# Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb

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## Abstract

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## 1. Introduction

## 2. Forward Dynamics of the Artificial Lower Limb Movement

- Suppose, Joint position (Knee or ankle) of the artificial lower limb = $\varnothing $
- Joint force of the lower limb = $\tau $
- Joint velocity of the lower limb = $\dot{\varnothing}$
- Joint acceleration of the lower limb = $\ddot{\varnothing}$

- $\frac{d}{dt}\frac{\partial L}{\partial \dot{\varnothing}}=$time derivative of partial derivative of L with respect to $\dot{\varnothing}$
- $\frac{\partial L}{\partial \varnothing}=$partial derivative of L with respect to $\varnothing $
- $\mathsf{\tau}=$ power produced by the joints related to $\left(\varnothing ,\dot{\varnothing}\right)$.

- ${m}_{1}$ = mass of the joint2,
- ${v}_{1}=$ velocity of the joint2.$$K.{E}_{2}=\frac{1}{2}{m}_{1}{v}_{1}^{2}=\frac{1}{2}{m}_{1}\left({\dot{a}}_{1}^{2}+{\dot{b}}_{1}^{2}\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{2}{m}_{2}\left(\left({x}_{1}^{2}+2{x}_{1}{x}_{2}\mathrm{cos}{\varnothing}_{2}+{x}_{2}^{2}\right){\dot{\varnothing}}_{1}^{2}+2\left({x}_{2}^{2}+{x}_{1}{x}_{2}\mathrm{cos}{\varnothing}_{2}\right)\dot{{\varnothing}_{1}}\dot{{\varnothing}_{2}}\phantom{\rule{0ex}{0ex}}+{x}_{2}^{2}{\dot{\varnothing}}_{2}^{2}\right)$$

- $M(\varnothing )\ddot{\varnothing}=n\times n$ mass matrix shown in Equation (20)
- $C\left(\varnothing ,\dot{\varnothing}\right)$ = velocity product term shown in Equation (21)
- $g(\varnothing )=$ gravity term, i.e., gravity due to springs in the mechanical model for potential energy is shown in Equation (22).$$M(\varnothing )\ddot{\varnothing}=\left[\begin{array}{cc}{m}_{1}{x}_{1}^{2}+{m}_{2}\left({x}_{1}^{2}+2{x}_{1}{x}_{2}\mathrm{cos}{\varnothing}_{2}+{L}_{2}^{2}\right)& {m}_{2}\left({x}_{1}{x}_{2}\mathrm{cos}{\varnothing}_{2}+{x}_{2}^{2}\right)\\ {m}_{2}\left({x}_{1}{x}_{2}\mathrm{cos}{\varnothing}_{2}+{x}_{2}^{2}\right)& {m}_{2}{x}_{2}^{2}\end{array}\right]$$$$C\left(\varnothing ,\dot{\varnothing}\right)=\left[\begin{array}{c}-{m}_{2}{x}_{1}{x}_{2}\mathrm{sin}{\varnothing}_{2}\left(2\dot{{\varnothing}_{1}}\dot{{\varnothing}_{2}}+{\dot{\varnothing}}_{2}^{2}\right)\\ {m}_{2}{x}_{1}{x}_{2}{\dot{\varnothing}}_{1}^{2}\mathrm{sin}{\varnothing}_{2}\end{array}\right]$$$$g(\varnothing )=\left[\begin{array}{c}({m}_{1}+{m}_{2}){x}_{1}g\mathrm{cos}{\varnothing}_{1}+{m}_{2}g{x}_{2}\mathrm{cos}\left({\varnothing}_{1}+{\varnothing}_{2}\right)\\ {m}_{2}g{x}_{2}\mathrm{cos}\left({\varnothing}_{1}+{\varnothing}_{2}\right)\end{array}\right]$$

## 3. Experimental Methods

#### 3.1. Stability Analysis of the Non-Linear System

#### 3.2. Workflow of the System Stability Analysis

## 4. Results and Discussions

- Step 1: Representation of polynomials in characteristics equation form.
- Step 2: Determination of the equilibrium point of the system.
- Step 3: Generation of Jacobian matrix to implement in the characteristics equation.
- Step 4: Determination of roots from the characteristics equation.
- Step 5: Verifying the stability conditions from the roots.
- Step 6: Declaration of Asymptotic stability condition.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 9.**Graphical representation of the step response of the open loop transfer function of the system.

**Figure 11.**Graphical representation of the closed loop transfer function of the system using an advanced control system designer.

**Figure 14.**Graphical representation of the characteristics of the dead zone of the lower limb system.

**Figure 15.**Graphical representation of the characteristics of the saturation of the lower limb system.

Proportional Constant (k_{p}) | Integral Constant (k_{i}) | Derivative Constant (k_{d}) | Stability Status |
---|---|---|---|

0.92214 | 2.8732 | 0 | Stable |

Rise Time (s) | Settling Time (s) | Overshoot (%) | Peak (s) | Stability Status |
---|---|---|---|---|

0.626 | 7.77 | 10.3 | 1.1 | Stable |

Rise Time (s) | Settling Time (s) | Peak Amplitude (s) | Stability Status |
---|---|---|---|

0.00191 | 5.05 | 0.965 | Stable |

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**MDPI and ACS Style**

Das, S.; Nandi, D.; Neogi, B.; Sarkar, B.
Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb. *Symmetry* **2020**, *12*, 1727.
https://doi.org/10.3390/sym12101727

**AMA Style**

Das S, Nandi D, Neogi B, Sarkar B.
Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb. *Symmetry*. 2020; 12(10):1727.
https://doi.org/10.3390/sym12101727

**Chicago/Turabian Style**

Das, Susmita, Dalia Nandi, Biswarup Neogi, and Biswajit Sarkar.
2020. "Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb" *Symmetry* 12, no. 10: 1727.
https://doi.org/10.3390/sym12101727