Underwater Robots - Gianluca Antonelli.pdf

174 7. Dynamic Control of UVMSs
L v =blockdiag { I 3 , 20I 3 , 160, 160, 900} ,
L p =blockdiag { 5000I 3 , 10 5 I 3 , 10 3 , 10 3 , 2 · 10 3 } ,
K p =blockdiag { 400I 3 , 500I 3 , 1500I 3 } ,
K v =blockdiag { 4000I 3 , 4000I 3 , 400I 3 } .
Adigital implementation ofthe control laws has been considered. The
sensor update rate is 100Hzfor the joint positions and 20 Hz for the vehicle
position and orientation, while the control inputs to the actuators are updated
at 100Hz, i.e., the control law iscomputed every 10 ms. The relatively high
update rate for the vehicle position and orientation measurements has been
chosen soastoachieve asatisfactory tracking accuracy.
Quantization effects have been introduced in the simulation by assuming
a16-bit A/D converter onthe sensors outputs. Also, Gaussian zero-mean
noise has been added to the signals coming from the sensors.
Figure 7.13 shows the time history of the norm ofthe tracking and estimation
errors obtained with the control laws (7.41),(7.42) and (7.62), respectively.
Figures 7.14 to7.16 show the corresponding control forces, moments and
torques. It can be recognized that good tracking is achieved in both cases,
although the performance in terms of tracking errors is slightly better for the
control law (7.62), where numerical derivatives are used. Onthe other hand,
the presence ofmeasurement noise and quantization effects results in chattering
of the control commands to the actuators; this ismuch lower when the
control law (7.41),(7.42) isadopted, as compared to control law (7.62). An indicator
of the energy consumption due tothe chattering atsteady state is the
variance of the control commands reported inTable 7.4 for each component;
this data clearly show the advantage of using the controller-observer scheme.
Of course, the improvement becomes clear when the noise and quantization
effects are larger than acertain threshold. The derivation ofsuch athreshold
would require astochastic analysis ofanonlinear system, which isbeyond
the scope ofthe present work. Moreover, it can beeasily recognized that such
abound is strongly dependent on the characteristics ofthe actuators.
Second case study. In this case study the same task as above is executed
by adopting the control law (7.43),(7.44) and its counterpart using numerical
differentiation ofthe measured position/orientation, i.e.,
u = B † � M � ζ ˙
r + K v ( ζ r − ζ )+K p e d + g ( q , R I �
B ) , (7.63)
wherethe same parameters as in theprevious case study have been used.Also,
the same measurement update rates, quantization resolution and sensory
noise have been considered in the simulation.
The results are shown in Figure 7.17 interms of tracking and estimation
errors. It can be recognized that the errors are comparable to those obtained
with the control scheme (7.41),(7.42) inspite ofthe extremely simplified con-