Comparative stable walking gait optimization for small-Sized biped robot using meta-heuristic optimization algorithms

Vietnam Journal of Mechanics, VAST, Vol. 40, No. 4 (2018), pp. 407 – 424 DOI: https://doi.org/10.15625/0866-7136/12294 COMPARATIVE STABLE WALKING GAIT OPTIMIZATION FOR SMALL-SIZED BIPED ROBOT USING META-HEURISTIC OPTIMIZATION ALGORITHMS Tran Thien Huan1, Ho Pham Huy Anh2,∗ 1Ho Chi Minh City University of Technology and Education (HCM-UTE), Vietnam 2Ho Chi Minh City University of Technology, VNU-HCM, Vietnam ∗E-mail: hphanh@hcmut.edu.vn Received April 19, 2018 Abstract. This paper proposes a new way to optimize the biped walking gait design for biped robots that permits stable and robust stepping with pre-set foot lifting magnitude. The new meta-heuristic CFO-Central Force Optimization algorithm is initiatively applied to optimize the biped gait parameters as to ensure to keep biped robot walking robustly and steadily. The efficiency of the proposed method is compared with the GA-Genetic Algorithm, PSO-Particle Swarm Optimization and Modified Differential Evolution algo- rithm (MDE). The simulated and experimental results carried on the prototype small-sized humanoid robot demonstrate that the novel meta-heuristic CFO algorithm offers an effi- cient and stable walking gait for biped robots with respect to a pre-set of foot-lift height value. Keywords: biped robot, meta-heuristic optimization algorithm, Central Force Optimization (CFO) algorithm. 1. INTRODUCTION The walking gesture of human up to now still contains many such sophisticated concepts that the humanoid robot can not fully demonstrate. Thus the research for biped robot walking mechanism is being developed in different directions. Several standards have been applied to human robots to ensure stable and natural gait. Static walking is the first applied principle, in which the vertical projection of center of mass (COM) of the ground is always in the supporting foot. In other words, humanoid robots can stop at any times when walking without falling apart. By its very nature, this principle applies to slow-speed robots whereby dynamic effects can be ignored [1, 2]. Researchers then began to focus on developing dynamic walking [3]. This method allows the human robot to speed up the pace. However, during the robot process, the robot may fall due to environmental interference and can not stop abruptly. Therefore, ZMP-based walking is proposed to help control and manipulate inertia [4, 5]. c© 2018 Vietnam Academy of Science and Technology 408 Tran Thien Huan, Ho Pham Huy Anh Recently, several studies have focused on improving the performance of humanoid robot walking gesture. Huang in [6] introduces a stable gait using the gaiter to use the in- terpolation function. The method developed by D. Huan, through the GA algorithm, op- timizes the gait generator to help robot move steadily with the least amount of energy [7]. Dip et al. [8] exhibit steady gait at constant velocity using the sine wave generator. Max- imo et al. [9] introduced a new stable and fast model-free gait with arms movement for humanoid robots. Khusainov et al. (2018) [10] successfully combined kinematic and dynamic approaches in gait optimization for humanoid robot locomotion. Intelligent al- gorithms are applied in this method to optimize the gait generator for humanoid robots such as genetic algorithm (GA) [8], algorithm for optimal swarm (PSO) [11], modified dif- ferential algorithm (MDE) [12,13]. Shaffi in [14] introduces the humanoid robot achieved a stable gait by using the Fourier series gait generator. These methods used intelligent al- gorithms to develop human robot walking movement, such as bee swarm algorithm [15], fuzzy TS controller [16], evolutionary technique [17,18], Ant-Colony optimization (ACO) optimized by recurrent neural networks [19], Pontryagin’s maximum principle [20] and so on. Among various meta-heuristic approaches, the powerful potential of the CFO- Central Force Optimization algorithm has not yet being applied to optimize the biped robot walking gait generator. To overcome this gap, the paper proposes a robot outlier based on the Central Force Optimization (CFO) algorithm, based on the dynamic walking method, and use ZMP Standard to maintain stability. The simulation and empirical results show that using the CFO algorithm allows optimal optimization of the gait parameters for the robot to reach steady gait with accurate foot lifting magnitude. Small-sized humanoid robot HUBOT-5 is used to verify the experimental results. The rest of this paper is arranged as follows. Section 2 introduces the original HUBOT- 5 biped robot. Section 3 presents the new stable gait generation fot small-sized biped robot HUBOT-5. Section 4 proposes the novel gait parametric optimization using CFO technique. Section 5 presents and analyses the simulation and experiment results. Finally conclusion is presented in Section 6. 