Proving Feasibility of a Docking Mission: A Contractor Programming Approach
Abstract
:1. Introduction
1.1. Context
1.2. State of the Art
1.3. Contribution
2. Formalization
2.1. Formalizing Robots as Dynamical Systems
 T is the system’s time set in which the time parameter t evolves;
 $\mathcal{S}$ is the system’s state space containing the system’s state $\mathbf{x}$;
 $\varphi \left(\right)open="("\; close=")">0,\xb7$ is the identity function, $\forall \mathbf{x}\in \mathcal{S},\phantom{\rule{0.166667em}{0ex}}\varphi \left(\right)open="("\; close=")">0,\mathbf{x}$;
 $\varphi \left(\right)open="("\; close=")">{t}_{1},\varphi \left(\right)open="("\; close=")">{t}_{2},\mathbf{x}=\varphi \left(\right)open="("\; close=")">{t}_{1}+{t}_{2},\mathbf{x}$ for any ${t}_{1},{t}_{2}\in T$ and for any $\mathbf{x}\in \mathcal{S}$.
2.2. Accounting for System’s Uncertainties
2.3. Tube Arithmetic
2.4. Constraint Programming
2.5. Tube Implementation and the Codac Library
3. Guaranteed Integration: A Constraint Programming Approach
3.1. Lohner Algorithm
 Find a global enclosure $\left(\right)$ for the system’s trajectories over the time interval $\left(\right)$;
 Using $\left(\right)$, find an enclosure for the state at time ${t}_{k+1}$, i.e., $\left(\right)$.
 Given an initial state enclosure $\left(\right)$, Algorithm 2 can compute a sequence of boxes $\left(\right)$ that contains the system’s state at time ${t}_{k}=k\delta $;
 For all $k\ge 0$, the enclosure $\left(\right)$ is actually represented by a tilted box in Algorithm 2: $\left(\right)open="["\; close="]">{\mathbf{x}}_{k}$, where both $\left(\right)$ and $\left(\right)$ represent the system’s state enclosure, respectively, in the canonical basis and in the one defined by the orthogonal matrix ${\mathbf{B}}_{k}$;
 Algorithm 2 computes at each time step k the global enclosure $\left(\right)$, such that for all $t\in \left(\right)open="["\; close="]">{t}_{k},{t}_{k+1}$ and for all ${\mathbf{x}}_{k}\in \left(\right)open="["\; close="]">{\mathbf{x}}_{k}$, $\varphi \left(\right)open="("\; close=")">t{t}_{k},{\mathbf{x}}_{k}$.
Algorithm 1GlobalEnclosure (in: $\left(\right)$, $\u03f5$, $\mu $, ${i}_{max}$, out: $\left(\right)$, inout: $\delta $)  
 
 
 
 
 Increase the number of iterations 
 
 Reset the iteration counter 
 Reduce the time step by a factor$\mu $ 
Reset the a priori estimates  
 
 
 
 Inflate the a priori estimate 
 Compute the new a priori estimate 
 

Algorithm 2SimpleLohner (in: $\left(\right)$, $\delta $, N, out: $\left(\right)$)  
 initialisation 
 
 
 
 
 
 
 
 
 
 orthogonal part of the QR factorisation 
 
 
 
 

3.2. Lohner Contractor
 $\left[\mathbf{x}\right]\left(\right)open="("\; close=")">{t}_{0}$ can be seen as the input gate of slice 0 and the system’s initial condition, i.e., forming an IVP;
 $\left[\mathbf{x}\right]\left(\right)open="("\; close=")">{t}_{k}$ can be interpreted as the kth slice’s input gate, the $\left(\right)$th slice’s output gate, and the system’s state enclosure at time ${t}_{k}$;
 $\left[\mathbf{x}\right]\left(k\right)$ is the kth slice of $\left[\mathbf{x}\right]\left(\xb7\right)$, thus containing all the system trajectories over the time interval $\left(\right)$. $\left[\mathbf{x}\right]\left(k\right)$ can therefore be seen as a global enclosure for the system over $\left(\right)$.
 Assume that we initialize the algorithm with the initial box $\left[\mathbf{x}\right]\left(\right)open="("\; close=")">{t}_{0}$;
 Then, all the system trajectories during the time interval $\left(\right)$ must be contained both in the slice $\left[\mathbf{x}\right]\left(k\right)$ and in the global enclosure $\left(\right)$;
 Consider the kth slice: at time ${t}_{k}$, the system’s state is enclosed by $\left[\mathbf{x}\right]\left(\right)open="("\; close=")">{t}_{k}$, and at time ${t}_{k+1}$, it is enclosed both in the slice’s output gate $\left[\mathbf{x}\right]\left(\right)open="("\; close=")">{t}_{k+1}$ and in the box $\left(\right)$ computed by Algorithm 2;
 Finally, we have $\left(\right)open="["\; close="]">{\mathbf{x}}_{k}$.
Algorithm 3${\mathcal{C}}_{Lohner}$ (inout: $\left[\mathbf{x}\right]\left(\xb7\right)$)  
 
