Scale effects on locomotion:
how the optimal gait changes
according to dimensions
Umberto Scarfogliero, Cesare Stefanini, Paolo Dario
Scuola Superiore Sant’Anna, CRIM Lab
IMT Lucca Institute for Advanced Studies
How scale effects influence locomotion?
•
•
Investigate scale effects on locomotion in
different sized animals.
Build a robotic platform for scientific
investigation on jumping in small animals
Scale effects in animals
Starting from the observation that smaller fast animals take longer
step length compared to their leg length, a locomotion strategy
for small legged robots has been investigated [1]
Despite the energetic cost per gram per stride is almost constant
along different sized animals running at equivalent speed, smaller
animals are generally less efficient than bigger ones. This because
the specific energetic cost is proportional to stride frequency [2]
Robotics?: when decreasing dimensions
friction becomes more relevant respect to
mass-related forces such as inertial forces
[1] D.F.Hoyt, S.J. Wickler, E.A. Cogger, Time of contact and step length: the effect of limb length, running speed, load
carrying and incline, J. Exp. Biol., 203, 221–227 (2000)
[2] N.C. Heglund, C.R. Taylor, Speed, stride frequency and energy cost per stride: how do they change with body size
and gait?, J. Exp. Biol., 138, 301-318, (1988)
Volume related forces and surface forces scales
differently, the former proportional to l3 the
latter to l2. This influence the bone diameter,
the posture and many other gait characteristics,
such as maximum exerted force respect to
body weight (BW)
As both the maximum yield stress and safety factor in bones are
about constant for different animals, in small animals supporting and
moving the body is not a critical issue. Thus they can easily achieve
high relative running speed. This performance would be impossible for
an elephant due to structural bone and muscles limitations.
Relative running speed [3] [body length s-1]
Elephant (Loxodonta Africana)
1.40
Rodent (Dipodomys merriami)
80.31
[3] J.I. Diaz, Different scaling performance in small and large terrestrial mammals, J. Exp. Biol., 205, 2897-2908, (2002)
Animal
Maximum force in
body weight [5]
Human
3
Kangaroo rat
8
Locust
13
Froghoppers
400
Fleas
135
A force comparable to body weight means small variation in vertical
momentum, and thus a short airborne phase. Elephants can reach
speeds of 6 m/s, without engaging an airborne phase [4], while
rodents difficultly move at low speeds [2].
Gait frequency is triggered by gravity: relatively high vertical
forces naturally makes it easier for small animals to jump several
body length within one step (man on the moon)
[4] S.A. Frank, M.A. Nowak, Are fast-moving elephants really running?, Nature, 422, 493-494, (2003)
[5] R.M. Alexander, Principles of animal locomotion, Princeton University Press, (2003)
A key factor in considering scale effects on locomotion is the
Froude number Fr .
v2
Fr 
gl
m
l
The Froude number takes into account the ratio between
kinetic and potential energy during the stance phase
Fr can be also considered as the ratio between centrifugal
and gravitational force acting on a stance leg.
Theoretically contact is lost for Fr>1
It is observed that different sized animals switch from walk to run
at the same Fr≈0.4 [5],[6].
At the same Fr locomotion in different sized animals is dynamically
similar, which means that decreasing dimensions has the same
effect as increasing the square of the speed
[5] R.M. Alexander, Principles of animal locomotion, Princeton University Press, (2003)
[6] C.T. Farley, C.R. Taylor, A mechanical trigger for the trot–gallop transition in horses, Science, 253, 306–308, (1991)
Bio-inspiration
Taking into account scale effects, small robots
would have a longer airborne phase respect to
bigger ones, leading to a jumping gait as a
possible efficient solution for locomotion
Jumping vs. hopping
•
Contact forces >> mg
•
Long airborne phase
•
Short contact phase
•
High peak power at take off
•
Relevant
changes
potential energy
in
cm
In jumping frogs, peak power output is from 1.5 up
to 7 times greater than the maximum muscle
power [9]. This is due to the presence of elastic
tendons and an inertial click mechanism. The
muscle stretch the tendon that releases the energy
stored when the click mechanism is relished.
In the robot, this allows to reduce the mass of
actuation and power supply.
[9] T.J. Roberts, R.L. Marsh, Probing the limits to muscle-powered accelerations: lessons from jumping bullfrogs, J. Exp.
Biol., 206, 2567-2580, (2003)
Jumping implies high changes in center-of-mass (CM) potential
energy and recovering impact energy when landing is important in
order to preserve efficiency
Ek 
l
v
Fc
h
m gl
2 sin 20 
E p  mgh  Ek sin2 0 
For
0=45°
potential energy is
half the kinetic energy needed
for the jump
Passive forelegs store and release impact
energy in their elastic recoil, and in continuous
gait remarkably increase the robot performances.
The model that we obtain in this way is similar to
the Raibert hopper.
Due to the short contact phase and high contact forces, flight
stability can be achieved by using wings or aerial passive
appendages.
Passive rotation of the wings
let them orientate during take off
and flight phase
Steering can be achieved
airborne rotating the wings
using a four bar mechanism
actuated by a small motor
The concept: a jumping mini robot
Based on these observations, a jumping robot was designed with:
•
Active rear legs: the thrust for the jump is provided by springs
loaded by a tiny motor
•
Passive forelegs: store and release impact energy in their elastic
recoils
•
Spiny foot: foot-ground contact is ensured during the whole leg
extension by a spiny foot
•
Wings are used to stabilize the gait and let the robot steer
Fx
Fground
Fy
Experimental platform
2.
1.
3.
6.
4.
3.
4.
5.
1. Magnetic click
mechanism
2. Rear legs actuation
springs
3. DC motor
4. 3V Battery
5. Springy forelegs
6. Eccentric cam
2.
5.
Dimensioning rear legs according to jumping
energetic considerations
performances:
1. Kinetic energy needed at take off
Ek 
1 2
m gl
mv 
2
2 sin20 
m gl  2 Ed
K2 
(x22  Δx12 ) sin2α 
2. Thrust effectiveness: rear feet contact
xT   x2e acotan 
sinacotan()  cosacotan()


