3LP: A linear 3D-walking model including torso and swing dynamics

Abstract

Abstract-In this paper, we present a new model of biped locomotion which is composed of three linear pendulums (one per leg and one for the whole upper body) to describe stance, swing and torso dynamics. In addition to double support, this model has different actuation possibilities in the swing hip and stance ankle which could be widely used to produce different walking gaits. Without the need for numerical time-integration, closed-form solutions help finding periodic gaits which could be simply scaled in certain dimensions to modulate the motion online. Thanks to linearity properties, the proposed model can provide a computationally fast platform for model predictive controllers to predict the future and consider meaningful inequality constraints to ensure feasibility of the motion. Such property is coming from describing dynamics with joint torques directly and therefore, reflecting hardware limitations more precisely, even in the very abstract high level template space. The proposed model produces human-like torque and ground reaction force profiles and thus, compared to point-mass models, it is more promising for precise control of humanoid robots. Despite being linear and lacking many other features of human walking like CoM excursion, knee flexion and ground clearance, we show that the proposed model can predict one of the main optimality trends in human walking, i.e. nonlinear speedfrequency relationship. In this paper, we mainly focus on describing the model and its capabilities, comparing it with human data and calculating optimal human gait variables. Setting up control problems and advanced biomechanical analysis still remain for future works.

Scalable Closed-Form Trajectories for Periodic and Non-Periodic Human-Like Walking

“…In this work, we propose a method that can cover a wide range of walking conditions generated by physics-based simulations [Coros et al 2010;Mordatch et al 2010;Yin et al 2007] while offering 2-3 orders of magnitude faster simulation speeds. We use symbolic equations of a linear simplified model (3LP) [Faraji and Ijspeert 2017] that has closed-form solutions. While physics-based simulations need submillisecond integration times and hardly reach real-time factors, we use closed-form solutions of 3LP (as fast as microseconds only) to update the state only at display frames (e.g., 30 frames per second).…”

“…In these case, we scaled the walking speed proportionally. The gait kinematics in 3LP is independent of the body mass however [Faraji and Ijspeert 2017].…”

Bipedal walking and push recovery with a stepping strategy based on time-projection control

“…High-Level Stabilization: On top of the 3LP gaits, one can derive transition equations like Poincaré maps and use an infinite horizon discrete LQR (DLQR) controller for footstep adjustment Faraji and Ijspeert (2017a). This architecture offers the same optimality as our MPC, but due to a discrete nature, it is sensitive to intermittent short disturbances.…”

“…The 3LP model, as its name implies Faraji and Ijspeert (2017a), is composed of three linear pendulums to simulate legs and torso dynamics in walking. These pendulums are connected through a mass-less rigid pelvis of a certain width which stays in a constant-height plane, similar to the LIP model.…”

“…Figure 1 demonstrates the 3LP model conceptually with the three pendulums, limb masses, fixed-height planes, state variables, Figure 1. A schematic of 3LP model used in this paper for walking gait generation Faraji and Ijspeert (2017a). This 3D model is composed of three linear pendulums connected with a massless pelvis of a certain width to simulate natural lateral bounces.…”