# Structural and Material Optimization for Automatic Synthesis of Spine-Segment Mechanisms for Humanoid Robots with Custom Stiffness Profiles

^{*}

## Abstract

**:**

## 1. Introduction

#### Literature Review

- A mechanism with angular stiffnesses comparable to the L4–L5 IJ;
- A mechanism with geometry comparable to the IJ, but increased stiffness in flexion;
- A mechanism with geometry comparable to the IJ, but significantly reduced range of motion in lateral bending and axial rotation.

- Successfully applying sGA to simultaneous optimization of structural and material parameters of flexible parallel structures for use in humanoid robots, based on actual human body joints;
- Obtaining custom mechanisms with geometry close to that of the IJ, but with significantly different responses to static loads.

## 2. Materials and Methods

#### 2.1. Solving Three-Dimensional, Elastostatic Problems

**p**:

**R**= the rotation matrix from the moving to the stationary platform reference frame, sα = sinα, cα = cosα. The sequence was assumed after [32,33], while the angles α, β, γ corresponded to the flexion, the lateral bending, and the axial rotation of the moving platform. The forces and the moments acting on the moving platform and caused by linear springs were computed using the following equations:

**F**(

_{s}**M**) = the force (moment) generated by the linear spring, k

_{s}_{s}= the stiffness parameter for the spring element, Δl

_{s}= the change of the spring element length, and

**a**(

_{s}**b**) = the position vector of the spring element attachment to the lower (upper) platform.

_{s}**F**(

_{s}**M**) = the forces (moments) generated by the linear springs,

_{s}**F**(

_{c}**M**) = the forces (the moments) generated by the linear cables,

_{c}**F**(

_{ext}**M**) = the external force (moment) acting on the upper platform, and n (m) = the number of the springs (cables).

_{ext}#### 2.2. The Objective Function

**x**, also referred to as a solution, and an objective function, which takes in the vector

**x**and rates it. In this study, the vector

**x**contained the full information required to describe a mechanism (see the next section). To rate the mechanisms, based on their static behavior, the following objective function was used in minimization:

**w**= the weight i (here,

_{i}**w**= 1 (i = 1.3), w

_{i}_{4}= 2), not_passed = the number of loads, for which the solver did not converge to a solution in 200 iterations, and:

_{α}= the flexion displacement indicator; Δα = the angular displacement obtained from the mechanism at flexion moment, M

_{αi}from 0 Nm to 10 Nm (ΔM

_{αi}= 1 Nm), and Δα

_{ref}= the reference angular displacement measured on the actual joint or obtained from a verified joint model at flexion moment, M

_{αi}from 0 Nm to 10 Nm (ΔM

_{αi}= 1 Nm).

#### 2.3. Obtaining the Bounds for the Optimization Procedure

#### 2.4. The Optimization Procedure

- The type (a binary value; 1 corresponded to a spring, while 0 to a cable);
- The stiffness (N/mm);
- The coordinates of the attachment to the lower platform (mm) (2 if the link was in the xy plane or 3 if not);
- The coordinates of the attachment to the upper platform (mm) (2 if the link was in the xy plane or 3 if not);
- The free length (mm).

## 3. Results

#### 3.1. The Reference Angular Stiffnesses for the Optimization

#### 3.2. Case #1—A Mechanism to Substitute the L4–L5 IJ

#### 3.3. Cases #2 and #3—Mechanisms with Custom Stiffness Profiles

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) The L4–L5 intervertebral joint (IJ); (

**b**) the ligament system of the IJ (ligaments substituted with cables).

**Figure 2.**The force-elongation curves for an actual ligament, a linearized cable used in generating bounds for the optimization procedure, and a cable after optimization.

**Figure 3.**A sample structure with its corresponding decision variable vector in structured Genetic Algorithm (sGA). The image of the platform mechanism based on [30].

**Figure 5.**The obtained mechanism with the L4 and the L5 vertebrae for reference: (

**a**) Sagittal-plane view; (

**b**) Transverse-plane view.

**Figure 6.**Simulation results for Case #1. Graphs of moment loads acting on the system and angular displacements they cause (black “·” marker = the data obtained from the mechanism, grey “*” marker = the reference data), where α = the flexion, β = the lateral bending angle, and γ = the axial rotation angle.

**Figure 8.**Simulation results for (

**a**) Case #2; (

**b**) Case #3. Graphs of moment loads acting on the system and angular displacements they cause (black “·” marker—the data obtained from the mechanism, grey “*” marker—the reference data); α—the flexion, β—the lateral bending angle, γ—the axial rotation angle.

Case Number | Δα [deg] | Δβ [deg] | Δγ [deg] |
---|---|---|---|

Case #1 | as in Reference [3] | as in Reference [3] | as in Reference [3] |

Case #2 | $\frac{log(\raisebox{1ex}{${M}_{\alpha}$}\!\left/ \!\raisebox{-1ex}{$5.40$}\right.+1)}{0.34}$ | as in Reference [3] | as in Reference [3] |

Case #3 | as in Reference [3] | $\frac{log(\raisebox{1ex}{${M}_{\beta}$}\!\left/ \!\raisebox{-1ex}{$101.70$}\right.+1)}{0.19}$ | $\frac{log(\raisebox{1ex}{${M}_{\gamma}$}\!\left/ \!\raisebox{-1ex}{$36.11$}\right.+1)}{0.49}$ |

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**MDPI and ACS Style**

Ciszkiewicz, A.; Milewski, G.
Structural and Material Optimization for Automatic Synthesis of Spine-Segment Mechanisms for Humanoid Robots with Custom Stiffness Profiles. *Materials* **2019**, *12*, 1982.
https://doi.org/10.3390/ma12121982

**AMA Style**

Ciszkiewicz A, Milewski G.
Structural and Material Optimization for Automatic Synthesis of Spine-Segment Mechanisms for Humanoid Robots with Custom Stiffness Profiles. *Materials*. 2019; 12(12):1982.
https://doi.org/10.3390/ma12121982

**Chicago/Turabian Style**

Ciszkiewicz, Adam, and Grzegorz Milewski.
2019. "Structural and Material Optimization for Automatic Synthesis of Spine-Segment Mechanisms for Humanoid Robots with Custom Stiffness Profiles" *Materials* 12, no. 12: 1982.
https://doi.org/10.3390/ma12121982