Mathematical Model of Human Walking: A Theoretical Study Based on Anthropometric Data

Abstract

Background. Walking is a fundamental human activity, vital for daily living, social connection, employment, etc. Methods. In the current study, we present a mathematical model of it, based on the planar double pendulum system influenced by gravity. For parameters of the pendulum, i.e., the characteristic of the limbs (thigh + shank), we use realistic mass–inertial parameters. The model incorporates anthropometric and inertial data specific to the average Bulgarian, Russian, German, and American male, including segment masses, centres of mass, as well as densities of the segments taken from experimental studies. Results. We derive the corresponding nonlinear differential equations governing the model. We solve them analytically, when possible, and, in the general case, numerically. For moderate initial angles (from the frontal plane) and angular velocities of the thigh and shank, the pendulum exhibits motion closely resembling natural human gait. The results for all nationalities considered are very close to each other. For comparatively slow walking speeds, the model provides realistic results. Conclusions. Our approach highlights how a relatively simple biomechanical model can capture essential features of human locomotion and provides a foundation for further refinement and comparison with more complex gait modelling techniques. Such modifications are outlined.

1. Introduction

This article demonstrates that the plane double gravitational pendulum can be helpful in matters of biomechanics related to the problems of human walking.

Our model is based on actual data for the average Bulgarian [1], Russian [2], German [3], and American [4] male. The Bulgarian “average male” is obtained by calculating the mean value over the anthropometric data measured on 2435 individuals. We took into account the experimentally determined densities of the limb segments. We stress that specific limitations exist because the objects of study are living humans with lymphatics, blood, etc., floating within the body. Note that measuring even a local mass distribution within a segment is a problem that, up to now, is not possible to solve without some damage to the studied person. The limb density in living man has been, e.g., measured using radiation on soldiers in the Soviet Army by Zatsiorsky [2,5]. For details in obtaining the mass–inertial characteristics of the limbs, like inertial moments, centres of mass, the masses of the segments, etc., see Ref. [6]. This data is needed to define the biomechanical model. We decomposed the body into segments according to the procedure recommended in [2]. As we will see later, proceeding in the way described, we derive values for the average stride lengths, cadence, and walking speed which are reasonably close to those reported in the literature. We admit and stress that we have achieved that despite the obvious simplicity of the model. Its improvement can be looked for in many directions. This will be discussed shortly in the Conclusion section of this article. We believe, however, that it is noteworthy to know how far one can go with such a type of a simple model in obtaining results to be compared with those reported in the literature. This, we suppose, should be conducted before trying more complicated and elaborate models, and it is what we do in the current article.

Mathematically, the system’s behaviour is described by a set of coupled, nonlinear, second-order ordinary differential equations that defy, in general, analytical solution [7]. Numerical simulation provides a powerful tool for exploring its dynamics, but requires careful attention to numerical errors and energy conservation. Visualisation tools, such as animations and phase space plots, are crucial for analysing and interpreting the simulation results [8]. In addition to numerically solving the equations, we have also used these methods. The numerical exploration of the double pendulum’s dynamics reveals its most compelling characteristic: its sensitivity to initial conditions. This sensitivity, a hallmark of chaotic systems, means that small differences in the initial angles or velocities of the pendulums can lead to different trajectories over time [9,10,11]. The chaotic nature of the double pendulum can be most appealingly visualised through phase space plots. These plots, which map the system’s state over time, reveal intricate and often fractal-like structures. Unlike simpler systems with predictable periodic motion that result in closed, repeating loops in phase space, the double pendulum exhibits trajectories that wander seemingly randomly through the phase space, never repeating themselves exactly.

The human locomotive gait is inherently complex and has nonlinear dynamics which received particular attention in recent years. In particular, to design and conduct motion planning for bipedal robotics and gait-assistive exoskeletons, we must understand human gait dynamics by explaining the oscillatory moment of the centre of mass and the corresponding ground reaction forces [12]. It is well known that walking has two main distinct phases: the single support phase (SSP) and the double support phase (DSP). The SSP assumes a much longer portion of the gait cycle where one leg acts as the swing (moving) leg, and the other one is stationary (stance leg) [8]. Frequently, the swing leg is modelled as a double pendulum [8,13,14]. In addition to the double pendulum model, many more advanced approaches have been suggested, such as (i) multibody dynamic gait modelling [15,16]; (ii) nonlinear biomechanical models [17]; and (iii) via applications of fractional-order dynamics in human motion [18]; for the definition of variable-order fractional operators, one can consult, e.g., Ref. [19]. We believe, however, that one should use an evolutionary approach, making a model more complicated step by step.

As a general reference to studies of human motion, we refer the reader to Ref. [20].

Using our model, we show that, within the typical range of parameters of human motion,

• For moderate initial angles (from the frontal plane) and angular velocities of the thigh and shank, the pendulum exhibits motion closely resembling natural human gait.

  −

  When these parameters are small, it shows regular motion, i.e., it does not attest essential chaotic behaviour; the walking speed is then low.

  −

  When they are larger: the walking speed increases, but elements of chaos appear and increase with their increment, thus requiring in such regimes of motion much finer tailoring with the function of the muscles, nervous system, joints, etc.

  Our findings support that human movement requires intricate control [21,22,23,24]. It is a medically well-established fact that Parkinson’s patients, in whom the brain control of movement is disturbed, demonstrate walking patterns as described in small stride lengths and walking speed—see, e.g., Refs. [25,26,27,28].
• We obtain estimations for the average walking speed of healthy humans, which are in agreement with the data reported in the literature [29,30,31].

As for Parkinson’s disease, we are very aware that this is a very complicated matter. It is fundamentally asymmetric in the brain and in the body. It is customarily believed that the disease usually starts with one hemisphere being more affected. Thus, it seems to us natural, see also [32], that when the brain does not optimally function, the gait will be more rigid, with smaller variability and small steps and speed—something like the body adapting to move on autopilot without the brain controlling the moments when muscles, joints, and nerves function together as in a healthy person. From our point of view, this is why the body chooses to move, as much as possible, avoiding the chaotic elements coming from mechanical dynamics. This is what our model leads to for small initial angles. The nonlinear dynamics features can be used to discriminate between Parkinson’s patients and healthy individuals, as discussed in Refs. [33,34]. It is clear that to better understand the gait disturbance in Parkinson’s disease (say, the factors that contribute to falls), it is useful to include different approaches and methods, one of them being the use of mathematical models and analysis. Hence, the current paper is one small step in this direction. A recent comprehensive review on the detection and assessment of Parkinson’s disease based on gait analysis is given in Ref. [35].

The above remarks are, of course, related to the much broader issues of optimal movement variability and the coordination rigidity connected to it. Contemporary nonlinear motor-control theory proposes that healthy movement emerges from an optimal amount of movement variability, characterised by a complex, chaotic structure that reflects adaptability and robustness [32]. Too little variability produces overly stable, inflexible, and stereotyped movement patterns, whereas too much variability yields noisy, unstable, and poorly controlled behaviour. The concept of optimal movement variability was formalised by Stergiou, Harbourne, and Cavanaugh, who argued that mature motor skills occupy a middle region between rigidity and randomness, where variability is structured and exhibits long-range correlations [32]. In contrast, coordination rigidity refers to abnormally low variability. Such rigidity is often observed in, say, neurological disorders or musculoskeletal injury. The concepts of variability and rigidity provide a theoretical basis for evaluating movement quality in gait analysis and rehabilitation via nonlinear metrics such as entropy and Lyapunov exponents, and detrended fluctuation analysis can quantify the temporal structure of variability. Additional information on the topic can be found in [36,37,38].

