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Inverted pendulum models are commonly used to study the bio-mechanics of biped walkers. In its simplest form, the inverted pendulum consists of a point mass attached to two straight mass-less legs. Most works constrain the motion of the mass to the sagittal plane,
*i.e.* the plane perpendicular to the ground that contains the direction toward the biped is walking. In this article, we remove this constrain to study the oscillations, the mass experiences in the direction perpendicular to the sagittal plane as the biped walks. While small, these lateral oscillations are unavoidable and of importance in the understanding of balance and stability of walkers, as well as walkers induced oscillations in pedestrian bridges.

When a human walks, the sagittal plane refers to the plane perpendicular to the ground that contains the direction toward the person is walking. As the person walks, its center of mass oscillates, both in the vertical direction (perpendicular to the ground) and lateral direction (perpendicular to the sagittal plane). One of our goals is to contribute to the understanding of the lateral oscillations.

The lateral oscillations play an important role in the balance and stability of individuals as they walk. Thus, their understanding is of interest in the field of bio-mechanics. These oscillations are also of interest in the field of robotics, since their understanding and control are likely to help improve the design of stable biped robots. While small, these lateral oscillations are the cause of some observed undesired and unexpected motions of pedestrian bridges when too crowded [

The use of inverted pendulum models to study the bio-mechanics of walking is a common practice. In its simplest form, the inverted pendulum consists of a point mass, which models the center of mass of the individual, attached to two straight mass-less segments, the legs. Works that use inverted pendulum (in its simplest or more sophisticated forms) or similar models to study aspects of the mechanics of biped walkers or related toys or biped robots include [

Different simple models are surveyed in [

Most works using the simplest inverted pendulum model constrain the motion of the center of mass to the sagittal plane. In this article, we remove this constrain. As a consequence, we are able to use this unconstrained inverted pendulum model to study the lateral oscillations the mass experiences as the person walks, i.e. the oscillations in the direction perpendicular to the sagittal plane. We believe and hope that the model and techniques described in this article will be adopted by other researchers and will prove useful in the study of different aspects of the mechanics of biped walkers.

In the next section the model is introduced. In the following section we describe the equations governing the dynamics of the mass while one foot is off the ground. Subsequently, we identify the solutions to the governing equations that correspond to periodic walking. We then report results of numerical simulations. We further explore our model by restricting our attention to the parameter regime of slow walkers. We then also study the short steps parameter regime. We finish the article with a discussion.

We model a human as a point mass

Assumption 1. At all times, either one or both feet are touching the ground.

The fact that each step takes the same time leads to our next assumption.

Assumption 2. Let

During the time interval

Assumption 3. The left leg is the stance leg during the time intervals

Assumption 4. During the time interval

Observation 1. During the time interval

As it makes contact with the ground at time

Assumption 5. Let

Observation 2. Since the legs and feet are mass-less, the motion of the swing leg does not affect the motion of the mass

Assumption 6. When both feet are touching the ground, at

One could wonder if, when both feet are touching the ground, the mass

In

Recall that

Assumption 7.

We now proceed to describe the motion of the mass during the time interval

where primes denote derivatives with respect to

Let

where dots denote derivatives with respect to

Let

where

We will solve these equations during the first step, i.e. in the time interval

for some constant

for some

The initial azimuthal angle,

then

Assume the parameters

In this section we assume that the parameters

Condition 1. If

Note that

Given any function

Let

Condition 2. If

In the above condition

Observation 3. If Conditions 1 and 2 are necessary and sufficient for periodic walking.

Observation 4. If Condition 1 is satisfied, so is Condition 2.

The proof of this observation is simple. One shows that, if Condition 1 is satisfied,

The azimuthal angle

Note

Due to symmetry, the Condition 1 is equivalent to: The minimum azimuthal angle

that

On one hand, given

Since

We summarize the findings in this section, and add to that, in the following observation.

Observation 5. (1) Let

(2) If the pair

(3) Let

(4) Let

(5) While we were not able to prove it, we have evidence to believe that, if

In what follows we will study aspects of the periodic walking of the model biped. The parameters that determine the motion are

Let

Let

For the parameter values

Let

Let

With the parameter values

In

We remind the reader that the dimensionless energy satisfies the constrains

The choice of the square in

In the Appendix 2 we show that the asymptotic value of the minimum polar angle

In the Appendix 3 we show that the asymptotic value of the dimensionless time of one step,

In the above equation, we mean that

Note that

Given Equation (19), we can obtain the asymptotic value of the amplitude of the lateral oscillations from Equation (16):

Compare

In this section, we explore a different parameter regime. Namely, we restrict our attention to small values of the initial azimuthal angle

This corresponds to the biped taking steps that are much shorter than the lengths of its legs, a realistic parameter regime.

In the Appendix 4, we outline the steps required to get the asymptotic value of the minimum azimuthal angle

and

The asymptotic value of the dimensionless time required by the biped to take one step,

Thus,

(see Equation (15)).

Making use of Equations (24) and (16) we get the asymptotic value of the amplitude of the lateral oscillations

Note that

On the other hand, the amplitude of vertical oscillations goes to zero quadratically as

As previously defined,

Using the initial conditions

On the other hand, Equation (7) leads to

Note that

In this article, we use a very simple inverted pendulum model to explore aspects of the mechanics of biped walkers. The novelty of this article is that we do not restrict the motion of the mass of the pendulum to the sagittal plane. As a consequence, we were able to study the lateral oscillations of the center of mass as the biped walks. These oscillations were beyond the capability of the simplest inverted pendulum models when the motion of the mass was restricted to the sagittal plane.

We performed numerical simulations and explore different parameter regimes with the use of asymptotic techniques. Our analysis shows that the inverted pendulum model remains simple enough to study, even when the mass is not restricted to move in the sagittal plane. We believe and hope the approach introduced in this paper will prove useful and be adopted by other researchers to study different aspects of the dynamic of biped walkers.

Goldsztein, G.H. (2017) Modeling Walking with an Inverted Pendulum Not Constrained to the Sagittal Plane. Numerical Simulations and Asymptotic Expansions. Applied Mathematics, 8, 57-76. http://dx.doi.org/10.4236/am.2017.81006

1. Proof of Point (3) in Observation 5

Let

that satisfies

and solving for

since

It also easily follows from the definition of

Our findings regarding

Observation 6. Fix

is a continuous and increasing function of

Consider

Assume

Since

Let

Since

Using Equations (41) and (42), plus

Make now the change of variable

The limit

Observation 7.

Assume now that

Let

Since

Making the change of variables

Note that

which tends to zero as

Observation 8.

Observations 7 and 8 show that, if

2. Calculations leading to Equation (19)

Consider

Recall that we are in the parameter regime of Equation (18), i.e.

We first note that

because

Next, we compute the asymptotic value of

Thus, we have the following approximation for

Making the change of variable

This last integral can be computed analytically to get

Given Equation (50), we have

This equation can be easily solved to give Equation (19).

3. Derivation of Equation (20)

We now proceed to compute the asymptotic value of the dimensionless time of one step, i.e.

Let

where

We first note that

Next, we compute the asymptotic value of

Thus, we have the following approximation for

Making the change of variable

In the above equation we have used the facts that

4. Derivation of Equations (24) and (25)

Let

Let

Note that

from where we get

Next, we observe that

From the last two equations we conclude that

Further calculation and expanding in powers of

Note that

Further elementary operations and expansions in powers of