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A quadratic programming based simultaneous impact model (QPSIM) for mechanisms

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Abstract

Multi-body systems, like robotic mechanisms, may frequently encounter impacts while interacting with the environment. These non-linear impacts are often modeled as instantaneous events using discretized time impulse-based approaches. Although, momentum and energy conservation laws are sufficient to determine a solution for frictionless single point impulse problems, additional assumptions are required to solve simultaneous impacts. We propose a novel method (called QPSIM) to solve such simultaneous collision problems in links connected by frictionless joints by generalizing the quadratic programming (QP) based simultaneous impact model for unconstrained rigid bodies presented in the work of Rakshit and Chatterjee [35]. Two constraint equations are derived, which when included in the QP problem, results in a solution containing both collision impulses as well as joint reaction impulses and impulsive moments. Additional constraints depicting the physical laws of contact mechanics are also used in the simultaneous impact problem. A generalized matrix based derivation of the constraints and the impact model is presented which is applicable to both closed-loop and open-chain mechanisms. The solution is formed by the impulse set that minimizes the system’s net change in kinetic energy. QPSIM does not require an impulse propagation sequence to be assumed and is a computationally efficient alternative to the modeling of collisions in a force-based domain. Results for simultaneous collision scenarios in two planar and one spatial mechanisms are presented in this paper. In addition, this method is applied to solve a simultaneous impact problem on a top-hammer drill machine which is often used in mines and quarries. Moreover, the results are compared with results obtained using Linear Complementarity and ADAMS software simulations. The results show that QPSIM never results in an increase in kinetic energy and is able to predict contact separation between bodies having zero pre-impact relative velocity of approach. The solutions also exhibit an acceptable correlation with ADAMS software simulations.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology, Madras, India

    Koushik Kabiraj & Sourav Rakshit

Authors
  1. Koushik Kabiraj
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  2. Sourav Rakshit
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Correspondence to Sourav Rakshit.

Appendix

Appendix

1.1 Data for four-bar linkage colliding with two-link manipulator

See tables 2, 3, and 4.

Table 2 Contact point coordinates in the BCS of reference body for the two-link manipulator and four-bar linkage system shown in figure (2).
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Table 3 Initial values of the independent coordinates for the two-link manipulator and four-bar linkage system shown in figure (2).
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Table 4 Initial velocities of the independent coordinates for the two-link manipulator and four-bar linkage system shown in figure (2).
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1.2 Data for two-link manipulator colliding with multiple balls

See tables 5 and 6,

Table 5 Contact point coordinates in the BCS of the reference body for the two-link and two balls system shown in figure (5).
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Table 6 Initial values of the independent coordinates for the two-link and two balls system shown in figure (5).
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1.3 Data for spatial slider-crank colliding with multiple balls

See tables 7, 8, and 9

Table 7 Inertia properties of the slider-crank mechanism shown in figure (9).
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Table 8 Joint and contact coordinates in the BCS of the reference body for the spatial slider-crank mechanism shown in figure (9).
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Table 9 Parameters used in ADAMS for simulating the the slider-crank mechanism shown in figure (9).
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1.4 Data for top-hammer drill machine

See tables 10, 11, and 12.

Table 10 Inertia properties of the linkages in the top-hammer drill shown in figure (12b).
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Table 11 Initial orientation of linkages in top-hammer drill shown in figure (12b).
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Table 12 Joint and contact point coordinates in the BCS of the reference body for the top-hammer drill shown in figure (12b).
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Kabiraj, K., Rakshit, S. A quadratic programming based simultaneous impact model (QPSIM) for mechanisms. Sādhanā 49, 194 (2024). https://doi.org/10.1007/s12046-024-02525-9

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