It is worth mentioning that the pendulum systems are well-known classical models of nonlinear dynamics [39]. We would like to stress that, generally, the study of the double pendulum is not merely an academic exercise; it holds significance in various scientific and engineering fields [23,40,41,42,43,44,45,46]. It is helpful for apprehending questions related, as stated, to the physics of human motion, but also climate modelling [47,48], fundamental physics, etc.

1.1. On Some Classical Applications of the Double Pendulum

In addition to what was already stated above, we shall mention studying entertaining and sport activities, like in a golf swing [49], in games with a bat [50], and to describe the action of upper or lower limb segments in activities such as throwing [51,52,53], running [54], and kicking [53,55].

The planar double pendulum has played a central role in the development of simplified biomechanical models of human movement for more than half a century. Early conceptual treatments of multi-segment limb dynamics using double-pendulum mechanics appeared in the sports-physics literature, where Daish [56] demonstrated that many ballistic human actions—such as throwing, striking, and kicking—can be approximated by a two-segment linkage driven by sequential torque generation at the proximal and distal joints. This framework captured the essential physics of energy transfer from the proximal to the distal segment, a mechanism fundamental to high-velocity limb movements in sports and daily activities.

During the 1970s and 1980s, Jorgensen and colleagues formalized the double-pendulum model as the dominant theoretical tool for analysing the golf swing. Their work demonstrated that coordinated timing of segmental release—specifically, delaying the distal segment’s acceleration until after the proximal segment reaches peak angular velocity—maximizes endpoint speed at impact [57]. These studies established the double pendulum as a powerful explanatory model for multi-segment coordination in skilled human movement. A comprehensive review on the use of the model in describing golf activities is presented in Ref. [49].

Some synthesis of this literature was provided by Cross [50], who combined theoretical analysis with experimental validation to examine the double-pendulum swing in baseball and racquet sports. Cross reviewed the model’s historical development, clarified its assumptions, and demonstrated how it explains key performance variables, such as bat or racquet speed, impact efficiency, and timing strategies. His work also highlighted the model’s limitations, including its restriction to planar motion and its simplified representation of neuromuscular control.

Although originally developed for sports biomechanics, the planar double pendulum has since been adopted widely in robotics, nonlinear dynamics, and human-movement science. Its appeal lies in its balance between simplicity and expressiveness: the model is analytically tractable yet capable of exhibiting rich dynamical behaviour, including chaotic motion, resonance phenomena, and complex energy-transfer patterns. Modern studies continue to use the double pendulum as a benchmark system for understanding coordination, control, and stability in multi-segment human motion. Together, these works form a coherent historical trajectory: from early conceptual models of limb mechanics [56], through sport-specific applications (Jorgensen), to modern synthesis and experimental validation [50]. The planar double pendulum remains a foundational tool in biomechanics because it captures the essential physics of coordinated multi-segment motion while remaining mathematically transparent and computationally efficient.

1.2. On Some Modern Applications of the Double Pendulum

Contemporary research continues to employ double-pendulum and double-inverted-pendulum models as compact yet expressive representations of human movement dynamics. In gait analysis, double-pendulum formulations have been used to study inter-limb coordination, unilateral impairments, and compensatory strategies. For example, Rasouli and Reed [58] demonstrated that a planar double-pendulum model can reproduce key asymmetries observed in pathological gait and can be manipulated to explore biomechanical equivalence between impaired and unimpaired limbs. Their work highlights the utility of low-dimensional models for probing how altered segmental dynamics influence whole-body motion.

Double-inverted-pendulum models have also become central in the study of human balance, fall recovery, and reactive postural control. Cerda-Lugo et al. [59] developed a continuous-time double-inverted-pendulum framework to investigate neuromechanical strategies during fall recovery, showing that multi-segment coordination is essential for stabilizing the centre of mass after perturbations. Similarly, Hettich, Fennell, and Mergner [60] used a double-inverted-pendulum representation to model reactive stance control, demonstrating how trunk–leg coupling contributes to maintaining upright posture under unexpected disturbances.

Beyond gait and balance, double-pendulum models remain influential in studies of rhythmic and quasi-periodic movements. Brenière and Ribreau [61] employed a double-inverted-pendulum formulation to analyse stepping-in-place dynamics, revealing how postural control adapts to changes in stepping frequency. These models continue to serve as benchmark systems for exploring nonlinear dynamics, resonance phenomena, and energy-transfer mechanisms in human movement.

Overall, modern applications of the double pendulum extend well beyond its original use in sports biomechanics. Its combination of analytical tractability, rich dynamical behaviour, and clear physical interpretation makes it a versatile tool for investigating coordination, stability, and control across a wide range of human-movement contexts.

In the current article, we present a mathematical model of human walking. It is a theoretical study, but it is based on anthropometric data taken from the relevant literature. We do not use any fitting parameters.

The structure of this article is as follows. In Section 2, we present the definition of our model. The derivation of the equations of motion, obtained from it, is explained in Section 3. Up to that point, there are no specific data used. The presentation just makes clear which data are needed to make specific predictions from our general framework. Such data for Germans, US Americans, Russians, and Bulgarian males with European descent are given in Section 4. They are all sourced from the existing literature. There are no fitting parameters. Section 5 provides the results—the analytical ones in Section 5.1 and the numerical ones in Section 5.2. The comparison of our findings with the experimental ones reported in the literature is given in Section 6. In Section 7, we describe some important issues related to the gait asymmetry, where we believe our model can be helpful. Currently, we do not have the data needed to accomplish them, but they can definitely be realized by people in the biomechanical community or by us in the future. This article ends with some concluding remarks, given in the corresponding Section 8.

2. The Model

In this section, we describe a model of the human limbs consisting of two segments. Note that the parts of the human body must be properly defined and somehow segregated from one another in order to be able to take into account their geometric and inertial properties. This is known as the “segmentation” of the human body (this is realized not only in the classical studies of on biomechanics like Dempster [62], Clauser [63], Chandler [64], Zatsiorsky [2,5], de Leva [65], Challis [66], and Dumas et. al. [67], but also when working on practical applications—see, e.g, Ref. [68]). The goal of segmentation is to approximately represent the body as a kinematic multi-link (branched) chain with links with known mass–inertia characteristics. In a first approximation, individual body parts are assumed to move like rigid bodies that can rotate about joint axes located at the ends of these segments. Let us emphasise that in body decomposition, we are talking about “segments”, which are real parts of the body with their specific geometric shape, and in kinematic chains for “links”, idealised solid bodies with a regular geometric shape, which in kinematic schemes we mark as straight lines, considering their mass as concentrated in the gravitational centre. When performing segmentation, we must realise that the human body is continuous and any separation into segments is a matter of judgement and, to some extent, definition. The parts of the body are actually rather complex, consisting mainly of soft and rigid tissues, as well as liquids. In addition, muscles and other soft tissues pass from one part to another, so boundary lines can only be drawn conditionally. Regrettably, individual authors use different segmentation protocols in their studies. Consequently, it is important to compare data from various studies carefully [2].

