REPORTS

ammonia would react directly with compounds 4a and 4b, and a dissociative path D,
in which ammonia would react with a 14electron complex formed after dissociation
of olefin (Scheme 5). The rates of decay of
the pentene complex 4b were measured by
31P NMR spectroscopy with varied amounts
of olefin and ratios of ammonia to olefin
(data and plot are shown in figs. S1 and S2
and table S1). The observable rate constants,
kobs, predicted for reaction by associative
path C (Eq. 1) and dissociative path D (Eq.
2), were derived with the steady state
approximation. For path C, the observed rate
constant would be independent of the concentration of olefin, but for path D, a plot
of 1/kobs versus the ratio of olefin to ammonia
is predicted to be linear with a nonzero
intercept. The reactions were clearly slower
at higher concentrations of olefin, and a plot
of 1/kobs versus the ratio of olefin to ammonia
was found to be linear with a positive slope
(0.20
104 T 0.01
104 s1) and a nonzero
y intercept (0.65
104 T 0.16
104 s).
These data suggest that olefin dissociation
is the first step in the reaction, and, if so, the
y intercept of this double reciprocal plot
would correspond to the inverse of the rate
constant for dissociation of olefin.
1
kj1
1

1
0
kobs
k1 Eammonia^
k1 k2 Eammonia^
1
kj1 Epentene^
1

0
kobs
k1 k2 Eammonia^
k1

Because substitution reaction of squareplanar d8 complexes typically proceed associatively, and because the reactions could occur
by more complex pathways with multiple
equilibria preceding N-H bond cleavage, we
conducted further experiments to test whether
the reaction was initiated by dissociation of
olefin. The pentene in complex 4b is displaced by ethylene to form ethylene complex
4c. If the reactions of 4b occur dissociatively,
then the rate constants for dissociation of
pentene obtained from the reaction of 4b with
ethylene and from the reaction of 4b with
ammonia should be the same.
Consistent with dissociative reactions of
4b, the reaction of 4b with ethylene was
independent of the concentration of ethylene
or 0.03 to 0.3 M added pentene; all reactions
occurred with rate constants within 3% of the
mean value of 1.6
10j3. Moreover, this
mean value is well within experimental error
of the value of k1 (1.5
10j3) measured for
the reaction of ammonia with 4b.
The identification of an iridium complex
that undergoes oxidative addition of ammonia
and the elucidation of key thermodynamic
and mechanistic aspects of the reaction
advance our understanding of how to cleave

1082

N-H bonds under mild conditions. We anticipate that this understanding will accelerate
the development of catalytic chemistry that
parallels the existing reactions of hydrogen,
hydrocarbons, silanes, and boranes but begins
with oxidative addition of the N-H bond of
abundant and inexpensive ammonia.
References and Notes
1. G. B. Kauffman, in Coordination Chemistry: A Century
of Progress. (American Chemical Society Symposium
Series, Washington, DC, 1994), vol. 565, pp. 334.
2. Two reactions of ammonia are among the top 10
challenges for catalysis listed in this article: J. Haggin,
Chem. Eng. News. 71, 23 (1993).
3. For a reaction of an Ir(I) complex with ammonia to
form an insoluble product containing hydrides and
bridging amides, as characterized by solid-state IR
and 1H NMR spectroscopy and derivatization, see
A. L. Casalnuovo, J. C. Calabrese, D. Milstein, Inorg.
Chem. 26, 971 (1987).
4. For the protonolysis of a hydride ligand on a group
IV metal complex, presumably after coordination of
the ammonia ligand, see G. L. Hillhouse, J. E. Bercaw,
J. Am. Chem. Soc. 106, 5472 (1984).
5. For an example of a reaction of ammonia with a
cluster to form a product in low yield that was
formulated by 1H NMR spectroscopy to contain a
bridging hydride and bridging amide, see E. G. Bryan,
B. F. G. Johnson, J. Lewis, J. Chem. Soc. Dalton Trans.
1977, 1328 (1977).
6. F. L. Joslin, M. P. Johnson, J. T. Mague, D. M. Roundhill,
Organometallics 10, 2781 (1991).
7. A. W. Kaplan, J. C. M. Ritter, R. G. Bergman, J. Am.
Chem. Soc. 120, 6828 (1998).
8. J. Campora, P. Palma, D. del Rio, M. M. Conejo, E.
Alvarez, Organometallics 23, 5653 (2004).
9. D. Conner, K. N. Jayaprakash, T. R. Cundari, T. B.
Gunnoe, Organometallics 23, 2724 (2004).
10. H. E. Bryndza, W. Tam, Chem. Rev. 88, 1163 (1988).
11. J. R. Fulton, A. W. Holland, D. J. Fox, R. G. Bergman,
Acc. Chem. Res. 35, 44 (2002).
12. J. R. Fulton, S. Sklenak, M. W. Bouwkamp, R. G.
Bergman, J. Am. Chem. Soc. 124, 4722 (2002).
13. D. J. Fox, R. G. Bergman, Organometallics 23, 1656
(2004).
14. M. Kanzelberger, B. Singh, M. Czerw, K. Krogh-Jespersen,
A. S. Goldman, J. Am. Chem. Soc. 122, 11017 (2000).

