Foot function enabled by human walking dynamics

Bipedal walking, the habitual gait for man, is rather unique in nature and poses particular challenges for balance and propulsion. The characteristic double-humped ground reaction force profile has been widely observed but not put into functional context. We propose a mathematical model that captures the dynamics of the human foot in walking including the characteristic motion of the center of pressure. Using this model, we analyze the functional interplay of all essential biomechanical contributors to foot dynamics in walking. Our results demonstrate the intricate interplay of a self-stabilizing mechanism which allows extending a leg's stance phase while simultaneously powering rapid swing by condensing the essentials of foot dynamics into a reductionist, biomechanical model. A theory is presented which identifies the foot to be the key functional element and which explains the global dynamics of human walking. The provided insights will impact gait therapy and rehabilitation, the development of assistive devices, such as leg prostheses and exoskeletons, and provide guidelines for the design and control of versatile humanoid robots.

Figure 1

Kinematic setup. Sagittal marker positions were recorded at the hip (greater trochanter, Trc), the knee (lateral knee joint gap, Kne), the toe (5th metatarsal joint, Mt5), and the ankle (lateral malleolus, Ank). The center of mass (CoM) of the HAT segment was derived from sex-, height-, and weight-specific regression curves [22]. The foot segment was defined between Mt5 and Ank, the shank segment between Ank and Kne, the thigh segment between Kne and Trc, and the HAT segment between Trc and ${\mathrm{CoM}}_{\mathrm{HAT}}$. Ankle angle ${\phi }_{\text{Ank}}$, knee angle ${\phi }_{\text{Kne}}$, and hip angle ${\phi }_{\text{Hip}}$, were defined as inner joint angles between two adjacent segments.

Figure 2

Evaluated and discarded steps per subject. Bars indicate the number of recorded steps per subject with the total number of recorded steps listed above each bar. Colors indicate evaluated and discarded steps (see legend). The average residual (in Nm) of a function fit to the foot acceleration signal (see Appendix pp2-s4 for fitting details) is listed for each subject.

Figure 3

A schematic of the foot model with all dynamic actors. The foot is assumed as a body defined by its center of mass ${\text{CoM}}_f$, its moment of inertia $I_f$ and the location of the ankle joint. The marker at the 5th metatarsal joint ($\text{Mt}5$, see also Fig. 1) is used as a reference point on the foot (Fig. 4). The gravitational force ${(0,-m_{f}g)}^T$ acts on the center of mass, ankle force ${\vec{F}}_{\text{Ank}}$ and ankle torque ${\mathcal{T}}_{\text{Ank}}$ act at the ankle joint. The reflection of primary forces, i.e., the vector ${\vec{F}}_{\text{GRF}}$ of the ground reaction force (GRF), acts on the foot at the position ${\vec{R}}_{\text{CoP}}={(R_{\text{CoP},x},0)}^T$ of the moving center of pressure (CoP), which is the projection (see Sec. pp2-s1) of the experimentally determined center of pressure (${\text{CoP}}_{\text{exp}}$) along the instantaneous ${\vec{F}}_{\text{GRF}}$ into the plane of the center of rotation (CoR). The positions of ankle, ${\text{CoM}}_f$ and CoP relative to the CoR are given by the respective vectors. The symbol $\mathcal{R}$ indicates the lever for the resulting torque caused by the ankle force and its reflection. Subject to the resulting dynamics, the foot executes a pure tilt ($\dot{\phi },\ddot{\phi }$) around the fixed CoR (Fig. 4).

Figure 4

CoR variability and lever length error. For each step, the position of the CoR ($x_c, y_c$, the origin of our model's coordinate system), determined by optimization (Appendix pp2-s4), is indicated by a “$+$,” given in coordinates relative to the ground (vertical) and the respective most anterior Mt5-marker (see Figs. 3 and 1) position while the foot is planted flat on the ground. CoRs for left and right steps are indicated individually as some subjects exhibit a clear left/right asymmetry, resulting in CoR clustering. Gray circles indicate a distance in steps of $5\mathrm{mm}$ from the median CoR of a subject respectively ($2\mathrm{cm}$ for the outermost ring). Histograms show the number of samples for which each instantaneous lever arm $|{\vec{R}}_{\text{CoM},f}|$ and $|{\vec{R}}_{\text{Ank}}|$ deviates a certain percentage from its optimized fixed lever arm length $R_{\text{CoM},f}$ or $R_{\text{Ank}}$, respectively. Each gray box indicates an error margin of one percent. Blue bars include only samples of the fit interval, red samples include the entire intervals beginning with the onset of single support and ending with the foot's takeoff. The bar graph at the bottom left indicates the total number of samples considered for either interval in each subject. The intervals are illustrated in the plot at the bottom right of the figure, with the full circle symbolizing an entire gait cycle.

