Lateral oscillations of the center of mass of bipeds as they walk. Inverted pendulum model with two degrees of freedom

When a human walks in a straight direction, the sagittal plane refers to the plane that contains the direction perpendicular to the ground and the direction toward the person is walking to. 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). While small, these lateral oscillations are the cause of some observed, undesired and unexpected motions of pedestrian bridges when being crossed by large crowds. The physics behind the wobbling of these bridges is understood to some degree, but not completely. The following is a brief summary of what the relevant physics are believed to be.

Since the center of mass of pedestrians oscillate laterally as they walk over the bridge, the bridge must be exerting lateral forces on the pedestrians and thus, the action-reaction principle implies the pedestrians exert lateral forces on the bridge. These forces are oscillating and close to periodic.

Assume N pedestrians are crossing the bridge. They all may walk with similar frequencies, but not in phase (they are not synchronized) and thus, the forces the pedestrians apply to the bridge partially cancel. However, the Central Limit Theorem of probability asserts that all those forces do not add up to zero. Instead, they add up to a force that would result from having of the order of $N$ pedestrians walk in phase, i.e. a force of the order of magnitude $N$ times the force due to one pedestrian.

When N is large large enough, and the frequency of the walkers is close enough to the natural frequency of the bridge, the force described in the last paragraph is large enough to cause the bridge to begin to wobble.

As the bridge wobbles, the pedestrians react in an attempt to keep their balance. As a consequence of their response, the frequency of the walkers becomes closer to the frequency of oscillation of the bridge and the pedestrians walk in phase. Thus, the forces they exert on the bridge no longer partially cancel. Instead, they act constructively for the most part, and thus, the net force they exert on the bridge is larger than when the bridge is not moving and the pedestrians walk out of phase, causing the amplitude of the oscillations of the bridge to increase even further.

Motivated by the above discussion, in this article we study the lateral oscillations of the center of mass of pedestrians as they walk on a non-moving flat ground. More precisely, we develop the simplest possible model (we could think of) that is able to capture these oscillations and the physics behind them. This model is able to relate the amplitude of the lateral oscillations of the center of mass of pedestrians as they walk with the other parameters, such as length of the pedestrian legs, the length of each step, how wide apart the feet are, and how fast the pedestrian walks.

The lateral oscillations of the center of mass of pedestrians as they walk 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 is likely to help improve the design of stable biped robots. Thus, the results of this article is relevant in other fields beyond the problem of pedestrians induced oscillations in bridges that was our original motivation.

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 attached to two thin straight mass-less legs. By thin legs we mean that they are infinitely thin, i.e. uni-dimensional segments. Most works using the simplest inverted pendulum model constrain the motion of the mass and legs to the sagittal plane. While the inverted pendulum models constrained to the sagittal plane have proved to be very powerful tools to study several aspects of the dynamics of walking, the lateral oscillations of the center of mass of pedestrians as they walk are beyond the capabilities of these models. The reason is simple. These oscillations are in the direction perpendicular to the sagittal plane.

In this article, we remove the constrain described in the previous paragraph. In other words, we still consider a simple inverted pendulum model consisting of a point mass attached to two thin straight mass-less legs, but we do not constrain its motion to any plane. We use our model to study the lateral oscillations of the center of mass of walkers. In doing so, we illustrate that the analysis of this unconstrained inverted pendulum remains relatively simple. Thus, it is our 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.

This article is organized as follows. In section II we describe work by other researchers relevant to our studies. In section III we introduce our model.

While the wobbling of pedestrian bridges induced by walkers motivates the studies of this article, we are here concerned with the dynamics of walking in non-moving platforms. Thus, we will not go into any details on the work that has been done to understand the pedestrians-bridges interaction, but we refer the interested reader to the articles.^1–11 

Regarding inverted pendulum models, as mentioned in the introduction, they have been widely used to study the mechanics of biped walking, but for the most part, the models constrain the legs and the center of mass to the sagittal plane. In fact, all the models in the articles mentioned in this section satisfy this constrain. In this section, we give a very brief summary of a portion of relevant work available in the scientific literature.

Some of the earliest works on inverted pendulum models to study the mechanics of walking that pave the way for future studies are Refs. 12–16.

McGeer’s is one of the most influential works in the field.¹⁷ He studied the dynamics of what he called passive walkers. These are bipeds that are able to walk downhill when the slope of the ground is small. These walkers are powered by gravity alone. He used an inverted pendulum type model with sections of circles as feet and with the legs having some mass. McGeer continued his influential work in Ref. 18. Numerical investigations of passive walkers extending the ideas introduced by McGeer can be found in Refs. 19 and 20.

