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Static Balancing and Inertia Compensation of a Master
Manipulator for Tele-operated Surgical Robot
Application
Karthik Chandrasekaran
Dept of Engg Design,
IIT Madras,
Chennai, India.
karthik86.c@gmail.com
Sakthivel Sivaraman
Dept of Mech. Engineering,
NIT Suratkal
Karnataka, India
sakthi2793@gmail.com
Asokan Thondiyath
Dept of Engg Design,
IIT Madras,
Chennai, India.
asok@iitm.ac.in
ABSTRACT
Tele-operated robotic surgery is becoming popular and there is a
need for design of better robotic arms for increased dexterity and
user convenience. In the master-slave configuration of surgical
robots, master manipulator plays an important role in ensuring
user comfort and dexterity. The design and analysis of a master
arm with 6 DOF with a wrist decoupled configuration is presented
here. Primary focus of the design is to reduce the number of
balancing masses required. By strategically positioning the joint
axes, the balancing requirement for some of the axes has been
eliminated. Also, through proper design modifications, remote
placement of balancing masses has been achieved which reduces
the net inertia of the system. Static balancing of the arm is
analysed and an optimal design to achieve the balancing is
presented. The residual imbalance is corrected through inertia
compensation using feed forward control. The design details of
the arm, its kinematic analysis, and balancing are presented in this
paper.
Keywords
Master-slave manipulation; robotic surgery; gravity
compensation; static balancing; feed forward control; inertia
compensation.
1. INTRODUCTION
Teleoperation in robotics is gaining worldwide recognition and
acceptance. In teleoperated robotic surgery a slave arm duplicates
the surgeon’s command at a master manipulator [1]. Typical tele-
operated systems electronically transmit surgeon’s input at the
master control manipulator to the slave arm, similar to a drive-by-
wire technology used in automotive systems. A surgeon typically
holds a pair of graspers at the distal end of the master manipulator
and moves it in space to actuate the slave arm [2]. As most
surgeries take very long time, constant exertion would lead to
premature fatigue and tremor for the surgeon.
The master manipulator should hold its position within its
workspace to feel ethereal. Therefore, the master manipulator
should have least inertia and should be statically balanced [3]
within the entire working range for ease of operation. Static
balancing can be achieved by the use of counterweights.
1.1 Design Considerations for Master Arm
Most of the master manipulators will have 6 DOF, where the
positioning and orienting the grasper is achieved through 3 DOF
each. The major performance requirements for a master
manipulator are good dexterity, static balancing and minimal
inertia of the links [4]. A serial chain of 3 links will be generally
used for positioning and a gimbal assembly with 3DOF will be
used for orientation [5]. All the three axes of the gimbal are
orthogonal and have a common intersection p oint. T he surgeon’s
grasper will be placed at this common intersection point. When a
typical wrist decoupled spatial 3R manipulator, as shown in
Figure 1, is used as a master manipulator, any changes in the
angles θ1 and θ2 of the joints of links 2 and 3 affect the angular
alignment of the orienting links 4, 5 & 6. Therefore, special
arrangements have to be made to gravity compensate the
individual links of the gimbal assembly at the wrist point.
If some kinematic constraints make link 4 of the manipulator
always perpendicular to the ground, any rot ation α of the joint 4
will not alter the potential energy of the subsequent gimbal links
and therefore balancing is not required for this joint. Similarly no
balancing is required for rotation γ if uniform mass distribution
Figure 1. A 3R spatial master arm with decoupled
wrist joints
Figure 3. Frame assignments for the master manipulator
about that rotational axis is ensured. In such a situation, balancing
is required only for the masses that rotate through t he angle β as
far as the gimbal is concerned. Based on these constraints, a
kinematic configuration as shown in Figure 2 is chosen for the
master arm. This configuration is adopted from a palletizing robot
normally used for industrial applications. A palletizing robot has
parallelogram linkages that keep the orientation of the first link of
the wrist constant.
A conceptual rendering of the master manipulator with its
reachable workspace is shown in Figure 2.
2. KINEMATIC ANALYSIS OF THE ARM
The manipulator is a fractionated mechanism and has several
parallel closed loop chains. The forward kinematics of the
mechanism cannot be solved directly. Initially the mechanism is
treated as a serial linkage [6] as shown in Figure 3, even though
the mechanism has several closed loops. The frame assignments
for the master manipulator are given in Figure 3 and the DH
parameters are shown in Table 1. The forward kinematics problem
was solved using these parameters.