Figure 5

Main actor dynamics during stance phase. The ankle force norm ($|{\vec{F}}_{\text{Ank}}|=F_{\text{Ank}}$), as determined by inverse dynamics, is shown in panel (A). The experimentally determined force ($F_{\text{Ank},\exp }$) for each step is shown in dark gray, its mean is shown in blue, and the corresponding equilibrium force ($F_{\text{Ank},\mathrm{eq}}$), calculated from Eq. (3) with the left-hand side set to zero is depicted as a dashed cyan line. The dashed black line indicates the combined weight of all segments proximal to the foot on the ground ($m_{p}g$). Gray shadings of the background indicate single support (dark) and double support (light). Panel (${\text{A}}_1$) shows the difference of both signals. The vertical lines indicate important gait events, namely averages of (i) the instance the ankle force exceeds $m_{p}g$ ($\mathrm{orange}$), (ii) the instance the ankle power turns positive indicating the onset of push-off and alleviation phase [26] ($\mathrm{gray}$), (iii) the instance the ankle force reaches its second maximum ($\mathrm{green}$), (iv) the instance the ankle torque reaches its minimum ($\mathrm{brown}$), (v) the instance the trajectory of the resulting torque transitions into its bell-shaped part [(Fig. 8 in Appendix pp3), $\mathrm{pink}$], and (vi) the onset of the launching phase defined by the jerk maximum of the ankle joint angle [26] ($\mathrm{olive}$). The respective color for each event remains the same in all following figures. Analogously, in panel (B) the experimental ankle torque (${\mathcal{T}}_{\text{Ank},\exp }$) is indicated in red, the equilibrium torque (${\mathcal{T}}_{\text{Ank},\mathrm{eq}}$) in dashed cyan and panel (${\text{B}}_1$) shows the difference of their norms. Panel (C) shows the relation of experimental and equilibrium force/torque in blue and red respectively over time, with 1 indicating equilibrium and values above 1 that the measured property exceeds the equilibrium torque. It can be clearly seen that force and torque depart from the equilibrium quite synchronously but in opposite directions indicating a loss of equilibrium in favor of the protagonist and forward rotation of the foot.

Figure 6

Three torque contributions according to Eq. (3). Panel (A) shows the components of all individual steps in gray and means for antagonist (${\mathcal{T}}_{\text{ant}}$), protagonist (${\mathcal{T}}_{\text{pro}}$), and changing gravitational torque (${\mathcal{T}}_{\text{chg}}$, very small in comparison). Gray shadings of the background indicate single support (dark) and double support (light). The vertical lines indicate averages of (i) the instance the ankle force exceeds $m_{p}g$ ($\mathrm{orange}$), (ii) the instance the ankle power turns positive indicating the onset of push-off and alleviation phase [26] ($\mathrm{gray}$), (iii) the instance the ankle force reaches its second maximum ($\mathrm{green}$), (iv) the instance the ankle torque reaches its minimum ($\mathrm{brown}$), (v) the instance the trajectory of the resulting torque transitions into its bell-shaped part [(see Fig. 8 in Appendix pp3) $\mathrm{pink}$], and (vi) the onset of the launching phase defined by the jerk maximum of the ankle joint angle [26] ($\mathrm{olive}$). Panel (B) shows a comparison of the protagonistic torque with the inverted antagonistic torque. Panel (C) shows the trajectory of the changing gravitational torque again, as well as the sum of torques (${\mathcal{T}}_{\text{pro}}+{\mathcal{T}}_{\text{ant}}$) without the changing contribution. All three measured torque components (${\sigma }_{\text{exp}}$), the linear part of the fit [Eq. (B29)], as well as the combined fit of linear and Gaussian parts [${\sigma }_{\text{fit}}$, Eq. (B29)] are depicted (see Appendix pp2-s4 for fitting details).

Figure 7

Comparison of calculated and measured CoP progression. Gray lines indicate the CoP for each step, calculated for a dynamic foot equilibrium [$\ddot{\phi }{I}_{f,\text{CoR}}=0$, Eqs. (B27) and (B28)] and compared to measured data. Gray shadings of the background indicate single support (dark) and double support (light). Vertical colored lines indicate the timing of events as explained in the caption of Fig. 5 (see Appendix pp3 for detailed description of events). In the instant when the equilibrium is lost ($t_{\epsilon }$), the measured CoP deviates on average by $(0.47\pm 0.33)\mathrm{mm}$ from its equilibrium.

Figure 8

Timing of important gait events. All histograms indicate the number of steps for which an event occurs at a specific sample, the $x$ axis indicating time in % GC. Distribution of absolute event timing for all analyzed steps is shown on the diagonal. The events analyzed are (i) ankle force norm exceeds proximal body weight ($t_{F>m_{p}g}$), (ii) push-off onset ($t_{\text{push}\text{−}\text{off}}$), (iii) maximum of ankle force ($t_{\text{max}\left(F_{\text{Ank}}\right)}$), (iv) minimum of ankle torque ($t_{\text{min}\left({\mathcal{T}}_{\text{Ank}}\right)}$), (v) loss of equilibrium ($t_{\epsilon }$), and (vi) onset of launching phase [26] ($t_{\text{launch}}$). Gray shading indicates single (dark) and double (light) support phases. The touchdown of the contralateral leg ($t_{\text{TDc}}$, i.e., the right edge of the dark gray area occurs at $(50.45\pm 0.16)$ % GC. The determination of all events is explained in detail in Appendix pp3. The mean event time (in % GC) and standard deviation are reported in numbers and indicated with a vertical line and horizontal error bar. The distribution of relative event timing is shown in the upper half for all analyzed steps. Each column shows the timing relative to all other key events as indicated in the subfigures' titles. Zero (marked with a blue vertical bar) indicates the number of steps for which the two events occur simultaneously; negative numbers indicate that the event of the respective row precedes the event of the respective column. The bottom row shows example trajectories of the respective characteristic quantities from which each event has been determined, with the mean event time indicated by a vertical dashed line and error bar.

Figure 9

Main actor dynamics during stance phase for all subjects. Details as shown in Fig. 5 for all 21 subjects. While shapes and magnitudes differ for all subjects, the general dynamics are consistent throughout.

Figure 10

Three torque contributions according to Eq. (3) for all subjects. Details as shown in Fig. 6 for all 21 subjects. While shapes and magnitudes differ for all subjects, the general dynamics as shown in panels C are consistent throughout.