arXiv:1208.6289v1  [physics.class-ph]  30 Aug 2012
Lift-offdynamics in a simple jumping robot
Jeffrey Aguilar
School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
Alex Lesov, Kurt Wiesenfeld, and Daniel I. Goldman*
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
(Dated: October 29, 2018)
We study vertical jumping in a simple robot comprising an actuated mass-spring arrangement.
The actuator frequency and phase are systematically varied to find optimal performance. Optimal
jumps occur above and below (but not at) the robot’s resonant frequency f0. Two distinct jumping
modes emerge: a simple jump which is optimal above f0 is achievable with a squat maneuver, and
a peculiar stutter jump which is optimal below f0 is generated with a counter-movement. A simple
dynamical model reveals how optimal lift-offresults from non-resonant transient dynamics.
Introduction – Organisms [1, 2] and robots [3–5] that
inhabit the terrestrial world must run, crawl, and jump
over a diversity of substrates and do so by effective de-
formations of appendages and bodies. While both simple
[6, 7] and complex [8] models have been created to study
optimal movement patterns, simple models have the abil-
ity to be fully analyzed and can thus provide guidance
for simplifying control of more complex devices, and even
reveal principles of biological locomotion [9].
Jumping is an important behavior for animals and
robots and is interesting, since it involves a transient
burst of activity. Biological studies have revealed mecha-
nisms of jumping in a diversity of organisms [8, 10–13]. In
robotics, biologically inspired legged jumping robots have
been constructed as an alternative to wheeled robots to
better traverse rough terrain [14–18]. The initial move-
ment strategies for optimal jumping are typically chosen
by empirical tuning for steady state hopping [17, 19, 20]
or squat jumps [16, 18, 21]. Systematic studies of the dy-
namics of transient behaviors, critical to issues of lift-off,
are relatively scarce.
In this paper we perform a detailed study of a sim-
ple jumping robot, a 1D mass-spring system with an
actuated mass; this model was originally developed as
a template for hopping in steady-state[22] and encapsu-
lates the leg compliance and an organism’s ability of self-
deformation via leg actuation. Systematic variation of
forcing parameters reveals complex dynamics which are
sensitive to amplitude, phase and frequency. Contrary to
our initial expectation, optimal jumping does not occur
at resonance. We introduce a reduced “piecewise linear”
equation of motion for the robot to analyze the transient
dynamics; another non-linearity appears as a result of
ground collisions. The model reveals a richness in behav-
ior governed by interplay of forced and free motion.
Experiment and model – The robot (total mass m =
1.18 kg) consisted of a linear motor actuator (Dunker-
motoren ServoTube STA11) with a series spring (k =
5.8 kN/m) rigidly attached to the bottom end of the
actuator’s lightweight thrust rod.
The actuator was
mounted to an air bearing which allowed for 1D, and
−1
0
0
1
2
 
 
 
 
−1
0
1
N (forcing cycles)
xp (mm)
xa
Sensor Voltage
xp
xa
Lift-off 
sensor
actuator
Moving 
thrust rod
0.276g
h
(a)
(b)
(c)
A
xp (mm)
FIG. 1. Jumping robot (a) schematic diagram. (b,c) Actuator
position xa and video tracked position of the thrust rod xp,
and the sensor voltage for A = 0.19 mm in which robot does
not lift-offand A = 0.30 mm in which the robot lifts off.
nearly frictionless, motion.
Due to power limitations
in the actuator, the bearing was inclined at 15 ◦rela-
tive to the horizontal, reducing gravitational accelera-
tion to 0.276g. The position of the actuator relative to
the bottom of the thrust rod, xa, was controlled such
that xa(t) = A sin(2πft + φ), where the amplitude, A,
frequency, f, and initial phase offset, φ are constant
during a jump.
The natural frequency of the robot,
f0 =
1
2π
p
k/m = 11.13 Hz. Video tracking of the de-
cay of the free oscillations of the robot (with fixed actu-
ator position and spring on the ground) indicated that
damping was small (damping ratio, ζ ≈0.01), and thus
the resonant frequency was virtually equal to the natural
frequency.
The jumping platform consisted of a rigid aluminium
plate; the coefficient of restitution of the robot with the
plate was 0.8 ± 0.06 over the range of relevant collision
speeds. To detect lift-off, a continuity sensor attached to
2
the bottom of the metal spring measured an open circuit
when the spring left the ground. Time to lift-offfrom
the onset of actuator activation and time of flight were
determined to 1 msec and jump height was calculated
from time of flight.
In concert with the experiments, we studied a simple
dynamical model of the robot, with equation of motion
¨xp = −¨xa
ma
m −α

