An in silico central pattern generator: silicon oscillator, coupling, entrainment, and physical computation.

Abstract

In biological systems, the task of computing a gait trajectory is shared between the biomechanical and nervous systems. We take the perspective that both of these seemingly different computations are examples of physical computation. Here we describe the progress that has been made toward building a minimal biped system that illustrates this idea. We embed a significant portion of the computation in physical devices, such as capacitors and transistors, to underline the potential power of emphasizing the understanding of physical computation. We describe results in the exploitation of physical computation by (1) using a passive knee to assist in dynamics computation, (2) using an oscillator to drive a monoped mechanism based on the passive knee, (3) using sensory entrainment to coordinate the mechanics with the neural oscillator, (4) coupling two such systems together mechanically at the hip and computationally via the resulting two oscillators to create a biped mechanism, and (5) demonstrating the resulting gait generation in the biped mechanism.

Full text

Page 1

Abstract. In biological systems, the task of computing a
gait trajectory is shared between the biomechanical and
nervous systems. We take the perspective that both of
these seemingly different computations are examples of
physical computation. Here we describe the progress
that has been made toward building a minimal biped
system that illustrates this idea. We embed a significant
portion of the computation in physical devices, such as
capacitors and transistors, to underline the potential
power of emphasizing the understanding of physical
computation. We describe results in the exploitation of
physical computation by (1) using a passive knee to
assist in dynamics computation, (2) using an oscillator to
drive a monoped mechanism based on the passive knee,
(3) using sensory entrainment to coordinate the me-
chanics with the neural oscillator, (4) coupling two such
systems together mechanically at the hip and computa-
tionally via the resulting two oscillators to create a biped
mechanism, and (5) demonstrating the resulting gait
generation in the biped mechanism.
1 Introduction
Locomotion is a fundamental activity in both animals
and robots. Recent work in biology and robotics has
clearly shown that the computation of the gait trajectory
during locomotion is shared between the biomechanical
and nervous systems (Taga et al. 1991; Taga 1995a,b;
Pratt and Pratt 1998; Kimura et al. 1999; Pratt and Pratt
1999; Dickinson et al. 2000; Kimura et al. 2001a,b). In
this paper, we demonstrate a silicon-circuit embodiment
of a central pattern generator (CPG) locomotor con-
troller for a robotic monoped and biped. In both
monopedal and bipedal systems, the passive dynamics
contribute substantially to the computation of the
overall leg trajectories. By explicitly sharing the compu-
tation of gait trajectory between a CPG, instantiated in
silicon, and the mechanics, instantiated in robotic
hardware, we have created a robot with a very low-
complexity control system.
When fully developed, this chip could be embedded in
a robot such as Tekken (Kimura et al. 2001a,b) to create
a remarkably elegant and compact robot control system.
This technology may also have alternative uses in the
rehabilitation of patients with various neurological def-
icits, since the circuit could be adapted for correction,
augmentation, and generation of locomotor patterns in
such patients.
1.1 Background
Locomotion in animals is strongly periodic. The ani-
mal’s nervous system modulates its locomotor behavior
in response to altered sensory input and the volition of
the animal. It is generally agreed that the neural
architecture necessary to support the control of loco-
motion is structured as follows: (1) Neural circuits in the
spinal cord called the CPG are capable of generating
the neural oscillations necessary for locomotion. (2) The
CPG interacts strongly with sensory feedback from the
muscles and joints. (3) Descending modulatory influ-
ences from the brain produce adaptive locomotor
movements based on volition and exteroceptive senses,
e.g., vision (Grillner 1981; Armstrong 1986; Cohen and
Boothe 1999).
The CPG is composed of intrinsic oscillators em-
bedded in a distributed system of circuits (Grillner
1981). These circuits enforce constraints among them-
selves, which gives rise to phenomena such as patterns of
phase relationships between oscillators. This allows
flexibility in generating the necessary phase relationship
for a variety of gaits. This is evident in quadrupeds that
can walk, trot, canter, gallop, pronk, and pace, pre-
sumably by modifying the connections between the
intrinsic oscillators to generate the required movement
Correspondence to: M.A. Lewis
Biol. Cybern. 88, 137–151 (2003)
DOI 10.1007/s00422-002-0365-7
? Springer-Verlag 2003
An in silico central pattern generator: silicon oscillator, coupling,
entrainment, and physical computation
M. Anthony Lewis1, Ralph Etienne-Cummings2, Mitra J. Hartmann3, Zi Rong Xu5, Avis H. Cohen4
1Iguana Robotics, Inc., P.O. Box 628, Mahomet, IL 61853, USA (e-mail: tlewis@iguana-robotics.com)
2Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
(e-mail: etienne@ece.jhu.edu)
3Jet Propulsion Laboratory, 303-300, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
(e-mail: hartmann@brain.jpl.nasa.gov)
4Department of Biology, Neuroscience and Cognitive Science, University of Maryland, College Park MD 20742, USA
(e-mail: ac61@umail.umd.edu)
5Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
Received: 31 October 2001 /Accepted in revised form: 17 September 2002

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patterns. A computational model of biological nervous
systems must have a similarly distributed architecture.
The mechanics also ‘‘compute’’ a great deal of the
walking movement, as has been shown by Ruina and
colleagues in their work on passive walking bipeds
(Garcia et al. 1998a,b; Collins et al. 2001) and by Pratt
and Pratt, in the use of lower-leg-segment swing in an
actively controlled robot (Pratt and Pratt 1998, 1999).
By physical computation, researchers understand that
the physics of devices (whether Newtonian mechanics or
device physics of circuits) can be a substrate for com-
putation. Exploiting the physics of computation may
lead to vastly more efficient designs. This is a common
thread in the work presented here as well.
The contribution of this current work is to demon-
strate the minimal system that satisfies the principles
above (i.e., distributed CPG, computation using passive
dynamics) that can control a bipedal walking mecha-
nism. The work gives compelling support for the hy-
pothesis that systems designed based on such principles
can be realized using minimal computational resources.
This study does not address balance and postural
control. To our knowledge, integration of the CPG with
balance and postural control in a real bipedal machine
has not been addressed. We plan to reserve this topic for
future work, but we are confident that it will be com-
patible with the current approach.
1.2 Comparison to previous CPG chip work
CPG chips and circuits have been created before. For
example, Still reports on a VLSI (very large scale
integration) implementation pattern generator used to
drive a small robot (Still 2000; Still et al. 2000). This
circuit captured some of the basic ideas of a CPG, but it
did not incorporate a motor neuron output stage, and
the system did not provide for adaptation via sensory
input. However, the work did demonstrate control of a
rudimentary walking machine.
