Fractional Modeling of Robotic Systems
A mathematical model helps represent the complex behavior of a dynamical system into a set of mathematical formulations to capture the essential characteristics and to characterize the impact of different constraints on the optimal system response. It hel
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Fractional Modeling of Robotic Systems
A mathematical model helps represent the complex behavior of a dynamical system into a set of mathematical formulations to capture the essential characteristics and to characterize the impact of different constraints on the optimal system response. It helps to analyze the system response mathematically, perform simulations and provides a tool for testing theoretical phenomena. This chapter introduces mathematical modeling of robotic systems. The systems utilized for modeling are 2D Gantry Crane System, Double Inverted Pendulum on a Cart, Missile Launching Vehicle (MLV) or Pad System and Pendulum on a Cart. It is possible to extend this notion to many other systems such as ball and beam system, snake-type systems, locomotive systems, swimming robots, aircraft, acrobatic robots, helicopters, underwater vehicles, and surface vessels. Various methods of modeling are available such as Newtonian mechanics, Hamiltonian mechanics, Lagrangian mechanics. In this chapter, EulerLagrangian (E-L) formulation is used to model the system under consideration. These modeling equations are used for further analysis. In classical mechanics, problems related to the basic movements of objects are examined which has led to the designing of various fascinating models. There are numerous approaches which can be used to derive models from these essential systems. One of the most accepted approach is to utilize the Euler-Lagrange formulation [1, 2], for which the system energy needs to be calculated. The energy of a system can be classified as (i) Potential Energy (PE), that is, stored energy; for example, when spring is compacted, or an object lifted at a certain height, and (ii) Kinetic Energy (KE), which gets from the motion of the object. Let P and V are potential and kinetic energy of the system; then the EL equation can be expressed as d dt
∂(T − V ) ∂ x˙
−
∂(T − V ) = F, ∂x
© Springer Nature Switzerland AG 2021 A. P. Singh et al., Fractional Modeling and Controller Design of Robotic Manipulators, Intelligent Systems Reference Library 194, https://doi.org/10.1007/978-3-030-58247-0_2
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2 Fractional Modeling of Robotic Systems
where x is position and v is velocity. To find the equations of motions for a given system, one must follow these steps: (i) Find the values of PE and KE to compute the Lagrangian L = T − V . (ii) Compute ddqL . (iii) Compute dd qL˙ and dtd dd qL˙ . It is important that q˙ be treated as a complete variable in its own right, and not as a derivative. (iv) Equate ddqL + F(t) = dtd dd qL˙ . (v) Solve the differential equation obtained in the preceding step. At this point, q˙ is treated “normally”. Note that the above equation might be a system of equations and not simply one equation.
2.1 Mathematical Modeling Models portray how a particular system works [3, 4]. In mathematical modeling, we make a translation of those behaviors into equations due to the following merits: • Mathematics helps understand system properties and basic assumptions. • Mathematics
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