In its simplest version, the model can be reduced to a plane double gravitational pendulum, i.e., we consider the corresponding limb of the human body to embrace two segments—see Figure 1. Obviously, the actual movements of the limbs of a human are not entirely two-dimensional. This is one assumption, which greatly simplifies the analytical treatment. We will see that, even then, the model leads to a system of two coupled nonlinear differential equations that can be treated analytically only in some limiting cases. In general, one has to rely on numerical procedures for solving them.

To determine the Lagrangian of the system, we obtain first the total kinetic energy K:

$$K=K_1+K_2,$$

(1)

where $K_i$ and $i=1,2$ are the kinetic energies of the segments, respectively.

For $K_1$, one has

$$K_1={\displaystyle \frac{1}{2}}{\int }_{\left\{S_1\right\}}{\rho }_1\left(s_1\right)A\left(s_1\right)v_1^2(s_1,t)d{s}_1,$$

(2)

where $\left\{S_1\right\}$ is the set of allowed values of $s_1\in [0,L_1]$, $A\left(s_1\right)$ is the area of the cross-section of segment 1 at position $s_1$ measured from the beginning of this segment, and $v_1^2(s_1,t)$ is its velocity. We assume that the densities in a given segment are uniform, i.e., ${\rho }_i\left(s_i\right)={\rho }_i$, $i=1,2$. This accurately reflects the situation with the experiment: the mass density of a specific segment is dependent on that segment, yet the variations within the segment remain unavailable. This is why they are assumed as homogeneous. Then, taking that into account,

$$v_1^2(s_1,t)={\dot{x}}_1^2(s_1,t)+{\dot{y}}_1^2(s_1,t),$$

(3)

and that with

$$x_1(s_1,t)=s_{1}sin\left({\theta }_1\left(t\right)\right),\mathrm{and}{y}_1(s_1,t)=-s_{1}cos\left({\theta }_1\left(t\right)\right),$$

(4)

we obtain

$${\dot{x}}_1(s_1,t)=s_{1}cos\left[{\theta }_1\left(t\right)\right]{\dot{\theta }}_1\left(t\right),\mathrm{and}{\dot{y}}_1(s_1,t)=s_{1}sin\left[{\theta }_1\left(t\right)\right]{\dot{\theta }}_1\left(t\right),$$

(5)

and, therefore

$$v_1^2(s_1,t)={\dot{x}}_1^2(s_1,t)+{\dot{y}}_1^2(s_1,t)=s_1^2{\dot{\theta }}_1^2\left(t\right),$$

(6)

and one derives

$$\begin{array}{c}\hfill K_1={\displaystyle \frac{1}{2}}{\rho }_1{\int }_{\left\{S_1\right\}}A\left(s_1\right)v_1^2(s_1,t)d{s}_1={\displaystyle \frac{1}{2}}{\rho }_1{\dot{\theta }}_1^2\left(t\right){\int }_{\left\{S_1\right\}}A\left(s_1\right)s_1^{2}d{s}_1={\displaystyle \frac{1}{2}}{\dot{\theta }}_1^2\left(t\right)I_1.\end{array}$$

(7)

Here

$$I_1={\rho }_1{\int }_0^{L_1}A\left(s_1\right)s_1^{2}d{s}_1$$

(8)

is the moment of inertia of segment “1” about an axis perpendicular to it and centred at its top.

For the kinetic energy $K_2$ of the second segment in the limb, one derives

$$\begin{array}{ccc}\hfill x_2(s_2,t)& =& L_{1}sin{\theta }_1\left(t\right)+s_{2}sin{\theta }_2\left(t\right),\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill y_2(s_2,t)& =& -L_{1}cos{\theta }_1\left(t\right)-s_{2}cos{\theta }_2\left(t\right),\hfill \end{array}$$

(10)

where $s_2\in [0,L_2]$ is the coordinate along this segment. With

$$\begin{array}{ccc}\hfill {\dot{x}}_2(s_2,t)& =& L_{1}cos{\theta }_1\left(t\right){\dot{\theta }}_1\left(t\right)+s_{2}cos{\theta }_2\left(t\right)\dot{{\theta }_2}\left(t\right),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\dot{y}}_2(s_2,t)& =& L_{1}sin{\theta }_1\left(t\right){\dot{\theta }}_1\left(t\right)+s_{2}sin{\theta }_2\left(t\right)\dot{{\theta }_2}\left(t\right),\hfill \end{array}$$

(12)

we arrive at

$$v_2^2(s_2,t)=L_1^2{\dot{\theta }}_1^2\left(t\right)+s_2^2{\dot{\theta }}_2^2\left(t\right)+2L_1{s}_{2}cos[{\theta }_1\left(t\right)-{\theta }_2\left(t\right)]{\dot{\theta }}_1\left(t\right){\dot{\theta }}_2\left(t\right).$$

(13)

Thus, for $K_2$, we derive the following:

$$\begin{array}{cc}\hfill K_2& ={\displaystyle \frac{1}{2}}{\int }_{\left\{S_2\right\}}{\rho }_2\left(s_2\right)A\left(s_2\right)v_2^2(s_2,t)d{s}_2\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\rho }_2{\int }_{\left\{S_2\right\}}\left[L_1^2{\dot{\theta }}_1^2\left(t\right)+s_2^2{\dot{\theta }}_2^2\left(t\right)+2L_1{s}_{2}cos[{\theta }_1\left(t\right)-{\theta }_2\left(t\right)]{\dot{\theta }}_1\left(t\right){\dot{\theta }}_2\left(t\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{L}_1^2{\dot{\theta }}_1^2\left(t\right)\left[{\int }_{\left\{S_2\right\}}{\rho }_{2}A\left(s_2\right)d{s}_2\right]+{\displaystyle \frac{1}{2}}{\dot{\theta }}_2^2\left(t\right)\left[{\int }_{\left\{S_2\right\}}{\rho }_{2}A\left(s_2\right)s_2^{2}d{s}_2\right]\hfill \\ \hfill & +L_{1}cos[{\theta }_1\left(t\right)-{\theta }_2\left(t\right)]{\dot{\theta }}_1\left(t\right){\dot{\theta }}_2\left(t\right)\left[{\int }_{\left\{S_2\right\}}{\rho }_{2}A\left(s_2\right)s_{2}d{s}_2\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{m}_2{L}_1^2{\dot{\theta }}_1^2\left(t\right)+{\displaystyle \frac{1}{2}}{I}_2{\dot{\theta }}_2^2\left(t\right)+L_1{m}_2{C}_{2}cos[{\theta }_1\left(t\right)-{\theta }_2\left(t\right)]{\dot{\theta }}_1\left(t\right){\dot{\theta }}_2\left(t\right),\hfill \end{array}$$

(14)

where $\left\{S_2\right\}\equiv [0,L_2]$,

$$\begin{array}{cc}\hfill m_2& ={\rho }_2{\int }_{\left\{S_2\right\}}A\left(s_2\right)d{s}_2,\hfill \\ \hfill I_2& ={\rho }_2{\int }_{\left\{S_2\right\}}{s}_2^{2}A\left(s_2\right)d{s}_2,\hfill \\ \hfill C_2& ={\displaystyle \frac{{\int }_{\left\{S_2\right\}}{s}_{2}A\left(s_2\right)d{s}_2}{{\int }_{\left\{S_2\right\}}A\left(s_2\right)d{s}_2}}.\hfill \end{array}$$

(15)

Obviously, $C_2$ is the coordinate of the centre of mass of the second segment, measured from its beginning.

