# An Inverse Dynamics-Based Control Approach for Compliant Control of Pneumatic Artificial Muscles

## Abstract

**:**

## 1. Introduction

- 1.
- In many applications, the controlled range of motion (ROM) is quite limited, for instance −20 to +20 degrees which is not enough for many rehabilitation exercises.
- 2.
- Compliance with external forces and load variation is very specific to implementations.
- 3.
- PAM behavior is estimated as direct input–output, single-input single-output (SISO) models in terms of pressure and force but muscle contraction length is interpreted in model parameter approximations.

- 1.
- Larger controlled ROM up to 90 degrees.
- 2.
- Better compliance to external forces and load variation.
- 3.
- Estimation of PAM dynamic behavior, independent of application mechanism.

## 2. Materials and Methods

#### 2.1. DynamicsBehaviorAnalysis for PAM

_{s}. It is important to express that in data acquisition, muscle full contraction (25%) and full extension (5%) ranges havebeen reached. The hardware structure of the testbed is illustrated in Figure 5. Explanations for components and range of physical quantities are given in Table 1 and Table 2 successively.

_{w}) is the sum of the applied load and load arm reflected weight. In order to estimate the net effective load force applied to PAM (F

_{L}), we use the moment calculation around the load arm rotation joint. Moments around the rotation joint link are equal to each other as given in (1), (2), and (3).

#### 2.2. Inverse Dynamics Estimator-Based Feedback Control Scheme

#### 2.3. NARX for Inverse Dynamics Estimation

#### 2.4. MISO NARX Structure for Inverse Dynamics Learning

#### 2.5. Training Method for MISO NARX

_{1}and y

_{2}are column vectors of measured output and estimated MISO NARX output, respectively. Both vectors have a length of N, as data size. y

_{2}is also a function of NARX parameter vector w. In order to minimize the cost function E(w), a gradient operator G is defined as follows:

#### 2.6. Software Implementation of MISO NARX

#### 2.7. The Control System Implementation

_{p}= 150, T

_{i}= 0.24, T

_{d}= 0.06. After tuning pressure loop PID, step and sinusoidal tracking responses are obtained which is given in Figure 13. Position loop PID controller is tuned using the Ziegler–Nichols cyclic oscillation method. The PID parameters are estimated as K

_{p}= 0.6, T

_{i}= 0.05, T

_{d}= 0.01. Parameters were kept fixed during operations.

## 3. Results

#### 3.1. Regular Trajectory Tracking Results with NARX Estimation Active Control

#### 3.2. Position Trajectory Tracking Responses at Different Frequencies

#### 3.3. Performance Comparison Results: PID-Only versus NARX Estimation Active

#### 3.4. Results Occurred in Transition between NARX Deactivation and Reactivation Modes

#### 3.5. Results Demonstrating Compliant Operation Capability of the System

## 4. Discussion

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Festo pneumatic artificial muscle and dynamic characteristics [9].

**Figure 13.**Pressure loop PID responses after tuning: (

**a**) step response and (

**b**) 0.25 Hz sinusoidal reference.

**Figure 14.**Overview of NARX estimation-based control response at 0.1 Hz sinusoidal trajectory tracking operation. I: isotonic extension, II: isotonic flexion, and III: quasi-isometric contraction. For I F = 229.5 N, II F = 263.1 N, and III F = 315.7 N.

**Figure 15.**Overview of NARX estimation-based control response at step reference trajectory tracking operation.

**Figure 16.**Position trajectory responses when NARX estimation-based control is active: (

**a**) step reference, (

**b**) 0.1 Hz sinusoidal reference, (

**c**) 0.25 Hz sinusoidal reference, and (

**d**) 0.50 Hz sinusoidal reference.

**Figure 18.**Position errors for sinusoidal trajectories when NARX estimation-based control is active.

**Figure 19.**Performance comparison results for 0.25 Hz sinusoidal reference position tracking: (

**a**) PID-only operation and (

**b**) NARX active operation.

**Figure 20.**Performance comparison results for step reference position tracking: (

**a**) PID-only operation and (

**b**) NARX active operation.

**Figure 21.**Control system response during NARX activation–deactivation transitions at 0.25 Hz sinusoidal position trajectory.

**Figure 22.**Dynamic external load addition and removal responses at 0.10 Hz sinusoidal position trajectory.

**Figure 23.**Dynamic external load addition and removal performance comparison results for 0.25 Hz sinusoidal reference position tracking: (

**a**,

**b**) PID-only operation and (

**c**,

**d**) NARX active operation responses.

**Figure 24.**External load addition and removal performance comparison results: (

**a**,

**b**) for 0.50 Hz sinusoidal reference position tracking and (

**c**,

**d**) for step reference position tracking.

**Figure 25.**Impulsive force load application and overall system response at 0.1 Hz sinusoidal trajectory tracking operation.

