# Tackling Modeling and Kinematic Inconsistencies by Fixed Point Iteration-Based Adaptive Control

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## Abstract

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## 1. Introduction

- The Lyapunov function-based design is not a simple algorithm that can be learned. It requires creative, mathematically well-educated designers. From the failure to find the appropriate Lyapunov function, no conclusions can be drawn regarding the solvability of the given problem, though for a large class of problems, appropriate Lyapunov function candidates are suggested for use (e.g., [15]).
- It usually guarantees normal or asymptotic convergence of a scalar norm made of various error components that individually have physical interpretations and significance and should be driven to zero monotonically. However, the various components of this composite norm do not converge to zero monotonically, even if this norm itself monotonically vanishes.
- Normally, complicated model terms must be precisely computed or at least estimated when this method is used.

## 2. Related Work

**which is a simple structural issue**, the following paper provides a good example. In [32], a relatively simple system of the form of $\dot{x}=Ax\left(t\right)+bu\left(t\right)+bf(x\left(t\right),t)$ with state vector $x\in {\mathbb{R}}^{n}$, constant matrices with unknown elements $A\in {\mathbb{R}}^{n\times n}$, $b\in {\mathbb{R}}^{n\times 1}$, and an unknown bounded function $f(x,t)$ that represents the system non-linearities, model uncertainties, and the external disturbances was controlled. The authors provided a Lyapunov function-based solution using three special assumptions, five remarks, and two theorems, the mathematical details of which were expressed over pages 976–984 (approximately along 9 pages), and only the rest contained simulation results for a simple 2-degree-of-freedom system. Different Lyapunov functions were added in the complicated proof, which is a typical technique. The paper also serves as an example that normally, “observers” have to be developed for a Lyapunov function-based approach (Shortcoming 3).

**structural issue**. Instead of prescribing the behavior of some function $V\left(t\right)\in \mathbb{R}$, the time-dependence of certain components ${e}_{i}\left(t\right)$ is directly controlled. (There is no need for using and computing any Lyapunov function).

## 3. Problem Formulation and Development of the Design Structure

**the actual desired value**${x}^{des}\equiv {q}^{\left(order\right)des}$;

**the deformed value in the previous control cycle**${q}^{\left(order\right)def}$; and the

**the response obtained for the previously applied control input**${q}^{\left(order\right)}$ that are available in the given time instant due to the delay. Because, in general, it is difficult to determine the appropriate value of parameter $\alpha $, various constructions were suggested for the deformation in [16,42,43]. They have various parameters by the setting of which the convergence of the iteration can be guaranteed if the system model allows it mathematically.

**modeling and kinematic inconsistencies**concerning the application of the fixed point iteration-based adaptive control for underactuated systems. These problems are revealed and tackled in the sequel.

#### 3.1. Suggested Controller Structures for Underactuated Systems without and with Increased Relative Order

#### 3.2. Control of a Corrupted 3D Puma-Type Robot Arm (Problem without Increasing the Relative Order of the Controller)

- Fill it in with the parameters of the available approximate model and the actually measured (noise-filtered) values of the variables ${q}_{1}^{s}$, ${q}_{2}^{s}$, ${q}_{3}^{s}$, and ${\dot{q}}_{1}^{s}$, ${\dot{q}}_{2}^{s}$, ${\dot{q}}_{3}^{s}$;
- Substitute ${Q}_{3}=0$ and ${\ddot{q}}_{3}^{des}$ into Equation (8d) and calculate ${\ddot{q}}_{2}^{des}$;
- Substitute ${\ddot{q}}_{2}^{des}$ and ${\ddot{q}}_{3}^{des}$ into Equation (8c) and calculate ${Q}_{2}^{appr}$;

#### 3.2.1. Simulations without Measurement Noise

#### 3.2.2. Simulations with Measurement Noise

#### 3.3. Control of Coupled Non-Linear Springs Increased to Relative Order 3

**can be set independently of**${q}_{2}\left(t\right)$, ${\dot{q}}_{2}\left(t\right)$, and ${\ddot{q}}_{2}\left(t\right)$. In our case ${\stackrel{\u20db}{q}}_{2}\left(t\right)$

**cannot be set independently of these values**: the system’s dynamic model determines ${\stackrel{\u20db}{q}}_{2}\left(t\right)$ by its lower order derivatives, and by ${Q}_{1}$.

