Skip to main content
SpringerLink
Log in

Constraint-wrench analysis of robotic manipulators

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

The constraint-wrench analysis of mechanisms, with focus on parallel robots, is the subject of this paper. Although the method proposed here can be generalized for parallel robots with multiple-loop kinematic chains, here, single-loop chains are targeted. To this end, a novel representation of the constraints imposed by the kinematic pairs is introduced. With this representation, the constraint matrix of a mechanism is readily derived. For the calculation of the constraint wrenches, by means of the constraint matrix and based on the Newton–Euler formulation, a new procedure is introduced. As a case study, the constraint wrench analysis of the McGill Schönflies Motion Generator (SMG), while undergoing a test cycle adopted by the industry, is conducted.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

Notes

  1. This mode of representation of twist and wrench arrays is crucial here, for otherwise the inner product of Eq. (7) would be meaningless.

  2. As screw arrays do not admit a norm, they neither admit the concept of orthogonality; what they admit is the concept of reciprocity, as per Eq. (8).

References

  1. Staicu, S., Carp-Ciocardia, D.C.: Dynamic analysis of Clavel’s delta parallel robot. In: Proc. of the 2003 IEEE Int. Conference on Robotics and Automation, Taipei, Taiwan, September 14–19 (2003)

    Google Scholar 

  2. Choi, H.B., Konno, A., Uchiyama, M.: Inverse dynamic analysis of a 4-dof parallel robot h4. In: Proc. of 2004 IEEE/RJS Int. Conference on Intelligent Robots and Systems, Sendai, Japan, September 28–October 2 (2004)

    Google Scholar 

  3. Miller, K.: Dynamics of the new uwa robot. In: Proc. 2001 Australian Conferences on Robotics and Automation, Sydney, Australia, November 14–19 (2001)

    Google Scholar 

  4. Pang, H., Shahinpoor, M.: Inverse dynamics of a parallel manipulator. J. Robot. Syst. 11(8), 693–702 (1994)

    Article  MATH  Google Scholar 

  5. Li, Y., Qingsong, X.: Dynamic analysis of a modified delta parallel robot for cardiopulmonary resuscitation. In: Proc. 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2005), Edmonton, Canada, August 2–6 (2005)

    Google Scholar 

  6. Staicu, S.: Matrix modeling of inverse dynamics of spatial and planar parallel robots. Multibody Syst. Dyn. 27(2), 239–265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Khalil, W., Ibrahim, O.: General solution for the dynamic modeling of parallel robots. J. Intell. Robot. Syst. 49, 19–37 (2007)

    Article  Google Scholar 

  8. Angeles, J., Lee, S.: The modelling of holonomic mechanical systems using a natural orthogonal complement. Trans. Can. Soc. Mech. Eng. 13, 81–89 (1989)

    Google Scholar 

  9. Angeles, J.: Fundamentals of Robotic Mechanical Systems Theory, Methods, and Algorithms. Springer, New York (2007)

    Book  MATH  Google Scholar 

  10. Roth, B.: Screws, motors, and wrenches that cannot be bought in a hardware store. In: Brady, M., Paul, R.P. (eds.) Robotics Research. The First International Symposium, pp. 679–693. MIT Press, Cambridge (1984)

    Google Scholar 

  11. Klein, F.: Ueber liniengeometrie und metrische Geometrie. Math. Ann. V, 257–303 (1871)

    Google Scholar 

  12. Hartenberg, S., Denavit, J.: Kinematic Synthesis of Linkages. McGraw-Hill, New York (1964)

    Google Scholar 

  13. Hervé, J., Sparacino, F.: Star, a new concept in robotics. In: Proc. 3rd Int. Workshop on Advances in Robot Kinematics, Ferara, Italy, September 7–9, pp. 176–183 (1992)

    Google Scholar 

  14. Wohlhart, K.: Displacement analysis of the general spatial parallelogram manipulator. In: Proc. 3rd Int. Workshop on Advances in Robot Kinematics, Ferara, Italy, September 7–9, pp. 104–111 (1992)

