Functional Design of a 6-DOF Platform for Micro-Positioning

Round 1

Reviewer 1 Report

In general, the paper makes an impression of an engineering study rather than research. The design is well known as well as kinematic modeling and optimization approach. Authors should better describe their main contribution and make the paper more scientifically sound.  Is it new objective function (22) based on indexes (21)? How is it better compared to classical ones like described, e.g. in [Hollerbach, J.M., Wampler, C.W.: The calibration index and taxonomy for robot kinematic calibration methods. Int. J. Robot. Res. 15(6), 573–591 (1996)]? 

Some technical questions and remarks:

1. Authors talk about accuracy in entire workspace. However, they define quite a limited one of 5deg in xyz rotations.
2. Talking about 2^6 IPK solutions, do you consider holonomic constraints between P_i due to upper platform shape?
3. How the suggested multistart optimization procedure prevents sticking to a local minima? M.b. consider others global optimization techniques.  
4. There is no complete optimization problem statement including free variables, hard and soft  constraints specification.
5. Fig.7 shows determinant values.  I suggest to show objective function values as well (in Fig. and Table 2).

Author Response

The Authors want to thank the Reviewers for their valuable suggestions. The Authors made every effort to answer the questions and requests posed by the Reviewers.

• In general, the paper makes an impression of an engineering study rather than research. The design is well known as well as kinematic modeling and optimization approach. Authors should better describe their main contribution and make the paper more scientifically sound.  Is it new objective function (22) based on indexes (21)? How is it better compared to classical ones like described, e.g. in [Hollerbach, J.M., Wampler, C.W.: The calibration index and taxonomy for robot kinematic calibration methods. Int. J. Robot. Res. 15(6), 573–591 (1996)]?

The work is focused on the functional design of a parallel mechanism for positioning tasks at the micro scale. The aim of the work is showing the kinematic performance of the Hexaglide architecture at the micro scale when real design specifications are imposed. The sections of the paper do not introduce new formulations but an adaptation of known algorithms with the aim of obtaining a manipulator able to provide the design requirements. Moreover, even if the Hexaglide is known in the literature, the paper reveals a new potential use of its architecture, still not valued today.

The Authors added some references to classical functions typically used in robotic optimization problems. Calibration is presently under study, but it is not investigated in the paper. Only an evaluation of the positioning errors due to actuators resolution has been taken into account at this stage of the work.

• Some technical questions and remarks:
• Authors talk about accuracy in entire workspace. However, they define quite a limited one of 5deg in xyz rotations.

With the term whole workspace the Authors referred to the one mentioned in the design specifications section. The term whole has been neglected and more details have been given in the new version of the text.

• Talking about 2^6 IPK solutions, do you consider holonomic constraints between P_i due to upper platform shape?

In the inverse kinematics problem the pose of the moving platform is given, therefore the holonomic constraints between the points due to the shape of the upper platform are intrinsically imposed. 2^6 are all the theoretical combinations that come out from the mathematical problem of having six legs that can have two configurations each one.

• How the suggested multistart optimization procedure prevents sticking to a local minima? M.b. consider others global optimization techniques.

Actually the Authors were looking for local minima, because a preliminary discrete screening of the effect of different parameters values on the objective function had already been made in order to find the mentioned pre-sizing values. Starting from them (gross solution) a local minima is searched with the mentioned algorithm. More details have been given in the text.

• There is no complete optimization problem statement including free variables, hard and soft  constraints specification.

Partially the answer is given in the previous point. All the optimization parameters of table 2 are free variables in their domain of variation. Hard and soft constraints are indicated in the new version of the text.

• Fig.7 shows determinant values.  I suggest to show objective function values as well (in Fig. and Table 2).

Many variables are involved in the study and showing for each of them the map of the function keeping constant the other parameters does not give actually a better understanding of the function behaviour. Moreover the objective function is not evaluated for every parameter value, but only at discrete values. A map would require a finer knowledge. The text has been improved to clarify the procedure and justify the results. In table the associated values of the objective function have been introduced.

Reviewer 2 Report

This manuscript presents the optimization of the kinematic parameters of a parallel platform for micro-positioning.

The platform has 6 DOF's and is based on a known configuration called "hexaglide", where all the actuators are linear and arranged in parallel.
The kinematic model has been parametrized based on the Study's quadric, and the best parameter values have been found using a search according to a Global Condition Index.
A particular benefit of this configuration is that the displacements on the x-axis can be performed by the entire mechanism, without inner relative motions. This feature gives a smaller set of parameters and reduces the complexity of the kinematic and performance analysis.
The design has some constraints provided by the manufacturing company requirements and the performance of the available actuators. For this reason, this work is an excellent example of a practical application of mathematical methods for the optimization and kinematic performance evaluation of a parallel manipulator.
This manuscript is very well written. It includes a clear introduction to the problem and a progressive description of the parametrization of the model. The inverse kinematic problem is provided in a closed form equation, and the inverse kinematic model has been solved by using a numeric approach based on the Newton-Raphson method. After the derivation of the Jacobian matrix, the optimization function is presented. The index is based on the ponderated sum of the eigenvalues of the linear and rotational parts of the manipulator's Jacobian.
Section 4.1 provides a valuable discussion about selecting the initial values of the parameters based on the specifications. However, the description of the optimization process is barely described but includes a discussion part. It also includes a section about performance evaluation with another discussion. However, the Conclusions section is little more than a summary of the contents with insights on a prototype with a more compact link arrangement.

