Optimal nature-walking gait for humanoid robot using Jaya optimization algorithm

Abstract

This article proposes a new method used to optimize the design process of nature-walking gait generator that permits biped robot to stably and naturally walk with preset foot-lift magnitude. The new Jaya optimization algorithm is innovatively applied to optimize the biped gait four key parameters initiatively applied to ensure the uncertain nonlinear humanoid robot walks robustly and steadily. The efficiency of the proposed Jaya-based identification approach is compared with the central force optimization and improved differential evolution (modified differential evolution) algorithms. The simulation and experimental results tested on the original small-sized biped robot HUBOT-4 convincingly demonstrate that the novel proposed algorithm offers an efficient and stable gait for humanoid robots with precise height of foot-lift value.

Keywords

Uncertain nonlinear humanoid robot Jaya optimization algorithm central force optimization algorithm modified differential evolution algorithm nature-walking gait generator gait optimization for humanoid robot zero-moment point concept inverse and forward kinematics

Introduction

The research for advanced biped robot walking mechanism is increasingly improved in various aspects. Several new standards have been applied to humanoid robots to ensure more stable and natural gait gesture. Static walking is the first applied principle, in which the vertical projection of center of mass (CoM) on the ground is always kept within the supporting foot surface. In other words, humanoid robots can stop and move at any times during walking without falling apart. Due to its very simple manner, this principle is only successfully applied to slow-speed biped robots in which dynamic effects can be ignored.^1,2 Researchers then began to focus on developing dynamic walking.³ This method allows the humanoid robot to speed up the pace. However, during the robot locomotion process, the biped may fall due to environmental interference and cannot stop abruptly. Therefore, zero-moment point (ZMP)-based biped walking is proposed to help stable walking control and to handle biped inertia.^4,5

Numerous interesting studies have focused on improving the performance of humanoid robot walking gesture. Huang et al.⁶ introduced a stable gait generator based on a newly proposed gait pattern in using the interpolation function. The method developed by Dau et al.,⁷ through genetic algorithm (GA), optimized the gait generator parameters as to help biped robot move steadily with the least amount of energy. Dip et al.⁸ exhibited steady gait for biped at constant velocity using the sine wave generator. Maximo et al.⁹ introduced a new stable and quick model-free gait with specific arm movement for biped robots. Khusainov et al.¹⁰ successfully combined kinematics and dynamics approaches in biped gait optimization for humanoid robot locomotion. Moreover, nowadays, intelligent algorithms are being increasingly applied in this domain to optimize the gait generator parameters for humanoid robots such as GA,⁸ particle swarm optimization (PSO) algorithm,¹¹ modified differential evolution (MDE) algorithm,¹² and central force optimization (CFO) algorithm.¹³ Shafii et al.¹⁴ introduced a humanoid robot that achieved a stable gait using the Fourier series–based gait generator. Furthermore, novel approaches continually applied different intelligent algorithms to develop human robot with stable and nature-walking movement, such as bee swarm algorithm,¹⁵ T-S fuzzy controller,¹⁶ evolution techniques,¹⁷ and recurrent neural networks optimized by ant-colony optimization (ACO).¹⁸ These intelligent algorithms have shown their effectiveness in gait generator for humanoid robot. However, there are still numerous areas for further improvement, especially those related to the computational cost of meta-heuristic algorithms.

Recently, among newly proposed meta-heuristic optimization approaches, the powerful potential of the Jaya optimization algorithm has not yet being applied to optimize the biped robot walking gait generator. An enhanced optimization method, such as Jaya, initiatively introduced by Rao,¹⁹ proves to be compact and efficient to use with great benefit, which does not require any particular control coefficients. Thus, it has been quickly used and improved for optimally solving different problems in various sectors such as mechatronics,^20,21 neural model identification,²² electronics,²³ optimal civil problems,^24–26 and control system implementation,^27,28 among others. Enhancing with promising applications of the Jaya optimization algorithm, this article intends to improve the Jaya approach for nature-walking gait optimization for humanoid robot.

In this article, a new nature-walking pattern generation (N-WPG) is proposed, which consists of gait generator for the two-foot trajectory, hip trajectory, and the inverse kinematics in the full walking periodic sequence which can be seen as combination of three sub-sequences: Starting step, Periodic steps, and Ending step. A combined objective function is newly introduced by linking two separate objective functions for humanoid robots that allows stable stepping with preset foot-lift magnitude. The first one is based on ZMP criterion to stable stepping, and the second is based on the difference between the magnitude of the foot-lift parameter and the foot-lift preset value. The Jaya algorithm is used to optimally identify the biped gait parameters applied to reach steady nature-walking gait with accurate preset foot-lift magnitude. The simulation and experimental results of the proposed algorithm applied on the small-sized biped robot demonstrate the performance of the novel algorithm, allowing the biped robot to move steadily and stably with an effectively reduced training time. Furthermore, the optimal nature-walking gait results obtained through the proposed algorithm are comparatively tested with those obtained by various well-known optimization approaches such as the MDE and the CFO algorithms.

The rest of this article is arranged as follows. Section “Proposed N-WPG methodology” presents the new N-WPG for biped robot. In section “Optimal nature-walking gait for humanoid robot using Jaya algorithm,” optimal identification of the nature gait parameters based on Jaya algorithm is presented. Section “Simulation and experimental results” presents and analyses the simulation and experimental results. Finally, concluding remarks are given in section “Conclusion.”

