Σ
Σ Σ
Kp
Kp/Ti
1/Tt
f
1-z1-1
fmax
-fmax
(a)
(b)
force controller
-ks
f
xf x
vm vext fc
F Ferr
Fref controllerforce nonlinearmotor
sensor-tendon complex force
compensation
Ferr
Figure 64: Force control architecture. Theforce compensationandlinear mo-torblocks in (a) represent the compensated motor model from Fig.62(c). The detailed view of the PI force controller with an antiwindup scheme (gray-shaded region) is shown in (b).
linear encoder
force sensor drive rod
tendon extension spring (a)
(b)
m
vext vs
ks
xs
ks kt
kteff vm
resting length
Figure 65: The walking motor as a force generator. In (a) a computer ren-dering of the motor equipped with a linear encoder and a force sensor connected to a series elasticity in form of an extension spring and a tendon is shown. In (b) the different series elastic-ities are depicted as a spring-mass model of the sensor-tendon complex. The force sensors consists of a massmwith two liner springs having the stiffnessks.
Dyneema (Royal DSM, The Netherlands) tendon used in this setup.
kte f f is the effective spring constant of the elastic elements connecting the force sensor to an external object like a wall or the pulley of a joint (see next chapter). vm corresponds to the movement velocity of the motor drive rod while vext is an external velocity input. For a fixed wall,vext is obviously 0. ps is an auxiliary state variable (momentum of the force sensor). The state-space equations of this model have been derived using thebond graphs methodology. The derivation process is described in appendixD. The final equations take the following form
˙ xs x˙te f f
˙ pm
=
0 0 −m1
0 0 m1
ks −kte f f 0
xs xte f f
pm
+
1 0
0 −1
0 0
vm
vext
(145) and
xf = 1 1+1ks kt
0
xs
xte f f pm
(146) wherex-variables correspond to the amount of elongation of the elastic elements with corresponding subscripts. Additionally,xf is the overall elongation of the force sensor (both elastic connectors having the spring constantks). When multiplied with 0.5ks(series connection of the connectors), this product delivers the magnitude of the measured force. The force controller used in the architecture of Fig.64is a PI type
with an additional back-calculation antiwindup control scheme [215].
In the design process, the following set of parameters was obtained for the controller:
Kp = 1975, Ti = 0.05, Tt = 15
with the above constants being the proportional gain, the integral and the tracking time constants, respectively.
There are theoretical limits on the performance of the controller. As force generation depends on the displacement of the drive rod x in combination with the effective stiffness K of the tendon the motor pulls on, the controller can only track forces which do not require it to move faster than the maximal velocity vmaxof the motor. In the following, these theoretical limits are derived and the actual performance of the force controller is evaluated. For this purpose, the consecutive reference force signal is used in the experimental setup shown in Fig.40(a) with the controller from Fig.64(b):
Fre f(t) = A
2 sin(2πνt) + A
2 (147)
The signal spans a range A of forces between its minimal and maximal value and contains a bias term because only pulling (positive) forces can be generated in the arrangement of Fig.40(a) or a tendon-driven robot joint. The rate of the signal is varied by the frequencyνand its period is T=1/ν. The distance which the drive rod needs to travel in order to generate the reference force is given by
x(t) = Fre f(t)
K (148)
for a given effective spring constant K. Note, that the effective spring constant is a simplification of the sensor-tendon complex as introduced above in order to simplify the mathematical derivation of the limit on the performance of the force controller as presented below. This theoretical limit is defined by the following equation
vmax≥ max
t∈[0,T](|x˙(t)|) = max
t∈[0,T](|F˙re f(t)|
K ) (149)
which states that the rate of change in the reference signal cannot exceed the maximal velocity of the motor at any point and resolves into
vmax ≥ πνAK . (150)
The theoretical limit from (149) can be replaced with a softer condition vmax ≥ 1
T Z T
0 |x˙(t)|dt= 1 KT
Z T
0 |F˙re f(t)|dt (151)
1Hz 2Hz 5Hz 10Hz
20Hz 10
9 8 7 6 5 4 3 2 1
00 2 4 6 8 10 12 14 16 18 20
effective spring constant [10 N/m]3
forcesinerange[N]
K1 K2 K3
A
Figure 66: The theoretical limits of the range of a sine-shaped reference force signal in dependency of tendon stiffness and signal frequency according to (153). Dashed lines indicate the three effective spring constants (K1, K2 and K3) and the force sine range A used to evaluate the performance of the controller experimentally – see Fig.67.
