Newton-Euler Equations Research Articles

Parallel processing of robot-arm control computation on a multimicroprocessor system

A parallel-processing scheme is described for robot-arm control computation on any number of parallel processors. The scheme employs two multiprocessor scheduling algorithms called, respectively, depth first/implicit heuristic search (DF/IHS) and critical path/most immediate successors first (CP/MISF); these were recently developed by the authors. The scheme is applied to the parallel processing of dynamic control computation for the Stanford manipulator. In particular, the proposed algorithms are applied to the computation of the Newton-Euler equations of motion for the Stanford manipulator and implemented on a multimicroprocessor system. The test result was so successful that the use of six processor pairs in parallel could attain the processing time of 5.37 ms. It is also shown that the proposed parallel-processing scheme is applicable to an arbitrary number of processors.

Unifying Robot Arm Control

A unified approach to three stages of robot arm control is presented based on the Newton-Euler equations of motion. The stages unified are resolved motion, gross motion, and fine motion. Apart from the conceptual advantage, this unification can require less computation than if the three stages are computed separately. In particular, fewer computations are required for arms with no more than about a dozen joints (a number unlikely to be exceeded for most arms, at least in the near future). Computation times are estimated assuming the computing elements are fabricated from current (very large-scale integrated circuit) (VLSIC) technology. It is also shown how friction can be incorporated into the unified approach. The concept of pseudo-force is introduced to relate fine motion to the Newton-Euler equations.

An adaptive control strategy for mechanical manipulators

This paper focuses on the study of an adaptive perturbation control which tracks a desired time-based trajectory as close as possible for all times over a wide range of manipulator motion and payloads. The proposed adaptive control is based on the linearized perturbation equations in the vicinity of a nominal trajectory. The controlled system is characterized by feedforward and feedback components which can be computed separately and simultaneously. The feedforward component computes the nominal torques from the Newton-Euler equations of motion to compensate all the interaction forces among the various joints. The feedback component consisting of recursive least-square identification and an optimal adaptive self-tuning control algorithm for the linearized system computes the perturbation torques which reduce the position and velocity errors of the manipulator along the nominal trajectory. A computer simulation study was conducted to evaluate the performance of the proposed adaptive control.

Resolved Motion Adaptive Control for Mechanical Manipulators

This paper presents the development of a resolved motion adaptive control which adopts the ideas of “resolved motion rate control” [8] and “resolved motion acceleration control” [10] to control a manipulator in Cartesian coordinates for various loading conditions. The proposed adaptive control is performed at the hand level and is based on the linearized perturbation system along a desired hand trajectory. The controlled system is characterized by feedforward and feedback components which can be computed separately and simultaneously. The feedforward component resolves the specified positions, velocities, and accelerations of the hand into a set of values of joint positions, velocities, and accelerations from which the nominal joint torques are computed using the Newton-Euler equations of motion to compensate all the interaction forces among the various joints. The feedback component consisting of recursive least square identification scheme and an optimal adaptive self-tuning controller for the linearized system computes the perturbation torques which reduce the manipulator hand position and velocity errors along the nominal hand trajectory. The feasibility of implementing the proposed adaptive control using present day low-cost microprocessors is explored.

On the Equivalence of Lagrangian and Newton-Euler Dynamics for Manipulators

Recently, there has been considerable interest in efficient formulations of manipulator dynamics. The inefficiency of the classic Lagrangian formulation is well known, leading several researchers to a new formulation based on the Newton-Euler equations. This formulation is highly ef ficient, but there may be some confusion as to the source of this efficiency. This paper shows that there is in fact no fundamental difference in computational efficiency between Lagrangian and Newton-Euler formulations. The efficiency of the above-mentioned Newton-Euler formula tion is due to two factors: the recursive structure of the computation and the representation chosen for the rota tional dynamics. Both of these factors can be achieved in a Lagrangian formulation. Recursive Lagrangian dy namics has been discussed previously by Hollerbach. This paper compares the representations that have been used and shows that with a proper choice the Lagrangian formulation is indeed equivalent to the Newton-Euler formulation.

Some aspects of Euler-Newton equations of motion

Euler-Newton's equations for a system of connected rigid bodies, written in a special state space form, provide a systematic method of arriving at the differential equations of the system. This method is amenable to programming and symbolic algebraic manipulation. The elimination of some or all forces of constraint is by projection, implementing the principle of virtual work, and is done by inner products. The computation of these forces requires symbolic inversion of a matrix for which an iterative scheme is proposed here. A method for construction of Lyapunov functions for stability of such systems in the vicinity of an arbitrary operating point is proposed. This construction may be achieved by symbolic manipulations and supplements applications of the Euler-Newton method.

Comment on “multiport representation of inertia properties of kinematic mechanisms”

This paper describes the history of the bond graph description of rigid body rotation dynamics and resolves a paradox that resulted from the common Euler Junction Structure (EJS) description of the exterior product in the Newton–Euler equation describing rigid body rotation [D.C. Karnopp, R.C. Rosenberg, Analysis and Simulation of Multiport Systems – The Bond Graph Approach to Physical Systems Dynamics, MIT Press, Boston, 1968; D.C. Karnopp, The energetic structure of multibody dynamic systems, J. Franklin Inst. 306 (2) (1978) 165–181]. The proposed alternative representation that resolves this paradox allows direct insight into the stability of rotations around the principal axes of a rigid body. It eliminates the bond loop and it is shown to be a canonical decomposition of the corresponding nonlinear 3-port gyrator, whereas the EJS is not canonical. The consequences of these differences are demonstrated, in particular the increased expressive power of this notation, which is particularly important in education.

Dynamic Analysis of Multi-Rigid-Body Systems

A method is presented for formulating and solving the Newton-Euler equations of motion of a system of interconnected rigid bodies. The digital simulation may involve numerical integration of the kinematic equations as well as the dynamic equations. The reaction forces and torques resulting from rigid constraints imposed at the connecting joints are also determined. The derivation of kinematic expressions for first and higher derivatives is demonstrated based on direct differentiation of the rotation matrix in the spirit of the classical vector approach. A representative problem in spatial mechanism analysis is solved and illustrated with numerical results.