The work of DeWeerth and colleagues (Patel et al.
1998) captures the neural dynamics on a much more
detailed level than has been achieved here. However,
there are great difficulties in applying such a system to the
controlofarobot.Primarily,parametersensitivitymakes
such circuits difficult to tune. To address this issue,
DeWeerth and collaborators have implemented neurons
that self-adapt their firing rate (Simoni and DeWeerth
1998). The adaptation, however, is independent of ex-
ternal inputs from sensors. While detailed neural models
are difficult to work with in silicon, we will undoubtedly
learn a great deal from these efforts in the future.
Ryckebusch and colleagues (1994) created a VLSI
CPG chip based on observations in the thoracic circuits
controlling locomotion in locusts. The resulting VLSI
chip was used as a fast simulation tool to explore un-
derstanding of the biological system. Their system did
not use feedback from sensors, nor was it connected to a
robotic system. However, their objective – of modeling a
particular biological circuit – was different than the
objective described in this paper.
Our work differs from the previous work in several
respects. First, we allow adaptation based on sensory
input. Adaptation is shown as a phase resetting of the
CPG based on certain sensory triggers. Second, we make
use of integrate-and-fire neurons for the output motor
neurons. Our abstraction is at a higher level than other
reported work (Patel et al. 1998; Simoni and DeWeerth
1998). We believe that by using a higher level of
abstraction we will be able to more easily implement
on-chip learning. In systems based on numerous inter-
related parameters, it is not apparent how learning at the
level of behavior can be coupled to low-level-parameter
changes.
1.3 Current approach
The long-range objective of the current work is to
develop a CPG chip that can control a biped or
quadruped robot. This chip would adapt based on
sensory feedback as well as on input from higher centers.
Physical computation – both in the mechanics and in the
control system – will be maximally exploited. In the
initial work, described here, we develop a minimal
system that incorporates the major principles of biped
locomotion, with the exception of the incorporation of
higher-level input. The key components we have iden-
tified are:
(1) Creating a monoped where the lower limb segment
‘‘computes’’ a significant portion of the gait trajec-
tory. Below we will begin by analyzing the dynamics
of the lower-limb segment and understanding the
dynamics of this system through mathematical
analysis.
(2) Driving this monoped with a single oscillator.
(3) Using entrainment to close the loop between the
mechanical system and the neural oscillator. We give
results from an analysis of the role of sensory feed-
back in entraining the oscillator and we use geo-
metric methods to visualize this effect. We also show
the results of ‘‘lesion’’ experiments in which we
demonstrate the effect of reversibly removing sen-
sory feedback to the oscillator. We can now use this
single leg to build a biped.
(4) Together the two oscillators for each half monoped.
Coupling of oscillators is a nontrivial problem. First,
we give mathematical results of coupling using pulse
trains and determine that for our system multispike
coupling gives superior performance over single-
spike coupling. This was a major result of the anal-
ysis. Now we demonstrate that the effect can be seen
in the geometry of the map function describing the
convergence of the two oscillators. Finally, we per-
form experiments on the chip to empirically deter-
mine the map function of the chip. We find that the
model gives an excellent prediction of the chip be-
havior.
(5) Demonstrating the complete biped. Here we record
the overall gait trajectories to show that the gait is
smooth and resembles a natural gait.
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2 Experimental apparatus
2.1 Robot mechanism
Mechanical elements. Two robotic mechanisms were
used in the experiments described here. The first was a
mechanism with a single leg. The leg consisted of a small
(14 cm high, foot to hip), two-degree-of-freedom (DOF)
mechanism. The upper joint was driven by a hobby-type
servomotor Futaba MDL 9303. For all experiments
described only the hip joint was driven, and the ‘‘knee’’
remained completely passive. The knee rotated on a low-
friction ball-bearing joint and was prevented from going
into hyperextension by a hard mechanical stop at zero
degrees relative to the thigh. The passive knee is a critical
element of the robotic setup, as we exploit the dynamics
of the knee to compute the lower-limb trajectories.
The leg was suspended by a harness that served to
constrain the mechanism’s hip rotational axis, as well as
mediolateral (side to side) movements. The monoped
could move forward and up and down to a small extent.
In the bipedal device used in the walking experiments,
two legs were attached to a ‘‘hip’’ assembly that in turn
was attached to a rotating boom. For the running ex-
periments, the boom was also used; in some cases a
linear sliding mechanism was used. In all cases, the hip
was restricted rotationally. Because of this constraint,
the effect of one leg on the other was primarily through
translational movements (up/down and forward/back-
ward). The bipedal setup is shown in Fig. 1. The pitch
angle of the hip, specified by the harness, was found to
be critical during running (see Sect. 2 in the Appendix
B). If the harness were not set up with a correct angle,
the system would not run well. The harness also limited
how far the leg could fall. This limitation allowed the
robot to start running from a standstill. Without this
limitation, the robot would probably not have been able
to raise itself into a running posture on its own.
Sensors. Each robotic leg had three sensors on it. Two
custom-built inductive LVDT-type (linear variable dif-
ferential transformer) sensors monitored the position of
the knee and hip joints. LVDT sensors were used
because they introduce minimal friction and have
infinite resolution. Having minimal friction minimized
interference with the natural dynamics of the passive
knee. A miniature load-cell sensor (Sensotec Model 13)
on the foot was used to monitor ground forces. The
units of the load cell are uncalibrated in all figures.
Sensor signal processing.
sensor at the hip monotonically increased as a sigmoid
function of hip angle over the swing range used in our
experiments. The output of the sensor was sampled
digitally and interval-coded. Two intervals were selected
as representing the extreme front or back movement of
the hip. When these extremes were reached, the corre-
sponding interval became active. When the leg swung
forward past the point designated as the front extreme,
the front interval went active. When the leg swung
backward past the point designated as the back extreme,
the back interval went active. These intervals thus served
as binary signals that indicated the extremes of hip
movement and were thus used to entrain or adapt the
CPG chip. The analog values of the sensors indicating
the joint angle were used for analysis but were not fed
back to the controller.