˙xp
c
m + xp
k
m

−g
(1)
where k, c, and g are stiffness, damping and grav-
ity, respectively.
The total mass m is the sum of the
actuator/air-bearing mass, ma = 1.003 kg, and the mass
of the thrust rod, mp = 0.175 kg. The piece-wise con-
stant, α = 1, if xp < 0 and 0, if xp ≥0. A constant co-
efficient of restitution of 0.8 (measured from experiment)
modeled the collision of the spring with the ground.
 0
4
8
12
16
0
5
10
15
20
25
30
 
35
xa(t)
φ (rad)
f (Hz)
h (mm)
(a)
Single
(d)
Jump
xp
xA
Stutter Jump
(c)
(b)
No Jumps
MJ
S
S
ST
f0
ST
0
π/2
π
3π/2
FIG. 2. Jump height and jumping modes. (a) Experimental
jump height, h, as a function of f and φ with illustrated
actuator trajectories (top) at different φ with A = 4 mm.
White lines (derived from model) separate different jumping
modes, with ST indicating stutter jump, S for single jump,
and MJ for multi-jump. (b) inset shows model (Eq. 1) with
same parameters. (c,d) illustrate trajectories for the stutter
and single jumps. Robot is airborne (white robots) when the
rod position, xp > 0. Global actuator position, xA, not to
scale with rod length.
Lift offand jump height – In all experimental runs,
at t = 0 the actuator was commanded to move from
rest (Fig. 1b,c). For fixed frequency, below a minimum
amplitude Amin(f), no lift-offwas detected and above
Amin(f) the robot was able to jump; Amin(f) was deter-
mined by an iterative procedure in which a binary search
was implemented until Amin(f) was determined to within
0.00625 mm, the resolution of the actuator encoders. As
expected, when the actuator was continuously activated,
the absolute minimum of Amin over all frequencies oc-
curred at the resonant frequency f0, and was independent
of φ. However, since we were interested in rapid jumps
from rest, actuator forcing was then restricted to only
one cycle (N = 1). To our surprise, things were quali-
tatively different: the smallest Amin(f) did not occur at
f0, and varied with phase offset φ.
We next systematically examined jumping height for
N = 1. We fixed A = 4 mm, which was above Amin(f)
for f > 3.5 Hz, and studied how jump height h depended
on f and φ. The mean h from three trials was recorded.
Variation in h from jump to jump was small; the standard
deviation of h was less than 0.5 mm, or approximately
1% of mean h, past approximately 4 Hz. Certain frac-
tions of f0 below 4 Hz exhibited significant variance due
to small multi-jumps that occurred as a result of sub-
resonant harmonics (max heights seen in MJ section of
Fig. 2a). These frequencies were perfectly timed to allow
multiple complete oscillations during motor actuation to
precede a larger final jump.
Fig 2 shows the results of 6720 × 3 experiments. For
fixed φ, above a critical f the robot was able to lift-off.
Two broad maxima in h were observed, neither occurring
at f0. Integration of Eq. 1 quantitatively reproduced the
experiments, see Fig. 2b inset.
The two local maxima correspond to two distinct
modes of jumping: a “single jump” and a “stutter jump”.
In the single jump mode, the robot compressed the spring
and was propelled into the air. In the stutter jump mode,
the robot performed a small initial jump followed by a
larger second jump, see Fig. 2c. We used the model to
determine the boundaries of the regions of the φ−f plane
of the different modes. For large φ single jumps predom-
inate while stutter jumps occurred at lower f and φ.
The emergence of the stutter jump was unexpected.
To understand its presence, consider (for example) the
case φ = π/2, so that the initial actuator acceleration
is negative. This causes the less massive thrust rod to
be accelerated upward before moving down to compress
the spring and then lift offagain. Interestingly, the stut-
ter jump was observed even for phases somewhat larger
than π (Fig 2), for which the initial actuator acceleration
is expected to progress positively from 0. The reason lies
in the physical constraint that the actuator must start
from rest, regardless of phase offset.
Thus, any phase
offset corresponding to a non-zero initial actuator veloc-
ity causes an initial impulse acceleration (i.e. the initial
actuator trajectory is not an ideal sine wave). For a phase
such as π, the initial, brief actuator acceleration is large
and negative, causing an intermediate hop.
We next examined various characteristics associated
with optimal jump height, see Fig. 3. At the optimal f of
each phase φ, maximum h was determined and displayed
two broad maxima (Fig 3a); the maximum h values were
3
insensitive to f and were nearly 10× larger than A. The
f which gave maximum h was less than f0 for stutter
jumps and greater than f0 for single jumps (Fig 3b). The
time to lift-off(Fig.
3c) was smaller for single jumps
than stutter jumps. As we will show below, peak power
expended in deforming the system Pdef scaled like f 3 and
thus increased dramatically for single jumps.
Max h (mm)
 
