Bezier Curve Optimization for Passive Neural Control

+410 - PROCESS

5.3 Bezier Curve Model for Efficient Locomotion Model This section presents Bezier curve based passive neural control applied in bio-inspired

5.3.3 Bezier Curve Optimization for Passive Neural Control

The angle value of the joints are generated from the effect of output neuron signal.

There-fore, the value of hip-x, hip-y, and knee joints are affected by the input from neuron signal in

each leg. The value of angle joint in the ith leg (0,, where p E {hip-x, hip-y, knee}) depends

on the neuron value of neuron 4 Active neuron signal is generated by mutual inhibition of

coupled neuron while angle joint is generated by an equation that describes the relationship

between signal neuron component and certain joint angle. In order to acquire the relationship

model, we record the required data from the previous model of locomotion, where all joints

5.3. BEZIER CURVE MODEL FOR EFFICIENT LOCOMOTION MODEL ¹³³

are represented by coupled neuron [178]. Its computational cost will be reduced by using this proposed model. The relationship graph is tabulated in Fig. 5.33.

Bezier model in locomotion is usually used as the kinematics [73]. It is also proposed as the method to solve the kinematic problem of hand-eye coordination, and more specifically, tool-eye recalibration of humanoid robots [205]. Bezier curve model generates smooth signal which is easy to be adjusted into the graph reference [73]. In this proposed model, we used a 2-dimensional quadratic Bezier curve model as the optimization function in order to get the relationship equation between the .angle value of joints and motor neuron. The mathematical model of Bezier curve model is presented in Eq. (5.44), where the Bezier curve point in time t is denoted by B(t); t is from 0 to 1; n represents the number of degrees in Bezier curve and Pi is the ith selection point.

In Bezier curve, the axis value (x axis and y axis) is resulted from 2 Bezier point notations which are Bx (t) representing neural signal and By (t) representing joint angle with t as the timing control. In this case, we have to find y axis By (passive neuron) by determining the value of x axis B. Parameter t is required for relating both parameters. Therefore, in order to acquire the value of t, we invert the equation of Bx(t) to become t(Bx). Since Bx(t)

parameter is a third order polynomial equation, we used Cardano formula to solve the inverse

equation. After inversion, the Bezier point in y axis can be denoted as By (t(Bx ) ) . B(t)_ti(1—tr-1Pi(5.44)

z—o

One relationship graph represented as 3 quadratic Bezier curves is illustrated in Fig. 5.29.

We use time parameter in order to facilitate and support the model and separate it into 3 curves. In the graph generated from t = 0 to t = tmax, when 0ct < t1 curve 1 is generated, when t1 <t < t2 curve 2 is generated, when t2 Ct < t.max curve 3 is generated.

Based on the model, we have 12 two-dimensional points to be optimized. From the pre-liminary studies, we used Bacterial Memetic Algorithm (BMA) to optimize the Bezier point.

More details about the algorithm can be found in [25]. The bacterium structure has 8 genes divided into 2 parts, genes representing the point in the line of reference (G(16}, G(2b) ) and

genes representing the point supporting the curve (G(3b} - G(8b) }. G11)} represents points P3 and P4, and G(2.b) represents points P7 and P8. Other genesG3b},4), G. G(6b), G(7b),G(8b) repre-

sent points P1, P2, P5, P6, Po, P10, respectively. In the encoding process which is presented

in Eq. (5.45), parameter G(ib) and 4) aredecided depending on the point in reference graph

with certain time cycle t. Therefore, the limitation point is from 1 to the number of time cycles tmarx. Other genes are decided by the position of the point in reference line. G(3b)and

Ggb} depend on the point position of P0, G(4b)and 4} depend on the point position of P3,

134 CHAPTER 5. LOCOMOTION GENERATOR MODEL

Curve 2

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Figure 5.29: The proposed Bezier curve model. There are 3 quadratic Bezier curves, where 4

points are required in each curve, Po — P3, P4 — F7, P8— P1 i represent 1st curve, 2nd curve, 3rd curve, respectively. P3, P7, P11 are equal to P4, P8, F0, respectively. Po and P11 are defined as the first point of the joint trajectory. Therefore, there are 8 genes to be optimized which consist

of 2-D parameters.

and G`(6b) and G.1') depend on the point position of P7.