Proposed N-WPG methodology

The biped robot model with upper torso and two legs is described in Figure 1. Each leg consists of three parts: foot-link, shank-link, and thigh-link. The joint between the foot-link and shank-link is the ankle, the joint between the shank-link and thigh-link is the knee, and meanwhile, the one between the thigh-link and waist-link is the hip. The investigated biped robot contains 10 degrees of freedom (DOFs) including 2 DOFs at the hip, 1 DOF at the knee, 2 DOFs at ankle, and 5 DOFs in each leg. Each DOF corresponds to an independent actuator. This article focuses only on the biped robot for straight nature-walking gait. Then, it fixes the biped upper body and exploits the biped lower body containing 10 joint-angles for the two legs $(θ_{1}, θ_{2}, θ_{3}, θ_{4}, θ_{5}, θ_{6}, θ_{7}, θ_{8}, θ_{9}, θ_{10})$ which are adequately defined as shown in Figure 1.

Figure 1.

Humanoid robot structure.

The Cartesian coordinates of the points represent the position of DOFs placed at the joints of two leg of biped robot P_i (P_ix, P_iy, P_iz) as shown in Figure 1. P₁ and P₂ represent the position of 2 DOFs at the ankle of left leg. P₃ represents the position of DOF at the knee of left leg. P₄ and P₅ represent the position of 2 DOFs at the hip of left leg. P₉ and P₁₀ represent the position of 2 DOFs at the ankle of right leg. P₈ represents the position of DOF at the knee of right leg. P₆ and P₇ represent the position of 2 DOFs at the hip of right leg.

For humanoid robot to be able to obtain nature-walking, it needs to plan an N-WPG in the full walking sequence. The biped walking pattern is a set of time series of joint-angles for desired walking, and to create it, we use the proposed N-WPG. The N-WPG is composed of a gait generator of the two-foot trajectory, hip trajectory, and biped inverse kinematics.

Generated trajectories of two feet and hip for humanoid robot nature-walking

As illustrated in Figure 2(a), the nature-walking sequential process is analyzed in three sub-sequential steps as follows: Starting step is where the robot begins from a full stop and gets to the first step, preparing to the periodic phase. In this starting step, the biped changes its posture from two legs in parallel erecting shape to moving the right leg forward with two knees flexed. Periodic steps represent the steps that the biped repeats walking periodically. Ending step is the ending phase in which the biped realizes a full stop after the periodic phase. In this step, the biped robot changes its posture from two knees flexed raised up to two legs in parallel erecting posture.

Figure 2.

(a) Full three nature-walking phases of humanoid robot; (b) timeline of a step as to bring the back leg forward.

A step period of humanoid robot walking (0 – T) is composed of three intervals. The first interval is a double support phase (DSP) whose range is (0 – T₁], in which the humanoid robot sways its hips toward the supporting leg to move the center of gravity and prepares to lifts its swing leg moving forward. The second phase is a single support phase (SSP) whose range is [T₁ – T₂], in which the humanoid robot lifts its swing leg moving forward. The third interval is a DSP whose range is [T₂ – T), in which the humanoid robot lands its swing leg and sways back its hips. It is important to note that T₁ and T₂ are times that humanoid robot starts and stops on the swing leg at the beginning and the ending of SSP. Figure 2(b) shows the timeline of these three intervals, which includes bringing the back swing leg to the forward position.

The nature-walking gait is expressed in terms of the following parameters: step length S, bending height h, maximum lifting height H, and maximum frontal shift n, which are all clearly described in Figure 2(a). In that, d₀ represents the length of the upper body, d₁ is the distance between two points P₁ and P₂, d₂ is the distance between two points P₂ and P₃, d₃ is the distance between two points P₃ and P₄, d₄ represents the distance between two points P₄ and P₅, and w represents the distance between two legs. According to Huang et al.,⁶ if both the feet trajectories (P₁ and P₁₀) and the hip trajectory (P₅) are known, all joint trajectories of the biped robot can be determined by kinetic constraints. P₁ and P₅ are the reference points at ankle and hip of left leg, and P₁₀ is the reference point at ankle of right leg, as shown in Figure 2(a). To build the trajectory of P₁, P₁₀, and P₅ for humanoid robots, Shih et al.²⁹ assign the trajectory of P₁ and P₁₀ as the sinusoidal form and use an iterative method to find the trajectory of P₅; Huang et al.⁶ rely on the third-order interpolation function to find the trajectories of P₁ and P₁₀ and iterative methods to find the trajectory of P₅; and Dip et al.,⁸ based on human gait, assume the trajectory of P₁, P₁₀, and P₅ as the sinusoidal form depending on four key parameters.

Therefore, the humanoid robot nature-walking motion is generated by choosing an appropriate time function for the three reference points: $P_{1} = [P_{1 x}, P_{1 y}, P_{1 z}]$ , $P_{5} = [P_{5 x}, P_{5 y}, P_{5 z}]$ , and $P_{10} = [P_{10 x}, P_{10 y}, P_{10 z}]$ . $P_{1}$ is stationary for 0 ≤ t ≤ T and 2T < t ≤ 3T and acts as a reference for $P_{5}$ and $P_{10}$ . For T < t ≤ 2T, the positions of $P_{1}$ and $P_{10}$ are interchanged. Based on the human walking pattern as depicted in Whittle’s³⁰ and Dip’s⁸ methods, we assumed that the trajectories of three reference points $P_{1}$ , $P_{5}$ , and $P_{10}$ were generated by the gait functions as trigonometric functions with kinetic constraint (biped’s waist links are always parallel to the flat surface when walking straight on a flat surface, or the height of the hips is always equal $(z_{l} (t) = z_{r} (t))$ , as illustrated in Figure 3). $P_{1}$ and $P_{10}$ are selected according to the desired leg movement for a specific activity. The choice of $P_{5}$ depends on the mechanical structure of the biped as it involves shifting of ZMP from one foot to another. In a nature-walking sequence (0 ≤ t ≤ 3T), $P_{1}$ and $P_{10}$ for straight walking are provided by equations (1) and (2), while $P_{5}$ for straight walking is given by equation (3)

