Human balance control in 3D running based on virtual pivot point concept

Human leg morphology and motor control have evolved over thousands of years to perform various locomotion tasks over a wide repertoire of surfaces. Inspired by biological evolution, roboticists try to achieve similar capabilities in legged robots (Kim and Wensing, 2017). In that respect, simple template models such as the spring-loaded inverted pendulum (SLIP) (Blickhan, 1989), which can predict human leg function in running, have been used to design and control legged robots (Ahmadi and Buehler, 1999; Hubicki et al., 2016). Further, quadruped robots such as the MIT Cheetah can outperform humans concerning performance (speed) and efficiency (Seok et al., 2014). However, human capabilities in agile movement tasks such as running are still far beyond those of bipedal robots (Tajima et al., 2009).

Among different fundamental elements of legged locomotion, namely locomotor subfunctions (Sharbafi et al., 2017), upper-body balance is one of the most challenging. Bipeds are dynamically unstable and need to keep their trunk upright, especially during agile gaits such as running. The increased complexity may result from the placement of more than 50% of total body mass in the upper body having significantly large inertia (Thorstensson et al., 1984). Because of that, small deviations in trunk position can significantly affect whole-body stability (Drama and Badri-Spröwitz, 2019). Nevertheless, humans benefit from the trunk's inertia to facilitate locomotion and can simply keep balance in running with minimal energy consumption (Bramble and Lieberman, 2004). Therefore, understanding balance control in running can simplify bipedal robot control (Sharbafi et al., 2016) and improve gait assistance with exoskeletons (Nasiri et al., 2018; Sugar et al., 2017).

The mechanics of balance can be understood in terms of how the ground reaction force (GRF) acting at the center of pressure (CoP) changes the body's angular momentum about its center of mass (CoM) (Popovic et al., 2005). Several methods have been developed to quantify stability based on the relationship between GRF, CoM and CoP such as zero moment point (ZMP) (Kajita et al., 2007), foot rotation indicator (FRI) (Goswami, 1999) and centroidal moment pivot (CMP) (Popovic et al., 2005). In 2010, a new method was introduced by Maus et al. (2010), which could describe balancing in biological legged locomotion with a virtual pendulum (VP) model. They showed that the GRFs intersect near a virtual pivot point (VPP) above the CoM in the sagittal plane during human and animal walking (Maus et al., 2010). This method provides a template model for posture control, which does not need to measure body orientation with respect to the ground. Therefore, balancing can be achieved using proprioceptive sensory signals, i.e. trunk to leg angle (Maus et al., 2010).

The VPP concept was utilized not only to predict biological gaits but also for the control of robots, e.g. ATRIAS robot walking (Peekema, 2015). Because of the bioinspired nature of the VPP concept, it has also been used to control assistive devices (Zhao et al., 2017, 2019; Firouzi et al., 2021). In previous studies, generating a VPP above the CoM was used as the control goal to achieve postural stability in different gaits in simulations and robots (Sharbafi et al., 2013a,b; Maus et al., 2010). However, just a single study experimentally reported a VPP above the CoM for running gait, and this was limited to a single trial of a single subject without any statistical analysis (Blickhan et al., 2015). Recently, a simulation study using template models showed that intersecting GRFs below the CoM (resulting in negative VPP) could better explain running gait in the sagittal plane concerning the trunk angle (Drama and Badri-Spröwitz, 2019). Another study demonstrated similar negative VPP in human running at 5 m s⁻¹ (Drama et al., 2020).

However, the VPP method was applied only to analyze running in the sagittal plane whereas lateral balance is even more critical in bipedal gaits (Kuo, 1999; MacKinnon and Winter, 1993). Humans prefer a narrow step width in running gait, which may make maintaining lateral balance more challenging (Arellano and Kram, 2011). From a broader perspective, although the majority of research on human gait has focused on the sagittal plane, movements in the frontal plane have particular importance for balance control (Krebs et al., 2002). In that respect, lateral stability in running gait needs further investigation.

