Curvature based mobility analysis and form closure of smooth planar curves with multiple contacts

• Geometric approach used osculating circles for curvature based mobility analysis.
• Analytical approach used 2nd order Taylor expansion of rotation matrix.
• Vector representation of contact enabled mobility-Boolean for multi-contact cases.
• Unified framework for mobility, from closure, and synthesis of kinematic pairs
• Synthesis of planar kinematic pairs is shown to require second-order results.

This paper presents a simple second-order, curvature based mobility analysis of planar curves in contact. The underlying theory deals with penetration and separation of curves with multiple contacts, based on relative configuration of osculating circles at points of contact for a second-order rotation about each point of the plane. Geometric and analytical treatment of mobility analysis is presented for generic as well as special contact geometries. For objects with a single contact, partitioning of the plane into four types of mobility regions has been shown. Using point based composition operations based on dual-number matrices, analysis has been extended to computationally handle multiple contacts scenario. A novel color coded directed line has been proposed to capture the contact scenario. Multiple contacts mobility is obtained through intersection of the mobility half-spaces. It is derived that mobility region comprises a pair of unbounded or a single bounded convex polygon. The theory has been used for analysis and synthesis of form closure configurations, revolute and prismatic kinematic pairs.