# A Dynamic Pole Motion Approach for Control of Nonlinear Hybrid Soft Legs: A Preliminary Study

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## Abstract

**:**

## 1. Introduction

## 2. Analysis of a Motorized Hybrid Soft Leg: Dynamic Routh’s Stability Criterion for Nonlinear Systems

_{M}, an actuator with viscous damping B¬

_{M}, a joint actuator with a relative angular displacement θ

_{M}, a motor shaft with a torque τ

_{M}, and an end effector with a relative displacement θ

_{L}. The joint flexibility is modeled by a linear torsional spring with stiffness k. Defining τ

_{M}= r, the Euler–Lagrange equation can represent the dynamics of the hybrid soft leg as

_{i}, (i = 1, 2, 3, 4) can be derived as

_{i}($i\in \left. [1,7\right]$) and b equal to 1 in Equation (3) as

- (a)
- For stability of the system, all the elements in the first column of the dynamic Routh’s array must be positive non-zero values. Thus, we find that $\psi >0$ from all the conditions of $\psi +4>0$, ${\psi}^{2}+2\psi +8>0$, and $\psi >0$ from each row, which means the condition of $sin\left({x}_{3}\left(t\right)\right)/\left({x}_{3}\left(t\right)\right)>0$ should be met to make the system stable. The stability region of $sin\left({x}_{3}\left(t\right)\right)/\left({x}_{3}\left(t\right)\right)$ is graphically represented in Figure 5.
- (b)
- Zero value at any rows in the first column of the dynamic Routh’s array represents that oscillatory dynamic poles are located on the imaginary axis of the g-plane, which indicates instability of the system. Zero value exists only if $sin\left({x}_{3}\left(t\right)\right)=0$, which occurs periodically.
- (c)
- As the conventional Routh’s stability criterion, the dynamic Routh’s stability criterion can indicate the number of dynamic poles on the right-hand plane (RHP) of the g-plane by the number of sign (+ or −) changes in the first column of the dynamic Routh’s array. From the array, it can be found that one sign change could occur, which represents that one dynamic pole could be located in RHP of the g-plane when the system is not stable. Without a sign change, no dynamic poles are located in RHP of the g-plane, and the system is stable.

## 3. Error-Based Adaptive Controller (E-BAC) for the Motorized Hybrid Soft Leg

‘As error decreases from a large value to a small value, ${K}_{p}\left(e,t\right)\left(={\omega}_{n}^{2}\left(t\right)\right)$ is continuously decreased from a very large value to a small value, and simultaneously, ${K}_{v}\left(e,t\right)\left(=2\zeta \left(t\right){\omega}_{n}\left(t\right)\right)$ is increased from a small value to a large value’.

- (a)
- For the stability of the hybrid soft leg system, the dynamic poles should be always located on LHP on the g-plane for all values of ${x}_{3}\left(t\right)$.
- (b)
- For achieving the fast response time, the system must have a large bandwidth for large errors and small bandwidth for small errors. Thus, the position feedback as the bandwidth parameter must be a function of the system error $e\left(t\right)$.
- (c)
- For no overshoot in the system response, damping should be adjusted continuously as a function of $e\left(t\right)$. ${K}_{p}\left(e,t\right)$ and ${K}_{v}\left(e,t\right)$ are designed such that they yield a small damping ratio with a large bandwidth for large errors, and a large damping ratio with small bandwidth for small errors.

## 4. Simulation Study and Results

## 5. Discussion and Conclusions

- ○
- The dynamic pole motion approach based on the g-plane is effective to control the NLTV hybrid soft leg systems.
- ○
- The dynamic Routh’s stability criteria can quickly confirm the instability of the NLTV hybrid soft leg system.
- ○
- The E-BAC can control an unstable state of the NLTV hybrid soft leg system to quickly get back to a stable state of the system without any overshoot.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Comparison of six-legged insect and robot: (

**a**) a biological six-legged ant and schematics of its hinder leg, (

**b**) schematics of a rigid hexapedal robot and the structure of rigid leg with motors, and (

**c**) schematics of a hybrid soft hexapedal robot and the structure of its hybrid soft leg with a motor and a spring. The same colors in 2D diagram represent the same segments of the leg (some segments in a biological insect leg are excluded in a robot leg). The blue arrow and the gray column at coxa joint in 2D diagram represent the direction of motion and the body, respectively.

**Figure 2.**A schematic diagram of the mechanism of a motorized hybrid soft leg illustrated in Figure 1c. The mechanical structure consists of a single link manipulator and a flexible spring joint.

**Figure 3.**Block diagram of the hybrid soft leg with rigid links and a flexible joint (spring). The system has both linear and nonlinear feedback.

**Figure 4.**Having the stability region, a three-dimensional g-plane consists of a real part $\sigma \left(t\right)$, imaginary part $j\omega \left(t\right)$, and time-dependent part. The time-dependent part can be time $t$, error $e\left(t\right)$, or state variable $\mathit{x}\left(t\right)$.

**Figure 5.**Graphical representation of the stable region of the dynamic characteristic equation, ${g}^{4}\left(t\right)+2{g}^{3}\left(t\right)+\left(3+\psi \left(t\right)\right){g}^{2}\left(t\right)+\left(2+\psi \left(t\right)\right)g\left(t\right)+\psi \left(t\right)=0$. For simplicity of graphical representation, a specific time t is applied for $\psi \left(t\right)$.