 
0
0.5
1
1.5
2
3
9
15
Opt f (Hz)
 
f0
Stutter Jumps
Single Jumps
(b)
(a)
Phase φ (π radians)
0.1
0.2
Lift-Off t (s)
(c)
0
(d)
10
20
30
0.5
1
|Pdef| (W)
FIG. 3. Experimental (solid) and simulation (dashed) results
of (a) maximum jump height at each phase, (b) the corre-
sponding optimal frequency for each phase, (c) time to lift-
off, and (d) deformation power at optimal frequency; initial
transients are omitted in this calculation.
Theory of transient mixing – At first glance, Eq.(1)
looks completely tractable. Unfortunately, the disconti-
nuity associated with the factor α renders the equation
“piecewise linear”, which is to say nonlinear.
Indeed,
simulations of Eq. 1 show a wide variety of behaviors
(including bifurcations, hysteresis, chaos). The situation
is reminiscent of other piecewise linear dynamical sys-
tems which display complex dynamics, including the tent
map [23] and the bouncing ball [24]. Nevertheless, using
analysis and numerics, Eq. 1 allows us to gain insight
into the experimental observations. We are particularly
interested in why optimal jumps occur only offresonance.
Consider first the peak labeled S in Fig. 2a, represent-
ing the highest single jumps. This peak occurs at actu-
ator phases near φ = 3π/2. For a relatively low thrust
rod mass, jump height is proportional to the square of
the absolute actuator velocity, ˙xA(t) = ˙xp + ˙xa, at take-
off. Neglecting damping and collisional loss, at φ = 3π/2,
this velocity is (solving Eq. 1 with α = 1, c = 0, ma = m)
˙xA(t) = 2πAf 2
f 2
0 −f 2 f0 (−sin 2πf0t + (f0/f) sin 2πft)
(2)
The take-offvelocity is thus a prefactor times the sum
of two sinusoids (the one at frequency f0 represents the
transient response, which mixes with the steady state
contribution). The prefactor generally favors f near f0,
but destructive interference suppresses ˙xA too close to
resonance.
Moving offresonance, the prefactor favors
higher f over lower, so the optimum f lies somewhat
above f0. This argument holds regardless of A.
Understanding the optimality of the stutter jump is
more complicated. The key is to consider the system en-
ergetics, and in particular the conditions that maximize
the total work done during the drive cycle. The instanta-
neous power input is P = Fextvm, where Fext is the total
external force (including gravity and spring forces) and
vm is the center-of-mass robot velocity, which is to good
approximation the absolute actuator velocity, ˙xA. The
total work done by external forces is maximized when
˙xA both (1) is large in magnitude, and (2) has the same
sign as the Fext.
Absolute Motor Velocity (m/s)
 
