Abstract
Many mobile robots such as wheeled robots, boats, or plane are described by nonholonomic differential equations. As a consequence, they have to satisfy some differential constraints such as having a radius of curvature for their trajectory lower than a known value. For this type of robots, it is difficult to prove some properties such as the avoidance of collisions with some moving obstacles. This is even more difficult when the initial condition is not known exactly or when some uncertainties occur. This paper proposes a method to compute an enclosure (a tube) for the trajectory of the robot in situations where a guaranteed interval integration cannot provide any acceptable enclosures. All properties that are satisfied by the tube (such as the non-collision) will also be satisfied by the actual trajectory of the robot.
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Notes
- 1.
A trajectory \(\mathbf {x}\), which is a function from \(\mathbb {R}\) to \(\mathbb {R}^{n}\), can be denoted equivalently \(\mathbf {x}\left( t\right) \) or \(\mathbf {x}\left( \cdot \right) \). When no ambiguity may exist, i.e., when t is already used in the same paragraph, we shall often prefer \(\mathbf {x}\left( t\right) \), for simplicity.
References
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Le Bars, F., Sliwka, J., Reynet, O., Jaulin, L.: State estimation with fleeting data. Automatica 48(2), 381–387 (2012)
Berz, M., Makino, K.: Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliable Comput. 4(3), 361–369 (1998)
Bethencourt, A., Jaulin, L.: Solving non-linear constraint satisfaction problems involving time-dependant functions. Math. Comput. Sci. 8(3), 503–523 (2014)
Blanchini, F., Miani, S.: Set Theoretic Methods in Control. Birkhauser, Boston (2008)
Bouissou, O., Chapoutot, A., Djaballah, A., Kieffer, M.: Computation of parametric barrier functions for dynamical systems using interval analysis. In: IEEE CDC, Los Angeles, United States (2014)
Chabert, G., Jaulin, L.: Contractor programming. Artif. Intell. 173, 1079–1100 (2009)
Chapoutot, A., Alexandre dit Sandretto, J., Mullier, O.: Dynibex. ENSTA (2015). http://perso.ensta-paristech.fr/~chapoutot/dynibex/
Fernandes, M.L., Zanolin, F.: Remarks on strongly flow-invariant sets. J. Math. Anal. Appl. 128, 176–188 (1987)
Jaulin, L., Le Bars, F.: An interval approach for stability analysis: application to sailboat robotics. IEEE Trans. Robot. 27(5), 282–287 (2012)
Kapela, T., Zgliczynski, P.: A lohner-type algorithm for control systems and ordinary differential inclusions. Discrete Continuous Dyn. Syst. 11(2), 365–385 (2009)
Langson, W., Chryssochoos, I., Rakovic, S.V., Mayne, D.Q.: Robust model predictive control using tubes. Automatica 40(1), 125–133 (2004)
Lhommeau, M., Jaulin, L., Hardouin, L.: Capture basin approximation using interval analysis. Int. J. Adap. Control Sig. Process. 25(3), 264–272 (2011)
Lohner, R.: Enclosing the solutions of ordinary initial and boundary value problems. In: Kaucher, E., Kulisch, U., Ullrich, C.H. (eds.) Computer Arithmetic: Scientific Computation and Programming Languages, pp. 255–286. BG Teubner, Stuttgart (1987)
Le Menec, S.: Linear differential game with two pursuers and one evader. In: Breton, M., Szajowski, K. (eds.) Advances in Dynamic Games, vol. 11, pp. 209–226. Birkhauser, Boston (2011)
Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)
Nedialkov, N.S., Jackson, K.R., Corliss, G.F.: Validated solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105(1), 21–68 (1999)
Raissi, T., Ramdani, N., Candau, Y.: Set membership state and parameter estimation for systems described by nonlinear differential equations. Automatica 40, 1771–1777 (2004)
Ratschan, S., She, Z.: Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions. SIAM J. Control Optim. 48(7), 4377–4394 (2010)
Revol, N., Makino, K., Berz, M.: Taylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY. J. Logic Algebraic Program. 64, 135–154 (2005)
Stancu, A., Jaulin, L., Bethencourt, A.: Stability analysis for time-dependent nonlinear systems: an interval approach. Internal report, University of Manchester (2015)
Wilczak, D., Zgliczynski, P.: Cr-Lohner algorithm. Schedae Informaticae 20, 9–46 (2011)
Yorke, J.A.: Invariance for ordinary differential equations. Math. Syst. Theor. 1(4), 353–372 (1967)
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Jaulin, L. et al. (2016). Computing Capture Tubes. In: Nehmeier, M., Wolff von Gudenberg, J., Tucker, W. (eds) Scientific Computing, Computer Arithmetic, and Validated Numerics. SCAN 2015. Lecture Notes in Computer Science(), vol 9553. Springer, Cham. https://doi.org/10.1007/978-3-319-31769-4_17
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