Whole-Body Dynamic Telelocomotion: A Step-to-Step Dynamics Approach to Human Walking Reference Generation

The H-LIPM [21], an extension of the LIPM, is a two-domain hybrid system that encapsulates the hybrid dynamics of point-foot walking. The two domains of walking are a single support phase (SSP) and a double support phase (DSP). The domain-specific continuous dynamics of the H-LIPM are governed by

where $x$ is the center of mass (CoM) position relative to the stance foot, $\omega=\sqrt{\frac{g}{h}}$ is the natural frequency under the gravity field $g$, and $h$ is the CoM height. The H-LIPM describes the discrete transitions between SSP (S) and DSP (D) through smooth hybrid system reset maps, which are defined as follows:

where the control input $\ell$ is the step length and the superscripts $+$ and $-$ denote CoM states after and before each discrete state reset, respectively. The closed-form solution of the hybrid system is derived by solving Eqs. (1) - (2). This solution defines the time evolution of the H-LIPM dynamics, as shown below:

where time in each phase is denoted by $t\in[0,T_{SSP}]$ or $t\in[0,T_{DSP}]$. $T_{SSP}$ and $T_{DSP}$ are the duration of SSP and DSP, respectively. The constants $c_{1}=\frac{1}{2}(x^{+}+\frac{1}{\omega}\dot{x}^{+})$ and $c_{2}=\frac{1}{2}(x^{+}-\frac{1}{\omega}\dot{x}^{+})$ are constants of integration.

A mapping of human stepping motion to desired walking motion was introduced in [15]. The two assumptions made are that 1) the desired human walking gait corresponds to a stable walking gait of the canonical LIPM and 2) the human pilot’s (h) stepping frequency and CoM position along our human-machine interface (HMI) are sufficient for the pilot to intuitively generate these HWR ($\overline{\textsc{h}}$) gaits.

From a technical perspective, these two assumptions allow mapping the human pilot’s stepping frequency, $T_{s}^{-1}$, and CoM position, $x_{\textsc{h}}$, to a stable walking gait. The walking gait chosen to represent nominal human walking is characterized as satisfying $\left({x}^{+},\dot{x}^{+}\right)=\left(-{x}^{-},\dot{x}^{-}\right)$ during a single step. This one-step periodic orbit, known as a P1 orbit, can be generated in real-time from $T_{s}$ and $x_{\textsc{h}}$. The core idea is to map $x_{\textsc{h}}\rightarrow\xi_{x\overline{\textsc{h}}}^{-}$, i.e., the human CoM position along the HMI equates to some desired pre-impact (end-of-step) value of the walking reference’s divergent component of motion (DCM). The DCM, defined $\xi_{x}:=x+\frac{\dot{x}}{\omega}$, is the key state tracked in our telelocomotion framework to synchronize human and robot locomotion as it captures the unstable component of motion of the pendulum dynamics [9].

It can be shown that $\xi_{x}^{-}$ and $T_{s}$ can completely characterize a P1 orbit given some necessary and sufficient conditions are satisfied. Full details can be found in [15]. In general, from human stepping motion, the desired pre-impact CoM states of a stable walking gait can be computed as following:

where $\sigma_{1\textsc{h}}$ is the orbital slope of the human LIPM, defined $\sigma_{1\textsc{h}}:=\omega_{\textsc{h}}\>\coth\Big(\frac{T_{s}}{2}\omega_{\textsc{h}}\Big{missing}).$ We can extend this mapping to stable walking gaits of the H-LIPM.

The key idea is to abstract the HWR as a H-LIPM with a control policy that captures the desired human pilot’s walking behavior while obeying the step-to-step dynamics of walking. In our telelocomotion framework, this will enhance the robot controller’s capability to consistently track the motion of the HWR when scaled to robot proportions.

