This paper applies geometric design principles to planar mechanical systems, attempting to explain complex motion in mechanism–object/environment interaction for realizing multiple kinematic tasks in the vicinity of a contact location. The latter is a critical feature not fully captured in existing design methodologies. This is achieved through the development of a general planar geometric model that allows the derivation of multiple velocity and acceleration specifications compatible with mechanism–object curvature constraints in the vicinity of the contact. By incorporating these higher-order kinematic specifications into the design task formulation, contemporary planar kinematic synthesis is generalized allowing robust designs. The results are illustrated by the kinematic synthesis of two planar revolute–revolute (RR) linkages that are able to push and roll-slide along the object’s curvature in the vicinity of a contact location, a hybrid hand that incorporates a four-bar prosthetic finger that is able to grasp objects in two different modes, and a six-bar orthotic wheelchair for disabled canines that integrates body tilt and climbing motions.

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