In this study, we analyzed human running in 3D, based on the VPP concept. We show that by choosing an appropriate coordinate frame, the VPP concept can also explain the relationship between GRF, CoM and CoP in the frontal plane and consequently lateral balance control in human running experiments. Further, we explored the relationship between VPP position and motion speed. Then, we extended the template SLIP-based model of running with a trunk segment (called TSLIP) to employ the VPP to stabilize 3D running gait. The simulation and experimental results were compared, revealing the potential for future applications of this method to develop agile bipedal robots or assistive devices supporting human balance in running gaits.

In the first part of this study, we analyzed human running gait with the VPP landscape in the sagittal and the frontal plane. Our gait analysis involved humans running on a treadmill over a range of speeds. In the second part, we performed a simulation study using a template model to support our experimental findings in the 3D environment. We compared the results of our simulation with those from the human experiment to investigate the ability of our model to predict human kinematics and kinetics during running gait. Also, we examined the robustness of our simulation against perturbation in CoM velocity (representing an external push) when the position of the VPP was considered as a control target.

This section describes the experimental data and explains how to verify the VPP in steady-state running gaits in the frontal and sagittal planes. Then, we present the method of applying VPP for control to generate stable running in a template-based 3D model.

For studying VPP in human running, we borrowed the dataset from Hamner and Delp (2013). The experiments involved treadmill running of 10 male individuals (age: 29±5 years, height: 177±4 cm, mass: 70.9±7 kg) at four running speeds. Each subject was an experienced long-distance runner who reported running at least 50 km per week. The positions of 54 reflective markers were measured at 100 Hz using eight Vicon MX40+ cameras. In addition to motion capture data, GRFs and moments were measured at 1000 Hz using a Bertec Corporation instrumented treadmill. Each subject ran between 15 and 20 steps at each speed. Marker positions and GRFs were filtered by a 15 Hz low-pass filter with a zero-phase fourth-order Butterworth filter and a critically damped filter, respectively. The Stanford University Institutional Review Board approved the experimental protocol and subjects provided informed consent to participate. More details about the dataset can be found in Hamner and Delp (2013).

To analyze human balance control, we utilized a 12 segment, 29 degrees of freedom generic musculoskeletal model (Hamner et al., 2010) in OpenSim software (Delp et al., 2007). First, the model was scaled to match each subject's anthropometry based on experimentally measured markers placed on anatomical landmarks. Then, joint angles were calculated using an inverse kinematic (IK) tool. The output of the IK tool together with the GRF data was employed to compute the CoM and the VPP.

The first step for finding the VPP is choosing an appropriate (reference) coordinate frame with respect to the human body. We define the reference frame in the sagittal plane with the origin at the body CoM while the upper-body direction (the hip to shoulder vector) determines the vertical axis (Maus et al., 2010) (Fig. 1A). Also, we skipped the data for the first 25% of the stance phase to remove the impact effect. After transferring the GRF data from world frame to the defined reference frame, the VPP was calculated as instructed in Maus et al. (2010).

The reference frame in the frontal plane is also centered at body CoM, whereas the pelvis orientation defines the horizontal axis (Firouzi et al., 2019) (Fig. 1B). We tested several different orientations for the reference frame in the frontal plane (e.g. lumbar line) and the pelvis orientation provided the simplest and most accurate description of the GRF direction. Previous studies (MacKinnon and Winter, 1993) also presented the significant contribution of the pelvis to lateral balance control in the single support phase.

To predict human balance control in running, we developed a 3D template model by extending the SLIP model with an additional trunk and damper. This model consists of a massless parallel spring–damper arrangement representing the virtual leg, beside a rigid trunk for the upper body with mass m and moment of inertia J (see Fig. 1C,D). Generating stable running with an upright trunk (90 deg) using TSLIP (trunk+SLIP) is possible (Sharbafi and Seyfarth, 2014). However, to mimic human inclined upper-body posture with the template model, injected energy in the trunk should be dissipated. For this, we add a damper to the leg.

In our model, the virtual hip joint represents a point in the middle of the human left and right hip joints. The virtual leg is a segment that connects the virtual hip to the foot. The trunk orientation, defined by a line connecting the hip to the CoM, is characterized by the corresponding angles in the sagittal (φ_s), frontal (φ_f) and horizontal (φ_h) planes. By controlling hip torque in the sagittal and frontal planes in order to pass GRFs through VPP in these planes and choosing an appropriate leg adjustment strategy, stable human-like running can be achieved in 3D space. The equations of motion are presented in the Appendix.