Summarising the above, we conclude that the total kinetic energy $K=K_1+K_2$ is equal to

$$K={\displaystyle \frac{1}{2}}\left[I_1+m_2{L}_1^2\right]{\dot{\theta }}_1^2\left(t\right)+{\displaystyle \frac{1}{2}}{I}_2{\dot{\theta }}_2^2\left(t\right)+L_1{m}_2{C}_{2}cos[{\theta }_1\left(t\right)-{\theta }_2\left(t\right)]{\dot{\theta }}_1\left(t\right){\dot{\theta }}_2\left(t\right).$$

(16)

Now, we turn to the potential energy of the system V. One has

$$V=V_1+V_2,$$

(17)

where $V_1$ and $V_2$ are the potential energies of the segments. From the geometry depicted in Figure 1, one obtains

$$\begin{array}{cc}\hfill V_1& =g{\int }_0^{L_1}{\rho }_1\left(s_1\right)A\left(s_1\right)y\left(s_1\right)d{s}_1\hfill \\ \hfill & =-g{\rho }_1{\int }_0^{L_1}A\left(s_1\right)s_{1}cos\left({\theta }_1\right)d{s}_1=-g{\rho }_{1}cos\left({\theta }_1\right){\int }_0^{L_1}A\left(s_1\right)s_{1}d{s}_1\hfill \\ \hfill & =-gcos\left({\theta }_1\right)m_1{C}_1,\hfill \end{array}$$

(18)

where

$$C_1={\displaystyle \frac{{\int }_0^{L_1}{\rho }_1{s}_{1}A\left(s_1\right)d{s}_1}{{\int }_0^{L_1}{\rho }_{1}A\left(s_1\right)d{s}_1}}={\displaystyle \frac{{\int }_0^{L_1}{s}_{1}A\left(s_1\right)d{s}_1}{{\int }_0^{L_1}A\left(s_1\right)d{s}_1}}.$$

(19)

In a similar way, for $V_2$ one derives

$$\begin{array}{cc}\hfill V_2& =g{\int }_0^{L_2}{\rho }_2\left(s_2\right)A\left(s_2\right)y\left(s_2\right)d{s}_2\hfill \\ \hfill & =g{\int }_0^{L_2}{\rho }_2\left(s_2\right)A\left(s_2\right)\left[-L_{1}cos\left({\theta }_1\right)-s_{2}cos\left({\theta }_2\right)\right]d{s}_2\hfill \\ \hfill & =-g{L}_{1}cos\left({\theta }_1\right){\int }_0^{L_2}{\rho }_2\left(s_2\right)A\left(s_2\right)d{s}_2-gcos\left({\theta }_2\right){\int }_0^{L_2}{\rho }_2\left(s_2\right)s_{2}A\left(s_2\right)d{s}_2\hfill \\ \hfill & =-g{m}_2\left[L_{1}cos\left({\theta }_1\right)+cos\left({\theta }_2\right)C_2\right].\hfill \end{array}$$

(20)

From Equations (18) and (20), we arrive at

$$V=V_1+V_2=-gcos{\theta }_1(C_1{m}_1+L_1{m}_2)-gcos{\theta }_2{C}_2{m}_2.$$

(21)

Once we know the kinetic K and the potential V energies, we derive the Lagrangian $\mathcal{L}$ of the system

$$\begin{array}{cc}\hfill \mathcal{L}=K-V=& {\displaystyle \frac{1}{2}}\left[I_1+m_2{L}_1^2\right]{\dot{\theta }}_1^2+{\displaystyle \frac{1}{2}}{I}_2{\dot{\theta }}_2^2+L_1{m}_2{C}_{2}cos[{\theta }_1-{\theta }_2]{\dot{\theta }}_1{\dot{\theta }}_2\hfill \\ \hfill & +g(C_1{m}_1+L_1{m}_2)cos{\theta }_1+g{C}_2{m}_{2}cos{\theta }_2.\hfill \end{array}$$

(22)

One can also determine the Hamiltonian $\mathcal{H}$ of the system

$$\begin{array}{cc}\hfill \mathcal{H}=K+V=& {\displaystyle \frac{1}{2}}\left[I_1+m_2{L}_1^2\right]{\dot{\theta }}_1^2+{\displaystyle \frac{1}{2}}{I}_2{\dot{\theta }}_2^2+L_1{m}_2{C}_{2}cos[{\theta }_1-{\theta }_2]{\dot{\theta }}_1{\dot{\theta }}_2\hfill \\ \hfill & -g(C_1{m}_1+L_1{m}_2)cos{\theta }_1-g{C}_2{m}_{2}cos{\theta }_2.\hfill \end{array}$$

(23)

Since we did not introduce explicit dependence on the time t, the Hamiltonian of the system is conserved and equal to the stationary energy of the system. According to the standard Hamiltonian formulation, the state of a system is characterised not only by its positions (generalised coordinates) ${\theta }_1$ and ${\theta }_2$, but also by its momenta $p_1$ and $p_2$. Hence, the Hamiltonian of the system depicted in Figure 1 is as follows:

$$\begin{array}{ccc}\hfill H& =& K+V={\displaystyle \frac{I_2}{2{\Delta }_0}}{p}_1^2+{\displaystyle \frac{I_1+m_2{L}_1^2}{2{\Delta }_0}}{p}_2^2-{\displaystyle \frac{m_2{L}_1{C}_2\cos \left({\theta }_1-{\theta }_2\right)}{{\Delta }_0}}{p}_1{p}_2\hfill \\ & & -g\left[\left(m_1{C}_1+m_2{L}_1\right)\cos {\theta }_1+m_2{C}_2\cos {\theta }_2\right],\hfill \end{array}$$

(24)

where

$$p_1={\displaystyle \frac{\partial K}{\partial {\dot{\theta }}_1}},p_2={\displaystyle \frac{\partial K}{\partial {\dot{\theta }}_2}},\mathrm{and}{\Delta }_0=I_1{I}_2+m_2{I}_2{L}_1^2-m_2^2{L}_1^2{C}_2^2{\cos }^2\left({\theta }_1-{\theta }_2\right).$$

(25)

3. The Equations of Motion

From Equation (22), one obtains the corresponding Euler–Lagrange equations (this procedure is well known; for general details on the subject, see, e.g., Refs. [2] (pp. 394, 434–435), [20] (p. 273), [41] (p. 109), [69] (p. 58), etc.)