However, there are some minor document tips that I have to point out.

- The sentence in lines 34-36 is hard to understand. Please clarify.

- The "Hexaglide" name is used from the beginning and associated with reference [9], but it would be welcome to define its origin.

- Check that the style of the design specifications (bullets) matches the journal style. Also, check if they are requirements instead. 

- The references to "Study Quadric" and "Normalization Equation" are written in cursive style in line 113 and as regular text in other parts. Please unify the style. Similar happens to "Zero Configuration."

- The sentence in lines 121-123 is hard to understand. Please clarify.

- Lines 144 and following:

• Using xk as Cartesian position can be confused with the x0 .. x3 used for the matrix parameters.
• Xk is introduced to present the Newton Raphson iterations (line 144 and before) but equation (16) uses xi. i has been used to iterate over degrees of freedom. May be confusing at first glance.

- In line 180 X is defined as [WT, VT]T (with angular velocity on top) but line 169 specifies the top part of the Jacobian as the linear part and the bottom half as the rotational part. Please check this.

- Please, justify the sentence of line 183.

- In line 220, rotations are limited to +-5 degrees. However, the captions of figures 6, 7, and 8 are mentioning higher angles.

- Consider joining Table 3 as an additional row of Table 2 to make it more compact. 

Author Response

• Section 4.1 provides a valuable discussion about selecting the initial values of the parameters based on the specifications. However, the description of the optimization process is barely described but includes a discussion part. It also includes a section about performance evaluation with another discussion. However, the Conclusions section is little more than a summary of the contents with insights on a prototype with a more compact link arrangement.

More insight about the optimization procedure has been given in the new version of the paper and conclusions have been improved and rearranged.

• However, there are some minor document tips that I have to point out.
• The sentence in lines 34-36 is hard to understand. Please clarify.

The sentence has been reviewed.

• The "Hexaglide" name is used from the beginning and associated with reference [9], but it would be welcome to define its origin.

The origin of the Hexaglide architecture has been cited in the text.

• Check that the style of the design specifications (bullets) matches the journal style. Also, check if they are requirements instead.

The conventional latex command \itemize has been used, with the style.css inherited from MDPI template.

• The references to "Study Quadric" and "Normalization Equation" are written in cursive style in line 113 and as regular text in other parts. Please unify the style. Similar happens to "Zero Configuration."

The idea was to present key words in italic font when they occured for the first time in the text. As suggested by the reviewer normal font style has been assigned to them in the new version of the paper, except for the word Hexaglide that has been emphasized only once when a particular attention is to be given to its name.

• The sentence in lines 121-123 is hard to understand. Please clarify.

The sentence has been reviewed and improved.

• Lines 144 and following:

• Using xk as Cartesian position can be confused with the x0 .. x3 used for the matrix parameters.
• Xk is introduced to present the Newton Raphson iterations (line 144 and before) but equation (16) uses xi. i has been used to iterate over degrees of freedom. May be confusing at first glance.

More detail has been introduced in the text to clarify the procedure. The choice of x_k as Cartesian position has been kept because representing the Study’s parameters, as previously defined in the text and coherently with the content of vector x = (x_0, …x_3; y_0,… y_3).

• In line 180 X is defined as [WT, VT]T (with angular velocity on top) but line 169 specifies the top part of the Jacobian as the linear part and the bottom half as the rotational part. Please check this.

There was an error in the order of the terms, expressions (20) and the following sentence have been reviewed.

• Please, justify the sentence of line 183.

The sentence has been reviewed.

• In line 220, rotations are limited to +-5 degrees. However, the captions of figures 6, 7, and 8 are mentioning higher angles.

The sentence has been improved in order to clarify the reason of values higher than 5°.

• Consider joining Table 3 as an additional row of Table 2 to make it more compact.

The two table have been joined as suggested.

Round 2

Reviewer 1 Report

Reviewer's previous comments were properly addressed. The main lacoon remains lack of scientific novelty of the paper. However, as a nice engineering study, the paper can be accepted.

Author Response

The Authors want to thank the Reviewer for his/her suggestions. The Authors did their best to verify the text and correct it from typos and errors. The text is now completely reviewed.