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In Eq. (5.45), Pl(x'y) is a 2-dimensional point of Bezier model, where, if k equals 3 and 8 then 1 equals 0, if k equals 4 and 5 then 1 equals 3, and if k equals 6 and 7, then 1 equals 7. R(t) is a 2-dimensional coordinate of reference graphic with input t. In Eq. (5.45), r is a random value

from 0 to 1; gmin and gmax are the minimum and maximum value from {G(36) , , Gg} } set;

a and ,8 are constant values where a << 8. :

In the evaluation process, we minimize the error of the graph in each curve (Ed), where the c parameter represents the index of the cth curve as depicted in Eq. (5.46). The error ratio (E1 : E2 : Es) determines the probability of BMA operation, where e.g., if the percentage error of the 1st curve is high, then the genes representing the 1st curve has high probability for modification. The total fitness is accumulated from E1, E2. and E3

Ec= E Ij

_(5.46)

5.3. BEZIER CURVE MODEL FOR EFFICIENT LOCOMOTION MODEL 135

5.3.4 Experimental Result and Discussion

In the experimental result, we analyze the walking pattern of cat [10] and record the joint angles which are generated based on orbital function based swing trajectory proposed in [169]. We represent the cat having 3 joints (hip-x, hip-y, and knee) in each leg as depicted in Fig. 7.7, where the 4 sequences for 1 walking cycle is as follows: first swing is Leg 2, second is Leg 4, third is Leg 3, and the forth swing is Leg I. The reference joint angles for each leg can be seen in Fig. 5.34.

5.3.4.1 Neural Oscillator Optimization

Here, we reduce the number of neurons required for the walking pattern from 24 neurons to 4 neurons. In order to form the neural signal appropriate with the reference of the joint trajectory and to have the same phase difference, the synaptic weight values of 4 neurons representing the active signal are optimized by using BMA that has proven its capability in the preliminary tests.

The aim of this optimization is to form the suitable signal for 4 legged robot locomotion based on reference hip-x joint signal acquired from cat walldng. In this neural oscillator optimization one bacterium has 6 genes denoted by G(n) representing the neural oscillator

parameters.C1, 07.) and G3n)represent T, T1, and Tf, respectively. The other genesG4n},

Gn}, G(64) representthe synaptic weights tabulated in Table 5.9.

Table 5.8: Parameter Values of Optimization ParamsNVgf nNbac Ardcinge Ii3Ib tl t2 t3

Valuet 40( 1 100 20 I 0.0051.0 2.f 24 49 74

BMA parameters and the result of optimization are tabulated in Tables 7.1 and 5.9, re-spectively, where in Table 7.1, Ngen, NbQ,c, and Nczone are the number of generations, number of bacteria, and the number of clones respectively, b is the rate of the adaptation value of neural oscillator. We set parameter t1 — t3 based on the preliminary test. The neural

oscilla-tor signal is depicted in Fig. 5.30. Since the walking cat trajecoscilla-tory has 7r/2 phase difference in one walking cycle, the neuron oscillator also has ir/2 phase difference as designed in the

optimization evaluation. This phase difference can be changed depending on the feedback sensor from sensory neurons.

136 CHAPTER 5. LOCOMOTION GENERATOR MODEL

Table 5.9: Representation Parameters and Results

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Figure 5.30: Neural oscillator signal as the input of the passive neural system

5.3.4.2 Bezier Curve Optimization

After optimizing the neuron structure, we optimize the walking model. This walking model requires 3 optimization processes in each leg which represents the relationship of neu-ral signal and joint trajectories (hip-y, hip--x, and knee). There are 12 points that are required to be optimized for one optimization process. We use the same optimization parameters pre-sented in Table 7.1 as in the previous optimization process.