which only requires that the average rate of change in the reference signal does not exceed vmax. Because of the periodicity and symmetry of the cosine, it follows that
1 KT
Z T
0 |F˙re f(t)|dt←→x˙=cos 4 KT
Z T/4
0
F˙re f(t)dt (152) and the inequality (151) leads to
vmax≥ 4 KT
h
Fre f(t)i
T 4
0 = 2νA
K . (153)
Fig.66 is the illustration of the theoretical force controller limits as defined in (153) for the PiezoLegs-motor due to its maximal velocity.
It shows the maximal range of the reference sine-shaped force signal that a perfect controller could track on average for a given spring constant K and signal frequency ν.
The performance of the controller was evaluated both in computer simulations and in real experiments. For the purpose of the experi-ments, the motor was employed in the setup from Fig.40(a). Three linear springs of increasing spring constants were used resulting in three effective stiffnesses K1 =500, K2= 2400 and K3= 10000 N/m.
As a reference sine-shaped force signals of the form shown in (147) were used. The reference signal was biased and spanned the range of 5N (A =5) in order to be positive and remain in the well-compensated range of the motor. Seven different frequencies νfrom 0.1 to 20Hz were used in the real experiments. The measured force was then compared with the reference signal. The results of the simulations and
0 5 10 15 20 frequency [Hz]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
output-inputratio
K1
K2
K3
0 5 10 15 20
frequency [Hz]
phaseshift[degree]
-180 -160 -140 -120 -100 -80 -60 -40 -20 0
K1
K2
K3
(a)
(b)
(c)
(d)
0 2 4 6 8
10 1214 16 18 20 frequency
[Hz]
effect.
springconst.
[103N/m]
output-inputratio
0 0.2 0.4 0.6 0.8 1 1.2
0 2 4 6 8
10 12 14 1618 20 frequency
[Hz]
effect.
spring const.
[10 3N/m]
0 -20 -40 -60 -80 -100 -120 -140 -160 -180
phaseshift[degree]
K1
K1
K2
K2
K3
K3 0 24 68 10 12 14
1820 16
0 24 68 10 12 14
1820 16
Figure 67: Three-dimensional Bode plots of force controller performance in dependency of tendon stiffness [(a) and (b)] computed from simulation data. (c) and (d) show the experimental data obtained for three stiffness levels (circles: K1<diamonds: K2<x-markers:
K3) and seven different frequencies. Thick curves are simulation results copied from (a) and (b).
experiments are summarized in Fig.67 with the help of Bode-like dia-grams.1 The simulations were able to predict most of the experimental results. In the magnitude plots (a and c) the measured data deviates from the simulated only for the highest stiffness K3. Surprisingly, in this case the real system performs better than expected from the simu-lations. One explanation for this are possibly nonlinear characteristics of the Dyneema tendon used in the experiments. When a soft spring is employed in series with the relatively stiff tendon, the first has a stronger influence on the elongation of the spring-tendon complex than the latter. When the spring is stiff, the nonlinear effects in the tendon gain on importance. In the phase plots (b and d) experimental data deviates from the simulated mostly for high-frequency inputs.
Also in this case, the reason could be attributed to nonlinear stretch effects in the Dyneema tendon. In the simulations, the tendon is modelled as a linear spring with a constant stiffness.
1For a true bode diagram, the gain in the magnitude plot should be depicted in dB.