Simulation Fitting Method for Setting Motion Parameters of Robotic ...

Simulation Fitting Method for Setting Motion Parameters of Robotic Fish
Feng Guo, Gang Peng
Abstract Based on analyses of motion control on biomimetic robot fish, this paper promotes an
innovative theory of curve fitting which is an idea employed from Physics, and applies
it on calculation of link-based robot fish’s motion parameters by simulation. Its
efficiency and availability is proved via analyzing of the results.
Keywords Biomimetic robot fish, curve fitting, multi-link, physical simulation
1 Preface
Fish is one of the oldest vertebrate. In the long term of evolution, fish have got many splendid
characters in aquatic motion. They are able to keep low power consumption and high efficiency, as
well as high flexibility, at a high velocity for a long time [1] . Traditional propeller generates backset
on its side and therefore reduces efficiency and velocity [2] . These unique performances are not
achieved by current artificial aquatic robots.
The excellent performance of fish has drawn many scientists’ attention. In 1994, Triantafyllou
in MIT made the first biomimetic robot fish—Robotuna. In the second year, MIT made
RoboPike [3~4] . Also in 1994, Jianyu Cheng and Reinhard Blickhan in Universität des Saarlandes
promoted elongated body theory [5] . NMRI in Japan made PF series and UPF series. Sony and
Mitsubishi developed a variety toy robot fish [6] . In 2005, Huosheng Hu in Essex, UK, made the
first self-control robot fish.
2 Fishes’ swimming style & link-based robotic fish
A basic thought for biomimetic robots is to imitate biology’s behavior and performance. The
following is fish’s swimming style and a kind of robot fish’s structure.
2.1 The way fish swim
In 1960, Lighthill and his colleagues promoted that the wave of a swimming fish could be
described by a formula:
,
(1)
In which stands for the offset of fish body from its axes, which is a function of axially
coordinates and time ; c ,c are coefficients of envelop curves; k is wave number on fish
body; ω is the frequency of fish wave. The formula describes a waving curve with the lapse of t.
We call it fish wave. One static fish wave is shown in Figure 1.
Actually, different types of fish swing in different scales and to different extents. Some may
wave part of their body while others wave the whole. For example, eel wave from their head to tail
when they swim, but carangiform fish swing merely tails and caudal fin. That is, fish wave starts
from different points on different fishes. Besides, some may cover a bigger part of the wave while
others do smaller part. For example, there can be more than one wave crest in eel, but sometime
none in carangiform fish. In other words, the whole curve in Figure 1 represents eel’s wave, but
the former part, perhaps from 0 to 1.4 , represents carangiform fish’s already.
2.2 Link-based robotic fish
Among various structures of biomimetic robot fish, link-based robot fish is a significant one
(as in Figure 2). This kind of structure is simple and reliable. Besides, there is no need of special
flexible material for the fish, and a mere powerful steering gear at the joint is enough.
In order to

4
3
Fish wave
Envelope curve
2
Lateral Displacement
1
0
-1
-2
-3
-4
0 1 2 3 4 5 6 7
Longitude Distance from Head
Figure 1 Static fish wave
Figure 2 Link-based robot fish
make the machine swim like a fish, we should set parameters, such as angle at the joint, proportion
of each rod, one segment of link, so that the link mimics fish’s shape as well as possible.
3. Traditional strategy in fitting fishcurve with link
Traditional method is widely used. It is simple, but there are also some flaws in it.
3.1 Preparation for fitting fish curve
Suppose the robot fish acts N times in a period with equal interval. Then, a period should be
discretized into N time points. So at each point, fish wave degenerate into N static curves. We may
call them fish curves.
For 2/;i=0,1...N-1; formula of fish curve i should be:

