Bipedal Walking on Constrained Footholds:
Momentum Regulation via Vertical COM Control

Humans can traverse a variety of terrain types with ease, including: flat surfaces, ascending and descending stairs, and discrete stepping stones with height variations. One of the central goals of the bipedal walking robot community is to develop humanoid robots that can locomote on these diverse terrain types with the ease and dynamic stability displayed by humans—this requires locomoting in environments with constrained footholds. In the context of fully-actuated humanoids, such problem can be realized by specifying desired center of pressure (COP) trajectories utilizing reduced order zero-moment-point models [1, 2, 3, 4] under the assumption of ankle actuation. However, these methods do not generalize to walking with limited contact area [5] on the ground, where the COP cannot be changed even in the presence of the ankle actuation. The result is that the robot becomes underactuated [6, 7, 8], thus motivating the study of underactuated walking with constrained footholds.

Underactuated walking is primarily described as a periodic motion where orbital stability is characterized by the eigenvalues of Poincaré map [9, 10] associated with the limit cycle. Using this definition of stability, many approaches for generating stable walking behaviors have used offline optimization [11, 12]. One example is the hybrid zero dynamics (HZD) framework [9, 7, 13] where the “stepping stones” problem has been studied. Previous work with a gait library [14], and control barrier function (CBF) [15, 16, 17] and learning [18] to modulate the foot placement [19] has had some success. However, these methods work favorably for small perturbations of the nominal periodic orbits that are designed to be exponentially stable within their regions of attraction. Despite interpolating among gait library and implementing CBF constraints improved the success rate notably, it perturbs the system away from the HZD manifold and thus breaks the formal guarantee of stability.

A philosophically different approach to generate underactuated walking is based on the approximation of the underactuated dynamics with reduced order models (ROMs) [20, 21, 22, 23, 24, 25], which stabilize the robot COM dynamics largely using foot placement planning. For instance, [24, 26] apply Linear Inverted Pendulum (LIP) [27] model to approximate the underactuated robot dynamics. Stepping controllers [23, 28, 26] that change the desired step sizes are then synthesized in closed-form to stabilize the horizontal COM states. However, these approaches can not be applied for walking with constrained footholds.

The non-periodic nature of walking on constrained footholds, and the need to precisely place the feet on these footholds in a dynamic fashion, prevents the application of existing methods. The goal is to develop a new methodology that enables dynamic constrained-foothold locomotion, with reactive online planning, by combining the strengths of the two aforementioned approaches into a unified framework. The key idea in achieving this goal is to stabilize the underactuated dynamics of the robot—expressed in the COM horizontal coordinates—and control the hybrid dynamics via the discrete dynamics corresponding to the footstrike.

To embed the walking behaviors generated on the underactuated dynamics, we use the underactuated LIP model to approximate the continuous dynamics of the robot during stance, which determines the step duration and desired momentum at the beginning of each step. The desired vertical COM velocity is then realized via online recursive optimization on the vertical COM trajectory. The desired trajectories of the torso configuration and the swing foot are constructed with closed-form polynomials. Finally, a task-space quadratic program based feedback controller is formulated for tracking the desired trajectory. To demonstrate the generality of the proposed approach, it is realized on two robots with different morphology—AMBER and Cassie—wherein both robots can walk stability in various scenarios with constrained footholds.

This section provides the technical preliminaries on the underactuated robot dynamics.

Bipedal walking considered in this work is characterized as a single-domain hybrid control system [3, 7]. The continuous swing phase is modeled as a single support phase (SSP), assuming that the robot is subject to contact holonomic constraints with the stance foot and the ground. The Lagrangian dynamics of the robot can be written as:

where $D(q)\in\mathbb{R}^{n\times n}$, $H(q,\dot{q})\in\mathbb{R}^{n}$, $B\in\mathbb{R}^{n\times m}$ are the inertia matrix, the collection of centrifugal, Coriolis and gravitational forces, and the actuation matrix, respectively. $\tau\in\mathbb{R}^{m}$ is the input torque, $J_{c}(q)\in\mathbb{R}^{n\times h}$ is the Jacobian matrix of the holonomic constraint associated with contact, and $F\in\mathbb{R}^{h}$ is the corresponding constraint wrench.

The impact model assumes instantaneous lift-off of the pre-impact stance foot; thus, contact forces are only applied at the pre-impact swing foot. The angular momentum about the swing foot is conserved. As the stance and swing leg alternates during impact, post-impact $L_{y}^{+}$ is the angular momentum w.r.t. the swing foot before impact. Rearranging Eq. (6) and adding the reset map, we have the following hybrid model for the robot’s underactuated dynamics:

where the dependencies on $q$, $\dot{q}$ are dropped for simplicity.

Remark: This LIP model is the canonical LIP [2] without ankle actuation, and the Hybrid-LIP [26] without the double support phase. It was also used in the capture point approach [23]. The use of the LIP in this paper is most similar to [26, 23, 35], which is for approximating the continuous dynamics of a walking robot.