P_{1 x} (t) = {\begin{matrix} 0, 0 \leq t < T \\ \frac{S}{2} (1 - \sin (\frac{w}{2} t - \frac{π}{2})), T \leq t \leq 2 T \\ S, 2 T < t \leq 3 T \end{matrix}

(1a)

P_{1 y} (t) = 0, 0 \leq t \leq 3 T

(1b)

P_{1 z} (t) = {\begin{matrix} 0, 0 \leq t < T \\ H \sin (\frac{π}{2} (\sin (\frac{w}{2} t - \frac{π}{2}) + 1)), T \leq t \leq 2 T \\ 0, 2 T < t \leq 3 T \end{matrix}

(1c)

P_{10 x} (t) = {\begin{matrix} \frac{S}{4} (1 + \sin (\frac{w}{2} t - \frac{π}{2})), 0 \leq t < T \\ \frac{S}{2}, T \leq t \leq 2 T \\ \frac{S}{4} (3 + \sin (\frac{w}{2} t - \frac{5 π}{2})), 2 T < t \leq 3 T \end{matrix}

(2a)

P_{10 y} (t) = - w, 0 \leq t \leq 3 T

(2b)

\begin{matrix} P_{10 z} (t) = \\ {\begin{matrix} H \sin (\frac{π}{2} (\sin (\frac{w}{2} t - \frac{π}{2}) + 1)), 0 \leq t < T \\ 0, T \leq t \leq 2 T \\ H \sin (\frac{π}{2} (\sin (\frac{w}{2} t - \frac{5 π}{2}) + 1)), 2 T < t \leq 3 T \end{matrix} \end{matrix}

(2c)

P_{5 x} (t) = {\begin{matrix} \frac{S}{8} (1 + \sin (\frac{w}{2} t - \frac{π}{2})), 0 \leq t < T \\ \frac{S}{2} [1 + \frac{1}{2} \sin (\frac{w}{2} t - \frac{3 π}{2})], T \leq t \leq 2 T \\ \frac{S}{8} (7 + \sin (\frac{w}{2} t - \frac{5 π}{2})), 2 T < t \leq 3 T \end{matrix}

(3a)

\begin{matrix} P_{5 y} (t) = \\ {\begin{matrix} n \sin (\frac{π}{2} \sin (\frac{w}{4} t + 1)) \sin (\frac{w}{2} t), 0 \leq t < T \\ n \sin (\frac{π}{2} (\sin (\frac{w}{4} t - \frac{π}{2}) + 1)) \sin (\frac{w}{2} t), T \leq t \leq 2 T \\ n \sin (\frac{π}{2} (1 - \sin (\frac{w}{4} t))) \sin (\frac{w}{2} t), 2 T < t \leq 3 T \end{matrix} \end{matrix}

(3b)

\begin{matrix} P_{5 z} (t) = \\ {\begin{matrix} k_{1} + h \sin (\frac{π}{2} \sin (\frac{w}{4} t + 1)) \sin (\frac{w}{2} t + \frac{π}{2}), 0 \leq t < T \\ k_{1}, T \leq t \leq 2 T \\ k_{1} + h \sin (\frac{π}{2} \sin (\frac{w}{4} (3 T - t) + 1)) \sin (\frac{w}{2} (3 T - t) + \frac{π}{2}), 2 T < t \leq 3 T \end{matrix} \end{matrix}

(3c)

where T is the time taken by the humanoid robot to perform a step and $k_{1} = d_{1} + d_{2} + d_{3} + d_{4} - h$ . Based on equations (1)–(3), not only trajectory of P₅ but also trajectories of P₁ and P₁₀ are all applied to ensure nature-walking gait for the biped robot.

Figure 3.

Parameters for inverse kinematics problem.

Inverse kinematics of humanoid robot

In this section, we discuss how to calculate the rotation angles of the biped robot $(θ_{1}, θ_{2}, θ_{3}, θ_{4}, θ_{5}, θ_{6}, θ_{7}, θ_{8}, θ_{9}, θ_{10})$ after we know the position of two legs $(P_{1} = [P_{1 x}, P_{1 y}, P_{1 z}], P_{10} = [P_{10 x}, P_{10 y}, P_{10 z}])$ and hip $(P_{5} = [P_{5 x}, P_{5 y}, P_{5 z}])$ as described in Figure 1.