The approach in [15] forced the pilot to learn the robot’s dynamics of transitioning between different stable walking gaits. The HWR was assumed to be a LIPM whose dynamics always evolved along a P1 orbit. In order to aid the pilot, a haptic virtual spring was implemented to give them an acceleration sensation similar to that of the HWR. This allowed for successful HWR motion generation and dynamic synchronized walking between human and bipedal robot, but it was not sustainable for prolonged periods.

The non-trivial ability to traverse between these stable walking gaits can be more accurately explained by the step-to-step (S2S) dynamics of bipedal walking [20]. The S2S dynamics of walking are perfectly highlighted by the H-LIPM. As shown in Eq. (3), under the H-LIPM approximation, the dynamics of walking are purely passive throughout the continuous domains. From Eq. (2b), the control input, $\ell$, is applied only discretely. This motivates viewing the dynamics of walking at the step level. The result is a discrete linear dynamical system,

that maps step $k$ pre-impact CoM states, $\mathbf{x}_{k}$, to step $k+1$ pre-impact CoM states, $\mathbf{x}_{k+1}$, via the step length $u_{k}=\ell$. The state transition matrix, $A$, and control input mapping vector, $B$, are defined by combining Eqs. (2) - (3). Thus, stabilization of $\mathbf{x}$ to a desired $\mathbf{x}^{*}$ in Eq. (5) can be enforced with feedback control. The case of particular interest is when $\mathbf{x}^{*}$ corresponds to a P1 orbit. In this case, the deadbeat controller from [20] can optimally drive $\mathbf{e}:=(\mathbf{x}-\mathbf{x^{*}})\rightarrow\mathbf{0}$ in two steps:

In this work, Eq. (6) is the control policy of the HWR. The control objective is to achieve the desired human locomotion strategy captured by Eq. (4) on the dynamics of the HWR. The dynamics of the HWR evolve according to Eq. (3). By modeling the HWR as a H-LIPM with step-to-step dynamics given by Eq. (5), the hybrid and underactuated nature of walking is considered in the motion generation of the HWR as outlined in Algorithm 1.

The 15.8 kg, 16 degree-of-freedom (DoF) bipedal robot Tello has the following full-body dynamics:

where $q$ is the robot generalized coordinates vector, $M$ is the mass matrix, $C$ is the Coriolis matrix, $G$ is the gravity vector, $S$ is the actuated DoF selection matrix, $J_{c}$ is the contact Jacobian matrix, $\tau$ is the joint torque vector, and $f$ is the ground reaction wrench vector.

A popular approximation of Eq. (7) is the single rigid-body model (SRBM) [23, 24]. The SRBM is a simplified model of legged robots with low centroidal inertia isotropy (CII) [22], which neglects the inertial effects introduced by the robot’s limbs. Thus, the torso’s translational and rotational dynamics are only considered. The states of the torso (SRB) are defined as $s:=\left[p,\dot{p},R,^{B}\omega\right]^{\top}$, where $p$ is the CoM position vector, $R$ the rotation matrix of the body frame relative to the world frame, and ${}^{B}\omega$ is the angular velocity vector expressed in body frame, and $\{B\}$ is the body frame. The SRBM state-space dynamics are the following:

where $m$ is the robot mass, ${}^{B}I$ is the torso inertia matrix in the body frame, $a_{g}$ is the gravity vector, $F$ and $\tilde{\tau}$ are translational and rotational components of the net wrench acting on the CoM, respectively, and $\hat{\left(\cdot\right)}$ is the operator that maps an element of $\mathbb{R}^{3}$ to the 3D rotation group $SO(3)$. The net wrench, $\mathcal{F}$, arises from the ground reaction forces (GRFs) generated at active contact points, i.e.,

where $n$ is the number of active contact points, $\mathbb{I}$ is the identity matrix, $r_{i}$ represents the vector from the CoM to the $i$-th active contact point, while $\tilde{f}_{i}$ denotes the GRF vector at that contact point. For Tello, which has line feet, we assume two contact points at the toe and heel of each foot [24].

