 Abstract—It is a challenging problem to achieve fast and realistic six degree-of-freedom (DOF) haptic simulation of scenarios involving large number of multi-region contacts. In this paper, we propose an optimization-based constrained method enhanced by parallel quadratic programming to solve the rendering problem. Hierarchical sphere-tree models are used to represent the moving haptic tool and its surrounding static objects. Given a moving graphic tool as the avatar of the haptic tool in the virtual environment, we compute its quasi-static motion by solving a configuration-based optimization. Instead of using traditional active-set method, we transform the original optimization problem into its dual problem and solve the optimum about the graphic tool using a parallel quadratic programming method. Our algorithm has been implemented with a 6-DoF Phantom Premium 3.0. We validate the proposed algorithm in several benchmarks involving complex, large-region contacts. The results demonstrate that the proposed method can achieve a two to three times speed improvement than the active-set method. A further speed-up for haptic rendering may be achieved by the parallel implementation on parallel processor such as graphic processing units. I. I NTRODUCTION 6-DoF haptic rendering, defined as the process of computing and generating forces and torques in response to user interactions with virtual objects, can greatly benefit many applications involving medical training tasks and dexterous engineering manipulation, such as dental surgical simulation and virtual assembly [1-5]. In these applications, large area/volume contacts are commonly to be interacted, such as probing the periodontal pocket depth to diagnose whether it’s the healthy periodontium or with inflammation in Fig.1. When a dentist inserts the dental probe into the bottom of pocket, the working end of the probe will totally be wrapped with the tooth and gingival. Another example is a virtual assembly task as shown in Fig.2. The collision occurs between the outer surface of the splined shaft with numbers of keys and the inner surface of the splined hole with the same number of keyways, which leads to a rather large contact region. In these cases, the performances of stability and real-time for force and torque output become particularly important to simulate haptic-enabled tasks realistically. *Research supported by the National Natural Science Foundation of China Ge Yu, Dangxiao Wang and Yuru Zhang are with the State Key Lab of Virtual Reality Technology and Systems, Beihang University, Beijing, 100191, China. Dangxiao Wang (corresponding author to provide e-mail: hapticwang@buaa.edu.cn); Ge Yu; Yuru Zhang. A. 6-DoF Haptic Rendering Approaches Several 6-DoF haptic rendering approaches have been proposed in recent years, which can be generally classified into two groups according to the collision response as penalty-based and constraint-based methods. Most of existing methods are penalty-based in which the graphic tool as a dynamic object governed by Newtonian principle [2-7]. These algorithms may allow interpenetration between virtual tool and virtual environments such as the graphic tool can traverse through thin objects, or introduce some forms of virtual coupling [8] to keep feedback force stable, which results in a reducing perception of geometric details by filtering the changes of forces orientation. Compared with penalty-based methods, constraint-based methods can eliminate the haptic and visual artifacts by obtaining an exact contact configuration of the tool. Duriez et al. modeled the non-penetration and friction contact as a linear complementary problem (LCP) to solve 6-DoF haptic rendering problem [9], [10]. Based on the 3-DoF god-object approach, Ortega et al. [11] extended it into 6-DoF haptic rendering, making the generalized acceleration of the graphic tool as the variable and simulating the motion of the god-object by solving the Gauss’s projection problem. These approaches could allow a more realistic force, but the high computational cost will lead to a low efficiency for detailed objects. In our previous work [12], [13], a novel configuration-based 6-DoF haptic rendering method used sphere-trees to model arbitrary objects and formulated non-penetration constraints for active-set based optimization. For a common model with 4681 spheres (an octree with 4 levels), if more than 100 pairs of spheres are detected being intersected, the rate of haptic loop will decrease far away from real-time feedback of 1kHz which is also not appropriate for simulating large area/volume contacts in real time. The time cost of optimization becomes the bottleneck to achieve the realistic haptic feedback. Accelerating Optimization-based Haptic Rendering by Parallel Quadratic Programming Method Ge Yu, Dangxiao Wang, and Yuru Zhang, Senior Member, IEEE (a) (b) Figure 1. (a) Healthy periodontium Figure 2. Virtual assembly of splined (b) Periodontium with inflammation shaft and hole 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) November 3-7, 2013. Tokyo, Japan 978-1-4673-6357-0/13/$31.00 ©2013 IEEE 4499