**Figure 26.**Impulsive force load application and system responses at 0.25 Hz (

**a**,

**b**)and 0.50 Hz sinusoidal trajectory tracking (

**c**,

**d**).

Item | Specifications |
---|---|

Control Unit and I/O Interface Circuits | Atmel, Arm Cortex-M3 CPU, Signal Amplifier, and Valve Drive Circuits |

Encoder for Arm Position Angle (Enc) | Bourne, AMS22S5A1, 0.1 deg resolution |

Pneumatic Artificial Muscle (PAM) | Festo, DMSP 20–250, 0–6 bar, L: 250 mm |

Pressure Sensor (PS) | Honeywell, 24PCFF, 0–100 psi |

Proportional Directional Control Valve | Festo, MPYE-5-M5 |

Force Sensor (LD) | Zemic, H3-P3 load cell, 0–100 kg |

Item | Description | Range/Value |
---|---|---|

θ_{m} | PAM testbed load arm position angle | 120–230 deg |

θ_{min} | Testbed load arm minimum angle | 120 deg |

L_{m} | PAM dynamic length | 200–257 mm |

L_{min} | PAM minimum operation length | 200 mm |

L_{mcontr} | PAM contraction length: (L_{m}−L_{min}) | 0–57 mm |

Ω | Radial coefficient for testbed load arm | 29.5 |

P_{m} | PAM-applied pressure | 0–690 kPA |

F_{m} | PAM testbed measured total force | 0–1000 N |

F_{w} | Total weight (applied load + arm) | 2.83–137.5 N |

F_{L} | Effective load force applied to PAM | 20.6–1000 N |

K_{g} | Testbed load arm lever gain: l_{2}/l_{1} | 7.28 |

γ | Load vertical inclination angle | 0–50 deg |

Symbol | Meaning |
---|---|

r(k) | Position control loop reference value |

θ_{m}(k)
| PAM testbed arm position angle |

e_{1}(k)
| Position control loop error |

p_{r}(k)
| Pressure loop reference value from position controller |

p_{est}(k)
| Pressure feedback value from inverse dynamics estimator |

e_{2}(k) | Pressure control loop error |

u(k) | Pressure controller output |

y(k) | PAM testbed output |

p_{m}(k)
| PAM-applied pressure |

f_{m}(k)
| PAM testbed total force (PAM generated + load) |

Symbol | Meaning |
---|---|

u_{1}(k)
| First exogenous input from system via measurement (muscle dynamic length) |

u_{2}(k)
| Second exogenous input from system via measurement (muscle total force) |

y_{1}(k)
| Output feedback from system via measurement (pressure) |

y_{2}(k)
| NN-NARX estimated output to be used for control system feedback (pressure) |

q^{−1} | Time delay operator |

x_{i} | ith element of input vector |

b_{1} | Bias vector for hidden layer nodes |

Σ | Node summation function |

ψ | Activation function for hidden layer |

w^{1}_{j,i} | Weight between input nodei and hidden layer node j |

λ_{j} | Output of hidden layer node j |

w^{2}_{o,j} | Weight between hidden node j and output layer node o |

b_{2} | Bias vector for output layer node |

Φ | Activation function for output layer |

**Table 5.**Numerical values for critical points marked in Figure 21.

Item | Point X | Point Y |
---|---|---|

Time (sec) | 13.83 | 45.88 |

Force (N) | 290.3 | 270.9 |

Muscle Length (mm) | 238.9 | 216.2 |

Applied Pressure (kPa) | 241.2 | 559.4 |

Pressure Control Setpoint (kPa) | 241.3 | 558.0 |

Position Angle (deg) | 195.6 | 158.3 |

Position Control Setpoint (deg) | 180.5 | 157.1 |

**Table 6.**Numerical values for critical points marked in Figure 22.

Item | Point X | Point Y |
---|---|---|

Time (sec) | 20.59 | 49.24 |

Force (N) | 269.9 | 303.5 |

Muscle Length (mm) | 227.7 | 242.1 |

Applied Pressure (kPa) | 299.2 | 154.7 |

Pressure Control Setpoint (kPa) | 317.2 | 147.9 |

Position Angle (deg) | 173.8 | 201.5 |

Position Control Setpoint (deg) | 169.6 | 205 |

**Table 7.**Numerical values for critical points in Figure 25.

Item | Point A | Point B | Point C | Point D |
---|---|---|---|---|

Time (sec) | 21.85 | 37.85 | 53.51 | 78.56 |

Force (N) | 497.5 | 507.9 | 504.7 | 515.8 |

Muscle Length (mm) | 223.2 | 241.3 | 221.7 | 233.1 |

Applied Pressure (kPa) | 482.7 | 245.7 | 493.5 | 343.4 |

Pressure Control Setpoint (kPA) | 530.9 | 248.2 | 551.6 | 351.6 |

Position Angle (deg) | 161.5 | 197.5 | 156.4 | 183.5 |

Position Control Setpoint (deg) | 147.5 | 195.6 | 156.1 | 184.1 |

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**MDPI and ACS Style**

Baysal, C.V.
An Inverse Dynamics-Based Control Approach for Compliant Control of Pneumatic Artificial Muscles. *Actuators* **2022**, *11*, 111.
https://doi.org/10.3390/act11040111

**AMA Style**

Baysal CV.
An Inverse Dynamics-Based Control Approach for Compliant Control of Pneumatic Artificial Muscles. *Actuators*. 2022; 11(4):111.
https://doi.org/10.3390/act11040111

**Chicago/Turabian Style**

Baysal, Cabbar Veysel.
2022. "An Inverse Dynamics-Based Control Approach for Compliant Control of Pneumatic Artificial Muscles" *Actuators* 11, no. 4: 111.
https://doi.org/10.3390/act11040111