The balance controller computes the GRF vectors required to reactively track a desired SRBM trajectory. A PD task-space controller is implemented,

to compute the desired net wrench, $\mathcal{F}^{d}$, to track the desired task-space trajectory, $q_{s}^{d}$ and $\dot{q}_{s}^{d}$. The task-space coordinate vector, $q_{s}$, of the SRB is defined as $q_{s}:=[p,\Theta]^{\top}$, where $\Theta$ is the euler-angle representation of $R$. Appropriate coordinate transformations are made to construct $q_{s}$ and $\dot{q}_{s}$ from $s$. $K_{p}$ and $K_{d}$ are diagonal proportional and derivative gain matrices, respectively.

Once $\mathcal{F}^{d}$ is determined using Eq. (10), a force distribution problem is formulated as a quadratic program (QP). This QP aims to calculate the corresponding GRF vectors at the active contact points, satisfying Eq. (9). Furthermore, the GRF vectors undergo projections from task-space to both the friction cone and the motor-space, where linear inequality actuation constraints are enforced. This accounts for both the parallel actuation scheme of Tello [22] and the effects of friction. GRF vectors are also constrained to not exert a pulling force on the ground. Thus, the actuation-aware force distribution QP takes the following form:

where $\tilde{f}$ is the collection of GRF vectors at active contact points, $\tilde{f}_{0}$ is the previous value of $\tilde{f}$, $w_{1}$ and $w_{2}$ are scalar weighting factors, $R_{BW}$ is a matrix of rotation matrices $R^{\top}$ that map $\tilde{f}$ from the world frame to the body frame, $J_{JT}$ is a matrix of contact Jacobians that map task-space forces to joint torques, $J_{MJ}$ is a matrix of topology Jacobians that map joint torques to motor torques, $\tau_{m}$ is a vector of configuration and speed-dependent motor torque limits, $q_{j}$ and $\dot{q}_{j}$ are leg joint positions and velocities, respectively, and $\mu$ is the friction coefficient. The second term of the cost function in Eq. (11) acts as a solution filter.

Finally, joint torques for legs in stance are set using the SRBM approximation of Eq. (7). However, to maintain the task-space configuration of the robot feet, joint-space posture control is appended to the optimization-based balance controller from Eq. (11). Consequently, the final expression for the joint torques of legs in stance, $\tau_{st}$, is the following:

where $\tau_{po}$ is vector of posture control joint torques. A sigmoid function is utilized to smoothly transition between stance-leg joint torques and swing-leg joint torques. The computation of $\tau$ from Eq. (7) is the summation of smoothed stance-leg and swing-leg joint torques.

During SSP, when one of the robot’s legs is in swing, corresponding joint torques cannot be computed using Eq. (11). This is because the leg is no longer in contact with the ground and as a result a GRF cannot be generated. Instead, the goal of the swing-leg controller is to map the robot’s foot position at the start of SSP to a location by the end of SSP that achieves the desired step placement.

This mapping of the robot’s foot during SSP is accomplished via smooth task-space trajectories similar to in [19]. On the transverse plane, specifically the $x$ or $y$ axis, the function,

maps the robot’s initial foot position, $p^{i}_{sw}$, to a final desired foot position, $p^{f}_{sw}$, along each axis. The phase variable, denoted as $s\in[0,1]$, represents the normalized progression of time during a step. On the other hand, $z$-position trajectories are defined differently. The objective is to re-establish contact with the ground at time $s=1$. The following function,

characterized by a maximum step height, $z_{cl}$, attains the intended foot z-position motion.

In this work, swing-leg joint torques, $\tau_{sw}$, are computed using joint-space PD control. First, utilizing inverse kinematics, Eqs. (13) - (14), along with their derivatives, are transformed into desired joint positions, $q^{d}_{j}$, and corresponding desired joint velocities, $\dot{q}^{d}_{j}$, trajectories. Then, the joint-space trajectories are tracked using the PD control law,

where $K^{sw}_{p}$ and $K^{sw}_{d}$ are diagonal proportional and derivative gain matrices, respectively.

