Hyper-Jerk Analysis

Abstract

The hyper-jerk, also known as snap or jounce, is the time rate of the jerk and is more than an academic pursuit; for example, Wohlhart (2010) showed that any planar parallel manipulator at a singular configuration could be architecturally mobile according to hyper-jerk analysis. In this chapter the hyper-jerk analysis of rigid bodies is approached using screw theory. A few decades ago the theory of screws introduced by Ball (1900) seemed to be an “old-fashioned” mathematical tool limited to elementary kinematic analyses, such as the displacement and velocity analyses of rigid bodies. Although the robustness of the definition of reduced acceleration state of a rigid body was introduced by Brand (1947), the representation in screw form of the acceleration analysis of kinematic chains was formally published half a century later by Rico and Duffy (1996). Before that contribution, most kinematicians commonly accepted that screw theory could not handle higher-order analyses of mechanisms, which today we recognize to be a wrong assumption. The escape from singular configurations of serial manipulators, the characterization of singularities of closed chains, as well as the acceleration analysis of parallel manipulators have been the most recurrent subjects since the formulation introduced by Rico and Duffy (1996) was proved successfully; see, for instance, Rico et al. (1995), Rico and Gallardo (1996), Gallardo-Alvarado et al. (2007, 2010), Gallardo et al. (2008). Furthermore, the elucidation of the acceleration analysis in screw form opened the possibility to extend it to the jerk analysis (Rico et al. 1999; Gallardo-Alvarado and Rico-Martinez 2001), which was applied in the higher-order kinematic analyses of parallel manipulators (Gallardo-Alvarado 2003, 2012; Gallardo-Alvarado and Camarillo-Gomez 2011). In that concern, as shown by Lipkin (2005), screw theory is not limited by the order of the kinematic analysis. Without a doubt, the higher-order analysis is a subject of growing interest covering topics like path-planning control (Kyriakopoulos and Saridis 1994), improvement of maneuvers of mobile robots (Oyadiji et al. 2005), high-speed machining (Zhang et al. 2011), jerk dynamics (Maccari 2011), smooth path planning for conventional vehicles (Villagra et al. 2012), optimal time-jerk algorithms for trajectory planning (Gasparetto et al. 2012), and so forth.