Bipedal Safe Navigation over Uncertain Rough Terrain: Unifying Terrain Mapping and Locomotion Stability

TL;DR

A hierarchical locomotion-dynamics-aware sampling-based navigation planner is proposed for bipedal robot navigation in complex environments with uncertain and rough terrain and a novel trajectory evaluation metric is developed to minimize motion deviation and maximize information gain of the terrain elevation map.

Abstract

We study the problem of bipedal robot navigation in complex environments with uncertain and rough terrain. In particular, we consider a scenario in which the robot is expected to reach a desired goal location by traversing an environment with uncertain terrain elevation. Such terrain uncertainties induce not only untraversable regions but also robot motion perturbations. Thus, the problems of terrain mapping and locomotion stability are intertwined. We evaluate three different kernels for Gaussian process (GP) regression to learn the terrain elevation. We also learn the motion deviation resulting from both the terrain as well as the discrepancy between the reduced-order Prismatic Inverted Pendulum Model used for planning and the full-order locomotion dynamics. We propose a hierarchical locomotion-dynamics-aware sampling-based navigation planner. The global navigation planner plans a series of local waypoints to reach the desired goal locations while respecting locomotion stability constraints. Then, a local navigation planner is used to generate a sequence of dynamically feasible footsteps to reach local waypoints. We develop a novel trajectory evaluation metric to minimize motion deviation and maximize information gain of the terrain elevation map. We evaluate the efficacy of our planning framework on Digit bipedal robot simulation in MuJoCo.

Figures (10)

• Figure 1: (Top) The bipedal robot Digit navigates through an environment with rough terrain in our MuJoCo simulation. (Bottom) Snapshots of the trajectory of the bipedal robot at various time instants as it navigates towards the goal (yellow star). The white line depicts the traversed trajectory, and the orange dot is the current targeted local waypoint.
• Figure 2: Prismatic Inverted Pendulum Model (PIPM) model for our robot Digit for traversing over uncertain and uneven terrain.
• Figure 3: Overall block diagram of the local approximation method for GP prediction with NN kernel. Whenever the global elevation map undergoes re-estimation, all input locations in the terrain are split into k clusters via K-mean clustering algorithm. Each cluster consists of a centerpoint $c_i$ and the subset input locations $\{\bm{\xi}\}_i$. Iterating through each of the k clusters, the centerpoint is passed to the KD-Tree method to find n nearest neighboring training datapoints denoted as $D_{nearest}^{i} = \{(\bm{\xi}^j,y^j)\}^{i}_{j\in\{1,...,n\}}$. Given n closest training datapoints to the centerpoint (i.e., $D_{nearest}^{i}$) and the set of test points in that cluster to be estimated (i.e., $\{\bm{\xi}\}_i$), we then have all of the information to make GP predictions of the terrain heights at all input locations within the $i^{th}$ cluster region. With this local approximation method, GP prediction with NN kernel is only computed k times, thereby reducing the computational load as compared to making a separate prediction for all input locations.
• Figure 4: Overall block diagram of the proposed global-local planning framework for bipedal navigation over rough terrain.
• Figure 5: (a) Illustration of the smoothing algorithm. The goal is to find the smoothest path between the start $\boldsymbol{w}_0$ and the target $\boldsymbol{w}_4$. Connecting $\boldsymbol{w}_0$ and $\boldsymbol{w}_4$ directly is invalid because the line between them passes through an obstacle. Connecting to $\boldsymbol{w}_3$ is also invalid because the resultant heading angle exceeds the limit $\Delta\theta_{{\rm safe}}.$ Thus, the smoothing algorithm connects $\boldsymbol{w}_0$ to $\boldsymbol{w}_2$ to $\boldsymbol{w}_4$. (b) Illustration of LDA-L-RRT* vertex selection criteria. The left red vertex is invalid because it lies in an obstacle, and the right red vertex is impossible because the heading angle change exceeds $\Delta\theta_{{\rm safe}}$. (c) Illustration of the local navigation trajectory parameters. (d) Illustration of the LDA-G-RRT* safety criteria. The solid green lines are the locomotion safety barriers, and the shaded green area is their convex hull. Starting from vertex $v_0$, the connection to $v_4$ is invalid because $v_4$ is outside the locomotion safety barriers. The connection from $v_2$ to $v_4$ is invalid as it crosses a locomotion safety barrier. The connection from $v_2$ to $v_3$ is valid, as the dashed green line is not a locomotion safety barrier.

Theorems & Definitions (4)

• Definition 1: Gaussian Process Regression
• Definition 2: Local Navigation Trajectory
• Definition 3: Locomotion-dynamics-aware Local RRT*
• Definition 4: Locomotion-dynamics-aware Global RRT*