Higher-order derivatives of rigid body dynamics with application to the dynamic balance of spatial linkages

J.J. de Jong*, A. Müller, J.L. Herder

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

10 Citations (Scopus)
190 Downloads (Pure)


Dynamic balance eliminates the fluctuating reaction forces and moments induced by high-speed robots that would otherwise cause undesired base vibrations, noise and accuracy loss. Many balancing procedures, such as the addition of counter-rotating inertia wheels, increase the complexity and motor torques. There exist, however, a small set of closed-loop linkages that can be balanced by a specific design of the links' mass distribution, potentially leading to simpler and cost-effective solutions. Yet, the intricacy of the balance conditions hinder the extension of this set of linkages. Namely, these conditions contain complex closed-form kinematic models to express them in minimal coordinates. This paper presents an alternative approach by satisfying all higher-order derivatives of the balance conditions, thus avoiding finite closed-form kinematic models while providing a full solution for arbitrary linkages. The resulting dynamic balance conditions are linear in the inertia parameters such that a null space operation, either numeric or symbolic, yield the full design space. The concept of inertia transfer provides a graphical interpretation to retain intuition. A novel dynamically balanced 3-RSR spatially moving mechanism is presented together with known examples to illustrate the method.

Original languageEnglish
Article number104059
JournalMechanism and machine theory
Publication statusPublished - Jan 2021


  • Dynamic balance
  • Higher-order derivatives
  • Momentum
  • Multipole representation
  • Parallel mechanisms
  • Parameter-linear form
  • Rigid body dynamics
  • Screw theory


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