The VPP model has already been utilized to predict human-like hip torque and posture control in different gaits in the sagittal plane (Sharbafi et al., 2013a,b; Maus et al., 2008). Here, we use this concept for posture control in 3D running using the template-based model described in ‘Simulation model’, above, and in Fig. 1C,D. In the following, we first describe the VPP-based posture control in 2D, and then the extension to 3D. Finally, the adaptation of the VPP position using an optimal and robust controller is explained.

It is worth mentioning that the two VPP points in the frontal and sagittal planes are not related as they are controlled by different actuators. As a result, the 3D VPPC is designed by separate adjustment of the VPP positions in the two mentioned planes using the corresponding control parameters (γ and r).

As described in Sharbafi et al. (2013a,b), the VPPC can generate stable running and hopping in 2D. However, to provide robustness against perturbations, the VPP position needs to be adapted. This controller adaptation is also a prerequisite for generating a stable and robust running gait in 3D. Inspired by our previous study (Sharbafi et al., 2013a,b), we utilized a linear quadratic regulator (LQR) as an optimal and robust controller to adjust the VPP position once per step. In this higher control level, the new VPP position is computed at each apex (highest CoM height during one step) for the next stance phase (see Fig. 2).

In the coordinate frame shown in Fig. 1D, the 3D template model configuration in the flight phase can be given by the position of the CoM and the orientation of the trunk denoted by the kinematic vector X=[x,y,z,φ_s,φ_f,φ_h]^T. By detecting the foot contact knowing the attack angle at touchdown, vector X suffices to determine body configuration in the stance phase. Thus, the system state can be defined by S=[X^T, X^T]^T. Because of the definition of the apex, is equal to zero, and it can be omitted from the state vector. In steady-state running, the horizontal position of the CoM (x) is considered irrelevant and can be omitted. Therefore, the reduced state space at the apex is given by ⁠.

The importance of the error and energy consumption can be tuned by setting the weighting matrices R and Q.

This section first investigates the existence of the VPP in human running in the sagittal and frontal planes and the variations concerning gait speed. Then, the simulation results of VPP-based balance control in 3D running are presented.

To verify the existence of VPP at each experimental trial, we found the VPP point as described in Materials and Methods, ‘Experimental data analysis’, above, and calculated the R² value from Eqn 1. As mentioned above, human balance control matches the VPP concept if R²≥70%. Further, we analyzed the VPP position variation versus running speed.

To find the VPP, the GRF vectors were transferred to a CoM-centered coordinate frame that was aligned with the upper-body orientation (Fig. 1A). Fig. 3 shows the GRF and the VPP point for one sample running step at 4 m s⁻¹. To remove the impact effect, the GRF data in the first 25% of the stance phase (shown by dashed lines in Fig. 3A) were not considered to calculate the VPP. Fig. 3B depicts the CoM, the VPP and the GRF vectors, plotted from the CoP in the selected coordinate frame. A clear intersection point at the VPP (green circle) is visible in this sample step. In this case, the VPP is placed below the CoM (red circle).

The position of the VPP in the selected coordinate frame and R² coefficients are depicted for each step in Fig. 4. Light and dark green illustrate the corresponding information for the left and right legs, respectively. Except for a few samples, the VPP position in the sagittal plane was below the CoM for all subjects at different speeds. The coefficient of R² for all subjects was higher than 90%, which indicates that the VPP can successfully predict experimental GRFs. Also, there was no significant difference between the right and left legs for the VPP position (P=0.42).

VPP position is not fixed at each speed, but changes around a line with a negative slope, shown with the same colors used for drawing the VPPs of different legs. The identified regions for the VPP positions at various speeds were comparable (Table 1). The slope of the fitted lines is reduced by increasing the gait speed. On average, the position of the VPP is in front of the CoM at all speeds. This property is also observed in the average CoP position during running gait (see Fig. 3B as an example). Mean±s.d. VPP location and R² values are presented in Table 1. On average, the VPP in the sagittal plane was located about 19 cm below and about 1.2 cm anterior to the CoM. Also, horizontal and vertical deviation of the VPP from the CoM increased by raising the gait speed, except from 4 to 5 m s⁻¹.