2. HUMANOID ROBOT MODEL Small-sized humanoid robot (HUBOT-5) consists of the upper torso and two legs as described in Fig. 1. Each leg consists of 3 parts, which are femoral, legs, and foot with a total of 6 DOF (degree of freedom), including 3 DOF at the hip, 1 at the knee and 2 at the foot. The HUBOT-5 can mimic the walking gesture of human with respect to the front size interface (YZ - Frontal view) and the side view (XZ - Sagittal view). Total weight of the HUBOT-5 is about 1.5 kg, including the dynamic actuator, sensor, controller, amplifier and it has a height of about 50 cm. The HUBOT-5 is innovatively designed to ensure full dynamic structure, with each dof is corresponding with 1 independent actuator. The Servo DC engine HD-1501 is used as the actuator. The significant advantage of HD-1501 servo motor is small, compact and light (60 g) with twisted momentum 17 kg. The control signal supplied to the servo via MATLAB/Simulink using the RS-485 transmission standard. Comparative stable walking gait optimization for small-sized biped robot using meta-heuristic optimization algorithms 409 Section 5 presents and analyses the simulation and experiment results. Finally conclusion is presented in Section 6. 2. Humanoid Robot Model (a) (b) Figure 1. HUBOT-5 humanoid robot set-up with 12 DOF Small-sized humanoid robot (HUBOT–5) consists of the upper torso and two legs as described in figure 1. Each leg consists of 3 parts, which are femoral, legs, and foot with a total of 6 DOF (degree of freedom), including 3 DOF at the hip, 1 at the knee and 2 at the foot. The HUBOT–5 can mimic the walking gesture of human with respect to the front size interface (YZ – Frontal view) and the side view (XZ – Sagittal view). Total weight of the HUBOT-5 is about 1,5 kg, including the dynamic actuator, sensor, controller, amplifier and it has a height of about 50 cm. The HUBOT–5 is innovatively designed to ensure full dynamic structure, with each dof is corresponding with 1 independent actuator. The Servo DC engine HD–1501 is used as the actuator. The significant advantage of HD – 1501 servo motor is small, compact and light (60 g) with twisted momentum 17 kg. The control signal supplied to the servo via MATLAB / Simulink using the RS–485 transmission standard. This paper focuses on the straight walking gait parametric optimization of the humanoid robot, as the upper torso is fixated and then only the 10 engines with respect to ten DOF are being controlled and defined as presented in figure 2. The bounds of these 10 rotary angles depend on the real set-up of the HUBOT-5 biped and are tabulated in Table 1. Fig. 1. HUBOT-5 humanoid robot set-up with 12 DOF Figure 2. Humanoid HUBOT-5 structure Table 1: Ten angular limitations Angle Plane Leg Joint Value YZ Right Ankle -200 to 200 XZ Right Ankle -300 to 300 XZ Right Knee -300 to 300 XZ Right Hip -300 to 300 YZ Right Hip -200 to 200 YZ Left Hip -200 to 200 XZ Left Hip -300 to 300 XZ Left Knee -300 to 300 XZ Left Ankle -300 to 300 YZ Left Ankle -200 to 200 3. Human Gait Generation fot Biped HUBOT-5 Four most important variables of the humanoid robot that play an essential role in stable gait generation, including S – walking step length, H – Leg Lifting [m], h – Leg kneeling [m] and n – Hip swinging, are clearly described in Figure 3. In which, d0 represents the height of the torso, d1 is the distance between the 2 dof at the knee joints, d2 is the length of the leg, d3 is the length of the femoral and d4 represents the distance between 2 hips. 1q 2q 3q 4q 5q 6q 7q 8q 9q 10q Fig. 2. Humanoid H -5 structure This paper focuses on the straight walking gait parametric optimization of the hu- manoid robot, as the upper torso is fixated and then only the 10 engines with respect to ten DOF are being controlled and defined a pres nted in Fig. 2. The bounds of these 10 rotary angles depend on the real set-up of the HUBOT-5 biped and are tabulated in Tab. 1. Table 1. Ten angular limitations Angle Plane Leg Joint Value θ1 YZ Right Ankle −20◦ 20◦ θ2 XZ Right Ankle −30◦ to 30◦ θ3 XZ Right Knee −30◦ to 30◦ θ4 XZ Right Hip −30◦ to 30◦ θ5 YZ Right Hip −20◦ to 20◦ θ6 YZ Left Hip −20◦ to 20◦ θ7 XZ Left Hip −30◦ to 30◦ θ8 XZ Left Knee −30◦ to 30◦ θ9 XZ Left Ankle −30◦ to 30◦ θ10 YZ Left Ankle −20◦ to 20◦ 3. HUMAN GAIT GENERATION FOT BIPED HUBOT-5 Four most important variables of the humanoid robot that play an essential role in stable gait generation, including S - walking step length, H - Leg Lifting [m], h - Leg kneeling [m] and n - Hip swinging, are clearly described in Fig. 3. In which, d0 represents 410 Tran Thien Huan, Ho Pham Huy Anh the height of the torso, d1 is the distance between the 2 dof at the knee joints, d2 is the length of the leg, d3 is the length of the femoral and d4 represents the distance between 2 hips. Figure 3. Four variables influence the human walking gait of humanoid robot HUBOT-5 As described in Figure 3, the total three trajectories of biped, including hip trajectory and ankle trajectory of the supporting leg, and ankle trajectory of the moving legs, will depend on 4 variables (S, H, h, n) with respect to both of the frontal (YZ-Frontal View) and sagittal (XZ-Sagittal View) interface. The three selected trajectories , , are considered as sine-time dependent, and described in the equation (1), (2) và (3) (see additional works at [19]). 