15. D. Morales-Morales, D. W. Lee, Z. Wang, C. M.

Jensen, Organometallics 20, 1144 (2001).
16. M. Kanzelberger et al., J. Am. Chem. Soc. 125, 13644
(2003).
17. K. Krogh-Jesperson et al., J. Am. Chem. Soc. 124,
10797 (2002).
18. N. A. Al-Salem, H. D. Empsall, R. Markham, B. L.
Shaw, B. Weeks, J. Chem. Soc. Dalton Trans. 1979,
1972 (1979).
19. M. A. McLoughlin, R. J. Flesher, W. C. Kaska, H. A.
Mayer, Organometallics 13, 3816 (1994).
20. Materials and methods, including the details of the
synthesis of the olefin complexes, are available as
supporting material on Science Online.
21. J. F. Riehl, Y. Jean, O. Eisenstein, M. Pelissier, Organometallics 11, 729 (1992).
22. J. F. Hartwig, R. G. Bergman, R. A. Andersen, J. Am.
Chem. Soc. 113, 3404 (1991).
23. J. Ruiz, V. Rodriguez, G. Lopez, P. A. Chaloner, P. B.
Hitchcock, J. Chem. Soc. Dalton Trans. 1997, 4271
(1997).
24. M. Kanzelberger et al., J. Am. Chem. Soc. 125, 13644
(2003).
25. P. L. Holland, R. A. Andersen, R. G. Bergman, J. K. Huang,
S. P. Nolan, J. Am. Chem. Soc. 119, 12800 (1997).
26. H. A. Y. Mohammad et al., Organometallics 21, 5775
(2002).
27. H. D. Empsall et al., J. Chem. Soc. Chem. Commun.
1977, 589 (1977).
28. C. Crocker et al., J. Chem. Soc. Dalton Trans. 1982,
1217 (1982).
29. This method would detect deuterium in the ligand if
it were present in 10% of the sample.
30. We thank the Department of Energy for funding.
Structural data for compounds 4a and 5 have been
deposited in the Cambridge Crystallographic Data
Centre under CCDC 260224 (4a) and 260225 (5),
and can be obtained free of charge at www.ccdc.cam.
ac.uk/conts/retrieving.html. We thank L. Bienen for
editing of the manuscript.
Supporting Online Material
www.sciencemag.org/cgi/content/full/307/5712/1080/
DC1
Materials and Methods
Figs. S1 to S5
Tables S1 to S6
References and Notes
1 December 2004; accepted 12 January 2005
10.1126/science.1109389

Efficient Bipedal Robots Based on

Passive-Dynamic Walkers
Steve Collins,1 Andy Ruina,2* Russ Tedrake,3 Martijn Wisse4
Passive-dynamic walkers are simple mechanical devices, composed of solid
parts connected by joints, that walk stably down a slope. They have no
motors or controllers, yet can have remarkably humanlike motions. This
suggests that these machines are useful models of human locomotion;
however, they cannot walk on level ground. Here we present three robots
based on passive-dynamics, with small active power sources substituted for
gravity, which can walk on level ground. These robots use less control and less
energy than other powered robots, yet walk more naturally, further
suggesting the importance of passive-dynamics in human locomotion.
Most researchers study human locomotion
by observing people as they walk, measuring
joint angles and ground reaction forces (1).
Our approach is different: We study human
locomotion by designing and testing walking
machines that we compare to humans in
terms of morphology, gait appearance, energy use, and control.

18 FEBRUARY 2005 VOL 307

SCIENCE

Previous bipedal robots with humanlike

forms have demonstrated smooth, versatile
motions (25). These impressive robots are
based on the mainstream control paradigm,
namely, precise joint-angle control. For the
study of human walking, this control paradigm is unsatisfactory, because it requires
actuators with higher precision and frequen-

www.sciencemag.org

REPORTS
cy response than human muscles have (6)
and requires an order of magnitude more
energy. To address these issues, passivedynamic walkers (Fig. 1) were proposed as
a new design and control paradigm (7). In
contrast to mainstream robots, which actively
control every joint angle at all times, passivedynamic walkers do not control any joint
angle at any time. Although these walkers
have no actuation or control, they can walk
downhill with startlingly humanlike gaits (8).
To demonstrate that the humanlike properties of passive-dynamic machines are not
dependent on gravitational power, but rather
extend to level-ground walking, we built
three powered walking robots (Fig. 2) at
three institutions, substituting gravitational
power with simple actuation. The Cornell
biped (Fig. 2A) is based on the passive
device in Fig. 1D and is powered by electric
motors with springs that drive ankle pushoff. It has five internal degrees of freedom
(two ankles, two knees, and a hip), each arm
is mechanically linked to the opposite leg,
and the small body is kinematically constrained so that its midline bisects the hip
angle. The Delft biped (Fig. 2B) has a similar morphology, but it is powered by pneumatic hip actuation and has a passive ankle.
The Massachusetts Institute of Technology
(MIT) learning biped (Fig. 2C) is based on
the simpler ramp-walkers in Fig. 1, A and B.
It has six internal degrees of freedom (two
servo motors in each ankle and two passive
hips), each arm is mechanically linked to the
opposite leg, the body hangs passively, and it
uses reinforcement learning to automatically
acquire a control policy. The supporting
online movies show these robots walking
and the supporting online text describes their
construction details (9).
The Cornell biped is specifically designed for minimal energy use. The primary
energy losses for humans and robots walking
at a constant speed are due to dissipation
when a foot hits the ground and to active
braking by the actuators (negative work).
The Cornell design demonstrates that it is
possible to completely avoid this negative
actuator work. The only work done by the
actuators is positive: The left ankle actively
extends when triggered by the right foot
hitting the ground, and vice versa. The hip
joint is not powered, and the knee joints only
have latches. The average mechanical power
1
Mechanical Engineering, University of Michigan, Ann
Arbor, MI 48104, USA. 2Theoretical and Applied
Mechanics, Cornell University, Ithaca, NY 14853,
USA. 3Brain and Cognitive Sciences and Center for Bits
and Atoms, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA. 4Mechanical Engineering,
Delft University of Technology, NL-2628 CD Delft,
Netherlands.

*To whom correspondence should be addressed.

E-mail: ruina@cornell.edu

(10) of the two ankle joints is about 3 W,

almost identical to the scaled gravitational
power consumed by the passive-dynamic
machine on which it is based (8). Including
electronics, microcontroller, and actuators,
the Cornell biped consumes 11 W (11).
To compare efficiency between humans
and robots of different sizes, it is convenient to
use the dimensionless specific cost of transport, ct 0 (energy used)/(weight
distance
traveled). In order to isolate the effectiveness
of the mechanical design and controller from
the actuator efficiency, we distinguish between the specific energetic cost of transport,
cet, and the specific mechanical cost of
transport, cmt. Whereas cet uses the total
energy consumed by the system (11 W for
the Cornell biped), cmt only considers the
positive mechanical work of the actuators
(3 W for the Cornell biped). The 13-kg Cornell biped walking at 0.4 m/s has cet , 0.2
and cmt , 0.055. Humans are similarly
energy effective, walking with cet , 0.2, as
estimated by the volume of oxygen they
consume (VO2), and cmt , 0.05 (1214).
Measurement of actuator work on the Delft
biped yields cmt , 0.08. Based on the small
slopes that it descends when passive, we

estimate the MIT biped to have cmt Q 0.02.