For investigating VPP in the frontal plane, the GRFs were transferred to a CoM-centered coordinate frame aligned based on the pelvis orientation as shown in Fig. 1B. As expected, the VPP position in the frontal plane was different for the left and right legs (see two sample steps of running at 4 m s⁻¹ in Fig. 5). The focused intersection point of GRFs in Fig. 5 shows that the VPP concept can be extended to the frontal plane by choosing an appropriate coordinate frame. Unlike the sagittal plane, the VPP in the frontal plane was placed above the COM and on the right (left) side of the CoM during the right (left) foot contact. Also, the distance of the VPP to the CoM was not significantly different between the right and left leg (P=0.47).

The VPP positions with R² values above 70% are shown in Fig. 6A. The distribution of the VPPs for the left and right steps in the frontal plane was almost symmetrical. To quantify this qualitative observation, we depicted the best linear approximation in this figure (dark and light green lines for right and left leg, respectively). Statistical comparison between different speeds is reported in Table 1. On average, the VPP in the frontal plane was located about 35 cm higher than the CoM with about 5 cm horizontal distance on the same side of the stance foot. To speed up the running gait, the VPP moves downward (towards the CoM) and horizontally farther from the CoM.

Fig. 6B illustrates the R² values for both legs of different subjects at four different speeds. Out of 80 different cases (2 legs×4 speeds×10 subjects), VPP existed in 70 cases in the frontal plane. The remaining 12.5% of the experimental trials in which VPP was not clearly identified were related to subjects s1, s6 and s7. For s7, clear VPPs with R² above 90% (instead of 70%) were found in both legs for all trials except the left leg steps in running at 5 m s⁻¹. VPPs with R² values above 70% could be identified in both legs in most running steps of subject s6 at 2 and 3 m s⁻¹, and in right leg steps at 4 m s⁻¹. It is fair to say that subject s1's running patterns did not show meaningful VPPs in most trials.

In some steps (especially for subject s1), R² had a negative value. A negative value for R² can be achieved when GRFs are almost parallel. These unacceptable VPPs are not shown in Fig. 6, but the trials experiencing such conditions are illustrated by the pink regions in Fig. 6B. The average R² values for all 10 subjects are presented in Table 1. We also calculated the mean R² values at each speed without considering subjects s1 and s6, as shown in Table 1. This new calculation supports the existence of VPP in the frontal plane for 8 subjects out of 10.

In this section, we present our simulation results in 3D environments.

The model was able to generate stable running at different speeds. Here, we detail the outcomes for running at 4 m s⁻¹. We set the VPPC control parameter to r_s=22 cm and γ_s=−174 deg for the VPP position below the CoM in the sagittal plane and r_f=30 cm and |γ_f|=4 deg for the VPP position above the CoM in the frontal plane when the right leg is in the stance phase. Indeed, the sign of γ_f changes by changing the supporting leg (e.g. from right to left) as observed in Fig. 6 for human running. While selecting VPP in the frontal plane, we need to consider the mapping between the model virtual hip and the real hip in humans. For example, by increasing the vertical position of the VPP in humans, the horizontal distance from VPP to hip decreases. Therefore, to compensate for the increase in r_f in the frontal plane, we need to decrease γ_f. Also, we applied a perturbation at the touch-down moment to the CoM speed to investigate the robustness of the system. As stated in Materials and Methods, ‘Robust VPPC’, above, event-based control of the VPP position allows us to improve the system behavior and robustness. Controllability of the pair [J_S,J_U] is the only necessary condition easily met in this problem. In addition, using D-LQR for the selection of the gain vector K (Eqn 8), we can use the weight matrix Q and R in Eqn 6 to determine the importance of the state variable error and control inputs. We set Q=diagonal matrix [10,1,1,1,5,1,1,1,1,1] to devote higher importance to track the forward speed and trunk angle in the sagittal plane because of the significance of keeping the speed and balance control, especially in the face of the introduced perturbation. Also, the larger weight of the states in the sagittal plane relates to the unstable behavior of the negative VPP (explained in Discussion, ‘Experiments versus simulation’, below). By setting R=diagonal matrix [1,1,1,1] we equalize the importance of all control inputs.

