Improved DAE formulation for inverse dynamics simulation of cranes

Abstract

Cranes are underactuated systems with less control inputs than degrees of freedom. Dynamics and control of such systems is a challenging task, and the existence of solution to the inverse dynamics simulation problem in which an r-degree-of-freedom system with m actuators, m < r, is subject to m specified motion task (servo-constraints) is conditioned upon the system is differentially flat (all the system states and control inputs can be algebraically expressed in terms of the outputs and their time derivatives up to a certain order). The outputs are often designed as specified in time load coordinates to model a rest-to-rest maneuver along a trajectory in the working space, from the initial load position to its desired destination. The flatness-based methodology results then in the required control inputs determined in terms of the fourth time derivatives of the imposed outputs, and the derivations are featured by substantial complexity. The DAE formulation motivated in this contribution offers a more convenient approach to the prediction of dynamics and control of cranes executing prescribed load motions, and only the second time derivatives of the specified outputs are involved. While most of the inverse simulation formulations, both flatness-based and DAE ones, are performed using independent state variables, the use of dependent coordinates and velocities may lead to substantial modeling simplifications and gains in computational efficiency. An improved DAE formulation of this type is presented in this paper.

Simulation of multibody systems with servo constraints through optimal control

“…5.4]. A well-known approach based on a projection of the dynamics was introduced in [8]; see also [6]. For this, we have to compute time-dependent projection matrices in order to split the dynamics of the underactuated system into constrained and unconstrained parts.…”

Simultaneous stabilization and trajectory tracking of underactuated mechanical systems with included actuators dynamics

“…Vector function f q (x q ) ∈ R 2n and matrix function B q (x q ) ∈ R 2n×na are defined in expression (9). Space variables vector x q ∈ R 2n of mechanical system is defined as…”

“…Relation between the functionγ ac (x ac , x q ) and the input vector u using expressions from (9) and (12) …”

“…second-order nonholonomic chained systems is given in [16], control of UNMS with servoconstraints is presented in [7][8][9]22], while control of holonomic and nonholonomic systems using sliding mode is presented by the direct Lyapunov approach [15]. Specific examples of simultaneous stabilization and/or trajectory tracking of UNMS are given in [3,14,27,33].…”

Analysis of servo-constraint problems for underactuated multibody systems

“…It allows a straightforward formulation of the servo-constraint problem and simplifies significantly the numerical solution of the arising DAEs. This method has also been applied to differentially flat multibody systems with mixed geometric and servo-constraints as reported by Betsch et al (2008) and Blajer and Kolodziejczyk (2011). In the coordinate transformation approach the servoconstraint problem is reformulated in new coordinates containing the output.…”