# Using nodal coordinates as variables for the dimensional synthesis of mechanisms

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## Abstract

The method of the lower deformation energy has been successfully used for the synthesis of mechanisms for quite a while. It has shown to be a versatile, yet powerful method for assisting in the design of mechanisms. Until now, most of the implementations of this method used the dimensions of the mechanism as the synthesis variables, which has some advantages and some drawbacks. For example, the assembly configuration is not taken into account in the optimization process, and this means that the same initial configuration is used when computing the deformed positions in each synthesis point. This translates into a reduction of the total search space. A possible solution to this problem is the use of a set of initial coordinates as variables for the synthesis, which has been successfully applied to other methods. This also has some additional advantages, such as the fact that any generated mechanism can be assembled. Another advantage is that the fixed joint locations are also included in the optimization at no additional cost. But the change from dimensions to initial coordinates means a reformulation of the optimization problem when using derivatives if one wants them to be analytically derived. This paper tackles this reformulation, along with a proper comparison of the use of both alternatives using sequential quadratic programming methods. In order to do so, some examples are developed and studied.

## Keywords

Nodal coordinates Dimensional synthesis SQP Deformed energy error function Minimum distance position problem## 1 Introduction

The synthesis of mechanisms is a problem of high practical interest and, thus, it has been the scope of many research jobs. Synthesis problems can be classified in different types such as structural synthesis, geometrical synthesis, design synthesis, configuration synthesis, type synthesis, position synthesis, dimensional synthesis, kinetostatic synthesis, kinetic synthesis, kinematic synthesis, rectified synthesis, optimal synthesis, or probabilistic synthesis. The contributions presented in this paper will be centred in the kinematic dimensional synthesis, where the dimensions of the links of a mechanism are searched for while trying to fulfil certain position kinematic requisites defined in this case as synthesis points.

Actually, many methods have been used to accomplish the study of the synthesis of mechanisms and here a short resume will be presented. Some of these methods are heuristic and some fall into the numerical type of techniques. Between the first main group of techniques, there are the Genetic Algorithms [1, 2, 3, 4, 5, 6], the Simulated Annealing [7, 8, 9], the Ant Colony Optimization [10, 11], the Particle Swarm Optimization [12, 13, 14, 15], and some others like the Tabu Search [16, 17]. Among the second main group of techniques, there are the Sequential Quadratic Programming (SQP) methods where the most common ones are based on the method of Newton, or Quasi-Newton approaches. In order to introduce restrictions, if they are of linear nature, the Null Subspace method should be appropriate, and a good analysis of the different alternatives is exposed in [18]. If they are non-linear, the methods of the Penalty Function, of the Lagrange Multipliers, or the Augmented Lagrangian Function should be used. In the case of linear inequality restrictions, the methods of Karmarkar or the Primal-Dual should be adequate. Finally, for non-linear inequality restrictions, the methods of the Slack Variables or the Logarithmic Barrier Function can be used. [19, 20, 21].

A common way of classifying the types of synthesis problems is path generation, function generation, rigid solid guide and mixed synthesis. Path generation tries to obtain the best possible correlation between the path described by the joints of a mechanism during the solid rigid motion, together with some other previously specified path. Function generation studies the coordination or synchronization of the positions of the input and output links of a mechanism. Rigid solid guide is the part of the mechanism synthesis that studies the problem of locating a floating element (coupler) of a mechanism along a series of given positions. The mixed synthesis, in its turn, is a combination of some of the aforementioned types of synthesis in the same problem. In this paper a new alternative for a method for dimensional synthesis is presented, which has been under continuous development and accurate improvement for the last thirty years, since in 1982, in reference [22] for the first time the concept of the deformed position problem was presented. The main idea being to obtain the minimum energy position of the elements of a mechanism when one or more of its joints is obliged to fulfil certain geometrical restrictions out from the possible motions as rigid solid of the mechanism. The mechanism is considered composed of deformable elements with a linear elastic behaviour. Thus, the initial position problem was solved by means of the minimization of an energetic function, defined as the summation of the difference between deformed and undeformed squared lengths for each link in the mechanism. The same methodology was employed for the definition of the finite displacements problem, the deformed position problem and the static equilibrium problem. It was also suggested to solve the optimum synthesis based on these same ideas by summing the minimum deformation energy in each synthesis point. Later on, this idea was applied in [23], using the dimensions of the mechanism as variables. Exact derivatives were obtained for the deformed problem, but for the synthesis the length in each iteration was obtained via the arithmetical average of the deformed lengths of each of the deformed position problems. In 1989, the algorithm’s convergence was improved by using approximate derivatives by means of finite differences in the synthesis instead of the arithmetical average [24]. The possibility of considering dimensional restrictions was also introduced by means of a stiffness that increases as long as the dimension gets nearer to the restricted value, which can be considered as a penalty function method. In 1993, in [25], the algorithm was improved by obtaining the exact first and second derivatives of every term. Furthermore, special elements with three joints were introduced to solve the low stiffness problems that appear whenever those joints are aligned. Here was also introduced a method to consider the fixed joints positions as variables. In order to do so, it is supposed that these are joined in the ends of a spring in direction of x and another spring in direction of y to a fixed point. In 2000, a preliminary study of the application of genetic algorithms to the synthesis of mechanisms and other mechanical problems is presented in [26]. Here it is demonstrated that the deformed position method is not very appropriate to be used together with the GA, due to the problem of the high aptitude of low stiffness mechanisms combined with the explorative nature of GAs. As a result, in 2004, another error function based on the minimum distance between the mechanism joint and the synthesis point [27] is applied, which happens to be valid to be applied with GAs.

