Asymptotically Stable Gait Generation and Instantaneous Walkability Determination for Planar Almost Linear Biped with Knees
Authors: Fumihiko Asano, Ning Lei, Taiki Sedoguchi
Organizations: Graduate School of Advanced Science and Technology, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan
A class of planar bipedal robots with unique mechanical properties has been proposed, where all links are balanced around the hip joint, preventing natural swinging motion due to gravity. A common property of their equations of motion is that the inertia matrix is a constant matrix, there are no nonlinear velocity terms, and the gravity term contains simple nonlinear terms. By performing a Taylor expansion of the gravity term and making a linear approximation, it is easy to derive a linearized model, and calculations for future states or walkability determination can be performed instantaneously without the need for numerical integration. This paper extends the method to a planar biped robot model with knees. First, we derive the equations of motion, constraint conditions, and inelastic collisions for a planar 6-DOF biped robot, design its control system, and numerically generate a stable bipedal gait on a horizontal plane. Next, we reduce the equations of motion to a 3-DOF model, and derive a linearized model by approximating the gravity term as linear around the expansion point for the thigh frame angle. Through numerical simulations, we demonstrate that calculations for future states and walkability determination can be completed in negligible time. By applying control inputs to the obtained model, performing state-space realization, and then discretizing it, instantaneous walkability determination through iterative calculation becomes possible. Through detailed gait analysis, we discuss how the knee joint flexion angle and the expansion point affect the accuracy of the linear approximation, and the issues that arise when descending a small step.
Figures & tables
Fig. 1: Model of planar, 6-DOF, almost linear biped with knees
Fig. 2: Simulation results of asymptotically stable walking of kneed biped
m1
1.0
kg
m2
1.0
kg
L1
0.5
m
L2
0.5
m
r1
0.25
m
r2
0.25
m
TABLE I: Physical and control parameters
Fig. 3: Evolution of gait descriptors in asymptotically stable gaits of nonlinear and CLRed models where β=0.5 [rad]
Fig. 4: Gait descriptors with respect to β for seven values of κ where Tset=0.7 [s] of nonlinear and CLRed models
Fig. 5: Evolution of gait descriptors during generated gaits involving descending 2.0 [cm] step for seven values of Tset′ where β=0.7 [rad]
This paper presents the kinematic and dynamic modeling, trajectory generation, and stability analysis of an 8-degree-of-freedom (DOF) biped robot walking on flat and inclined terrain. Denavit-Hartenberg (DH) parameters and homogeneous transformations are used to derive the forward kinematics, while closed-form inverse kinematics maps the desired hip and swing-foot Cartesian trajectories, generated with cubic splines, to joint angles. Joint torques are computed using the Newton-Euler iterative algorithm, and dynamic stability is evaluated using the zero moment point (ZMP) criterion. A genetic algorithm (GA) optimizes the hip height, maximum swing-foot lift, and frontal-plane tilt angle by minimizing the work done by the joints subject to a ZMP feasibility penalty. Simulation results in MATLAB show that the nominal 8-DOF model remains ZMP-stable for step completion times down to 0.5 s and for slope inclinations up to 22.5 degrees with the given foot geometry. Beyond these limits, the ZMP leaves the support polygon, and either the foot dimensions or the trajectory parameters must be modified. The results also show that ZMP stability is governed by the mass distribution among the links rather than the total mass of the robot.
Madhav Rijal
Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India
In this paper, we explore the impulsive dynamics common to single-joint, two-link models of walking and brachiating gaits with respect to slope and switching time. In particular, we investigate how the stability of a gait and bifurcations encountered within a family of gaits change under time-based and state-based switching of the impulsive dynamics.
Alan Estrada Flores, Nelson Rosa
Illinois Institute of Technology, Chicago IL 60616, USA
Foot slip is a major source of instability in bipedal locomotion on low-friction or uncertain terrain. Standard control approaches typically assume no-slip contact and therefore degrade when slip occurs. We propose a control framework that explicitly incorporates slip into the locomotion model through virtual nonholonomic constraints, which regulate the tangential stance-foot velocity while remaining compatible with the virtual holonomic constraints used to generate the walking gait. The resulting closed-loop system is formulated as a hybrid dynamical system with continuous swing dynamics and discrete impact events. A nonlinear feedback law enforces both classes of constraints and yields a slip-compatible hybrid zero dynamics manifold for the reduced-order locomotion dynamics. Stability of periodic walking gaits is characterized through the associated Poincaré map, and numerical results illustrate stabilization under slip conditions.
Centre for Automation and Robotics (CSIC-UPM), Ctra. M300 Campo Real, Km 0,200, Arganda del Rey - 28500 Madrid, Spain. · the Department of Applied Mathematics, Comillas Pontifical University, Madrid, 28015 - Madrid, Spain. · School of Science and Technology, IE University, Madrid, Spain. +1