力触覚の提示と計算研究委員会　第 14回研究会 (2015年 3月)

PBDによる変形物力覚レンダーリングの改善

Improvement of Deformable objects Haptic Rendering based on Position based dynamics

丁海陽 1), 三武 裕玄 2), 長谷川　晶一 3)

Ding Haiyang, Hironori Mitake and Shoichi Hasegawa

1) 東京工業大学 大学院総合理工学研究科 知能システム科学専攻

（〒 226-8503 神奈川県横浜市緑区長津田町 4259 R2-626, teikaiyo@haselab.net）

2) 東京工業大学 精密工学研究所

（〒 226-8503 神奈川県横浜市緑区長津田町 4259 R2-624, mitake@pi.titech.ac.jp）

3) 東京工業大学 精密工学研究所, 独立行政法人 科学技術振興機構

（〒 226-8503 神奈川県横浜市緑区長津田町 4259 R2-624, hase@pi.titech.ac.jp）

概要:

We propose an improvement for stable haptic rendering method for deformable objects. The method

we use for deformation calculation is called Oriented Particle (OP). Since OP is based on position

based dynamics (PBD) simulation, we calculate user-input of virtual coupling by a PBD constraint

in OP. This constraint solves haptic input by displacement to match the PBD feature. We call it

”haptic constraint”. The instability problem occurred in explicit integration haptic simulation is

solved by this method and we achieved a great alleviation of oscillation during the interaction. Our

main contribution is an improved of haptic rendering system even using high spring value or low

model stiffness. This work is achieved by taking advantage of PBD, especially for deformable model

interaction.

キーワード： Haptic rendering, Position based Dynamics, Oriented Particle

1. INTRODUCTION

The dynamic simulation of haptic interaction with solid

or deformable objects has been investigated for more

than two decades. Applications in virtual prototyping,

creative modeling and the medical area proved the value

of haptics in virtual reality systems. In order to build a

deformable object haptic system of rich reality, our re-

search focuses on the Oriented Particle (OP) method,

which is faster and more robust than high cost methods

such as the general Finite Element Model (FEM).

Haptic rendering requires a haptic interface, a virtual

environment and a control law linking them. Generally

the device end-point is connected to a virtual environ-

ment proxy through the control law, called virtual cou-

pling. The conventional virtual coupling scheme uses a

spring and a damper in parallel between the device end-

point and the proxy. Forces and moments resulting from

user inputs are then able to be calculated from the vir-

tual coupling, and applied to both the virtual model and

the haptic interface.

Since OP is implemented on Position Based Dynamics

(PBD), user inputs of the previous step are used to start

the calculation of the current step. However, in this gen-

eral PBD approach, haptic user inputs cause oscillations

on the contact position when using low model stiffness or

a high virtual coupling spring value.

We propose a improved stable haptic rendering method

following the PBD concept. This method works in a sim-

ilar way to the PBD constraint and can be seen as a PBD

constraint design for haptics. We will describe this in de-

tail in section III. Our implementation work is depicted

in section IV, while evaluation follows in section V.

2. RELATED WORK

Over the past decades, much attention has been de-

voted to research on haptic simulation with deformable

objects. Recently, Barbic[1] succeeded in presenting a

two object real-time haptic interaction with 1 kHz hap-

tic rate. He achieved this with a reduced FEM, named

St.Venant-Kirchhoff Deformable Model, by using dimen-

sional model reduction. Galoppo[2] gave a haptic sim-

ulation environment that can handle multiple objects.

These objects are simulated by a rotational FEM method

but it is still inevitable to use the GPU to achieve real-

time simulation.

As a geometric deformation calculation, Muller[4] pre-

sented the shape-matching method. Muller[6] extended

his method, introducing OP as a sparse arranged struc-

ture based on shape-matching. Since it is a fast and

robust method, OP is easier to implement and simplifies

the simulation of large models compared to general FEM

without reduction or the support of GPU. Our idea is

taking advantage of these features, as we use OP for our

deformation calculation.