Although the MIT and Delft bipeds were
not specifically designed for low-energy use,
both inherit energetic features from the
passive-dynamic walkers on which they
are based. By contrast, we estimate the
state-of-the-art Honda humanoid Asimo to
have cet , 3.2 and cmt , 1.6 (15). Thus
Asimo, perhaps representative of joint-angle
controlled robots, uses at least 10 times the
energy (scaled) of a typical human.
Control algorithms for state-of-the-art,
level-ground walking robots are typically
complex, requiring substantial real-time
computation. In contrast, the Delft and Cornell bipeds walk with primitive control algorithms. Their only sensors detect ground
contact, and their only motor commands are
on/off signals issued once per step. In addition
to powering the motion, hip actuation in the
Delft biped also improves fore-aft robustness
against large disturbances by swiftly placing
the swing leg in front of the robot before it
has a chance to fall forward (16, 17).
The MIT biped (Fig. 2C) is designed to
test the utility of motor learning on a passivedynamic mechanical design. The goal of the
learning is to find a control policy that

Fig. 1. Ramp-walking,
A
C
D
downhill, unpowered,
or passive-dynamic
machines. Our powered
bipeds are based on these
passive designs. (A) The
Wilson Walkie (27).
(B) MITs improved version (28). Both (A) and
(B) walk down a slight
ramp with the comical,
awkward, waddling gait
of the penguin (27).
(C) Cornell copy (29)
of McGeers capstone
B
design (7). This fourlegged biped has two
pairs of legs, an inner
and outer pair, to prevent falling sideways. (D) The Cornell passive biped with arms [photo: H. Morgan]. This walker has
knees and arms and is perhaps the most humanlike passive-dynamic walker to date (8).
Fig. 2. Three levelground powered walking robots based on the
ramp-walking designs
of Fig. 1. (A) The Cornell biped. (B) The Delft
biped. (C) The MIT
learning biped. These
powered robots have
motions close to those
of their ramp-walking
counterparts as seen
in the supporting online movies (movies S1
to S3). Information on
their construction is in
the supporting online
text (9).

www.sciencemag.org

SCIENCE

VOL 307

18 FEBRUARY 2005

1083

REPORTS
stabilizes the robot_s trajectory on level terrain
using the passive ramp-walking trajectory as
the target. The robot acquires a feedback
control policy that maps sensors to actions
using a function approximator with 35 parameters. With every step that the robot takes, it
makes small, random changes to the parameters and measures the change in walking
performance. This measurement yields a
noisy sample of the relation between the
parameters and the performance, called the
performance gradient, on each step. By means
of an actor-critic reinforcement learning algorithm (18), measurements from previous
steps are combined with the measurement
from the current step to efficiently estimate
the performance gradient on the real robot
despite sensor noise, imperfect actuators, and
uncertainty in the environment. The algorithm
uses this estimate in a real-time gradient descent optimization to improve the stability of

the step-to-step dynamics (Fig. 3). The robot_s

actuators are mounted so that when they are
commanded to their zero position, the robot
imitates its passive counterpart. Starting from
this zero policy, the learning system quickly
and reliably acquires an effective control
policy for walking, using only data taken from
the actual robot (no simulations), typically
converging in 10 min or 600 steps. Figure 3
illustrates that the learned control policy not
only achieves the desired trajectory but is also
robust to disturbances. The robot can start,
stop, steer, and walk forward and backward at
a small range of speeds. This learning system
works quickly enough that the robot is able to
continually adapt to the terrain (e.g., bricks,
wooden tiles, and carpet) as it walks.
Each of the robots here has some design
features that are intended to mimic humans.
The Cornell and Delft bipeds use anthropomorphic geometry and mass distributions in

n+1

Fig. 3. Step-to-step
1.6
dynamics of the MIT
biped walking in place
1.4
on a level surface,

before (q) and after

1.2
(x) learning. Shown is
the roll angular velocity when the right
1
foot collides with the
ground (q 0 0, q 9 0)
0.8
at step n 1 versus .
step n. Intersections
0.6
of the plots with the
solid identity line are
fixed points. The hor0.4
identity
izontal dashed line is
desired
the theoretical ideal;
0.2
before learning
the robot would reach
after learning
q 0 0.75 sj1 in one
0
step. This ideal cannot
0.2
0.4
0.6
0.8
1.2
1.4
1.6
0
. 1
.
be achieved due to
n at = 0, > 0
limitations in the controllability of the actuation system. On a level surface, before learning, the robot loses energy on every step (q n1 G qn),
eventually coming to rest at q 0 0. After learning, the robot quickly converges near q 0 0.75 sj1 for
0 e q0 e 1.7 sj1.

their legs and demonstrate ankle push-off and

powered leg swinging, both present in human
walking (14, 19). They do not use high-power
or high-frequency actuation, which are also
unavailable to humans. These robots walk
with humanlike efficiency and humanlike
motions (Fig. 4 and movies S1 to S3). The
motor learning system on the MIT biped uses
a learning rule that is biologically plausible at
the neural level (20). The learning problem is
formulated as a stochastic optimal feedback
control problem; there is emerging evidence
that this formulation can also describe biological motor learning (21).
The Cornell and Delft bipeds demonstrate
that walking can be accomplished with extremely simple control. These robots do not
rely on sophisticated real-time calculations or
on substantial sensory feedback such as from
continuous sensing of torques, angles, or attitudes. This implies that steady-state human
walking might require only simple control as
well; the sequencing of human joint-angles in
time might be determined as much by morphology as by motor control. We note that no
other robots have done particularly better at
generating humanlike gaits even when using
high-performance motors, a plethora of sensors, and sophisticated control.
In theory, pushing off just before heelstrike requires about one-fourth the energy of
pushing off just after heel-strike (22, 23), so
the Cornell robot was initially designed with
this preemptive push-off strategy. Initial
push-off resulted in both higher torque demands on the motor and a high sensitivity to
push-off timing that our primitive control
system could not reliably stabilize. Humans
seem to solve both of these problems without
a severe energy penalty by using a double
support phase that overlaps push-off and heelstrike. These issues must also be addressed
in the design of advanced foot prostheses.
The success of the Delft robot at balancing using ankles that kinematically couple

Fig. 4. Two sets of

video stills of the Cornell ankle-powered biped walking on a level
surface next to a person. A little less than
one step is shown at
7.5 frames/s. Both the
robot and the person
are walking at about
1 step/s. The stick figure indicates the leg
angles for the corresponding video stills;
the right arm and leg
are darker than the left.