Table 1. DH Parameters for master manipulator
i
1i
1i
a
i
d
i
1
0
0
0
ѱ
2
-90°
0
0
2
3
0
2
a
3
d
3
4
0
3
a
4
d
4
5
-90°
4
a
5
d
α
6
90
0
0
β
7
-90°
0
0
γ
The absolute orientation of link 4 does not change due to the
parallel links which keep its orientation constant within the
workspace. This kinematic constraint for link 4 must be explicitly
app lied. This is done by defining θ 4 from Table 1, the relative
angle between link 3 and link 4 of the mechanism, as a function of
θ2 and θ3. From Figure 3 the constraint for the dependent angle θ4
is given by
θ4 = π – φ+ θ2+ θ3 (1)
Therefore the mechanism despite being fractionated can be treated
as a serial mechanism accounting for the kinematic constraint
imposed by the parallel links on link 4.
2.1 Sizing of the links
The sizing of the positioning and gimbal links were carried out to
accommodate the entire workspace of the slave manipulator,
whose workspace is assumed to be constructed by spherical
coordinates with radial distance, polar angle and azimuth angle as
60mm, 90° and 90° respectively. For the master manipulator to
have true isotropic point within the workspace, the ratio of the
link 3 to link 2 have been taken to be 0.707. This was shown by
Salisbury and Craig [7]. The link lengths of the gimbal assembly
are based on mechanical constraints. The effective length and
offset for link 4 will be from coordinate frame 4 to the intersection
point of the gimbal assembly.
The mass of every link is calculated on the assumption that the
links are hollow rods with a diameter of 10mm and the centre of
gravity at the midpoint of the link. The detailed kinematic
structure of the manipulator is shown in Figure 4. The links and
joints for the positioning and orienting part are marked. The
physical lengths and masses considered for analysis is listed in
Table.2
Table 2. Link masses and lengths
Link
l1
l2
l3
l4
l5
l7
l8
G1
G4
a
b
y
mm
70
25
70
70
69.5
49.5
12
31
15
12
12
3
3. PASSIVE BALANCING
One of the requirements of a master arm is that the changes in
potential energy of the arm should be negligible within the
operating range [8]. This ensures that the manipulator left in any
Mass
m1
m2
m3
m4
m5
m6
m7
m8
m9
m10
m11
g
45
30
45
45
100
140
35
10
5
5
5
Gimbal
assembly
Counter weight
Positioning
linkages
Reachable workspace
Figure 2. A conceptual rendering of the master
manipulator with its reachable workspace
Figure 5. Objective function 3D plot
Figure 4. Passive balancing by counterweights
position and orientation within the workspace will hold that pose
and helps the surgeon to easily manipulate the arm and thus
reduce fatigue. Gravity balancing is one of the ways to achieve
this. This is achieved by using counterweights as shown in Figure
4. Remote placement of the counterweights has the advantage of
reduced system inertia than placing the counterweights at every
joint of the serial manipulator. Gravity compensation is
independently required for both the positional and rotational axes
of the manipulator. It can be observed from Figure 4 that any
change in input angle ѱ, i.e. the out of plane spatial rotation, does
not change the potential energy of the manipulator. However,
rotations ϕ1 & ϕ2 considerably change the potential energy of the
system and therefore requires to be balanced. Change in the
gimbal angle α does not change the potential energy since
orientation of link 8 which is along Y axis does not change for
any position of the manipulator due to the kinematic constraint
imposed by the parallelogram configuration. Mass distribution is
uniform for rotation about rotation angle γ so this axis needs no
balancing. Only change in angle β alters the center of gravity of
the subsequent links and needs gravity compensation. The
potential energy of the system as a function of the input angles ϕ1,
ϕ2 & β can be obtained by vectorially adding the potential energy
contribution of each link and is given as:
(2)
It is difficult to achieve 100% static balancing within the working
range of the manipulator by using counterweights alone, as
shown in Figure 4. Hence an optimal balancing need to be arrived
at by posing the balancing as an optimisation problem [9]. The
objective here is to identify the counterweight mass M1 & M2 and
the distance L1 & L2 at which they should be placed from the
origin of the reference axes. L1 is aligned with link l1 and L2 is
oriented along link l2. The optimisation problem can be stated as
minimise (F), where the objective function F is obtained from the
potential energy function by integrating the change in potential
energy for the input design variables as given in Eq.3. :
(3)
The limits for the variables are taken as follows;
10 ≤ L1 ≤ 50 mm (4)
10 ≤ L2 ≤ 50 mm (5)
0.1 ≤ M1 ≤ 0.5 Kg (6)
0.1 ≤ M2 ≤ 1 Kg (7)
ϕ1 = 35°~140° (8)
ϕ2 = 100°~190° (9)
β = 0°~ 90° (10)
The limits for ϕ1 and ϕ2 were chosen so as to keep the manipulator
away from singularity.