Our research starts from three-degree-of-freedom (3-

DOF) haptic interaction. Zilles[8] proposed the god-

object as the first application of Lagrange multipliers for

contact in 3-DOF haptic rendering. The proxy method[7]

published by Ruspini is an extension of god-object by us-

ing a spherical shape and an updating algorithm similar

to robotic arm motion. We implement this method in

our system. Other methods, such as the ideal haptic in-

terface point (IHIP)[3], which limits the intersection test

only around neighbor primitives, are not considered at

this time.

3. METHOD

In this section, we will begin with a short explanation

of PBD. Then we will introduce our new method.

3.1 Basic position based dynamics

Since any time integration generally computes veloci-

ties first from the accelerations and then positions from

velocities, PBD uses accelerations and velocities of the

previous simulation step to predict positions. Then, af-

ter revising the positions by constraints, the velocities

of the current step are calculated from the difference of

positions. This is why PBD is called position based.

For the sake of a clear view of PBD flow, the main

PBD simulation loop is as follows:

• Initialize xi,vi for all vertices.

• Main Loop:

– Compute velocity for all vertices:

vi(t0) = vi(t−1) + ∆tfi

– Predict position for all vertices:

pi = xi(t0) + ∆tvi(t0)

– Iteration Loop:

∗ Constraints projection for all vertices:

gi = (C1(pi), C2(pi), ...)

pi = gi

– End iteration loop

– Update velocity for all vertices:

vi(t1) = (gi − xi(t0))/∆t

– Apply position for all vertices:

xi(t1) = gi

• End main loop

First, the vertex position xi is predicted according to

the velocity vi, computed from the the acceleration ob-

tained from the external force fi. Then the predicted

position pi is corrected in the constraint projection step

such that it satisfies a set of constraints (C1(pi), C2(pi), ...),

such as collision or deformation. This step is looped one

or more times. Finally, the corrected predicted position,

called goal position gi, is applied to the vertex position xi

to be displayed while the velocity vi is updated from the

position difference. By this way of simulation, vertex i is

directly projected to a valid location, under the control

of constraints.

3.2 Haptic constraint

As we described before, haptic rendering needs virtual

coupling to link a dynamic model of the virtual proxy

with a haptic interface. Conventionally, the relative mo-

tion of the virtual proxy and the haptic interface are

linked by a spring and a damper. Then, user inputs are

transformed to forces and moments and applied to the

model. Finally, the output displayed on the haptic inter-

face is equal and opposite to these forces and moments.

In PBD, force based simulation accumulates forces and

moments at the beginning of each simulation loop. That

means user inputs applied to model are generally involved

in the position prediction step. Since they are calculated

from the previous simulation loop, the effect of user in-

puts occurs one loop later than the other constraints.

This explicit way of integration becomes unstable when

the user input introduces a relatively large effect com-

pared to the effect of PBD constraints. Examples that

can lead to such behavior are the use of a high virtual

coupling spring value or low stiffness deformable models

without damping.

By following the concept of PBD, we design a new vir-

tual coupling as a constraint to build a non-explicit inte-

gration. This constraint is also a part of the constraint

projection loop. Here, user inputs are introduced to the

PBD solver as a spring constraint and projected together

with other constraints. This concept was first proposed

Figure1: Calculation of haptic constraint: Dis-

tance from proxy position to haptic device end-

point is projected to particle translation and ro-

tation to compute the goal position of haptic con-

straint

by Anitescu[9] in a spring model for implicit integration

of dynamic simulation. In PBD, a stiffness coefficient

is used to replace the effect of the spring. We will give

an explanation on how to calculate this constraint in the

next section.

On the other hand, the feedback force to the user is

calculated in the same way as the conventional method.

The same spring coefficient as the spring constraint is

used here. Therefore the haptic simulation scheme is no

longer explicit, and the unstable interactions generated

by explicit integration are eliminated. This is the main

advantage of our proposal. We named this method ”hap-

tic constraint”.

4. IMPLEMENTATION

In this section, we will describe how our method is

implemented in a PBD system especially for OP config-

urations.

4.1 Calculation of Haptic Constraint

The haptic constraint is used to apply haptic external

effects on a virtual model. Specifically for PBD, this

constraint works by pushing the model vertices forward.