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{{\theta }_i}}}-{\displaystyle \frac{\partial \mathcal{L}}{\partial {\theta }_i}}=0,i=1,2$$

(26)

governing the behaviour of the two segments of the considered limb. In this way, we obtain a set of two coupled, second-order, nonlinear differential equations:

$$\begin{array}{ccc}\hfill & & gsin\left[{\theta }_1\right](C_1{m}_1+L_1{m}_2)+\left(I_1+L_1^2{m}_2\right){\ddot{\theta }}_1+C_2{L}_1{m}_2\left\{{\dot{\theta }}_1{\dot{\theta }}_{2}sin[{\theta }_1-{\theta }_2]\right.\hfill \\ & & \left.+{\ddot{\theta }}_{2}cos[{\theta }_1-{\theta }_2]-{\dot{\theta }}_2\left[{\dot{\theta }}_1-{\dot{\theta }}_2\right]sin[{\theta }_1-{\theta }_2]\right\}=0\hfill \end{array}$$

(27)

and

$$\begin{array}{ccc}\hfill & & g{C}_2{m}_{2}sin\left[{\theta }_2\right]+I_2{\ddot{\theta }}_2\hfill \\ & & +C_2{L}_1{m}_2\left\{\ddot{{\theta }_1}cos[{\theta }_1-{\theta }_2]-{\dot{\theta }}_1^{2}sin[{\theta }_1-{\theta }_2]\right\}=0.\hfill \end{array}$$

(28)

Equation (27) can be rewritten in the simpler form

$$\begin{array}{ccc}\hfill & & gsin\left[{\theta }_1\right](C_1{m}_1+L_1{m}_2)+\left(I_1+L_1^2{m}_2\right){\ddot{\theta }}_1\hfill \\ & & +C_2{L}_1{m}_2\left\{{\ddot{\theta }}_{2}cos[{\theta }_1-{\theta }_2]+{\dot{\theta }}_2^{2}sin[{\theta }_1-{\theta }_2]\right\}=0.\hfill \end{array}$$

(29)

Equations (28) and (29) form the system of two coupled nonlinear differential equations. When modelling the process of walking, one shall keep in mind that ${\theta }_2\le {\theta }_1$ when ${\theta }_1>0$. The constraint comes from the usual properties of the joint at the knee [70].

In the form of a dynamical model, this system is

$$\begin{array}{ccc}\hfill & & {\ddot{\theta }}_1=a_1-a_2{\dot{\theta }}_1^2-a_3{\dot{\theta }}_2^2,\hfill \\ & & {\ddot{\theta }}_2=-a_4+a_5{\dot{\theta }}_1^2+a_6{\dot{\theta }}_2^2,\hfill \end{array}$$

(30)

where

$$\begin{array}{ccc}\hfill & & a_1={\displaystyle \frac{g}{{\Delta }_0}}\left[m_2^2{C}_2^2{L}_{1}cos\left({\theta }_1-{\theta }_2\right)sin{\theta }_2-I_2\left(C_1{m}_1+L_1{m}_2\right)sin\left({\theta }_1\right)\right],\hfill \\ & & a_2={\displaystyle \frac{m_2^2{C}_2^2{L}_1^{2}cos\left({\theta }_1-{\theta }_2\right)sin\left({\theta }_1-{\theta }_2\right)}{{\Delta }_0}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{m_2^2{C}_2^2{L}_1^{2}sin\left[2\left({\theta }_1-{\theta }_2\right)\right]}{{\Delta }_0}},\hfill \\ \hfill & & a_3={\displaystyle \frac{m_2{C}_2{I}_2{L}_{1}sin\left({\theta }_1-{\theta }_2\right)}{{\Delta }_0}},\hfill \\ & & a_4=-{\displaystyle \frac{g{C}_2{m}_2}{{\Delta }_0}}\left\{L_1\left[C_1{m}_{1}sin\left({\theta }_1\right)cos({\theta }_1-{\theta }_2)+L_1{m}_{2}cos\left({\theta }_1\right)sin({\theta }_1-{\theta }_2)\right]-I_{1}sin\left({\theta }_2\right)\right\},\hfill \\ & & a_5={\displaystyle \frac{m_2{C}_2{L}_{1}sin\left({\theta }_1-{\theta }_2\right)\left(L_1^2{m}_2+I_1\right)}{{\Delta }_0}},\hfill \\ & & a_6={\displaystyle \frac{1}{2}}{\displaystyle \frac{m_2^2{C}_2^2{L}_1^{2}sin\left[2({\theta }_1-{\theta }_2\right)]}{{\Delta }_0}},\hfill \\ & & {\Delta }_0=I_1{I}_2+m_2{L}_1^2\left[I_2-m_2{C}_2^2{cos}^2\left({\theta }_1-{\theta }_2\right)\right].\hfill \end{array}$$

(31)

Note that $a_2=a_6$ for any ${\theta }_1$ and ${\theta }_2$.

Let us perform dimensional analysis for the quantities shown in Equation (31). One has that $\left[{\Delta }_0\right]={(\mathrm{kg}\times {\mathrm{cm}}^2)}^2$. Then $\left[a_2\right]=\mathcal{O}\left(1\right)$, $\left[a_3\right]=\mathcal{O}\left(1\right)$, $\left[a_5\right]=\mathcal{O}\left(1\right)$, and $\left[a_6\right]=\mathcal{O}\left(1\right)$, i.e., these constants are dimensionless. Finally, $\left[a_1\right]={\mathrm{s}}^{-2}$ and $\left[a_4\right]={\mathrm{s}}^{-2}$; in addition, $g=9.8\mathrm{m}/{\mathrm{s}}^2=980\mathrm{cm}/{\mathrm{s}}^2$.

The data needed for the evaluation of the above quantities are presented in

Table 1. The table provides biomechanical data from different sources available in the literature. According to Ref. [6], ${\rho }_1=1062$ kg/m³ for the thigh, while for the shank ${\rho }_2=1088$ kg/m³. For the case of Germans, see Ref. [3]; the data reported here for $m_1,m_2,L_1,L_2$ and for the inertial moments are not directly given there but can be calculated using the regression equations, the coefficients of which they provided in [3], for any of the quantities discussed here.

Study	n	Age [Years]	Body Height [cm]	Body Mass [kg]	Lengths [cm]	Mass [kg]	Centre of Mass [%]	Principal Inertial Moments [kg.cm2]
Bulgarian males Nikolova and Toshev [6]	2435	30–40	171	77.7	$L_1$ = 51.0 $L_2$ = 37.2	$m_1$ = 11.0 $m_2$ = 3.3	$c_1$ = 41.4 $c_2$ = 44.6	$I_{\mathrm{xx}}^{\left(1\right)}=1564.0$ $I_{\mathrm{yy}}^{\left(1\right)}=1564.0$ $I_{\mathrm{zz}}^{\left(1\right)}=307.7$ $I_{\mathrm{xx}}^{\left(2\right)}=231.9$ $I_{\mathrm{yy}}^{\left(2\right)}=231.9$ $I_{\mathrm{zz}}^{\left(2\right)}=34.0$
Russian males Zatsiorsky [2]	100	23.8	174	73.0	$L_1=52.0$ $L_2=39.3$	$m_1=10.4$ $m_2=3.2$	$c_1=45.0$ $c_2=40.5$	$I_{\mathrm{xx}}^{\left(1\right)}=1999.4$ $I_{\mathrm{yy}}^{\left(1\right)}=1997.8$ $I_{\mathrm{zz}}^{\left(1\right)}=413.4$ $I_{\mathrm{xx}}^{\left(2\right)}=371.0$ $I_{\mathrm{yy}}^{\left(2\right)}=385.0$ $I_{\mathrm{zz}}^{\left(2\right)}=64.6$
German males Shan and Bhon [3]	25	25.9	181	78.4	$L_1$ = 46.3 $L_2=43.8$	$m_1$ = 10.8 $m_2$ = 3.7	$c_1$ = 32.2 $c_2$ = 46.4	$I_{\mathrm{xx}}^{\left(1\right)}=1940.9$ $I_{\mathrm{yy}}^{\left(1\right)}=1942.4$ $I_{\mathrm{zz}}^{\left(1\right)}=375.7$ $I_{\mathrm{xx}}^{\left(2\right)}=500.2$ $I_{\mathrm{yy}}^{\left(2\right)}=284.6$ $I_{\mathrm{zz}}^{\left(2\right)}=96.6$
The US american males McConville et al. [4]	31	27.4	177	77.3	$L_1$ = 46.5 $L_2=40.7$	$m_1$ = 8.4 $m_2$ =5.4	$c_1$ = 57.1 $c_2$ = 37.0	$I_{\mathrm{xx}}^{\left(1\right)}=1551$ $I_{\mathrm{yy}}^{\left(1\right)}=1633$ $I_{\mathrm{zz}}^{\left(1\right)}=419$ $I_{\mathrm{xx}}^{\left(2\right)}=570$ $I_{\mathrm{yy}}^{\left(2\right)}=578$ $I_{\mathrm{zz}}^{\left(2\right)}=66$

Here, n is the number of persons subject to the study. SD denotes standard deviation. All the people studied, for whom data are presented, are of European descent.

