Soft Bilinear Inverted Pendulum: a Model to Enable Locomotion with Soft Contacts

Introduction

Legged robots are designed to traverse a wide variety of natural terrains, including soft ground conditions such as snow, mud, and sand. However, the majority of existing control strategies do not account for the terrain compliance, leading to reduced efficiency and performance.

This work presents a control strategy for legged robots that considers the softness of the terrain. It adopts a two-control-blocks architecture composed of a Motion Planner (MP) and a Tracking Controller (TC). For the first time, both blocks consider the contact compliance. The MP uses the novel Soft Bilinear Inverted Pendulum (SBIP) model to compute a feasible trajectory for the robot. The TC employs considers the full robot dynamics and the soft contact model, and computes the optimal joint torques using Hierarchical Optimization (HO). The proposed approach is validated through simulations and experiments on multiple quadrupedal robots. Additionally, the proposed approach is compared with the state-of-the-art and the effects of including the compliance in the MP and/or TC are evaluated.

Results

Key Performance Indicators

The heading and lateral Root Mean Square Error (RMSE) are

where $v_{\mathrm{h}, i}$ and $v_{\mathrm{h}, i}^\mathrm{des}$ are respectively the measured and commanded base velocity in the heading direction at the $i$th instant, and similarly for the lateral direction; while $N$ is the total number of measurements.

The slippage along a trajectory is computed as the integral of the feet’s horizontal movements while in contact with the terrain

where $c_f$ is equal to $1$ if the $f$th foot is in contact with the terrain, and $0$ otherwise; the velocity of the $f$th foot at the $i$th instant is $v_{f, i}$.

The Cost of Transport (CoT) is computed by assuming that most of the energy is lost due to thermal effects and not due to the mechanical work of the actuators. It is computed with

where $\vec{v}_{\mathrm{hor}, i}$ and $\vec{v}_i^\mathrm{des}$ are respectively the horizontal velocity and horizontal reference velocity at the $i$th instant.

Simulations

Controller Behavior

Performance Comparison

Fall Performance Comparison

Experiments

Solution Approach

Full Dynamics Model

The full dynamics model of a legged robot is

where ${q}$ is the generalized coordinates vector, ${v}$ is the generalized velocity vector, ${M}({q})$ is the mass matrix, ${h}({q}, {v})$ is the vector of Coriolis and centrifugal forces, ${S}$ is the selection matrix, ${\tau}$ is the joint torques vector, ${J}_\mathrm{c}({q})$ is the contact Jacobian matrix, and ${f}_\mathrm{c}$ is the contact forces vector.

Soft Bilinear Inverted Pendulum

The SBIP consists in a LIP that lies on a soft surface, modeled as a linear spring-damper system in the normal direction. The dynamic equations are

where $x_\mathrm{CoM}$, $y_\mathrm{CoM}$, and $z_\mathrm{CoM}$ are the coordinates of the Center of Mass (CoM) of the robot base, $x_\mathrm{ZMP}$ and $y_\mathrm{ZMP}$ are the coordinates of the ZMP, $g$ is the module of the gravity acceleration, $m$ is the total mass of the robot, and $h_\mathrm{CoM}$ is the height of the CoM relative to the terrain. Finally, terrain deformation in the SBIP contact point is $\delta$, while $k_\mathrm{p}$ and $k_\mathrm{d}$ are the terrain stiffness and damping, respectively.

Motion Planner

The Motion Planner uses Model Predictive Control (MPC) to compute a feasible trajectory for the robot. It employs a simplified model (i.e. the SBIP) and optimized over a “large” time horizon.

The MPC computes the optimal ZMP positions along a sliding window. The base trajectory is derived from the ZMP positions and the SBIP model dynamics. The feet trajectories are computed using a simple heuristic that utilized the ZMP positions.

Whole-Body Controller

The Whole-Body Controller (WBC) is a Tracking Controller that computes the optimal joint torques to achieve motion tracking. It considers the full robot dynamics model over a “short” time horizon.

A legged robot mush satisfy a wide range of constraints with different importances and potentially conflicting with each other. Hierarchical Optimization solves this problem by solving a set of conflicting tasks while respecting an assigned priority order.

The set of tasks and their priorities are

Each task is formulated as a set of linear equality and inequality constraints, as follows

Here, $x = \begin{bmatrix} \dot{v}^T & f_c^T & \delta_d^T \end{bmatrix}^T$ is the optimization vector, where $\dot{v}$ is the generalized velocity vector, $f_c$ are the contact forces, and $\delta_d$ are the desired terrain deformations. Conversely, $s_{\mathrm{e}, p}$ and $s_{\mathrm{i}, p}$ are the equality and inequality slack variables, respectively.

Conclusion

This paper presented a novel approach for quadrupedal locomotion in the presence of soft contacts. The proposed control architecture consists of two blocks: the Motion Planner (MP) and the Tracking Controller (TC). The benefits of including the compliance in the MP and TC were evaluated through simulations and experiments on multiple quadrupedal robots.

BibTeX citation

    @article{DeBenedittis2025Soft,
  author = {De Benedittis, Davide and Angelini, Franco and Garabini, Manolo},
  journal = {IEEE Transactions on Systems, Man, and Cybernetics: Systems}, 
  title = {Soft Bilinear Inverted Pendulum: A Model to Enable Locomotion With Soft Contacts}, 
  year = {2025},
  volume = {55},
  number = {2},
  pages = {1478-1491},
  keywords = {Legged locomotion;Quadrupedal robots;Foot;Vectors;Optimization;Computational modeling;Trajectory;Tracking;Planning;Jacobian matrices;Contacts;legged locomotion;optimal control;predictive control;quadratic programming},
  doi = {10.1109/TSMC.2024.3504342}
}