Research paperMethods for dimensional design of parallel manipulators for optimal dynamic performance over a given safe working zone
Introduction
Design of parallel manipulators is a difficult task, due to many reasons, such as the complex shapes of their workspaces, existence of singularities inside these workspaces, link interferences, etc. While designing a manipulator, firstly, it is essential to ensure that the manipulator is free of all these issues over the user-specified ranges of motion. This paper adopts the concept of the safe working zone (SWZ), detailed in [1], [2], to define the desired linear as well as angular ranges of motion. A candidate design of the manipulator is considered feasible if its SWZ encompasses the requisite ranges of motions. In theory, there can be infinitely many feasible designs for a given set of specifications. Therefore, it is customary to define certain performance indices, based on which the best design(s) can be identified.
Several kinematic performance indices have been used in design optimisation, namely, the global condition index [7], the global transmission index [8], the kinematic isotropy index [9], the dexterity index [10], and the measure of manipulability [11]. The kinematic indices suffer from two major limitations. Firstly, they consider only the geometry and the configuration of the manipulator, without any regard for the actuator efforts2 necessary to produce the stipulated motions. Secondly, the frequently occurring dimensional inhomogeneity of the velocity Jacobian matrix, when the end-effector exhibits both translational as well as rotational motions, makes the definition of most of these indices ambiguous [12]. Monte Carlo simulation-based approach has been adopted in [13] to design parallel manipulators with a prescribed workspace, while ensuring the well-conditioning of certain Jacobian matrices, as well as the constraints on joint motions, inside the said workspace. Researchers have also analysed the stiffness of the end-effector over its range of operation [14] in the context of machining robots, where there is a need to resist the reaction forces in certain directions. Gao et al. have considered a multi-objective optimisation problem with mean stiffness of the manipulator and its variance over the workspace as the dual objectives in the problem of design in [15]. Tong et al. have proposed two dynamic indices using the inertia matrix and the stiffness matrix to arrive at a dynamically isotropic design for a class of generalised symmetric Gough-Stewart parallel manipulators in [16].
Another potential candidate for a desirable design objective is the minimisation of the actuator efforts required to produce the specified motions of the end-effector. This problem is perhaps the most relevant in applications such as vehicle simulators, pick and place operations, contour tracking, and computer controlled machining. In all these applications, the ranges of translational and rotational motions, maximum speeds and accelerations of the end-effector (i.e., the moving platform) are known a priori. The designer is faced with the problem of determining the dimensions of the links of the manipulator such that the actuator efforts required for executing various tasks are reduced, in general. This objective is directly related to the sizing of the actuators, and the overall bulk as well as the cost of the manipulator. The research works published in this regard are relatively sparse. Asada has extended the notion of inertia ellipsoid of a single rigid body to the generalised inertia ellipsoid (GIE) of a robot manipulator to visualise its dynamic characteristics and improve its design in [3], [4]. Yoshikawa has introduced the concept of dynamic manipulability and demonstrated its utility in the design of a 2-R planar serial manipulator in [17]. Gregorio and Parenti-Castelli have introduced two performance indices indicative of the dynamic isotropy and swiftness of the motion of the end-effector in [18]. Ma et al. have introduced the dynamic conditioning index to quantify the coupling effects among the actuators of serial robots and used the same for optimal trajectory planning of welding robots in [19]. More recently, Kilaru et al. have developed similar indices for the optimal dynamic design of parallel manipulators using a multi-objective optimisation strategy with constraints on the size of the SWZ in [20].
Unal et al. [21] have considered the global isotropy index and the global dynamic index as the two objectives to obtain designs of a five-bar manipulator with best worst-case kinematic and dynamic performance, inside a chosen region of interest. However, the constraints considered in their formulation do not ensure that this region is free from the gain-type singularities3, which is one of the important criteria in the design of parallel manipulators. Wu et al. [23] have proposed a multi-objective optimisation approach for the design of a three degrees-of-freedom (DoF) spherical parallel manipulator, in which the total mass of the links and the global conditioning index are considered as objectives, while the singularity avoidance, minimum stiffness criterion, and torque limits, are treated as the constraints. All of these computations were performed specifically on two predefined trajectories in the task-space of the manipulator, and the resulting designs were also tested on the same set of trajectories used in the design process. Such a design methodology may suffer from a serious drawback: a 3-DoF manipulator is expected to possess a three-dimensional workspace, which cannot be adequately represented by two paths, let alone two specific trajectories defined on those two paths. The performance of the manipulator on other paths, or even other trajectories on the said paths, cannot be logically predicted from the results obtained. Zhao [24] has proposed a methodology for the optimum design of the Delta robot considering three different objectives, namely, torque, power, and closeness to the gain-type singularity manifold, with constraints on the size of the workspace and motion of the joints. The proposed optimisation problems were solved via the sequential quadratic programming technique, which required a “good” initial guess to converge at an optimal design. Since the chosen manipulator belongs to the class χ03 (see [25]) and has three translational DoF, it might have been possible to come up with a feasible design as an initial guess with some preliminary analysis. However, in a general scenario where a manipulator possesses both rotational as well as translational DoF, it is very difficult, if not impossible, to come up with feasible designs intuitively.