1084

18 FEBRUARY 2005 VOL 307

SCIENCE

www.sciencemag.org

REPORTS
leaning to steering hints that humans could
similarly use a simple coupling between lean
and lateral foot placement to aid balance.
Furthermore, simulations used in the development of the Delft robot showed that the
swift swing-leg motion not only increased
fore-aft stability but also increased lateral
stability. Indeed, the physical robot was not
able to balance laterally without sufficient
fore-aft swing-leg actuation. This highlights
the possible coupling between lateral and
sagittal balance in human walking.
The MIT biped shows that the efficiency
of motor learning can be strongly influenced
by the mechanical design of the walking
system, both in robots and possibly in
humans. Previous attempts at learning control
for bipedal robots have required a prohibitively large number of learning trials in
simulation (24) or a control policy with
predefined motion primitives on the robot
(25). By exploiting the natural stability of
walking trajectories on the passive-dynamic
walker, our robot was able to learn in just a
few minutes without requiring any initial
control knowledge. We also found that it
was possible to estimate the walking performance gradient by making surprisingly small
changes to the control parameters, allowing
the robot to continue walking naturally as it
learns. This result supports the use of actorcritic reinforcement learning algorithms as
models for biological motor learning.
The conclusion that natural dynamics may
largely govern locomotion patterns was already suggested by passive-dynamic machines.
A common misconception has been that
gravity power is essential to passive-dynamic
walking, making it irrelevant to understanding
human walking. The machines presented here
demonstrate that there is nothing special about
gravity as a power source; we achieve successful walking using small amounts of power
added by ankle or hip actuation.
We expect that humanoid robots will be
improved by further developing control of
passive-dynamicsbased robots and by
paying closer attention to energy efficiency
and natural dynamics in joint-controlled
robots (26). Whatever the future of humanoid robots, the success of human mimicry
demonstrated here suggests the importance
of passive-dynamic concepts in understanding human walking.
References and Notes
1. D. A. Winter, The Biomechanics and Motor Control of
Human Gait (Univ. of Waterloo Press, Waterloo,
Ontario, Canada, 1987).
2. I. Kato et al., Proc. CISM-IFToMM Theory and Practice of Robots and Manipulators (Udine, Italy, 1973),
pp. 1224.
3. Y. Sakagami et al., Proc. IEEE/Robotics Society of
Japan (RSJ) Int. Conf. Intell. Robots Syst. (IEEE/RSJ,
Lausanne, Switzerland, 2002), pp. 24782483.
4. C. Chevallereau et al., IEEE Control Syst. Mag. 23, 57
(2003).
5. F. Pfeiffer, K. Loffler, M. Gienger, Proc. IEEE Int. Conf.

6.
7.
8.
9.
10.

11.
12.
13.
14.
15.

16.
17.
18.
19.
20.

Robotics Automation (IEEE, Washington, DC, 2002),

pp. 31293135.
F. Zajac, Crit. Rev. Biomed. Eng. 17, 359 (1989).
T. McGeer, Int. J. Robotics Res. 9, 62 (1990).
S. H. Collins, M. Wisse, A. Ruina, Int. J. Robot. Res.
20, 607 (2001).
Supporting online movies and text are available at
Science Online.
Mechanical power is defined here as net positive
mechanical work at the joints 0 XT0 S [wi Mi] dt/T
where T is the period of one step, wi is the relative
angular velocity at one joint, Mi is the torque across
that joint, [x] 0 x if x 9 0 and 0 otherwise, and the
sum is over all the joints. Because only the ankle
does positive work on the Cornell robot, this can be
measured by measuring the foot force as the ankle
extends during push-off.
For the Cornell robot, total power was measured by
averaging the voltage across a 1-ohm resistor put in
series with the battery.
E. Atzler, R. Herbst, Pflueg. Arch. Gesamate Physiol.
215, 291 (1927).
N. H. Molen, R. H. Rozendal, W. Boon, Proc. K. Ned.
Akad. Wet. Ser. C 75, 305 (1972).
J. M. Donelan, R. Kram, A. D. Kuo, J. Exp. Biol. 205,
3717 (2002).
Hondas ASIMO can walk at a variety of speeds, kick
balls, and even climb stairs. It weighs 510 N, can
walk at speeds up to 1.6 km hourj1, and drains a
38.4-V, 10A-hour battery in about 30 min (http://
world.honda.com/ASIMO/). Using these numbers, we
estimate cet , 3.2 and, assuming a 50% drive train
efficiency, cmt , 1.6.
M. Wisse, J. van Frankenhuyzen, Proc. Conf. Adaptive
Motion Anim. Machines (Kyoto, Japan, 2003).
M. Wisse, A. L. Schwab, R. Q. van der Linde, F. C. T.
van der Helm, IEEE Trans. Robot., in press.
R. Tedrake, T. W. Zhang, M. Fong, H. S. Seung, Proc.
IEEE/RSJ Int. Conf. Intell. Robots Syst. (IEEE/RSJ,
Sendai, Japan, 2004).
T. McGeer, J. Theor. Biol. 163, 277 (1993).
H. Seung, Neuron 40, 1063 (2003).