**This easily may lead to inconsistent kinematic requirements if the model parameters are not exactly known and, e.g., the stiffness of the non-linear spring in our case can be divergent function of the spring’s length.**Because of this, the controller’s structure outlined in Figure 13 is suggested to tackle this problem.

#### 3.3.1. Simulations for the Affine Model without Measurement Noise

- While the tracking error remained in the same range, the illusion of the MRAC control, i.e., that on the basis of the kinematic prescriptions the dynamics of the affine model was controlled was well provided by the solution. The affine model was used by the external kinematic loop for the computation of the necessary control force, and precise trajectory tracking was achieved. The affine model’s force need was approximately in the range $[-120,-30]\phantom{\rule{0.166667em}{0ex}}\mathrm{N}$ following the transient phase, while the actual control force varied within the range $[-120,5]\phantom{\rule{0.166667em}{0ex}}\mathrm{N}$.
- Due to the adaptivity the duration of the initial “transient swinging phase” of the control force was considerably reduced.
- The duration of the initial transient swinging in the position of mass point 1 was considerably reduced due to the adaptivity.
- Furthermore, the application of adaptivity considerably reduced the amplitude of the control force in the initial transient phase of the motion from the range $[-4000,4500]\phantom{\rule{0.166667em}{0ex}}\mathrm{N}$ to $[-3000,2500]\phantom{\rule{0.166667em}{0ex}}\mathrm{N}$.

#### 3.3.2. Simulations for the Affine Model with Measurement Noise

#### 3.3.3. Simulations for the Complex Order 3 Model without Measurement Noise

`function Q_3rdOrdModel(q2_pppDes,q1,q2,q1_p,q2_p,q2_pp)`

`local casual`

`casual=m2a*q2_pppDes-F2xa(q2-q1)*(q2_p-q1_p)`

`casual-=F2ya*(q2_pp-ga-F1a(q1,q1_p)/m1a+F2a(q2-q1,q2_p-q1_p)/m1a)`

`return -m1a*casual/F2ya`

`end`

`q1`and

`q2`stand for ${q}_{1}$ and ${q}_{2}$,

`q1_p`and

`q2_p`is in the role of ${\dot{q}}_{1}$ and ${\dot{q}}_{2}$, and

`q2_pp`represents ${\ddot{q}}_{2}$,

`m1a`,

`m2a`, and

`ga`denote the approximate parameter values of ${m}_{1}$, ${m}_{2}$, and g,

`F2xa`and

`F2ya`correspond to the functions ${F}_{2x}$ and ${F}_{2y}$ in Equation (12) with the approximate model parameters. In a similar manner, the functions

`F1a`and

`F2a`are the counterparts of the functions in Equation (11).

#### 3.3.4. Simulations for the Complex Order 3 Model with Measurement Noise

#### 3.4. Summary of the Innovation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The structure of the higher order Fixed Point Iteration-based Adaptive Controller for fully actuated system (after [16]).

**Figure 2.**Symbolic description of the abstract rotations in the adaptive deformation (after [44]).

**Figure 3.**The scheme of the Fixed Point Iteration-based MRAC controller for fully actuated systems (after [46]).

**Figure 4.**The kinematic structure of the 3D robot arm in the “home position” (after [50]).

**Figure 6.**The nominal and realized motion for ${\mathsf{\Lambda}}_{1}={\mathsf{\Lambda}}_{3}=4.0\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$: (

**a**) Settling ${\dot{q}}_{1}\left(t\right)$. (

**b**) Small fluctuations in ${\dot{q}}_{1}\left(t\right)$. (

**c**) Settling ${q}_{3}\left(t\right)$. (

**d**) Small fluctuations in ${q}_{3}\left(t\right)$.

**Figure 7.**Results for ${\mathsf{\Lambda}}_{1}={\mathsf{\Lambda}}_{3}=4.0\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$: (

**a**) The control force ${Q}_{1}\left(t\right)$. (

**b**) The control force ${Q}_{2}\left(t\right)$. (

**c**) The motion of axle ${q}_{2}\left(t\right)$. (

**d**) Angle of the adaptive abstract rotation.

**Figure 8.**The nominal and realized motion for ${\mathsf{\Lambda}}_{1}={\mathsf{\Lambda}}_{3}=8.0\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$: (

