Sandipan Bandyopadhyay

This paper presents the computation of singularity-free regions in the position and orientation workspaces of Stewart platform manipulators (SPMs). Notwithstanding several previous publications in this regard, certain issues persist, such as the unavailability of analytical solutions and the consequent lack of accurate estimates of the numbers of solutions in various cases; the improper use of an incompatible metric while finding such regions inside SO(3) or SE(3). In this paper, the singularity-free regions have been envisaged as spheres in either R3 or SO(3). An analytical formulation for identifying such singularity-free spheres (SFSs) is presented. Several algebraic methods are explored to solve the resulting set of polynomial equations, leading to valuable insights into the numbers of (finite, complex) solutions to this problem, in both the position and the orientation workspaces. A physically meaningful and mathematically sound metric is used while computing the SFS in the orientation workspace. The formulation and computational algorithms have been demonstrated by identifying the SFSs (in R3 as well as SO(3)) for multiple architectures of the SPM through implementations in the C language.

This paper presents a generic geometric-algebraic framework to solve the forward kinematic problem (FKP) of 6-6 Stewart platform manipulators (SPM) in which both the fixed and moving platforms can be either planar or non-planar/general. The FKPs of the SPMs having some of these architectures, e.g., one with a planar moving and general fixed platform, have not been solved explicitly in the existing literature, to the best of the knowledge of the authors. Moreover, some of the existing algorithms applicable to some other classes of SPMs are seen to fail in this case. The algorithms proposed in this paper solves the FKP of this class SPMs, and also that of the 6-RSS manipulator, by relating it to the former. Apart from generating these novel results, the algorithm is amenable to fast and reliable implementation in a generic programming language, such as C, and is able to solve a typical FKP in about 0.7 ms on an ordinary PC. The formulation is illustrated via applications to a number of SPMs and 6-RSS manipulators of different architectures.

The knowledge of singularity-free regions (SFRs) inside the workspace of a manipulator is helpful in its path-planning and design. To identify such regions analytically, the singularity manifold needs to be determined first. This paper presents the derivation of the gain-type singularity manifold in the task-space of the -RPS manipulator, which is further utilised to compute its SFR, in the form of a singularity-free cylinder (SFC), which is free of gain-type singularities. The problem of identifying the longest SFC for a given constant radius and a chosen base circle is posed as a constrained optimisation problem, which reduces to a 21-degree univariate polynomial in the length of the SFC. The formulation is illustrated via a numerical example.

This paper presents the computation of the safe working zone (SWZ) of a parallel manipulator having three degrees of freedom. The SWZ is defined as a continuous subset of the workspace, wherein the manipulator does not suffer any singularity, and is also free from the issues of link interference and physical limits on its joints. The proposed theory is illustrated via application to two parallel manipulators: a planar 3-RRR manipulator and a spatial manipulator, namely, MaPaMan-I. It is also shown how the analyses can be applied to any parallel manipulator having three degrees of freedom, planar or spatial.

This paper presents a novel six-degrees-of-freedom spatial hybrid manipulator and the analysis of its forward kinematics. The proposed manipulator has three limbs of identical architecture connecting a hexagonal fixed platform to a moving platform in the form of an equilateral triangle. The limbs are spaced symmetrically w.r.t. the fixed platform. Each limb consists of a planar five-bar mechanism at its base, the tip of which connects to the moving platform via a passive RS link. The forward kinematic problem of the manipulator is reduced to that of a circle intersecting an oct-circular curve of degree 16 in its own plane. As expected, this leads to a maximum of 16 isolated (complex) solutions, the reals among which lead to poses with a certain mirror-symmetry. Consequently, a polynomial of only degree 8 has to be solved eventually. Coefficients of this polynomial have been found as closed-form symbolic expressions in the architecture parameters and input variables. The theoretical results are demonstrated and validated numerically using an example.

