Rendering stiffer walls: A hybrid haptic system using continuous
and discrete time feedback
1
Hari Vasudevan,
2
Manohar B.S and
1,3
M.Manivannan
1
Indian Institute of Technology Madras, Chennai, India, hvasudevan@smail.iitm.ac.in
2
Indian Institute of Science, Bangalore, India, bs.manohar@gmail.com
3
Massachusetts Institute of Technology, Cambridge MA, USA, mani@mit.edu
Abstract
Instability in conventional haptic rendering destroys the perception of rigid objects in virtual
environments. Inherent limitations in the conventional haptic loop restrict the maximum stiffness
that can be rendered. In this pap er we present a method to render virtual walls that are much
stiffer than those achieved by conventional techniques. By removing the conventional digital haptic
lo op and replacing it with a part-continuous and part-discrete time hybrid haptic loop, we were
able to render stiffer walls. The control loop is implemented as a combinational logic circuit on an
FPGA. We compared the performance of the conventional haptic loop and our hybrid haptic loop
on the same haptic device and present mathematical analysis to show the limit of stability of our
device. Our hybrid method removes the compute intensive haptic loop from the CPU, this can free a
significant amount of resources that can be used for other purposes such as graphical rendering and
physics modeling. It is our hope that in the future, similar designs will leads to a haptics processing
unit or HPU.
keywords: Haptics, Instability ,Passiveness, Continuous Time
1 INTRODUCTION
Haptics has traditionally followed the developments and algorithms in the field of computer graphics
to solve various issues that are common to both disciplines. However there are certain problems that
are unique to haptics, one of them is the stability of contact interactions. Haptics by its very nature
involves bidirectional interaction with the user: forces rendered by the haptic device and those exerted
by the user are interdependent. Forces are displayed to the user by a “haptic loop”. The haptic loop is
a closed loop control system that determines the nature of interaction of the haptic device with the user
based on the position and velocity of the end effector. Such a control system describes an impedance
based haptic device.
1
Most of the haptic devices use a digital haptic loop, which inherently suffers from quantization
both temporally and spatially. This limits the maximum stiffness of the virtual surface that can be
rendered. On contact with the virtual surface, the force rendered is very often controlled by the force
law F = k∆x − Bv. ∆x is the depth of penetration into the surface and ‘v’ is the velocity of the haptic
device. The multiplier ‘k’ controls the stiffness of the virtual surface being rendered. In order to render
a rigid surface, we would like as high a stiffness possible. However arbitrarily increasing the gain ‘k’
results in non-passivity of the haptic loop. This manifests itself as buzzing of the haptic device and
destroys the realism of the environment.
We present new method of implementation of the haptic loop that increases the maximum stiffness
that can be achieved by the haptic device. We eliminate the temporal quantization present in the
conventional haptic loop by running it in continuous time. Another advantage of our hybrid system
is that it frees the computer from the typical update rate(1Khz) of the rendered haptic force. This
significantly frees the computational resources for other tasks such as graphics.
In the next section we present a brief review of the literature on the stability haptic loops. We then
describe our system and compare its design with the conventional haptic loop. We then proceed to
describe the implementation of the system and present the experimental results. This is followed by a
short discussion on results and future work planned.
2 Previous Work
Literature on the s tability of haptic devices has been mostly limited to the analysis and modification of
digital haptic loops. [1] investigated the stability of Haptic interaction and derived a condition for the
stability of the haptic device based on considerations of sampling rate of the controller. [2] extended
this criterion to include the effects of position quantization. Their treatment of the instability problem
involved the coulomb friction present in the haptic device.[3] recognize that passive analog control laws
need not necessarily translate to stable digital control laws and derive conditions for such an analog
control law to successfully be transformed into a stable digital one. [4] describe energy leaks caused due
to the Zero Order Hold inherent in digitally sampled control systems and present control strategies to
create the illusion of a passive system to the user.[5] presented a theoretical analysis of the passivity of the
stiff wall and in 2004, [6] provided a condition the haptic device must satisfy to exhibit passive behavior.
[7] present a “Passivity Observer” and “Passivity Controller” method to track energy movements in
haptic interactions with the user and to dissipate the excess energy if it tends to cause active behavior
in the system. [8] explore the use of a multi-rate controller to reduce the ZOH effect in haptic devices
and present a mathematical analysis of the same. A more recent paper by [9] accounts for inertia,
viscous, and Coulomb friction of the device to the amplifier delay, sampling rate, encoder resolution,
and controller stiffness. It also delineates areas of passive, locally stable, limit cycles and unstable
behavior of the haptic device. The work on continuous time haptic devices are far and few in between,
[10] desc ribe a haptic interface device with an analog circuit which exerts continuous-time impedance
2
Figure 1: Diagram of our haptic control loop
within the sampling period of a conventional haptic loop. Though a continuous time circuit is used, it
serves only to enhance the conventional digital loop. The continuous time loop is implemented using
noise prone analog amplifiers and p otentiometers. In contrast, our device makes use of asynchronous
digital circuitry eliminating most noise problems. Our hybrid control loop also has the advantage that it
eliminates the conventional haptic loop from the CPU. [11] seek to exploit the electrical characteristics
of a DC motor to render virtual surfaces and interface it to virtual environments by means of “wave
variables” and analog circuits alone. In our paper instead of exploiting the electrical characteristics of a
DC motor, we use as ynchronous digital circuits along with synchronous digital circuits which are readily
available to obtain a comparable virtual wall stiffness.