The work in Ref. 21, is concerned with energetic. The metabolic energy required for walking is discussed. The analysis in that article uses the simplest inverted pendulum model as described in the introduction of the present article. In Ref. 22 the author studies the transition from walking to running as the speed of the biped increases. He assumes the biped will select the most energetically efficient mode of locomotion between walking and running. He uses an inverted pendulum type model where he assumes the legs are compliant and have some mass. Compliant legs are needed in this study because he author considers running. Running is fundamentally different than walking. While walking, at least one foot is in contact with the ground at all times. Both feet are in the air at times during running. Running can be though of as a sequence of jumps, thus the need of compliant legs for the proper modeling of running. A later work that also studies the transition from walking to running, and also with an inverted pendulum model with compliant legs is Ref. 23.

Inverted pendulum models were used to obtain a relationship between speed and step length in bipeds based on energy considerations.^24,25 The article²⁶ mentions the inverted pendulum analogy as one of the two most prominent theories of walking, the other being the six determinants of gait. The author compares both theories in that article and discusses the importance of the energy exchanges that occur as a foot strikes the ground and the other pushes off the ground.

In Ref. 27, the simplest inverted pendulum model is used to quantify the work done by each leg separately while walking. Most of the work occurs as a leg strikes the ground, or as a leg pushes off the ground. They compare their analysis with experiments. An other article concerned with the energetic of walking is Ref. 28, where the authors study the energetic advantage of having the feet with the shape of a sections of circles as oppose to point feet. The authors conduct experiments with humans wearing specially designed boots, and compare the results obtained with the predictions from a inverted pendulum type model with sections of circles as feet.

On the experimental side, seminal the work of McGeer¹⁷ has inspired the construction of toys that can not stand, but can walk stably down a slope.^29,30

We conclude this section by mentioning review articles where the reader can find more details. The article³¹ is a review of inverted pendulum models, as well as other simple models to study walking, running, and other human motions. A more recent review of models of walking, that includes the inverted pendulum, is Ref. 32. Two recent reviews on locomotion of human and animals are Refs. 33 and 34.

We model a human as a point mass m with two straight mass-less thin legs. Each leg is of length L. The mass m is the common end point of the two legs. The other end point of each leg is its foot. Only the feet can touch the ground. We will call this model of a human the model biped. The model biped walks symmetrically (with respect to its left and right), at a constant pace (i.e. each step takes the same time), and all its steps have the same size. Next, we describe our modeling assumptions. In the statements below k is any integer.

In contrast to running, when both feet are in the air at times, at least one foot is touching the ground at all times. This leads to our first modeling assumption.

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

During the time interval $( t ̄ k , t ̄ k + 1 )$⁠, the foot of only one leg is touching the ground. This leg is called the stance leg. The other leg is called the swing leg. For definiteness, we assume the left and right leg alternate being the stance and swing legs as follows.

Frictional forces prevent the foot of the stance leg to slip. These forces allow us to make our next assumption.

As mentioned above, frictional forces play an important role, they prevent the model biped to slip and fall. These are forces on the stance leg. In the next Observation, we describe the forces acting only on the mass m, not on any of the legs. We only need the forces acting on m to describe its dynamics. Accordingly, and to keep Figure 1 as simple as possible, we display in that Figure only the forces acting on m, not the frictional forces or any other force acting on the legs.

As it makes contact with the ground at time $t ̄ = t ̄ k$⁠, the leg that was the swing leg during the time period $( t ̄ k − 1 , t ̄ k )$⁠, exerts an impulse on the mass m, changing its momentum, and preventing the mass from falling to the ground. Simultaneously, the leg that was the stance leg during the time period $( t ̄ k − 1 , t ̄ k )$⁠, exerts an other impulse on the mass m, changing its momentum further, and giving the mass enough energy to take its next step. This last impulse corresponds to the human pushing off the ground with the foot of the leg that is changing from being the stance leg to being the swing leg. The impulse each leg exerts on the mass must be parallel to that leg and pointing from the mass away from its foot.

The discussion of these last paragraph leads to the next assumption.

In fact, we will find later in this article that α_k = β_k. We are not imposing this condition at this point, but it will result from requiring that the motion be periodic in the sense of Assumption 3.9, which is detailed at the end of this section. We also mention that, while not imposed as a condition, we will find that energy is conserved during the motion. This is partially understood from Figure 1, while one foot is not touching the ground. During that period of time, the only non-conservative force on the mass is the one due to the stance leg, but that force is perpendicular to the trajectory of the mass and thus, does not do work on the mass. We will also find that α_k = β_k implies conservation of energy while both feet are touching the ground.

