Open AccessArticle

Development of a Reduced-Degree-of-Freedom (DOF) Bipedal Robot with Elastic Ankles

Sharafatdin Yessirkepov

Michele Folgheraiter

Arman Abakov

and

Timur Umurzakov

Department of Robotics, School of Engineering and Digital Sciences, Nazarbayev University, Kabanbay Batyr Ave. 53, Astana 01000, Kazakhstan

Author to whom correspondence should be addressed.

Robotics 2024, 13(12), 172; https://doi.org/10.3390/robotics13120172

Submission received: 27 September 2024 / Revised: 20 November 2024 / Accepted: 21 November 2024 / Published: 4 December 2024

(This article belongs to the Section Intelligent Robots and Mechatronics)

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Abstract

One of the most challenging aspects of designing a humanoid robot is ensuring stable walking. To achieve this, the kinematic architecture must support 3D motion and maintain equilibrium, particularly during single-foot support. Without proper configuration, the robot may experience unbalanced weight distribution, significantly increasing the risk of falling while walking. While adding redundant degrees of freedom (DOFs) can enhance adaptability, it also raises the system’s complexity and cost and the need for more sophisticated control strategies and higher energy consumption. This paper explores a reduced-DOF bipedal robot, which, despite its limited number of DOFs, is capable of performing 3D motion. It features an inverted pendulum and elastic ankles made of thermoplastic polyurethane (TPU), enabling effective balance control and attenuation of disturbances. The robot’s electromechanical design is introduced alongside the kinematic model. Momentum equilibrium in a pseudo-static mode is considered in both the frontal and sagittal planes, taking into account the pendulum and the swinging leg during the single support phase. The TPU ankle’s performance is assessed based on its ability to resist external bending forces, highlighting challenges related to the robot’s weight equilibrium stability and ankle inversion. Experimental results from both Finite Element Analysis (FEA) and real-world tests are compared. Lastly, the joint movements of the inverted pendulum-based biped robot are evaluated in both a virtual environment and a physical prototype while performing lateral tilting and various gait sequences.

Keywords:

reduced-DOF bipedal robot; minimally actuated robot; inverted pendulum; robot balance equilibrium; bipedal locomotion; elastic ankle; ankle inversion–eversion

1. Introduction

Designing an optimized bipedal robot presents a complex research challenge. It involves adjusting the configuration and placement of the joints, along with optimizing the link geometry to achieve the desired movements. Most humanoid robots capable of performing 3D gait have 12 degrees of freedom (DOFs), with 6 DOFs for each leg. This design enables mobility similar to that of a human, but it adds complexity and cost. In this scenario, each leg may include knee, ankle, and hip components with one, two, and three joints, respectively, mimicking the structure of human legs [1,2,3].

Conversely, reduced-DOF bipedal robots use fewer motors, resulting in movements that do not closely resemble those of humans [4,5,6,7]. Certain tasks, such as maintaining a standing position, making minor adjustments, or moving feet minimally to operate stationary machinery or monitor a designated area, may not require intricate human-like mobility. For applications where conserving energy is crucial, such as extended monitoring or patrolling [8], a robot may adopt a more energy-efficient walking style [9,10], quasi-static motion, or waddling, instead of more complex and energy-demanding human-like gaits.

Some research works have been dedicated to advancing bipedal robot development. Key research on walking biped robots includes topics such as trajectory generation using an inverted pendulum model [11,12,13,14,15], stability control [16,17,18,19,20], foot slip mitigation [21,22] and adaptation on uneven surfaces [23,24,25].

