J. C. Sprott

Introduction

What follows is the derivation of a set of differential
equations describing the dynamics of walking and running. The model is
extremely simple. It assumes the legs are stiff and oscillate at their
natural resonant frequency. Newton’s second law is solved assuming the
mass (*m*) of the body is concentrated at the top of the legs (length
*L*)
and moves forward with a constant horizontal velocity (*v*_{x}).
The transition from walking to running occurs when both feet lose contact
with the ground.

The Model

The mass *m* moves vertically in response to
the superposition of two forces, gravity *mg* (*g* = 9.8 m/s^{2})
in the downward direction, and an upward normal force *N* exerted
by the ground on the foot. Thus by Newton’s second law, the vertical component
of its velocity *v*_{y} obeys the equation:

*m*d*v*_{y}/d*t* = *N*
– *mg*

As long as the foot maintains contact with the ground,
the mass moves along a circular trajectory with a downward acceleration
of approximately *v*_{x}^{2}/*L*, from which
the required normal force can be calculated as *N* = *mg* – *mv*_{x}^{2}/*L*.

Note that the normal force is constant and less than
*mg*
throughout the circular portion of the trajectory and there is a value
of *v*_{x} at which *N* reaches zero. For larger values
of *v*_{x}, the foot leaves the ground and the mass moves
along a parabolic trajectory rather than a circular trajectory with gravity
as the only force. This condition corresponds to the transition from walking
to running in this model. The speed at which the transition occurs is *v*_{x}
= (*gL*)^{1/2}, or about 3 m/s (~ 7 MPH) for a person of average
size.

Although the normal force is constant and less that
*mg*
throughout the downward curving portion of the trajectory, there must by
a large impulsive upward force greater than *mg* at the cusps where
the trajectory abruptly changes from downward to upward. This impulse must
equal the integral of the downward force over the remaining interval so
that there is no net acceleration of the mass averaged over a cycle. Hence
*I* = (*mg* – *N*)*T*/2, where *T* is the natural
period of oscillation of a leg. This impulse changes the vertical component
of the momentum by *m*D*v _{y}
= I*, from which the initial value of

From Newton’s second law the dynamical equations can thus be derived:

*dy*/*dt = v _{y}*

*dv _{y}*/

The transition from walking to running is smooth at the value calculated
above, and there is no qualitative change in the trajectory of the mass.

Power Requirement

The average power required to move forward at velocity
*v*_{x}
can be estimated by assuming that one has to supply the energy *mgh*
to raise the center of mass through a height *h* each half cycle,
and that none of this energy is recovered. The average power is thus *P*
= 2*mgh* / *T*. The height *h* can be calculated from *h*
= *v*_{y}(0)*T*/8, from which the power can be calculated:

*P* = *mgv*_{y}(0)/4 = (*g*
– max[0, *g* – *v*_{x}^{2}/*L*])*mgT*/4

For walking at low speed, the power is *P*_{W}
= *v*_{x}^{2}*mgT*/4*L*, whereas for running
the power is *P*_{R} = *mg*^{2}*T*/4. At
the transition speed, the power required for *m* = 50 kg and *T*
= 0.5 s is about 600 W. Walking is generally performed with a period close
to the natural resonance of the leg, *T* = 2p(2*L*/3*g*)^{1/2},
whereas for running, one can increase the period to achieve higher speed,
with a corresponding proportional increase in the required power. At this
optimal period, the initial vertical velocity when walking is *v*_{y}(0)
= p*v _{x}*/

*P* = 1.283 *mgv _{x} *min[1,

Finally the general expression for the initial vertical velocity subject to the above assumptions is

*v _{y}*(0) = 1.283

This initial condition along with the dynamical equations above provide
a complete description of the motion for all values of *v _{x}*
within the assumptions of this rather idealized model.