Conventionally, humanoid inverse kinematics is successfully solved by calculus or numerical algorithms. For analytical methods, more calculations must be performed than numerical methods. Solving the inverse kinematics problem by numerical methods is a combination of forward kinematics and iterative method. The Newton–Raphson method gives accurate and fast results by applying the Jacobian inverse matrices. However, this method does not work correctly in singular posture because it does not guarantee a stable calculation of inverse kinematics at the singular posture. When biped has a singular posture, the Jacobian matrix becomes a singular matrix and we cannot calculate its inverse. By using the singularity robust (SR) inverse matrices, we can ensure the stable calculation of inverse kinematics at the singular posture. The iterative method using SR inverse matrices is often called the Levenberg–Marquardt method.³¹

However, this section presents geometric methods.³² The parameters of inverse kinematic model are as follows: $x_{l} (t)$ , $y_{l} (t)$ , $z_{l} (t)$ , $x_{r} (t)$ , $y_{r} (t)$ , $z_{r} (t)$ , $l_{l} (t)$ , $l_{r} (t)$ , $θ_{A} (t)$ , $θ_{B} (t)$ , $θ_{C} (t)$ , $θ_{D} (t)$ , which are clearly defined in Figure 3 and equation (4). It is important to note that, $l_{1}$ represents the distance between $P_{2}$ and $P_{4}$ , and $l_{r}$ represents the distance between $P_{9}$ and $P_{7}$ ; $d_{1}$ , $d_{2}$ , $d_{3}$ , and $d_{4}$ are adequately illustrated in Figure 2

{\begin{matrix} x_{l} = P_{5 x} - P_{1 x}, y_{l} = P_{5 y}, z_{l} = P_{5 z} - P_{1 z} \\ l_{l} = \sqrt{{(P_{4 x} - P_{2 x})}^{2} + {(P_{4 y} - P_{2 y})}^{2} + {(P_{4 z} - P_{2 z})}^{2}} \\ x_{r} = P_{6 x} - P_{10 x}, y_{r} = P_{5 y}, z_{r} = P_{6 z} - P_{10 z} \\ l_{r} = \sqrt{{(P_{7 x} - P_{9 x})}^{2} + {(P_{7 y} - P_{9 y})}^{2} + {(P_{7 z} - P_{9 z})}^{2}} \\ θ_{A} = \arccos (\frac{d_{2}^{2} + d_{3}^{2} - l_{l}^{2}}{2 d_{2} d_{3}}), θ_{B} = \arccos (\frac{d_{3} \sin (θ_{A})}{l_{l}}) \\ θ_{C} = \arccos (\frac{d_{2}^{2} + d_{3}^{2} - l_{r}^{2}}{2 d_{2} d_{3}}), θ_{D} = \arccos (\frac{d_{3} \sin (θ_{C})}{l_{l}}) \end{matrix}

(4)

The coordination of $P_{6} (x, y, z)$ is computed using $P_{5} (x, y, z)$ , and the coordination of [ $P_{2} (x, y, z)$ , $P_{4} (x, y, z)$ , $P_{7} (x, y, z)$ , $P_{9} (x, y, z)$ ] is computed using [ $P_{1} (x, y, z)$ , $P_{5} (x, y, z)$ , $P_{6} (x, y, z)$ , $P_{10} (x, y, z)$ ] and the rotary angle $[θ_{1}, θ_{5}, θ_{6}, θ_{10}]$ . Equation (5) is used to determine $P_{2}, P_{4}, P_{6}, P_{7}, P_{9}$

{\begin{matrix} P_{2 x} = P_{1 x}, P_{2 z} = d_{1} \cos (θ_{1}), P_{2 y} = P_{2 z} \sin (θ_{1}) \\ P_{4 x} = P_{5 x}, P_{4 z} = P_{5 z} - d_{4} \cos (θ_{1}), P_{4 y} = P_{5 y} - (P_{5 z} - P_{4 z}) \sin (θ_{1}) \\ P_{6 x} = P_{5 x}, P_{6 y} = P_{5 y} - w, P_{6 z} = P_{5 z} \\ P_{7 x} = P_{6 x}, P_{7 z} = P_{6 z} - d_{4} \cos (θ_{10}), P_{7 y} = P_{6 y} - (P_{6 z} - P_{7 z}) \sin (θ_{10}) \\ P_{9 x} = P_{10 x}, P_{9 z} = P_{10 z} + d_{4} \cos (θ_{10}), P_{9 y} = P_{10 y} + (P_{9 z} - P_{10 z}) \sin (θ_{10}) \end{matrix}

(5)

where w is clearly illustrated in Figure 2. Eventually, the angle of rotation of the two legs of biped is defined in equation (6)

{\begin{matrix} θ_{1} (t) = \arctan (\frac{y_{l} (t)}{z_{l} (t)}), θ_{5} (t) = - θ_{1} (t) \\ θ_{10} (t) = \arctan (\frac{y_{r} (t)}{z_{r} (t)}), θ_{6} (t) = - θ_{10} (t) \\ θ_{3} (t) = π - θ_{A} (t), θ_{8} (t) = π - θ_{C} (t) \\ θ_{4} (t) = \frac{π}{2} - θ_{A} (t) + θ_{B} (t) - \arcsin (\frac{x_{l} (t)}{l_{l} (t)}) \\ θ_{7} (t) = \frac{π}{2} - θ_{C} (t) + θ_{D} (t) - \arcsin (\frac{x_{r} (t)}{l_{r} (t)}) \\ θ_{2} (t) = θ_{3} (t) - θ_{4} (t), θ_{11} (t) = θ_{9} (t) - θ_{7} (t) \end{matrix}

(6)

In summary, based on equations (4)–(6), the eventual 10 trajectories of the joint-angles lying on the two legs of the humanoid robot in a nature-walking sequence (0 ≤ t ≤ 3T) are exactly calculated to ensure precise and effective control of the humanoid robot with nature-walking gait. Thus, the set of four parameters, H, h, s, and n, needs to be optimally selected so that the resulting ZMP parameter ensures that the biped robot can walk steadily with the preset foot-lift value. In this article, the Jaya algorithm is used to satisfactorily solve this task.

