507 Constraint-based haptic rendering of a parametric surface N Zafer Department of Mechanical Engineering, Eskisehir Osmangazi University, Bati Meselik Kampusu, Eskisehir, 26480, Turkey. email: nzafer@ogu.edu.tr The manuscript was received on 12 May 2006 and was accepted after revision for publication on 20 November 2006. DOI: 10.1243/09596518JSCE297 Abstract: 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. Keywords: robotics, haptic surface, virtual manipulator, stability 1 INTRODUCTION restoring forces are always proportional to the gradient of the simulated surface at the contact [7], Haptic display of an object is mostly accomplished and several closest-point algorithms have been by providing the user via a haptic device (HD) with developed for haptic contact with polygonal models the ability for pointwise exploration of the object’s [7, 8], implicit models [9], and NURBS (non-uniform surface. The system is designed to generate contact rational B-spline) models [10]. In reference [11], forces to prevent penetration into the surface, as the Thompson et al. presented another kind of closest- collision between the user’s hand (HD end-effector) point algorithm, in which the closest point was and the object is detected. This approach is usually continuously updated (using the Newton iteration carried out by employing a penalty-based rendering method), based on the motion of the HD end- method [1], in which a closest point (proxy) [2] on effector. Later on, a more general iterative approach the surface is located to obtain the extension of a for surface-to-surface tracking was developed by virtual coupler [3]. The interaction with the surface is Nelson et al. [12] for general NURBS with polyhedral modelled through the coupler by means of feedback approximation, making use of surface tangents and a normal of each surface. The direct approach forces generated as a measure of the penetration depth. One drawback to the approach is that it developed by Patoglu and Gillespie [13] was based on control theory in maintaining the extremal distance produces the virtual environment mostly using non-passive algorithms (causing instabilities [4]) between parametric surfaces without polyhedral approximations. The motion along a (convex) surface and clever algorithmic techniques [5, 6] and must therefore be designed for complex geometries tangent direction was produced by an impedance- type controller, and integrated to determine the (including surface detail). Another drawback is that the HD end-effector is allowed to penetrate the closest point. Because global convergence can only be achieved [14] without resorting to polyhedral surface, before any action is taken, and this violates the real-world constraints. By this approach, the approximations, this penalty-based (non-iterative) JSCE297 © IMechE 2007 Proc. IMechE Vol. 221 Part I: J. Systems and Control Engineering