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A desired total orientation workspace for a parallel manipulator is usually an essential requirement in a practical application. At present, for the multiobjective optimization method of 6-DOF parallel manipulator for desired total orientation workspace, it is needed to predefine maximal and minimal lengths of the legs to serve as the constraint, and then the numerical method is used to solve the length of the legs and judge whether the solved maximal and minimal leg lengths meet the constraint. Predefining maximal and minimal length of the legs limits of the optimal range, the numerical method has heavy calculation burden and low calculating accuracy. In this paper, a hybrid method for solving the maximal and minimal lengths of the legs of 6-DOF parallel manipulator with desired total orientation workspace is proposed, and the actuator stroke length is calculated according to the maximal and minimal leg lengths. By judging whether the actuator stroke length can be solved to serve as the constraint, without the predefined maximal and minimal leg lengths to serve as the constraint, the optimal range is enlarged. Aiming at the physical size of the parallel manipulator and the proposed desired workspace condition index (DWCI), the optimization of the geometric parameters of the parallel manipulator is conducted based on the multiobjective optimization algorithm (NSGA-II), which is subject to the actuator stroke length. Stewart platform is set as the example; the geometric parameters of the platform whose workspace contains the desired total orientation workspace are optimized and the hybrid method is proved to be more accurate and efficient compared to the traditional numerical method. This method provides the optimization guidance for engineering designers to design the parallel manipulator for desired total orientation workspace.

Parallel manipulator has the advantages of compact structure, strong bearing capacity, high precision of motion, and low inertia and therefore is widely used in the flight simulator, ship motion simulation, ship stabilized platform, space docking manipulator, parallel machine tools, and robot wrist [

At present, in the design of the parallel manipulator for the workspace requirements, there are two types [

However, maximizing the workspace blindly does not meet the engineering requirements, since a desired regular shape workspace is usually an essential requirement in practice. The other type is to obtain the geometric parameters of a parallel manipulator whose workspace contains a desired workspace. Liu [

At present, there are few studies on multiobjective optimization of 6-DOF parallel manipulator for desired total orientation workspace. In general, numerical method is used to discretize the six- dimensional desired total orientation workspace to solve the length of the legs; however, it consumes a lot of time. Furthermore, the solved length of the legs should be subject to the predefined maximal and minimal length, which limits the optimization design of the geometric parameters. Therefore, a hybrid method for solving the maximal and minimal length of the legs of 6-DOF parallel manipulator with desired total orientation workspace is proposed in this paper, and it is more accurate and efficient compared to the numerical method. And the actuator stroke length is calculated according to the maximal and minimal leg lengths, by judging whether the actuator stroke length can be solved to serve as the constraint; without the predefined maximal and minimal leg lengths to serve as the constraint, the optimal range is enlarged. And then aiming at the physical size of the parallel manipulator and the proposed DWCI, the optimization of the geometric parameters of the parallel manipulator is conducted based on the multiobjective optimization algorithm (NSGA-II), which is subject to the actuator stroke length. Stewart platform is set as the example; the geometric parameters of the platform whose workspace contains the desired total orientation workspace are optimized and the hybrid method is proved to be more accurate and efficient compared to the traditional numerical method. This method provides the optimization guidance for engineering designers to design the parallel manipulator for desired total orientation workspace.

The structural sketch of 6-DOF parallel manipulator is shown in Figure

The structure sketch of 6-DOF parallel manipulator.

Letting_{i} (_{i} (_{a} is the origin of the reference coordinate frame _{1} and_{6}, and the direction of_{i}, and_{b} is the center of the symmetrical hexagon, the point_{b} is the origin of the reference coordinate frame _{1} and_{6}, and the direction of_{a} is the origin of _{b} is the origin of _{i} and_{i} in

The total orientation workspace of the 6-DOF parallel manipulator can be denoted as

According to the position P and posture Q in the workspace and the transformation matrix

The vector diagram for the

Then the length of the

The most usual types of workspace of 6-DOF parallel manipulator are reachable workspace, constant orientation workspace, orientation workspace, total orientation workspace, etc. Total orientation workspace is all the locations that may be reached with all the orientations among a set defined by ranges on the orientation angles [

Numerical method [

Equation (

Solving process of function of m(

The solving process of function of k variables is shown in Figure

The numerical method takes much time to calculate the maximal and minimal leg lengths, due to the large calculation quantity on the leg length for each discretized pose. And the precision is affected by the step lengths; usually a smaller step length leads to the higher precision and more time. The algebraic method takes less time to calculate the explicit maximal and minimal leg lengths. However, it is difficult to obtain the extremal points of the six-element function, and the extremal points of the function can be only obtained with the specified orientation variables (

Therefore, a hybrid method including the numerical method and algebraic method is proposed, where the numerical method is used for orientation variables and the algebraic method is used for translation variables. Firstly, the three-dimensional posture workspace is discretized and the discretized postures are obtained. Then the functions of 3 variables (

Equation (

Solving process of function of n(

_{TP} discretized postures have been calculated.

_{TP} discretized postures.

The maximal and minimal leg length solved for the desired total orientation workspace should be between the maximum and minimum of the actuator length. The maximal and minimal values of the actuator can be determined by dead length and stroke length of the actuator. The dead length is the part which is not in the extension part of the cylinder, and the stroke length is the part in the extension part of the cylinder [_{d} denotes the dead length of actuator, and_{s} denotes the actuator stroke length. Hydraulic cylinders, pneumatic cylinders, or electric cylinders are generally used as actuators, and the dead length_{d} usually can be determined according to the actual condition.