 
 
 
 
 
 
 
 
 see Algorithm 2 
 
 
 
 
 orthogonal part of the QR factorization 
 
 
 
 
 contracts$\left(\right)$ 
 adjusts the center of the tilted box 
 contracts uncertainties in tilted frame 
 contracts the slice and the output gate 

4. Applications
4.1. Simple Illustrating Example
4.2. Underwater Docking Mission Feasibility
4.2.1. Modeling the Robot as a Dynamical System
4.2.2. Computing ${\mathcal{D}}^{+}$
Algorithm 4(in: $\left(\right)$, $\left[\mathbf{d}\right]$, $\u03f5$)  
 
 
 
 
 
 pop a box from the temporary list 
 initialize the tube$\left[\mathbf{x}\right]\left(\xb7\right)$with$\left[\mathbf{x}\right]$ 
 contract the initial tube with${\mathcal{C}}_{Lohner}$ 
 
 
 
 
 
 bisect$\left[\mathbf{x}\right]$to obtain a thinner trajectory 
 
 
 
 

4.2.3. Numerical Results
4.2.4. Discussion
5. Conclusions and Outlook on Future Work
5.1. Summary
5.2. Future Recommendations
 The first contractor that could be of use in robotics is the one enforcing a differential constraint of the form $\dot{\mathbf{x}}\left(t\right)=\mathbf{f}\left(\right)open="("\; close=")">\mathbf{x}\left(t\right),\mathbf{u}\left(t\right)$, where $\mathbf{u}\left(t\right)$ is not computed using the robot’s state but another tube of trajectories in the command space, possibly without analytical expression. This would come down to dealing with nonautonomous and timevarying differential equations $\dot{\mathbf{x}}\left(t\right)=\mathbf{f}\left(\right)open="("\; close=")">\mathbf{x}\left(t\right),t$, which can be useful, for example, when the command is simply measured or when the system is operated in an openloop mode and the mission is to compute the actual robot’s trajectory using this command and the available sensor data;
 A second contractor could extend Poincaré maps to the world of contractor programming. A Poincaré map returns the impact point, where the trajectory of a dynamical system transversely crosses an arbitrary surface (we refer the reader to [24,46,47] for more details). This contractor could help solve the docking problem presented in this article, as the trajectories would not be limited by the time constraint modeled by T anymore: any trajectory “impacting” the entrance of the docking station could be considered valid. The method for stability analysis, extended to Poincaré maps, can also be used to analyze more general dynamic systems such as systems with hybrid dynamics in which the state transitions result from eventtriggered control approaches.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Sample Availability
Abbreviations
PID  Proportional, Integral, Derivative 
IVP  Initial Value Problem 
ROV  Remotely Operated Vehicle 
CAPD  ComputerAssisted Proofs in Dynamics 
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Bourgois, A.; Rohou, S.; Jaulin, L.; Rauh, A. Proving Feasibility of a Docking Mission: A Contractor Programming Approach. Mathematics 2022, 10, 1130. https://doi.org/10.3390/math10071130
Bourgois A, Rohou S, Jaulin L, Rauh A. Proving Feasibility of a Docking Mission: A Contractor Programming Approach. Mathematics. 2022; 10(7):1130. https://doi.org/10.3390/math10071130
Chicago/Turabian StyleBourgois, Auguste, Simon Rohou, Luc Jaulin, and Andreas Rauh. 2022. "Proving Feasibility of a Docking Mission: A Contractor Programming Approach" Mathematics 10, no. 7: 1130. https://doi.org/10.3390/math10071130