The evolutionary algorithm successfully decreased tle error as illustrated in Fig. 5.31.

The evolution of the Bezier curve model in certain generations can be seen in Fig. 5.32.

The comparison graph between the reference and the optimized result is shown in Fig. 5.33, where neural signal is shown as x-axis and joint angle value as y-axis. Since 3 joints have to be optimized, thus 3 graphs are shown. Based on the result in Fig. 5.33, the Bezier based optimization successfully forms the graph similar to the reference graphics.

5.3. BEZIER CURVE MODEL FOR EFFICIENT LOCOMOTION MODEL 137

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5.3. BEZIER CURVE MODEL FOR EFFICIENT LOCOMOTION MODEL 139

After the relationship graphs are optimized, the inversion process is required in order to invert t parameter as output in the Bezier equation for x-axis. Therefore, by giving the neuron signal value, we get the t parameter, then the t value is used as input in Bezier equation for y-axis. Indirectly the output of angle joint is acquired from the value of motor neuron.

After inversion process, the comparison graphs between joint references and the output joint of Bezier model were recorded. In Fig. 5.34 x-axis is the time cycle and y-axis is the joint angle values. The representation of neuron ID to leg ID is decided based on preliminary

analysis. The output of hip-y, hip-x, and knee joint is generated based on the input of neuron as the signal generator. In time based graph, the joints comparison between the reference and the output of the proposed system is not much different. Only some small ripples are occurring, such as in the knee output joint in second, third, and fourth leg. These ripples do not affect the performance of the locomotion.

In order to prove the effectiveness of the proposed model, we applied the proposed system in a small 4 legged robot which has only 16 MHz microcontroller. This neuro-locomotion is successfully implemented in the robot. We run the robot through rough terrain, in the sloped terrain with 8' and 16° slope which are shown in Fig. 7.18. The recorded oscillation body tilt robot in pitch and roll direction can be seen in Fig.5.36 and the effect of ground sensor can be seen in Fig. 5.37. In the computational cost evaluation, this proposed system can be significantly reduced by 63% from the previous neuro-locomotion design [178] from 9.744.10-3 seconds to 3.603 .10-3 seconds for 1000 times looping. Therefore, this proposed locomotion model is able to be applied to low cost 4 legged robot locomotion.

5.3.5 Discussion

The sensor system integration is required in order to realize the dynamic locomotion.

The influence of the feedback sensor to the locomotion system is designed based on the pre-liminary studies. We used simple sensor integration between body tilt angle, ground sensor, and the output of joint angle. In future, advance and complex sensor integration is required in order to reach better performance. It is important in neuro-based locomotion in order to generate the dynamic signal and to enable walking on unstructured terrain. However the com-plexity of the sensor integration will increase the computational cost. Timing control of neu-ral signal for responding sensor feedback will be implemented. Parameter gz in Eq. (5.68) is required in order to fit into the relationship graph based on Bezier model. The proposed pas-sive neural control based neuro-locomotion can be extended to 3-dimensional Bezier model, such that the signal from ground and tilt sensor can cause the changes of neural signal and can affect the joint trajectory as well.

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Neuron 2 (c) third leg generated by Neuron 4 (d) fourth leg generated by Neuron 3

5.3. BEZIER CURVE MODEL FOR EFFICIENT LOCOMOTION MODEL 141

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Neuron 4 (d) fourth leg generated by Neuron 3

142 CHAPTER 5. LOCOMOTION GENERATOR MODEL

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on the rough terrain.

ドキュメント内 Azhar Aulia Saputra Intelligent System Synthesis for Dynamic Locomotion Behavior in Multi-legged Robots (ページ 166-177)