ULT-model: Towards a one-legged unified locomotion template model for forward hopping with an upright trunk

TL;DR

The paper tackles the problem of formulating a unified, compact template model for upright-trunk locomotion that spans both stance and swing. It extends a trunk SLIP framework by adding a non-negligible leg mass and a point-foot, along with a leg-extension actuator, and enforces a phase-based switching control to realize a complete gait with stable limit cycles. The key contributions are a non-discretized dynamic model $oldsymbol{S}$ describing dynamics $oldsymbol{D}$ and environment $oldsymbol{E}$ under a phase-independent controller $oldsymbol{C}$, a VPP-based stance controller, a simple swing controller with leg retraction and angle-of-attack adaptation, and orbital stability validation via a Poincaré map yielding Floquet multipliers inside the unit circle. This work provides a tractable path toward closed-loop, anchor-matching gait control for upright-trunk locomotion, with potential implications for both neuromuscular understanding and robust legged-robot control; future steps include a unified control law across the gait cycle and more sophisticated contact models.

Abstract

While many advancements have been made in the development of template models for describing upright-trunk locomotion, the majority of the effort has been focused on the stance phase. In this paper, we develop a new compact dynamic model as a first step toward a fully unified locomotion template model (ULT-model) of an upright-trunk forward hopping system, which will also require a unified control law in the next step. We demonstrate that all locomotion subfunctions are enabled by adding just a point foot mass and a parallel leg actuator to the well-known trunk SLIP model and that a stable limit cycle can be achieved. This brings us closer toward the ultimate goal of enabling closed-loop dynamics for anchor matching and thus achieving simple, efficient, robust and stable upright-trunk gait control, as observed in biological systems.

Figures (10)

• Figure 1: Unified locomotion template model (ULT-model) that combines all locomotor subfunctions Seyfarth2015: The repulsive leg function and postural stabilization during stance (inspired by the SLIP and VPP models) as well as the forward leg swing function (derived from RHM and SLP model) and leg retraction function (similar to SLP model) in the flight phase.
• Figure 2: The ideal ULT-model is composed of a compact system model $\mathbb{S}$ that is able to describe the system dynamics $\mathbb{D}$ and interaction with the environment $\mathbb{E}$ during both stance and swing as well as a phase-independent controller $\mathbb{C}$. Here, the current and desired system state are denoted $\boldsymbol{x}$ and $\boldsymbol{x}^d$, respectively, and the control effort is represented by $\boldsymbol{u}$. The environment exerts friction and ground reaction forces $\boldsymbol{F}_E$, when the system is in contact.
• Figure 3: Coordinate definition and force diagram of the proposed dynamics model. The positions and orientations are defined with respect to a common world frame $\mathcal{E}_W$. A torque $\boldsymbol{\tau}$ that acts on the hip corresponds to a certain force $\boldsymbol{F}_{\tau}$ that acts on the foot and is perpendicular to the leg axis.
• Figure 4: System architecture. The two desired inputs that drive the system behavior are leg retraction, shaped by the desired resting length $l_0^d$, and system forward velocity, defined by $\dot{x}_c^d$. The ground reaction and friction forces exerted by the environment are denoted as $\boldsymbol{F}_E$.
• Figure 5: Virtual Pivot Point Concept. With an appropriate hip torque, the left model behaves exactly like the right one. The offset angle from the body axis to the VPP is denoted as $\delta$ and the angle between the vector pointing from foot to hip and from foot to VPP is denoted as $\gamma$.

Theorems & Definitions (2)

• Definition II.1
• Definition II.2