Fig. 7A shows the trunk angle, and forward and medio-lateral speed of the CoM, as well as the VPP parameters (angle and radius), when perturbation occurs in the sagittal plane. Here, perturbation is defined as a sudden 10% increase in the forward speed (+0.4 m s⁻¹). Before applying the perturbation, the system is stable, and the model predicts running at 4 m s⁻¹. Before the perturbation, the mean trunk inclination in the sagittal plane is about 5 deg, and it oscillates ±4 deg in the frontal plane. After the perturbation, the trunk bends backward, and the controller decreases the VPP angle to compensate for the perturbation. According to Fig. 7A, there is a clear correlation between the trunk's bending angle and the VPP angle after perturbation. The perturbation in the sagittal plane also affects trunk orientation in the frontal plane as a result of the coupling between two planes. By changing the VPP parameters to recover from perturbation, the D-LQR controller can stabilize the system after a few steps, and system states converge to the states before perturbation.

The robustness of the system against lateral perturbation was investigated by examining the reaction of the VPPC to a sudden increase in the medio-lateral speed (100% of maximum medio-lateral speed added, +0.05 m s⁻¹). Fig. 7B demonstrates that the perturbation has a negligible impact on the forward speed and trunk angle in the frontal plane, while the medio-lateral speed and the trunk angle in the sagittal plane need a few steps for recovery.

We selected the VPP concept to analyze postural stability in running in a 3D space. The quality of the identified VPP was utilized as a measure to evaluate the VPP method in predicting human running. Moreover, replicating the experimental results in a template-based simulation model confirmed the proposed method's functionality. In addition to discussing the presented outcomes, a detailed comparison between simulation and human running is presented below.

Previous studies reveal that appropriate control of whole-body angular momentum (WBAM) is critical to maintaining dynamic balance during locomotion (Begue et al., 2019). WBAM can be computed by integrating the net external moment produced by the measured GRFs around the CoM. Experimental evidence shows that WBAM remains small during steady-state human walking (Herr and Popovic, 2008). Further, the prediction ability of WBAM regulation decreases as the locomotion task becomes more dynamic (from standing and walking to running) (Popovic et al., 2002). As a more general postural control measure, the VPP concept not only can predict WBAM variations (Gruben and Boehm, 2012) but also can describe human balance control in more dynamic locomotion tasks. In this regard, we analyzed human running and the identified VPP in the sagittal and frontal planes at different speeds.

As stated in Materials and Methods, the R² value indicates the amount of agreement between experimentally measured GRFs and force vectors predicted by the VPP method. The distinguished VPP (R²≈98%) in the sagittal plane demonstrates the remarkable matching of the experimental measurements and the predicted posture control. Similar high R² values were reported by Drama et al. (2020) at 5 m s⁻¹ level ground running. Here, the VPP was placed below the CoM (negative VPP) at all speeds (except in rare samples), which means the WBAM component in the sagittal plane has a negative mean value. This result is in line with previous findings showing that WBAM drops in the first half of the stance phase and rises in the second half of the stance phase (Sepp et al., 2019; Hinrichs, 1987). Also, our results confirm the negative VPP found in Drama et al. (2020). The negative VPP creates GRFs that pass behind the CoM in the first half of the stance phase, exerting a flexion moment on the body, which causes the drop in WBAM. The subsequent rise in WBAM during the second half of the stance phase results from the GRF vectors passing the negative VPP and, consequently, in front of the CoM (Hinrichs, 1987). The negative VPP increases the acceleration and deceleration in the fore–aft direction compared with the positive VPP, observed in human walking (Drama and Badri-Spröwitz, 2019; Maus et al., 2010). Furthermore, a simulation study shows that the negative VPP reduces the leg loading and net hip work (Drama and Badri-Spröwitz, 2019).

