal the nature of the variation of actuator forces in the Stewart platform and justify the dynamic modelling for control.

A discrete state perspective of manipulator workspaces Vorstellungen von diskreten zuständen von manipulator-arbeitsräumen

The finite resolution of joint drives or sensors imparts a discrete nature to the joints of a manipulator. Because of this an arbitrary point in the workspace cannot be reached without error even in ideal mechanical environment. This paper investigates the effect of this discrete nature of the joints on the accuracy of performance of a manipulator and develops a method to select the joint states to reach a point with least error. It is shown that the configuration leading to least error cannot, in general, be found from configuration space, especially when there is large variation in the link lengths or joint resolutions or both. The anomaly becomes severe when the gross motion of the end-effector approaches the local resolution of the workspace. The paper also shows how to distinguish two workspaces which may be identical so far as the boundary points are concerned, taking the joint resolutions into account. Finally, the concepts have been extended to define continuous space global and local performance indices for general multi degree of freedom manipulators.

A magnetic micro-manipulator for application of three dimensional forces

Magnetic manipulation finds diverse applications in actuation, characterization, and manipulation of micro-and nano-scale samples. This paper presents the design and development of a novel magnetic micro-manipulator for application of three-dimensional forces on a magnetic micro-bead. A simple analytical model is proposed to obtain the forces of interaction between the magnetic micromanipulator and a magnetic micro-bead. Subsequently, guidelines are proposed to perform systematic design and analysis of the micro-manipulator. The designed micro-manipulator is fabricated and evaluated. The manipulator is experimentally demonstrated to possess an electrical bandwidth of about 1 MHz. The ability of the micro-manipulator to apply both in-plane and out-of-plane forces is demonstrated by actuating permanent-magnet micro-beads attached to micro-cantilever beams. The deformations of the micro-cantilevers are also employed to calibrate the dependence of in-plane and out-of-plane forces on the position of the micro-bead relative to the micro-manipulator. The experimentally obtained dependences are found to agree well with theory. (C) 2015 AIP Publishing LLC.

Three-degree-of-freedom Parallel Manipulator to Track the Sun for Concentrated Solar Power Systems

In concentrated solar power(CSP) generating stations, incident solar energy is reflected from a large number of mirrors or heliostats to a faraway receiver. In typical CSP installations, the mirror needs to be moved about two axes independently using two actuators in series with the mirror effectively mounted at a single point. A three degree-of-freedom parallel manipulator, namely the 3-RPS parallel manipulator, is proposed to track the sun. The proposed 3-RPS parallel manipulator supports the load of the mirror, structure and wind loading at three points resulting in less deflection, and thus a much larger mirror can be moved with the required tracking accuracy and without increasing the weight of the support structure. The kinematics equations to determine motion of the actuated prismatic joints in the 3-RPS parallel manipulator such that the sun's rays are reflected on to a stationary receiver are developed. Using finite element analysis, it is shown that for same sized mirror, wind loading and maximum deflection requirement, the weight of the support structure is between 15% and 60% less with the 3-RPS parallel manipulator when compared to azimuth-elevation or the target-aligned configurations.

A Jacobian-Based Algorithm for Planning Attitude Maneuvers Using Forward and Reverse Rotations

Algorithms for planning quasistatic attitude maneuvers based on the Jacobian of the forward kinematic mapping of fully-reversed (FR) sequences of rotations are proposed in this paper. An FR sequence of rotations is a series of finite rotations that consists of initial rotations about the axes of a body-fixed coordinate frame and subsequent rotations that undo these initial rotations. Unlike the Jacobian of conventional systems such as a robot manipulator, the Jacobian of the system manipulated through FR rotations is a null matrix at the identity, which leads to a total breakdown of the traditional Jacobian formulation. Therefore, the Jacobian algorithm is reformulated and implemented so as to synthesize an FR sequence for a desired rotational displacement. The Jacobian-based algorithm presented in this paper identifies particular six-rotation FR sequences that synthesize desired orientations. We developed the single-step and the multiple-step Jacobian methods to accomplish a given task using six-rotation FR sequences. The single-step Jacobian method identifies a specific FR sequence for a given desired orientation and the multiple-step Jacobian algorithm synthesizes physically feasible FR rotations on an optimal path. A comparison with existing algorithms verifies the fast convergence ability of the Jacobian-based algorithm. Unlike closed-form solutions to the inverse kinematics problem, the Jacobian-based algorithm determines the most efficient FR sequence that yields a desired rotational displacement through a simple and inexpensive numerical calculation. The procedure presented here is useful for those motion planning problems wherein the Jacobian is singular or null.