Optimal nature-walking gait for humanoid robot using Jaya algorithm

Objective function specification

To evaluate human gait parameters of the humanoid robot, one must define the objective function. The goal of humanoid robot is to achieve a stable gait with preset foot-lift value. For this purpose, the ZMP point is always ensured to lie within the foot area.³³ If the ZMP is within the area of the supporting leg, the robot does not fall.³⁴ The calculation of the ZMP of biped robots in walking is shown in Appendix 1. The stability of the humanoid robot is quantified by the distance between the ZMP and the foot center in the step cycle. Walking gait with maximum stability is obtained by minimizing the function, f₁, in equation (7)⁸

f_{1} = \int_{0}^{T} \sqrt{x_{ZMP}^{2} + y_{ZMP}^{2}} . dt

(7)

where T denotes the step cycle and $(x_{ZMP}, y_{ZMP})$ denotes the coordination of ZMP point in the robot’s process of stepping which is away from the quadrant in the center of the foot. In this article, equation (7) represents the first objective function.

In addition, for the humanoid robot to follow the preset foot-lift height value, $H_{ref}$ , the difference between the magnitude of the foot-lift value, $H$ , and the foot-lift preset value, $H_{ref}$ (see equation (8)), represents the second objective function

f_{2} = | H_{ref} - H |

(8)

Thus, in order for biped robot to obtain a steady gait with the foot-lift set up in advance, we find the minimum value of the two objective functions $f_{1}$ and $f_{2}$ or similarly to find the minimum of the function $f$ as

{\begin{matrix} f = λ (\int_{0}^{T} (\sqrt{x_{ZMP}^{2} + y_{ZMP}^{2}}) dt) + (1 - λ) | H_{ref} - H | \\ - F_{x 1} \leq x_{ZMP} \leq F_{x 2}; - F_{y 1} \leq y_{ZMP} \leq F_{y 2} \end{matrix}

(9)

where $F_{x 1} + F_{x 2}$ and $F_{y 1} + F_{y 2}$ are the length and width of the biped robot foot, respectively, and $λ (0 \leq λ \leq 1)$ can be used to satisfactorily select the priority between the walking stability ( $λ$ increased) and the variance with the desired foot-lift magnitude ( $λ$ decreased).

Jaya algorithm

Rao¹⁹ proposed a variant of the teaching-learning-based optimization (TLBO) algorithm called Jaya algorithm. Unlike the traditional TLBO algorithm, Jaya is much simpler due to its two advantages: (1) using only one phase instead of two phases of the traditional TLBO algorithm; (2) requiring only popular identification parameters like population size and amount of generations. The effectiveness and robustness of the method are demonstrated through unconstrained and constrained benchmark functions and constrained mechanical design problems. The numerical results revealed that Jaya algorithm proved superior to or competitive with other well-known optimization algorithms for the problems considered. The details of Jaya algorithm are presented below.

First, an initial population of N solutions is created by means of using randomly sampling from the design space. Each ith solution of the population is a vector containing D design variables, $x_{j} = (x_{1}, x_{2}, \dots, x_{D})$ , and is generated by

\begin{matrix} x_{j, i} = x_{j, i}^{l} + rand (x_{j, i}^{u} - x_{j, i}^{l}), \\ i = 1, 2, \dots, N; j = 1, 2, \dots, D \end{matrix}

(10)

where $x_{j}^{l}$ and $x_{j}^{u}$ are the lower and upper bounds of $x_{j}$ , respectively; rand is a uniformly distributed random number in the interval [0,1]; N is the population size; and D is the number of design variables.

Second, corresponding to each ith solution $x_{i}$ , a new solution $u_{i}$ is generated using the information of the current solution $x_{i}$ , the best solution x_best, and the worst solution x_worst of the population through the operation

\begin{matrix} u_{j, i} = x_{j, i} + ran d_{1, j} (x_{j, best} - | x_{j, i} |) \\ - ran d_{2, j} (x_{j, worst} - | x_{j, i} |) \end{matrix}

(11)

where $ran d_{1, j}$ and $ran d_{2, j}$ are the random numbers within the interval [0, 1], and $| x_{j, i} |$ is the absolute value of the solution $x_{j, i}$ . In equation (11), the component $ran d_{1, j} (x_{j, best} - | x_{j, i} |)$ has a trend to move the new solution closer to the best solution, while the component $ran d_{2, j} (x_{j, worst} - | x_{j, i} |)$ has a trend to move the solution far away from the worst solution. The random numbers $ran d_{1, j}$ and $ran d_{2, j}$ will guarantee good exploration of the search space.

Finally, using the objective function, the vector $u_{i}$ is compared to the target vector $x_{i}$ . The better one having lower objective function value will survive to the next generation

x_{i} = {\begin{matrix} u_{i} if f (u_{i}) \leq f (x_{i}) \\ x_{i} otherwise \end{matrix}

(12)

Proposed Jaya training approach

This section introduces the fully principal steps in order to employ the Jaya optimization method as presented in Algorithm 1.