The signal from the LVDT
Walking and running surfaces. In all experiments with
the monoped, the running surface consisted of a rotating
drum that was free to rotate under the contact forces of
the leg. In the walking biped experiments, a flat,
carpeted surface was used. The running surface for the
legs consisted of a powered conveyor belt that served as
a treadmill (Dormer Corp., model 2100M). A custom-
built speed controller allowed the treadmill to establish
and maintain a given rate.
2.2 The CPG chip
A CPG chip was fabricated in silicon using a 1.2-micron
CMOS process. The chip contains electronic analogs of
a biological neuron: inhibitory and excitatory ‘‘synapses,’’
‘‘cell membrane,’’ ‘‘axon hillock,’’ and ‘‘axon.’’ In
addition, the chip contains dynamic analog memories
that can be used with synapses to modulate weights or to
modulate the membrane conductance. Using these
components, nonlinear oscillators, integrate-and-fire
spiking neurons, and graded response neurons can be
constructed (Fig. 2). In addition, in some cases the
hysteretic comparator was replaced by a linear buffer
circuit to obtain a graded response. The resulting circuit
was used as part of adaptive circuits for the motor
neurons.
Two integrate-and-fire neurons, based on the neuron
drawn schematically in Fig. 2a, are included on the chip.
Fig. 1. RunningMan Biped on treadmill. Mechanism is about 14 cm
tall. Hips are actuated. Knees are passive. The foot is a round pad
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The integrate-and-fire neuron can also be made to act as
a pacemaker envelope neuron.
We model the integrate-and-fire neuron of the chip
usingthefollowingsetofnonlineardifferentialequations:
Cmem
i
dVmem
i
dt
¼ Ispon? ST
iIdis
þ I0
X
j
Wþ
ijST
j?
X
j
W?
ijST
j
!
ð1Þ
STþe
i
¼
1 ifðST
if ðST
i¼ 1 ^ Vmem
i¼ 0 ^ Vmem
i
> V?
< Vþ
TÞ _ ðVmem
TÞ _ ðVmem
i
> Vþ
< V?
TÞ
TÞ
0
ii
(
ð2Þ
where Cmem
the state of the hysteretic comparator as well as the
neuron output (0 or 1) at time T þ e. The currents Idis
and Isponare discharge and spontaneous charging rates,
respectively. The current I0 is the nominal synaptic
currents from incoming excitatory and inhibitory spikes;
ST
j2 0;1
applied to the incoming spikes, and Vmem
‘‘membrane’’ voltage. W?
atory weight factors, respectively.
i
is the ‘‘membrane’’ capacitance and STþe
i
is
½? represents the state of the weighting factors
ij;Wþ
i
is the
ijare inhibitory and excit-
Relationship of CPG circuit to biological neurons.
neuron is a lumped parameter membrane model of a
biological neuron. A single voltage, Vmem
state of the membrane voltage at the dendrites and cell
body. The state, ST
i, represents the state of the axon
(firing or not firing).
The synapse strength consists of a ‘‘weight’’ that is
represented as a transistor that controls the maximum
current flow onto the membrane capacitor. A constant,
applied voltage sets the current flow, or ‘‘weight.’’ In
principle, this voltage could be placed in an analog
memory on board the chip, but we did not do that here.
This weight determines the quanta of charge placed on
the cell membrane with each incoming spike. In contrast
to biological neurons, this ‘‘quanta’’ of charge is rela-
tively independent of the cell-membrane voltage.
The axon hillock, which in biological neurons is the
site of axon spike generation (Kandel et al. 1991), is
represented as a hysteretic comparator and a ‘‘spike
reset’’ transistor, representing a voltage-gated ion
channel. In contrast to a biological neuron, we use
‘‘hysterisis’’ and a ‘‘spike reset’’ gate as a rough substi-
tute for a time-inactivation mechanism (Hille 1984). The
hysteretic comparator and ‘‘spike reset’’ transistor work
in the following fashion: the hysteretic comparator has
upper Vþ
at design time and cannot be changed after fabrication.
When the membrane potential rises to exceed the Vþ
the output of the comparator goes high. The output will
stay high until the membrane potential falls below V?
The output of the comparator is fed back to a gate at the
input to the comparator (‘‘spike reset’’). This resets the
membrane potential by draining charge from Cmem.
Because of the hysteresis, the draining continues until
the membrane potential is below V?
tude of the discharge current Idis>Ispon by design, the
net charge is drained from Cmem. This organization in-
troduces a second state variable needed to create an
oscillatory system. This two-state system is in the spirit
of the neuronal models of FitzHugh (1961) and Nagumo
et al. (1962).
The
i
, represents the
tand lower V?
tthresholds, both of which are set
t,
t.
t. Since the magni-
Creating spiking and envelope pacemaker neurons. Note
that in this circuit, the interspike interval is determined
by the magnitude of Ispon. By setting Idis? Ispon we
obtained a spiking output. If Idis? 2 ? Ispon, we obtain a
‘‘pacemaker’’ output with roughly symmetric high and
low periods. By varying the ratio of the Ispon and Idis
currents, the neuron can behave as an integrate-and-fire
neuron or as a pacemaker (oscillator) neuron. In the case
Fig. 2. Schematic of neurons implemented on the CPG chip. a
Pacemaker and integrate-and-fire neural elements. b Interneuron with
linear amplification
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of the pacemaker, although the output is square-wave
and nonspiking, it can be considered the time-averaged
output of a spike train.
Interneuron. By using a linear buffer in place of the
hysteretic comparator and eliminating the discharge
path, we can create a graded response neuron that does
not spike.
2.3 Enhancement to pacemaker neuron
In some experiments reported below, we used an
enhanced pacemaker neuron. A small PICTM(Micro-
Chip 16C877) processor was used to generate a brief
burst of pulses upon transition from the low to the high
state. As will be discussed below, the chip was originally
designed for continuous coupling between neurons to
achieve the alternating burst pattern seen in motor
neurons across the midline. However, we found it
necessary to use pulse coupling for these neurons (cf.
Sect. 3.4 for further explanation of this problem and its
solution). This change resulted in a shortage of silicon
neurons. As a consequence, we used a small processor to
generate the bursting pattern needed for pulse coupling.
We feel that this change did not materially affect our
results, as this processor can be replaced in future work
with a neuronal equivalent.
2.4 Interface to robot
A PICTM(MicroChip 16C877) processor was used to
convert the pulses from the motor neurons to a pulse-
width-modulated signal for the motors. The PIC imple-
mented a pure integrator. This integrated signal was
then set as the position for the motor, which has its own
position-control circuit on board. A position command
is given to the motor, and the motor is driven toward the
desired set point. We plan to use force-level control of
the hip in future iterations.
