# Event-Based, Intermittent, Discrete Adaptive Control for Speed Regulation of Artificial Legs

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background and Related Work

## 3. Methods

#### 3.1. Overview of Control

#### 3.2. Event-Based, Intermittent, Discrete Control

#### 3.3. Adaptive Control

#### 3.4. Control of a Pendulum

#### 3.4.1. Model

#### 3.4.2. Discrete-Time Model

#### 3.5. Linearized Models and Adaptations per Swing

#### 3.5.1. One Model/One Measurement/One Adaptation (1Mo-1Me-1Ad) per Period

#### 3.5.2. Two Models/Two Measurements/One Adaptation (2Mo-2Me-1Ad) per Period

#### 3.5.3. Two Models/Two Measurements/Two Adaptations (2Mo-2Me-2Ad) per Period

## 4. Results

#### 4.1. Computer Simulations

#### 4.2. Hardware Setup

#### 4.3. Hardware Experiments

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Key idea behind event-based discrete control: The nominal value of the performance variable ${z}_{0}$ is tracked at event k and $k+1$ spread over time. The nominal values for the control parameters are ${U}_{0}$. When the system is perturbed, the performance variable for the perturbation is ${z}_{k}=z\ne {z}_{0}$. Our goal is to use the measurement at k, $z-{z}_{0}$ to find the correction in control ${U}_{0}+\delta U$ such that at the event $k+1$, the performance variable $z(k+1)={z}_{0}$. Note that the time between events $k+1$ and k is typically of the order of the time scale of the system.

**Figure 2.**Simple pendulum: This consists of a massless rod of length ℓ with a point mass m at the end G. The pendulum is controlled by a torque T at the hinge joint H. Gravity g points downward. The angle between the vertical and pendulum is $\theta $ measured in the counter-clockwise direction.

**Figure 3.**A single swing of a simple pendulum. The pendulum’s motion may be divided into four distinct instants. (

**a**) Instant $i,i+1,\dots $, the pendulum is in the vertically downward position with angular velocity in the counter-clockwise direction, ${\dot{\theta}}_{i},{\dot{\theta}}_{i+1}$, (

**b**) instant $i+0.25,i+1.25,\dots $, the pendulum is in the extreme position ${\theta}_{i+0.25},{\theta}_{i+1.25},\dots $, (

**c**) instant $i+0.5,i+1.5,\dots $, the pendulum is in the vertically downward position with angular velocity in the clockwise direction, ${\dot{\theta}}_{i+0.5},{\dot{\theta}}_{i+1.5}$, and (

**d**) instant $i+0.75,i+1.75,\dots $, the pendulum is in the extreme position ${\theta}_{i+0.75},{\theta}_{i+1.75},\dots $

**Figure 4.**Comparison of adaptive to non-adaptive control: (

**a**) 1Mo-1Me-1Ad, (

**b**) 2Mo-2Me-1Ad, (

**c**) 2Mo-2Me-2Ad, and (

**d**) all adaptive controllers, where Mo is the number of models per swing, Me is the number of measurements per swing, and Ad is the number of adaptations per model.

**Figure 5.**Adaptations of the parameters (

**a**) $\widehat{a}$ and (

**b**) $\widehat{b}$ as a function of the swing number for all three adaptations.

**Figure 7.**Comparing error and torque for five trials for the adaptive and non-adaptive controllers: (

**a**,

**c**) are for added mass of $0.3$ kg; (

**b**,

**d**) are for added mass of $0.5$ kg. The band shows one standard deviation for five trials and the line shows the mean.

**Figure 8.**The control parameters as a function of time. (

**a**,

**c**) show parameters for added mass of $0.3$ kg and (

**b**,

**d**) show parameters for added mass of $0.5$ kg. The solid lines show the adaptive control parameters, where each line corresponds to a trial, and the dashed lines show the parameters for the non-adaptive controller.

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**MDPI and ACS Style**

Echeveste, S.; Hernandez-Hinojosa, E.; Bhounsule, P.A.
Event-Based, Intermittent, Discrete Adaptive Control for Speed Regulation of Artificial Legs. *Actuators* **2021**, *10*, 264.
https://doi.org/10.3390/act10100264

**AMA Style**

Echeveste S, Hernandez-Hinojosa E, Bhounsule PA.
Event-Based, Intermittent, Discrete Adaptive Control for Speed Regulation of Artificial Legs. *Actuators*. 2021; 10(10):264.
https://doi.org/10.3390/act10100264

**Chicago/Turabian Style**

Echeveste, Salvador, Ernesto Hernandez-Hinojosa, and Pranav A. Bhounsule.
2021. "Event-Based, Intermittent, Discrete Adaptive Control for Speed Regulation of Artificial Legs" *Actuators* 10, no. 10: 264.
https://doi.org/10.3390/act10100264