In a simple PBD vertex case, if the proxy tries to push

the vertex, the pushed distance is from proxy position l

to the location of the haptic device end-point u together

with a stiffness kc like this:

gi = pi + kc(u− l), (1)

where gi is the goal position of the vertex in general PBD

and pi is the predicted position of the same vertex. This

equation corrects the vertex position to the goal position

of the haptic constraint.

Since the PBD constraint is iterated multiple times,

the effect of kc is non-linear. Here we follow the stiffness

transformation equation[6] of PBD to linearize the effect

of kc by k′c = 1 − (1 − kc)

1/ns . Here ns stands for the

iteration time. Therefore we are able to use kc in a linear

form similar to a conventional spring.

As in OP, vertices are controlled by particles and those

particles are corrected in the PBD constraint loop. We

realize the push by applying a weighted distance value

to surrounding particles resulting in a peak shaped force

effect. Since replacing a force by a distance applied on

particles will cause both translation and rotation in dy-

namics, we compute both for each particle, as shown in

figure 1. They can be formulated as follows:

The push direction n is computed from the proxy po-

sition l and the haptic device end-point u:

n =u− l

d, (2)

Here, d is the total pushed distance,

d = ||u− l||. (3)

We start the deduce from the force integration scheme of

dynamics equations:

mv =

∫f dx, (4)

Iω = r×∫

f dx. (5)

Besides equation 3 the pushing distance of proxy d can

be also defined for a single particle as:

d =

∫V · ndx. (6)

=

∫(v + ω × r) · ndx. (7)

V is the total velocity of this movement. Then, by sub-

stituting the equation 4 and equation 5, the following

deduction is:

d =

∫(1

m

∫f dx+ (I−1r×

∫f dx)× r) · ndx (8)

=

∫(n

m

∫f − I−1r× n× r ·

∫f) · ndx (9)

= (n

m− I−1r× n× r) · n ·

∫∫fdx (10)

So, ∫∫fdx =

d

( 1mn+ I−1r× n× r) · n

(11)

By integrating the equation 4 and the equation 5 by time

we get

mt = n ·∫∫

fdx (12)

Ia = r× n ·∫∫

fdx (13)

Use equation 11 to substitute∫∫

fdx in the equation

above, we get the final equation for a single particle.

t =1

mn

d

( 1mn+ I−1r× n× r) · n

, (14)

a = I−1r× nd

( 1mn+ I−1r× n× r) · n

. (15)

Since the pushed distance is weighted for each particle

according to the length from proxy to the particle center,

we use these equation in our implemented system:

tp =1

mn

wpd

( 1mn+ I−1r× n× r) · n

, (16)

ap = I−1r× nwpd

( 1mn+ I−1r× n× r) · n

, (17)

Finally, the PBD goal position and orientation of the

haptic constraint are calculated for each particle, by the

following equations:

gp = pp + k′ctp, (18)

Rp = Qp + k′cAp, (19)

where gp and Rp are the particle goal position and ori-

entation matrix, pp and Qp are the predicted particle

position and orientation matrix, while Ap is the rotation

matrix transformed from the rotation vector ap. Thus

we are able to apply our haptic constraint on OP.

After the constraint loop, the final goal position and

orientation are calculated and the proxy position is set-

tled down. The settled distance from proxy to device

end-point is then passed to the haptic interface to calcu-

late the device force feedback.

5. STABILITY EVALUATION

This section describes our method evaluation. We

will give a stability comparison of our haptic constraint

method to the conventional force based method after the

explanation of how we set-up our experiments.

5.1 Set up

To implement the conventional force based method,

we treat the user input as the external effect in general

PBD flow, which is calculated in the prediction step. For

a valid and direct comparison, we want to identify the

same spring effect for both the force based method spring

and the haptic constraint spring. Since PBD’s spring

coefficient is a constraint gradient, the coefficient is not

the same as the ordinary force based spring. However,

the linearly corrected coefficient kc is proportional to a

force based spring coefficient. Therefore, we made an

identification simulation for both methods as follows:

Figure2: Evaluation. The force based simulation

with increasing spring value.

Figure3: Evaluation. The haptic constraint with

increasing spring value.

1. Use a determinate spring value kf for the force

based method;

2. Let the proxy push the particle to deform a dis-

tance hf , then record hf ;

3. Adjust the stiffness kc of the haptic constraint and

make the same proxy motion to push the particle

deform a distance hc.