1.1. Trajectory Generation Using an Inverted Pendulum Model

One of the major challenges in bipedal robots is generating an optimized trajectory that ensures stable, energy-efficient, and natural walking motions. Integrating an inverted pendulum into the robot’s kinematic architecture simplifies its dynamics, making it easier to control the center of mass (COM). In [11], an inverted pendulum was attached to the top of a turning biped robot to create a smooth trajectory, while in [12], a Spring-Loaded Inverted Pendulum (SLIP) model was used to optimize the spring stiffness in each leg to achieve stable walking. A novel touchdown return map was also introduced to control the stability of the foot placement using a COM-velocity controller. In [13], the Linear Inverted Pendulum Model (LIPM), and the Linear Pendulum Model (LPM) were characterized to produce a flexible gait sequence. This approach considered variable dynamic equations and planned trajectories for scenarios such as transitions between LIPM and LPM models, periodic walking, adjusted walking speed, distributed state recovery, and terrain-blind walking. An online motion control system for a biped robot running on uneven terrain was developed in [14], incorporating a dual inverted pendulum model. Linear Model Predictive Control (MPC) and a Quadratic Problem solver were implemented. A single inverted pendulum model proves inadequate for rough terrain due to the terrain’s nonlinearity. In contrast, the dual inverted pendulum model addresses this issue by using the first pendulum to manage horizontal walking sequences and the second pendulum to handle vertical adjustments.

1.2. Stability Control

To maintain weight equilibrium effectively, stability control methods were implemented in [16,17,18]. In [16], the swinging behavior of the hip and knee joints was tested by comparing models developed in both real and virtual environments. Joint positions and torques of a Seven-Link biped robot were controlled using a PD controller. In [17], and a Passive Dynamic Autonomous Control (PDAC) concept was proposed to stabilize the robot’s gait sequence, considering foot contact constraints. Here, walking direction and speed were controlled as conservative quantities. In [18], a bivariate-stability-margin optimization algorithm was introduced to achieve locomotion stability for a 12-DOF robot, along with the development of a Random Vector Function-link Neural Network (RVFLNN) mechanism to minimize zero moment point (ZMP) error.

Similarly, balance control method in humanoid robots was achieved through calculations based on the Center of Mass (COM) and zero moment point (ZMP) [19,20]. The 3D trajectory of the COM and Center of Pressure (COP) was generated using a convex boundedness approach [19]. In [20], a biped robot with a small contact point on each foot was developed. To control ZMP effectively, a pole-placement control method was employed. A key challenge encountered during the experiments, was the presence of noise interference at high frequencies. To overcome this drawback, the authors proposed using the third derivative of position (jerk) that helps minimize vibration.

1.3. Foot Slip Mitigation

During the single support phase, foot slip is a common challenge [21,22]. A fast turning method was developed for the biped robot while standing on one foot [21]. In this case, both the time and space required to turn the robot in the yaw orientation were minimized while mitigating foot slip. Additionally, a friction constraint approach was explored to reduce rotational slip of the biped legs during fast walking [22].

1.4. Adjustment on Rough Terrains

Gait planning on uneven surfaces presents a significant challenge in bipedal locomotion [23,24,25]. Omnidirectional walking on an inclined terrain was tested in [23], where three types of gait sequences were classified: longitudinal, transverse, and oblique walks. The double support phase (DSP), a part of a walking sequence when a bipedal robot stands on two feet, was examined to prevent acceleration jitters at the COM. In addition, the foot trajectory was generated by considering walking speed, step length, and facing direction. In [24], a foot adjustment algorithm was developed to achieve a stable walking trajectory on rough and uneven terrain. Sensors were installed at the bottom of each foot to measure ground contact points. The velocity stability and walking trajectory of a reduced-DOF robot were controlled by analyzing its motion on varying ground stiffness [25]. A spring-damping model was utilized to predict the behavior of foot contact.

Most of the humanoid robots capable of moving in 3D space are equipped with 12 joints at the lower limb structure [13,15,26,27,28]. However, the drawbacks of a 12-DOF bipedal robot include complexity, high costs, sophisticated control requirements, increased energy consumption, and significant mechanical backlash, which results in reduced accuracy at the end-effector position. To mitigate these issues, reduced-DOF robots can be considered as a solution. The following challenges associated with reduced-DOF robots have been addressed in recent studies: investigation of asymptotically-stable planar walking robot employed with Poincare’s method [4], absorption of external disturbance forces with MABEL robot and gait performance on uneven ground [5], development of a trajectory planning strategy and pulley based parallelogram mechanism [6] and time-invariant gait planning strategy [7].

