# Trajectory Planning for Mechanical Systems Based on Time-Reversal Symmetry

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## Abstract

**:**

## 1. Introduction

#### Background to the Study

## 2. Materials and Methods

#### 2.1. Motivational Case Study: Mass-Spring-Damper Model

^{−1}] is a damping coefficient; k [N·m

^{−1}] is a spring stiffness; $y(t)$ is the output of the dynamic system (position of the mass), $\dot{y}(t)$, $\ddot{y}(t)$ are first and second derivative of $y(t)$ respectively (velocity and acceleration). Assuming no external force, the mass-spring-damper model can be described by the following 2

^{nd}order ordinary differential equation

#### 2.1.1. State-Space Description of Mass-Spring-Damper Model

^{−1}]—velocity; $u(t)$ [N]—force.

#### 2.1.2. Trajectory Planning for Mass-Spring-Damper Model

- Obtaining a response to initial conditions ${x}_{2}(0)$, ${x}_{1}(0)$ whose values correspond to the predefined final state ${x}_{2}(T)=0$, ${x}_{1}(T)=1$ are supposed to reach at time $t=T$ by application of so far unknown control signal $u(t)$ brought to the input of the system according to Figure 2. Note that the input $u(t)$ is absent at the moment. Resulting waveform is depicted in Figure 3.
- As the output signal shown in Figure 3 is too oscillatory to represent a good candidate for a trajectory, the scheme depicted in Figure 2 is modified by adding an artificial damping to the system and stores the signal referred to as ${u}_{aux}(t)$ is provided, where ${u}_{aux}(t)=-2.5\xb7{x}_{2}(t)$. The value of the damping parameter was adjusted to keep the stability of the system and obtain the sufficient system’s response in time and frequency domains. This modified scheme is depicted in Figure 4 and its simulation leads to the waveform shown in Figure 5 which is now considered as an appropriate candidate for a state trajectory.
- The compensating damping is illustrated with the Figure 4 through an output drawn from ${x}_{2}(t)$ to be stored in ${u}_{aux}(t)$. This system is then simplified as presented with Figure 6 and continues to maintain time reversal symmetry. The simulation result from systems described in Figure 4 and Figure 6 give out identical waveform as shown in Figure 5.
- Reversing the time flow of the control signal ${u}_{aux}(t)$ depicted in Figure 6 in time presented in Figure 7 into ${u}_{rev}(t)$ using the relation ${u}_{rev}(t)={u}_{aux}(\vartheta )={u}_{aux}(T-t)$. The initial conditions applied in Figure 7 will correspond to the final values reached in previous phase at the time $t=T$. Supposing adequate time range, in this case the values will be very close to zero, ${x}_{2}(0)=0$, ${x}_{1}(0)=\epsilon \approx 0$, respectively. Resulting waveform is depicted in Figure 8.
- Going back to the original model described in Figure 1. The effect of damping will be eliminated by subtracting the damping term from control signal ${u}_{rev}(t)$ which results in a control signal ${u}_{revF}(t)$ which is stored for further use as shown in Figure 9, where ${u}_{revF}(t)={u}_{rev}(t)-0.5\xb7{x}_{2}(t)$.

#### 2.2. Primary Case Study: Swing-Up of the Inverted Pendulum on the Cart

^{−2}]—acceleration (control signal); g [m·s

^{−2}]—gravity constant; b [s

^{−1}]—a shear friction coefficient.

#### Time-Reversal Symmetry (Reversibility) of the System

#### 2.3. Methodology: Time-Reversal Symmetry Applied to the Inverted Pendulum Model

#### 2.3.1. Expert Choice of $g(\circ )$ Function

#### 2.3.2. Calculation of $g(\circ )$ Function Based on Numerical Optimization Procedure

- Term ${J}_{c}$ penalizes violation of basic constraints placed on state trajectories and control;
- Term ${J}_{{x}_{1}}$ penalizes error between actual trajectory ${x}_{1}$ and predefined state trajectory ${x}_{1ref}$;
- The other terms ${J}_{{x}_{2}}$, ${J}_{{x}_{3}}$, ${J}_{{x}_{4}}$ penalize error of actual trajectories and predefined zero values at the final point of time interval;
- The last term ${J}_{u}$ is a stabilizing term assuring a converging solution, it represents energy minimization.

## 3. Results for Primary Case Study

#### 3.1. Results Based on Expert Choice of $g(\circ )$ Function

#### 3.2. Results Based on the Numerical Optimization Procedure for $g(\circ )$ Function

- ${A}_{0}=0.5107$;
- ${A}_{1}=-1.1854$;
- ${\omega}_{1}=1.3606$;
- ${\vartheta}_{1}=4.6795$;
- ${A}_{2}=-1.7894$;
- ${\omega}_{2}=1.8575$;
- ${\vartheta}_{2}=4.0269$.

- ${A}_{0}=0.2469$;
- ${A}_{1}=-2.9825$;
- ${A}_{2}=-0.1663$;
- ${A}_{3}=1.3260$;
- ${A}_{4}=-0.2716$.

## 4. Discussion

## 5. Conclusions

#### Further Research Plans

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Simulation experiment: a response to initial conditions according to Figure 2.

**Figure 4.**Simulation experiment: Adding an artificial damping to the system and storing signal ${u}_{aux}(t)$.

**Figure 10.**Simulation experiment: application of the computed control signal ${u}_{revF}(t)$ to the original system.

**Figure 15.**Simulation experiment: application of the reference control signal $u(t)$ to perform the swing-up.

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**MDPI and ACS Style**

Ozana, S.; Docekal, T.; Kawala-Sterniuk, A.; Mozaryn, J.; Schlegel, M.; Raj, A.
Trajectory Planning for Mechanical Systems Based on Time-Reversal Symmetry. *Symmetry* **2020**, *12*, 792.
https://doi.org/10.3390/sym12050792

**AMA Style**

Ozana S, Docekal T, Kawala-Sterniuk A, Mozaryn J, Schlegel M, Raj A.
Trajectory Planning for Mechanical Systems Based on Time-Reversal Symmetry. *Symmetry*. 2020; 12(5):792.
https://doi.org/10.3390/sym12050792

**Chicago/Turabian Style**

Ozana, Stepan, Tomas Docekal, Aleksandra Kawala-Sterniuk, Jakub Mozaryn, Milos Schlegel, and Akshaya Raj.
2020. "Trajectory Planning for Mechanical Systems Based on Time-Reversal Symmetry" *Symmetry* 12, no. 5: 792.
https://doi.org/10.3390/sym12050792