## Motivation

Legged robots are typically built with rigid bodies that incorporate multiple degrees of freedom and joints driven by hydraulic pumps or electric motors, resulting in a mechanically complex and bulky structure. Such designs, while facilitating advanced movements, increase safety risks; for example, Atlas weighs approximately 89 kg and could cause harm in case of controller malfunctions or falls, particularly when navigating slopes or stairs where it may roll uncontrollably. In contrast, soft robots, known for their lighter weight and compliance, maneuver effectively in tight spaces and adapt to environmental changes with enhanced safety. Their deformable bodies naturally absorb impacts, which protects their core components but sacrifices speed and precision, making them less suitable for tasks requiring intricate locomotion. These contrasting characteristics of rigid and soft robots highlight a significant challenge in the field: balancing mobility, environmental perception, and safe interaction within diverse environments. The integration of rigid robots’ agility and precision with the adaptability and safety of soft robots is crucial, propelling forward the next wave of robotic innovation.

## Objective

**The core objective of this project is to theoretically formulate and experimentally validate a novel robotic design that combines soft and rigid elements, enhancing stability, mobility, and safety in legged robotic systems.** This design focuses particularly on improving fall mitigation and recovery capabilities on steep inclines and stairs. Unlike previous models, our approach addresses the robustness and performance of legged robotic systems as interconnected, yet distinct, components. We intend to develop a methodology that mathematically models the benefits of integrating soft and rigid components, thereby equipping legged robots with enhanced capabilities.

These capabilities include utilizing tunable friction and smart materials to adeptly navigate complex environments and manage steep inclines and narrow passages. Additionally, the robots are designed to proactively respond to unexpected falls or impacts with a multi-functional soft manipulator, significantly enhancing safety measures. Furthermore, the design ensures inherent safety in proximity to humans, even in scenarios where humans are undetected by traditional sensors.

This figure depicts the robot’s hybrid soft-rigid design, engineered for agility and precision, while ensuring adaptability and safety in human-occupied environments. Key features include a multi-functional soft manipulator, contact-sensitive soft inflated cushions, and innovative soft robot shoes with adjustable friction. This unique design, complemented by specialized algorithms, ensures all interactions with environmental objects are inherently safe.

## Approach

\( \renewcommand{\vec}[1]{\mathbf{\boldsymbol{#1}}} \require{color} \) Specifically, we will implement this control strategy as an optimal problem. Recognizing the brief nature of fall events, we utilize a Single Rigid Body (SRB) approximation to model the robot’s dynamics, excluding the leg dynamics to focus on rapid and efficient countermeasures. This approximation also applies to the manipulator since the bulky valve bank is rigidly connected to the torso of the robot and the soft inflated manipulator is light-weighted. The governing dynamics under this simplified model are expressed as follows: \begin{align} m \left( \vec{\dot{v}} – \vec{g} \right) &= \sum_{i=1}^{n} \vec{f_{i}} + \textcolor{purple}{\vec{f_{M}}}, \\ \notag \vec{I} \vec{\dot{\omega}} + \vec{\omega} \times \vec{I}\vec{\omega} &= \sum_{i=1}^{n} \vec{r_{i}} \times \vec{f_{i}} + \vec{r_{M}} \times \vec{f_{M}}, \end{align} where $m$ is the mass of the system including robot and mass of the valve bank, $\vec{\dot{v}}$ and $\vec{g}$ denote the acceleration of the robotâ€™s center of mass and gravitational acceleration, respectively, $\vec{f_{i}}$ and $\vec{f}_{M}$ represent the ground reaction forces exerted by the robotâ€™s legs and the soft manipulator, respectively, $\vec{I}$ and $\vec{\omega}$ are the inertia matrix and angular velocity of the robot, respectively, and $\boldsymbol{r}^i$ represents the displacement from foot to CoM within the world coordinate frame. Upon determining the desired motion for the CoM, the desired GRFs are computed by solving a Quadratic Program (QP). The optimization employs the following cost function to minimize both the magnitudes of the forces and the deviations from their values in the preceding iteration subject to constraints that ensure these forces fall within the limits defined by the friction cone $\vec{F_{\mu}}$: \begin{align} \label{eq:QP} \text{Min} \, J &= \frac{1}{2} \boldsymbol{f}^{\text{T}}\left(\boldsymbol{A}^{\text{T}} \boldsymbol{S} \boldsymbol{A} + \boldsymbol{W} + \boldsymbol{U}\right) \boldsymbol{f} + \left(-\boldsymbol{b}_d^{\text{T}} \boldsymbol{S} \boldsymbol{A} – \boldsymbol{f}_{\text{pre}}^{\text{T}} \boldsymbol{U}\right) \boldsymbol{f}, \quad \text{subject to} \quad \vec{F_{\mu}} \boldsymbol{f} \ge \vec{0} \\ \notag & \text{where} \quad \boldsymbol{A} = \begin{bmatrix} \boldsymbol{1} & \boldsymbol{1} & \boldsymbol{1} & \boldsymbol{1} & \boldsymbol{1}\\ {\left[\boldsymbol{\vec{r}}_{\text{FR}}\right]_{\times}} & {\left[\boldsymbol{\vec{r}}_{\text{FL}}\right]_{\times}} & {\left[\boldsymbol{\vec{r}}_{\text{RR}}\right]_{\times}} & {\left[\boldsymbol{\vec{r}}_{\text{RL}}\right]_{\times}} & {\left[\boldsymbol{\vec{r}}_{\text{M}}\right]_{\times}} \end{bmatrix} \quad \text{and} \quad \boldsymbol{b}_d = \begin{bmatrix} m(\vec{\dot{v}} – \vec{g} ) \\ \vec{R} \boldsymbol{I} \vec{R}^{\text{T}} \dot{\boldsymbol{\omega}} \end{bmatrix}. \notag \end{align} $\vec{R}$ is the rotation matrix transitioning from the body-fixed coordinate system to the world frame. $\boldsymbol{f} \in \mathbb{R}^{15}$ encapsulates the GRFs $\boldsymbol{f}_{i}$ for each leg and the manipulator force $\vec{f_{M}}$. $\boldsymbol{S}$, $\boldsymbol{W}$ and $\boldsymbol{U}$ are adjustable weight matrices. $\boldsymbol{f}_{\text{pre}}$ references the GRFs computed in the preceding iteration. These equations will be implemented in ROS and solved by QuadProg++.

This figure illustrates the robot’s fall mitigation strategy when navigating uneven slopes. The multi-functional soft arm can adapt its mode to manage various scenarios: it braces for impacts when descending from high ground (a), cushions and stabilizes during rolling motions and acts as an air jack if the robot is upside-down (c), and prevents further rolling during fall recovery (d).