Conventional wisdom would have it that moving mechanical systems that dissipate energy by Coulomb friction have no relationship between force and average speed. One could argue that the work done by friction is constant per unit of distance travelled, and if propulsion forces exceed friction, the net work is positive, and the system accumulates kinetic energy without bound. We present a minimalistic model for legged propulsion with slipping under Coulomb friction, scaled to parameters representative of single kilogram robots and animals. Our model, amenable to exact solutions, exhibits nearly linear (R2 > 0.96) relationships between actuator force and average speed over its entire range of parameters, and in both motion regimes, it supports. This suggests that the interactions inherent in multilegged locomotion may lead to governing equations more reminiscent of viscous friction than would be immediately obvious.

References

References
1.
Coulomb
,
C. A.
,
1785
, “
Theorie des Machines Simples
,”
Memoirs de Mathematique et de Physique de l’Academie Royale
.
2.
Anh
,
L.
, and
Belyaev
,
A.
,
2012
,
Dynamics of Mechanical Systems with Coulomb Friction
,
Springer Science & Business Media
,
Heidelberg, Germany
.
3.
Reynolds
,
O.
,
1886
, “
On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower’s Experiments, Including an Experimental Determination of the Viscosity of Olive Oil
,”
Philos. Trans. R. Soc. London
,
177
, pp.
157
234
, IV.
4.
Olsson
,
H.
,
Åström
,
K. J.
,
de Wit
,
C. C.
,
Gäfvert
,
M.
, and
Lischinsky
,
P.
,
1998
, “
Friction Models and Friction Compensation
,”
Eur. J. Control
,
4
(
3
), pp.
176
195
.
5.
Alexander
,
R.
,
1985
, “
The Maximum Forces Exerted by Animals
,”
J. Exp. Biol.
,
115
(
1
), pp.
231
238
.
6.
Walter
,
R. M.
, and
Carrier
,
D. R.
,
2007
, “
Ground Forces Applied by Galloping Dogs
,”
J. Exp. Biol.
,
210
(
2
), pp.
208
216
.
7.
Weyand
,
P. G.
,
Sternlight
,
D. B.
,
Bellizzi
,
M. J.
, and
Wright
,
S.
,
2000
, “
Faster Top Running Speeds are Achieved With Greater Ground Forces not More Rapid Leg Movements
,”
J. Appl. Physiol. Respir. Environ. Exerc. Physiol.
,
89
(
5
), pp.
1991
1999
.
8.
Gregorio
,
P.
,
Ahmadi
,
M.
, and
Buehler
,
M.
,
1997
, “
Design, Control, and Energetics of an Electrically Actuated Legged Robot
,”
IEEE Trans. Syst. Man. Cybernet. Part B
,
27
(
4
), pp.
626
634
.
9.
Gabrielli
,
G.
,
1950
, “
What Price Speed?
,”
Mech. Eng.
,
72
(
10
), pp.
775
781
.
This content is only available via PDF.
You do not currently have access to this content.