**a**) Settling ${\dot{q}}_{1}\left(t\right)$. (

**b**) Small fluctuations in ${\dot{q}}_{1}\left(t\right)$. (

**c**) Settling ${q}_{3}\left(t\right)$. (

**d**) Small fluctuations in ${q}_{3}\left(t\right)$.

**Figure 9.**Results for ${\mathsf{\Lambda}}_{1}={\mathsf{\Lambda}}_{3}=8.0\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$: (

**a**) The control force ${Q}_{1}\left(t\right)$. (

**b**) The control force ${Q}_{2}\left(t\right)$. (

**c**) The motion of axle ${q}_{2}\left(t\right)$. (

**d**) Angle of the adaptive abstract rotation.

**Figure 10.**The nominal and realized motion under measurement noises: (

**a**) Settling ${\dot{q}}_{1}\left(t\right)$. (

**b**) Small fluctuations in ${\dot{q}}_{1}\left(t\right)$. (

**c**) Settling ${q}_{3}\left(t\right)$. (

**d**) Small fluctuations in ${q}_{3}\left(t\right)$.

**Figure 11.**(

**a**) The control force ${Q}_{1}\left(t\right)$. (

**b**) The control force ${Q}_{2}\left(t\right)$. (

**c**) The motion of axle ${q}_{2}\left(t\right)$. (

**d**) Angle of the adaptive abstract rotation.

**Figure 14.**Control of the affine model with $\mathfrak{A}=7.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ without adaptivity: (

**a**) Trajectory tracking. (

**b**) Trajectory tracking error. (

**c**) The motion of mass point 1. (

**d**) The control force.

**Figure 15.**Comparison of the motion of ${q}_{1}$ for the adaptive and the non-adaptive control for $\mathfrak{A}=7.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$. (

**a**) Non-adaptive motion of ${q}_{1}$ in the first second. (

**b**) Non-adaptive motion of ${q}_{1}$ in the rest of the trajectory. (

**c**) Adaptive motion of ${q}_{1}$ in the first second. (

**d**) Adaptive motion of ${q}_{1}$ in the rest of the trajectory.

**Figure 16.**Comparison of the control force ${Q}_{1}$ for the adaptive and the non-adaptive control for $\mathfrak{A}=7.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$. (

**a**) Non-adaptive motion in the first second. (

**b**) Non-adaptive motion in the rest of the trajectory. (

**c**) Adaptive motion in the first second. (

**d**) Adaptive motion in the rest of the trajectory.

**Figure 17.**Control of the affine model with $\mathfrak{A}=7.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ and adaptivity: (

**a**) Trajectory tracking. (

**b**) Trajectory tracking error.

**Figure 18.**Comparison of the motion of ${q}_{1}$ for the adaptive and the non-adaptive control for $\mathfrak{A}=5.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$. (

**a**) Non-adaptive motion of ${q}_{1}$ in the first second. (

**b**) Non-adaptive motion of ${q}_{1}$ in the rest of the trajectory. (

**c**) Adaptive motion of ${q}_{1}$ in the first second. (

**d**) Adaptive motion of ${q}_{1}$ in the rest of the trajectory.

**Figure 19.**Comparison of the control force ${Q}_{1}$ for the adaptive and the non-adaptive control for $\mathfrak{A}=5.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$. (

**a**) Non-adaptive motion in the first second. (

**b**) Non-adaptive motion in the rest of the trajectory. (

**c**) Adaptive motion in the first second. (

**d**) Adaptive motion in the rest of the trajectory.

**Figure 20.**Control of the affine model with $\mathfrak{A}=5.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ (

**a**) Non-adaptive trajectory tracking. (

**b**) Non-adaptive trajectory tracking error. (

**c**) Adaptive trajectory tracking. (

**d**) Adaptive trajectory tracking error.

**Figure 21.**Comparison of the motion of ${q}_{1}$ for the adaptive and the non-adaptive control for $\mathfrak{A}=0.4$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ (

**a**) Non-adaptive motion of ${q}_{1}$ in the first two seconds. (

**b**) Non-adaptive motion of ${q}_{1}$ in the rest of the trajectory. (

**c**) Adaptive motion of ${q}_{1}$ in the first three seconds. (

**d**) Adaptive motion of ${q}_{1}$ in the rest of the trajectory.

**Figure 22.**Comparison of the control force ${Q}_{1}$ for the adaptive and the non-adaptive control for $\mathfrak{A}=0.4$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ (

**a**) Non-adaptive motion in the first two seconds. (

**b**) Non-adaptive motion in the rest of the trajectory. (

**c**) Adaptive motion in the first two seconds. (

**d**) Adaptive motion in the rest of the trajectory.

**Figure 23.**Control of the affine model with $\mathfrak{A}=0.4$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ (

**a**) Non-adaptive trajectory tracking. (

**b**) Non-adaptive trajectory tracking error. (

**c**) Adaptive trajectory tracking. (

**d**) Adaptive trajectory tracking error.

**Figure 24.**Control of the affine model with $\mathfrak{A}=7.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ without adaptivity under measurement noises: (

**a**) Trajectory tracking. (

**b**) Trajectory tracking error. (

**c**) The motion of mass point 1. (

**d**) The control force (zoomed in excerpt).

**Figure 25.**Control of the affine model with $\mathfrak{A}=7.0$$\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ with adaptivity under measurement noises: (

**a**) Trajectory tracking. (

**b**) Trajectory tracking error. (

**c**) The motion of mass point 1. (

**d**) The control force (zoomed in excerpt).

**Figure 26.**The control structure of the underactuated coupled springs using the complex order 3 model: the input of the “Complex Reference Model” corresponds to the function in Equation (19). For the sake of clarity the figure does not contain each input.

**Figure 27.**Control of the complex model without measurement noises: (

**a**) Non-adaptive trajectory tracking. (

**b**) Non-adaptive trajectory tracking error. (

**c**) Adaptive trajectory tracking. (

**d**) Adaptive trajectory tracking error.

**Figure 28.**Comparison of the control force ${Q}_{1}$ for the adaptive and the non-adaptive control for the complex model without measurement noises. (

**a**) Non-adaptive motion in the first second. (

**b**) Non-adaptive motion in the rest of the trajectory. (

**c**) Adaptive motion in the first second. (

**d**) Adaptive motion in the rest of the trajectory.

**Figure 29.**Comparison of the motion of ${q}_{1}$ for the adaptive and the non-adaptive control for the complex model without measurement noises (

**a**) Non-adaptive motion of ${q}_{1}$ in the first second. (

**b**) Non-adaptive motion of ${q}_{1}$ in the rest of the trajectory. (

**c**) Adaptive motion of ${q}_{1}$ in the first second. (

**d**) Adaptive motion of ${q}_{1}$ in the rest of the trajectory.

**Figure 30.**Control of the complex model with adaptivity under measurement noises: (

**a**) Trajectory tracking. (

**b**) Trajectory tracking error. (

**c**) The motion of mass point 1. (

**d**) The control force (zoomed in excerpt).

**Table 1.**The kinematic and dynamic parameters of the robot arm (after [50]).

Parameter | Measurement Unit | Exact Value | Approx. Value |
---|---|---|---|

Length of Link 2: ${L}_{2}$ | [m] | 1.0 | 1.0 |

Length of link 3: ${L}_{3}$ | [m] | 2.0 | 2.0 |

Mass of link 2: ${m}_{2}$ | [kg] | 10.0 | 15.0 |

Mass of link 3: | [kg] | 20.0 | 25.0 |

Gravitational acceleration g | $\left[\mathrm{m}\xb7{\mathrm{s}}^{-2}\right]$ | 9.81 | 9.81 |

Moment of inertia of link 1: $\mathsf{\Theta}$ | $\left[\mathrm{kg}\xb7{\mathrm{m}}^{2}\right]$ | 50.0 | 60.0 |

Parameter | Measurement Unit | Value |
---|---|---|

Digital time resolution: $\delta t$ | [s] | 10${}^{-3}$ |

Noise filtering parameter ${\lambda}_{s}$ | $\left[{\mathrm{s}}^{-1}\right]$ | 10^{3} |

Adaptive interpolation parameter ${\lambda}_{a}$ | [nondimensional] | 0.9 |

Trajectory tracking parameter ${\mathsf{\Lambda}}_{1}$ | $\left[{\mathrm{s}}^{-1}\right]$ | 4.0 and 8.0 |