21. E. Todorov, Nature Neurosci. 5, 1226 (2002).

22. A. D. Kuo, J. Biomech. Eng. 124, 113 (2002).
23. A. Ruina, J. Bertram, M. Srinivasan, J. Theor. Biol., in
press.
24. H. Benbrahim, J. A. Franklin, Robot. Auton. Syst. 22,
283 (1997).
25. A. Kun, W. T. Miller III, Proc. IEEE Int. Conf. Robotics
Automation (IEEE, Minneapolis, MN, 1996).
26. J. Pratt, thesis, Massachusetts Institute of Technology
(2000).
27. J. E. Wilson, U.S. Patent 2,140, 275; available at
www.tam.cornell.edu/ruina/hplab/.
28. R. Tedrake, T. W. Zhang, M. Fong, H. S. Seung, Proc.
IEEE Int. Conf. Robotics Automation (IEEE, New
Orleans, LA, 2004).
29. M. Garcia, A. Chatterjee, A. Ruina, Dyn. Stabil. Syst.
15, 75 (2000).
30. The Cornell robot was developed by S.C. with suggestions from A.R.; the Delft robot was developed by
M.W. and J. van Frankenhuyzen on an Stichting
Technische Wetenschappen grant, with help from A.
Schwab; and the MIT robot was developed by R.T. and
T. Weirui Zhang with help from M.-f. Fong and D. Tan
in the lab of H. Sebastian Seung. A.R. and S.C. were
funded by an NSF Biomechanics grant. R.T. was funded
by the Packard Foundation and the NSF. The text was
improved by comments from N. Agnihotri, C. Atkeson,
J. Burns, A. Chatterjee, M. Coleman, J. Grizzle, P.
Holmes, I. ten Kate, A. Kun, A. Kuo, Y. Loewenstein,
S. van Nouhuys, D. Paluska, A. Richardson, S. Seung, M.
Srinivasan, S. Strogatz, and N. Sydor.
Supporting Online Material
www.sciencemag.org/cgi/content/full/307/5712/1082/
DC1
Materials and Methods
SOM Text
Movies S1 to S3
References and Notes
22 November 2004; accepted 26 January 2005
10.1126/science.1107799

Terrestrial Gamma-Ray Flashes

Observed up to 20 MeV
David M. Smith,1* Liliana I. Lopez,2 R. P. Lin,3
Christopher P. Barrington-Leigh4
Terrestrial gamma-ray flashes (TGFs) from Earths upper atmosphere have
been detected with the Reuven Ramaty High Energy Solar Spectroscopic
Imager (RHESSI) satellite. The gamma-ray spectra typically extend up to 10
to 20 megaelectron volts (MeV); a simple bremsstrahlung model suggests
that most of the electrons that produce the gamma rays have energies on
the order of 20 to 40 MeV. RHESSI detects 10 to 20 TGFs per month, corresponding to 50 per day globally, perhaps many more if they are beamed.
Both the frequency of occurrence and maximum photon energy are more than
an order of magnitude higher than previously known for these events.
Terrestrial gamma-ray flashes (TGFs) were unexpectedly detected from Earth_s atmosphere
by the Burst and Transient Source Experiment
1
Physics Department and Santa Cruz Institute for
Particle Physics, University of California, Santa Cruz,
1156 High Street, Santa Cruz, CA 95064, USA.
2
Astronomy Department and Space Sciences Laboratory, University of California, Berkeley, Berkeley, CA
94720, USA. 3Physics Department and Space Sciences
Laboratory, University of California, Berkeley, Berkeley, CA 94720, USA. 4University of British Columbia,
2329 West Mall Vancouver, BC V6T 1Z4 Canada.

*To whom correspondence should be addressed.

E-mail: dsmith@scipp.ucsc.edu

www.sciencemag.org

SCIENCE

VOL 307

(BATSE) on the Compton Gamma-Ray Observatory (CGRO), a NASA satellite in low-Earth

orbit between 1991 and 2000. Each BATSE
TGF (1) lasted between a fraction of a millisecond and several milliseconds, shorter than
all other transient gamma-ray phenomena observed from space. Since they were first
detected, it has also been noticed that TGFs
had a harder energy spectrum (higher average
energy per photon) than any of these other
phenomena (1).
Fishman et al. (1) immediately interpreted
the TGFs as high-altitude electrical discharges
and found a correlation with thunderstorms.

18 FEBRUARY 2005

1085

SUPPORTING ONLINE MATERIAL

for
Efficient bipedal robots based on passive-dynamic walkers
Steven H. Collins1 , Andy Ruina2 , Russ Tedrake3 , Martijn Wisse4
1 Mechanical

Engineering, University of Michigan, Ann Arbor, MI 48104, USA

& Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
3 Brain & Cognitive Sciences and Center for Bits and Atoms,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
4 Mechanical Engineering, Delft University of Technology, NL-2628 CD Delft, NETHERLANDS
2 Theoretical

Correspondence

to: ruina@cornell.edu.

February 11, 2005

This supporting material includes

The controller kicks the robot into random initial conditions between learning trials. After
a few minutes, the robot is walking well in
place, so we command it to walk in a circle.
Finally, we show the robot walking down the
hall, on tiles and outside; this footage is taken
from a single trial where the robot adapted to
each change in the terrain as it walked.

1. Materials and Methods. Details about the

robots construction and control.
2. An analogy. A description of the parallels
(in content, not in significance) between first
powered flight and these robots.
In addition we hope readers will look at the videos:

More material and other videos are available

S1 Cornell powered biped. This movie shows through
videos of the robot walking on flat ground. A http://tam.cornell.edu/ruina/powerwalk.html .
slow-motion segment shows the ankle pushoff actuation.

1 Materials and Methods

S2 Delft pneumatic biped. This movie shows the

robot walking down a hall with views from the Details about the three robots are presented here.
front, side, and back.

1.1 Cornell powered biped.

S3 MIT learning biped. This movie begins with

the powered robot imitating passive walking
down a 0.9 degree slope, from three camera
angles. Then it shows the robot learning to
walk on flat terrain with foam protective pads.