In this paper the use of initial coordinates is explored for the synthesis of mechanisms using SQP and the deformation energy error function. The use of these kind of variables for the synthesis is not new, and has also been used using GAs and a distance error function with success, but the use of them in a deformation energy error function has not been yet studied. In this paper the relevant mathematical developments are presented and the analysis of several examples is exposed.

The motivation behind this change is that the use of dimensions does not include any information on the assembly configuration and, thus, the search space is somehow reduced. In the former formulation, an immutable set of coordinates were introduced by the user which were used to solve the position problem, and these coordinates had decisive influence on the deformed position problem solution.

This change has a main drawback though: an infinite set of initial coordinates define the same mechanism and, therefore, the optimization problem is always underdefined. This means that the optimization solver needs to be able to solve underdetermined systems.

The paper is organized as follows. First, a review of the deformed energy method is exposed. Afterwards, the choice of using initial coordinates as synthesis variables along with the deformed energy method is reasoned. Then, the energetic error function using initial coordinates is developed and the analytic expressions are presented, and the method for introducing boundary conditions is exposed. After that, some remarks on the optimization method are commented. Finally, some results are presented and some conclusions are driven.

## 2 The optimization of mechanisms using the deformed energy method

In the synthesis error function, the deformed position problem is solved for each one of the precision positions, so that a set of coordinates is obtained for each of them, which, in term, define the deformed lengths \(l_{ji}(x_{i})\).

The optimization of the synthesis function has been approached in two ways. The uncoupled approach, which is based in discarding the effect of the modification of a dimension in the deformed position problem [23] and the coupled approach, which takes into account this effect [24, 25].

## 3 Reasons for the use of initial coordinates as parameters for the optimization of mechanisms

To keep the formulation simple, this paper will only include the definition of mechanisms composed by R-Joints and modeled by simple truss elements. No higher order elements or other joints will be considered, although the ideas exposed here can easily be generalized for developments such as those published in [25] or [28].

## 4 Error function

*P*is the number of precision points;

*B*is the number of trusses defining the mechanism; \(L_{j}\) is the dimension of the

*j*-th truss (and, thus, the design vector,

*L*, is of dimension

*P*); \(x_{i}\) are the set of coordinates which minimize the deformation energy of the mechanism for the requirements in the precision point

*i*, and \(l_{ji}\) is the length of the truss

*j*as defined by the set of coordinates \(x_{i}\); this is, the deformed length of the

*j*-th truss. It is important then to state that each of the \(x_{i}\) vectors are obtained by an optimization process whose objective is to yield the lower deformation energy of the mechanism (considered as deformable) in the

*i*-th precision point.

## 5 Computing the derivatives

*L*. Taking the derivative of the expression in Eq. (1) with respect to \(L_{j}\), one can write Eq. (4):

*i*. Thus, one can write Eq. (6):

*L*a similar development to that presented in Eqs. (6) and (7) could be performed in this case, leading to similar conclusions, this is, the first derivatives are not affected by the use of coupled and uncoupled hypothesis. One can write the expression in (8).

## 6 Boundary conditions

*g*and

*H*. If one considers Eq. (15):

*g*can be expressed as the summation of those components obtained in the previous section and an additional term. In order to use the same finite element approach described before, one can write Eq. (16):

*j*, joining joints

*k*and

*l*:

*k*in truss

*j*is fixed and 0 if it is not; \(f_{l}\) equals 1 if node

*l*in truss

*j*is fixed and 0 in the other case. It has also been used here the Eq. (20):

## 7 Optimization algorithm

As exposed before, the chosen optimization algorithm is an in-house developed SQP method, which, in our case, has full Hessian analysis. This is necessary because, as exposed before, when using initial coordinates as parameters of the synthesis, the Hessian matrix should always be underdetermined. This algorithm is applied both to the synthesis problem and the inner deformed position function, which, as exposed before, is itself an optimization problem. The Hessian matrices and gradient vectors are built via assembly of the trusses matrices and, afterwards, linear restrictions (required for the inner function) are introduced via direct manipulation of these matrices. The resultant linear system is afterwards diagonalized by means of the method presented in [30], which is able to solve underdetermined systems.