’ ,
(2)
When N=12, the 12 curves are shown in Figure 3.
2
1.5
1
Lateral Displacement
0.5
0
-0.5
-1
-1.5
-2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Longitude Distance from Head
Figure 3 Fish curves on 12 time points
After time discretization, we can fit the curves one by one. The fishcurve with 0 phase, that is
the first fishcurve, is picked for the following work.
3.2 Joint Location Method
Just as stated before, because of different swing styles, we must know which part of a fish
curve is to be fitted by robot fish. Suppose the part is from 0 to 2π*R(R>0). The size of R affects
fish pushing ability and speed significantly. In this article, set R=0.7 according to our robot fish.
Current fitting strategy that is commonly seen is to locate every joint on the fish curve, and
ensure the link’s length matches the curve. Then we get the result. We may call this method “Joint
Location Method” (JLM).
JLM is simple and reliable, but there are some flaws in it. When the derivative of a fish curve
remains the same sign, that is, when it’s convex or concave, this method will cause certain
accumulative error, and the link we get will distribute on one side of the curve. As a result, the gap
between link and curve is relatively large. As in Figure 4, the blue dashed line represents a fish
curve with 0 phase, and the red line represents the link we get with this method. We can see that
almost the whole link locates beneath the fish curve. So it is the situation with other phases. This
result not only distorts the shape of the curve, but also decreases its amplitude. And therefore may
affect robot fish’s swimming quality.
3.3 Other way of fitting
An easily thought of way to fix the problem is to integrate the difference between fish wave

0.5
Fish curve
Link
0
Lateral Displacement
-0.5
-1
-1.5
-2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Longitude Distance from Head
Figure 4 Result by JLM
and link and make the outcome minimum. However, there are even more problems in this method.
First of all, there is no obvious guide for adjustment in intermediate states. So there is only one
way to find the result, searching thoroughly. Then it causes another problem: high computational
complexity, especially when there are many segments in the link. So we need to figure out another
method.
4 Simulation Fitting Method
It is necessary to find a method which can compromise between computational complexity
and precision. Just as we borrow the structure of real fish to design robot fish, we can also borrow
ideas from physics to reach our goal.
4.1 Principle of the method
The basic thought of SFM is simple: view the fishcurve and link as real objects, imagine a
uniform force field between them and let physic laws do the rest.
As in Figure 5, a series of elastic ropes (shown as black lines) are attached between the link
and fish curve. Suppose the ropes always satisfy Hook Law and tend to contract. Then, in an
unstable state, rods are subjected to certain force and torque which drive them to translate and
rotate. Under such effect, link moves to its target position and finally stops in a stable state. Now,
the shape of the link is right for robot fish. All we need to do is to build a mathematic model,
simulate the process and record the result.
4.2 Physical modeling and analysis
With the set scene, it is easy to analyze the system.
4.2.1 Analyze single rod

0.5
0
Lateral Displacement
-0.5
-1
-1.5
Fish curve
Link
-2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Longitudinal Distance from Head
Figure 5 Simulation Fitting Method
According to their moving pattern, the rods should be divided into 2 categories: (1) head static
but tail free, that is, the first rod; (2) both head and tail free, that is, all rods except the first.
Analyze the 2 categories separately. Suppose a rod with dip α is subjected to erect force with
distribution density at x (as in Figure 6). The acceleration of the head is a and its angular
acceleration is .
Figure 6 Force analyses on rod
(1) For head static, but tail free rod.
Because of its static head, rod can just rotate. Then a=0. Let’s take a look into β. According to
law of rigid-body rotation:
(3)

