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Research Papers

# A Penalty Formulation for Dynamics Analysis of Redundant Mechanical Systems

[+] Author and Article Information
Bilal Ruzzeh

Department of Mechanical Engineering, and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montréal, QC, H3A 2K6, Canadabilal.ruzzeh@mail.mcgill.ca

József Kövecses

Department of Mechanical Engineering, and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montréal, QC, H3A 2K6, Canadajozsef.kovecses@mcgill.ca

The condition of invertibility of the augmented matrix in Eq. 6 is generally known to be the independence of the constraints; i.e., the constraint Jacobian has to be of full row rank.

By contact group, it is meant to be a set of mobile objects with a chain of contacting mobile objects between any two members of the group.

A good initial guess can be the minimizer of the objective function $g(x)$ in unconstrained optimization.

For further illustration, detailed derivations, and proofs, the reader can consult specialized books on mathematical optimization such as Ref. 21.

It will be shown in the next subsection that the actuating constraints are always holonomic. The calculation of the different quantities in the actuating constraint equations at the acceleration level will also be illustrated.

It is noteworthy that algorithms, which depend on model-based parameters to calculate an appropriate penalty factors, are not available in literature yet, but it might be possible to develop them in the future.

The star $(∗)$ superscript signifies the state of not yet converged or nonfinal solutions.

The insertion point is on the part with a smaller contact surface between the muscle and the bone, while the attachment point (or origin) is on the part with a larger contact or attachment area.

The first four constraint equations are associated with the redundant link 3. The other remaining four constraint equations associated with the second redundant link 4 exhibit the same behavior.

The mathematical algorithm may not succeed in achieving the required precision or tolerance in the presence of a singularity due to the limitation of the approximate estimation of the Lagrange multipliers (21).

J. Comput. Nonlinear Dynam 6(2), 021008 (Oct 28, 2010) (12 pages) doi:10.1115/1.4002510 History: Received December 09, 2009; Revised June 29, 2010; Published October 28, 2010; Online October 28, 2010

## Abstract

Redundancy in the constraining of mechanical systems achieves more stability and larger load capacity for the system, while in actuation it provides better robustness against singularities and higher maneuverability. Few techniques have been developed with the aim to handle redundancy and singularities in dynamics analysis, and further research is still needed in this area. In this paper, we illustrate the concept of actuating and passive constraints. Then, we expand on the existing penalty techniques by incorporating the concept of actuating and passive constraints to present a penalty formulation that is capable of efficiently handling singularities and redundancy in constraining and actuation and can carry out either forward or inverse dynamics analysis of mechanical systems. As such, the proposed approach is referred to as the actuating-passive constraints penalty approach.

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## Figures

Figure 1

Example of a translational actuator

Figure 2

Five-bar mechanism (redundantly constrained)

Figure 14

The driving moments at joints L and B and the corresponding vertical components of accelerations of the centers of mass GL and GU of the model

Figure 3

Plots of θ1 (rad), x3 (m), and x4 (m)

Figure 4

Violations of acceleration level constraints

Figure 5

Violations of configuration level constraints

Figure 6

Individual constraint forces for μ=107I, where I is the m×m identity matrix with suitable mass units and m is the number of kinematic constraints

Figure 7

Individual constraint forces for μ=1010I, where I is the m×m identity matrix with suitable mass units and m is the number of kinematic constraints

Figure 8

Sagital view of the simplified human lower limb with two actuators

Figure 9

The driving forces represented as individual driving constraint reactions

Figure 10

The generalized driving forces represented as joint torques

Figure 11

Plot of configuration coordinates α,β

Figure 12

The deviation of β̈ from the reference solution

Figure 13

The individual actuating efforts λa,1 and λa,2 are associated with the translational actuators a1 and a2; λa,3 is associated with a motor actuator at joint L

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