This allows one to obtain the increment vector in a decoupled system, where one can verify the signs of each variable to check if it leads to a maximization or a minimization, or an inflexion point. The underdetermined nature of the problem will lead to, at least, an stationary point in one direction. After this procedure, unidimensional optimization techniques are applied.

The optimization algorithm chosen is of the exploitative type. This means that it is very effective when one wants to improve an initial mechanism with an acceptable quality. If it is desired to find an appropriate mechanism from a fresh start, it would be more logic to use one explorative algorithm such as the Genetic Algorithm. In such case, it would not be possible to use this deformation energy based error function because, as it is demonstrated in reference [27], this type of functions leads to mechanisms with low stiffness and, therefore, of low usefulness.

## 8 Experimental results

In the left-handed picture the problem along with the starting guess is shown. The obtained result is shown in the right-handed picture, where it can be observed the optimized position of the fixed node A. The result is obtained in about 8 iterations to a precision in the order of \(10^{-31}\).

Being this a numerical algorithm its sensitivity is determined by the size of the floating point used and other factors such as the preciseness of the criteria of convergence. In this case floating point of double precision have been worked with and criteria of convergence have been adjusted to obtain results with at least 5 significant numbers.

It is important to point out that although in the results presented the precision points are achieved in the specified order, this is due to the fact that they belong to feasible paths for the mechanisms of the considered typology. That is, in the optimization process it has not been introduced any condition to verify this order. However, in the proposed algorithm constraints could be introduced to force the mechanism to follow a certain order. In any case, these constraints could cause the lack of convergence towards a quality solution.

The result is obtained in a similar number of iterations, with an increased cost for each of them. These results show that the algorithm is able to deal with both exact and approximate synthesis. In both of these examples, due to their particular nature, coupled and uncoupled formulations coincide. Now more complex problems will be addressed.

Initial coordinates of the fourbar

\(X_A\) | \(Y_A\) | \(X_B\) | \(Y_B\) | \(X_C\) | \(Y_C\) |
---|---|---|---|---|---|

− 5.7114 | 2.5202 | − 3.8503 | − 0.4130 | − 2.1952 | − 0.5217 |

\(X_D\) | \(Y_D\) | \(X_E\) | \(Y_E\) | ||
---|---|---|---|---|---|

− 2.0260 | − 3.2762 | − 2.8596 | 0.8072 |

Coordinates of the 9 precision points to be followed by the fourbar

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

− 2.6301 | 1.0126 | − 0.2139 | 2.2690 |

− 2.1589 | 1.0488 | 0.0882 | 2.8610 |

− 1.6757 | 1.1213 | 0.2443 | 3.5135 |

− 1.2408 | 1.3630 | 0.2931 | 4.1358 |

− 0.6850 | 1.7254 |

Final coordinates of the fourbar

\(X_A\) | \(Y_A\) | \(X_B\) | \(Y_B\) | \(X_C\) | \(Y_C\) |
---|---|---|---|---|---|

− 9.3343 | 3.7231 | − 2.2052 | 0.4771 | − 1.2526 | 2.9409 |

\(X_D\) | \(Y_D\) | \(X_E\) | \(Y_E\) | ||
---|---|---|---|---|---|

− 6.7509 | − 0.5369 | − 3.8337 | 1.4035 |

Obviously, with the dimensional approach, one cannot include the basement locations as optimization variables without introducing complex modifications, as explained before. In order to compare methods, the same problem will be solved including restrictions so the fixed nodes are not part of the optimization.