On the other hand, most reduced-DOF bipedal robots, as noted in previous research [4,5,6,7], primarily exhibit planar motion due to the absence of servomotors in the yaw direction. Furthermore, the following additional challenges arise when using fewer than six DOFs per leg: customized control strategy requirements [29], reduced adaptability to diverse tasks, limited agility, performance and robustness. Notably, there is potential to design reduced-DOF robots with yaw-oriented actuators, which can rotate about the z-axis to enable spatial motion. In other words, in order to decrease the number of DOFs and still be able to perform a 3D gait, at least two joints per leg must be oriented differently.

Moreover, most state-of-the-art research works exhibit point feet [30,31,32], spring-based feet [7,33,34], and rigid flat feet [6,17] structures. This raises the scientific question of whether to design and test a bipedal robot featuring an optimized elastic ankle structure that functions as a damper to maintain posture stability.

Despite having a limited number of motors in the whole structure, the proposed reduced-DOF bipedal robot with two DOFs per leg can effectively perform locomotion. The joints of the proposed mechanism are mounted in the roll and yaw orientations, which can rotate about the x and z axes, respectively. This design provides several advantages, including operational simplicity, compact size, and reduced backlash at each foot. Conversely, equipping each leg with six DOFs would increase both programming and electrical cost and result in a larger robot. Additionally, the numerous servo motors and their connections could introduce significant mechanical play at the feet, potentially compromising foot stability. In other words, a biped robot with fewer actuators is less likely to experience walking failures and is quicker to assemble, reducing both electrical and financial costs.

This paper also investigates the behavior of the elastic ankle’s inclination in the frontal plane. Although ankle bending in the sagittal plane was examined in [35], this study focuses on maintaining the robot’s balance and stability in the frontal plane. It aims to prevent falls by optimizing the geometry of the TPU-based ankle structure, especially during single support phases or lateral movements. The risk of balance failure is greater in the coronal plane than in the sagittal plane due to the narrower foot width compared to its length. Therefore, this research characterizes the kinematic and dynamic properties of TPU-based ankles and their effects on a biped robot with a five-degree-of-freedom (DOF) structure, including its pendulum actuator. This is achieved by comparing different ankle shapes through two distinct experiments:

The ankle inclination angle is measured by applying external forces. Predictions from Finite Element Analysis (FEA) and real-world models are compared.
The stability of the robot’s central link and the roll motor of the lifting hip during the single support phase is assessed.

In the final stage, the biped robot operates using an optimized ankle structure derived from the previous experiments, as described in the third experiment. Here, the joint parameters of the robot are analyzed based on measurements obtained from both physical prototypes and virtual models in CoppeliaSim 4.3.1. In particular, the roll- and yaw-joint-based bipedal robot executes the following three different actions by considering a pseudo-static mode: lifting a swinging leg by 45 degrees at SSP configuration, forward movement, and lateral movement.

The remainder of this paper is organized as follows: Section 2 discusses the electromechanical design of the robot along with its kinematic and dynamic calculations. Section 3 focuses on the experimental results, while Section 4 interprets the key findings from these experiments and their corresponding outcomes. The study’s results will be compared with those from other leading research. Finally, Section 5 concludes the research and outlines future work to be undertaken.

2. Robot Design and Computations

When the number of joints per leg is fewer than six, the robot is unable to perform human-like movements. In this situation, the ability to control the orientation of the end-effector frame may be lost. In other words, as the number of revolute joints in the lower limb of the bipedal robot decreases, its motion becomes more limited. However, spatial movement can still be achieved by utilizing two revolute joints per leg, which helps reduce energy consumption, operational costs, and maintenance expenses compared to a traditional humanoid robot structure. In this scenario, an additional pendulum balancer should be mounted on the upper limb part to assist in lifting the swinging leg.