The experimental data (Table 1) shows that VPP in the sagittal plane is located anterior to the CoM at all speeds (P<0.001). This observation can be explained using the trunk inclination and WBAM behavior in the sagittal plane. As the net external moment acting on the CoM in the flight phase is zero, WBAM remains constant. Hence, for a periodic motion such as running (considering left and right leg symmetry in the sagittal plane), it is necessary to create a restoring extension moment in the second half of the stance phase to compensate for the flexion moment in the first half of the stance phase. In other words, the external moment resulting from the GRF must become balanced around the CoM (Fig. 3). The counterbalance results in forward inclination of the upper body.

The horizontal deviation of the VPP from the CoM increases with running speed as a faster gait needs a more inclined trunk. In addition to increasing upper-body inclination, distancing VPP from the CoM could increase both braking and propulsive forces. To further analyze the connection between the VPP position and the gait speed, the general trend of calculated VPPs was approximated by a linear relationship. As shown in Fig. 4 and Table 1, the faster the running speed, the lower the slope of the fitted line. This means that for faster motions, the horizontal adjustment of the VPP is employed more than the vertical adjustment and vice versa.

WBAM in the frontal plane has a greater range than that in the sagittal and horizontal planes during running (Sepp et al., 2019; Hinrichs, 1987). According to Table 1, the mean R² value of the VPP in the frontal plane is about 80%, meaning that the VPP concept is also valid for analyzing lateral balancing. The existence of the VPP could not be verified for 2 out of 10 subjects. Skipping the data from these two subjects, the average R² will be about 90%, which concurs with VPP-based lateral balance control for 80% of the dataset. Unlike in the sagittal plane, positive VPP (above the CoM) was found in the frontal plane, at all speeds, except for sporadic samples. Also, the VPP was placed on the same side of the CoM as the support leg. Compared with the CMP (Herr and Popovic, 2008), this feature of VPP placement helps better predict WBAM and balance external momentum around the CoM at each gait cycle in the frontal plane (Hinrichs, 1987). WBAM in the frontal plane rises when the left leg is in contact with the ground and drops when the right leg is in contact with the ground. In other words, the runner's body axis always tends to lean toward the swing leg. It seems that the VPP position could be a compromise between hip torque minimization and WBAM minimization.

When increasing the speed up to 4 m s⁻¹, the VPP tends to move towards the stance leg hip joint (equivalently, increasing the distance from the COM). This pattern in VPP placement was observed at speeds up to the preferred running speed (4 m s⁻¹ as stated in Kong et al., 2012). Thus, from slow to moderate running, the priority of hip abduction torque minimization is higher than WBAM minimization. However, a faster gait enforces a shorter step duration and, as a result, the increase in WBAM does not yield higher upper-body inclination. Roughly speaking, tuning the distance between VPP and CoM can be used to control WBAM by adjusting the lever arm of the GRF around the CoM. Therefore, the upper body keeps balance by fast switching between two unstable modes (tending to fall toward the swing leg) (Firouzi et al., 2019). Finally, increasing the horizontal distance between the VPP and the body's CoM decreases the lever arm of the GRF around the knee joint center in the frontal plane and consequently reduces the varus moment. This may be a mechanism to reduce injury risk during running (Powers, 2010).

The VPP below the CoM (negative VPP), found in human experiments in the sagittal plane, represents an inverted pendulum behavior that is inherently unstable (Firouzi et al., 2019; Müller et al., 2017). Therefore, such a negative VPP is not robust against even small perturbation except using an adaptation method (e.g. D-LQR). This finding is supported by the simulation results (Fig. 7), showing that the perturbation impact on the trunk angle is more significant in the sagittal plane than in the frontal plane. In this study, we used perturbation to investigate the robustness of our simulation when the position of the VPP was considered as a control target. If the VPP is the key for balance control, it should also have a solution for perturbation recovery. Our simulation results show that the 3D model can easily recover from perturbation using VPP position readjustment. These results are in line with the finding of Drama et al. (2020) that the horizontal position of the VPP in level running differs from that in perturbed running. This observation also reveals the potential of VPP adjustment to be used in robot posture control.

