1. M. Zareinejad

2.  What ’ s Virtual Proxy? ◦ A substitute for the probe in the VE ◦ An extension of the ‘ God-Object ’ ◦ A finite sized massless sphere that runs after the probe

3.  Why sphere? ◦ To solve the ‘ fall-through ’ problem of the God- Object method ◦ For easy collision-detection

4.  ‘ Fall-through ’ of the God-Object

5.  Virtual Proxy ’ s behavior in the same situation

6.  Example

7.  Check whether a line-segment, specified by the proxy and the probe, falls within one radius of any obstacle in the environment  This line-segment checking method can successfully render thin objects

8.  Configuration space obstacle ◦ A mapped obstacle to the configuration space ◦ In our problem, it consists of all points within one proxy radius of the original obstacle  Constraint plane ◦ Where the line-segment intersects the configuration space obstacle

9.  The proxy moves to the probe until it makes a contact with a C-obstacle  If the proxy makes a contact, it moves to the closest position to the probe on the constraint plane

10.  A sub-goal can be represented by minimize ∥x-p∥ subject to n i x ≥ 0, 0 ≤ i ≤ m ◦ p is the vector from the current proxy to the probe ◦ x is the sub-goal ◦ n i, 0 ≤ i ≤ m, are the unit normals of the constraint planes  The problem can be solved using a standard quadratic programming package, or a similar method that the God-Object method uses

11.  the force exerted on the proxy by the user can be estimated by f = k p (p-v) ◦ k p is the proportional gain of the haptic controller ◦ p is the position of the proxy ◦ v is the position of the probe

12.  If ∥ f t ∥ ≤ μ s ∥ f n ∥, proxy is not moved ◦ f is the estimated force exerted on the proxy ◦ f n is the vertical element of f on the constraint plane ◦ f t is the horizontal element of f on the constraint plane ◦ μ s is static friction parameter of constraint surface

13.  The motion of one dimensional object is ◦ μ d is the dynamic friction parameter of the surface ◦ m is the mass of the object ◦ x ’’ is the acceleration of the object ◦ x ’ is the velocity of the object ◦ b is the viscous damping parameter

14.  Because the mass of the proxy is 0, the previous equation can be rewritten as  This equation can be used to bound the amount that the proxy can move in one clock cycle

15.  Stiffness of a surface can be modeled by reducing the position gain of the haptic controller  But changing the position gain is not desirable  Solve this problem by repositioning the proxy

16.  ◦ p is the position of the proxy ◦ p’ is the new position of the proxy ◦ v is the position of the probe ◦ s is the stiffness parameter of the surface, 0≤s≤1  p’ is used for the haptic control loop  p is retained for surface following

17.  D. Ruspini, K. Kolarov, and O. Khatib, "The Haptic Display of Complex Graphical Environments," in Computer Graphics Proceedings (ACM SIGGRAPH 97), 1997, pp. 345-352.