The schematic diagram of actuator.

According to (

The Elitist Nondominated Sorting Genetic Algorithm version II (NSGA-II) is one of the most classical and popular multiobjective evolutionary algorithms, which is proposed by Deb [

The detailed process of NSGA II.

By the determined volume of the desired total orientation workspace, the larger physical size of the parallel manipulator leads will meet the workspace requirements better. So the physical size of the parallel manipulator needs to be optimized to achieve higher economy and practicability. In general, the maximal leg length [

One of the kinematics and dynamics indexes should be selected as the target to ensure the good performance of the manipulator. The global performance indexes such as the global condition index (GCI) [

When the Monte Carlo method is used to randomly generate m poses for the desired workspace, the DWCI can be obtained by (

The range of

The actuator stroke length of parallel manipulator for desired total orientation workspace is taken as the constraint. If (

Calculate the maximal and minimal leg lengths by the hybrid method proposed in this paper, and then calculate the stroke length. The specific process is shown in Figure

The proposed optimization method is used to optimize the ship motion simulation platform. The ship moves in six DOF, which includes heave (along

The desired total orientation workspace.

| | | |
---|---|---|---|

Translation Amplitude(mm) | ±35 | ±35 | ±50 |

Rotation Amplitude(°) | ±10 | ±8 | ±5 |

The Stewart parallel manipulator is chosen as the ship motion simulation platform, whose joint position is shown in Figure _{i}, which is inscribed in a circle with the radius of Ra around the point_{a}._{a} denotes the length of the shorter edge of the symmetrical hexagon and_{a} denotes the central angle corresponding to the short side_{a}. Also, a symmetrical hexagon is constructed by the point_{i}, which is inscribed in a circle with the radius of_{b} around point_{b}. lb denotes the length of the shorter edge of the symmetrical hexagon and_{b} denotes the central angle corresponding to the short sides_{b}.

Positions of the joints on the base and moving platform.

The central angles_{a} and_{b} can be obtained by (_{a},_{b}

_{ a },_{b}

The volume of the Stewart parallel manipulator and the DWCI are chosen as targets of the optimization algorithm. The volume of the Stewart parallel manipulator can be got by (

The cost function can be expressed as

All the parameters of NSGA-II are presented in Table

The parameters of NSGA-II.

parameters | values |
---|---|

Population size | 50 |

Number of iterations | 300 |

Crossover probability | 0.9 |

Mutation probability | 1/3 |

Distribution index for SBX | 20 |

Distribution index for polynomial mutation | 100 |

The Pareto front, which is the optimization solutions including 50 sets of geometric parameters, is shown in Figure

The optimization solutions.

A set of geometric parameters is selected to verify the hybrid method proposed in this paper, as shown in Table

The set of geometric parameters for verifying the hybrid method.

Design variables | Targets | |||
---|---|---|---|---|

| | | | |

522.71mm | 322.91mm | 853.95mm | 7.33×108mm^{3} | -0.25917 |

The numerical method and the hybrid method are, respectively, used to obtain the maximal and minimal lengths of 3 legs, and the results are shown in Table

The maximal and minimal length of 3 legs.

Step length | Numerical method | Hybrid method | |
---|---|---|---|

5mm, 1° | 2mm, 1° | 1° | |

Maximum(mm) | 1041.0121 | 1041.0121 | 1041.0121 |

pose | (35,-35,50,5,8,10) | (35,-35,50,5,8,10) | (35,-35,50,5,8,10) |

Minimum(mm) | 796.3186 | 796.3167 | 796.3160 |

pose | (-20,-35,-50,-5,-8,-10) | (-23,-35,-50,-5,-8,-10) | (-22.01,-35,-50,-5,-8,-10) |

time/s | 457.47 | 8425.23 | 0.33 |

The total orientation workspace defined by the set of geometric parameters of the Stewart parallel manipulator and the desired total orientation workspace shown in Table

The desired total orientation workspace and total orientation workspace.

A hybrid method for solving the maximal and minimal leg lengths of 6-DOF parallel manipulator with desired total orientation workspace is proposed. The maximal and minimal leg lengths are obtained by the hybrid method, which consists of the numerical method used for orientation variables and algebraic method used for translation variables. The extremal points can be obtained by the hybrid method based on the three-element function without solving six-element function. And compared with the numerical method, it is more accurate and efficient.

The actuator stroke length is obtained to be the constraints, which is calculated by the maximal and minimal length of the legs, which enlarges the optimal range without the predefined maximal and minimal length of the legs.

Aiming at the physical size of the parallel manipulator and the proposed DWCI, the optimization of the geometric parameters of the parallel manipulator is conducted based on the multiobjective optimization algorithm (NSGA-II), which is subject to the actuator stroke length. Stewart platform is set as the example; the geometric parameters of the platform whose workspace contains the desired total orientation workspace are optimized and the hybrid method is proved to be more accurate and efficient compared to the traditional numerical method. This method provides the optimization guidance for engineering designers to design the parallel manipulator for desired total orientation workspace.

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research is supported by the National Natural Science Foundation of China (no. 51875499) and Innovative Research Assistant Support Project of Hebei Province (no. CXZS201801).