. In Ref. [6], one also provides data for principal moments of inertia $I_{xx},I_{yy}$, and $I_{zz}$, where the coordinate system is centred at the centre of mass of the corresponding segment; the z-axis is along the segment, while the x- and y-axes form a plane orthogonal to it. Due to the supposed symmetry, $I_{xx}=I_{yy}$. One way of acting, in order to determine $I_1$ and $I_2$, is to realise they are equal to

$$I_i=(I_{xx}^{\left(i\right)}+I_{yy}^{\left(i\right)}-I_{zz}^{\left(i\right)})/2+m_i{c}_i^2,$$

(32)

where we have used Steiner’s theorem.

4. Biomechanical Data

The thigh length $L_1$ is the distance between anthropometric points iliospinale—tibiale. The shank length $L_2$ is the distance between anthropometric points tibiale—sphyrion. Such data for Bulgarians, Russians, Germans, and the US Americans are shown in Table 1. All the people studied, for whom data are presented, are of European descent. As is clear from the table, the set of data with the best statistics, encompassing 2435 individuals, is available for Bulgarians. That is why our basic calculations will be based on these data.

Using data shown in Table 1, and the ones for the densities reported above, from Equation (31), one obtains values for the parameters ${\Delta }_0,\cdots ,a_6$ as functions of ${\theta }_1$ and ${\theta }_2$. Thus, one is ready to attack numerically, or analytically, the system Equation (30), or, equivalently, Equations (28) and (29), of two mutually coupled second order nonlinear differential equations of variables ${\theta }_1$ and ${\theta }_2$.

Table 2. The table provides the hip, knee, and ankle joint angles for each of the eight phases of the human gait cycle.

Gait Phases of the Human Gait Cycle	Initial Contact	Loading Response	Mid Stance	Terminal Stance	Pre Swing	Initial Swing	Mid Swing	Terminal Swing
Hip	20°	20°	0°	−20°	−10°	15°	25°	20°
Knee	0°–5°	20°	0°–5°	0°–5°	40°	60°–70°	25°	0°–5°
Ankle joint	0°	5°–10°	5°	10°	15°	5°	0°	0°

guides us to the reasonable values of these angles. We will present solutions for them, as a function of time, in the next section.

The human gait is usually considered as a cycle decomposed into eight phases, see Figure 2 and Refs. [71,72]. The gait is typically thought of as a continuous series of periodic activities; the gait cycle is normally defined as the time interval between the first heel strike contact of one foot (for example, the right foot) and the subsequent heel strike contact of the same foot. Following one of the widely accepted categorisations reported in the literature, we use the so-called new gait terms involving the following eight phases: (1) initial contact, (2) loading response, (3) midstance, (4) terminal stance, (5) pre swing, (6) initial swing, (7) mid swing, and (8) late swing.

5. Analytical and Numerical Results

5.1. Analytical Results

An analytical solution of the equations Equations (28) and (29), governing the system, i.e., of the functions ${\theta }_1\left(t\right)$ and ${\theta }_2\left(t\right)$, is only possible when these angles are small enough, so that an expansion in terms of ${\theta }_1\left(t\right)$ and ${\theta }_2\left(t\right)$ in them is possible.

Retaining only the linear terms, we obtain

$$\ddot{{\theta }_1}\left(t\right)=g{\displaystyle \frac{I_2\left[C_1{m}_1+L_1{m}_2\right]{\theta }_1\left(t\right)-C_2^2{L}_1{m}_2{\theta }_2\left(t\right)}{L_1^2{m}_2\left[C_2^2{m}_2-I_2\right]-I_1{I}_2}}$$

(33)

and

$$\ddot{{\theta }_2}\left(t\right)=g{m}_2{C}_2{\displaystyle \frac{{\theta }_2\left(t\right)\left[I_1+L_1^2{m}_2\right]-L_1{\theta }_1\left(t\right)\left[C_1{m}_1+L_1{m}_2\right]}{L_1^2{m}_2\left[C_2^2{m}_2-I_2\right]-I_1{I}_2}}.$$

(34)

These two equations describe two linearly coupled oscillators. Looking for a solution in the form

$${\theta }_1\left(t\right)=A_{1}exp\left[i\omega t\right],\mathrm{and}{\theta }_2\left(t\right)=A_{2}exp\left[i\omega t\right]=A_{1}rexp\left[i\omega t\right],\mathrm{with}r=A_2/A_1,$$

(35)

we obtain

$$\omega =\pm \sqrt{g}{\displaystyle \frac{\sqrt{C_1{m}_1+L_1{m}_2}}{\sqrt{L_1{m}_2(C_2+L_1)+I_1}}},$$

(36)

and

$$r={\displaystyle \frac{(C_1{m}_1+L_1{m}_2)(C_2{L}_1{m}_2+I_2)}{C_2{m}_2\left[L_1{m}_2(C_2+L_1)+I_1\right]}}={\displaystyle \frac{{\omega }^2}{g}}{\displaystyle \frac{C_2{L}_1{m}_2+I_2}{C_2{m}_2}}.$$

(37)

Plugging in Equations (36) and (37), the values of all quantities involved, we have

$$\begin{array}{ccc}\hfill \omega & =& \pm 4.709\mathrm{and}r=1.618\mathrm{for}\mathrm{Bulgarians},\hfill \\ \hfill \omega & =& \pm 4.641\mathrm{and}r=1.642\mathrm{for}\mathrm{Russians},\hfill \\ \hfill \omega & =& \pm 4.503\mathrm{and}r=1.409\mathrm{for}\mathrm{Germans},\hfill \\ \hfill \omega & =& \pm 4.512\mathrm{and}r=1.417\mathrm{for}\mathrm{US}\mathrm{Americans}.\hfill \end{array}$$

(38)

As we see, the difference in the results for the different nationalities is quite small.

The obtained analytical solution of Equations (28) and (29) for ${\theta }_1\left(t\right)$ and ${\theta }_2\left(t\right)$ is visualised in Figure 3. It is purely periodic. There are no elements of chaos in this case. This is the most elementary movement under walking of a human; as we will see below, it corresponds to slow motion and small steps, as observed in patients with Parkinson’s disease—see, e.g., Refs. [25,26,27].

