# 2 reduced order model of an articulated robot system

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2. REDUCED-ORDER MODEL OF AN ARTICULATED ROBOT SYSTEM 2.1. Equations of motion As a typical example of the robot system, an ar- ticulated robot shown in Fig. 1 is taken up. This sys- tem can be regarded as a 3-mass system composed of a motor rotor, a gear reducer’s input shaft and a driven machine part, and it is controlled by the veloc- ity control loop using the PI control. Equations of motion of this geared system are written as ( ) ( ) , θ θ θ θ θ + + = ±± ± ± m m s m g s m g m J C K T (1) ( ) ( ) { ( ) ( )}/ 0, g g s g m s g m g g g l g g g l g J C K C R K R R θ θ θ θ θ θ θ θ θ + + + + = ±± ± ± ± ± (2) ( ) ( ) 0, θ θ θ θ θ + + = ±± ± ± l l g l g g g l g g J C R K R (3) where m θ = angular rotation of the motor, g θ = angular rotation of the gear reducer’s input shaft, l θ = angular rotation of the driven machine part, T m = output torque of the motor, J m = moment of inertia of the motor rotor, J g = moment of inertia of the reducer’s input shaft, J l = moment of inertia of the driven machine part, R g = reduction ratio of the gear reducer, K s = torsional stiffness between the motor rotor and the gear reducer’s input shaft, K g = torsional stiffness of the gear reducer, C s = damping factor between the motor rotor and the gear reducer’s input shaft, C g = damping factor of the gear reducer. Further, when the motor speed is controlled by the PI control, equations related to the motor armature are expressed as ( ) , ω = + v c v cb e m i K di L K K e edt K i Ri K dt T (4) . ω ω = cmd m e (5) The output torque of the motor is expressed as , = m t T K i (6) where ω cmd = velocity command, ω m = rotating speed of the motor, ω g = rotating speed of the gear reducer’s input shaft = rotating speed of the wave generator, ω l = rotating speed of the driven machine part, e = error, i = current of the armature, R = motor armature resistance, L = motor armature inductance, K t = torque constant, K e = voltage constant, K c = current loop gain, K cb = current feedback gain, K v = proportional gain of the PI control, T i = integral time constant of the PI control. According to (1)-(6), a block diagram of the geared mechanical system can be expressed as Fig. 2. 2.2. Reduced-order model of mechanical part Most of the geared mechanical systems can be grouped into two classes due to the insufficiency of the torsional stiffness. The first case is that the stiff- ness of a geared stage is much higher than those of shafts. (a) Robot arm. (b) Model. Fig. 1. A robot arm and its analytical model. 