In which, , each represents the head and tail of the rod; J represents moment inertia, and in
this situation where rigid even rod rotate surrounding its end, 。
Then:
(2) For both head and tail free rods.
According to Newton second law:
And work energy relationship:
(4)
(5)
Then we get:
(6)
(7)
(8)
If we simulate w ith equal step, in one step, rod’s shift is proportional to and its rotate angle to .
4.2.2 Analyze the link
An obvious fact is that the link must not be broken, so the tail of a former rod should keep
pace with the head of the latter one. In this way, the plain shift of the latter
one will cause rotation
on the former one. Since the shift in every step is much smaller than the rod’s length, the rotation
angle is proportional to of the latter one. Suppose ∆ is the modifying rotation angle caused
by this effect, then:
∆ ∆ /
(9)
In which ∆ represents the plain shift of the latter rod; represents the length of the former. So
far we have calculated all crucial parameters of the model.
4.3 Experimental modeling and simulation
According to the physical model above, it’s easy to program and simulate with equal step
method in Matlab. When the precision comes to certain range, result is got. Flowchart of
simulation program is shown as Figure 7. The result of simulation is shown as Figure 8.
With SFM, we can repeat the simulation and get link shape at other time points one by one
and finally get every shape.
5 Comparison and Analysis
JLM performs well when the second derivative of the curve varies slightly, that is when curve
is close to straight line. Nevertheless, in general, accumulated error is unavoidable, especially
when the curve is concave or convex function.
SFM avoids that kind of error theoretically, and gets similar amplitude with the original curve.
Of course, different factors always violate each other. When the curve is close to straight line, the
link we get doesn’t stick tightly to the curve; instead, there is a small gap, which, in fact, can be
neglected. Besides, the computational complexity of SFM is a little higher than that of JLM.
Table 1 gives 2 kinds of error by the two methods. Linear error (err) is the integral of

Figure 7 Simulation flow chart
0.5
Fish curve
Link
0
teral Displacement
La
-0.5
-1
-1.5
-2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Longitude Distance from Head
Figure 8 Result by SFM
difference between link and fish curve, calculated with the following formula:
(9)
In which represents the shift of fish curve, and is the shift of link. Square error (serr) is the
integral of the second power of the difference, calculated with the following formula:
(10)

Table 1 Error comparison
Linear error
Square error
JLM SFM JLM SFM
1 282.7539 -31.6493 117.5796 74.4139
2 299.8585 -18.6237 135.8531 86.8413
3 244.7837 -10.0649 116.8211 75.6017
4 169.971 11.6239 71.1852 42.3516
5 59.8206 40.3002 33.7386 18.1548
6 45.0723 0.574 39.2045 20.1163
7 8.744 -4.4178 34.7495 15.3578
8 -63.6139 21.0387 22.7306 8.5813
9 -101.151 50.1628 24.9156 11.4755
10 -150.719 56.1411 36.8875 19.145
11 -176.029 53.2019 53.8563 32.5745
12 -241.814 42.7809 85.025 52.4728
13 -282.754 30.299 117.5796 74.5112
14 -299.859 18.684 135.8531 86.8327
15 -244.784 10.0424 116.8211 75.6027
16 -167.971 -11.6179 71.1852 42.3517
17 -59.8206 -40.2987 33.7386 18.1547
18 -45.0723 -0.5738 39.2045 20.1164
19 -8.744 4.4176 34.7495 15.3578
20 63.6139 -21.04 22.7306 8.5813
21 101.1514 -50.1645 24.9156 11.4757
22 150.7191 -56.146 36.8875 19.1454
23 176.029 -53.1993 53.8563 32.5744
24 241.8141 -42.7759 85.025 52.4725
From table 1, we can see, on 24 discrete time points, the errors with simulation fitting method
are all smaller than those with JLM. The linear error of the form er ranges 1.27%~67.37% of the
latter, with an average of 26.26%; the rate with square error i s 7.75% ~64.71% , with an average of
54.89%。
6 Conclusion
This article promotes an innovativ e method, simulation fitting method, in orde r to reduce
the difference betwee n link and fish curve. Two points feature this me thod : first, it is borrowed
from physics phenomenon, so we need n’t to worry about its astringency an d stability; second,
and give out adjusting direction in simulation, and therefore avoid unacceptable calculation
amount. Mo reover, this method can also be used whenever l ink is used to tak e the form of curve.
Nevertheless, we didn’t test its performance with fluid mechanics, but most likely, the result

will show its excellence only i n high speed swimming. I n further study, a test on swimming
performance of the re sult should be made, including both simulation a nd real robot fish test.
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