Algorithm 1. Jaya training algorithm
1: Create the initial population 2: Assess the fitness of each individual in the population 3: while (delta > tolerance and MaxIter is not reached) do 4: for i = 1 to NP do 5: Determine x_best, x_worst 6: for j = 1 to D do 7: if x_j,i is variable $r_{f}^{(k)}$ 8: $u_{j, i} = x_{j, i} + ran d_{1, j} (x_{j, best} - \| x_{j, i} \|) - ran d_{2, j} (x_{j, worst} - \| x_{j, i} \|)$ 9: elseif x_{j, i} is variable $t^{(k)}$ 10: $u_{j, i} = x_{j, i} + ran d_{1, j} (x_{j, best} - \| x_{j, i} \|) - ran d_{2, j} (x_{j, worst} - \| x_{j, i} \|)$ 11: $u_{j, i} = round (u_{j, i}) to set t_{D}$ 12: end 13: end for 14: Evaluate the new solution u_i 15: if $E_{N} (u_{i}) \leq E_{N} (x_{i})$ 16: $x_{i} = u_{i}$ 17: end 18: end for 19: Determine f_best, $f_{mean} = \frac{1}{N} \sum_{i = 1}^{N} f_{i}$ , and $delta = \| \frac{f_{mean}}{f_{best}} - 1 \|$ 20. end while

Algorithm 1. Jaya training algorithm

1: Create the initial population
2: Assess the fitness of each individual in the population
3: while (delta > tolerance and MaxIter is not reached) do
4: for i = 1 to NP do
5: Determine x_best, x_worst
6: for j = 1 to D do
7: if x_j,i is variable

r_{f}^{(k)}

u_{j, i} = x_{j, i} + ran d_{1, j} (x_{j, best} - | x_{j, i} |) - ran d_{2, j} (x_{j, worst} - | x_{j, i} |)

9: elseif x_{j, i} is variable

t^{(k)}

10:

u_{j, i} = x_{j, i} + ran d_{1, j} (x_{j, best} - | x_{j, i} |) - ran d_{2, j} (x_{j, worst} - | x_{j, i} |)

11:

u_{j, i} = round (u_{j, i}) to set t_{D}

12: end
13: end for
14: Evaluate the new solution u_i
15: if

E_{N} (u_{i}) \leq E_{N} (x_{i})

16:

x_{i} = u_{i}

17: end
18: end for
19: Determine f_best,

f_{mean} = \frac{1}{N} \sum_{i = 1}^{N} f_{i}

, and

delta = | \frac{f_{mean}}{f_{best}} - 1 |

20. end while

Here, tolerance represents the preset value which allows for termination and MaxIter represents the maximal number of iterations. Based on Algorithm 1, Jaya algorithm will stop the searching process either in case the delta is less than or equal to the tolerance or in case the maximal number of iteration, MaxIter, is attained.

Simulation and experimental results

The simulation and experimental results were fully tested on the small-sized HUBOT-4 biped robot. Figure 4 illustrates the photographs of the small-sized biped robot (HUBOT-4). The values of physical parameters of the biped robot model presented in Figure 2 and applied to the biped HUBOT-4 are shown in Table 1.

Figure 4.

Photograph of small-sized humanoid robot (HUBOT-4).

Table 1.

Physical parameters of biped HUBOT-4.

Parameters	Values
$d_{0}$	6.0 cm
$d_{1}$	4.5 cm
$d_{2}$	6.0 cm
$d_{3}$	6.0 cm
$d_{4}$	4.5 cm
$w$	6.6 cm
$F_{y 1}$	2.0 cm
$F_{y 2}$	4.0 cm
$F_{x 1} = F_{x 2}$	2.4 cm
${\begin{matrix} m_{i} \\ i = 1, \dots, 10 \end{matrix}$	70 g

The biped robot HUBOT-4 will walk stably with the preset foot-lift value. The coefficient $λ$ used in the objective function of equation (9) is optimally selected to be equal to 0.4, and this $λ$ value will be used throughout the comparative testing process using CFO, MDE, and the proposed Jaya optimization algorithms. For optimal test based on CFO, MDE, and Jaya optimization algorithms, the four key gait parameters derived from the biped HUBOT-4 are limited as given in Table 2.

Table 2.

Bounds for four key walking gait parameters of humanoid HUBOT-4.

Parameters	Lower bound	Upper bound
S, step length	8 cm	12 cm
H, foot lifting	0.1 cm	5 cm
h, kneeling	1.1 cm	1.5 cm
n, hip swinging	0.1 cm	6 cm

The mathematical properties of CFO, MDE, and proposed Jaya optimization algorithms are meta-heuristic algorithms, so each algorithm will perform 10 different training times, with each training repeated 200 times (N = 200) using the same population size (NP = 32) and the same number of variables (D = 4). Table 3 eventually presents the optimally selected parametric values of CFO, MDE, and Jaya optimization algorithms. The optimal process is performed by MATLAB version 2014b on Intel Core i53210m computer with a clock rate of 2.5 GHz and 8GB RAM.

Table 3.

Parameters of CFO, MDE, and Jaya algorithms.

Methods	Parameters	Values
CFO	Alpha	0.25
	Beta	0.35
	Gamma	0.95
	Frep	0.5
	deltaFrep	0.05
MDE	Mutation value (F)	Random [0.4; 1.0]
	Crossover probability (CR)	Random [0.7; 1.0]
Jaya	No parameters	No

CFO: central force optimization; MDE: modified differential evolution.

The foot-lift height of biped HUBOT-4 is $H_{ref} = 2 cm$ . Table 4 shows the optimum gait value and the best value for the target function of 10 runs corresponding to the CFO, MDE, and the proposed Jaya algorithms. Figure 9 illustrates the mean value of the target function after 10 runs of each algorithm (CFO: green, MDE: blue, Jaya: red).

Table 4.

Optimized gait value and best target value through 10 runs.