An eigenproblem approach to classical screw theory

This paper presents a novel algebraic formulation of the central problem of screw theory, namely the determination of the principal screws of a given system. Using the algebra of dual numbers, it shows that the principal screws can be determined via the solution of a generalised eigenproblem of two real, symmetric matrices. This approach allows the study of the principal screws of the general two-, three-systems associated with a manipulator of arbitrary geometry in terms of closed-form expressions of its architecture and configuration parameters. We also present novel methods for the determination of the principal screws for four-, five-systems which do not require the explicit computation of the reciprocal systems. Principal screws of the systems of different orders are identified from one uniform criterion, namely that the pitches of the principal screws are the extreme values of the pitch.The classical results of screw theory, namely the equations for the cylindroid and the pitch-hyperboloid associated with the two-and three-systems, respectively have been derived within the proposed framework. Algebraic conditions have been derived for some of the special screw systems. The formulation is also illustrated with several examples including two spatial manipulators of serial and parallel architecture, respectively.

A real-time algorithm for simulation of flexible objects and hyper-redundant manipulators

Flexible objects such as a rope or snake move in a way such that their axial length remains almost constant. To simulate the motion of such an object, one strategy is to discretize the object into large number of small rigid links connected by joints. However, the resulting discretised system is highly redundant and the joint rotations for a desired Cartesian motion of any point on the object cannot be solved uniquely. In this paper, we revisit an algorithm, based on the classical tractrix curve, to resolve the redundancy in such hyper-redundant systems. For a desired motion of the `head' of a link, the `tail' is moved along a tractrix, and recursively all links of the discretised objects are moved along different tractrix curves. The algorithm is illustrated by simulations of a moving snake, tying of knots with a rope and a solution of the inverse kinematics of a planar hyper-redundant manipulator. The simulations show that the tractrix based algorithm leads to a more `natural' motion since the motion is distributed uniformly along the entire object with the displacements diminishing from the `head' to the `tail'.

A geometric approach for determining inner and exterior boundaries of workspaces of planar manipulators

In this paper a method to determine the internal and external boundaries of planar workspaces, represented with an ordered set of points, is presented. The sequence of points are grouped and can be interpreted to form a sequence of curves. Three successive curves are used for determining the instantaneous center of rotation for the second one of them. The two extremal points on the curve with respect to the instantaneous center are recognized as singular points. The chronological ordering of these singular points is used to generate the two envelope curves, which are potentially intersecting. Methods have been presented in the paper for the determination of the workspace boundary from the envelope curves. Strategies to deal with the manipulators with joint limits and various degenerate situations have also been discussed. The computational steps being completely geometric, the method does not require the knowledge about the manipulator's kinematics. Hence, it can be used for the workspace of arbitrary planar manipulators. A number of illustrative examples demonstrate the efficacy of the proposed method.

An algebraic formulation of static isotropy and design of statically isotropic 6-6 Stewart platform manipulators

In this paper, we present an algebraic method to study and design spatial parallel manipulators that demonstrate isotropy in the force and moment distributions. We use the force and moment transformation matrices separately, and derive conditions for their isotropy individually as well as in combination. The isotropy conditions are derived in closed-form in terms of the invariants of the quadratic forms associated with these matrices. The formulation is applied to a class of Stewart platform manipulator, and a multi-parameter family of isotropic manipulators is identified analytically. We show that it is impossible to obtain a spatially isotropic configuration within this family. We also compute the isotropic configurations of an existing manipulator and demonstrate a procedure for designing the manipulator for isotropy at a given configuration. (C) 2008 Elsevier Ltd. All rights reserved.