5 5 5 5, ,x y zP P P Pé ù= ë û 11 1 1, ,x y zP P P Pé ù= ë û 10 101 00 1, ,x y zP P P Pé ù= ë û 1P 5P 10P ( ) ( ) ( ) ( ) ( ) 1 1 1 1 sin . .[ ( 2 ) ( )] 2 2 .[ ( 2 ) ( )] 1 sin . 0.5 .[ ( 2 ) ( )] p p ì é ùæ öï = - - - -ç ÷ê úï è øë û ï = - - -í ï é ùæ öï = + - - -ê úç ÷ï ê úè øë ûî x y x z S TP t t u t T u t T T P t w u t T u t T P t P t H u t T u t T S Fig. 3. Four variables influence the human walking gait of humanoid robot HUBOT-5 As described in Fig. 3, the total three trajectories of biped, including hip trajectory P5 = [ P5x, P5y, P5z ] and ankle trajectory P1 = [ P1x, P1y, P1z ] of the supporting leg, and ankle trajectory P10 = [ P10x, P10y, P10z ] of the moving legs, will depend on 4 variables (S, H, h, n) with respect to both of the frontal (YZ-Frontal View) and sagittal (XZ-Sagittal View) interface. The thr s lected trajectories P1, P5, P10 are consi red as sine-time dependent, and described in Eqs. (1), 2) an (3) (see additional works at [21]). P1x (t) = S 2 sin [ pi T ( t− T 2 )] [u(t− 2T)− u(t− T)], P1y (t) = w[u(t− 2T)− u(t− T)], P1z (t) = H sin [ pi ( P1x (t) S + 0.5 )] [u(t− 2T)− u(t− T)], (1)  P10x (t) = S 2 sin [ pi T ( t− T 2 )] [u(t)− u(t− T)], P10y (t) = −w[u(t)− u(t− T)], P10z (t) = H sin [ pi ( P10x (t) S + 0.5 )] [u(t)− u(t− T)], (2) Comparative stable walking gait optimization for small-sized biped robot using meta-heuristic optimization algorithms 411  P5x (t) = S 4 sin ( pi T ( τ − T 2 )) , P5y first half cycle (t) = n sin (pi T τ ) [ u (τ)− u ( τ − T 2 )] + n cos ( pi T ( τ − T 2 )) [ u ( τ − T 2 ) − u (τ − T) ] , P5y (t) = P5y first half cycle (t) [u(t)− u(t− T)] −P5y first half cycle (t) [u(t− 2T)− u(t− T)], P6z (t) = (d1 + d2 + d3 + d4 − h) . (3) In which, T represents the time to perform a step of the humanoid robot, w represents the distance between 2 legs, τ = { t if 0 ≤ t ≤ T t− T otherwise and u (t) = { 0 if t < 0 1 otherwise . From Eqs. (1)–(3), both of hip and ankle trajectories of the supporting leg and ankle trajectory of the moving leg are used to generate walking gait for the humanoid robot. Finally, the trajectories of the ten angular joints located at the 2 legs in one walk- ing interval cycle can be defined from P1 = [ P1x, P1y, P1z ] , P5 = [ P5x, P5y, P5z ] and P10 =[ P10x, P10y, P10z ] , and based on the biped inverse kinematics. The biped inverse kinemat- ics can be conventionally solved by calculus or numerical methods. However, in this section, the geometric method based on the HUBOT-5 robot rotary joint will be shown, as described in the Eq. (4). Figure 4. Variables defined in formula (4). ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 5 1 10 6 10 3 8 4 7 2 3 4 11 9 7 arctan , , arctan , , , , arcsin , 2 arcsin , 2 , q q q q q q q p q q p q pq q q pq q q q q q q q q ì æ ö = = -ï ç ÷ç ÷ï è ø ï æ öï = = -ç ÷ç ÷ï è øï ï = - = -ï í æ öï = - + - ç ÷ï ç ÷ è øï æ ö = - + - ç ÷ç ÷ è ø = - = -î l l r r A C l A B l r C D r y t t t t z t y t t t t z t t t t t x t t t t l t x t t t t l t t t t t t t ( )4 ï ï ï ï ï ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5 1 5 1 5 1 22 2 4 2 4 2 4 2 6 10 6 10 6 10 22 2 7 9 7 9 7 9 2 2 2 32 3 2 3 2 2 3 , , , , , , , sin arccos , arccos , 2 arccos q q q q = - = - = - = - + - + - = - = - = - = - + - + - æ öæ ö+ - = = ç ÷ç ÷ è ø è ø + = l x x l y y l z z l x x y y z z r x x r y y r z z r x x y y z z Al A B l C x P P y P P z P P l P P P P P P x P P y P P z P P l P P P P P P dd d l d d l d d ( ) ( ) 2 2 3 2 3 5 sin , arccos . 2 q q ì ï ï ï ï ï ï í ï ï ï ï ï æ öæ ö- = ç ÷ç ÷ï è ø è øî Cr D l dl d d l Fig. 4. Variables defined in formula (4) 412 Tran Thien Huan, Ho Pham Huy Anh In which, yl (t) , zl (t) , yr (t) , zr (t) , θA (t) , θB (t) , θC (t) , θD (t) , xl (t) , xr (t) , ll (t) , lr (t) at specified time t, are defined as in Fig. 4 and Eq. (5); represents the distance be- tween P2 and P4, represents the distance between P9 and P7. θ1 (t) = arctan ( yl (t) zl (t) ) , θ5 (t) = −θ1 (t) , θ10 (t) = arctan ( yr (t) zr (t) ) , θ6 (t) = −θ10 (t) , θ3 (t) = pi − θA (t) , θ8 (t) = pi − θC (t) , θ4 (t) = pi 2 − θA (t) + θB (t)− arcsin ( xl (t) ll (t) ) , θ7 (t) = pi 2 − θC (t) + θD (t)− arcsin ( xr (t) lr (t) ) , θ2 (t) = θ3 (t)− θ4 (t) , θ11 (t) = θ9 (t)− θ7 (t) , (4)  xl = P5x − P1x, yl = P5y − P1y, zl = P5z − P1z, ll = √ (P4x − P2x)2 + ( P4y − P2y )2 + (P4z − P2z)2, xr = P6x − P10x, yr = P6y − P10y, zr = P6z − P10z, lr = √ (P7x − P9x)2 + ( P7y − P9y )2 + (P7z − P9z)2, θA = arccos ( d22 + d 2 3 − l2l 2d2d3 ) , θB = arccos ( d3 sin (θA) ll ) , θC = arccos ( d22 + d 2 3 − l2r 2d2d3 ) , θD = arccos ( d3 sin (θC) ll ) . (5) In which d1, d2, d3 and d4 are illustrated in Fig. 3. The coordination P6(x, y, z) is calculated based on P5(x, y, z), and the coordination of [P2(x, y, z), P4(x, y, z), P7(x, y, z), P9(x, y, z)] is calculated based on [P1(x, y, z), P5(x, y, z), P6(x, y, z), P10(x, y, z)] and the ro- trary angle [θ1, θ5, θ6, θ10]. Eqs. (6) below are used to determine, P2, P4, P6, P7, P9. P2x = P1x, P2z = d1 cos (θ1) , P2y = P2z sin (θ1) , P4x = P5x, P4z = P5z − d4 cos (θ1) , P4y = P5y − (P5z − P4z) sin (θ1) , P6x = P5x, P6y = P5y − w, P6z = P5z, P7x = P6x, P7z = P6z − d4 cos (θ10) , P7y = P6y − (P6z − P7z) sin (θ10) , P9x = P10x, P9z = P10z + d4 cos (θ10) , P9y = P10y + (P9z − P10z) sin (θ10) . (6) In summary, using Eqs. (4)–(6), the ten trajectories of the rotary angles located at the 2 legs of biped HUBOT-5 in one interval walking cycle are computed to accurately and efficiently control the biped walking gait. Thus the set of four parameters H, h, s and n need to be optimally selected so that the resulted ZMP parameter ensures that the biped robot can walk steadily with the preset foot-lift value. In this paper the CFO-Central Force Optimization algorithm is used to satisfactorily solve this task. Comparative stable walking gait optimization for small-sized biped robot using meta-heuristic optimization algorithms 413 4. GAIT PARAMETRIC OPTIMIZATION USING CFO 4.1. CFO algorithm The CFO was first introduced in 2007 by R. A. Formato [22]. The algorithm is a somewhat new optimization technique which is nature-based, multidimensional, meta- heuristic, and based heavily on the gravitational kinematics concept, for example, object fly through space. Unlike other evolutionary algorithms (GA, PSO), CFO does not con- tain random characteristic, and that makes the CFO deterministic. Concretely the proposed CFO algorithm consists of following steps: (1) Calculate initial posision of probe, evaluate their value and their accelerations, then compute their mass. (2) Based on the evaluated acceleration, find new position of the probes. (3) Check if each probe mass and position is located inside the decision space, or whether is it converged. (4) Update the fitness value with new position. (5) Update acceleration. (6) Loop until stopping criterion is satisfied. It is also important to notice the followed CFO notable parameters: • Alpha coefficient represents the influence of probes to each other, using the dif- ference of mass. The lower alpha is the less effect that probe with approximate mass will influence each other. • Beta, as distance extended, determines the remaining influence of probes to each other in space. The lower beta is the more effect one probe can have to others as great distance. • Gamma, simply determines accelerations, the higher gamma is, the faster the probe accelerate. This can be good when the user want to quickly find the opti- mal solution. However, this may have a bad effect on the convergence charac- teristics. 4.2. Specification of the objective function The objective function must be defined to evaluate gait parameters of the humanoid robot. The goal of the HUBOT-5 biped robot is to achieve a stable gait with preset foot- lifting value. For this purpose, the ZMP point projection on the foot area based on the ZMP principle will be used. When the feet touch the ground, the area of the supporting foot is the area between the two feet of the human robot, and when one foot touches the ground, the foot area is the surface of the foot touching the ground. The supporting foot area in the two cases is illustrated in Fig. 5. The zero moment point ZMP is a point in the plane where the total external torque applied to the humanoid robot at this point is zero. Then if the ZMP is within the area of the supporting leg, the robot does not fall. The calculation of the ZMP trajectory of biped robots in walking is shown in Section 4.3. 414 Tran Thien Huan, Ho Pham Huy Anh 5. Update acceleration. 6. Loop until stopping criterion is satisfied. It is also important to notice the followed CFO notable parameters: • Alpha coefficient represents the influence of probes to each other, using the difference of mass. The lower alpha is the less effect that probe with approximate mass will influence each other. • Beta, as distance extended, determines the remaining influence of probes to each other in space. The lower beta is the more effect one probe can have to others as great distance. • Gamma, simply determines accelerations, the higher gamma is, the faster the probe accelerate. This can be good when the user want to quickly find the optimal solution. However, this may have a bad effect on the convergence characteristics. 