The optimum kinematic design of a 6 DOF micro parallel robot.
Key words: workspace, parallel robot, genetic algorithms.
In the literature, various methods to determine workspace of a parallel robot have been proposed using geometric or numerical approaches. Parallel robots have become a large area of interest in the field of robotics. Parallel robots generally have larger load capacities, faster and more accurate motions and a larger stiffness throughout their workspace as compared to the serial ones (Hesselbach, 2004).
Early investigations of robot workspace were reported by (Merlet, 1995). Other works that have dealt with robot workspace are reported by (Agrawal, 1990; Ceccarelli, 1995). Agrawal determined the workspace of in-parallel manipulator system using a different concept namely, when a point is at its workspace boundary, it does not have a velocity component along the outward normal to the boundary.
Configurations are determined in which the velocity of the end-effector satisfies this property. In (Stan, 2003) was presented a genetic algorithm approach for multi-criteria optimization of PKM.
The workspace of a robot is defined as the set of all endeffector configurations which can be reached by some choice of joint coordinates.
As the reachable locations of an end-effector are dependent on its orientation, a complete representation of the workspace should be embedded in a 6-dimensional workspace for which there is no possible graphical illustration; only subsets of the workspace may therefore be represented
In this paper, the optimization workspace index is defined as the measure to evaluate the performance of a 6 degree of freedom parallel micro robot. Another contribution is the optimal dimensioning of the Hexapod model for the largest workspace.
2. OPTIMAL DESIGN
2.1 Six DOF micro parallel robot
The micro parallel robot is a 6 DOF parallel manipulator comprising a fixed base platform and a payload platform, linked together by six independent, identical, open kinematic chains (Fig. 1). Kinematics of this structure is presented in Refs. (Stan, 2003).
[FIGURE 1 OMITTED]
2.2 Workspace index
One of the most important issues in the process of design of robot is their workspace. For parallel robots, this issue may be more critical since parallel robots will sometimes have a rather limited workspace. Closed loop nature of the parallel robots limits their workspace. Also, in the context of design, the workspace determination procedure should be simple enough to be included in an optimization algorithm.
Because of this, applications involving these parallel robots require a detailed analysis and visualization of the workspace of these robots. The algorithm for visualization of workspace needs to be adaptable in nature, to configure with different dimensions of the parallel robot's links. The workspace is discretized into square and equal area sectors. A multi-task search is performed to determine the exact workspace boundary. Any singular configuration inside the workspace is found along with its position and dimensions. The volume of the workspace is also computed. A type of parallel robot, namely Hexapod-type six-degree of freedom robot is considered to demonstrate the effectiveness of the algorithm.
The workspace is the volume in the space case where the tool centre point (TCP) can be controlled and moved continuously and unobstructed. The workspace is limited by singularities. At singularity poses it is not possible to establish definite relations between input and output coordinates. Such poses must be avoided by the control. Workspace is another significant design criterion for describing the kinematics performance of parallel robots. Parallel robots use volume to evaluate the workspace ability. However, is hard to find a general approach for identification of the workspace boundaries of the parallel robots.
[FIGURE 2 OMITTED]
This is due to the fact that there is not a closed form solution for the direct kinematics of these parallel robots. That's why instead of developing a complex algorithm for identification of the boundaries of the workspace, it's developed a general visualization method of the workspace for its analysis and its design. The possible workspace of the robot is of a great importance for optimization of the parallel robots. The general analysis of the workspace consists in workspace determination using the described discretization method.
2.3 Design optimization
The design of the robot can be made based on any particular criterion. The paper presents a genetic algorithm approach for workspace optimization of six-dof parallel micro robot. For simplicity of the optimization calculus a symmetric design of the structure was chosen.