 
 
 
0
0.4
0
0.05
0.1
0.15
0.2
0.25
0
4
Power Input (mW)
 
 
f/f0 = 1
0
0.4
0
0.4
0.8
0
4
0
4
8
Time (s)
f/f0 = 0.54
f/f0 = 0.72
FIG. 4. Simulated time trajectories of absolute actuator ve-
locity (black) and power input by external forces (red) for
f/f0 = 0.54, 0.72 (optimal), and 1, at φ = π/2. The vertical
dotted line indicates when the actuator stops. Light gray ar-
eas: aerial state (xp > 0); dark gray: negative force ground
state (mg/k < xp ≤0); white: positive force ground state
(xp ≤(mg/k)).
The situation is illustrated in Fig.
4, for φ = π/2,
i.e. when the stutter jump is most effective. At f = f0
(lower panel), the actuation is too fast and the actuator
turns offwell before lift-off.
At a low f (top panel),
the actuation is too slow: the actuator turns offwell
after lift-off, so much of the power stroke is wasted. The
optimal drive (middle panel) lies somewhere in between.
In addition, an optimal stutter jump depends not only on
the phasing of competing sinusoids while on the ground
(just as for single jumps), but also on the proper timing
of ground and aerial states. The latter varies with φ and
4
does not generally occur at f0. This sensitivity to proper
timing explains the narrow frequency bandwidth required
to achieve optimal jump heights using the stutter jump
mode. A further consequence is a strong dependence of
optimal f with respect to A: larger A produce lower
optimal f, and smaller A produce higher optimal f. In
contrast, the optimal f for the single jump mode does
not show a strong dependence on A.
Assuming small damping and no collisional losses, de-
formation power (defined as Pdef = mp¨xp ˙xp + ma¨xA ˙xA)
was calculated as Pdef = 4π3mamp
m
A2f 3. Thus the stut-
ter jump is energetically advantageous since it has a lower
optimal f than the single jump. In fact, the stutter jump
uses nearly an order of magnitude less power to achieve
comparable jump height to the single jump.
Conclusion– We have analyzed the dynamics of lift-off
in the simplest hopping robot and found that the per-
formance is quite rich and remarkably sensitive to start-
ing phase trajectory, largely a result of the transient dy-
namics in a linear-mass spring system. Unlike in steady
state, in the transient regime, optimization occurs at non-
resonant frequencies. The system becomes hybrid for cer-
tain parameters as a stutter jump emerges. This mode
achieves comparable jump height but uses less power.
Analysis of a simplified model reveals that impulse accel-
erations and discontinuous transitions to the aerial state
are essential ingredients in understanding the dynamics.
Our model provides insight which more complex and
multi-functional robots can use to execute rapid jumps
and starts. Biologically, our model is in accord with a
previous model of bipedal jumping which predicted that
counter-movement achieves greater jump height than the
squat jump [25].
A quick single jump that resembles
a squat jump is beneficial when a fast escape is essen-
tial, while a slower stutter jump similar to a counter-
movement can achieve comparable jump height.
Pri-
mates like Galagos (bushbabies) have been documented
to perform this double jump behavior to reach a higher
platform [26]. Based on the power arguments above, we
hypothesize that this mode is advantageous.
It would be interesting to investigate how other factors,
intrinsic and environmental, affect optimal performance.
A non-sinusoidal actuation could improve jump height,
take offtime, or efficiency. Animals jump offcompliant
surfaces (like tree branches) and from deformable sub-
strates (like sand).
Systematic studies of the jumping
robot under analogous conditions could yield insight into
optimal strategies in these and other (man-made) actors.
Acknowledgements– We thank Harvey Lipkin, Paul
Umbanhowar, Nick Gravish, and Yang Ding for discus-
sion, as well as Andrei Savu for apparatus construction.
This work was supported by the GEM Fellowship, the
Burroughs Wellcome Fund and the ARL MAST CTA.
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