5.2 Loop shaping controller design
5.2.2 Stance phase
5 - Controller design
-150 -100 -50 0 50
Magnitude (dB) Loop function
Sensitivity function
Complementary sensitivity function
100 101 102 103
-540 -360 -180 0 180
Phase (deg)
Position loop
Frequency (rad/s)
-360 -270 -180 -90 0
-50 -40 -30 -20 -10 0 10 20 30 40 50
-40 dB 6 dB
3 dB 1 dB
0.5 dB 0.25 dB
0 dB
-1 dB
-3 dB -6 dB
-12 dB -20 dB Loop function
II order prototype loop function Position loop
Open-Loop Phase (deg)
Open-Loop Gain (dB)
Figure 5.6: Upperleg flight phase closed loop Bode diagram and Nichols chart
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Magnitude (dB) Loop function
Sensitivity function
Complementary sensitivity function
100 101 102 103
-540 -360 -180 0 180
Phase (deg)
Position loop
Frequency (rad/s)
-360 -270 -180 -90 0
-50 -40 -30 -20 -10 0 10 20 30 40 50
6 dB 3 dB 1 dB
0.5 dB 0.25 dB
0 dB
-1 dB
-3 dB -6 dB
-12 dB
-20 dB
-40 dB Loop function
II order prototype loop function Position loop
Open-Loop Phase (deg)
Open-Loop Gain (dB)
Figure 5.7: Modified upperleg flight phase closed loop Bode diagram and Nichols chart
5 - Controller design
Lowerleg
Starting from the mathematical model derived in section 4.3, the transfer function for the lowerleg in stance phase is computed adding the bearing friction information.
θ2(s) + θ1(s) Tmotor(s) = 1
s
108.36
s + 0.17 (5.17)
Referring to the generic closed loop system model shown in Figure 5.3, the direct branch G is computed in function of the unknown lowerleg flight phase controller function Gc,ll,s.
G(s) = Gc,ll,s(s) 6.99 106(s + 4403)
s(s + 1.16 106)(s + 4201)(s + 0.17) (5.18) The feedback branch H is composed by the encoder filter function stated in section 3.4.
H(s) =
1 1 + 1.414s
25 + s 25
2
2
(5.19)
Due to the fast phase lag introduced by the filtering action, the proposed controller is based on a phase-lead action. The controller is designed trying to shape the closed loop function like a prototype second order function on the Nychols chart. The used prototype is slightly under-damped, in order to have a fast response (ζ = 0.6), with the crossover frequency as fast as possible.
Gc,ll,s(s) = 2.4(s + 0.18) 1 + s
50
(5.20)
As shown in Figure 5.8, the obtained closed loop function has a gain crossover frequency of ωcp = 10.67 rad/s, a phase crossover frequency of ωcg = 45.45 rad/s and a phase margin of ϕm = 67.00 deg.
The obtained controller is effective only for the analyzed model, but the real one has an higher dissipative component due to the power transmission system. Increasing the fric-tion parameter, in order to keep the loop shape unchanged, it is necessary to increase the controller zero frequency proportionally. Therefore, as for the flight phase, this parameter has to be tuned experimentally. An example is shown in Figure 5.9, where the friction parameter has been increased ten times and the controller zero has been modified to keep the loop shape as the one obtained during the controller design process.
The modified controller is Gc,ll,s,mod(s) = 2.4(s + 1.8) 1 + s
50 .
The final result obtained from the experimental tuning operation, compared to the model simulation, is discussed in chapter 6.