Thus kc matches the conventional spring value kf while

hf = hc. We do this three times to get the three matching

spring values for stability experiments.

The model we used for evaluation is a simple box model

of size 2cm×2cm×2cm and weight 8g. Particles are ran-

domly arranged on the box, totaling 125 vertices with 25

particles.

5.2 Simulation Result

Our system is able to simulate 1kHz haptic interation

in real-time by implementing a multi-rate method. We

use 0.01 second as a fixed time step similar to the general

OP system. We use Spidar-G6[10] as our haptic device

in both the evaluation section and the next section.

Figure4: Evaluation. The force based simulation

with decreasing model stiffness.

Figure5: Evaluation. The haptic constraint with

decreasing model stiffness.

The evaluation is done by the procedure: move device

end-point downwards, touch the model and move back

up.

In the first simulation, we compare different spring val-

ues for both the haptic constraint method and the force

based method. The three spring values kf are 100 N/m,

200 N/m and 300 N/m, and the corresponding kc are

0.075, 0.15 and 0.225. The Y coordinates of the feedback

force, the positions of the device end-point, the proxy

and the nearby particle are displayed in figure 2 and fig-

ure 3. Oscillation occurs for high virtual coupling spring

values in the force based simulation, while our method

stays stable.

In the second simulation, we compare the two methods

with decreasing model stiffness values from 0.5 to 0.3

with the spring value 300N/m. From figure 4 and figure

5 we can see that, when using lower stiffness values, the

oscillation is larger in the force based method, while our

method shows steady feedback.

Figure6: Demonstration. Screen-shots of haptic

interaction. Our system is able to handle stable

haptic interaction with large deformation. The

slime-rabbit model is dramatically deformed by

user’s haptic operation.

6. HAPTIC INTERACTION

We tried our system on a simplified stanford bunny

model. This model consists of 458 vertices and 80 par-

ticles. The sub index from 1 to 4 in figure 6 show the

screen-shots of interaction with the deforming bunny model.

The corresponding time frames are illustrated in figure

7 and figure 8. Figure 7 shows the rendering force and

the position of device, proxy and particle. The figure

shows that a stable haptic rendering is achieved. Figure

8 shows computation time for each simulation step. To-

tal computation times are less than 10ms on our Core

i5 laptop and simulations are performed in real-time. In

figure 8, ”Haptic” is the computation time for the col-

lision detection and the proxy update. ”Deform” is the

computation time for the simulation of OP. ”Blend” is

the computation time for the surface position updates

from OP. ”Total” includes the graphics rendering and all

other computations.

The user feels the bunny’s surface shape and the elas-

ticity smoothly without any jitter or oscillations. As we

do not employ force shading, the user can feel the edges

of each triangle. While the proxy update generally works

well, extreme deformation exceptionally causes falling of

the proxy through the model surface.

7. CONCLUSION

In this paper, we present an improved stable haptic

rendering method based on OP. This method is a rede-

veloped proxy method in a PBD simulation system. We

found that a force based implementation will cause oscil-

lation during simulation. This problem can be solved by

a new way of force calculation based on PBD constraint.

We realize this by designing a haptic constraint which is

similar to the other constraints in PBD. As a result of

Figure7: Demonstration. The force feedback,

proxy position, device position and nearby par-

ticle position during the interaction. The four

screen-shots are marked.

Figure8: Demonstration. The computation time

for each step during the interaction. The four

screen-shots are marked.

our evaluation, the oscillation is greatly alleviated with

large spring values or low model stiffness.

8. LIMITATION & FUTURE WORK

Since this work is only focused on three-degree-of-freedom

haptic interaction, further arrangements and variations

are needed for six-degree-of-freedom simulation. Although

our code and system are not yet optimized for practical

application, our research makes stable haptic implemen-

tation of the fast and robust OP deformation method

possible. Conventional haptic implemented virtual real-

ity systems are able to use OP as a simple and fast way

to simulate solids. We hope our work can be a reference

for future exploration.

ACKNOWLEDGMENT

This work is support by JSPS KAKENHI Grant Num-

ber 26280072.

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