This paper proposes two different methods for the design of symmetric parallel manipulators with good dynamic performance and evaluates the performance of the obtained designs by testing the resulting manipulators on arbitrary trajectories within their SWZ. The first one, termed as the extrinsic method, attempts to minimise the maximum actuator effort required to generate the desired motions within the user-specified workspace, at prescribed velocities and accelerations. However, quantification of this intent in terms of a mathematical objective function is fraught with several difficulties. Firstly, it is challenging, if not impossible, to find “representative” paths in the workspace, so as to ascertain the peak demands placed on the actuators in the entire workspace. This is further complicated by the fact that infinitely many trajectories can be defined on the same path. The best result, therefore, is always constrained to be subjective and uncertain, as it is possible, in principle, for two researchers to obtain very different results for the same problem, subject to the choices made by them, respectively. The second method, aims at improving the inherent dynamic characteristics of the manipulator, in such a manner, that it demands lower actuator inputs to perform the same task. This method, termed as the intrinsic method, does not need any considerations of the representative paths or trajectories at the design stage. Instead, it looks into the equation of motion directly, and attempts to modify the design in such a manner that the maximum inertia of the manipulator, as reflected at the actuators, is at a minimum. While this method has the advantage of being objective and unambiguous, it influences the ultimate objective, i.e., minimisation of the actuator efforts, only in an indirect manner. Therefore, one could expect this method to be less effective than the extrinsic one, which uses more information regarding the actual trajectories of the manipulators, and utilises these to evaluate the actuator efforts directly.
The focus of this paper is on studying the efficacies of the two methods, and understanding their relative merits and demerits. The design studies are conducted in the context of two 3-DoF parallel manipulators, namely, the planar 3-RRR manipulator and the spatial 3-RRS manipulator. The results show an interesting pattern, that in spite of the stark contrasts in the nature of the methods (the first one being subjective but direct, and the latter objective yet indirect), they seem to agree on the optimal results. The main contributions of the paper are summarised below.
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Two different optimal design methods are presented for the dimensional design of parallel manipulators, with applications to a planar and a spatial manipulator.
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The design methods optimise dynamic performance, while ensuring that the designed manipulator is free of singularities, link interferences, and joint limit issues over a specified subset of its workspace, designated as the safe working zone.
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In addition, individual links are designed to satisfy specified stiffness requirements, without resorting to a multi-objective optimisation problem involving the link stiffness as an objective.
The rest of the paper is organised as follows: the required mathematical and conceptual preliminaries are discussed in Section 2. The extrinsic and intrinsic design methods and their mathematical formulations are described in Section 3. The proposed methods are illustrated with case studies on the 3-RRR and 3-RRS manipulators in Section 4. Salient features of the design methods are listed in Section 5. Finally, the conclusions of this work are presented in Section 6.
Section snippets
Conceptual preliminaries
A brief discussion on the concept of the safe working zone (SWZ), and the actuator-space equation of motion is presented in this section.
Formulation and solution of the design optimisation problems
In the context of parallel manipulators, the specifications of the required motions are dictated by the applications for which the device is designed, while the cost is determined primarily by the actuators used to generate the specified motions. The extrinsic and intrinsic methods, briefly introduced in Section 1, exemplify the two different methods aimed at the common target of minimising the peak actuator forces, while ensuring that the desired SWZ, velocity and acceleration requirements are
Case studies
The design methods discussed in Section 3 are illustrated via applications to the 3-RRR planar and the 3-RRS spatial parallel manipulators. The 3-RRR planar parallel manipulator (PPM) has two translational and one rotational DoF and belongs to the class χ12 (see [25]). On the other hand, the 3-RRS spatial parallel manipulator has two rotational and one translational DoF, and therefore belongs to the class χ21.
Discussions on the design methods and possible extensions
The key observations and salient features of the proposed design methods are noted in the following.
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In both the extrinsic and intrinsic methods, the constraints on the dimensions of the SWZ specified should be achievable with the ranges [aj, bj] specified for the design variables (i.e., link lengths as well as the coordinate(s) needed for the placement of the SWZ). If the specified ranges are not realistic or too smallno feasible design may be identified by the optimiser. On the other hand, if
Conclusion
Two design methods (extrinsic and intrinsic) are proposed for the dimensional design of the parallel manipulators to meet the user-specified requirements on the ranges of motions, desired velocities, and accelerations, while minimising the actuator efforts required to achieve them. The concept of SWZ is adopted in this work to define the desired ranges of motions and the dimensions of the desired SWZ are treated as the constraints in both these methods. The auxiliary dimensions of the links are
Declaration of Competing Interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to acknowledge Mr. Murali K. Karnam and Mr. Saurav Agarwal, former Project Associates of the Robotics Laboratory, Department of Engineering Design, Indian Institute of Technology Madras, for their contributions in programming the initial versions of the proposed algorithms in the C++ environment. The first author also wishes to thank Mr. Teja Krishna Mamidi and Mr. Anirban Nag, PhD scholars in the same laboratory, for their kind help with accessing the computational
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This author contributed to this work while working as a Project Associate at the Robotics Laboratory, Department of Engineering Design, Indian Institute of Technology Madras.