An effective structural optimization method based on a sensitivity analysis is proposed to optimize the variable section of a slender robot arm. The structure mechanism and the operating principle of a polishing robot are introduced firstly, and its stiffness model is established. Then, a design of sensitivity analysis method and a sequential linear programming (SLP) strategy are developed. At the beginning of the optimization, the design sensitivity analysis method is applied to select the sensitive design variables which can make the optimized results more efficient and accurate. In addition, it can also be used to determine the scale of moving step which will improve the convergency during the optimization process. The design sensitivities are calculated using the finite difference method. The search for the final optimal structure is performed using the SLP method. Simulation results show that the proposed structure optimization method is effective in enhancing the stiffness of the robot arm regardless of the robot arm suffering either a constant force or variable forces.
Structure optimization is concerned with finding the optimal shape of a structure by the iterative process based on the structural response analysis and sensitivity calculation. Sensitivity analysis is used to determine how “sensitive” a model is to changes in the value of the parameters of the model and to changes in the structure of the model. This paper focuses on parameter sensitivity. The parameter sensitivity is to find the most sensitive parameters to the dynamic behavior of the slender robot arm, and such parameters the main analysis object will in the optimization design. This will reduce the difficulty and improve the efficiency. Remarkable progress has been achieved in the structure optimization during the past three decades [
It should be pointed out that most of the previous studies on sensitivity analysis were carried out based on the discrete approaches, in which the sensitivities of mechanical properties were considered as the design parameters. The sensitivity analysis results can be used to select the optimization variables, but these analysis results ignored the relationship among different mechanical properties. For example, when the structure stiffness increases, it may lead to the increase of mass. Therefore, it is necessary to improve the sensitivity analysis method in the future work to determine the final optimization variables. These variables must be more sensitive to the mechanical properties than the mass to ensure the accuracy of optimization results. Besides, there were very limited theories and studies on the structure optimization of slender robot arm. Nevertheless, the mechanical properties of slender robot arm (such as stiffness, equivalent stress, and inherent frequency, etc) are greatly affected by its own structure features, so further researches are very important, in both analytical and application aspects, toward the structure optimization of slender robot arm.
In this paper, a sensitivity analysis-based optimization program of ANSYS parametric design language (APDL) has been developed to perform structure optimization of a slender robot arm which belongs to a 3-DOF innerwall grinding robot of the solid-propellant rocket engines. Firstly, the main structure mechanism of the grinding robot is presented, and the stiffness model of the robot arm is established. Then, a sensitivity analysis method which has two functions for the structure optimization is proposed. The first function is to select the sensitive design variables by comparing the ratios of the mechanical performance sensitivities to mass sensitivity of the robot arm. The second one is to improve the convergence in the process of the optimization. Finally, the optimizing objective can be achieved using the SLD method.
As shown in Figure
Schematic of the grinding robot.
Experimental setup.
The big arm driven by the ball screw can move along the
In practice, the displacement of the slender robot arm due to its own gravity is much smaller than that induced by the force in the working process. Therefore, the impact of gravity on the displacement is omitted in order to simplify the process of force analysis. When the robot arm is contacted with the working surface, the end-effector will be subjected to two kinds of forces. One is normal pressure
Force condition of the end-effector.
The force diagram of the robot arm can be obtained after the force condition of the end-effector is applied to each joint. As shown in Figure
Force diagram of the robot arm.
Both big arm and forearm are variable cross-section cantilevers which are reducible to constant section cantilever beam for the approximate calculation. Let the height of section at the end of the big arm as
The rotation angles of big arm and forearm are
According to (
The displacement along the
The bending displacement
Then the displacement of the robot arm along the
In summary, the bending defection of the robot arm shown in (
Based on the above calculations, the function relationships among the rotation angles of the forearm and the wrist (
Design sensitivity analysis, that is, the calculation of quantitive information on how the response of a structure is affected by changes of the variables that define its shape, plays an important role in structural shape optimization. There are generally two approaches to calculate the sensitivities [
In this study, the first-order forward finite difference is used to calculate the design sensitivities of the objective functions and constraint functions.