The neurons of the CPG chip were interfaced to a
servomotor using a rudimentary muscle model, and the
muscle dynamics were simulated as a low -pass filter to
smooth the output of the spiking neurons. This was
followed by an integrator, implemented in software, to
convert the position signal to a position command
needed by the servomotor. A bias was intentionally
introduced into the chip to cause an asymmetry in the
backward and forward swing of the leg. This bias
might be typical of uncompensated parameters in a
chip.
An oscillator frequency was selected by hand to be
approximately 2–3 Hz. This frequency would excite the
mechanical structure of the robot and cause the leg to
run on a rotating drum. At lower frequencies, for ex-
ample 1 Hz, the foot would drag on the treadmill when
in swing phase. If the frequencies became too high, the
amplitude of the leg swing decreased significantly due to
the limited bandwidth of the hip actuator. We assume
that if the actuator had sufficient bandwidth, the leg
shank would not lock in place at the end of the swing
phase.
3 Analysis, simulation and empirical results
3.1 Analysis of passive knee
The passive knee can be modeled with the following
equations:
€ q q ¼
?sinðqÞ ? € y y ? cosðqÞ ? € x x ? sinðqÞ ? g
l
ðÞ
? k ? _ q q
ð3Þ
q < a
ð4Þ
where € x x; € y y
lower segment below the knee) angle, k is velocity
dampening, l is the shank center of mass relative to the
knee, g is the gravitational constant, and a is the hip
angle. By inspection, the only way to influence knee
trajectory is by accelerations at the knee cap. In the
absence of knee acceleration, the shank behaves as an
inverted pendulum. The most significant way to influ-
ence the frequency of the knee, and hence the running
frequency of the entire system, is to control the velocity
of the knee joint at the stance-swing boundary. Through
simulation analysis, (see Appendix B) we found that the
range over which the system could run could be slightly
extended by controlling the energy of the lower limb at
takeoff. But in all cases, the frequency of the swing phase
could not be significantly changed if running was to be
stable. This accords well with the observation that in
animals the stance phase is altered in the transition from
walking to running. It is not typically altered in the
swing phase, which remains constant. We anticipate that
very little control would be needed in the knee during
normal walking and gait transition. Appendix B gives
the complete analysis of the passive knee via simulation
results.
½? is the knee acceleration, q is the shank (the
3.2 The complete monoped circuit
Figure 3 shows the overall schematic for the circuitry to
control the monoped leg and a picture of the biped
mechanism. It is important to note that the circuitry
must be duplicated twice to control the biped shown.
For each leg, a pacemaker neuron drives two bursting
(spiking) neurons. In our system, we used two outputs
from the pacemaker: an inverted and a noninverted
output. When active, these signals inhibit the motor
neuron on which they synapse. This arrangement does
not allow both motor neurons to become active at one
time (i.e., no cocontraction is possible).
The spiking neuron receives self-limiting feedback.
This feedback limits the firing rate of the neuron.
Another circuit, built from an interneuron, supplies a
bias to increase the firing rate of the neuron, acting in
opposition to the self-limiting feedback. This bias is
reduced as the leg reaches the limits of travel. The
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circuit ensures that the stride length has the correct
width.
A feedback path goes from the nonlinear sensor to
the pacemaker cells. This feedback pathway resets the
pacemaker and allows the movement of the leg to
entrain or synchronize the pacemaker oscillator.
3.3 Effect of sensory feedback
3.3.1 General analysis of the effects of sensory feedback.
As mentioned in the introduction, sensory feedback is
critical in adapting a walking machine, either animal or
robotic, to the terrain. We performed a mathematical
analysis of the effect of sensory feedback on the
oscillator using map functions and stairstep diagrams
(Hale and Koc ¸ ak 1991).
Figure 4 considers the case of connecting a single os-
cillator to a leg of a monoped robot. The upper and lower
dashed lines in Fig. 4a and b are the ‘‘map functions.’’
These functions are created by solving the differential
equations (Eqs. 1–4 of Sect. 2.2) for hip position at every
half cycle of movement. The hip position at every half
cycle is then geometrically visible by connecting the main
diagonal with the two map functions (see caption for
details). Note that the choice for the hip position to range
between ? ?1 and ? 3 represents only arbitrary units.
Figure 4a clearly shows that in the absence of sensory
feedback, the hip position moves forward unbounded.
In the real robot, this would mean a rapid deterioration
of the gait. In contrast, the addition of sensory feedback
to entrain the CPG had the effect of forming a restric-
tion or ‘‘bottleneck’’ on the geometric surface (Fig. 5b).
This forced the leg to oscillate around a particular sen-
sory region, ensuring a stable gait. In contrast, in the
absence of sensory feedback, the leg continuously moves
forward, as will be described in Sect. 3.3.3; we were able
to confirm the results of this analysis through lesion
experiments in the real monopedal system. The stairstep
analysis presented above exactly confirmed the results of
running in the real robot.
Although the details of this analysis are complex and
highly specific to this chip, the stairstep method for vi-
sualizing the stability of the walking/running system is
effective.
3.3.2 Monoped ‘‘Running’’ with a passive knee: sensory
feedback intact. In this experimental setup, the CPG
circuit drove the actuator in the hip joint, and the
monoped was suspended above a rotating drum. Data
were collected from three sensors: foot pressure, knee,
and hip. Hip-sensor feedback from the hip was used as
feedback to the CPG. The foremost result was that the
Fig. 3. Interface schematic for RunningMan Biped. The CPG chip is
used to construct pacemaker and bursting neurons. The outputs of the
bursting neurons are summed. A conversion circuit (not shown)
converts the burst patterns to motor commands for the biped
mechanism. The filled circles represent inhibitory synapses
Fig. 4. The effect of sensory feedback is to alter the map function
describing the nominal position of the limb at each cycle. To compute
hip position and drift using these diagrams: (1) Choose a starting
point on the diagonal line (e.g., point ‘S’ in the figure). This is the
initial starting position of the hip. (2) Now move along a vertical line
to the upper dashed line, then horizontally to the diagonal line. This is
the position of the hip after one half cycle. (3) Repeat the process, this
time drawing the vertical line down to the lower dashed line. In a
perfectly balanced or symmetric system, the distance between the
dashed lines and the diagonal solid line are equal. In an unbalanced or
perturbed system, these lines have unequal distances. a In the absence
of sensory feedback, the lower dashed line is slightly closer to the
diagonal than the upper dashed line. The position of the limb
gradually drifts. b In this case, we add sensory feedback acting in a
phase-dependent manner. The effect is to create a bottleneck. This
bottleneck ‘‘traps’’ the trajectory of the limb, holding it in a stable
position. Lesioning (Sect. 3.2.3) transforms the geometry of the system
from b to that of a
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circuit adapts such that the passive knee joint has the
correct dynamics to enable running.
