Higher-order derivatives of rigid body dynamics with application to the dynamic balance of spatial linkages. (January 2021) - de Jong, J.J.; Müller, A.; Herder, J.L.
Highlights: A full description of the dynamic balance solution for arbitrary non-singular linkages. A recursive screw based algorithm computes the derivatives of the momentum equation up to arbitrary order. The algorithm performs solely linear matrix operations allowing symbolic or numeric inputs. A systematic partitioning and graphical interpretation of the solution promotes insight and design intuition. A dynamically balanced design for the 3-RSR mechanism. Abstract: 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.