# Abstract Rotations for Uniform Adaptive Control and Soft Modeling of Mechanical Devices

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## Abstract

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## 1. Introduction

- 1.
- For the free system, the two dimensional vectors ${[\ddot{q},0]}^{T}$ and ${[q,\dot{q}]}^{T}$ were augmented to three-dimensional ones of identical Frobenius norms ${[\ddot{q},0,{D}_{1}]}^{T}$ and ${[q,\dot{q},{D}_{2}]}^{T}$ exactly as it was performed in the rotations-based abstract deformations applied in [29]. In the case of the controlled model the three-dimensional vectors ${[Q,0,0]}^{T}$ and ${[q,\dot{q},{\ddot{q}}^{Des}]}^{T}$ were similarly augmented to produce the four dimensional ones of identical norms as ${[Q,0,0,{D}_{1}]}^{T}$ and $[q,\dot{q},{\ddot{q}}^{Des},{D}_{2}]$ in which ${D}_{1}$ and ${D}_{2}$ were the “dummy components” without any physical interpretation. Their role was to guarantee equal norms.
- 2.
- Coarse resolution grids were introduced in ${\mathbb{R}}^{2}$ for the $q,\phantom{\rule{0.166667em}{0ex}}\dot{q}$ values, and in ${\mathbb{R}}^{3}$ for the $q,\phantom{\rule{0.166667em}{0ex}}\dot{q},\phantom{\rule{0.166667em}{0ex}}{\ddot{q}}^{Des}$ values. In the center points of the grids, the appropriate $\ddot{q}$ and Q values were computed from the available exact model of the van der Pol oscillator. Following that, the abstract rotations defined in (7) were calculated that rotated ${[q,\dot{q},{D}_{2}]}^{T}$ into ${[\ddot{q},0,{D}_{1}]}^{T}$, and transformed $[q,\dot{q},{\ddot{q}}^{Des},{D}_{2}]$ into ${[Q,0,0,{D}_{1}]}^{T}$, respectively. Each grid cell was associated with a “neuron” that had the “activation function”. It executed the rotations according to (7), and had the following parameters:
- Its cell-limits as $\{{q}_{min},{q}_{max}\}$, $\{{\dot{q}}_{min},{\dot{q}}_{max}\}$ for the free motion, and $\{{q}_{min},{q}_{max}\}$, $\{{\dot{q}}_{min},{\dot{q}}_{max}\}$, $\{{\ddot{q}}_{min}^{Des},{\ddot{q}}_{max}^{Des}\}$ for the controlled system, respectively;
- The orthogonal unit vectors ${e}_{A}$ and ${e}_{B}$ of which the generator $\mathsf{\Omega}$ of the rotation in (7) can be computed, and the angle of the necessary rotation, $\phi $.