There are nine possible joint configurations for creating a reduced-DOF bipedal robot, with each leg having two of the following oriented actuators: RR, PP, YY, RP, PR, RY, YR, PY, and YP. In this case, R, P, and Y represent roll, pitch, and yaw-oriented joints, which can rotate about x, y, and z directions, respectively.

As demonstrated in Table 1, the proposed bipedal robot, referred to as the RRYY structure in the last row, is named for its roll and yaw actuators on each leg. Each reduced-DOF bipedal robot has its advantages and disadvantages depending on the posture equilibrium and stability control, foot sliding issues, complexity of the upper limb structure, and walking speed. If each lower limb has two actuators with the same orientations, the robot can perform a planar motion (RRRR, PPPP, and YYYY prototypes in Table 1). However, by employing two differently oriented servomotors in each leg combined with an upper limb pendulum balancer, the bipedal robot can achieve 3D gait (PPYY, YYPP, RRPP, PPRR, YYRR, and RRYY architectures).

Interestingly, having pitch joints in the lower limb requires the upper limb to have at least a two-DOF structure. This is because the bipedal robot must maintain weight balance in the yz (frontal) and xz (sagittal) planes while moving forward. In such a setup, it becomes more complex to control the upper limb with both roll and pitch joints simultaneously, as it requires posture stabilization at each step. However, this configuration allows the robot to achieve faster forward walking speed compared to designs without pitch actuators.

Another significant drawback is foot sliding, as the foot length is approximately 25 cm and thus three times greater than its width (see Figure 1). When the bipedal robot moves forward using the pitch and yaw joints, the legs tend to slide on the ground. This becomes impractical when a roll joint is absent in the lower limb structure.

Conversely, reducing the foot width helps minimize the sliding issue during lateral movements. But the risk of weight equilibrium failure becomes higher in the y direction. In this case, it is important to have an upper limb pendulum actuator that controls the robot’s equilibrium in the roll direction.

Another key criterion in terms of the bipedal robot design is to consider the locations of the joints in order to obtain an optimized 3D gait. The RRPP prototype outperforms the PPRR structure while maintaining a weight balance equilibrium in a single-support stance. Positioning the pitch actuator at the bottom of the leg makes it easier to rotate, as it has minimal impact on the robot’s center of mass (COM), unlike the PPRR prototype.

Similarly, gait execution with a PPYY configuration tends to be riskier compared to the YYPP structure due to the generation of an unstable posture when the pitch joint is activated at the robot’s hip section.

Although the proposed robot with an RRYY structure walks more slowly, it has advantages in terms of simplicity to control the posture stability in the single-support-phase (SSP) configuration, minimize foot sliding problems, and simplify the upper limb structure (see Table 1).

In Figure 1a, the CAD design of the proposed reduced-DOF bipedal robot is illustrated with an overall height of about 112 cm, length 25 cm, and width 28 cm. The width of each foot is 8.5 cm. With a longer foot length, the robot’s posture is more stable in the xz (sagittal) plane compared to the yz (frontal) plane. The overall weight of the robot is about 8 kg without the additional weight attached to the upper limb part. Each actuator was assembled, tested, and calibrated in-house, with a weight of approximately 0.8 kg. The actuation system incorporates a three-phase BLDC motor (Model: Gartt) with 200 W nominal mechanical power, a gearbox, and a magnetic encoder [36]. The load on each roll motor is less than 5 kg when lifting the opposite leg along with the central link. The details of the electrical system design are reported in Appendix A.

Overall, the proposed bipedal robot features five-degree-of-freedom (DOF) motion, which includes two differently oriented servo motors per leg and an additional servo motor mounted on the upper limb part as an inverted pendulum balancer (see Figure 1).

The yaw motor is located at the knee part and the roll motor at the hip part in order to make it easier to lift the whole body with an activated roll motor. Otherwise, if the roll actuator is located at the knee part and the yaw servomotor is at the hip part (see YYRR configuration in Table 1), this can increase the risk of the robot falling due to the additional inertia generated by the roll servomotor. In this structure, the yaw motor turns, which assists in performing a walking sequence combined with the roll joint.

