Trajectory Planning for Mechanical Systems Based on Time-Reversal Symmetry
Abstract
:1. Introduction
Background to the Study
2. Materials and Methods
2.1. Motivational Case Study: Mass-Spring-Damper Model
2.1.1. State-Space Description of Mass-Spring-Damper Model
2.1.2. Trajectory Planning for Mass-Spring-Damper Model
- Obtaining a response to initial conditions , whose values correspond to the predefined final state , are supposed to reach at time by application of so far unknown control signal brought to the input of the system according to Figure 2. Note that the input 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 is provided, where . 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 to be stored in . 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 depicted in Figure 6 in time presented in Figure 7 into using the relation . The initial conditions applied in Figure 7 will correspond to the final values reached in previous phase at the time . Supposing adequate time range, in this case the values will be very close to zero, , , respectively. Resulting waveform is depicted in Figure 8.
2.2. Primary Case Study: Swing-Up of the Inverted Pendulum on the Cart
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 Function
2.3.2. Calculation of Function Based on Numerical Optimization Procedure
- Term penalizes violation of basic constraints placed on state trajectories and control;
- Term penalizes error between actual trajectory and predefined state trajectory ;
- The other terms , , penalize error of actual trajectories and predefined zero values at the final point of time interval;
- The last term 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 Function
3.2. Results Based on the Numerical Optimization Procedure for Function
- ;
- ;
- ;
- ;
- ;
- ;
- .
- ;
- ;
- ;
- ;
- .
4. Discussion
5. Conclusions
Further Research Plans
Author Contributions
Funding
Conflicts of Interest
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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
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 StyleOzana, 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