4.2 Specification of the Objective Function The objective function must be defined to evaluate gait parameters of the humanoid robot. The goal of the HUBOT-5 biped robot is to achieve a stable gait with preset foot-lifting value. For this purpose, the ZMP point projection on the foot area based on the ZMP principle will be used. Figure 5. Area of the supporting foot ìn cases: (a) 2 foot on the ground, (b) 1 foot on the ground Fig. 5. Area of the supporting foot in cases: (a) 2 foot on the ground, (b) 1 foot on the ground The sum of the squared distance from the ZMP to the center of the supporting foot in 1 step walking of biped HUBOT-5, see Eq. (7), represents the first objective function. f1 = T∫ 0 √ x2ZMP + y 2 ZMP dt, (7) with T denotes stepping cycle and (xZMP, yZMP) denotes the coordination of ZMP in the process of the biped robot performs the step. The smaller f1 is the more stable the gait will become. Additionally, for the humanoid robot to follow the pre-set foot-lifting height value −Hre f , the difference between the magnitude of the foot-lift parameter - and the foot-lift preset value −Hre f (see Eq. (8)) represents the second objective function. Eq. (8) means that the smaller f2 is the more strictly the lifting magnitude H can follow Hre f . f2 = ∣∣Hre f − H∣∣ . (8) Thus, in order for biped HUBOT-5 to obtain a steady gait with the foot-lift set up in advance, we find the minimum value of the two objective functions f1 and f2, or similarly to find the minimum of the function f as f = λ  T∫ 0 √ x2ZMP + y 2 ZMPdt + (1− λ) ∣∣Hre f − H∣∣ . (9) Comparative stable walking gait optimization for small-sized biped robot using meta-heuristic optimization algorithms 415 In which, λ (0 < λ ≤ 1) is optimally selected as to prioritize between the walking sta- bility (λ increase) and the variance with the desired foot-lifting magnitude (λ decreased). 4.3. ZMP trajectory calculation The ZMP trajectory resulted from calculated ZMP point which is computed using Eq. (10) [6]  xZMP = ∑ni=1 mi (z¨i + g) xi −∑ni=1 mi x¨izi −∑ni=1 IiyΩ¨iy ∑ni=1 mi (z¨i + g) , yZMP = ∑ni=1 mi (z¨i + g) yi −∑ni=1 miy¨izi −∑ni=1 IixΩ¨ix ∑ni=1 mi (z¨i + g) . (10) In which, mi represents the weight of the ith stage and (xi, yi, zi) is the coordinates of the center of the ith stage in the Descartes coordinate system, Iix and Iiy denote properties of the inertia momentum, Ω¨ix and Ω¨iy are the acceleration properties of rotary angle around x and y at the center ith joint, g denotes gravitational acceleration, (xZMP, yZMP) represent the ZMP coordinations. For biped HUBOT-5, the weight of the links is concentrated on the mass of the joints, so the center of the joint is considered to be located at the end of the joint. Thus, the iner- tial moment term of Eq. (10) is considered zero, then the Equation for ZMP is calculated as (11)  xZMP = ∑ni=1 mi (z¨i + g) xi −∑ni=1 mi x¨izi ∑ni=1 mi (z¨i + g) yZMP = ∑ni=1 mi (z¨i + g) yi −∑ni=1 miy¨izi ∑ni=1 mi (z¨i + g) (11) In (11), the mass distribution mi and coordinates (xi, yi, zi) of the stages are defined in Fig. 6. In which, represents the weight of the ith stage và is the coordinates of the center of the ith stage in the Descartes coordinate system, and denote properties of the inertia momentu , and are the acceleration properties of rotary angle around x and y at the center ith joint, denotes gravitational acceleration, represent the ZMP coordinations. For biped HUBOT-5, the weight of the links is concentrated on the mass of the joints, so the center of the joint is considered to be located at the end of the joint. Thus, the inertial moment ter of the equation (10) is considered zero, then the Equation for ZMP is calculated as (11): In (11), the mass distribution and coordinates of the stages are defined in Figure 7. Figure 7: The mass distribution and coordinates of the stages ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 10 = = = = = = = = ì + - - W ï = ï +ï í + - - Wï =ï +ïî å å å å å å å å !!!!!! !! !!!!!! !! n n n i i i i i i iy iyi i i n i ii n n n i i i i i i ix ixi ZMP ZMP i i n i ii m z g x m x z I x m z g m z g y m y z I y m z g im ( ), ,i i ix y z ixI iyI W!! ix W!! iy g ( ),ZMP ZMPx y ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 11 = = = = = = ì + - ï = ï +ï í + -ï =ï +ïî å å å å å å !!!! !! !!!! !! n n i i i i i ii i n i ii n n i i i ZMP ZMP i i ii i n i ii m z g x m x z x m z g m z g y m y z y m z g im ( ), ,i i ix y z Fig. 6. The mass distribution and coordinates of the stages 416 Tran Thien Huan, Ho Pham Huy Anh The coordinates Pit (x, y, z) of the joint are determined from the 10 angles of rotation at one