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ps1 - EE 530 Reading assignment Ch 1 and Ch 2 The goal of...

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EE 530 PROBLEM SET 1 DUE: 16 OCT 2006 Reading assignment: Ch 1 and Ch 2 The goal of the first problem set is to: 1. Review basic concepts and mathematical tools from EE 428 and EE 527. 2. Become familiar with the dynamic model of the rotary inverted pendulum system that will be used in future problem sets to demonstrate adaptive control techniques. Problem 1: (25 points) Figure 1 shows a photograph of the rotary inverted pendulum system, while Figures 2 and 3 define several parameters and variables in the system model. Figure 1: The rotary inverted pendulum system. Figure 2: Top view of the rotary inverted pendulum sys- tem. Figure 3: Side view with pendulum in motion.
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Application of Lagrangian mechanics results in the non-linear ordinary differential equation model ( J eq + mr 2 ) ¨ θ - mLr cos( α α + mLr sin( α ) ˙ α 2 + B eq ˙ θ = T m (1) 4 3 mL 2 ¨ α - mLr cos( α ) ¨ θ - mgL sin( α ) = 0 . Figure 2 defines θ as the angular displacement of the rotating arm while Figure 3 defines α as the angular displacement of the pendulum from its upright position.The variable T m represents the output torque on the load from a DC motor located in the SRV02 unit shown in Figure 2 and coupled to the rotating arm through a gear reduction system. Table 1 defines the constant parameters appearing in equation (1). Symbol Description L length to pendulum’s center of mass m mass of pendulum arm r rotating arm length g acceleration due to gravity J eq equivalent moment of inertia of the motor system seen at the output of the gear reduction system B eq equivalent viscous damping coefficient seen at the output of the gear reduction system Table 1: Parameters for the rotary inverted pendulum system. The input to the inverted pendulum system is the armature voltage V m applied across the DC motor within the SRV02 system. The dynamic model of the DC motor simplifies by neglecting the armature inductance as the electrical time constant of the motor is more than two orders of magnitude smaller than the mechanical time constant. The resulting relationship between the output torque and armature voltage is T m = η m η g K t K g V m - K g K m ˙ θ R m , (2) where Table 2 defines the relevant constant parameters in the DC motor model.
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