This section presents our main contribution: an online planning and control methodology to stabilize $\mathcal{HZ}$ for walking on constrained footholds. We focus on the stabilization of the underactuated dynamics of walking via designing appropriate desired output trajectories. The high-level philosophy is similar to that of the HZD and these using ROMs in [26, 28]. Assuming sufficient control capabilities of the robot actuators, reasonable desired trajectories of the actuated DoFs or functions of them (e.g., selected outputs) can be tracked by a low-level controller. The hybrid underactuated dynamics are determined by the desired output trajectories, and the continuum of walking can be described as maintaining the pre- or post-impact underactuated states in certain viable regions in the state space.

This concept of viable regions can be illustrated using a reduced order model. For the LIP model, if the post impact $x_{\text{com}}^{+}<0$, $L_{y}^{+}>0$, and $E^{+}>0$ (indicating the post-impact state is in the blue shaded region in Fig. 3(a-LIP)) then the LIP is positioned such that it can cross its mid-stance position. If we describe the condition to enable walking continuation as the COM passes the mid-stance position, then the three inequality constraints are both sufficient and necessary for the LIP model to continue walking.

To transfer the condition to a robot, we first perform a change of coordinate for Eq. (7) using $z_{\text{com}}=\text{tan}(\alpha)x_{\text{com}}+\tilde{z}_{\text{com}}$ for the underactuated continuous dynamics:

Notice that if $\dot{\tilde{z}}_{\text{com}}\approx 0$ and $|L^{y}_{\text{com}}|\ll L_{y}$, the robot’s continuous underactuated dynamics resembles the continuous dynamics of the LIP model in Eq. (9). Indeed, it has been shown in the literature [24, 26, 28] that the state-space phase portrait for robot dynamics is topologically similar to that of the LIP model, and the centroidal momentum is small in magnitude. Given the similarity of the continuous dynamics between the underactuated LIP and the robot, we propose to use the post-impact orbital energy to approximate the condition for the robot to continue walking. If the post-impact orbital energy $E^{+}\in[E_{\text{min}},E_{\text{max}}]$ (for $E_{\text{min}},E_{\text{max}}>0$) along with $L_{y}^{+}>0$ and $x_{\text{com}}^{+}<0$, then the robot’s underactuated state takes a value such that next step can be taken. A physical interpretation for choosing $E_{\text{min}}>0$ is to overcome the bounded integrated model error, and that for having $E_{\text{max}}$ is to avoid dynamically infeasible impact time given bounded control input in reality.

The closed-form impact map for robot’s underactuated dynamics in Eq. (7) indicates that we can satisfy the specified post-impact conditions by regulating the pre-impact states. To ensure $x_{\text{com}}^{+}<0$, we propose to approximately control $x_{\text{com}}^{-}$ to some ratio $\epsilon\in(0,1)$ of the current step length using step duration $T_{s}$. For orbital energy, in this work we simply let $E_{\text{min}}=E_{\text{max}}=E^{*}$, which is a desired post-impact orbital energy we choose. We then regulate $L_{y}^{+}$ such that $E^{+}(x_{\text{com}}^{+},L_{y}^{+})=E^{*}$, which can be modulated with the pre-impact vertical velocity $\dot{z}_{\text{com}}^{-}$ as indicated in Eq. (7).

The desired walking behavior is encoded by the desired output trajectories. The outputs are $\mathcal{Y}=\mathcal{Y}_{a}-\mathcal{Y}_{d}$, where $\mathcal{Y}_{a}:TQ\to\mathbb{R}^{m}$, $\mathcal{Y}_{d}:R\to\mathbb{R}^{m}$ are the actual and desired outputs. Using a simple planar robot in Fig. 3(b) for illustration, the following outputs for the sagittal dynamics is picked:

where $\phi_{\text{other}}$ is an vector output for other angles with robot-dependent dimension. For instance, it can be torso angle alone for robots similar to Fig. 3(b) or also includes additional swing foot pitch angle for robots with feet. The superscript $d$ stands for the desired trajectory. $s$ is a monotonically increasing phasing variable defined as $s:=\frac{t}{T_{s}}$, where $t$ is the time measured from the beginning of the current step. $\alpha_{f}=[\phi^{d}_{\text{other}f},z^{d}_{\text{com}f},l_{des},h_{des}]$ represent the desired pre-impact posture. The desired trajectories are synthesized to 1) satisfy the final position of the swing foot, and 2) realize the chosen post-impact orbital energy $E^{*}$. A flowchart of the synthesis is provided in Fig. 4.

To conclude, we present an online planning and control framework to generate stable walking for underactuated bipedal walking on a variety of terrains with constrained foot locations. The proposed approach controls the angular momentum via the vertical COM state in the walking synthesis. We successfully realized the approach in simulation on two bipedal robots for walking on constrained foot locations with various settings, showing a strong premise to enable bipedal robots to locomote in challenging and real environments with discrete contact locations.