This robot is autonomous; it has no power lines

and no communication links to the outside. It consists of two 0.8 m long legs, each having knees, attached at a hip joint. The robot has curved-bottom
1

latch is released by a solenoid at the completion of

ankle push-off, at which point the knee is passive
until knee-strike. Ankle push-off restores energy
lost, mostly to heel-strike collisions. To minimize
the needed motor size, energy for ankle push-off is
stored in a spring between steps.
The control circuitry is located in the
hip/torso/head visible in the figure. A finitestate machine with eight binary inputs and outputs
is implemented in 68 lines of code on an Atmel
AT90S8515 chip running on an ATSTK500
standard development board. A second board
with relays and passive conditioning components
connects the board to the electromechanical and
sensory parts. During the first state, Left Leg
Swing, all actuators are unpowered and the left
knee latch passively locks at knee strike. When
ground-detection contact switches below the
left foot detect impending heel strike, the state
changes to Right Ankle Push-Off. This begins
a timed activation of the solenoids that release
the plantar-flexor spring of the right foot. When
switches detect full foot extension, the state
changes to Right Toe Return. During this state, a
9.5 Watt, 6.4 oz gear-reduced MicroMo R motor is
activated, slowly retracting the foot and restoring
spring energy. Also, a short time after detection of
impending left-foot heel-strike, a solenoid unlocks
the right knee. When a switch on the motor
indicates full foot retraction, the state changes to
Right Leg Swing, and the foot-retraction motor
is deactivated. The state machine then swaps
roles for the left and right legs and goes to the
initial state. Taking all sensing, including the
sensing of internal degrees of freedom (which
could in principle be made open loop), about 20
bits of information per step flows to the processor.
Environmental sensing, i.e., the instant of foot
contact, is about seven bits per step.
This machine has only one capability, walking
forward. It is designed to walk with minimal energy use. Its speed, path and joint motions are not
shaped or controlled but follow from its mechanical
design and primitive ankle push-off actuation. An-

Figure 1: The Cornell powered biped

feet, arms, and a small torso which is kept upright

by connection to the legs with an angle-bisecting
mechanism. Each arm carries a battery. The right
arm is rigidly attached to the left leg and vice versa,
reducing yaw oscillations (Fallis, 1888, Collins,
Wisse and Ruina 2001). The machine weighs 12.7
kg and has 5 internal degrees of freedom (one hip,
two knees, and two ankles). The thigh-to-shank
length and mass ratios are 0.91 to 1 and 3.3 to
1, respectively, which mimics human architecture
and seems important to the passive dynamics of the
system. The hip joint is fully passive. A latch at
each knee passively locks the shank to be parallel with its proximal thigh throughout stance. This
2

is 1/4 for push-off before heel-strike, d 0.4m

is the step length, df 0.2m is the foot length,
0.8m is the leg length, v = 0.4m/s is the
average velocity, and g 10 m/s2 is the gravity constant. In a dynamic 3-D model (adapted
from Kuo 1999) with geometry, mass distribution,
speed, and step length similar to this robot, without a hip spring or pre-emptive push-off, we found
the mechanical cost of transport to be 0.013. Using
the liberal collision reduction factor of 1/4 above,
this yields a theoretical minimum of 0.003. Spring
actuated leg swing, used by humans, could also
significantly reduce the mechanical work requirements for walking at this speed by reducing step
length. Thus, by a variety of estimates, the mechanical work of this robot walking at this speed,
small as it is, seems to have room for an order of
magnitude reduction.
The Cornell powered biped walked successfully
during a period of a few weeks starting in July
2003. This robot is a proof-of-concept prototype,
not a production-run machine. It was developed as
a one-shot attempt using a small ($10K) budget. As
is not unusual for experimental robots, the device
did not stand up well to long periods of testing; on
average about one mechanical component would
break per day of testing. For instance, the cables
connecting the motor to the primary ankle extension spring ran over a small radius pulley at the
knee and broke frequently. When the Cornell robot
was best tuned it would walk successfully at about
30% of attempts. Failed launches were due to inadequate matching of proper initial conditions, most
often ending with foot scuff of the swing leg. The
robot seems mildly unstable in heading, so once it
was launched, the primary failure mode was walking off of the (narrow) walking table or walking
into a wall. Uneven ground also lead to falls. Because it walked 10 or more steps many times, with
the end only coming from hitting a wall or cliff, the
gait is clearly stable (although not very) for both
(d df )2 v 2
0.0003
cet cmt J
lateral and sagital balance. However, the reader can
2
2gd
make his/her own judgments based on the videos
where J is the collision reduction factor, which which are the basic documentation of success.

kle extension occurs mostly after the opposite leg

has completed heel-strike collision, so in principle
the machine could be made to consume about four
times less energy by having ankle push-off before,
rather than after, the opposing legs foot-to-ground
collision (Kuo, 2002). However, push-off before
heel-strike seems to require more precise timing
and also requires greater ankle torques. We surrendered this possible gain in energy effectiveness
in trade for greater simplicity of control.
Low energy use was a primary goal in the design
of the Cornell robot. We measured its power consumption during walking trials using an off-board
digital oscilloscope connected with fine wires. At
500 samples per second, the scope measured battery voltage on one channel, and the voltage drop
across a 1 ohm power resistor in series with the
batteries on another. The product of the voltage
and current was, on average, 11 watts (yielding
cet = 0.2). Mechanical energy use was measured
in experimentally simulated push-off trials. The
force at each foot contact point was measured as
the ankle was slowly moved through its extension
range, and this force was integrated to estimate mechanical work per step, yielding an average over a
cycle of about 3 watts (cmt = 0.055). This is a
slight over-estimate of the mechanical energy used
for propulsion because some energy is lost at the
collision between the ankle and shank at full ankle
extension.
The theoretical lower limit for the cost of transport in walking models is cmt = 0. This can be
achieved by swaying the upper body with springs
in such a manner as to totally eliminate the collisional losses (Gomes and Ruina, 2005). Without
swaying the upper body, a motion that would have
significant energetic cost in humans, a rough lower
bound on energetic cost can be estimated from the
point-mass small-angle model of Ruina, Srinivasan
and Bertram (2005) as

A key aspect of the success, and also the touchiness, of this robot is the shape and construction of
the feet. The general issues related to feet for this
class of robots is discussed in Collins, Wisse and
Ruina (2001). We tried various support-rail curve
shapes and overall foot stiffnesses, and only one of
these led to successful walking.
The Cornell machine, which uses wide supporting feet for lateral stability, is not being maintained.
Rather, present efforts are aimed at developing a
machine which uses simple active control for lateral balance using foot placement. This is used
by humans during walking (Bauby and Kuo 2000)
and the idea is related to the kinematic lean-to-steer
mechanism of the Delft Biped and the steering used
by a bicycle rider for balance.