5 - Controller design
-150 -100 -50 0 50
Magnitude (dB) Loop function
Sensitivity function
Complementary sensitivity function
100 101 102 103
-540 -360 -180 0 180
Phase (deg)
Position loop
Frequency (rad/s)
-360 -270 -180 -90 0
-50 -40 -30 -20 -10 0 10 20 30 40 50
6 dB 3 dB 1 dB
0.5 dB 0.25 dB
0 dB
-1 dB
-3 dB -6 dB -12 dB
-20 dB
-40 dB Loop function
II order prototype loop function Position loop
Open-Loop Phase (deg)
Open-Loop Gain (dB)
Figure 5.8: Lowerleg stance phase closed loop Bode diagram and Nichols chart
-150 -100 -50 0 50
Magnitude (dB) Loop function
Sensitivity function
Complementary sensitivity function
100 101 102 103
-540 -360 -180 0 180
Phase (deg)
Position loop
Frequency (rad/s)
-360 -270 -180 -90 0
-50 -40 -30 -20 -10 0 10 20 30 40 50
6 dB 3 dB 1 dB
0.5 dB 0.25 dB
0 dB
-1 dB
-3 dB -6 dB -12 dB
-20 dB
-40 dB Loop function
II order prototype loop function Position loop
Open-Loop Phase (deg)
Open-Loop Gain (dB)
Figure 5.9: Modified lowerleg stance phase closed loop Bode diagram and Nichols chart Upperleg
From the mathematical model derived in section 4.3, the transfer function for the upperleg in stance phase is computed adding the bearing friction information.
θ1(s)
Tmotor(s) = 1 s
106.24
s + 0.28 (5.21)
Referring to the generic closed loop system model shown in Figure 5.3, the direct branch
5 - Controller design
G is computed in function of the unknown upperleg flight phase controller function Gc,ul,f. G(s) = Gc,ul,s(s) 6.85 106(s + 4403)
s(s + 1.16 106)(s + 4201)(s + 0.28) (5.22) The feedback branch H is composed by the encoder filter function stated in section 3.4.
H(s) =
1 1 + 1.414s
25 + s 25
2
2
(5.23)
As for the lowerleg, since the encoder filter introduces a fast phase lag, the proposed controller is based on a phase-lead action. The controller is designed trying to shape the closed loop function like a prototype second order function on the Nychols chart. The used prototype is slightly under-damped, in order to have a fast response (ζ = 0.6), with the crossover frequency as fast as possible.
Gc,ul,s(s) = 2.2(s + 0.3) 1 + s
40
(5.24)
As shown in Figure 5.10, the obtained closed loop function has a gain crossover frequency of ωcp = 9.58 rad/s, a phase crossover frequency of ωcg= 65.75 rad/s and a phase margin of ϕm = 69.54 deg.
-150 -100 -50 0 50
Magnitude (dB) Loop function
Sensitivity function
Complementary sensitivity function
100 101 102 103
-540 -360 -180 0 180
Phase (deg)
Position loop
Frequency (rad/s)
-360 -270 -180 -90 0
-50 -40 -30 -20 -10 0 10 20 30 40 50
6 dB 3 dB 1 dB
0.5 dB 0.25 dB
0 dB
-1 dB
-3 dB -6 dB -12 dB
-20 dB
-40 dB Loop function
II order prototype loop function Position loop
Open-Loop Phase (deg)
Open-Loop Gain (dB)
Figure 5.10: Upperleg stance phase closed loop Bode diagram and Nichols chart Again, since the real leg has the power transmission system friction that is unknown, the computed controller is effective only for the considered function. Increasing the friction parameter, in order to keep the loop shape unchanged, it is necessary to increase the controller zero frequency proportionally. Therefore, as for the flight phase, this parameter has to be tuned experimentally. An example is shown in Figure 5.11, where the friction parameter has been increased ten times and the controller zero has been modified to keep the loop shape as the one obtained during the controller design process.
5 - Controller design
-150 -100 -50 0 50
Magnitude (dB) Loop function
Sensitivity function
Complementary sensitivity function
100 101 102 103
-540 -360 -180 0 180
Phase (deg)
Position loop
Frequency (rad/s)
-360 -270 -180 -90 0
-50 -40 -30 -20 -10 0 10 20 30 40 50
6 dB 3 dB 1 dB
0.5 dB 0.25 dB
0 dB
-1 dB
-3 dB -6 dB -12 dB
-20 dB
-40 dB Loop function
II order prototype loop function Position loop
Open-Loop Phase (deg)
Open-Loop Gain (dB)
Figure 5.11: Modified upperleg stance phase closed loop Bode diagram and Nichols chart
The modified controller is Gc,ul,s(s) = 2.2(s + 3.0) 1 + s
40 .
The final result obtained from the experimental tuning operation, compared to the model simulation, is discussed in chapter 6.