Note that we are implicitly assuming that the person has the ability to move the swing leg and control where it will place its foot to take the next step, but that this motion of the swing leg does not affect the motion of the center of mass of the biped since the legs are mass-less. Of course, the legs of real persons are not mass-less, but their mass are small compared to the total mass of persons. Note also that we are not placing restrictions on the angle formed by the legs and having the mass as vertex. Real persons, or any mechanical biped, would have some restriction. While interesting, the discussion about these restrictions is beyond the scope of the present article. Nevertheless, the approximations or idealizations mentioned in this paragraph are very common and accepted practice in the field of biomechanics.^12,17,26

In Figure 2 we illustrate and introduce geometric parameters. The solid circles are the footprints. The footprints from the left foot are included in a dotted line. The footprints from the right foot are included in the other dotted line. These two lines are a distance w apart. Thus, w models how wide the steps are. The white circles are the orthogonal projection onto the ground of the mass when both feet are touching the ground. After each step, the center of mass advances a distance u. Thus, u models how long the steps are. e_f is the dimensionless unit vector that points in the direction the biped is walking. e_h is the dimensionless unit vector perpendicular to e_f, parallel to the ground and pointing to the right of the biped. e_v is the dimensionless unit vector perpendicular to the ground pointing upward (see Figure 1 also).

Recall that $T ̄ s$ is the time of one step and $r m = r m ( t ̄ )$ is the position of the mass m at time $t ̄$⁠. The symmetry and periodicity of the walk leads to the next assumption.

We now proceed to describe the motion of the mass during the time interval $( 0 , t ̄ 1 )$⁠. Recall that $t ̄ 1 = T ̄ s$⁠. During this period, the left leg is the stance leg and the right foot does not make contact with the ground. The left leg and the mass form an inverted pendulum.

According to Observation 1 and Newton’s third law

where primes denote derivatives with respect to $t ̄$⁠.

Let $r ˆ$ be the dimensionless unit vector pointing from the left foot to the mass, $r ˆ = ( r m − r ℓ ) /L$⁠. We define F = ‖F‖/(mg), where g is the acceleration of gravity and we use the notation ‖a‖ for the norm of any vector a, i.e. $‖a‖= a 1 2 + a 2 2 + a 3 2$⁠. We introduce the dimensionless time $t= g / L t ̄$⁠. Equation (1) becomes

where dots denote derivatives with respect to t.

Let θ be the polar angle and ϕ be the azimuthal angle,

(see Figure 1). In terms of θ = θ(t) and ϕ = ϕ(t), Equation (2) becomes

We will solve these equations during the first step, i.e. in the time interval (0, t₁), where $t 1 = T ̄ s g / L$⁠.

Equation (6) determines F = F(t). The dynamics of the mass m is given by Equations (4) and (5). Multiply Equation (5) by sinθ and integrate once to obtain

for some constant K. Use Equation (7) to eliminate $ϕ ̇$ from Equation (4) to get $θ ̈ − K 2 cos θ sin 3 θ =sinθ$⁠. This equation is integrated once after is multiplied by $θ ̇$ to get

for some E. E is the dimensionless mechanical energy.

The initial azimuthal angle, ϕ₀ = ϕ(0), and polar angle, θ₀ = θ(0), are related to the parameters displayed in Figure 2. More precisely, let

then

Given E and K, the system of Equations (7) and (8) subjected to the initial conditions (10) has a unique solution. However, this solution may not correspond to periodic walking in the sense of Assumption 3.9. In the next section, we list the two conditions for periodic walking, which will result in a relationship between E and K.

Note that t₁ is the dimensionless time of one step. θ(t₁) = θ₀ simply means that the height of the mass m at the start and the end of a step is the same. The need for ϕ(t₁) = π/2 − ϕ_w is illustrated in Figure 2 and is a consequence of the symmetry of the steps.

Given any function g = g(x) of one variable, we use the standard notations g(a⁺) and g(a⁻) for the limits of g(x) as x tends to a from the right (with values of x > a) and from the left (with values of x < a). Given two vectors a and b we denote by a ⋅ b = a₁b₁ + a₂b₂ + a₃b₃ their dot product.

Let $n ℓ ˆ$ be the unit vector from the left foot to the mass at time t = t₁ and let $n ˆ r$ be the unit vector from the right foot to the mass at time t = t₁. Given the Assumption 3.9, $r ˆ ̇ ( t 1 + ) = r ˆ ̇ ( 0 + ) −2 ( e h ⋅ r ˆ ̇ ( 0 + ) ) e h$ if $r ˆ ( t )$ correspond to periodic walking. This fact plus Assumption 3.6 leads to the second condition for periodic walking.

Lengthy calculations show that: if Condition 5.1 is satisfied, so is Condition 5.2. Thus, we only need to find the pairs of parameters (E, K) for which Condition 5.1 is satisfied. These pairs correspond to periodic walking.

