Natural ZMP trajectories for biped robot reference generation

The control of a biped humanoid is a challenging task due to the hard-to-stabilize dynamics. Walking reference trajectory generation is a key problem. Linear Inverted Pendulum Model (LIPM) and Zero Moment Point (ZMP) Criterion based approaches in stable walking reference generation are reported. In these methods, generally, the ZMP reference during a stepping motion is kept fixed in the middle of the supporting foot sole. This kind of reference generation lacks naturalness, in that, the ZMP in the human walk does not stay fixed, but it moves forward under the supporting foot. This paper proposes a reference generation algorithm based on the LIPM and moving support foot ZMP references. The application of Fourier series approximation simplifies the solution and it generates a smooth ZMP reference. A simple inverse kinematics based joint space controller is used for the tests of the developed reference trajectory through full-dynamics 3D simulation. A 12 DOF biped robot model is used in the simulations. Simulation studies suggest that the moving ZMP references are more energy efficient than the ones with fixed ZMP under the supporting foot. The results are promising for implementations

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

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 speed-frequency 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.Comment: Journal paper under revie

The Modeling and Stability Analysis of Humans Balancing an Inverted Pendulum

The control of an inverted pendulum is a classical problem in dynamics and control theory. Without active control, the inverted pendulum by itself is inherently unstable, thus serving as an ideal platform for control algorithms design and testing. This study utilizes an inverted pendulum setup to investigate the characteristics of manual control in executing a single-axial compensatory task. An inverted pendulum with sliding base on a single-axial rail was built for this purpose. Human subjects were asked to stabilize the pendulum by sliding the base on the rail. To mathematically quantify the characteristics of human manual control, a quasi-linear lead-lag with time delay model was chosen for the human operator. The mathematical model for the inverted pendulum was derived using the LaGrange\u27s method. Using these two models, a simulation of the closed-loop human-inverted pendulum system was built in Matlab/Simulink. The stability conditions of the closed-loop system were derived by applying the Routh-Hurwitz stability criterion to the system. This completes the modeling and simulation of the process of humans balancing an inverted pendulum. The Matlab simulation serves as a validation tool in this study. The data of the human subject\u27s input and the inverted pendulum\u27s output generated from the simulation were used to estimate the parameters assumed in the mathematical model for the human operator. The estimation algorithm employed is a Kalman filter. Results show that the estimations do converge very quickly to the parameters set in the simulated human controller and can stabilize the inverted pendulum when fed back into the simulation. This verifies the plausibility of the mathematical structure for the human operator and the validity of the estimator. Experimentally, the pendulum\u27s angle deflections from the vertical position and the human subjects\u27 hand positions were recorded using a motion capture system called VICON. Using the same estimator developed for processing the simulation data, the collected experimental data were processed to estimate the parameters in the model for the human operator when the human operator actually carries out the task of balancing the inverted pendulum. The estimated parameters from the experimental data were then fed into the simulation model. The characteristics of the human operator were analyzed using the estimated parameters