Forward dynamic analyses of cable-driven parallel robots (CDPRs) are performed accounting for the spatial motions of the cables while considering their mass, sag, elastic and damping properties. The winches feeding the cables are considered stationary. An efficient recursive forward dynamic algorithm is developed to perform the extremely demanding computations. As a part of this work, a solution to the kinetostatic problem of CDPRs is proposed, wherein starting with a non-equilibrium pose, the CDPR is allowed to evolve dynamically until attaining an equilibrium. This idea is demonstrated on a spatial 6-3 CDPR, the feed-support system of the Five-hundred-meter aperture spherical radio telescope (FAST), as well as the 8-8 CDPR, CoGiRo. Dynamic simulation of this nature using a full-scale model of the FAST manipulator is reported for the first time. The results are validated numerically, as well as against existing models, wherever feasible. Challenges involved in the modelling and computations at such a scale and the corresponding remedies are elaborated.

The double-wishbone (DWB) is a popular suspension system, particularly in the high-end automobiles. Simulation of its kinematics and dynamics are vital elements in the process of analysis and design of these complex mechanical systems. However, the simulation of the DWB suspension can be computationally demanding, due to the large number of nonlinear, coupled ordinary differential equations (ODEs) that arise from the motion of the multiple links present in the system. In this paper, a less common approach to the Lagrangian formulation is adopted, which is known as the embedded formulation or the actuator space formulation. In this formulation, the number of ODEs to be solved come down to only two-a number that equals the degree-of-freedom of the system. The joint variables associated with the unactuated links are computed through the forward kinematics of the system. This method has the advantage of reducing the computational burden in the numerical solution of the ODEs significantly, as the number of ODEs comes down to two, as opposed to eleven in the more commonly used configuration space formulation. Furthermore, the unactuated variables are determined from the kinematic constraints, as opposed to being computed from the numerical solutions to ODEs, which make them more accurate. The formulation is illustrated via numerical examples implemented in the Computer Algebra System (CAS), Mathematica. It is believed that such a formulation would aid in the understanding of the dynamics of the suspension systems, and help in the process of their design.

Accurate simulations of the motion of cable-driven parallel robots (CDPRs) are helpful in its modal analysis, testing robust control strategies, validating designs and estimating workspaces. On that account, a modular and computationally efficient framework to analyse the dynamics of CDPRs is developed in the present work. In contrast to the prior studies, the inertia, stiffness and damping properties of the cables, along with their temporal variations caused by feeding and retrieving, are included in the dynamic model. Further, the forward dynamics algorithm designed for this purpose is recursive in nature and has linear time complexity. Finally, the efficacy of the proposed framework is established with the help of the FAST manipulator, the largest existing CDPR. Also, its extensive application potential is established via the study of the CDPR CoGiRo, which, apart from being actuated redundantly, differs drastically from the former in terms of its mass and footprint.

Two optimal design methods are proposed for the dimensional design of parallel manipulators, considering their dynamic performance over the desired ranges of motions, velocities, and accelerations. The first method, termed as the extrinsic method, directly minimises the actuator forces/torques within the desired safe working zone of the manipulator, for typical velocities and accelerations of the end-effector. The second method, namely, the intrinsic method, minimises a measure of the manipulator's inertia, as reflected at the actuators, over the said safe working zone. Case studies on the design of 3-RRR and 3-RRS manipulators are presented to illustrate the proposed methods. Numerical studies show that the optimal link dimensions obtained through these conceptually disparate methods are fairly similar. Naturally, the dynamic performances of the resulting manipulators are also comparable, which are found to be significantly better than that of arbitrary designs respecting the same constraints.

This paper presents a novel geometric solution to the problem of finding singularity-free paths joining two arbitrary points in the constant orientation workspace of a semi-regular Stewart platform manipulator. The formulation builds upon the known closed-form expression for the gain-type singularity surface of the manipulator. Using a rational parametrisation of the surface, it computes the geodesic curve on this surface, connecting the projections of the two given points on this surface. A sequence of spheres is then constructed in such a manner that each sphere is tangential to a previous one as well as the singularity surface, at a point on the said geodesic curve. Thus the geodesic curve acts as a guide, over which the singularity-free sphere is rolled, till it reaches its destination. Multiple methods for computing such sequences of spheres are presented and compared with the help of a numerical example. Finally, a sequence of line segments connecting the centres of the spheres is constructed, which connects the two given points via a provably singularity-free path.