Figure 5 shows a phase plot of the hip and knee po-
sition and foot contact force. The bulk of the trajectory
describes a tight ‘‘spinning top’’ shape, while the few
outlying trajectories are caused by external disturbances.
After a disturbance the trajectory quickly returned to its
nominal orbit, implying that the system was stable.
3.3.3 Monoped running with a passive knee: sensory
feedback lesioning. We next established that sensory
feedback was critical to entrain the pacemaker neuron
and thereby ensure a centered stride, by performing
selective sensory feedback ‘‘lesions.’’ As described in
Sect. 2.4, we first added a chip bias so that there were
significant asymmetries in the forward and backward
swing of the limb. This accounts for the asymmetric map
functions of Fig. 4a. Figure 6shows the effect of lesioning
sensory feedback on the position of the hip and knee
joints and on the tactile input to the foot. When feedback
is intact, the circuit adjusts for the chip-induced asym-
metry, but when feedback is lesioned, the leg drifts
backwards. The leg returns to a stable gait only after
sensory input is restored.
We can relate these results to Fig. 4. The intact case
corresponds to Fig. 4b. The lesion case corresponds to
Fig. 4a. The effect of lesioning the robot is to transform
the geometry of the map function from a bottle neck
configuration to two parallel lines. These parallel lines
are sensitive to the slightest imbalance in the system. The
caption for Fig. 4 describes how to compute the position
of the hip through time.
3.4 Creating a minimal coupled CPG
The most basic property of the spinal CPG is coordi-
nation across the midline, which requires neurons to be
bidirectionally coupled. In our first approach to bipedal
coupling, we directly coupled two pacemaker neurons
together, with the output of one being fed directly to the
membrane capacitor of its complement. We called this
continuous coupling. We found that continuous cou-
pling resulted in patterns of oscillation that were very
unstable, since one neuron would disrupt the behavior of
the other.
In biological systems, however, coupling across the
midline in vertebrates is accomplished using spiking
neurons with decaying spike rates after the initial onset
of coupling (i.e., biological neurons use pulse coupling
with rate adaptation). This is quite different from the
continuously coupled nonspiking pacemaker neurons in
our system.
We therefore tried a second approach in which we
used spiking neurons to couple the two oscillators. This
approach revealed a significant advantage of using
spiking neurons in our system. The CPG oscillators were
particularly well behaved, and very stable, when con-
structed with spiking neurons to achieve pulse coupling.
Specifically, we found that the use of spiking neurons
improved the convergence of the phase locking.
It is generally agreed that the primary benefit of
spiking neurons is their tolerance to noise in the axon
(transmission line) and their ability to transmit over
large distances. In our case, however, we discovered
another advantage of using spiking neurons: the sys-
tem is amenable to discrete analysis using map func-
tions to describe their dynamics. The analysis and
simulation details are presented in Appendix A, and
we summarize only the major results in Sects. 3.4.1
and 3.4.2.
3.4.1 Geometric analysis of pulse coupled neurons. As in
the case of sensory feedback, we used geometrical
Fig. 5. Hip-, knee- and foot-contact phase diagram. Most of the
trajectory is in a tight bundle, while the outlying trajectories represent
perturbations
Fig. 6. The effects of lesioning sensory feedback. When the feedback
is lesioned (time 11–19 s and 31–42 s), the hip drives backward
significantly. As it does, the foot begins to lose contact with the
ground, and the knee stops moving. When the lesion is reversed at 19
and 42 s, the regularity of the gait is restored
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analysis to understand bidirectional pulse coupling
between two neurons. In this case, the map functions
(Hale and Koc ¸ ak 1991), describe the phase change of the
output of a spiking neuron due to pulse perturbations
from a second spiking neuron. As shown in Fig. 7, the
phase difference between the two oscillators does not
converge to an asymptotically stable fixed point, but
rather, convergence is bounded to a certain region. That
is, the phase difference will be bounded, but it is not
guaranteed to be zero. The details of this analysis can be
found in Appendix A.
We validated this mathematical analysis of spike
coupling by determining the real map function in the
CPG chip. As will be shown in Sect. 3.4.2, the map
function predicted by the analysis and the map function
empirically determined from the chip accord extremely
well.
3.4.2 Empirical determination of map function for multi-
pulse neuronal coupling. One of the major results of the
mathematical analysis derived in Appendix B is that
multiple spikes will result in better coupling charac-
teristics for our system. However, a key remaining
question is whether the mathematical model is a valid
description of the CPG chip.
To answer this question, we performed experiments
on the chip to determine the geometry of coupling. We
conducted an investigation in the case of a master os-
cillator driving a slave with either a single pulse or two
pulses for coupling.
For the driving oscillator, we used a very slow master
oscillator that had a period that was not an integer
multiple of the slave frequency. We then recorded the
state of the neuron output, S, for an extended period of
time (approximately 1–2 min).
Custom-built software extracted the period of the
pacemaker when there were no perturbations (control
case) and before and after a spike (or spikes) from a
second neuron perturbed the pacemaker. Figure 8a
shows the geometry of coupling based on a perturbation
with a single pulse. Figure 8b shows the geometry of
coupling based on perturbation with double spikes. On
these graphs, a fixed point is indicated by a map function
crossing the diagonal line.
In Fig. 8a, the slope becomes vertical at the fixed
point, manifesting itself as discontinuity when the phase
is about 0.46. On these graphs, a fixed point is indicated
by a map function crossing the diagonal line. In the case
of two pulses, the slope flattens out dramatically. This
indicates a region of asymptotic convergence for the
oscillators. Comparing the result to those predicted by
the model (see Figs. A1a and A3a), we find excellent
accord between the predictions of our model of the CPG
chip and the actual data collected. These results strongly
support the mathematical analysis presented in the
Appendix.