The simulation results showed that for stable running with a slightly inclined trunk, the VPP should be placed anterior to the CoM, which complies with the experimental results. Furthermore, to recover from the perturbation, the VPP shifts backward (in the selected coordinate frame), which decreases the trunk inclination (Fig. 7A).

We used the previously mentioned 3D running simulations at 4 m s⁻¹ to further validate the ability of the VPP concept and the proposed model in predicting human balance control and mimicking human running kinematic and kinetic behavior. Fig. 8, right, shows the trunk angle in the sagittal and frontal plane for simulations and experiments. The mean trunk inclination in the sagittal plane for the experimental data nicely matches our simulation results. However, the simulated trunk angle pattern differs from the measured angle patterns at the beginning of the stance phase. The observed forward movement in the early stance phase which is also consistent with previous studies (Thorstensson et al., 1984; Drama and Badri-Spröwitz, 2019) resulted from the negative VPP. The difference between the simulation and experiment data results from swing phase modeling imprecision; as in our template model, the swing phase is represented by a ballistic motion considering a constant angular velocity of the trunk. In contrast, the human trunk moves backward in the first half of the swing phase, followed by a forward movement. It is important to say that the human-like trunk movement in the stance phase is replicated by the model with a short delay, which is required to compensate for the missing trunk movement in the previous swing phase. One way to improve the similarity between the model and the human experiment is to add mass to the leg of our template model as implemented in the XTSLIP model presented in Sharbafi et al. (2013a,b). The trunk lateral angle (in the frontal plane) of the simulation model resembled the experimental data in the stance phase. Also, the mean pelvis inclination in the stance phase is quite similar for the experiment and simulation. The XTSLIP model may also predict upper body returning movement in the flight phase, which is missing in our model.

Fig. 8, right, shows hip torque in the sagittal (extension/flexion) and frontal (abduction/adduction) planes for both simulations and experiments. In human running, the stance phase starts with a small (negative) flexion hip torque followed by a large extension torque to support the desired forward movement of the body. The model predicts a larger hip flexion than what was found in the experiment to compensate for the swing phase movement. Then, the forward motion is generated by an extension torque which is larger and longer than the preceding flexion torque (similar to the experiment). As our model can control the trunk angle only in the stance phase, the hip joint first needs to resist the trunk's backward motion and then compensate for flexion torque exerted by the trunk weight. The simulation model can predict the hip adduction torque (in the frontal plane) in the stance phase.

In this study, we skipped the data at the first 25% of the stance phase to remove the impact effect. Impact duration differed among directions and foot-strike conditions. For example, impact phase duration varies from 15% to 30% of stance phase as a result of the running style (e.g. fore-foot strike, heel-strike or heel-toe running) (Nordin et al., 2017). In our dataset, most of the subjects identified as mid-foot to heel-strike runners which show longer impact duration (for more information see Hamner and Delp, 2013). The impact effect might be compensated for by intrinsic compliant element behavior in the leg, which does not undervalue the VPP-based posture control in most of the stance period. This limitation of our study can be further investigated in the future.

Moreover, we used an existing dataset (collected by another research group) that did not include women. Although the VPP concept was observed in locomotion in both men and women in the previous studies (Gruben and Boehm, 2012; Drama et al., 2020), excluding women in this study is a limitation that needs to be addressed in future studies.

In this study, we investigated the VPP in human running gait at a range of speeds in both the sagittal and frontal planes, and analyzed the implication of the observed VPP to balance control using a 3D template model. Our findings show that GRFs intersect near a point below the CoM in the sagittal plane and above the CoM in the frontal plane at all speeds. Also, replicating the kinematic and kinetic behavior of the experiments with the template-based simulation model supports the applicability of the VPP concept in predicting human posture control in running. Further, we showed that VPP adaptation at each step (e.g. using a D-LQR) can stabilize the gait and warrants the robustness against perturbations. Our synthesizing method can be used for developing model-based control approaches in humanoid robots and for gait assistance.

We are thankful for constructive discussions with Andre Seyfarth on the description of the results. The authors also thank Omid Mohseni for his support in revising the paper.