time in one step with the original coordinatation at the center of the suppoting foot by means of the geometric Equation (12) P1xt (t) = 0, P1yt (t) = 0, P1zt (t) = 0, P2xt (t) = P1xt, P2zt (t) = d1 cos [θ1 (t)] , P2yt (t) = P2zt (t) sin [θ1 (t)] , P3xt (t) = d2 sin [θ2 (t)] , P3yt (t) = P3zt (t) sin [θ1 (t)] , P3zt (t) = P2zt (t) + d2 cos [θ2 (t)] cos [θ1 (t)] , P4xt (t) = P3xt (t) + d3 sin [θ2 (t)− θ3 (t)] , P4yt (t) = P4zt (t) sin [θ1 (t)] , P4zt (t) = P3zt (t) + d3 cos [θ2 (t)− θ3 (t)] cos [θ1 (t)] , P5xt (t) = P4xt (t) + d4 sin [θ2 (t)− θ3 (t) + θ5 (t)] , P5yt (t) = P5zt (t) sin [θ1 (t)] , P5zt (t) = P4zt (t) + d4 cos [θ2 (t)− θ3 (t) + θ5 (t)] cos [θ1 (t)] , P6xt (t) = P5xt (t) , P6yt (t) = P5yt (t)− w, P6zt (t) = P5zt (t) , P7xt (t) = P6xt (t) , P7zt (t) = P6zt (t)− d4 cos [θ6 (t)] , P7yt (t) = P6yt (t)− [P6zt (t)− P7zt (t)] sin [θ6 (t)] , P8xt (t) = P7xt (t) + d3 sin [θ7 (t)] , P8zt (t) = P7zt (t)− d3 cos [θ7 (t)] cos [θ6 (t)] , P8yt (t) = P7yt (t)− [P7zt (t)− P8zt (t)] sin [θ6 (t)] , P9xt (t) = P8xt (t) + d2 sin [θ7 (t)− θ8 (t)] , P9zt (t) = P8zt (t)− d2 cos [θ7 (t)− θ8 (t)] cos [θ6 (t)] , P9yt (t) = P7yt (t)− [P7zt (t)− P9zt (t)] sin [θ6 (t)] , P10xt (t) = P11xt (t) + d1 sin [θ7 (t)− θ8 (t) + θ1 (t)] , P10zt (t) = P9zt (t)− d1 cos [θ7 (t)− θ8 (t) + θ9 (t)] cos [θ6 (t)] , P10yt (t) = P7yt (t)− [P7zt (t)− P10zt (t)] sin [θ6 (t)] , P0xt (t) = P5xt (t) + P6xt (t) 2 , P1yt (t) = P5yt (t) + P6yt (t) 2 , P0zt (t) = P5zt (t) + d0 2 . (12) In which, d0, d1, d2, d3 and d4 are illustrated in Fig. 3. Finally, the flow chart used to calculate the ZMP trajectory based on the optimal set of four gait parameters of the humanoid robot is illustrated in Fig. 7. 5. SIMULATION AND PRACTICAL EXPERIMENT RESULTS The simulated and experimental results are fully tested on the small-sized HUBOT- 5 biped robot. The physical parameters of the HUBOT-5 biped robot are presented in Tab. 2. In order to find the most appropriate value for the coefficients λ of the objective function in Eq. (9), based on CFO algorithm, it optimally selects λ = 0.4 which permits the HUBOT-5 biped robot attaining a steady gait with an adjustable foot-lift value, and this λ value will be used thorough the comparative testing process using GA, PSO, MDE and CFO. Comparative stable walking gait optimization for small-sized biped robot using meta-heuristic optimization algorithms 417Finally, the flow chart used to calculate the ZMP trajectory based on the optimal set of four gait parameters of the humanoid robot is illustrated in Figure 8. Figure 8. ZMP calculation flow chart 5. Simulation and Practical Experiment Results The simulated and experimental results are fully tested on the small-sized HUBOT-5 biped robot. The physical parameters of the HUBOT-5 biped robot are presented in Table 4. Table 4. Physical parameters of biped HUBOT-5 Parameters Value 6.000 cm 4.254 cm 9.109 cm 8.063 cm 9.345 cm 8.640 cm 60 gam 0d 1d 2d 3d 4d w 0 10 ì í =î ! im i Fig. 7. ZMP calculation flow chart Table 2. Physical parameters of biped HUBOT-5 Parameters Value d0 6.000 cm d1 4.254 cm d2 9.109 cm d3 8.063 cm d4 9.345 cm w 8.640 cm{ mi i = 0 . . . 10 60 gam Table 3. Bounds for 4 walking gait paramteres of hu- manoid HUBOT-5 Parameters Lower Bound Upper Bound S-Step length 1.5 cm 25 cm H-foot lifting 0.1 cm 10 cm h-kneeling 0.1 cm 1.5 cm n-hip swinging 0.1 cm 10 cm For optimal test based on GA, PSO, MDE and CFO optimization algorithms, the gait parameters derived from the biped HUBOT-5 are limited as given in Tab. 3. The mathematical properties of GA, PSO, MDE, and CFO optimization algorithms are meta-heuristic algorithms, so each algorithm will perform 10 different training times, with each training will repeat 500 times (N = 500) using the same population size 418 Tran Thien Huan, Ho Pham Huy Anh (NP = 32) and the same number of variables (n = 4). Tab. 4 eventually presents the GA, PSO, MDE and CFO selected parametric values. Table 4. Parameters of GA, PSO, MDE and CFO algorithm Method Paramters Value GA Mutation (F) 0.4 Crossover Probability (CR) 0.9 PSO Accelaration factor (C1) 2.0 Accelaration factor (C2) 2.0 Inertia Weight (w) [0.4; 0.9] MDE Mutation value (F) Random [0.4; 1.0] Crossover Probability (CR) Random [0.7; 1.0] CFO Alpha 0.25 Beta 0.35 Gamma 0.95 Frep 0.5 deltaFrep 0.05 Specify the foot-lifting height of HUBOT-5 being Hre f = 20 mm. Fig. 8 illustrates the mean value of the target function after 10 runs of each algorithm (GA: green, PSO: blue, MDE: red, CFO: reddish purple). Figure 9: Mean value of Derived from Table 7, the optimum set of parameters for the biped HUBOT-5 conformed to the objective of 10 runs per GA, PSO, MDE and CFO algorithms is shown in Table 7. Figure 10 shows resulted comparative ZMP and COM trajectories when HUBOT-5 steps along with a stepping cycle (T = 2s) with respect to the