3.4.3 Coupling the CPG to the runningman robot. We
next used the pulse-coupled pacemakers to control the
running biped. Figure 9 shows the circuit for the
complete CPG for the biped robot. The pacemaker
circuit drives motor neurons that interface the minimal
CPG to the robot, and the phase of the CPG and the
firing rate of the motor neurons are adjusted based on
sensory feedback from the leg. Because RunningMan
has passive knees, the trajectory of the lower limb
segment is implicitly ‘‘computed’’ as a physical compu-
tation.
3.5 Bipedal running
The biped mechanism was suspended above the tread-
mill using the harness, and the CPG was then started. A
digital video camera placed perpendicular to the tread-
mill recorded the movement of the legs. To facilitate
data reduction, LED markers were placed on the robot.
The approximate positions of the markers were: center
Fig. 7. Map functions of bidirectional coupled neurons. a Single pulse
case, different periods. The derivative at the fixed point is infinity.
Thus, with a single pulse it is not possible to achieve asymptotic
convergence with this neuron. b Multipulse case, equal periods. The
geometry begins to alter such that the line passing through the fixed
point is horizontal. Although the derivative is still infinity over a small
region, this region is much smaller
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of hip rotation, center of knee rotation, and middle of
foot.
Data analysis was done offline. Custom software was
written to track the x-y pixel position of each marker for
each frame and produce a table of results.
A second software program plotted the movement of
the legs. Figure 10 shows a single stride for the right leg.
The dark line shows the leg positions as it moves
backward. The foot hits the ground, and the knee locks.
When the leg changes direction (the leg is now drawn in
light gray), the knee breaks, and the leg swings forward.
When the leg reaches the end of travel, it hits an end
stop.
The light gray dashed line shows the trajectory of the
foot through time. The trajectory appears smooth and
somewhat natural. Notice that the frame of reference of
this video is the hip of the robot.
Notice also that the leg appears to swing forward
more than it does backward. Empirically, if the leg is
rotated more forward, the limb collapses under the ro-
bot. In practice, the lower limb would need to have a
mechanism to keep it locked in place. Pratt uses a
locking torque, applied at the knee, to accomplish this
(Pratt and Pratt 1999).
It is important to notice that if the knee had been
actuated, then all the positional and dynamical rela-
tionships between the hip and knee actuators of both
limbs would have to have been computed and imposed
by the control system. This would have required a
complicated controller, which is difficult to implement in
software and more difficult in hardware. A major result
of this work is that the ‘‘physical computation’’ per-
formed by the passive knees simplifies the biped control
system immensely. Furthermore, the minimal controller
and the natural dynamics of the passive knees accom-
plish very realistic (biologically speaking) running
action.
Experiments were also performed on the ground
where the robot was able to walk quickly around a cir-
cle. AVI clips showing the running biped, monoped, and
walking biped are available at http://www.iguana-
robotics.com.
4 Conclusions
In this work we presented analysis, simulation, and
empirical results in the use of a CPG chip to control a
monoped and a biped mechanism. We demonstrated
how, beginning with capacitors and transistors, we built
three different neural elements. We also showed how
these elements could be coupled together to produce a
simple circuit sufficient for creating a basic pattern of
movement in the hip joint of the mechanism. Through
analysis using geometric methods and simulation, we
Fig. 8. Results of empirical experiment supporting the CPG model
and analysis of coupling. a Single-spike coupling case. b Two-spike
coupling case
Fig. 9. Circuit for controlling the biped robot. The two halves of the
control system are coupled together using pulse coupling (multiple
spikes)
Fig. 10. The trajectory of one leg of the biped. See text for
explanation
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showed how pulse coupling could very efficiently couple
together basic neural elements to create a simple CPG.
We showed how sensory feedback influenced the
geometry of the system’s limit cycle.
In addition, we used a passive knee and showed how
this leg segment could participate in the computation of
the overall trajectory of the mechanism.
We conclude that with a reduced system of a few
transistors and capacitors (not counting the PICTM
processor, which could be replaced easily with a neuro-
nal equivalent in future work) and using insights from
biology, it is possible to generate basic movement pat-
terns in a mechanical system that bear a remarkable
overall resemblance to their biological counterparts.
The major challenge ahead is to incorporate balance
control and supraspinal control. The minor challenges
include the creation of a new chip with sufficient neurons
to implement all aspects of the design completely on a
self-contained chip and to create a new mechanism with
a locking knee for stability and a foot and ankle. Finally,
after the evolution of the chip is complete, it may be able
to control its biological counterparts, since it has been
designed from the beginning to be compatible with the
principles of biological systems.
Acknowledgements. The authors acknowledge support of Grant
No. N00014-99-0984 from ONR to Lewis & Etienne-Cummings,
NSF Career Grant #9896362 to Etienne-Cummings and NIH grant
MH44809 to Cohen. MAL thanks Suzanne Still for formative
discussions on the possibility of building a silicon CPG chip. The
authors would also like to thank Francesco Tenore for work in
constructing the experimental setup as well for as useful comments
on this manuscript. Many thanks go to Kaijen Hsiao and Chris
Milne for their suggestions and comments on the manuscript.
Appendix A. Analysis of pulse coupling
Here we describe the use of geometric analysis to
understand pulse coupling between our silicon neurons.
Consider two envelope pacemaker oscillators with
similar frequencies and one-way spike coupling. By
‘‘one-way’’ we mean that one oscillator will be the
‘‘master,’’ trying to entrain the ‘‘slave’’ oscillator. By
‘‘spike coupling’’ we mean that on each cycle, the master
oscillator will send a certain number of spikes (perhaps
only one) to the slave oscillator, instead of sending a
continuous entraining pulse, as would be the case in
‘‘continuous coupling.’’ Let us derive an equation
describing the change in phase of the slave oscillator
when the master sends a single spike to the slave oscil-
lator. The slave oscillator has equations governing its
behavior:
Cmem
i
dVmem
i
dt
¼ Isponþ K
ðA1Þ
Cmem
i
dVmem
i
dt
¼ Ispon? Idisþ K
ðA2Þ
for the charge and discharge cycle, respectively, where K
is the spike-coupling function, considered to be an
impulse function that adds charge q to the driven
oscillator. The effect of a single pulse on either the
charge or the discharge cycle is:
q
Cmem
i
Vþ? V?¼ðA3Þ
That is, the effect of a pulse is to raise the voltage by an
amount proportional to the charge in the pulse and
inversely proportional to the capacitor size. Here, the ?
and þ superscript indicate times just before and just
after the receipt of the impulse.
What is the effect of such a pulse on the phase of the
slave oscillator? Let the difference between the low and
high threshold of the oscillator circuit be DV . If we as-
sume a 50% duty cycle, then we quickly find that:
Dh ¼
q
2 ? DV ? Cmem
which is the equation we sought.