- 3.
- These neurons were arranged in a single layer in which each neuron obtained its input value for the “teaching process” as $\{q,\phantom{\rule{0.166667em}{0ex}}\dot{q},\phantom{\rule{0.166667em}{0ex}}\ddot{q}\}$ for the free system, and $\{q,\phantom{\rule{0.166667em}{0ex}}\dot{q},\phantom{\rule{0.166667em}{0ex}}{\ddot{q}}^{Des},\phantom{\rule{0.166667em}{0ex}}Q\}$ for the dynamic model. If the input signal belonged to the “range of competence” of the given neuron, ${e}_{A}$, ${e}_{B}$, and $\phi $ were computed. During the “normal operation” the neuron used the input for the free system modeling as $\{q,\phantom{\rule{0.166667em}{0ex}}\dot{q}\}$ and $\{q,\phantom{\rule{0.166667em}{0ex}}\dot{q},\phantom{\rule{0.166667em}{0ex}}{\ddot{q}}^{Des}\}$ for “the use for control mode”. If the input data belonged to its range of competence, it computed $O(\mathsf{\Omega},\phi )$, computed the rotated vector $O(\mathsf{\Omega},\phi ){[q,\dot{q},{D}_{1}]}^{T}$ for the free system, and $O(\mathsf{\Omega},\phi ){[q,\dot{q},{\ddot{q}}^{Des},{D}_{1}]}^{T}$ for the control application, and as its output, it provided the first component of the rotated vector that corresponded to the modeled value of $\ddot{q}$ and Q, respectively.
- 4.
- The last layer of the novel neural structure consisted of a single neuron that summarized the calculated outputs. Since the cells’ limits were determined in a way that the model had only disjoint cells, the output of the summarizing layer was the result of the “soft model”.
- 5.
- To reduce the effects of the jumps in the control signal ${Q}_{m}$ at the cell boundaries, the really applied generalized force ${Q}_{s}$ was smoothed by the tracking rule based on a positive constant $\lambda >0$ in (8)$$\left(\right)open="("\; close=")">\lambda +\frac{\mathrm{d}}{\mathrm{d}t}$$
- 6.
- The data representation made it possible to apply real-time modification (“step-by-step learning”) of the neuron’s previously learned parameters as the unit vectors ${e}_{A}$, ${e}_{B}$ and $\phi $ were refreshed according to a learning rule determined by the parameter $\alpha \in [0,1]$ as$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\phi}_{to\phantom{\rule{0.166667em}{0ex}}store}^{New}=\alpha {\phi}_{stored}^{Old}+(1-\alpha ){\phi}_{introduced}^{New}\phantom{\rule{5.0pt}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {e}_{A\phantom{\rule{0.166667em}{0ex}}to\phantom{\rule{0.166667em}{0ex}}store}^{New}=\alpha {e}_{A\phantom{\rule{0.166667em}{0ex}}stored}^{Old}+\beta {e}_{A\phantom{\rule{0.166667em}{0ex}}introduced}^{New}\phantom{\rule{5.0pt}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& S:=\sum _{\ell}{e}_{A\phantom{\rule{0.166667em}{0ex}}store{d}_{\ell}}^{Old}{e}_{A\phantom{\rule{0.166667em}{0ex}}to\phantom{\rule{0.166667em}{0ex}}stor{e}_{\ell}}^{New}\in [-1,1]\phantom{\rule{5.0pt}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \beta =-\alpha S+\sqrt{{\alpha}^{2}{S}^{2}+(1-{\alpha}^{2})}\ge 0\phantom{\rule{5.0pt}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {e}_{B\phantom{\rule{0.166667em}{0ex}}to\phantom{\rule{0.166667em}{0ex}}store}^{New}=\alpha {e}_{B\phantom{\rule{0.166667em}{0ex}}stored}^{Old}+\widehat{\beta}{e}_{B\phantom{\rule{0.166667em}{0ex}}introduced}^{New}\phantom{\rule{5.0pt}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \widehat{S}:=\sum _{\ell}{e}_{B\phantom{\rule{0.166667em}{0ex}}store{d}_{\ell}}^{Old}{e}_{B\phantom{\rule{0.166667em}{0ex}}to\phantom{\rule{0.166667em}{0ex}}stor{e}_{\ell}}^{New}\in [-1,1]\phantom{\rule{5.0pt}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \widehat{\beta}=-\alpha \widehat{S}+\sqrt{{\alpha}^{2}{\widehat{S}}^{2}+(1-{\alpha}^{2})}\ge 0\phantom{\rule{5.0pt}{0ex}}.\hfill \end{array}$$It must be noted that even if ${e}_{A\phantom{\rule{0.166667em}{0ex}}stored}^{Old}$ and ${e}_{B\phantom{\rule{0.166667em}{0ex}}stored}^{Old}$ were orthogonal to each other, the new unit vectors ${e}_{A\phantom{\rule{0.166667em}{0ex}}to\phantom{\rule{0.166667em}{0ex}}store}^{New}$ and ${e}_{B\phantom{\rule{0.166667em}{0ex}}to\phantom{\rule{0.166667em}{0ex}}store}^{New}$ will not be exactly orthogonal ones. Consequently, the new skew symmetric matrix ${\mathsf{\Omega}}^{New}:={e}_{A}^{New}{e}_{B}^{Ne{w}^{T}}-{e}_{B}^{New}{e}_{A}^{Ne{w}^{T}}$ can generate rotations in the form $exp\left(\right)open="("\; close=")">{\phi}^{New}{\mathsf{\Omega}}^{New}$, but, because ${\mathsf{\Omega}}^{Ne{w}^{3}}\ne -{\mathsf{\Omega}}^{New}$, instead of (7) we can state only that$$exp\left(\right)open="("\; close=")">{\phi}^{New}{\mathsf{\Omega}}^{New}$$

- 1.
- The controlled system is an underactuated two degree of freedom construction in which the directly controlled subsystem is dynamically coupled with a non-controllable one acting as “parasite dynamics”.
- 2.
- Instead of the simple CTC control and its robust variable structure/sliding mode-based correction (e.g., [88,89,90]), the “fixed point iteration-based adaptive control scheme” depicted in Figure 1 is applied to compensate the effects of the imprecisions of the coarse grid-based model with the application of the rotations-based adaptive controller announced in [29].
- 3.
- The effects of the measurement noises are investigated and reduced by a smoothing technique that is similar to the solution published in [91].
- 4.
- The computation time of the controller was measured for the hardware and software environment that was used in the simulations.

## 2. The Dynamic Model of the Controlled System

## 3. The Rotational Neural Model Structure Tailored to the Controlled System

## 4. Simulation Results

`Wheel_Applied_Sciences_Noisy.jl`”.

#### 4.1. Comparative Analysis of the Performances of the Neural and the Exact Models

#### 4.2. Estimation of the Computational Time of the Operations in the Control Cycles

`a+=@elapsed b=myfunc(-12.1)`” adds to the content of variable “

`a`” the computational time (in seconds) of the following steps: the function “

`myfunc`” is called with the input argument

`-12.1`and its output is stored in variable “

`b`”. The time-need of a bigger unit as e.g., “

`a+=@elapsed for...end`”, “

`a+=@elapsed while...end`”, etc. can be measured in similar manner.