The design sensitivities for objective functions can be written as
Though the bigger absolute value of the above sensitivities, the faster response of the corresponding variable is, only sensitivities may not be enough to evaluate the impact of the variables on the objective function or the constraint functions. This is due to the fact that if a variable is sensitive to both mechanical properties and mass, this variable may not necessarily be the key variable to the mechanical properties, because actually optimization may achieve the purpose of controlling the properties at the cost of increasing the mass of the structure. Based on the above consideration, the influence extent of each variable on the optimization objective can be assessed by comparing the ratios of the mechanical performance sensitivities to mass sensitivity of the robot arm. It can be calculated by the following equations:
The mathematical model for design optimization problem can be formulated as
In this study, sequential linear programming (SLP) is used to minimize the objective function with respect to the constraint equations. In the SLP algorithm, the objective and constraints functions are approximated with linear extensions at the current design point during the optimization iteration. Then the original problem is transformed into the following linear programming problems [
For this approach, the optimum solution is always obtained at one of the vertices formed by the design constraints. Since there is the finite number of such vertices in the feasible region, a systematic approach such as the simplex method is used to search for the optimum solution among these vertices.
In a SLP method, the result of each iteration is expected to be a better solution. Since the linear programming technique is used to find the optimum solution for a nonlinear problem, some measures are taken to improve the optimization process. In this study a heuristic iterative algorithm based on the design sensitivity is employed to control the changes in the design variables. At a given design stage, the design variables are updated iteratively as
The success of the optimization process depends on the scale of moving step which can improve the convergence to a large degree. For efficiency and effectiveness of the algorithm, the moving step
The basic algorithm for the structural shape optimization based on mathematical programming is shown in Figure
Flowchart of the structural optimization.
Establish the objective function
Build the finite element model of the structure.
Carry out a finite element analysis using the design variables.
Evaluate the sensitivities of the objective and constraint functions of the current design.
Calculate the scale moving step with (
Using a suitable optimization algorithm, such as SLP, generate a new structural shape which satisfies the constraints.
If the new structural shape is not optimum, update the model to Step
To realize the structure optimization task, different principles such as structural analysis, automatic mesh generation, finite element analysis, sensitivity analysis, and mathematical programming are interrelated. As shown in Figure
In order to validate the above analysis and optimization approach, the design of the robot arm of the inner-wall grinding robot of the solid-propellant rocket engines which was introduced at the beginning of the paper was performed.
When the robot arm is at the position shown in Figure
The model of the robot arm and design variables.
Due to the bigger the displacement of the robot arm, the smaller will be its stiffness, so the stiffness sensitivity can be replaced by the displacement sensitivity. The sensitivities of some significant mechanical properties are performed for the different design variables; the results are listed in Table
Variable sensitivity results.
Design variable | Initial value (mm) | Mass sensitivity ( |
Bending displacement sensitivity ( |
Equivalent stress sensitivity ( |
Modal strain energy sensitivity ( |
Inherent frequency sensitivity ( |
---|---|---|---|---|---|---|
|
5 |
|
|
0.114 | 0.169 |
|
|
5 |
|
|
|
0.110 |
|
|
5 |
|
0.226 |
|
|
|
|
6 |
|
|
|
|
|
|
6 | 0.127 |
|
|
|
|
|
7 | 0.158 |
|
|
|
|
|
180 |
|
|
|
|
|
|
90 | 0.106 |
|
0.155 |
|
|
|
150 | 0.109 | 0.231 |
|
|
|
|
75 |
|
|
|
|
|
The ratios of the mechanical performance sensitivities to mass sensitivity of the robot arm are demonstrated in Figure
Contrast of sensitivities.
Figure
Before the robot arm is optimized, the value of its properties is calculated by finite element analysis. The results are listed in Table
Initial properties value.