configurations based on GA, PSO, MDE and CFO algorithms, respectively. Table 7. Resulted parametric set for four comparative algorithms Href = 2 cm Agorithms Walking Gait Parameters value Best firness value f(cm) S (cm) H (cm) h (cm) n (cm) GA 15 1.99 0.82 6.99 14.87 PSO 15 2.00 0.8 6.89 14.87 MDE 15 2.00 0.8 6.89 14.87 CFO 15 2.00 0.89 6.99 14.87 fFig. 8. Mean value of f Comparative stable walking gait optimization for small-sized biped robot using meta-heuristic optimization algorithms 419 Derived from Tab. 5, the optimum set of parameters for the biped HUBOT-5 con- formed to the objective of 10 runs per GA, PSO, MDE and CFO algorithms is shown in Tab. 5. Fig. 9 shows resulted comparative ZMP and COM trajectories when HUBOT-5 steps along with a stepping cycle (T = 2 s) with respect to the configurations based on GA, PSO, MDE and CFO algorithms, respectively. Table 5. Resulted parametric set for four comparative algorithms Hre f = 2 cm Agorithms Walking Gait Parameters value Best firness S (cm) H (cm) h (cm) n (cm) value f (cm) GA 15 1.99 0.82 6.99 14.87 PSO 15 2.00 0.8 6.89 14.87 MDE 15 2.00 0.8 6.89 14.87 CFO 15 2.00 0.89 6.99 14.87 Figure 10: Resulted comparative ZMP and COM survey The optimal set of parameters for four comparative algorithms presented in Table 8 shows that the target is reached with respect to the preset foot-lift value. The ZMP and COM trajectories corresponding to each of the four comparative algorithms presented in Figure 10 show that they are always within the footprint and this means that biped HUBOT-5 are achieving steady-state stable and robust walking. Based on the results described in Figure 9, it is important to notice that the MDE algorithm searches for an optimal solution with an average value of 14.8706499 after about 119 generations, while the PSO algorithm is approximately 211 generations after the search, finding an optimal solution obtained an average value of 14.88034529, while the GA algorithm must need around 470 generations to find the optimal solution with an average value of 14.9039. The proposed CFO algorithm finds an optimal solution of 14.93174983 after an average of 89 generations. These results show that the CFO algorithm outperforms GA, PSO, and MDE algorithms in terms of convergence speed. Fig. 9. Resulted comparative ZMP and COM survey The optimal set of parameters for four comparative algorithms presented in Tab. 6 shows that the target is reached with respect to the preset foot-lift value. The ZMP and COM trajectories corresponding to each of the four comparative algorithms presented in Fig. 9 shows that they are always within the footprint and this means that biped HUBOT- 5 are achieving steady-state stable and robust walking. Based on the results described in Fig. 8, it is important to notice that the MDE algo- rithm searches for an optimal solution with an average value of 14.8706499 after about 119 420 Tran Thien Huan, Ho Pham Huy Anh generations, while the PSO algorithm is approximately 211 generations after the search, finding an optimal solution obtained an average value of 14.88034529, while the GA al- gorithm must need around 470 generations to find the optimal solution with an average value of 14.9039. The proposed CFO algorithm finds an optimal solution of 14.93174983 after an average of 89 generations. These results show that the CFO algorithm outper- forms GA, PSO, and MDE algorithms in terms of convergence speed. Tab. 6 demonstrates the optimized value of the walking gait parameters to ensure the biped HUBOT-5 to walk steadily with both cases corresponding to differnet preset foot-lift magnitude. (Hre f = 2 cm and Hre f = 4 cm) using CFO. Table 6. The optimized value of the walking gait parameters Hre f (cm) CFO optimization results S (cm) H (cm) h (cm) n (cm) 2.0 15 2.0 0.89 6.99 4.0 15 4.0 1.09 7.12 Fig. 10 illustrates the 2D gait in the X-Z plane of the HUBOT-5, corresponding to two cases with different foot lifts. Tab. 6 and Fig. 11 show that the biped HUBOT-5 has a pickup lift in the set value. Table 8 demonstrates the optimized value of the walking gait parameters to ensure the biped HUBOT-5 to walk steadily with both cases corresponding to differnet preset foot-lift magnitude. ( and ) using CFO. Tables 8. The optimized value of the walking gait parameters Href (cm) CFO optimization Results S (cm) H (cm) h (cm) n (cm) 2.0 15 2.0 0.89 6.99 4.0 15 4.0 1.09 7.12 Figure 11 illustrates the 2D gait in the X-Z plane of the HUBOT-5, corresponding to two cases ith differ nt foot lifts. Tables 8 and Fig. 12 show that t e biped HUBOT-5 has a pickup lift in the set value. Hình 11. Simulated 2D gait result of biped HUBOT–5 with different foot-lifting amplitudes 2=refH cm 4=refH cm Fig. 10. Simulated 2D gait result of biped HUBOT-5 wit iff rent f ot-lifting amplitudes Comparative stable walking gait