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Using infinitesimal symmetries for determining the first Maxwell time of geometric control problem on SH(2)
and 5th order. This method solves the differential us to identify the Maxwell points and, conse-
equations governing the system while adjusting quently, compute the first Maxwell time corre-
the time step based on the required precision. sponding to these symmetries, which, in turn, en-
This approach was chosen for its robustness and abled us to study the loss of optimality along a
efficiency in handling complex nonlinear systems. geodesic.
Using this method, we were able to obtain the The methods used in this study could
trajectories and detect the moments when the be particularly useful for analyzing other sub-
trajectories come sufficiently close, thus identify- Riemannian problems associated with a solv-
ing the Maxwell times. able Lie group. If the group is not solvable,
one can proceed using the nilpotent approxi-
Table 2. Maxwell times via numerical integration mation, which has been widely used in several
works. An interesting direction for future re-
Index Maxwell Time (Numerical) search would be to explore how the methodology
1 0.02013 developed for SH(2) can be extended to higher-
2 0.0401 dimensional Lie groups or more complex distri-
3 0.0201 butions.
4 0.0201 Moreover, the theoretical results obtained in
5 0.0401 this work have practical implications in robotics,
control engineering, and physics. Many real-world
systems, such as unicycle-type models, evolve
The Figures 2, 3, and 4 illustrate the tra-
jectory and its symmetry without numerical val- in non-Euclidean spaces, making geometric ap-
proaches essential. In particular, determining
ues, highlighting their intersections. Figures 5
Maxwell time enables real-time trajectory adjust-
and 6 introduce numerical parameters to visu-
ments, optimizing efficiency and energy consump-
alize the trajectory and its symmetric counter-
tion in applications like mobile robotics. While
part, while Figure 7 confirms the absence of in-
this study focuses on the theoretical aspects, fu-
tersections for strictly positive times. In the sec-
ture work could include numerical experiments to
ond case, Figure 10 reveals an intersection closer
further illustrate these benefits in concrete sce-
to 0 with a minimized value of k 0 . Numeri-
narios.
cally, Table 1 provides the Maxwell times for
each k 0 , whenever x t and y t approach 0, indicat-
ing an intersection between the trajectory and its Acknowledgments
symmetric counterpart which subsequently leads
None.
to a loss of optimality. Table 2 presents the
Maxwell times obtained through numerical in-
tegration. Our method, which leverages both Funding
geometric and algebraic properties, allows us to
determine the symmetry algebra of the system. None.
This structure enables us to efficiently identify
the points where the trajectory loses its optimal- Conflict of interest
ity, while reducing computational costs. More-
over, by comparing the two tables, we observe The authors declare that they have no conflict
that the Maxwell times in Table 1 are more precise of interest regarding the publication of this arti-
and inferred compared to those obtained numeri- cle.
cally.
Author contributions
Conceptualization: All authors
5. Conclusion
Formal analysis: All authors
In this work, we computed the infinitesimal sym- Methodology: All authors
metries of a sub-Riemannian problem under cer- Writing–original draft: All authors
tain conditions. We then applied this approach Writing–review & editing: All authors
to our sub-Riemannian problem on the special
hyperbolic Lie group SH(2), where we deter-
mined the infinitesimal symmetries of the geo-
Availability of data
metric control problem by leveraging the struc-
ture of the associated Lie algebra. This allowed Not applicable.
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