Properties | Mass (kg) | Bending displacement (mm) | Equivalent stress (Mpa) | Modal strain energy (mJ) | Natural frequency (Hz) |
---|---|---|---|---|---|
Value | 74.21 | 1.31 | 20.00 | 44.28 | 130.47 |
The constraint conditions are set as the following four parts.
The objective function is taken as the bending deflection. In this paper, it is replaced by the displacement
Figure
Optimization design results.
Design variables | Optimal value (mm) | Optimization results | ||||
---|---|---|---|---|---|---|
Mass (kg) | Bending displacement (mm) | Equivalent stress (MPa) | Modal strain energy (mJ) | Natural frequency (Hz) | ||
|
5.1842 | |||||
|
4.7505 | |||||
|
180.35 | 67.46 | 0.67 | 15.40 | 42.74 | 120.63 |
|
146.06 | |||||
|
78.562 |
Evolutionary history of the bending deflection in the design process.
It can be indicated from Tables
The bending displacement and equivalent stress of the robot arm in the optimum state are shown in Figures
Bending displacement of the robot arm.
Equivalent stress of the robot arm.
The above optimization results are obtained when the robot arm is suffering static force. However, in the practical working process, the load is changed with time due to the heterogeneity of the processing material, the vibration caused by the electromotor, and so on. The mechanical testing on the inner-wall grinding robot in practical working process indicates that the robot arm is subjected to a sinusoidal loading. When the robot arm is remodeled according to the optimal variables, its displacement with the above variable load is obtained by ANSYS and shown in Figure
Displacement response curves of the robot arm.
It can be demonstrated by Figure
By taking a 3-DOF inner-wall grinding robot of the solid-propellant rocket engines as an example, a structure optimization programme has been developed for a slender robot arm based on the finite element method and the sensitivity analysis strategy in this paper. The search for the optimal structure is performed using the SLP technique. The following conclusions are drawn. The sensitivity analysis method and a sequential linear programming (SLP) strategy are applicated and developed for the structure optimization of slender arm robot. The main sensitive design variables are selected by comparing the ratios of the mechanical performance sensitivities to mass sensitivity of the robot arm. The sensitivity analysis method can not only be used to reduce the number of the design variables before optimizing for the purpose of increasing efficiency and accuracy, it can also be used to determine the scale of moving step which will improve the convergency during the optimization process. The displacement of the robot arm with optimal structure is reduced significantly no matter it is suffering constant force or variable force. This study is application oriented and can be a useful example of structural design optimization for engineers.
Size of the area surrounded by the midline,
Simplified average width of big arm and forearm
Displacement and rotation variables of link
Elastic modulus of the material
Working frequency
Natural frequency
Normal pressure
Frictional force
Equivalent forces of
Shear modulus of the material
Height of section at the end of the big arm
Simplified average height of big arm and forearm
Rotational inertia of the section
Height-width ratio
Thickness of arm
Scale factor of rotational inertia,
Length of link
Number of constraints;
Mass of the robot arm before and after optimization
Variation range of the mass
Equi-moment of
Equi-moment of
Number of design variables
Origin of coordinates of
Bending deformation for arm
Displacement of the robot arm along the
Bending displacement matrix
Length of the midline of the section
Euclidean norm of the vector of the sensitivities
Design sensitivities for objective functions and constraint function
Safety factor
Rotational equi-moment of
Rotational equi-moment relative to
Strain energy
Limit value
A small perturbation in the variable
Vector of the design variables;
Lower and upper limit of the design variables,
Objective function
Constraint functions,
Derivative gradients of the objective function and constraint functions
Yield strength and maximum equivalent stress;
The twist angle,
Poisson’s ratio
Rotational angles of big arm and forearm,
Wall thicknesses of arm
Limit coefficient of the frequency range.
This work is supported by the National Science Foundation of China (51105064), the National Program on Key Basic Research Project (2012CB026000), and the Fundamental Research Funds for the Central Universities of China (N100403007).