The robot controller, composed of the optimization-based balance controller and swing-leg controller, is integrated into our whole-body dynamic telelocomotion framework (Fig. 4). Although the robot controller is capable of operating independently, it was designed to be purely reactive, with locomotion planning being entrusted to the human pilot.

The human locomotion strategy is represented by a 2D LIP trajectory. This trajectory is constructed using dynamic human motion data in the frontal plane, combined with the human walking reference generated from human stepping motion data along the sagittal plane. The trajectory is then scaled to robot proportions. The goal of the robot controller is to reproduce the resulting robot 2D LIP trajectory at the full-body (3D) dynamics level.

The reproduction of the robot’s 2D LIP trajectory is described next. The $x$-$y$ components of the desired robot CoM wrench from Eq. (11) are given by robot forces that track the desired scaled human motion [9]. Additionally, the LIPM assumptions of constant CoM height and the omission of angular dynamics are enforced by Eq. (10). The $x$-$y$ swing-leg trajectory from Eq. (13) is updated to keep dynamic synchronization between the human pilot and the robot at step transitions. Specifically, in the $x$-direction, the final desired robot foot position maintains tracking of the scaled HWR at the commencement of the next step. In the $y$-direction, the final desired robot foot position maintains equal scaled feet width between the human and robot.

The main challenge in tracking the desired robot 2D LIP trajectory is that the single support time $T_{SSP}$ is not known a priori. $T_{SSP}$ is normally fixed in bipedal walking controllers. Thus, to resolve this issue we leverage the consistency of human motion to implement a data-driven adaptive step timing strategy. In particular, we assume human swing-foot z-position motion approximately follows Eq. (14), parameterized by $s$ (normalized SSP time) and $z_{cl}$. During each step, we employ a damped least-squares method to continuously fit human swing-foot z-position data to Eq. (14). This allows the pilot to control the robot’s step height and vary the robot’s stepping frequency. This ensures the pilot and the robot will touch-down synchronously. The updated estimate of SSP duration is used to update the HWR and the $x$-$y$ swing-leg trajectories for more robust telelocomotion.

During telelocomotion simulation experiments, a trained human pilot utilizes our HMI to dynamically teleoperate the simulated bipedal robot Tello, whose dynamics and corresponding controller are running in the simulation PC in real-time. Communication between the HMI and the simulation PC is established via UDP. As shown in Fig. 5, the human pilot receives visual feedback through a computer monitor, which displays the rendered simulation. This is critical in both experiments as visual feedback is required for velocity target tracking and walking backwards a specific distance.

As illustrated in Fig. 5, our HMI consists of linear actuators, lower-body leg joysticks, a force plate, and a real-time embedded controller. The linear actuators perform dual functions of measuring the human CoM while simultaneously applying haptic forces to the human’s torso. The lower-body leg joysticks capture and measure spatial information pertaining to the locations of the human feet. The force plate measures the net contact wrench applied by the human. The actuators are controlled and the motion capture data is collected by a cRIO-9082 (National Instruments) computer.

The simulation of the full-body dynamics of the bipedal robot Tello, described by Eq. (7), is executed using the physics engine MuJoCo. The simulation is seamlessly integrated into a unified program alongside the Tello control software. This software is parallelized across nine distinct CPUs, with all threads running at a frequency of 1 kHz.

Using our framework, depicted in Fig. 4, along with visual feedback, a trained human pilot successfully completed both experiments. These experiments serve as a compelling demonstration, showcasing the synchronized locomotion between a human and a bipedal robot in tasks involving velocity target tracking and backward walking. The completion of these tasks show the effectiveness of our HWR motion generation method and our reactive robot controller.