Run	Algorithms	Nature-walking gait parameter value				Best fitness value, f (cm)
		S (cm)	H (cm)	h (cm)	n (cm)
1	CFO	11.9776	2.1089	1.1340	5.8702	104.227838693557
	MDE	11.9999	2.0000	1.1000	5.9999	101.856724029582
	Jaya	12.0000	2.0000	1.1000	6.0000	101.856724029549
2	CFO	11.8853	2.0000	1.1029	5.9911	102.917377396769
	MDE	12.0000	1.9999	1.1000	5.9999	101.856724029549
	Jaya	12.0000	1.9999	1.1000	6.0000	101.856724029549
3	CFO	11.9906	1.9999	1.1006	5.9397	102.056690154217
	MDE	11.9999	2.0000	1.1000	5.9999	101.856724029553
	Jaya	11.9999	1.9999	1.1000	6.0000	101.856724029548
4	CFO	12.0000	1.9999	1.1028	5.9918	102.020891026649
	MDE	11.9999	2.0000	1.1000	5.9999	101.856724029553
	Jaya	12.0000	1.9999	1.1000	6.0000	101.856724029549
5	CFO	11.9883	2.0000	1.1002	5.8400	102.213851101402
	MDE	12.0000	1.9999	1.1000	5.9999	101.856724029553
	Jaya	12.0000	1.9999	1.1000	6.0000	101.856724029549
6	CFO	11.9981	1.9999	1.1073	5.9951	102.275455603934
	MDE	11.9999	1.9999	1.1000	5.9999	101.856724029578
	Jaya	12.0000	2.0000	1.1000	6.0000	101.856724029549
7	CFO	11.9682	2.0000	1.1005	5.9805	102.152879788999
	MDE	11.9999	2.0000	1.1000	5.9999	101.856724029584
	Jaya	11.9999	1.9999	1.1000	6.0000	101.856724029548
8	CFO	11.9857	2.0000	1.1004	5.9286	102.101671823375
	MDE	11.9999	1.9999	1.1000	5.9999	101.856724029588
	Jaya	12.0000	1.9999	1.1000	6.0000	101.856724029549
9	CFO	11.9970	1.9999	1.1034	5.9158	102.198209311373
	MDE	11.9999	2.0000	1.1000	5.9999	101.856724029561
	Jaya	11.9999	2.0000	1.1000	5.9999	101.856724029548
10	CFO	11.9176	1.9999	1.1119	5.9766	103.194032541487
	MDE	11.9999	1.9999	1.1000	5.9999	101.856724029554
	Jaya	11.9999	1.9999	1.1000	6.0000	101.856724029548

CFO: central force optimization; MDE: modified differential evolution.

Note: Bold values represent the best estimated values.

Based on the results described in Figure 5, it is important to note that the Jaya algorithm searches for an optimal solution with an average value of 101.856724029549 after about 30 generations; the MDE algorithm finds an optimal solution with an average value of 101.856724029566 approximately 87 generations after the search; and the CFO algorithm needs around 68 generations to find the optimal solution with an average value of 102.535889744176. These results show that the Jaya algorithm outperforms CFO and MDE in terms of convergence speed.

Figure 5.

Mean value of fitness convergence $f$ .

Derived from Table 4, the optimum set of four key parameters for the biped HUBOT-4 conformed to the objective of 10 runs per CFO, MDE, and the proposed Jaya algorithms is shown in Table 5.

Table 5.

Resulting parameter set for three comparative algorithms.

H_ref = 2 cm
Algorithms	Walking gait parameter value				Best fitness valuef (cm)	Time to obtain optimal solutions for one run (s)
	S (cm)	H (cm)	h (cm)	n (cm)
CFO	12	1.9999	1.1028	5.9918	102.213851101402	2.210329809870977e + 02
MDE	12.0000	1.9999	1.1000	5.9999	101.856724029549	1.880558813712478e + 02
Proposed Jaya	11.9999	2.0000	1.1000	5.9999	101.856724029548	1.709196832045101e + 02

CFO: central force optimization; MDE: modified differential evolution.

Table 5 shows the comparative optimized results of CFO, MDE, and the proposed Jaya algorithms. The difference between the magnitude of the foot-lift value, $H$ , and the foot-lift preset value, $H_{ref} = 2 cm$ , of the proposed Jaya proves better than MDE and CFO with the value of about −0.005%. The offline optimization execution time for the proposed WPG has been optimally identified using the Jaya algorithm applied to HUBOT-4 with less consuming time than MDE and CFO algorithms. Using comparative results tabulated in Table 5, it is evident to conclude that the proposed Jaya algorithm proves to be the best precise and robust optimization algorithm in comparison with the MDE and CFO.

Using the gait parameter set which is optimally identified by the Jaya algorithm, a designed objective evaluation is performed. The four parameters, S, H, h, and n, are taken to propose N-WPG (see Section 2) to determine the orbit of the 10 joint-angles of the two legs of biped HUBOT-4 as illustrated in Figure 6.

Figure 6.

Resulting 10 joint-angle trajectories of biped robot HUBOT-4.

It is necessary to note that the resulting 10 trajectories are to be used to investigate the biped gait, including the CoM and ZMP trajectories in a nature-walking sequence through forward kinematics with respect to the configuration optimally identified by Jaya algorithms as illustrated in Figures 7 –9. Figure 7 shows that the biped HUBOT-4 obtains a pick-up foot-lift value with respect to preset value (H_ref = 2 cm). This figure clearly describes the stick diagram of robot simulation for a nature-walking sequence in the x-z plane and the y-z plane. In Starting step, the biped robot changes the posture of two legs from parallel erecting to moving the right leg forward with two knees flexed. In this process, the left leg plays the role of supporting leg and the right leg moves forward as shown in Figure 7(a)-[A]; the hip of the biped moves to the left of the left leg in the y-direction as shown in Figure 7(b)-[A]. In Periodic steps, the right leg is the lifting leg and moves forward as shown in Figure 7(a)-[B]; the hip of the biped moves to the left of the right leg as shown in Figure 7(b)-[B]. In Ending step, the biped robot changes its posture from two knees flexed in moving to two legs standing in parallel erecting posture. In this process, the left leg is the supporting leg and the right leg moves forward as shown in Figure 7(a)-[C]; the hip of the biped moves to the left of the left leg as shown in Figure 7(b)-[C].

