The transfer function of the low-pass filter for Ω ≤ Ωg is equal to H kork (Ω) = H k (Ω)/ H ∇k (Ω) =
k =1 Ω / sin Ω 2 = Ω / 2(1 − cos Ω) k =2 Ω3 /(−2 sin Ω + sin(2Ω)) k = 3.
(4)
The filter impulse response is the inverse Fourier transform of its frequency characteristic, thus:
hkork (n ) =
1 2π
Ωg
∫
H kork (Ω)e j Ωnd Ω .
(5)
∂k k ! jk H d k (Ω)H ∇k (Ω) |Ω= 0 = k . k ∂Ω ∆
(10)
3. A mathematical model of a robot manipulator In the next sections, the following problems will be solved: first, we will derive the equations for the DC motors, then, we will define the kinetic and potential energy of the system, and finally, we will symbolically derive the robot dynamic equations, using the second order Lagrange equations.
−Ωg
Unfortunately, integral (5) cannot be expressed by means of the analytic functions. It needs to be determined using some approximation. By expanding function H kork (Ω) into a Taylor series around the value Ω = 0 , we obtain:
Ω2 4 1 + 6 + O(Ω ) k = 1 Ω2 H kork (Ω) = 1 + + O(Ω4 ) k = 2 12 Ω2 + O(Ω4 ) k = 3. 1 + 4
(6)
The four-term approximation of the expansion appears to be fairly sufficient. The inverse Fourier transform of the function obtained by rejecting the terms of the higher orders is equal to:
12 6n 3π n Ωg cos(n Ωg ) + 2 2 2 (6 g 1 (2n Ωg cos(n Ωg ) + 3 hkork (n ) = 12n π (12n 2 n 2 2g g 1 (2n Ω cos(n Ω ) + g g 4n 3π 2 2 2 (4n n g g
(7)
Assume that the impulse response of the low-pass differential filter is:
hd k (n ) =
1
χk
hkork (n )WHarris (n ) ,
(8)
where WHarris (n ) is Harris window described by the following equation:
WHarris (n ) = 0,36 + 0, 49 cos(π n / M ) + +0,14 cos(2nπ / M ) + 0,01cos(3nπ / M ).
(9)
The parameter χ k should be selected in sum a way that the slope of the characteristic of the filter being designed at point Ω = 0 is the same as that of the ideal differential equation, thus:
Fig. 3. An electrically-driven manipulator Rys. 3. Manipulator z napędem elektrycznym
Let ϕ = [ϕ1 ϕ 2 ϕ 3 ] denote the vector of joint variables acting as generalized coordinates, mj – the mass, lj – the arm length, lc j – the distance from the centre of gravity and Sj – the motor of the link j. Using typical equivalent diagrams of DC motors available in the literature, e.g. Ref. [4], and the second Kirchoff law, we can write the following electrical equation of the DC motor:
U z j = U Rj + U Lj + Eej , for j = 1,2,3
(11)
where U z j is the voltage supplied to the rotor. Since an open-loop system may be difficult to control, it is essential that the identification be performed for a closed-loop system with properly selected PD controllers. Let us assume that the equations of the controllers have the following form:
U z j = K pj (ϕ z j (t ) − ϕ j (t )) − Kd j ϕ j (t ) ,
(12)
where: K pj , Kd j – the parameters of the controllers, ϕ z j (t ) – the control signals, ϕ j (t ) – the variables describing the position of the manipulator arms. The voltage drops across the rotor winding resistance and inductance are: Pomiary Automatyka Robotyka nr 12/2012
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