Estimating How Сlose Parallel Mechanisms are to their Singularities by Differentiating Constraint Equations

The paper presents an approach to computing velocity- and force-based criteria for determining how close a parallel mechanism is to its singularity by using inverse Jacobian matrix components derived via differentiating constraint equations. We show that, for this type of mechanisms, methods based on screw theory are not required to achieve the desired result; however, in this case certain assumptions are necessary. Let us consider a planar parallel mechanism. We limited our computation to linear output link velocity and a single force as external loading, which allows us to avoid normalising vectors containing quantities of dissimilar physical significance and to use this approach to analyse mechanisms with mixed degrees of freedom. Our computation yields maximum velocities and forces in actuated joints, which we compare to values computed for two velocity and external force vectors that have the same magnitude but whose direction is constant. We show that the approach suggested actually produces the maximum possible velocities and forces in actuated joints. At the same time, when the mechanism approaches its singularities, we note a significant increase in the deviation of the values computed via this method from those computed for output link velocities or external forces whose direction is constant

The study was conducted as part of a government scientific investigation assignment (grant no. 9.5309.2017/8.9)

References

[1] Hunt K.H. Special configurations of robot-arms via screw theory. Robotica, 1986, vol. 4, no. 3, pp. 171–179.

[2] Sugimoto K., Duffy J., Hunt K.H. Special configurations of spatial mechanisms and robot arms. Mech. Mach. Theory, 1982, vol. 17, no. 2, pp. 119–132. DOI: 10.1016/0094-114X(82)90042-8

[3] Gosselin C., Angeles J. Singularity analysis of closed-loop kinematic chains. IEEE Trans. Robot. Autom., 1990, vol. 6, no. 3, pp. 281–290. DOI: 10.1109/70.56660

[4] Glazunov V.A., Koliskor A.Sh., Kraynev A.F. Prostranstvennye mekhanizmy parallelnoy struktury [Spatial mechanisms of parallel structure]. Moscow, Nauka Publ., 1991.

[5] Zlatanov D., Fenton R.G., Benhabib B.A. Unifying framework for classification and interpretation of mechanism singularities. JMD, 1995, vol. 117, no. 4, pp. 566–572. DOI: 10.1115/1.2826720

[6] Laryushkin P.A. Classification and occurrence conditions of singularities in parallel mechanisms. Izvestiya vysshikh uchebnykh zavedeniy. Mashinostroenie [Proceedings of Higher Educational Institutions. Маchine Building], 2017, no. 1, pp. 16–23 (in Russ.). DOI: 10.18698/0536-1044-2017-1-16-23

[7] Merlet J.-P. Parallel robots. Springer, 2006.

[8] Hubert J., Merlet J.-P. Static of parallel manipulators and closeness to singularity. J. Mech. Robot., 2008, vol. 1, no. 1, art. 011011. DOI: 10.1115/1.2961335

[9] Glazunov V.A., Arakelyan V., Brio S., et al. Speed and force criteria for the proximity to singularities of parallel structure manipulators. J. Mach. Manuf. Reliab., 2012, vol. 41, no. 3, pp. 194–199. DOI: 10.3103/S1052618812030041

[10] Laryushkin P.A., Rashoyan G.V., Erastova K.G. On the features of applying the theory of screws to the evaluation of proximity to specific positions of the mechanisms of parallel structure. J. Mach. Manuf. Reliab., 2017, vol. 46, no. 4, pp. 349–355. DOI: 10.3103/S1052618817040100

[11] Gosselin C. Kinematic analysis optimisation and programming of parallel robotic manipulators. McGill University, 1988.

[12] Bonev I., Gosselin C. Singularity loci of planar parallel manipulators with revolute joints. Proc. 2nd Workshop on Computational Kinematics. Seoul, Korea, 2001, 10 p.