Constraint-based haptic rendering of a parametric surface

This paper presents a direct method for haptic rendering of a virtual object in which the object is represented as a virtual kinematic chain (virtual manipulator). The joint angles of the virtual manipulator (VM) are considered as parameters for the object's surface. The present algorithm is based on a closest-point approach that determines the joint angles (surface parameters) uniquely. The joint angles parameterize a point closest to the haptic device end-effector, and an impedance-type controller is designed for the haptic device that accounts for the haptic rendering algorithm. Within the control law, only the forces orthogonal to the object surface are rendered using the Jacobian of the VM, and the user feels a smooth surface whose stability (considering the coupled haptic device dynamics and closest-point algorithm kinematics) is guaranteed. Additional motion constraints on the virtual surface are also created by penalizing the joint angles of the VM, showing how this approach provides an efficient tool in designing a CAD (computer aided design) model.