Empty bucket

The right arm is

slaved to the left
thigh via this
drive belt

CO2 canister

The electronics,
provides on/off
signals for the
muscles

One of the four

artificial muscles

1.2 Delft pneumatic biped

This robot is also autonomous; all power sources
and computation are onboard. The robot weighs
8 kg, has 5 internal degrees of freedom (one hip,
two knees, two skateboard-truck-like ankles), has
an upper body, and stands 1.5 m tall. The swinging
arms do not add degrees of freedom; they are mechanically linked to the opposing thighs with belts.
The knees have mechanical stops to avoid hyperextension, and are locked with a controllable latch.
Two antagonist pairs of air-actuated artificial muscles (McKibben muscles) provide a torque across
the hip joint to power the walking motion.
The muscles are fed with CO2 from a 58
atm cannister, pressure-reduced in two steps to
6 atm through locally developed miniature pneumatics. Low-power, two-state valves from SMC
Pneumatics R connect the artificial muscles either
to the 6 atm supply pressure or to 0 atm. The calculation of cmt = 0.08 for the Delft biped, used in the
main paper, is based on actuator work (measuring
the force-length relation of the muscles at the operating pressure). It does not take into account the
huge (but inessential) losses from stepping down
the gas pressure. To find a value for cet , we calculated the decrease of available energy (or exergy)

The chains making up

the bisecting
mechanism that
keeps the
upper body upright

A latch at the knee

joint is locked through
stance and unlocked in
the swing phase

One foot
switch
underneath
each foot

Skateboard-trucklike ankle joints

add to lateral stability

Figure 2: Delft pneumatic biped.

for a pressure drop from the 58 atm saturated liquid state to atmospheric pressure. Available energy
represents the amount of work that could be done
with the pressurized gas if the both the gas expansion process and the simultaneous heat transfer process are reversible (i.e. lossless). In that hypothetical setting, one can use the enthalpy and entropy
values for the gas at the beginning and the end of
the expansion process. At a constant temperature
of 290 K, this amounts to a loss of available energy of 664 kJ per kg CO2 . A 0.45 kg canister
4

can power the 8 kg robot for 30 min of walking

at 0.4 m/s yielding cet = 5.3. This value has little meaning, however. First, even the best realworld gas-expansion systems can only use about
30% of the theoretically available energy, due to
irreversibility issues. More importantly, most of
the expansion loss would be eliminated if the CO2
had been stored at 6 atm. Unfortunately this would
require an impractically large storage tank. Thus
the discrepancy between cet = 5.3 and cmt = 0.08
is due to practical problems associated with using
compressed-gas energy storage.

sideways as a result of a disturbance, the ankle allows the foot to remain flat on the floor. Due to
the tilted joint orientation, the leaning is accompanied by steering. If the walker has sufficient forward velocity, this steering helps prevent it from
falling sideways, much like the turning of a bike
wheel into a fall helps prevent a bike from falling
down.
A Universal Processor Board from Multi
Motions R (based on the Microchip R PIC16F877
micro-controller) uses foot-contact switch signals
to open or close the pneumatic valves. The control
program is a state machine with two states: either
the left or the right leg is in swing phase. At the
beginning of the swing phase, the swing knee is
bent. Four hundred milliseconds after the start of
the swing phase, the knee latch is closed, waiting
for the lower leg to reach full extension through its
passive swing motion. Programmed in assembly,
this amounts to about 30 lines of code. The only
sensing is the time of foot contact, used once per
step. Taking account of the implicit rounding from
the processor loop time, we estimate the sensor information flow rate is about six bits per second.
The Delft powered biped first walked successfully in July 2004. When mechanically sound, most
of the manual launches (by an experienced person)
result in a steady walk. Falls can often be attributed
to disturbances from within the machine (a contact
switch that performs unreliably, or a cable that gets
stuck between parts), and occasionally to floor irregularities. Another problem is that the pneumatic
and mechanical systems (which were developed at
Delft for a proof-of-principle prototype rather than
an industrial-strength product) have frequent mechanical failures that often need a day or more to
fix. At present the machine is being kept working
so it can repeat the behavior shown in movie S2.

McKibben muscles have a low stiffness when

unactuated, leaving the joints to behave almost passively at zero pressure. At higher pressures, the
McKibben muscles behave as progressively stiffer
springs. By activating opposing muscles in different proportions, the relaxed angle of a joint can be
controlled. This is applied at the hip where the artificial muscles alternate in action. At the start of
each step, determined by a foot switch, one muscle is set to 6 atm and the other to 0 atm. The
swing leg is thus accelerated forward until the relaxed angle of the hip is reached, where it (approximately) stays due to damping in the muscles and in
the joint. If sufficient hip joint stiffness is obtained
from the hip muscles, stable walking similar to that
of McGeers four-legged machine can be obtained.
The upper body is kept upright via a kinematic restriction, a chain mechanism at the hip which confines the upper body to the bisection angle of the
two legs (Wisse, Hobbelen and Schwab, 2005).
Lateral stability in two-legged robots can be obtained in a number of ways (Kuo, 1999), and one
solution was tested in the Delft robot. The feet are
attached to the lower leg via special ankle joints
(Wisse and Schwab, 2005) which have a joint axis
that runs from above the heel down through the
middle of the foot, quite unlike the human ankle
but much like skateboard trucks. The mechanism
creates a nonholonomic constraint, which can enable stability without dissipation, as found in skateboards (Hubbard, 1979). If the robot starts to lean

1.3

MIT learning biped.