The azimuthal angle θ initially decreases and it attains its minimum at the time t = t^⋆ such that $θ ̇ ( t ⋆ ) =0$⁠. The value of this minimum azimuthal angle θ^⋆ = θ(t^⋆), can be obtained in terms of the constants of motion E and K from Equation (8) by setting $θ ̇ =0$

Note ϕ^⋆ = ϕ(t^⋆) can be computed as follows

Due to symmetry, the Condition 5.1 is equivalent to: The minimum azimuthal angle θ^⋆ is attained at the same time that the polar angle ϕ is zero, i.e. ϕ^⋆ = 0.

Since $d ϕ d θ = ϕ ̇ / θ ̇$⁠, setting ϕ^⋆ = 0, using the fact that ϕ₀ = − π/2 + ϕ_w, Equations (7) and (8), and simple manipulations, we get that Equation (12) becomes

This is Condition 5.1. Given θ₀, the pairs (E, K) that satisfy Equations (11) and (13) correspond to periodic walking. Next, we state a theorem that guarantees the existence of periodic walking solutions and tell us the parameter regime where we should look for those solutions.

Next, we study aspects of periodic walking. The parameters that determine the motion are ϕ_w, θ₀ and E. The angles ϕ_w and θ₀ determine the geometry of the biped, i.e. how wide apart the feet are and how long the steps are in relationship to how long the legs are.

Let T be the dimensionless time of one step. By symmetry, θ(T/2) = θ^⋆, the minimum azimuthal angle (see Equation (11)). Thus, $T/2= ∫ θ 0 θ ⋆ θ ̇ − 1 dθ$⁠. Using Equation (8) and simple manipulations we get

Let V be the dimensionless average velocity. From Figure 2, it can be shown that the dimensionless distance the biped covers in one step is 2sinθ₀cosϕ_w. Thus,

For the parameter values θ₀ = ϕ_w = π/6, Figure 3 shows V as a function of E. As expected, V is an increasing function of E. The more mechanical energy the mass has, the faster it moves in average. Our numerical calculations suggest that $lim E → 1 + V=0$⁠.

Let A_h be the dimensionless amplitude of the lateral oscillations. Note that $A h = e h ⋅ ( r ˆ ( 0 + ) − r ˆ ( T / 2 ) )$⁠. Thus,

Figure 3 shows A_h as a function of E when θ₀ = ϕ_w = π/6. Since V increases with E, note that A_h is a decreasing function of V. The faster the biped walks, the smaller the amplitude of the lateral oscillations.

In Figure 4 we show an example of the path traced by the mass in two full steps. The top figure shows a three dimensional plot of the path of the mass. The bottom figure shows a view from the top (the orthogonal projection of the path onto the ground). The lateral oscillations are seen in this figure. The thick solid line is the path of the mass. The thin lines are snap shoots of the stance leg. The swing leg is not shown. The parameters in that example are E = 1.1 and $ϕ w = θ 0 = π 6$⁠.

Next, we study the dynamics of slow walkers. This corresponds to values of the energy of the form

The asymptotic value of θ^⋆ (see Equation (11)) is

The asymptotic value of the dimensionless time of one step, T (see Equation (14)), and the dimensionless average velocity, V (see Equation (15)), are

Note that T → ∞ and V → 0 as ε → 0. Note also that the asymptotic formula for V is consistent with the plot in Figure 3, not only on the fact that V → 0 as ε → 0 (or E → 1), but also on how it approaches 0.

We obtain the asymptotic value of the amplitude of the lateral oscillations from Equations (18) and (16):

Figure 5 shows the orthogonal projection of the trajectory onto the ground when E = 1.01 (ε² = 0.01) and $ϕ w = θ 0 = π 6$⁠. Our numerical calculations show that the asymptotic formulas of Equations (19) and (20) are accurate. Figures 5 and 4 illustrate that the amplitude of the lateral oscillations increases as the velocity decreases.

The novelty of this article is that we do not restrict the motion of the mass to the sagittal plane and thus, we were able to study the lateral oscillations of the 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. Note that the standard inverted pendulum model constrained to the sagittal plane is obtained from our model by setting the setting the width of the step to zero, i.e. w = 0.

We mention that our goal was to introduce and study the simplest model possible able to capture the lateral oscillations. Thus, our focus on the simplest inverted pendulum model but not constrained to the sagittal plane. We acknowledge that this model has several simplifications, like the biped has no knees. While the vast literature over several decades using simple models and their success are plenty of evidence of the value of studying simple models as the one in this article, comparison of the results obtained here with more complex models, or the development of new more complex models motivated by this article, is certainly an interesting undertaking that we hope will be carried out in the future.

We performed numerical simulations and explored the slow walking parameter regime with the use of asymptotics. We show that the inverted pendulum model remains simple enough to be easily studied, even when the mass is not restricted to move in the sagittal plane. We believe the approach introduced in this paper will prove useful and be widely adopted to study different aspects of the dynamics of biped walkers.