`Wheel_Applied_Sciences_Noisy_Computing_Time_New.jl`” that utilizes this possibility (also available in the Web) was made in which the time needed for the essential computations within each control cycle was summarized. In these investigations, the parameter settings ${\lambda}_{a}=5\times {10}^{-2}$ and $\mathsf{\Lambda}=0.9\phantom{\rule{0.166667em}{0ex}}{s}^{-1}$ were used, all the other control and model parameters remained invariant. In this program, no computation was made for the operation of the “exact system"-based control. The nominal trajectory had time-dependent, increasing amplitude. During the calculations, only the console running Julia was in operation, a text editor in which the program file was kept opened, and the standard Internet connection was kept running. In the sequel, results obtained for the adaptive and non-adaptive controllers are discussed. Figure 17 reveals a few huge peaks that certainly cannot belong to the essential computations. The important information can be seen in the “zoomed-in excerpts” of the graphs displayed in Figure 18.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

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**Figure 1.**The structure of the fixed point iteration-based adaptive controller for a second-order dynamical system (after [21]).

**Figure 4.**Trajectory tracking for the maximal amplitude and circular frequency of the nominal trajectory for which the adaptive controller was able to use only the originally learned data. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 5.**Trajectory tracking error for the maximal amplitude and circular frequency of the nominal trajectory for which the adaptive controller was able to use only the originally learned data. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 6.**The phase trajectories for the maximal amplitude and circular frequency of the nominal trajectory for which the adaptive controller was able to use only the originally learned data (note that $\delta {q}_{1}\approx 0.628\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}$, and $\delta {\dot{q}}_{1}=2.0\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}\xb7{\mathrm{s}}^{-1}$). (

**a**): Non-adaptive phase trajectory for the controlled variable ${q}_{1}$, (

**b**): adaptive phase trajectory of the controlled variable ${q}_{1}$. (

**c**): Non-adaptive phase trajectory of the coupled mass point. (

**d**): Adaptive phase trajectory of the coupled mass point.

**Figure 7.**The control torque ${Q}_{1}$ for the maximal amplitude and circular frequency of the nominal trajectory for which the adaptive controller was able to use only the originally learned data. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 8.**The operation of the adaptive controller. (

**a**): The desired, deformed, and realized 2nd time-derivatives. (

**b**): The angle of the adaptive abstract rotation.

**Figure 9.**The operation of the adaptive controller in the lack of measurement noises. (

**a**): The desired, deformed, and realized 2nd time-derivatives. (

**b**): The angle of the adaptive abstract rotation.

**Figure 10.**Trajectory tracking for the small amplitude and high circular frequency of the nominal trajectory for which the adaptive controller was able to use only the originally learned data. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 11.**Trajectory tracking error for the small amplitude and high circular frequency of the nominal trajectory for which the adaptive controller was able to use only the originally learned data. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 12.**The control torque ${Q}_{1}$ for the small amplitude and high circular frequency of the nominal trajectory for which the adaptive controller was able to use only the originally learned data. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 13.**The operation of the adaptive controller for smaller amplitude and high circular frequency of the nominal motion. (

**a**): The desired, deformed, and realized 2nd time-derivatives. (

**b**): The angle of the adaptive abstract rotation.

**Figure 14.**Trajectory tracking for the small amplitude and circular frequency of the nominal trajectory. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 15.**Trajectory tracking error for the small amplitude and circular frequency of the nominal trajectory. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 16.**The control torque ${Q}_{1}$ for the small amplitude and circular frequency of the nominal trajectory. (

**a**): Non-adaptive, (

**b**): adaptive control.

**Figure 18.**The computational time of the control cycles in ms units (zoomed in excerpts). (

**a**): Adaptive. (

**b**): Non-adaptive.

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

Inertia momentum of the wheel $\mathsf{\Theta}$ | $\left(\right)$ | $100.00$ |

Inertia of the mass-point m | $\left[\mathrm{kg}\right]$ | $1.5$ |

Spring constant k | $\left(\right)$ | $1000.0$ |

Spoke length r | $\left[\mathrm{m}\right]$ | $1.0$ |

Gravitational acceleration g | $\left(\right)$ | $9.81$ |

Damping constant along the spoke d | $\left(\right)$ | $10.0$ |

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**MDPI and ACS Style**

Bitó, J.F.; Rudas, I.J.; Tar, J.K.; Varga, Á.
Abstract Rotations for Uniform Adaptive Control and Soft Modeling of Mechanical Devices. *Appl. Sci.* **2021**, *11*, 7939.
https://doi.org/10.3390/app11177939

**AMA Style**

Bitó JF, Rudas IJ, Tar JK, Varga Á.
Abstract Rotations for Uniform Adaptive Control and Soft Modeling of Mechanical Devices. *Applied Sciences*. 2021; 11(17):7939.
https://doi.org/10.3390/app11177939

**Chicago/Turabian Style**

Bitó, János F., Imre J. Rudas, József K. Tar, and Árpád Varga.
2021. "Abstract Rotations for Uniform Adaptive Control and Soft Modeling of Mechanical Devices" *Applied Sciences* 11, no. 17: 7939.
https://doi.org/10.3390/app11177939