Trajectory tracking parameter ${\mathsf{\Lambda}}_{3}$ | $\left[{\mathrm{s}}^{-1}\right]$ | 4.0 and 8.0 |

Norm of augmented vectors ${R}_{a}$ | [N · m] | 10^{4} |

$\sigma $ of Gaussian noise | [rad] | 0 and 10${}^{-5}$ |

Parameter | Measurement Unit | Exact Value | Approx. Value |
---|---|---|---|

Length of spring 1: ${L}_{1}$ | [m] | 1.0 | 0.8 |

Length of spring 2: ${L}_{2}$ | [m] | 1.5 | 1.3 |

Mass of point 1: ${m}_{1}$ | [kg] | 1.5 | 2.0 |

Mass of point 2: ${m}_{2}$ | [kg] | 2.0 | 2.5 |

Spring stiffness 1: ${k}_{1}$ | $\left[\mathrm{N}\xb7{\mathrm{m}}^{-1}\right]$ | 100.0 | 90.0 |

Spring stiffness 2: ${k}_{2}$ | $\left[\mathrm{N}\xb7{\mathrm{m}}^{-1}\right]$ | 200.0 | 190.0 |

Gravitational acceleration: g | $\left[\mathrm{m}\xb7{\mathrm{s}}^{-2}\right]$ | 9.81 | 9.81 |

Viscous damping 1: ${b}_{1}$ | $\left[\mathrm{N}\xb7\mathrm{s}\xb7{\mathrm{m}}^{-1}\right]$ | 2.5 | 2.0 |

Viscous damping 2: ${b}_{2}$ | $\left[\mathrm{N}\xb7\mathrm{s}\xb7{\mathrm{m}}^{-1}\right]$ | 3.0 | 2.5 |

Affine parameter 1: $\mathfrak{A}$ | $\left[\mathrm{N}\xb7{\mathrm{s}}^{3}\xb7{\mathrm{m}}^{-1}\right]$ | Not applicable | various |

Affine parameter 2: $\mathfrak{B}$ | $\left[\mathrm{N}\right]$ | Not applicable | −15.0 |

Parameter | Measurement Unit | Value |
---|---|---|

Digital time resolution: $\delta t$ | [s] | 10${}^{-3}$ |

Noise filtering parameter ${\lambda}_{s}$ | $\left[{\mathrm{s}}^{-1}\right]$ | 10^{3} |

Adaptive interp. param. ${\lambda}_{a}$ | [nondimensional] | 0.9, 0.05 |

affine, complex | ||

Trajectory tracking parameter $\mathsf{\Lambda}$ | $\left[{\mathrm{s}}^{-1}\right]$ | 4.0 |

Trajectory tracking parameter ${\lambda}_{k}$ | $\left[{\mathrm{s}}^{-1}\right]$ | 12.0 |

Norm of augmented vectors ${R}_{a}$ | [N] | 10${}^{-6}$ |

$\sigma $ of Gaussian noise | [m] | 0 and 10${}^{-6}$ |

Nominal trajectory rise time T | [s] | 3.0 |

Nominal trajectory amplitude ${A}_{o}$ | [m] | $0.2\xb7({L}_{1}+{L}_{2})$ (exact) |

Nominal trajectory deformation factor ${A}_{i}$ | [nondimensional] | 2.0 |

Nominal trajectory circular frequency $\mathsf{\Omega}$ | $\left[{\mathrm{s}}^{-1}\right]$ | 2.0 |

Nominal trajectory shift S | [m] | $({L}_{1}+{L}_{2})$ (exact) |

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## Share and Cite

**MDPI and ACS Style**

Atinga, A.; Tar, J.K.
Tackling Modeling and Kinematic Inconsistencies by Fixed Point Iteration-Based Adaptive Control. *Machines* **2023**, *11*, 585.
https://doi.org/10.3390/machines11060585

**AMA Style**

Atinga A, Tar JK.
Tackling Modeling and Kinematic Inconsistencies by Fixed Point Iteration-Based Adaptive Control. *Machines*. 2023; 11(6):585.
https://doi.org/10.3390/machines11060585

**Chicago/Turabian Style**

Atinga, Awudu, and József K. Tar.
2023. "Tackling Modeling and Kinematic Inconsistencies by Fixed Point Iteration-Based Adaptive Control" *Machines* 11, no. 6: 585.
https://doi.org/10.3390/machines11060585