In order to choose the robot dimensions L, [q.sub.1min], [q.sub.1max], [q.sub.2min], [q.sub.2max], [q.sub.3min], [q.sub.3max], [q.sub.4min], [q.sub.4max], [q.sub.5min], [q.sub.5max], [q.sub.6min], [q.sub.6max], we need to define a performance index to be maximized. The chosen performance index is W (workspace). One objective function is defined and used in optimization. It is noted as W, and corresponds to the optimal workspace. We can formalize our design optimization problem as the following equation:
Optimization problem is formulated as follows: the objective is to evaluate optimal link lengths which maximize (W). The design variables or the optimization factor is the ratios of the minimum link lengths to the base link length b, and they are defined by:
L (2) Constraints to the design variables are: 20<L<60 (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
For this example the lower limit of the constraint was chosen to fulfill the condition L[greater than or equal to]30. For simplicity of the optimization calculus the upper bound was chosen L[less than or equal to]60. During optimization process using genetic algorithm it was used the following GA parameters, presented in Table 1. A genetic algorithm (GA) is used because its robustness and good convergence properties.
The GA approach has the clear advantage over conventional optimization approaches in that it allows a number of solutions to be examined in a single design cycle. The traditional methods searches optimal points from point to point, and are easy to fall into local optimal point. Using a population size of 50, the GA was run for 100 generations. A list of the best 50 individuals was continually maintained during the execution of the GA, allowing the final selection of solution to be made from the best structures found by the GA over all generations.
We performed a kinematic optimization in such a way to maximize the workspace index W.
[FIGURE 3 OMITTED]
It is noticed that optimization result for Hexapod when the maximum workspace of the 6 DOF micro parallel robot is obtained for L=60 mm. The used dimensions for the 6 DOF micro parallel robot were: [q.sub.1min]=0 mm, [q.sub.1max]=100 mm. Maximum workspace of the mini parallel robot was found to be W= 45493 [mm.sup.3]. And the shape of the optimized workspace of the parallel micro robot is shown in Fig. 3. The results show that GA can determine the architectural parameters of the robot that provide an optimized workspace. Since the workspace of a parallel robot is far from being intuitive, the method developed should be very useful as a design tool. However, in practice, optimization of the robot geometrical parameters should not be performed only in terms of workspace maximization. Some parts of the workspace are more useful considering a specific application. Indeed, the advantage of a bigger workspace can be completely lost if it leads to new collision in parts of it which are absolutely needed in the application. However, it's not the case of the presented structure.
In this paper a mono-objective optimum design procedure for parallel robot was outlined by using optimality criterion of workspace and numerical aspects. A mono-objective optimization problem was formulated by referring to a basic performance of parallel robots. A kinematic optimization was performed to maximize the workspace of the 6 DOF micro parallel robot. Together with other optimization oriented toolboxes from MATLAB, the GAOT Toolbox provides a uniform environment for the mechanical engineer to experiment with and apply GAs to problems in optimization of parallel robots.
Hesselbach, H. Kerle, M. Krefft, N. Plitea, (2004) "The Assesment of Parallel Mechanical Structures for Machines Taking Account of their Operational Purposes". In: Proc. of the 11th World Congress in Mechanism and Machine Science-IFToMM 11, Tianjin, China.
S. Stan, Diplomarbeit, (2003), Analyse und Optimierung der strukturellen Abmessungen von Werkzeugmaschinen mit Parallelstruktur, IWF-TU Braunschweig, Germany.
J. P. Merlet. (1995), "Determination of the orientation workspace of parallel manipulators". Journal of intelligent and robotic systems, 13:143-160.
SK. Agrawal, (1990) "Workspace boundaries of in-parallel manipulator systems". Int. J. Robotics Automat, 6(3) 281-290.
M. Cecarelli, (1995) "A synthesis algorithm for three-revolute manipulators by using an algebraic formulation of workspace boundary". ASME J. Mech. Des. 1995; 117(2(A)): 298-302.
Jason J. Lee and Sun-Lai Chang, "On the kinematics of the UPS wrist for real time control", DBVol. 45, 22nd ASME Biennial Mechanisms Conference, Robotics, Spatial Mechanisms, and Mechanical Sysrems. Scorndale, Arizona, pp. 305-312, 1992.
Table 1. GA Parameters 1 Population 50 2 Generations 100 3 Crossover rate 0,08 4 Mutation rate 0,005
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|Title Annotation:||degrees of freedom|
|Author:||Stan, Sergiu; Maties, Vistrian; Balan, Radu; Hancu, Olimpiu|
|Publication:||Annals of DAAAM & Proceedings|
|Date:||Jan 1, 2007|
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