3 Stability of Haptic Devices
Current haptic devices implement a discretised version of the force law: F = k∆x − Bv. As [3] explain,
discretizing this force law need not result in a stable haptic loop. [1] express the stability of the discrete
force law as the upper limit on the maximum stiffness that can be achieved by a virtual wall. This
expression takes the form : k ≤
2B
T
where ‘T ’ is the sampling period of the control loop. [2] derive a
condition of passivity of the virtual wall considering the amplitude quantization in the measurement of
position and state the limit of stability as k ≤ min{
2B
T
,
2f
c
∆
} where ‘f
c
’ represents coulomb friction and
‘∆’ represents the m inimum measurable change in position.
The above conditions for stability apply only to digital control loops. We will be able to achieve
higher values of ‘k’ and hence stiffer walls if we employ an analog control system to control interactions
between the user and virtual environments. True analog control systems are continuous in both time and
amplitude. However the continuity in amplitude comes at a price - noise. Noise limits the performance
of any analog control system. It is seen that all continuous time implementation attempted so far [10]
and [11] involve analog circuitry. If we can eliminate the noise problems we might be able to develop a
3
high performance analog control loop for haptic devices.
We have designed and fabricated a rotational haptic device and we attempt to implement the control
law τ = k∆θ in continuous time but with discrete amplitude feedback. Due to the rotational nature
of the device, all linear parameters considered for the analysis of haptic devices have been replaced
by their rotational counterparts. τ , θ, ω represent the torque, angular position and angular velocity
respectively. The second part of the control law Bω is evaluated as done traditionally by sampling at
various frequencies. We then proceed to implement the full control law τ = k∆θ − Bω as a combination
of continuous and discrete time systems. It is seen that this system is capable of rendering much stiffer
than it is possible with the traditional sampled data control system. Figure 1 shows the block diagram
of the haptic control loop that has been implemented. E is a nonlinear operator given by:
E =
1, Input ≤ 0;
0, Input > 0
(1)
Here constants K, L, R are the Torque Constant, Inductance, Resistance of the motor respectively.
The quantities J, B, τ are the Moment of Inertia, Mechanical Damping and Torque of the motor
respectively.
4 Experimental Haptic Device
4.1 Our Haptic device
The haptic device consists of a Maxon RE29MAX - 226792 motor coupled to an 1024 cpr encoder. The
shaft of the motor drives a larger wheel giving us a seven fold torque magnification. A picture of the
experimental setup is shown in Figure 2.
Figure 2: Picture of the haptic device
4
The control loop is completed using an Altera-Flex10K20RC240-4 FPGA. The control equation is
evaluated in two parts both of which are evaluated simultaneously. We evaluate
τ
1
= k∆θ (2)
where ∆θ = (θ − θ
ref
) Here θ is the position of the motor shaft obtained from the quadrature signals
from the encoder. θ
ref
is the position of the virtual wall which is specified by the user. The second part
of the equation,
τ
2
= Bω (3)
is evaluated using simple backward differences. The Equations 2 and 3 are evaluated using single step(no
clock) digital multipliers. All numbers are held in 12 bit locations. The result of any multiplication is
therefore a 24 bit res ult. Since the motor control, pulse width modulator, has a maximum resolution of
12 bits, we truncate the result by eliminating the higher 12 bits. This does not affect the performance
of the device because it was seen that the result often occupied only the lower 12 bits.
It is important to note that Equation 2 is evaluated asynchronously and it is this mode of operation
that enables our device to render stiffer walls than is possible with conventional loops. In the course
of our investigation we found that it was not possible to evaluate a velocity term asynchronously from
position data. Hence Equation 3 requires sampled data. The velocity is therefore obtained by backward
differences by evaluating
ω = θ
n
− θ
n−1
(4)
The characteristic division by the sampling period T is not performed as Equation 4 is adequate to
obtain a velocity estimate. In a final step we combine the results of Equation 2 and Equation 3 to form
τ = τ
2
+ τ
3
(5)
= k∆θ − Bω (6)
The equation with the nonlinearity is:
τ
n
=
0, ∆θ < 0;
k∆θ − Bω, ∆θ ≥ 0.
(7)
We also note the fact that this equation is also bounded by a region of saturation, which means that
the final output torque is:
τ =
τ
n
, τ
n
<265mNm;
265mNm, τ
n
≥265mNm.
(8)
Here 265mNm corresponds to the maximum stall torque of the motor which is achieved at 100% duty
cycle. This occurs when the calculated value of τ
n
≥ 2
12
.
5