5.2. Numerical Results

Now we turn to the investigation of Equations (28) and (29) for ${\theta }_1\left(0\right)=5\pi /100=9^{\circ }$ and ${\theta }_1\left(0\right)=5\pi /36={25}^{\circ }$. These cases correspond to reasonable values of the initial parameters for standard walking. They can be attacked only numerically. Solving them, we obtain the results shown in Figure 4—for ${\theta }_1\left(0\right)=5\pi /100=9^{\circ }$—and Figure 5—for ${\theta }_1\left(0\right)=5\pi /36={25}^{\circ }$.

(1)

For relatively small initial angles ${\theta }_1\simeq 9^{\circ }$ and speeds, we obtain the following solutions for ${\theta }_1$ and ${\theta }_2$:

(2)

For initial angles ${\theta }_1\simeq {25}^{\circ }$ and slow speeds, we obtain the following solutions for ${\theta }_1$ and ${\theta }_2$:

The results shown in Figure 4 and Figure 5 demonstrate that, under the considered initial conditions, the movement is not purely periodic, but also contains an element of chaos in it: the curves in the phase diagram do not form closed loops. Furthermore, we observe that for ${\theta }_1\left(0\right)=5\pi /36={25}^{\circ }$ the spread of the phase diagram is about three times larger than for ${\theta }_1\left(0\right)=5\pi /100=9^{\circ }$, i.e., the larger the initial angles, the larger is the spread of the phase diagram, i.e., the stronger is the element of chaos in the system. The last implies that, for larger initial angles ${\theta }_1\left(0\right)$, one needs much finer tailoring of the muscles of the body, with the nerves governed by the brain of the human during walking. Additionally, the numerical results in Figure 4 and Figure 5 show that initial conditions significantly influence the model outputs (solutions), which is a necessary condition for chaotic behaviour.

The numerical solution of the system of two mutually coupled nonlinear differential equations of second order, Equations (28) and (29), has been performed using the Mathematica ^® program, version 14.3, of the Wolfram Corporation, Champaign, IL, USA. This has been realized in two equivalent ways: (1) directly as a system of two second-order equations; (2) as a system of four equations of first order, to which this system can be reformulated. Normally, the second approach is preferred for the Runge–Kutta methods. The results obtained in both ways are equivalent.

6. Comparison of Our Model Results to Experimental Data for Real Walking

In this section, we will briefly discuss how our results compare to real walking.

As already emphasised, walking is a fundamental human activity, essential for daily life, work, and social interaction. Its significance extends to well-being, enabling shopping, commuting, and maintaining connections with others. The study of human motion, with a particular focus on gait, has a long history, dating back to ancient times [73]. Currently, the research in this field encompasses diverse areas, including neuromuscular disorders, joint replacement, sports injuries, and assistive devices [74,75]. Furthermore, the increasing interest in humanoid and wearable robots [76] underscores the continuing importance of understanding human locomotion.

The results of our model of human walking indicate that for small initial angles (periodic movement), the period $T=2\pi /\omega \simeq 1.33$ s. One forward swing of the leg corresponds to a half cycle, which takes $T_{1/2}=0.67$ s. This is quite close to the number reported in Refs. [29,77], according to which at slow walk, $T_{1/2}\simeq 0.60$ s. $T_{1/2}=0.67$ s corresponds to about 2400 steps/h. If we assume the average step length to be approximately $0.6$ m, we obtain a speed of approximately $1.44$ km/h.

For the movements shown in Figure 4, $T_{1/2}=0.81$ s, while for Figure 5, $T_{1/2}=0.44$ s. These data correspond to $2.67$ km/h and $5.9$ km/h, respectively. The corresponding step lengths are, then, $2Lsin\left(\theta \right)$. Having that for $T_{1/2}=0.81$ s one has $\theta =2\times 5\pi /100\simeq {18}^{\circ }$, we obtain for the step length $(L_1+L_2)sin\left({18}^{\circ }\right)=0.27$ m and $(L_1+L_2)sin\left({50}^{\circ }\right)=0.68$ m for $T_{1/2}=0.44$ s. According to Ref. [29] (p. 127), see also Refs. [30,31], the speed for a slow walk is about 0.5 m/s = 1.8 km/h. There, one takes a step length of $0.3$ m, the time for a step $T_{1/2}\simeq 0.60$ s. For a fast walk, the speed is estimated about 2.0 m/s = 7.2 km/h, and the step length is about $0.7$ m, so the step time is at about $0.35$ s. As we see, all these numbers are quite close to the numbers we obtained within our model.

7. What Can the Model Be Good for in Addition?

One important issue of human walking is the gait asymmetry. For healthy people, the reader can consult the recent review [78]. Naturally, the situation is much more complicated for persons with certain diseases—see, e.g., Refs. [79,80,81].

We will delineate several avenues in which our model can contribute to research in this area.

7.1. For Studying the Phase Difference Between Left- and Right-Limb Kinematics

To obtain specific results within our model that lead to phase differences and asymmetry, we need the biomechanical characteristics reported in Table 1, separately for the left and right legs. Naturally, such calculations can be conducted for individuals, as well as using the average values of these quantities for a given set of individuals. Below, we briefly clarify how to do that. Such types of data can be found in, e.g., Refs. [82,83]. It should be noted that classical anthropometric studies contain separate data for left and right limbs (Dempster [62], Clauser [63], Chandler et al. [64], Zatsiorsky [2,5], de Leva [65], Challis [66], Dumas et al. [67]); however, the methods they use give an error of 2–5%, which is larger than that of typical biological asymmetry (1–3%) [84,85,86,87]. Several modern biomechanical and anthropometric studies do indeed measure left and right limbs separately, especially when using 3D body scanning, MRI (i.e., magnetic resonance imaging), DXA (i.e., dual-energy X-ray absorptiometry), or reaction-board methods. Some such data from measurements of the asymmetry and references related to the topic can be found, e.g., in Refs. [82,88]. A recent overview of how to obtain asymmetric inertial parameters can be found in [66]. Currently, to our knowledge, there are no widely accepted population-level tables that report different averages for left vs right thigh/shank/foot mass, centre of mass, or inertia.

To provide specificity, we will reference several particular left–right asymmetries and the corresponding studies.

7.2. Studying the Asymmetry of Walking Reflected by the Lengths of the Steps Made with the Left Versus the Right Leg

The simplest way to study asymmetry is to calculate the step lengths made via the left and right legs. We accept that the angular frequency $\omega$ is the same for both legs. Then, $\left[L_{1,R}+L_{2,R}\right]sin\left(2{\theta }_{1,\mathrm{R},max}\right)$ will determine the length of the step from the left leg, and $\left[L_{1,L}+L_{2,L}\right]sin\left(2{\theta }_{1,\mathrm{L},max}\right)$ via the right leg. The data needed for the corresponding calculations are easily experimentally measurable, say, via some sensors on the legs and a photo technique. This can easily be conducted and is, in fact, not using the essence of our model.

7.2.1. Studying the Phase Shift in Walking

Let us briefly explain how one can quantify the phase differences for walking mathematically.

(1)

Phase difference via cross-correlation

Let ${\theta }_{2,L}\left(t\right)$ and ${\theta }_{2,R}\left(t\right)$ denote the left- and right-leg joint-angle trajectories, respectively, expressed as continuous functions of normalised gait time $t\in [0,1]$. Each function represents one complete gait cycle, with $t=0$ corresponding to the instant of left heel-strike. The signals ${\theta }_{2,L}\left(t\right)$ and ${\theta }_{2,R}\left(t\right)$ are assumed to be approximately periodic and sufficiently smooth for correlation-based phase analysis.