Singularity and controllability analysis of parallel manipulators and closed-loop mechanisms

This paper presents a study of kinematic and force singularities in parallel manipulators and closed-loop mechanisms and their relationship to accessibility and controllability of such manipulators and closed-loop mechanisms, Parallel manipulators and closed-loop mechanisms are classified according to their degrees of freedom, number of output Cartesian variables used to describe their motion and the number of actuated joint inputs. The singularities in the workspace are obtained by considering the force transformation matrix which maps the forces and torques in joint space to output forces and torques ill Cartesian space. The regions in the workspace which violate the small time local controllability (STLC) and small time local accessibility (STLA) condition are obtained by deriving the equations of motion in terms of Cartesian variables and by using techniques from Lie algebra.We show that for fully actuated manipulators when the number ofactuated joint inputs is equal to the number of output Cartesian variables, and the force transformation matrix loses rank, the parallel manipulator does not meet the STLC requirement. For the case where the number of joint inputs is less than the number of output Cartesian variables, if the constraint forces and torques (represented by the Lagrange multipliers) become infinite, the force transformation matrix loses rank. Finally, we show that the singular and non-STLC regions in the workspace of a parallel manipulator and closed-loop mechanism can be reduced by adding redundant joint actuators and links. The results are illustrated with the help of numerical examples where we plot the singular and non-STLC/non-STLA regions of parallel manipulators and closed-loop mechanisms belonging to the above mentioned classes. (C) 2000 Elsevier Science Ltd. All rights reserved.

Redundancy resolution using tractrix -- Simulations and Experiments

Hyper-redundant robots are characterized by the presence of a large number of actuated joints, many more than the number required to perform a given task. These robots have been proposed and used for many applications involving avoiding obstacles or, in general, to provide enhanced dexterity in performing tasks. Making effective use of the extra degrees of freedom or resolution of redundancy has been an extensive topic of research and several methods have been proposed in literature. In this paper, we compare three known methods and show that an algorithm based on a classical curve called the tractrix leads to a more 'natural' motion of the hyper-redundant robot, with the displacements diminishing from the end-effector to the fixed base. In addition, since the actuators nearer the base 'see' a greater inertia due to the links farther away, smaller motion of the actuators nearer the base results in better motion of the end-effector as compared to other two approaches. We present simulation and experimental results performed on a prototype eight link planar hyper-redundant manipulator.

An algebraic formulation of static isotropy and design of statically isotropic 6–6 Stewart platform manipulators

In this paper, we present an algebraic method to study and design spatial parallel manipulators that demonstrate isotropy in the force and moment distributions. We use the force and moment transformation matrices separately, and derive conditions for their isotropy individually as well as in combination. The isotropy conditions are derived in closed-form in terms of the invariants of the quadratic forms associated with these matrices. The formulation is applied to a class of Stewart platform manipulator, and a multi-parameter family of isotropic manipulators is identified analytically. We show that it is impossible to obtain a spatially isotropic configuration within this family. We also compute the isotropic configurations of an existing manipulator and demonstrate a procedure for designing the manipulator for isotropy at a given configuration.

A Hydraulic Micromanipulator

For biological experiments requiring manipulations under a microscope, it is necessary to have remote control for the manipulator. Available systems offer the required accuracy at a high cost. Passive micromanipulators are economical but are deficient in performance, the most serious defects being the inability to attenuate operator-induced vibrations and lack of speed control The manipulator described in this paper provides versatile remote control and may be constructed economically.

A Hydraulic Micromanipulator

For biological experiments requiring manipulations under a microscope, it is necessary to have remote control for the manipulator. Available systems offer the required accuracy at a high cost. Passive micromanipulators are economical but are deficient in performance, the most serious defects being the inability to attenuate operator-induced vibrations and lack of speed control The manipulator described in this paper provides versatile remote control and may be constructed economically.