Figure 7.

Stick diagram of biped robot for a nature-walking sequence in the (a) x-z plane and (b) y-z plane—[A]: Starting step, [B]: Periodic steps, [C]: Ending step.

Figure 8.

Horizontal GCoM trajectory, ZMP trajectory, and footstep locations (ZMP always remains within the support region during the whole process).

Figure 9.

CoM trajectory of the biped robot in the frontal plane within biped nature-walking sequence: the biped robot first went through Starting step (green), then through Periodic steps (red), and finally ending with Ending step (blue).

Figure 8 shows the resulting ZMP and grounded CoM (GCoM) trajectories when biped HUBOT-4 steps along with a nature-walking sequence (0 ≤ t ≤ 3T), in which p₀, p₁, p₂, p₃, and p₄ represent the central points of foot placement and the rectangles (solid line) are successive footprints of the biped robot.

In Figure 8, p₀, p₁, p₂, p₃, and p₄ illustrate an example of successive foot placement, which can be represented using step length (S) along the walking direction and the distance between two legs (w) for the lateral y-direction.

Set p₀ and p₁ as the placements of two feet when the biped robot keeps two legs in the parallel erecting posture, and the corresponding coordinates are obtained as follows

{\begin{matrix} p_{0 x} = 0 \\ p_{0 y} = - w \end{matrix} and {\begin{matrix} p_{1 x} = 0 \\ p_{1 y} = 0 \end{matrix}

In Starting step, p₁ represents the place of the support foot, which is the left foot, and p₂ represents the place of the moving forward foot, which is the right foot, and the corresponding coordinates are fixed as

{\begin{matrix} p_{2 x} = \frac{S}{2} \\ p_{2 y} = - w \end{matrix}

In Periodic steps, p₂ represents the place of the support foot, which is the right foot, and p₃ represents the place of the moving forward foot, which is the left foot, and the corresponding coordinates are fixed as

{\begin{matrix} p_{3 x} = S \\ p_{3 y} = 0 \end{matrix}

Finally, in Ending step, p₃ represents the place of the support foot, which is the left foot, and p₄ represents the eventual place of the moving forward foot, which is the right foot, and the corresponding coordinates are fixed as

{\begin{matrix} p_{4 x} = S \\ p_{4 y} = - w \end{matrix}

The ZMP and GCoM trajectories in Figure 8 demonstrate that ZMP and GCoM positions are always within the footprint, and this means that biped HUBOT-4 achieves a steady-state stable and robust walking.

Figure 9 illustrates the biped CoM trajectory in the frontal view, in which the biped first went through Starting step (green), then through Periodic steps (red) during periodic walking, and finally through Ending step (blue) to return to the initial standing position. This figure shows in detail that within Starting step (green), the CoM position sways along with the hip in the y-direction and is lowered in the z-direction because the two legs from the erecting posture changed to flexed shape in walking. Then, the CoM position is raised and lowered periodically within Periodic steps (red) in both z- and y-directions and eventually, the CoM position is raised in the z-direction to move back to the initial standing position within Ending step (blue) since the two legs from flexed shape raised up to the parallel erecting posture of the initial standing position.

Conclusion

This article innovatively presents the new offline method for planning robust nature-walking patterns first applied to the small-sized biped robot HUBOT-4 in both sagittal and frontal planes. Human-like robust walking patterns are realized by analyzing human walk and then the desired step length, foot lift, and so on, are set according to this. The hip, knee, and ankle joint-angle trajectories are then planned according to the typical parameters of the biped robot and the ground conditions. Based on these principal parameters, a new study using the Jaya algorithm is proposed to optimize the parameters of N-WPG for biped robots that permits stable and robust stepping with preset foot-lift magnitude. Completed experiments are determined based on novel gait generator of the two-foot trajectory, hip trajectory, and the inverse kinematics in the humanoid robot walking. The stable gait generation of biped robot is determined based on four basic parameters: walking step length, leg lifting, leg kneeling, and hip swinging. The Jaya optimization algorithm is innovatively 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 value with respect to the preset foot-lift value. The simulated results of the proposed nature-walking gait generation algorithm applied on the small-sized biped HUBOT-4 demonstrate the perfect performance of novel algorithm, allowing the biped robot to move steadily and stably with an effectively reduced training time. After having done the offline optimization for N-WPG, for HUBOT-4 to perform the desired gait, a controller should be designed to control 10 motors located at the two legs of HUBOT-4 according to the corresponding 10-angle rotation from the output of the N-WPG. Future research will focus on designing the forward controller to control the HUBOT-4 gait in real time.

Footnotes

Appendix 1

Handling Editor: James Baldwin

Author note

Cao Van Kien is now affiliated with Faculty of Electronics Technology, Industrial University of Ho Chi Minh City (IUH), Vietnam.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was totally funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), under grant number MDT 107.01-2018.10.

ORCID iD

Ho Pham Huy Anh

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