The inter-limb phase difference (PD) quantifies the temporal offset between the left- and right-leg kinematic waveforms. It is defined as the temporal shift $\Delta$ that maximises the cross-correlation between the two signals:

$$\mathrm{PD}=arg\underset{\Delta }{max}\mathrm{corr}\left({\theta }_L\left(t\right),{\theta }_R(t+\Delta )\right),$$

where $\mathrm{corr}(·,·)$ denotes the correlation coefficient computed over the normalised gait cycle. For a healthy bipedal gait, the expected value is $\mathrm{PD}\approx 0.5$, reflecting the characteristic anti-phase alternation between the limbs.

(2)

Phase difference from heel-strike timing

When discrete gait events are available (e.g., heel-strike times from force plates, foot switches, or IMUs), the phase difference can be computed directly from event timing. Let $t_{\mathrm{HS},\mathrm{L}}$ and $t_{\mathrm{HS},\mathrm{R}}$ denote the times of the left and right heel-strike, respectively, and let $T_{\mathrm{stride}}$ be the stride duration (left heel-strike to subsequent left heel-strike). The phase difference is then given by the following equation:

$$\mathrm{PD}={\displaystyle \frac{t_{\mathrm{HS},\mathrm{R}}-t_{\mathrm{HS},\mathrm{L}}}{T_{\mathrm{stride}}}}.$$

Under normal gait conditions, this definition also yields $\mathrm{PD}\approx 0.5$, consistent with the cross-correlation approach.

(3)

Phase shift in harmonic decomposition

Each signal can be represented by a truncated Fourier series of the form

$$\theta \left(t\right)=a_0+{\displaystyle \sum _{k=1}^N}{A}_{k}cos\left(2\pi kt+{\varphi }_k\right),$$

where $A_k$ and ${\varphi }_k$ denote the amplitude and phase of the k-th harmonic, respectively, and $a_0$ is a time independent constant.

For each harmonic component, the inter-limb phase shift is defined as the difference between the harmonic phases of the right and left limbs:

$$\Delta {\varphi }_k={\varphi }_{k,R}-{\varphi }_{k,L}.$$

The fundamental harmonic ($k=1$) typically captures the primary flexion–extension pattern of the gait cycle. For a healthy bipedal gait, the limbs operate in near-perfect anti-phase, yielding

$$\Delta {\varphi }_1\approx \pi ,$$

while higher-order harmonics exhibit smaller and more variable phase differences. Deviations from these expected values reflect disruptions in inter-limb coordination and may indicate gait asymmetry or pathology.

7.2.2. Interpretation of the Phase Shift

A value of $\mathrm{PD}=0.5$ indicates perfect anti-phase coordination between the limbs. Deviations from this value reflect temporal asymmetries: $\mathrm{PD}<0.5$ indicates that the right limb leads the left, whereas $\mathrm{PD}>0.5$ indicates that the right limb lags. In healthy adults, variability is typically within $\pm 0.02$ of the ideal value, while larger deviations may indicate pathological gait patterns or compensatory strategies. Pathological gait often shows the following:

$$PD=0.45-0.60.$$

Obviously, approaches (1) and (3) are directly applicable in our model. To apply approach (3), one needs to define new discrete gait events—say, the moments $t_L$ and $t_R$, at which the left and the right legs, respectively, reach the maximum values of the angles ${max}_t{\theta }_{1,R}\equiv {\theta }_{1,R,max}$ and ${max}_t{\theta }_{1,L}\equiv {\theta }_{1,L,max}$.

7.3. For Studying Different Groups of People—Different Ages, Ethnicity, Etc.

An advantage of our model is that it can easily be scaled to study different groups (children, women, and adolescent persons) of human populations. It can also be tailored to specific, diverse ethnic populations, or even to individuals, making it a bridge between abstract physics and personalised biomechanical solutions. Thus, although the model is a theoretical construct, it has clear real-world applications across several domains:

• (a) Gait analysis and rehabilitation; it can simulate normal and pathological walking patterns, helping clinicians design targeted physiotherapy or post-surgery recovery plans.
• (b) Prosthetics and orthotics design; by adjusting parameters (mass, length, and inertia) to match a patient’s anatomy, engineers can optimise artificial limbs for more natural movement.
• It can be extended to issues like (c) injury prevention; understanding joint loads and motion dynamics can guide ergonomic recommendations for athletes and workers; (d) optimisation of human motion [89] and in determining its optimal controls [24].

We hope to return to some of the above-mentioned modifications of our model in the future.

8. Concluding Remarks

In the current article, we presented a mathematical model and an analysis based on it for walking using biomechanical data for an average Bulgarian, Russian, German, and US American male. Their data is reposted in Table 1. It turns out that the difference in the results is quite small—see, e.g., Equation (38). The estimation of the speed of walking of a person within our model depends mainly on (i) the time $T_{1/2}$ for forward swing of the leg; (ii) the maximum value of the angle ${\theta }_1>0$ (the larger the angle, the higher the speed); and (iii) on the sum of the lengths $L=L_1+L_2$ of the segments. The quantity of $T_{1/2}$ depends, of course, on all mass–inertial characteristics incorporated in the model, which appear as a solution of the system of two coupled nonlinear differential equations. As it turns out for values of ${\theta }_1\lesssim 9^{\circ }$ the agreement with the experimentally observed data for slow walking is excellent. Furthermore, the difference between the different sets of data for the nationalities considered turns out to be negligibly small. The situation changes for larger angles. When ${\theta }_1\simeq {25}^{\circ }$, which is about the maximal one for standard walking, the obtained walking speed, the maximum possible in our model, is about $3.0$ km/h. This clearly demonstrates that for faster walking the role of muscles is indispensable and ought to be considered. Taking into account the muscle torques acting on the segments of the pendulum will lead to source terms on the right-hand side of Equations (28) and (29), in lines of Refs. [21,90,91,92]. Understanding torques enables inferences about the muscles contracting at certain moments to cause the leg to swing in tandem with gravity [74]. The last will help in studying the complex control required for human movement [21,22,23,24]. One can also take into account the ground reaction forces, along the lines of Refs. [93,94,95,96]. Very important is the study of the role of joints, especially that of the knee [97,98].

One customarily states that the double pendulum is a cornerstone model for understanding how the human body moves, as it perfectly illustrates the mechanics of multi-segment chains. As we have seen, it delivers realistic results for human walking for relatively slow walking speeds. When achieving this, it is worthwhile mentioning that we used no fitting parameters in our model—they all reflect experimentally measured biomechanical ones, taken from the literature. Much of our gait (walking/running) can be explained by passive pendulum-like motion, which is incredibly energy-efficient. It gives us a good idea why, despite being capable of walking at speeds reaching up to about 9 km/h, humans naturally choose to walk at a typical speed of around $4.5$ km/h, known as the preferred walking speed [99]. This speed is not very far from the maximal speed within the double pendulum model of $3.0$ km/h, for which no active muscle work is necessary.

Let us also mention that we outlined, in Section 7, several possibilities for directly using our model for analysing the asymmetry in human walking, provided, of course, that some relatively easily measurable data for the quantities mentioned there is available.