For example, if DV =0.1, and the pulse changes the
voltage by 0.02, then the phase change will be 0.1.
Figure A1a shows a stairstep diagram for two same-
frequency oscillators coupled using single pulses. We see
a rapid convergence to a fixed point. Note that the final
position is not unique; rather, it is determined by the
initial conditions of the oscillators. Even so, the final
i
¼ðVþ? V?Þ
2 ? DV
ðA4Þ
Fig. A1a,b. Geometry of the convergence of two oscillators of the
same frequency. a Strong coupling; b Weaker coupling
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solution is confined to a small region, as shown in
Fig. A1b. If the coupling strength is reduced, the two
regions bounding the stairstep are squeezed together (see
Fig. A1b).
A.1 Coupling with different periods
We now analyze the more general case in which the
oscillators’ periods (and therefore frequencies) are
different. Consider two oscillators with periods related
as: T1¼ T2þ Dh, Dh > 0. Oscillator 2, with the time
constant T2, is the faster oscillator since its period is
shorter. Let oscillator 1 be the master oscillator.
The map function that describes the phase advance of
the slave oscillator due to the coupling of the master is
shown in Fig. A2. Basically, the introduction of Dh shifts
the graph vertically. In this figure, Dh ¼ 0:05 indicating
a slightly faster slave and q=C ¼ 0:1
Notice that we converge but experience an interesting
coupling phenomenon. We get a large jump forward in
phase followed by three small steps backward (for these
particular parameters). From this diagram, it is easy to
infer that if the periods are very different, we will not be
able to entrain the slave oscillator to the master. In that
case, the trajectory of the phase of oscillator will move
above the diagonal. At that point, entrainment becomes
impossible.
This phenomenon of dithering during locking is fairly
common. In simulations of the oscillators coupled
together, we often saw ‘‘jittering’’ in the coupling.
The case of bidirectional coupling is no more difficult.
It is very simple if we assume that pulses do not arrive at
exactly the same time. Under those conditions, we can
alternate back and forth, allowing each oscillator to be
the master and slave in turn.
By overlapping the two resulting map functions,
we can visualize bidirectionally coupled neurons. By
denoting the first oscillator’s map function as ‘‘odd’’ and
the second as ‘‘even’’ we realize the combined graph in
Fig. 7a.
A.2 Multipulse coupling
The more interesting case is when we consider multiple
pulses. There are three degrees of freedom in specifying
two pulses. The first degree of freedom is the strength of
the pulse. For the sake of comparison with the single-
pulse case, we will set the strength equal to qn¼ q=n,
where q is the charge of a pulse in the case of one pulse,
andnisthenumberofpulses.Aseconddegreeoffreedom
istherelativephaseofthesecondpulsewithrespecttothe
first pulse. We denote this relative phase difference as
DhR. The third degree of freedom is the absolute phase of
the first pulse. This third degree of freedom can be set
equal to zero without loss of generality. The only degree
of freedom that is not specified is DhR, the relative phase
of the first vs the second oscillator. To find this value,
there are two cases to consider:
Case 1: h < 0:5 ? ð2Dh þ DhRÞ or h > 0:5 ? 2Dh
In this case, the two pulses act like a single pulse because
both pulses influence either the rising capacitor voltage
or the falling capacitor voltage. Their effect is indistin-
guishable from a single pulse of weight 2*q.
Case 2: h > 0:5 ? ð2Dh þ DhRÞ and h < 0:5 ? 2Dh
In this case, the effects of the two pulses cancel each
other out since one acts on the rising capacitor voltage
and the other acts on the falling capacitor voltage. For
example, the first pulse advances the phase. Then, the
second pulse retards the phase by an equal amount, as
the two pulses have the same weighting. Thus, if we were
to plot a stairstep diagram for this canceling case, we
would find a region where there was no net change in the
phase. This would appear as a line segment on the main
diagonal.
If three pulses occur, then we have the following three
cases: (1) All three pulses could occur either before or
after h ¼ 0:5. In this case, all three pulses act either to
advance or retard the phase and produce results identical
to a single pulse. (2) One pulse advances the phase while
the other two retard the phase. (3) Two pulses advance
the phase while the third pulse retards the phase.
For the case of one pulse, we had two distinct regions
in the stairstep diagram. If we have two pulses, we have
three distinct regions (Fig. A3a). If we have three pulses
we have four distinct regions. By inference, if we have n
spikes, we will have n+1 distinct regions in the corre-
sponding stairstep diagram. This is illustrated in Fig.
A3, which shows the geometry of coupling given two
spikes (Fig. A3a) and ten spikes (Fig. A3b).
The point at which the map function crosses the di-
agonal line is a fixed point (see Fig. 7a). By Theorem 3.8
in Hale and Koc ¸ ak (1991), if ‘x’ is a fixed point, fðxÞ a
map function, and f0ðxÞ
ymptotically stable fixed point. If f0ðxÞ
unstable. In the case of a single-pulse coupling, we found
that
fðxÞ
stable.
When we tried multiple spikes, we found, interest-
ingly, that f0ðxÞ
jj < 1, then point ‘x’ is an as-
jj > 1, the point is
jj ¼ 1 and is therefore not asymptotically
jj converged to zero as more spikes were
Fig. A2. Entrainment of two oscillators with different frequencies
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added. This implies asymptotic stability. Thus, we
showed that multispike coupling is better than single-
spike coupling in our system.
It should be noted that the following factors would
also improve the stability: a single pulse with extended
spike duration is exactly equivalent to multiple single
spikes occurring on the same side of h ¼ 0...0:5. A
second stabilizing factor would be a strong nonlinearity
near the threshold of the neuron such that it became less
sensitive to firing. This would be the case in a biological
synapse where the current flow has a voltage-dependent
synapse. Simulation results (not published) show that
adding voltage dependency improves coupling charac-
teristics.
Appendix B: Passive knee analysis
B.1 Shank Dynamics
We make the following assumptions when modeling the
robot: (1) During stance phase, the robot leg is locked.
(2) If one leg is in stance, the other is in swing, except
during an instantaneous moment of dual support phase.
(3) The hip is held rotationally fixed. (4) The significant
dynamic element is the shank (lower portion of the leg)
of the swinging leg. (5) The criterion for a good ‘‘swing’’
is that the shank hits, or nearly hits, the knee lock in
swing phase the first time the hip of the corresponding
leg has near-zero velocity (i.e., the point where it is
beginning its return stroke). (6) The leg moves backward
the same distance that it moves forward.
