CS277 - Experimental Haptics Lecture 13. Six-DOF Haptic Rendering I

Transcription

1 CS277 - Experimental Haptics Lecture 13 Six-DOF Haptic Rendering I

2 Outline Motivation Direct rendering Proxy-based rendering - Theory - Taxonomy

3 Motivation 3-DOF avatar

4 The Holy Grail?

5 Tool-Mediated Interaction How many degrees of freedom do we need?

6 One Caveat

7 6-DOF Interaction 3-DOF Position/Translation 6-DOF + Orientation/Rotation

8 Avatars for 6-DOF Haptics 3-DoF Position/Translation Render Force 6-DoF + Orientation/Rotation + Render Torque

9 Impedance-Controlled Device position, orientation force, torque

10 Direct Rendering Analogue to force field rendering Must consider multiple contacts in different positions for 6-DOF rendering

11 Forces on a Body M 2 = r 2 F 2 F 2 r 2 r 1 o F 1 M 1 = r 1 F 1 Output to Device: F = X i F i = X i M i

12 Contact Model For each contact, you will need The contact position on the tool, and one of - a force vector (magnitude + direction), or - a contact normal and penetration depth ˆn d F = k p dˆn

13 Demo

14 Properties of Direct Rendering What are the advantages and disadvantages? [From B. Heidelberger et al., Vision Modeling and Visualization, 2004.]

15 Direct Rendering Summary Advantages - Easy to implement - Free space feels like free space Limitations - Object interpenetration - Pop-through - Force discontinuities - Unbounded stiffness!

16 Proxy-Based Rendering

17 "Virtual Coupling" / haptic display.passive Tool Simulation [From J. E. Colgate et al., Proc. IEEE/RSJ IROS, 1995.]

18 6-DOF Virtual Coupling Translational and rotational spring/ Haptic Handle damper coupling d k T - Force proportional to displacement b T F spring - Torque proportional to orientation difference k R m b R F spring Virtual walls again! Dynamic Object [From W. A. McNeely et al., Proc. SIGGRAPH, 1999.]

19 Proxy Simulation in 3-DOF surface avatar device

20 Proxy Simulation in 6-DOF avatar surface??? device

21 Proxy Simulation τ? F surface

22 Soft Constraints F 2 = k x 2 F 1 = k x 1 nx F net = F i + F vc i

23 Proxy Motion Numerically integrate the ODE over time to obtain x, the position of the avatar: F 2 = k x 2 mẍ = F net Do the same with moments to obtain orientation F 1 = k x 1 F net = nx i F i + F vc

24 Potential Problems? m F vc = k vc x

25 Quasi-Static Equilibrium surface F c avatar F net F vc

26 Quasi-Static Equilibrium surface F c F net F vc

27 Quasi-Static Equilibrium surface F c F net = 0 F vc

28 Quasi-Static Proxy Motion Solve directly for the position x for which the net force acting on the proxy is zero: nx i k x i + k vc x vc = 0 Do the same with orientation to obtain net moment of zero F net = F 1 = k x 1 nx i F 2 = k x 2 F i + F vc

29 Still Problems? avatar

30 Hard Constraints ˆn 2 ˆn 1 r 1 r 2 Generalized acceleration: Non-penetration constraint: a (~a, ~ ) ~a ˆn + ~ (r ˆn) 0

31 Proxy Simulation τ? F

32 Solve for Contact Forces ˆn 2 ˆn 1 r 1 r 2 F 2 = f 2ˆn 2 F 1 = f 1ˆn 1 Find fi which satisfy: With condition: a i = ~a ˆn i + ~ (r i ˆn i ) 0 f i a i =0

33 Solve for Contact Forces Write motion of contact points as: a = Af + b Express conditions in matrix form: Af + b 0, f 0 and f T (Af + b) =0 Solve linear complementarity problem for f Integrate ODE to obtain position as before [From D. Baraff, Proc. SIGGRAPH, 1994.]

34 Solve Directly for Motion ˆn 1 r 1 F, τ surface device

35 Gauss Principle The proxy s constrained motion is that which minimizes the acceleration energy: a c = arg min 1 a 2 (F Ma)T M 1 (F Ma) Subject to the contact constraints: J c a 0 Solution can be obtained via quadratic programming or point projection [From S. Redon et al., Proc. IEEE Intl. Conf. on Robotics and Automation, 2002.]

36 Solve Directly for Motion ˆn 1 r 1 surface F, τ device

37 Taxonomy Soft Constraints Hard Constraints Massless Proxy Quasi-Static Equilibrium Distance Minimization Proxy with Mass Penalty-Based Dynamics Constrained Dynamics [Adapted from M. A. Otaduy et al., Proceedings of the IEEE, 2013.]

38 Soft vs. Hard Constraints F 1 = k x 1 F net = F 2 = k x 2 nx i F i + F vc ˆn 2 ˆn 1 r 1 r 2 ~a ˆn + ~ (r ˆn) 0

39 Proxy With vs. Without Mass m F vc = k vc x F c F net = 0 F vc mẍ = F net n X i k x i + k vc x vc = 0

40 Demo

41 Summary Motivation for 6-DOF haptic rendering Direct rendering - Like force fields: not very good! Proxy-based rendering - Taxonomy of proxy-based methods On Thursday: - Study examples of 6-DOF rendering methods