The results of the velocity target experiment are shown in Figs. 6 - 7. Here, the human pilot controls the robot’s locomotion to ensure tracking of the target, as illustrated in Fig. 6(b). Notably, the pilot achieves 6.0 m of synchronized locomotion with the robot, as seen in Fig. 6(d), affirming the viability of sustained telelocomotion using our methods. Importantly, the pilot regulates the robot’s CoM velocity around the target velocity. The key lies in intuitively capturing the human pilot’s locomotion strategy from their stepping motion (Fig. 6(c)). Subsequently, the stepping motion is mapped to a desired walking gait, which serves as the basis for the HWR motion in Fig. 6(a). Indeed, as seen in Fig. 6(a), the abstraction of the HWR as a H-LIPM enables generating dynamically consistent walking references, allowing for smooth transitions between desired walking gaits by considering the step-to-step dynamics of walking. Lastly, the ability of the robot controller to reactively reproduce the desired robot 2D LIP trajectory is highlighted by the normalized DCM tracking performance along the sagittal plane (Fig. 7(a)) and along the frontal plane (Fig. 7(b)). As seen in Fig. 7(c), this leads to consistent and bounded haptic force feedback to the human pilot along the frontal plane. This feedback assists the human pilot in maintaining stepping synchronization with the robot. As discussed in our previous work [15, 9], this is evident from the normalized center of pressure (CoP) synchronization between the two systems, as shown in Fig. 7(d).

The results of the backward walking experiment are shown in Fig. 8. The human pilot dynamically synchronizes their backward stepping with bipedal robot backward walking, covering a distance of 2.0 m (Fig. 8(d)), before proceeding to walk forward to the initial position. Like in the first experiment, the human pilot’s stepping motion is utilized to generate a HWR motion trajectory, as seen in Fig. 8(a) and Fig. 8(c). However, by walking backwards, the pilot commands desired walking gaits characterized by negative orbital energy, as shown in Fig. 8(c). Additionally, to demonstrate the efficacy of our adaptive step time prediction approach, we analyze the first step from the human stepping data. As shown in Fig. 8(b), we continuously monitor the human z-foot position data and fit it to the presumed swing-leg z-trajectory curve. Thus, allowing us to predict both the duration ($T_{SSP}$) and the height of the step ($z_{cl}$). The prediction accuracy improves with the accumulation of more data, closely approximating the actual human z-foot position throughout the entire step, particularly towards the end. This accurate prediction ensures the dynamic synchronization between the human and bipedal robot at impact events and gives the human the capability to regulate the frequency of the synchronized stepping between the two bipedal systems.

The obtained results show promise in achieving robust telelocomotion of humanoid robots, enhancing their potential as physical avatars. These results also represent tangible progress from the work in [15], specifically in successfully realizing sustainable synchronization between human and bipedal robot locomotion in a full-body dynamics simulation. Previously, in a reduced-order dynamics simulation, only 1.28 m of forward walking (0.36 m/s max speed) was achieved. However, it is important to acknowledge the limitations inherent to this work. First, the walking speed of the robot is currently still limited to approximately 0.35 m/s, which, when considering a robot like Tello, corresponds to a human walking speed of 0.5 m/s. While it is possible for the robot to reach faster walking speeds, it becomes challenging for the pilot to sustain telelocomotion at these speeds. The most common failure is the robot lagging behind the HWR due to prolonged periods of DSP by the human that increases the error in normalized DCM between the two systems at the commencement of the next step. Second, it is essential to emphasize that significant training is required for a pilot to reach the demonstrated level of performance. For instance, in the velocity target tracking experiment, the pilot faced difficulties in abruptly stopping the robot from a walking speed of 0.3 m/s. Similarly, transitioning from backward walking to forward walking in the second experiment proved equally challenging. Lastly, it is worth noting that the telelocomotion behaviors enabled by our framework are presently constrained due to the utilization of linear pendulum models to abstract the locomotion of both the human and bipedal robot. To attain a broader spectrum of behaviors, including loco-manipulation tasks, it will be necessary to employ more dynamically rich reduced-order models that can fully exploit the human’s whole-body dynamics strategy.