If 1 through 4 are true, the position of the knee of the
leg in swing is determined by its kinematics. In particu-
lar, if the hip angle a is known, the knee position of the
swing leg is determined. This assumption will simplify
the analysis considerably.
The dynamics of the shank can be found by using the
Lagrange method (Asada and Slotine 1986). We model
the shank as a point mass concentrated at a length l3
from the knee.
We write down an equation for the position of this
point mass, differentiate, and give the equations for
kinetic and potential energy of the leg:
x ¼ xkneeþ l3sinðqðtÞÞðB1Þ
y ¼ yknee? l3cosðqðtÞÞðB2Þ
_ x x ¼ _ x xkneeþ l3cosðqðtÞÞ_ q qðtÞðB3Þ
_ y y ¼ _ y ykneeþ l3sinðqðtÞ_ q qðtÞÞ
KE ¼m
ðB4Þ
2? ðð_ x xÞ2þ ð_ y yÞ2ÞðB5Þ
PE ¼ yknee? cosðqðtÞÞ ? l3? g ? m
where g is the gravitational constant, l3the distance to
the center of mass of the shank, and m the mass of the
shank.
The Lagrangian is:
ðB6Þ
L ¼ KE ? PE
The dynamics are simply:
ðB7Þ
Fig. A3a,b. Effect of multiple pulses on coupling. a Two pulses; b 10
pulses. Notice that the convergence in the region of the fixed point
improves considerably
Fig. B1. Kinematic arrangement of the biped robot. The leading leg is
in stance phase. The trailing leg is in swing phase
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s ¼d
dt
@L
@ _ q q
??
?
@L
@q
??
ðB8Þ
where s is the joint torque. Since the link is passive,
s ? 0.
The knee has a lock to prevent it from hyperextend-
ing. This could be simulated as an elastic collision, in
which momentum and kinetic energy are conserved.
However, we have chosen to model the collision as a
very strong spring and damper that becomes active when
the leg locks. Thus, the leg will hit the end-of-joint lock
and then rebound elastically.
Plugging Eqs. B1–7 into B8 and simplifying we have:
€ q qðtÞ ¼ ð?sinðqðtÞÞ ? € y yknee? cosðqðtÞÞ ?€ x xknee
?sinðqðtÞ ? gÞ=l3? b ? r ? k ? _ q qðtÞ
where the b ? r term is the strong repelling spring
simulating the knee lock, r the distance past the lock,
and k ? _ q qðtÞ a damping element included to simulate
minor frictional effects.
Examining this equation, we see that in the absence of
knee acceleration and knee lock, the leg acts as damped
inverted pendulum, as would be expected. As we shall
see shortly, in our model, knee accelerations happen
principally at the joint limits. Therefore, for most of the
time, the system behaves like an inverted pendulum.
ðB9Þ
Fig. B2a–f. Trajectories and
joint angles of the leg for various
parameter conditions. (a and b)
Typical trajectory of the leg when
the parameters are set to a low
frequency. In a, the leg is moving
too fast, and thus the knee never
locks to late. (vmax¼ 3:0), while
in c the leg is moving too slowly
and the knee locks too soon. In e,
the parameters are set to a
‘‘good’’ combination, and the
knee locks at exactly the correct
point in time when the hip
changes direction. b, d, and f joint
angles (hip and knee) are shown
for the cases a, c, and e above.
The top trace in each graph is the
hip joint angle while the lower
trace is the knee angle
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We can also conclude that the frequency and period
of the leg are given as:
x ¼
ffiffiffi
g
l
r
:
ðB10Þ
t ¼
ffiffiffi
g
l
s
? 2p
ðB11Þ
Using rough measures from the robot, and assuming the
center of mass is about 2.54 cm below the knee, an
estimate for the frequency is about 3 Hz.
Figure B2 shows simulation results. Figure B2a, c,
and e are stick figures of the walking trajectory. Figure
B2b, d, and f are joint trajectories. Touching upper and
lower trajectories indicates that the shank has hit its
joint lock and is fully extended. The hip follows a
roughly symmetric, triangular trajectory. Figure B2b, d,
and f show the rising half of this triangular trajectory as
the hip moves forward. Ideally, the knee should lock
around the apogee of the swing (i.e., at zero velocity
when the hip reverses direction).
In Fig. B2a and b the lower segment hits the limit
prematurely. In Fig. B2c and d the hip swings too
quickly. In Fig. B2e and f the lower and upper limb are
perfectly synchronized so that the knee trajectory
intercepts the hip trajectory at the hip trajectories’
apogee at approximately 0.26 s.
B.2 Effect of initial conditions
The initial conditions can have a significant effect on the
behavior of the system as well. We noticed in empirical
experiments that the foot accelerated backward as it
unloaded. This is undoubtedly due to the conversion of
strain energy to kinetic energy as well as some rotational
energy (the entire leg is rotating), as there are no other
sources of energy in the shank. This initial condition of
the swing phase can significantly affect the timing of the
leg. In simulation experiments, we found that by varying
the initial leg velocity we could slow down the period of
the system by about 30% for reasonable initial veloc-
ities.
The overall posture is important as well. Given our
model, it is worth noting that a perfectly symmetric gait
on a perfectly level surface will cause the swing leg to
‘‘scuff’’ the floor as it swings forward. See Fig. B3 for a
typical case.
Notice that, as shown in Fig. B3, this system is very
sensitive to the endpoints (which determine the symme-
try of the gait) and the slope of the surface that the
system walks on. In even the best case, the actual range
of slopes over which we can walk, for a given set of
endpoints, is rather small.
In the results on the biped (Fig. 10), we note that the
robot appears to be ‘‘leaning backward’’ a bit as it runs.
This is not surprising given this analysis of the geometry.
To achieve a more robust gait, it is necessary to
consider either actuation of the knee and/or modulation
of the hip so as to introduce accelerations mid-stride.
These accelerations can be generated based on the
dynamics equations above.
Equation B9 indicates that if the joint is accelerated
backward, and the foot is lifted off the ground if the leg is
decelerated, the shank accelerates fast toward the knee.
Thus, we can, in principle, shape the knee trajectory by
actuating the hip to achieve a greater ground clearance.
This requires descending control in synchrony with the
limb.
The analysis of the effect of supraspinal input would
require an entire paper in itself.
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