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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The development of a new parallel mechanism based on simulation driven design is a rapid approach to discover the unique features or advantages of a conceptual model. In this research, one novel parallel mechanism which can generate three degrees-of-freedom translations is proposed. The kinematic model and Jacobian matrix is derived. The workspace generation and mapping is investigated based on simplified boundary searching method. The particle swarm algorithm is applied to search for the optimal volume of workspace.

The traditional parallel robots have proved their advantages in aspects of stiffness, rigidness, dexterity, reconfigurability, with the extensive applications in machine tools [

However, compared with the serial one, the drawbacks of the parallel robot is obvious [

For the scenario of medical micromanipulation, the requirement of precision and safety is high. The compliant mechanism is fabricated with very flexible hinge or other flexure joint. The integration of compliant mechanism and parallel mechanism can provide an effective solution especially in precision and dexterity [

Many restriction factors of the parallel robot, including degree-of-freedom and configuration constraint of various joints, mechanical collision of different components, actuators stroke and singular limitations, affect the performances of workspace. Many scholars have developed different approaches and algorithms to investigate the features of workspace, especially its volume [

In what follows, a new parallel mechanism that can generate three degrees-of-freedom translations is developed. Its kinematic model and Jacobian matrix is derived in Section 2. The workspace as the one of the most important indices of parallel mechanism is calculated and mapped in Section 3. The particle swarm algorithm based performance optimization is conducted to maximize the volume of workspace in Section 4. Section 5 gives the conclusions.

The novel parallel mechanism with three degrees-of-freedom translations is composed of a base structure, a moving platform and three legs connecting the base and platform. In each leg, a compliant revolute joint is attached to the moving platform. A four bar mechanism is hereafter connected to the revolute joint. The linear driven mechanism is embedded in the four bar mechanism which is actuated by a PZT. Another revolute joint which is perpendicular to the above-mentioned one is connected to the four bar mechanism and the base. The CAD model of the proposed 3-DOF parallel mechanism is shown in

Because of the nature of the compliant mechanism, generally speaking, the analysis of the parallel mechanism will be different to the conventional parallel mechanism. However, the detailed analysis will depend on the given case. Just like the traditional parallel manipulator, the proposed parallel mechanism has many performance indices,

A kinematics model of the manipulator is shown in _{i}_{i}_{1} = _{2} = _{3} = _{1} = _{2} = _{3} =

The position vector of _{i}

The position vector of _{i}

Thus, the position vector of _{i}

Thus, the inverse kinematics of the proposed parallel mechanism can be derived as:

Since the relationship of the differential for the input joints and the output displacements can be expressed as:

Thus, the Jacobian matrix of the proposed parallel mechanism is obtained as:

As was defined in [

The generation of workspace for the parallel mechanisms includes geometrical approach, numerical method and discretization method. With the integration of geometrical approach, discretization method and inverse kinematics model, a simplified boundary searching method (SBS) is developed to acquire the task workspace.

The calculating process of SBS method is described as follows:

_{min}, _{max} and _{min} and _{max} are the motion scopes of the PZT.

The features of workspace including shape and volume are related with the dimensions of the proposed parallel mechanism.

The traditional optimization methods usually adopt the local search by a convergent stepwise procedure which possibly falls into local optimal solution. If the complex function to be optimized does not possess convexity characteristics that essentially satisfy that the local extreme point is a global optimum, a global optimization algorithm is required. As an advanced computational intelligence method, particle swarm optimization (PSO) is inspired by simulating the swarm behavior such as bird flocking. Without the traditional evolution operators including crossover and mutation, PSO can be viewed as the extension and improvement of the working principle of genetic algorithm. Thus, the particle swarm algorithms will be used to search the overall optimal performance.

The general PSO algorithm is constituted with the following velocity and position [_{i}_{i}_{i}

The mechanism parameters chosen for optimization are _{min}, _{max} and

_{min} ∈ [12.5, 12.58]

_{max} ∈ [12.58, 12.62]

In this scenario, the common PSO with inertia is utilized to perform the optimization process. The maximal velocity divisor is 2, the particles number is 100.

Before optimization, the value of objective function is equal to 5.42×10^{−4} ^{3}, with the initial parameters as follows:

_{min} = 12.56

_{max} = 12. 6

After optimization with 35 epochs, the volume of workspace is 2.82×10^{−3} ^{3}, improved by a factor of 5.2. The best individuals of the five parameters are:

_{min} = 12.5

_{max} = 12.62

The main contributions of this paper are summarized as follows:

A new parallel mechanism which can generate three degrees-of-freedom translations is designed. The related kinematics modeling is investigated.

A general approach called simplified boundary searching method is developed to generate the reachable workspace of the proposed parallel mechanism.

Optimization of the dimensional parameters is conducted to obtain the maximal volume of workspace based on particle swarm algorithm.

This research focuses on developing a generic and simplified method for the modeling, mapping, calculation and optimization of workspace for a parallel mechanism. Currently, the modeling and optimization of parallel mechanisms is still one open issue for the scholars and engineers in the world who are interested in or already investigating this topic. The newly developed method about workspace generation and improvement is presented in a clear and operational way. Through case study, it can be proved that this approach is very efficient in modeling and computing time. The proposed method is feasible as one efficient solution for the open issue of workspace.

This study has endeavored in design optimization of the workspace for a novel moving stage which can generate three degrees-of-freedom translations. The kinematic model and Jacobian matrix is derived. The simplified boundary searching method to generate the workspace of the proposed parallel mechanism is generic. The particle swarm algorithm is applied to improve the volume of workspace. For the future work, the prototype will be fabricated based on the proposed modeling and optimization approach.

The proposed parallel mechanism; (

The reachable workspace; (

The mapping of workspace under different input parameters; (

The working principle of PSO.

The evolution process of the maximal workspace volume per epoch.

The envelope of the optimal workspace.

The authors would like to thank the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors gratefully acknowledge the financial support from Canada Research Chairs program, MITACS-NCE and ERA.