First we duplicated the Wilson design (Fig. 1a of

the main paper) using two rigid bodies connected
by a simple hinge. The kneeless morphology was
5

Before adding power or control, we verified that

this robot could walk stably downhill with the ankle joints locked.
The robots control code runs at 200Hz on an
embedded PC-104 Linux computer. The robot runs
autonomously; the computer and motors are powered by lithium-polymer battery packs, and communication is provided by wireless ethernet. This
communication allows us to start and stop the robot
remotely; all of the control algorithms are run on
the onboard computer.
The learning controller, represented using a linear combination of local nonlinear basis functions,
takes the body angle and angular velocity as inputs
and generates target angles for the ankle servo motors as outputs. The learning cost function quadratically penalizes deviation from the dead-beat controller on the return map, evaluated at the point
where the robot transfers support from the left
foot to the right foot. Eligibility was accumulated evenly over each step, and discounted heavily
( 0.2) between steps. The learning algorithm
also constructs a coarse estimate of the value function, using a function approximator with only angular velocity as input and the expected reward as
output. This function was evaluated and updated at
each crossing of the return map.
Before learning, outputs of both the control policy and the value estimate were zero everywhere
regardless of the inputs, and the robot was able to
walk stably down a ramp; because it is simulating
passive-dynamic walking, this controller runs out
of energy when walking on a level surface. The
robot kicks itself into a random starting position
using a hand-designed control script to initialize
the learning trials. The learning algorithm quickly
and reliably finds a controller to stabilize the desired gait on level terrain. Without the value estimate, learning was extremely slow. After a learning
trial, if we reset the policy parameters and leave the
value estimate parameters intact, then on the next
trial the learning system obtains good performance
in just a few steps, and converges in about two minutes.

Figure 3: The MIT learning biped

chosen to reduce the number of joints and actuators
on the robot, minimizing the combinatorial explosion of states and control strategies that the learning
algorithm needed to consider. The gait was iteratively improved in simulation by changing the foot
shape for a given leg length, hip width, and mass
distribution. The resulting ramp-walker (Fig. 1b
of main paper) walks smoothly down a variety of
slopes. The powered version uses tilt sensors, rate
gyros, and potentiometers at each joint to estimate
the robots state, and servo motors to actuate the
ankles. The completed robot weighs 2.75 kg, is
43cm tall, and has 6 internal degrees of freedom
(each leg has one at the hip and two at the ankle).
6

The resulting controller outputs ankle commands that are a simple, time-independent function
of the state of the robot, and does not require any
dynamic models. All learning trials were carried
out on the physical biped with no offline simulations. The learned controller is quantifiably (using
the eigenvalues of the return map) more stable than
any controller we were able to design by hand, and
recovers from most perturbations in as little as one
step. The robot continually learns and adapts to the
terrain as it walks.
The MIT biped, which was not optimized for energy efficiency, has cet = 10.5, as calculated by the
energy put back into the batteries by the recharger
after 30 minutes of walking. The cet for this robot
is especially high because the robot has a powerful
computer (700 MHz Pentium) on a light robot that
walks slowly.
The version of the MIT powered biped shown
here first walked successfully in January 2004. The
earliest powered prototype of this type at MIT first
walked successfully in June 2003.
The MIT biped is still working well, and is the
subject of active development and study. New
learning algorithms and new design elements (such
as different curvatures in the feet) are being tested
with the same hardware. A new version with knees
is mostly developed. The robot has walked for
a few one-hour on-the-treadmill energy-use trials
(the batteries would have lasted for about 90-100
minutes).

partially based on, walking toys.

The Wrights ideas about control of steer in aircraft were based on the relation between steer and
lean in bicycles. Our research in the passive balance of robots was inspired by the self-stability of
bicycles.
The Wrights worked for years developing gliders, planes powered by the release of gravitational
potential energy as they flew down a glide slope.
This was in contradiction to a common paradigm
of the time, which was to try to get a powered plane
to work, motor and all, all at once. Once they had
mastered gliding they were confident they could
master powered flight. On the second day they tried
the idea, adding a primitive engine to a glider design, they made their famous flight. Our development of passive-dynamic walkers, robots that walk
down gentle slopes powered only by gravity, was
by far the bulk of our efforts. Once we had those
working well we were confident that the machines
could walk on the level with a small addition of
power. The result that adding power to a downhill
machine works is one of the subjects of this paper.
The analogy above is not accidental. Tad
McGeer, the pioneer of passive-dynamic robotics,
was trained as an aeronautical engineer. McGeers
foray into robotics was directly and explicitly an
imitation of the Wright Brothers paradigm. It
worked for the Wrights after others failed at mastering power and flight all at once. Perhaps,
McGeer thought, it could work for the more pedestrian task of making an efficient walking robot.
McGeer put aside the project after making signifi2 Analogy with first powered
cant progress with passive machines (walking robot
gliders), returning to the world of airplane design.
flight
Our research has been, more or less, to pick up
On December 17, 1903 the Wright brothers first where McGeer left off, improve the gliders, and
flew a heavier-than-air man-carrying powered ma- then add simple power.
chine. There are various parallels between their
machine and the simply-powered low-energy-use
walking robots described here.
The analogy has its limits. Heavier-than-air
Starting from before the work began, the
Wrights were inspired by flying toys. The walk- powered flight was a well-defined major goal over a
ing machines here were also inspired by, and even long period of time with huge consequences. That

accomplishment swamps anything that might happen with robotics, including this research. The
Wright analogy does not extend to the significance
of our work, which is hugely less.

References for supplementary material

C. E. Bauby, A. D. Kuo, J. Biomech. 33, 1433
(2000).
S. H. Collins, M. Wisse, A. Ruina, Int. J. Robot.
Res. 20, 607 (2001).
G. T. Fallis, Walking toy, Patent, U.S.
Patent Office (1888).
Available at
http://www.tam.cornell.edu/.ruina/hplab.
M. Gomes, and A. Ruina, Phys. Rev. Letters E in
press (2005).
M. Hubbard, J. Appl. Mech. 46, 931 (1979).
A. D. Kuo, J. Biomech. Eng. 124, 113 (2002).
A. D. Kuo, Int. J. Robot. Res. 18, 917 (1999).
A. Ruina, M. Srinivasan, and J. Bertram, J. Theor.
Biol. in press (2005).
M. Wisse, A.L. Schwab, Int. J. Robot. Res., in
press (2005)
M. Wisse, D.G.E. Hobbelen, A.L. Schwab, IEEE
T. Robot., in press (2005)