**1. Introduction**

Humanoid robot is a complex machine with a series of joint and links. Biped motion asserts surpass advantage than the others due to flexibility and good adaption when moving on unpredictable surface. Difference from human beings, the humanoid robots have a limitation of structure and number of degree of freedom. Moreover, legged walking behavior requires an action of many active joints and is much more challenge in synthetic gait to keep balance. To solve this matter, the traditional approaches consider that ensuring Zero Moment Point within support polygon is important key. For example, Firstly, states of art is the works of Kajita [1–3] with the Linear Inverted Pendulum Model. Recently, Monje et al. [4] integrated the dynamic steadiness while moving in real time. Samadi and Moghadam-Fard [5] applied Gravity Compensated Inverted Pendulum Mode (GCIPM) with proposing that trajectory of ZMP is created by a first-order function. Kai Hu et al. [6] designed Compensative Zero-Moment Point Trajectory from the reference ZMP to decrease the effect of disturbances. Likewise, Yang et al. [7] presented bivariatestability-margin-based control model to compensate zero moment point and modeling error, opposition-based learning algorithm is applied to generated gait pattern.

In the other direction, a number of researchers concentrate on central pattern generator (CPG)-based walking. Alongside the very common studies in [8–10], we can mention some up-to-date prominent instances such that JimmyOr [11] proposed an approach based on combination of CPG and ZMP to control spine motion, it is promising way to enable natural walk of the robot. Wei Li et al. [12] developed a mechanism to generate muscle forces for biped motion, this method designed a feedback controller which is formed by a dynamic neural network with CPGs.

A optimization-based approach considering stability and walking speed was introduced by Goswami Dip et al. [13]. Likewise, In-sik Lim et al. [14] addressed a gait generation technique for legged walking up and down stairs, in which, genetic algorithm and the human motion data were used to produce the optimal trajectory. Newly, Ho Pham Huy Anh and Tran Thien Huan [15] optimized walking gait by modified Jaya optimization algorithm.

Other scholars are interested in model-predictive methods. For example, Zamparelli [16] et al. designed a stable model predictive control to generate CoM trajectory for the robot on uneven surface. Scianca [17] presented a prediction model with stability constraints. Hildebrandt et al. [18] proposed a model-predictive approach for generating walking gait of the robots with redundant kinematic design.

In this chapter, we proposed a novel gait generation method, in which, trigonometric function with randomized coefficients is used to produce trial gait data for conducting simulation. The result is collected to build approximation function of output: rotation angle value, lateral and walking distance. Then, we designed an optimization problem with constraints and applied an improve differential algorithm for solving it. Optimal coefficients is returned to trigonometric function and define it exactly. By setting up cycle time, walking gait data will be generated by explicit trigonometric function and can be used for reference data in control.

Next part of this chapter is organized into five divisions: The first section introduces the subject of this research named Kondo KHR-3. The second section assigns gait functions to actuator joints. Thirdly, optimization problem with constraint is built to optimize gait trajectory. The novel differential evolution algorithm with Gauss distribution is developed in Section 4 to solve optimization problem. Section 5 includes results and discussions. The final section summarizes achievements of this research.

### **2. Biped model of Kondo**

#### **2.1 Physical configuration**

The subject is presented in this chapter which is based on a humanoid robot named Kondo KHR-3 as depicted in **Figure 1a**. This robot consists of 22 degrees of freedom (DoF) However, we concentrated on lower body with 10 DoFs, in which 5 is the number of DoFs for each leg. The linkage configuration is shown in **Figure 1b**, where *θ*, *d*, *m* are rotation angle, length and weight of each link, respectively. Specially, the weigh of upper body is represented by *mo*. The structure parameters of the robot are described in **Table 1**.

based foot design to reduce weight and energy consumption, this research is specially meaningful for biped robot with limited physical parameters [26]. Our research presents a foot structure as shown in **Figure 2**, which is a combination of proposals mentioned in two papers [23, 26]. This sketch is built based on the following considerations: Firstly, all external forces acting on foot is at three areas consisting of heel, tip and big toe. Secondly, a passive toe joint mechanism with spring is applied to reduce impact of ground reaction force on foot. The material of

**Parameters Value** *do* 90.6 (mm) *d*<sup>1</sup> 38.5 (mm) *d*<sup>2</sup> 66.5 (mm) *d*<sup>3</sup> 62 (mm) *d*<sup>4</sup> 66 (mm) *d*<sup>5</sup> 168 (mm) *mo* 999.4 (g) *m*11,12 14 (g) *m*21,22 100 (g) *m*31,32 65 (g) *m*41,42 71.3 (g)

*Applying Improve Differential Evolution Algorithm for Solving Gait Generation Problem…*

Review on previous papers, we witness an arm swing mechanism which has been introduced and modeled using Adams in [27] as shown in **Figure 3**, the shoulder joint

is linked to the hip joint by two linear springs. This mechanism is expected to generate the reaction moment from the arms to the trunk which eliminates ground reaction torque [28]. It makes the robot stable in motion. Our research applies this structure for the robot and characteristic parameters of the linear spring are stiffness

foot is ABS plastic and design parameters are described in **Table 2a**.

of 0.8 ð Þ *newton=mm* and damping of 0.008 ð Þ *newton: sec =mm* .