Constrained non linear minimization algorithm was used for
optimisation and the optimised values for the lengths L1 & L2 were
found to be 48.7mm and 10.3 mm. The values for the counter
masses are 0.48 kg. and 0.1 kg. respectively. Since we have four
independent variables L1, L2, M1 & M2 visualizing the variation of
the dependent variable F is difficult. Therefore the variation of the
objective function F with respect to the product of L1M1 and L2M2
is plotted in Figure 5.
From Figure 5 it can be seen that the objective function F does not
vary along L2 M2 axis. A separate plot of F vs L2M2 at L1M1 =
23.37, the optimal point, is shown in Figure 6. From this plot, it
can be seen that eventhough there is a variation of F along
L2M2,the maximum value of F is considerably lower than the
maximum value of F along L1M1. This can be attributed to the fact
that the imbalance presented by the links 5, 7, 8 and the gimbal
assembly is partially counterbalanced by the link 2, 4 and part of
link 5.
Figure 8. Variation of potential energy with ϕ2
Figure 7. Variation of potential energy with ϕ
1
Figure 9. Feed Forward control scheme for inertia
cancellation
4. EFFECTS OF BALANCING
To study the effects of balancing on the change in potential
energy of the system, the variation of potential energy is plotted as
a function of joint angles ϕ1 and ϕ2 as shown in Figures 7 and 8. It
can be seen from the plots that after addition of the counter
masses, the potential energy of the entire system has come down
considerably. The potential energy of the system after balancing
has nearly become zero and the inset plots of Figures 7 and 8
depicts the variations in potential energy after balancing. Change
in potential energy with change in angle β was found to be very
minimal and can be easily balanced with a suitable hair spring.
5. INERTIA COMPENSATION
Balancing alone does not make the manipulator operation
comfortable to the surgeon. By adding counterweights, the net
inertia of the system has been increased [10] and this adversely
affects the ease of operation. Therefore, to reduce the net inertia
of the system, external motors can be used to provide inertia
cancellation torque.
A feedforward control mechanism is used for inertia
compensation. The control scheme is shown in Figure 9. The
compensation is done at each joint by estimating the effect of
velocity of the arm at that joint. Rotary encoders and torque
sensors at each joint can be used to obtain positional and torque
feedback, respectively. Feedforward control helps to anticipate the
errors and hence compensates for the deviation before they
happen. In this way, almost perfect compensation is possible with
feedforward controller, provided the system model is known
accurately. This is used with the basic control architecture of the
master arm which simultaneously updates the feed-forward
commands. It includes a feedforward term that estimates the
reflected inertia of the arm.
The reflected inertia at each joint is calculated by the acceleration
of the joint multiplied by the effective mass of the joint. The
effective mass of each joint depends on configuration of the arm.
The output of the model is motor torque, and inertia cancellation
is done by predicting the rate of intended acceleration by a
surgeon and supplying motor torque at the right amount to assist
the intended motion of the surgeon. This provides a force assist
for the surgeon and makes the arm feel weightless.
Figure 6. Objective function vs L2 M2
Figure 11. Torque provided by user at joint 1
Figure 12. Torque provided by user at joint 2
A simulation for the feedforward control in joint space has been
carried out for the joint angles ϕ1 and ϕ2 from Figure 4. The
velocity input to the system has been taken as a sine wave for a
time period of 4 seconds as shown in Figure 10.
The joint torque at the joints 1 and 2 corresponding to joint angles
ϕ1 and ϕ2 which is required to be supplied by the user for the given
input is shown in Figure 11 and 12 and it can be seen that there is
considerable decrease of input torque from the user after
implementation of feedforward control.
6. CONCLUSIONS
The design of a master manipulator based on the configuration of
a palletising robot and its balancing and inertia compensation
methods are presented in this paper. Static balancing is achieved
through counter masses which are optimised to reduce the static
imbalance. The effectiveness of the passive balancing has been
presented and it was found that the remote positioning of the
counter weights has brought significant reduction in imbalance. A
scheme for inertia cancellation through feedforward control is also
presented. Further work on the implementation of static balancing
using spring mechanism on a master arm as an alternate means to
counterbalance without increasing the net inertia of the
mechanism is in progress. Future work includes studying the
effect of changing the link lengths on isotropy of the manipulator.
7. REFERENCES
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surgery : From AESOP® and ZEUS® to da Vinci®,” J. Visc.
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[2] R. Cau, “Design and Realization of a M ast er-Slave System
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Interface : A Device for Probing Virtual Objects T hreee
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[6] J. J. Craig, Introduction to robotics: mechanics and control,
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[8] R. Hendrix, “Robotically assisted eye surgery: A hap tic
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[9] K. Deb, Optimization for engineering design: Algorithms
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Figure 10. Velocity input to joints
