18
Biologically Inspired Robotics
where n is the number of CPG modules in the network; m is the number of
neurons in one CPG module; s is the serial number of neurons connected to
the jth neuron; u j,i is the membrane potential of the jth neuron in the ith CPG
module; v j,i is the variable that represents the degree of adaptation; u 0,i is the
tonic driving input; τ 1,i and τ 2,i are the time constants; β is the adaptation coefficient; w is the weight between neurons; w ik is the connection weight of the
ith module from the kth module; y j,i is the output of the jth neuron in the ith
CPG module; y out,i is the output of the ith CPG module; w 0 is a constant value
for connection weight; and r is the number of the CPG in the network with
feedback connection.
The basic concept of this network is shown in Figure 2.2. Compared with
the unilaterally connected network, this network has the same unidirectional couplings between the oscillators. The difference is that the output of
the the rth CPG module is provided as feedback to the first module to form a
closed loop. The connection weight between CPGs w ik takes the same value
from Equation (2.3). In addition, the weight of the feedback w 1r from the rth
module to the first one should adopt a weight value, given by w 1r = w 0 .
Because the rth CPG module transmits the same value to the (r + 1)-th and
the first CPG module, the output waves of the (r + 1)-th and the 1st module
have completely identical shapes and phases. The (r + 1)-th module will also
transmit its output to the (r + 2)-th module as the first module transmits its
output to the second module. Thus, the (r + 2)-th module will generate the
same output as that of the second module. In the same way, the following
output of CPG modules will take the similar process mentioned above as
CPGs in the loop. Thus, the output of the CPG modules on the outside of the
loop can be represented as
y , + = y
p = , ,..., q = 1 2
out pr q
out q
,
1 2
, ,... r
(2.4)
To compare the output of a CPG network with a unilateral connection and
feedback connection, a network composed of six CPG modules is constructed.
For the network with feedback connection, the sixth module is selected to
provide feedback to the first CPG module. The results are simulated by use
of both the dual-neuron mutual inhibition model and the tri-neuron cyclic
inhibitory model. Rhythmic outputs of CPG models with respect to the set of
CPG parameters in Table 2.1 are shown in Figures 2.3 and 2.4.
As shown in Figure 2.3, the amplitude of the CPG outputs in the unilaterally connected network is not of uniform size and the waveforms cannot
form a perfect traveling wave to suit the control of the multilink robot. Thus,
it is necessary to make adjustments to obtain applicable rhythmic signals
for control of the robot. The output of CPGs in Figure 2.4 has a feedback
connection with the same amplitude and phase difference. Furthermore, it
is obvious that all of the output waves of the CPG modules in the loop are
Biologically Inspired Robotics
where n is the number of CPG modules in the network; m is the number of
neurons in one CPG module; s is the serial number of neurons connected to
the jth neuron; u j,i is the membrane potential of the jth neuron in the ith CPG
module; v j,i is the variable that represents the degree of adaptation; u 0,i is the
tonic driving input; τ 1,i and τ 2,i are the time constants; β is the adaptation coefficient; w is the weight between neurons; w ik is the connection weight of the
ith module from the kth module; y j,i is the output of the jth neuron in the ith
CPG module; y out,i is the output of the ith CPG module; w 0 is a constant value
for connection weight; and r is the number of the CPG in the network with
feedback connection.
The basic concept of this network is shown in Figure 2.2. Compared with
the unilaterally connected network, this network has the same unidirectional couplings between the oscillators. The difference is that the output of
the the rth CPG module is provided as feedback to the first module to form a
closed loop. The connection weight between CPGs w ik takes the same value
from Equation (2.3). In addition, the weight of the feedback w 1r from the rth
module to the first one should adopt a weight value, given by w 1r = w 0 .
Because the rth CPG module transmits the same value to the (r + 1)-th and
the first CPG module, the output waves of the (r + 1)-th and the 1st module
have completely identical shapes and phases. The (r + 1)-th module will also
transmit its output to the (r + 2)-th module as the first module transmits its
output to the second module. Thus, the (r + 2)-th module will generate the
same output as that of the second module. In the same way, the following
output of CPG modules will take the similar process mentioned above as
CPGs in the loop. Thus, the output of the CPG modules on the outside of the
loop can be represented as
y , + = y
p = , ,..., q = 1 2
out pr q
out q
,
1 2
, ,... r
(2.4)
To compare the output of a CPG network with a unilateral connection and
feedback connection, a network composed of six CPG modules is constructed.
For the network with feedback connection, the sixth module is selected to
provide feedback to the first CPG module. The results are simulated by use
of both the dual-neuron mutual inhibition model and the tri-neuron cyclic
inhibitory model. Rhythmic outputs of CPG models with respect to the set of
CPG parameters in Table 2.1 are shown in Figures 2.3 and 2.4.
As shown in Figure 2.3, the amplitude of the CPG outputs in the unilaterally connected network is not of uniform size and the waveforms cannot
form a perfect traveling wave to suit the control of the multilink robot. Thus,
it is necessary to make adjustments to obtain applicable rhythmic signals
for control of the robot. The output of CPGs in Figure 2.4 has a feedback
connection with the same amplitude and phase difference. Furthermore, it
is obvious that all of the output waves of the CPG modules in the loop are