optimization for small-sized biped robot using meta-heuristic optimization algorithms 421 Fig. 11 illustrates the resulted ZMP point trajectory and the projection of COM trajec- tory for two different foot-lifting amplitudes. This shows that the ZMP point is always in the supporting foot area and then it ensures that the HUBOT-5 biped robot keeps stable walking. Figure 12 illustrates the resulted ZMP point trajectory and the projecti n of COM trajectory for two different foot-lifting amplitudes. This shows that the ZMP point is always in the supporting foot area and then it ensures that the HUBOT-5 biped robot keeps stable walking. Figure 12. Resulted ZMP và COM trajectories Using ten rotary angular values to control the biped HUBOT-5, it performs two corresponding steps with two different foot-lift values. Figures 13 illustrates the photos of the HUBOT-5 biped robot in performing a stable and steady walking step with respect to the foot-lift value . Figure 14 demonstrates the ten rotary angular trajectorial errors in one stepping cycle of the two legs of experimental biped HUBOT–5 during walking with optimally resulted sets of gait parameters in term of pre-set foot-lifting value (tabulated in Table 8). 4=refH cm ( )1 2 3 4 5 6 7 8 9 10, , , , , , , , ,q q q q q q q q q q 2refH cm= Fig. 11. Resulted ZMP and COM trajectories Using ten rotary angular values to control the biped HUBOT-5, it performs two cor- responding steps with two different foot-lift value . Fig. 12 illustrates the photos of the HUBOT-5 biped robot in performing a stable and steady walking step with respect to the foot-lift value Hre f = 4 cm. Fig. 13 demonstrates the ten rotary angular trajectorial errors in one stepping cycle of the two legs of experimental biped HUBOT-5 (θ1, θ2, θ3, θ4, θ5, θ6, θ7, θ8, θ9, θ10) during walking with optimally resulted sets of gait parameters in term of pre-set foot-lifting value Hre f = 2 cm (tabulated in Tab. 6). The simulation and experiment results abovementioned convincingly ensure that the HUBOT-5 biped robo steadily walks without falling apart and keeping pace with desired foot-lift amplitude. Hence the proposed algorithm with gait parameters optimized by CFO algorithm is quite feasible. 422 Tran Thien Huan, Ho Pham Huy Anh Figure 13: Photos of experiment biped HUBOT-5 performing stable gait with foot-lifting amplitude of The simulation and experiment results abovementioned convincingly ensure that the HUBOT-5 biped robot steadily walks without falling apart and keeping pace with desired foot-lift amplitude. Hence the proposed algorithm with gait parameters optimized by CFO algorithm is quite feasible. 4=refH cm Fig. 12. Photos of experiment biped HUBOT-5 performing stable gait with foot-lifting amplitude of Hre f = 4 cm Figure 14: The ten rotary angular trajectorial errors of two legs of experiment biped HUBOT–5 ( =2cm) 6. Conclusion In this paper, we propose a new study using the CFO algorithm as to optimize the parameters of walking pattern generation for biped robots that permits stable and robust stepping with pre- set foot-lifting magnitude. Completed experiments are determined based on novel gait generator of the two-foot trajectory, hip trajectory and the inverse kinematics collected from the human robot walking. The stable gait generation of biped robot is determined based on four key biped walking parameters: walking step length, leg lifting, leg kneeling, and hip swinging. The CFO optimization algorithm is applied to find the best solution for human robotic gait parameters so that the ZMP distance to the center of the supporting foot attains the smallest with respect to the preset foot-lifting value. The simulated and experimental results of proposed algorithm applied refH Fig. 3. The ten rotary ngula t aject rial errors of two legs of experiment biped HUBOT-5 (Hre f = 2 cm) Comparative stable walking gait optimization for small-sized biped robot using meta-heuristic optimization algorithms 423 6. CONCLUSION In this paper, we propose a new study using the CFO algorithm as to optimize the parameters of walking pattern generation for biped robots that permits stable and robust stepping with pre-set foot-lifting magnitude. Completed experiments are determined based on novel gait generator of the two-foot trajectory, hip trajectory and the inverse kinematics collected from the human robot walking. The stable gait generation of biped robot is determined based on four key biped walking parameters: walking step length, leg lifting, leg kneeling, and hip swinging. The CFO optimization algorithm is applied to find the best solution for human robotic gait parameters so that the ZMP distance to the center of the supporting foot attains the smallest with respect to the preset foot- lifting value. 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