PRECISE PHASE PORTRAIT CONSTRUCTION FOR AN INVERTED PENDULUM USING CUBIC BEZIER CURVES
Abstract
A cubic Bezier curve formulation is used for generating phase portrait of an inverted pendulum system. Assuming that the cart and pendulum masses, pendulum length, and friction coefficient values in the inverted pendulum model are unknown, its finite number of experimentally obtained trajectories are utilized as a basis for generating the experimentally unavailable trajectories. It is shown that both the experimentally obtained and the remaining computationally generated trajectories can be formulated as an overlapping sequence of cubic Bezier curves in a computationally efficient manner. In the case of having no model of system dynamics, cubic Bezier curve formulation accurately relates experimentally obtained trajectories to the computationally generated trajectories corresponding to different initial conditions. A case study presented in this manuscript verifies that the computational approach using a cubic Bezier curves formulation has far better approximation performance compared to the ones obtained by the linear interpolation.
Keywords
References
- Stefani, R.T., Shahian, B., Savant, C.J., and Hostetter, G.H. (2002). “Design of Feedback Control Systems”. 4th ed. Oxford University Press.
- Chalupa, P., Bobal, V., (2008). “Modelling and predictive control of inverted pendulum”. 22nd European Conference on Modelling and Simulation; 3-6 June, Nicosia, Cyprus. pp. 531-537.
- Aström, K.J., Furuta, K., (2000). “Swinging up a pendulum by energy control”. Automatica, 36, 287-295.
- Slotine, J.J.E., Li, W., (1991), “Applied Nonlinear Control”. Prentice Hall.
- Farouki, R.T., (2012). “The Bernstein polynomial basis: A centennial retrospective”. Comput Aided Geom D 29, 379-419.
- Yamaguchi, F., (1988). “Curves and Surfaces in Computer Aided Geometric Design”. 5th ed. Springer Verlag.
- Riskus, A., (2006). “Approximation of a cubic Bezier curve by a circular arcs and vice versa”. Inf Technol Control, 35, 371-378.
- Zdesar, A., Skrjanc, I., Klanjar, G., (2013). “Visual trajectory-tracking model-based control for mobile robots”. Int J Adv Robot Syst 10, 323.