**2.3 Arm swing mechanism**

*Structure parameters of humanoid robot.*

**Figure 1.**

**Table 1.**

**43**

*Biped of Kondo: a) real robot; b) linkage model.*

*DOI: http://dx.doi.org/10.5772/intechopen.98085*

#### **2.2 Foot structure with toe mechanism**

Review on the works of foot structure for humanoid robots, we can list very old papers such as [19–21]. Recently, Sadedel et al. proposed a passive toe design. This mechanism improves energy efficiency of ankle and knee joints [22]. Likewise, Nerakae and Hasegawa simulated human-like foot structure which consist of big and tip toe to enhance biped walking gait in toe-off period [23]. In the other direction, Magistris et al. studied soft sole to minimize effect of the ground reaction force on stability during heel-strike period [24]. In [25], Daichi developed topology*Applying Improve Differential Evolution Algorithm for Solving Gait Generation Problem… DOI: http://dx.doi.org/10.5772/intechopen.98085*

**Figure 1.** *Biped of Kondo: a) real robot; b) linkage model.*


#### **Table 1.**

can mention some up-to-date prominent instances such that JimmyOr [11] proposed an approach based on combination of CPG and ZMP to control spine motion, it is promising way to enable natural walk of the robot. Wei Li et al. [12] developed a mechanism to generate muscle forces for biped motion, this method designed a feedback controller which is formed by a dynamic neural network with CPGs. A optimization-based approach considering stability and walking speed was introduced by Goswami Dip et al. [13]. Likewise, In-sik Lim et al. [14] addressed a gait generation technique for legged walking up and down stairs, in which, genetic algorithm and the human motion data were used to produce the optimal trajectory. Newly, Ho Pham Huy Anh and Tran Thien Huan [15] optimized walking gait by

Other scholars are interested in model-predictive methods. For example, Zamparelli [16] et al. designed a stable model predictive control to generate CoM trajectory for the robot on uneven surface. Scianca [17] presented a prediction model with stability constraints. Hildebrandt et al. [18] proposed a model-predictive approach for generating walking gait of the robots with redundant kinematic design. In this chapter, we proposed a novel gait generation method, in which, trigonometric function with randomized coefficients is used to produce trial gait data for conducting simulation. The result is collected to build approximation function of output: rotation angle value, lateral and walking distance. Then, we designed an optimization problem with constraints and applied an improve differential algorithm for solving it. Optimal coefficients is returned to trigonometric function and define it exactly. By setting up cycle time, walking gait data will be generated by explicit trigonometric function and can be used for reference data in control.

Next part of this chapter is organized into five divisions: The first section introduces the subject of this research named Kondo KHR-3. The second section assigns gait functions to actuator joints. Thirdly, optimization problem with constraint is built to optimize gait trajectory. The novel differential evolution algorithm with Gauss distribution is developed in Section 4 to solve optimization problem. Section 5 includes results and discussions. The final section summarizes achievements of

The subject is presented in this chapter which is based on a humanoid robot named Kondo KHR-3 as depicted in **Figure 1a**. This robot consists of 22 degrees of freedom (DoF) However, we concentrated on lower body with 10 DoFs, in which 5 is the number of DoFs for each leg. The linkage configuration is shown in **Figure 1b**, where *θ*, *d*, *m* are rotation angle, length and weight of each link, respectively. Specially, the weigh of upper body is represented by *mo*. The structure parameters

Review on the works of foot structure for humanoid robots, we can list very old papers such as [19–21]. Recently, Sadedel et al. proposed a passive toe design. This mechanism improves energy efficiency of ankle and knee joints [22]. Likewise, Nerakae and Hasegawa simulated human-like foot structure which consist of big and tip toe to enhance biped walking gait in toe-off period [23]. In the other direction, Magistris et al. studied soft sole to minimize effect of the ground reaction force on stability during heel-strike period [24]. In [25], Daichi developed topology-

modified Jaya optimization algorithm.

*Robotics Software Design and Engineering*

this research.

**42**

**2. Biped model of Kondo**

**2.1 Physical configuration**

of the robot are described in **Table 1**.

