Rigid Body Dynamics, Constraints, and Inverses

[+] Author and Article Information
Hooshang Hemami

Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210

Bostwick F. Wyman

Department of Mathematics, The Ohio State University, Columbus, OH 43210

J. Appl. Mech 74(1), 47-56 (Dec 19, 2005) (10 pages) doi:10.1115/1.2178359 History: Received December 03, 2004; Revised December 19, 2005

Rigid body dynamics are traditionally formulated by Lagrangian or Newton-Euler methods. A particular state space form using Euler angles and angular velocities expressed in the body coordinate system is employed here to address constrained rigid body dynamics. We study gliding and rolling, and we develop inverse systems for estimation of internal and contact forces of constraint. A primitive approximation of biped locomotion serves as a motivation for this work. A class of constraints is formulated in this state space. Rolling and gliding are common in contact sports, in interaction of humans and robots with their environment where one surface makes contact with another surface, and at skeletal joints in living systems. This formulation of constraints is important for control purposes. The estimation of applied and constraint forces and torques at the joints of natural and robotic systems is a challenge. Direct and indirect measurement methods involving a combination of kinematic data and computation are discussed. The basic methodology is developed for one single rigid body for simplicity, brevity, and precision. Computer simulations are presented to demonstrate the feasibility and effectiveness of the approaches presented. The methodology can be applied to a multilink model of bipedal systems where natural and/or artificial connectors and actuators are modeled. Estimation of the forces is accomplished by the inverse of the nonlinear plant designed by using a robust high gain feedback system. The inverse is shown to be stable, and bounds on the tracking error are developed. Lyapunov stability methods are used to establish global stability of the inverse system.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

A continuous nonlinear system H, and its inverse composed of a replica of H and two operators O1 and O2

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Figure 2

The free rigid body in rotation with input N1 and output Z1, (Eq. 13) in series with the nonlinear inverse with high gain feed forward and a model of the system in the feedback path

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Figure 3

The inverted pendulum in contact with a massless foot. An actuator with origin O on the pendulum and insertion point on the foot applies a force F to the pendulum along the line OI.

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Figure 4

The inverted pendulum with input N1 and output Z1 (Eq. 22) in series with the nonlinear inverse with high gain feed forward path and a model of the system in the feedback path. The estimate of total input torque is the output N2. The ground reaction force Λ (Eq. 23) is constructed from Z1 and Nt.

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Figure 5

The input torque En, the angles, and angular velocities as functions of time for one rigid body anchored at its center of gravity

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Figure 6

The estimated torques as outputs of the inverse system as functions of time. The error signals in Θ and Ω, as inputs to the high gain forward component of the inverse are also plotted as functions of time.

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Figure 7

The total applied torques to the inverted pendulum, and the resulting ground reaction force as functions of time

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Figure 8

The state trajectories of the inverted pendulum: Θ and Ω as functions of time

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Figure 9

The estimated total torque applied to the inverted pendulum, the estimated ground reaction forces as functions of time

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Figure 10

The state of the estimator as a function of time




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