    Google Scholar 

  15. Hervé, J.: The mathematical group structure of the set of displacements. Mech. Mach. Theory 29(1), 73–81 (1994)

    Article  Google Scholar 

  16. Morozov, A., Angeles, J.: The mechanical design of a novel Schönflies motion generator. Robot. Comput.-Integr. Manuf. 23, 82–93 (2007)

    Article  Google Scholar 

  17. Blajer, W., Schiehlen, W., Schirm, W.: A projective criterion to the coordinate partitioning method for multibody dynamics. Appl. Mech. 64, 86–98 (1994)

    MATH  Google Scholar 

  18. Bayo, E., Ledesma, R.: Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics. Nonlinear Dyn. 9, 113–130 (1996)

    Article  MathSciNet  Google Scholar 

  19. Zahariev, E., Cuadrado, J.: Dynamics of over-constrained rigid and flexible multibody systems. In: 12th IFToMM World Congress, Besançon, France, June 18–21 (2007)

    Google Scholar 

  20. Pahl, G., Beitz, W., Feldhusen, J., Grote, K.H.: In: Wallace, K., Blessing, W. (eds.) Engineering Design. A Systematic Approach. Springer, London (2007)

    Google Scholar 

  21. Gauthier, F., Angeles, J., Nokleby, S.: Optimization of a test trajectory for Scara systems. Adv. Robot Kinemat. Anal. Des. 4, 225–234 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Canada’s Natural Sciences and Engineering Research Council (NSERC) for providing funds to support this research via an Idea to Innovation Grant, which allowed the team to produce the experimental platform motivating the work reported here. Further work has been supported under NSERC’s Discovery Grants program and partly through a James McGill Professorship to the second author.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montreal, PQ, H3A 2K6, Canada

    Afshin Taghvaeipour & Jorge Angeles

  2. Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, PQ, H3A 2K6, Canada

    Larry Lessard

Authors
  1. Afshin Taghvaeipour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Angeles
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Larry Lessard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Afshin Taghvaeipour.

Appendices

Appendix A: Derivation of the twist shaping matrices

As shown in Eq. (52), the joint angle vector is mapped onto the twist vector of a robot by means of the twist shaping matrix. This matrix is composed of the twist-shaping matrices T i of all rigid bodies, namely,

(69)

which is a 6(n−1)×(d+p) matrix. The time derivative of the twist-shaping matrix is simply obtained by the derivation of its components with respect to time

(70)

In the sequel, the twist-shaping matrices of the seven rigid bodies of the McGill SMG, along with their time derivatives, will be needed. The parameters regarding the geometry of the McGill SMG that are used in the following formulas are depicted in Figs. 25 and 26.

where O n×m denotes the n×m zero matrix, and 0 m the m-dimensional zero vector.

with

with

with

with

with

Finally,

Fig. 25
figure 25

The McGill SMG dimensions

Full size image
Fig. 26
figure 26

The zoom-in of the MP and its dimensions

Full size image

Appendix B: Derivation of the angular velocity dyads

All the rigid bodies in the McGill SMG move under the same Schönflies displacement subgroup, which contains three translation and one rotation about the z-axis of Fig. 12. Therefore, the angular velocity of each rigid body of the Jth limb is the same, i.e., for the first three rigid bodies, which belong to limb I,

(71)

The last three rigid bodies also rotate with angular velocity \(\dot {\theta}_{\mathit{II}1}\), therefore,

(72)

The fourth rigid body, the MP, rotates through angle θ I1−θ I4, and hence,

(73)

Angeles [9] defined the angular-velocity dyad of an angular velocity vector as

(74)

in which Ω is the CPM of the angular velocity vector ω and O is the 3×3 zero matrix. It is important to note that since the Newton–Euler equations for a rigid body are cast in terms of the twist, which is a 6-dimensional array, Ω is defined as a matrix of 6×6. Thus, the seven angular velocity dyads corresponding to the robot rigid bodies are

with Ω i =CPM(ω), and hence,

(75a)
(75b)
(75c)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Taghvaeipour, A., Angeles, J. & Lessard, L. Constraint-wrench analysis of robotic manipulators. Multibody Syst Dyn 29, 139–168 (2013). https://doi.org/10.1007/s11044-012-9318-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-012-9318-7

Keywords

Search

Navigation