**2.2 Foot structure with toe mechanism**

*Structure parameters of humanoid robot.*

based foot design to reduce weight and energy consumption, this research is specially meaningful for biped robot with limited physical parameters [26]. Our research presents a foot structure as shown in **Figure 2**, which is a combination of proposals mentioned in two papers [23, 26]. This sketch is built based on the following considerations: Firstly, all external forces acting on foot is at three areas consisting of heel, tip and big toe. Secondly, a passive toe joint mechanism with spring is applied to reduce impact of ground reaction force on foot. The material of foot is ABS plastic and design parameters are described in **Table 2a**.

#### **2.3 Arm swing mechanism**

Review on previous papers, we witness an arm swing mechanism which has been introduced and modeled using Adams in [27] as shown in **Figure 3**, the shoulder joint is linked to the hip joint by two linear springs. This mechanism is expected to generate the reaction moment from the arms to the trunk which eliminates ground reaction torque [28]. It makes the robot stable in motion. Our research applies this structure for the robot and characteristic parameters of the linear spring are stiffness of 0.8 ð Þ *newton=mm* and damping of 0.008 ð Þ *newton: sec =mm* .

Where *θ<sup>i</sup>* is the angle assigned to joint *i*, *ai*, *bi*,*ci*, *di* are coefficients; *t*, *i* are time

0; *t* ¼ 0 *or t*≥3*:*6 �1*:*5; *t* ¼ 0*:*3&*t* ¼ 3*:*3 *θ*1ð Þ*t* ; 0*:*3 <*t*< 3*:*3

0; *t*≤0*:*3 *or t*≥ 3*:*6 *θ*2ð Þ *t* þ 0*:*6 ; 0*:*3<*t*< 3*:*3 15; *t* ¼ 3*:*3

0; *t*≤0*:*3 *or t* ≥3*:*6 *θ*3ð Þ *t* þ 0*:*6 ; 0*:*3<*t* <3*:*3 30; *t* ¼ 3*:*3

0; *t*≤0*:*3 *or t* ≥3*:*6 *θ*4ð Þ *t* þ 0*:*6 ; 0*:*3<*t* <3*:*3 15; *t* ¼ 3*:*3

0; *t* ¼ 0 *or t*≥ 3*:*3

0; *t* ¼ 0 *or t*≥ 3*:*3

0; *t* ¼ 0 *or t*≥ 3*:*3

15; *t* ¼ 0*:*3 *θ*2ð Þ*t* ; 0*:*3<*t*< 3*:*3

30; *t* ¼ 0*:*3 *θ*3ð Þ*t* ; 0*:*3<*t*< 3*:*3

15; *t* ¼ 0*:*3 *θ*4ð Þ*t* ; 0*:*3<*t*< 3*:*3

Response surface model (RSM) is a mathematical model which is used for an approximation of stochastic process and is firstly introduced by Box and Wilson [27]. Our research adopts 3*rd*-order RSM to predict output value of simulation such as lateral and walking distance, rotation angle. The expression of 3*rd*-order RSM is

*cpiix*<sup>2</sup>

Where *n* is a design variable number, *n* = 16, *xi* is a design variable, and *ap*, *bp*, *cp*, *dp* are the coefficients of terms. Number of sampling for initialization is calculated by

*ii* <sup>þ</sup> <sup>X</sup>*<sup>n</sup> ij i*ð Þ <*j* *cpijxix <sup>j</sup>* <sup>þ</sup>X*<sup>n</sup>*

*i*¼1

*dpiix*<sup>3</sup>

*ii:* (9)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

and index of joint, respectively; *ω* is an angular velocity, *ω* = 5.236 (*rad=s*)

*Applying Improve Differential Evolution Algorithm for Solving Gait Generation Problem…*

*θ*<sup>1</sup> ¼

8 ><

>:

8 ><

>:

8 ><

>:

8 ><

>:

8 ><

>:

8 ><

>:

*θ*<sup>5</sup> ¼

*θ*<sup>6</sup> ¼

*θ*<sup>7</sup> ¼

**4. Optimization problem with predictive function**

**4.1 Response surface methodology**

*P x* <sup>~</sup>ð Þ¼ *apo* <sup>þ</sup>X*<sup>n</sup>*

*i*¼1

*bpixi* <sup>þ</sup>X*<sup>n</sup>*

*i*¼1

displayed by Eq. (9).

Eq. 10.

**45**

*θ*<sup>2</sup> ¼

*DOI: http://dx.doi.org/10.5772/intechopen.98085*

*θ*<sup>3</sup> ¼

*θ*<sup>4</sup> ¼

8 ><

>:

**Figure 2.** *Novel foot design.*

**Figure 3.** *Principle of arm swing: a) mechanism with linear spring; b) integration in simulation model.*
