Abstract
Motivated by the path planning problem for robotic systems this paper considers nonholonomic path planning on the Euclidean group of motions SE(n) which describes a rigid bodies path in n-dimensional Euclidean space. The problem is formulated as a constrained optimal kinematic control problem where the cost function to be minimised is a quadratic function of translational and angular velocity inputs. An application of the Maximum Principle of optimal control leads to a set of Hamiltonian vector field that define the necessary conditions for optimality and consequently the optimal velocity history of the trajectory. It is illustrated that the systems are always integrable when n = 2 and in some cases when n = 3. However, if they are not integrable in the most general form of the cost function they can be rendered integrable by considering special cases. If the optimal motions can be expressed analytically in closed form then the path planning problem is reduced to one of parameter optimisation where the parameters are optimised to match prescribed boundary conditions. This reduction procedure is illustrated for a simple wheeled robot with a sliding constraint and a conventional slender underwater vehicle whose velocity in the lateral directions are constrained due to viscous damping.
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References
Baillieul, J.: Geometric Methods for Nonlinear Optimal control problems. Journal of Optimization Theory and Applications 25(4) (1978)
Biggs, J.D., Holderbaum, W.: Optimal kinematic control of an autonomous underwater vehicle. IEEE Transactions on Automatic Control 54(7), 1623–1626 (2009)
Biggs, J., Holderbaum, W.: Integrable quadratic Hamiltonians on the Euclidean group of motions. Journal of Dynamical and Control Systems 16(3), 301–317 (2010)
Biggs, J.D., Holderbaum, W., Jurdjevic, V.: Singularities of Optimal Control Problems on some 6-D Lie groups. IEEE Transactions on Automatic Control, 1027–1038 (2007)
Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, New York (2003)
Brockett, R.W.: Lie Theory and Control Systems Defined on Spheres. SIAM Journal on Applied Mathematics 25(2), 213–225 (1973)
Brockett, R.W., Dai, L.: Non-holonomic Kinematics and the Role of Elliptic Functions in Constructive Controllability. In: Nonholonomic Motion Planning, Kluwer Academic Publishers, Dordrecht (1993)
Bullo, F., Lewis, A.: Geometric Control of Mechanical Systems. Springer, NY (2005)
Canny, J.F.: The Complexity of Robot Motion Planning. MIT Press, Cambridge (1998)
Carinena, J., Ramos, A.: Lie systems in Control Theory. In: Nonlinear Geometric Control Theory. World Scientific, Singapore (2002)
Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, Cambridge (1997)
Justh, E.W., Krishnaprasad, P.S.: Natural frames and interacting particles in three dimension. In: Proceedings of 44th IEEE Conf. on Decision and Control and the European Control Conference (2005)
Lafferriere, G., Sussmann, H.: Motion Planning for Controllable systems without drift. In: Proc. IEEE Int. Conf. on Robotics and Automation, Sacramento, CA, pp. 1148–1153 (1991)
Latombe, J.C.: Robot Motion Planning. Kluwer, Boston (1991)
Leonard, N., Krishnaprasad, P.S.: Motion control of drift free, left-invariant systems on Lie groups. IEEE Transactions on Automatic Control 40, 1539–1554 (1995)
Luenberger, D.: Linear and Nonlinear Programming. Addison-Wesley Publishing Company, Reading (1984)
Murray, R., Li, Z., Sastry, S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994)
Murray, R., Sastry, S.: Steering nonholonomic systems using sinusoids. In: Proc. of 29th IEEE Conf. on Decision and Control, Honolulu, Hawaii, pp. 2097–2101 (1990)
Selig, J.M.: Geometric Fundamentals of Robotics. Monographs in Computer Science. Springer, Heidelberg (2006)
Sussmann, H.J.: An introduction to the coordinate-free maximum principle. In: Jakubczyk, B., Respondek, W. (eds.) Geometry of Feedback and Optimal Control, pp. 463–557. Marcel Dekker, New York (1997)
Sussmann, H.J., Liu, W.: Limits of highly oscillatory controls and approximations of general paths by admissible trajectories. In: Proc. of 30th IEEE Conf. on Decision and Control, pp. 437–442 (1991)
Zexiang, L., Canny, J.F.: Nonholonomic Motion Planning. Kluwer Academic Publishers, Dordrecht (1993)
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Biggs, J. (2011). Optimal Path Planning for Nonholonomic Robotic Systems via Parametric Optimisation. In: Groß, R., Alboul, L., Melhuish, C., Witkowski, M., Prescott, T.J., Penders, J. (eds) Towards Autonomous Robotic Systems. TAROS 2011. Lecture Notes in Computer Science(), vol 6856. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23232-9_19
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DOI: https://doi.org/10.1007/978-3-642-23232-9_19
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