1
2
T
acotan 
d
With =0.4 and xeq=0 x1
should be at least half x2
Dimensioning forelegs to maximize energy recovery: synchronize
fore and rear legs elongation
Forelegs: torsional and compression springs in
series. Consider a corresponding non-linear
compression spring of stiffness K’
x
----
Rear legs
Forelegs
Tactuation   fore   aft 

t
2
2d fore 1   fore


2
2d aft 1  aft
Gait simulations and robot model
0.1
PosX
0.1
grillo_050901.cmd
PosY
To Workspace
Foot_contact [mm]
To Workspace1
0.1
2+11+6+6+1+3=29
Parameter
Value
Mass
0.01 Kg
Rear leg stiffness K2
0.4 N/mm
Rear leg relative damping
0.1
Rear leg elongation _x
10 mm
Foreleg stiffness K1
0.2 – 1 N/mm
Foreleg relative damping
0.1- 1
Time delay Left [s]
0 in continuos gait
3
Rear_leg (11)
grillo_control_050721
13
Rear legs
S-Function
Scope X
14
Scope Y
Front_legs (6)
19
Front legs
1/z
20
adams_sub
Unit Delay
1/z
Scope X1
Unit Delay1
1/z
Unit Delay2
Scope X2
1/z
Unit Delay3
1/z
Unit Delay4
25
1/z
Unit Delay5
0.1
Time delay Right [s]
0 in continuous gait
27
Rotation_rear_legs (3)
29
Rotation rear legs
Simulations were run with Simulink and Adams on the jumping gait.
The mass distribution of the robot was given by the prototype design,
while linear springs modeled the legs. Spherical joints were chosen to
model the shoulders, constraining the rotations with torsional springs
of different stiffness according to the legs and to the plane of rotation
First, the model was modeled with a pitch stabilization, as would be
done by wings, in order to focus on the effects of the passive recoil
on the jumping gait. After few steps, the jumping height sensibly
increases despite the actuation remains the same at every step
Stiffness and damping parameters are
chosen so that forelegs contraction and
rear legs elongation are synchronized to
maximize jump thrust
When the robot was modeled as a free body, pitch rotation become
critical in determining the succeeding of the jumping sequence.
Before introducing an additional actuator, the possibility of a
triggered gait was investigated
Due to the relatively long airborne phase and high contact forces, the
best way to control the body rotation is during the flight phase. This
can be achieved using passive wings or through inertial forces
Conclusions
• Scientific lesson: jumping is an effective solution for achieving high
locomotion speeds within small dimensions
• Dedicated design solutions are needed for jumping:
 active and passive energy storing by means of elastic structures
 Stabilizing structures for the airborne phase
• Robotic platforms exploiting and proving the theoretical model can
be successfully developed
• Possible applications of jumping minirobots include exploration and
monitoring in unstructured environments
Future developments
•
Cam design and fabbrication: mechanism for spring loading
•
Study of the flight phase: air lift and grad can be studied
adopting small wings
•
Steering capability: force distribution at take off or airborne
turning could be explored
•
Ground contact: study foot-ground interaction