**Figure 6.**The sketch of dynamic pole motion (DPM) of the hybrid soft leg without a controller: (

**a**) two-dimensional representation with $\sigma \left(t\right)$- and $j\omega \left(t\right)$-axes and (

**b**) three-dimensional representation adding a ${x}_{3}\left(t\right)$-axis. The arrows in (

**a**) indicate the paths of pole movements.

**Figure 7.**Schematics of an error-based adaptive controller (E-BAC): ${x}_{4}\left(t\right)={\dot{x}}_{3}\left(t\right)$, and the change of the slopes of ${K}_{p}\left(e,t\right)$ and ${K}_{v}\left(e,t\right)$ curves. For various values of α and β, the directions of the arrows indicate the increasing values of $\alpha $ and $\beta $.

**Figure 8.**Dynamic pole motion of the controlled hybrid soft leg and the controlled system response by error change with the gains of ${K}_{pf}=1.1$, ${K}_{vf}=3.5$, $\alpha =1$, and $\beta =2$: (

**a**) 2D plot, (

**b**) 3D plot, and (

**c**) the system response to the step input. The arrows indicate the direction of pole motions.

**Figure 9.**Dynamic pole motions of the controlled hybrid soft leg and the controlled system response by error change with the gains of (

**a**) ${K}_{pf}=1.5$, ${K}_{vf}=3.5$, $\alpha =1$, and $\beta =2$, and (

**b**) ${K}_{pf}=0.1$, ${K}_{vf}=3.5$, $\alpha =1$, and $\beta =2$. The arrows indicate the direction of pole motions.

**Figure 10.**Dynamic pole motions of the controlled hybrid soft leg and the controlled system response by error change with the gains of (

**a**) ${K}_{pf}=1.5$, ${K}_{vf}=4$, $\alpha =1$, and $\beta =2$, and (

**b**) ${K}_{pf}=0.1$, ${K}_{vf}=0.3$, $\alpha =1$, and $\beta =2$. The arrows indicate the direction of pole motions.

**Figure 11.**Dynamic pole motions of the controlled hybrid soft leg and the controlled system response by error change with the gains of (

**a**) ${K}_{pf}=1.1$, ${K}_{vf}=0.3$, $\alpha =10$, and $\beta =2$, and (

**b**) ${K}_{pf}=1.1$, ${K}_{vf}=0.3$, $\alpha =0.1$, and $\beta =2$. The initial system responses were slightly distinct with different peaks, but the rest of the response remained similar. The arrows indicate the direction of pole motions.

**Figure 12.**Dynamic pole motions of the controlled hybrid soft leg and the controlled system response by error change with the gains of (

**a**) ${K}_{pf}=1.1$, ${K}_{vf}=0.3$, $\alpha =1$, and $\beta =20$, and (

**b**) ${K}_{pf}=1.1$, ${K}_{vf}=0.3$, $\alpha =1$, and $\beta =0.2$. The system responses were slightly distinct with different oscillating shapes (higher $\beta $ presented sharper peaks), but the rest of the response remained similar. The arrows indicate the direction of pole motions.

**Figure 13.**Dynamic pole motion of the controlled hybrid soft leg and the controlled system response by error change with the gains of ${K}_{pf}=0.25$, ${K}_{vf}=0.5$, $\alpha =2$, and $\beta =5$: (

**a**) 2D plot, (

**b**) 3D plot, and (

**c**) the system response to the step input. The arrows indicate the direction of pole motions.

**Figure 14.**Schematic block diagram of a compensated system of the motorized hybrid soft leg with a compensator and E-BAC.

**Figure 15.**Dynamic pole motion of the controlled compensated hybrid soft leg and the system response by error change with the gains of ${K}_{pf}=210$, ${K}_{vf}=31$, $\alpha =2$, and $\beta =0.5$: (

**a**) 2D plot, (

**b**) 3D plot, and (

**c**) the system response to the step input ($\zeta $: damping ratio and ${\omega}_{BW}$: bandwidth in Hz). The arrows in (

**a**,

**b**) indicate the direction of pole motions.

**Figure 16.**Variations of properties of the controlled compensated system: (

**a**) dynamic damping ratio $\zeta \left(t\right)$ (from 0.35 to 1) with respect to error and time (the dot line is the projection plot of $\zeta \left(t\right)$ on error-damping ratio plane), and (

**b**) 3D sketch of dynamic magnitude plot of the system and dynamic bandwidth ${\omega}_{BW}\left(t\right)$ (yellow curve, from 35.62 Hz to ~9 Hz) at each time interval (the dot line is the projection plot of ${\omega}_{BW}\left(t\right)$ on frequency–time plane).

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**MDPI and ACS Style**

Song, K.-Y.; Behzadfar, M.; Zhang, W.-J.
A Dynamic Pole Motion Approach for Control of Nonlinear Hybrid Soft Legs: A Preliminary Study. *Machines* **2022**, *10*, 875.
https://doi.org/10.3390/machines10100875

**AMA Style**

Song K-Y, Behzadfar M, Zhang W-J.
A Dynamic Pole Motion Approach for Control of Nonlinear Hybrid Soft Legs: A Preliminary Study. *Machines*. 2022; 10(10):875.
https://doi.org/10.3390/machines10100875

**Chicago/Turabian Style**

Song, Ki-Young, Mahtab Behzadfar, and Wen-Jun Zhang.
2022. "A Dynamic Pole Motion Approach for Control of Nonlinear Hybrid Soft Legs: A Preliminary Study" *Machines* 10, no. 10: 875.
https://doi.org/10.3390/machines10100875