final coordinates of the fourbar

\(X_A\) | \(Y_A\) | \(X_B\) | \(Y_B\) | \(X_C\) | \(Y_C\) |
---|---|---|---|---|---|

− 5.7114 | 2.5202 | − 5.2815 | − 2.0171 | − 1.8628 | − 11.183 |

\(X_D\) | \(Y_D\) | \(X_E\) | \(Y_E\) | ||
---|---|---|---|---|---|

− 2.0260 | − 3.2762 | − 3.1240 | − 0.8376 |

Final dimensions of the fourbar

\(L_0\) | \(L_1\) | \(L_2\) | \(L_3\) | \(L_4\) |
---|---|---|---|---|

2,2565 | 6,1263 | 5,4331 | 3,0782 | 4,1256 |

Initial coordinates of the double butterfly

\(X_A\) | \(Y_A\) | \(X_B\) | \(Y_B\) | \(X_C\) | \(Y_C\) |
---|---|---|---|---|---|

− 3.7300 | − 2.0300 | − 3.8200 | 1.8900 | − 2.4300 | 0.7300 |

\(X_D\) | \(Y_D\) | \(X_E\) | \(Y_E\) | \(X_F\) | \(Y_F\) |
---|---|---|---|---|---|

− 1.5400 | 1.6000 | 0.6800 | 1.8200 | − 0.2700 | 0.8500 |

\(X_G\) | \(Y_G\) | \(X_H\) | \(Y_H\) | \(X_I\) | \(Y_I\) |
---|---|---|---|---|---|

0.8500 | − 1.0400 | − 1.7300 | − 0.3500 | − 0.8400 | − 0.3900 |

\(X_J\) | \(Y_J\) | \(X_K\) | \(Y_K\) | ||
---|---|---|---|---|---|

− 1.3300 | − 1.0800 | − 0.7800 | 2.4800 |

Final coordinates of the double butterfly

\(X_A\) | \(Y_A\) | \(X_B\) | \(Y_B\) | \(X_C\) | \(Y_C\) |
---|---|---|---|---|---|

− 3.7300 | − 2.0300 | − 3.8213 | 1.9397 | − 2.1265 | 1.1661 |

\(X_D\) | \(Y_D\) | \(X_E\) | \(Y_E\) | \(X_F\) | \(Y_F\) |
---|---|---|---|---|---|

− 1.4903 | 2.3132 | 0.5738 | 1.8621 | − 0.5240 | 1.0219 |

\(X_G\) | \(Y_G\) | \(X_H\) | \(Y_H\) | \(X_I\) | \(Y_I\) |
---|---|---|---|---|---|

0.4528 | 1.5409 | 2.4686 | − 0.3562 | − 1.6774 | 0.2122 |

\(X_J\) | \(Y_J\) | \(X_K\) | \(Y_K\) | ||
---|---|---|---|---|---|

− 2.0583 | − 1.6033 | − 0.7114 | 3.4163 |

Obviously, in order to successfully apply these techniques to complex mechanisms like the present one, the starting solution is of most importance, because of the presence of a large amount of local optima and also because the energy function favours low stiffness mechanisms, which can be useless, but can reach any condition. In the case of the coordinate based approach, it can also yield to degenerated 2 dof mechanisms if the initial solution is too far from the desired optima. As exposed in [1], the use of distance based functions along with genetic algorithms can give good initial solutions in these situations.

The examples shown in this work have been run on an Intel Xeon E5645@2,4GHz and the code was not programmed for multithread. The execution times are very variable, where the fourbar examples lie under one second, although in cases of slow convergence it has been reached, very exceptionally, the 10 min. In the example of the double butterfly presented the execution time was of 33 s.

Comparing with synthesis methods based on dimensions, the use of initial coordinates presents a similar performance. This was to be expected as the number of unknowns does not increase in a considerable way.

## 9 Conclusions and future work

This paper has shown a new approach to the dimensional synthesis of mechanism which, although based in the same concepts as previous developments, introduces fundamental changes in its conception. The main contribution of this work is that thanks to the fact that the initial coordinates are used as optimization variables, the assembly configuration is included in the optimization process, which is of most importance in the definition of the mechanism. A second point of interest derives from the fact that the coordinates of the fixed points are also variables of the optimization and thus, one does not need to include workarounds to optimize them. A final advantage, directly derived from the first, is that all of the possible solution vectors define a mechanism which always can be assembled, which not always holds truth when using dimensions. These advantages come to some cost, namely the fact that the same mechanism can be defined in infinite ways, thus leading to an underdetermined optimization problem. This disadvantage can successfully be overcome with an appropriate optimization method. Experimentation has shown that, depending on the problem, the use of one or another of the methods can deliver different results, so the best bet is to use both or even combinations of them. In this paper an uncoupled approach has been used, which tends to be better at the initial stages, but is slower at the final iterations. In this paper the relevant algorithms and mathematical developments have been exposed and, although they have been limited to mechanisms composed by R-Joints, they can easily be generalized to P-joints and even three-dimensional problems. In any case, the new algorithm inherits not only the advantages of the former approach, but also some of its drawbacks, specially the problem of the low stiffness mechanisms. Further developments should tackle with this problem, possibly employing a minimum distance approach, which has already shown some good results along with genetic algorithms, but requires a complex development if SQP algorithms are to be applied. The use of coupled approaches could also be of interest.

## Notes

### Acknowledgements

The authors wish to thank the Spanish Ministry of Economy and Competitiveness for its support through Grant DPI2013-46329-P and DPI2016-80372-R. Additionally the authors wish to thank the Education Department of the Basque Government for ist support through grant IT947-16.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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