The RRYY robot shown in Figure 2a is prepared to execute a half gait sequence, where an upper limb pendulum actuator rotates by an angle

θ_{6}

(Figure 2b). The upper limb pendulum balancer assists both hip roll motors during the support phase to lift the swinging legs. The reason for implementing both pendulum and hip motors in the roll direction is to stabilize the robot’s posture in the frontal plane while performing a gait.

Subsequently, the robot stands at an SSP configuration by lifting its left leg using the right hip roll actuator at an angle of

θ_{2}

(Figure 2c). The yaw motor of the right leg turns the robot at an angle of

θ_{1}

to step forward (Figure 2d). In this case, the left swinging leg’s yaw actuator rotates at an angle of

- θ_{4}

against the position of the right leg yaw joint

θ_{1}

to face the left foot straight ahead. In Figure 2e,f, both the right hip roll and the upper limb pendulum actuator return to their home positions with angles of

- θ_{2}

and

- θ_{6}

, respectively. In the closeup view (Figure 2g), the left foot of the bipedal robot has moved forward by a distance of Δx compared to its initial position (see Figure 2a).

2.1. Kinematic Architecture

In the graphical design of the robot (see Figure 1c,d), each servomotor has its own frame combined with the feet and pendulum’s end-effector frames. Out of eight frames, frames 0 and 5 refer to the feet, followed by right and left leg servo motor frames 1, 2, 3 and 4. The pendulum’s actuator is controlled by frame 6, and the load applied to it is represented by frame 7. Frames 0, 5, and 7 remain fixed in their configurations due to the absence of servomotors, while frames 1, 2, 3, 4, and 6 have variable positions.

The proposed biped robot’s motion is hybrid and can be defined with the Grubler–Kutzbach Criterion [37] at the single support phase (SSP) and double support phase (DSP) configurations of the legs in Equation (1): F—degree of freedom (DOF) of the mechanism; λ—DOF of the space; l—number of links; n—number of joints; f_i—number of each joint’s degree of freedom.

F = λ (l - n - 1) + \sum_{i = 1}^{5} f_{i}

(1)

The parameters in Equation (1) are substituted with λ = 6, l = 6, n = 5 and f_i = 1 for SSP. Therefore, the biped robot has F = 5 DOFs by assuming it has a spatial movement.

But during the double support phase (DSP) configuration, it is assumed that both feet have contact with the ground without sliding. In this case, λ = 6, l = 5, n = 5 and f_i = 1. The robot has F = 1 DOF movement at a DSP configuration.

2.1.1. Forward Kinematics

The forward kinematics are calculated by starting from the right foot frame 0 and considering it as the initial frame.

Generally, two types of forward kinematics chains are solved. The first one refers to the frames 0, 1, 2, 3, 4 and 5 (see Figure 1c and Figure 3a), while the second kinematic chain includes frames 0, 1, 2, 6 and 7 (refer to Figure 1c and Figure 3b). Both chains intersect at the frames 0, 1 and 2 (refer to Figure 1c and Figure 3c), which correspond to the matrices

{}^{0}A_{1}

and

{}^{0}A_{2}

. This indicates that the term

{}^{0}A_{2}

is taken into account while solving the forward kinematics for both chains.

The forward kinematics computation for the first chain is expressed in matrix form in Equations (2) and (3):

{}^{0}A_{1} = [\begin{matrix} c θ_{1} & - s θ_{1} & 0 & 0 \\ s θ_{1} & c θ_{1} & 0 & 0 \\ 0 & 0 & 1 & l_{1} \\ 0 & 0 & 0 & 1 \end{matrix}], {}^{1}A_{2} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & c θ_{2} & - s θ_{2} & 0 \\ 0 & s θ_{2} & c θ_{2} & l_{2} \\ 0 & 0 & 0 & 1 \end{matrix}],

(2)

{}^{2}A_{3} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & c θ_{3} & - s θ_{3} & l_{3} \\ 0 & s θ_{3} & c θ_{3} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], {}^{3}A_{4} = [\begin{matrix} c θ_{4} & - s θ_{4} & 0 & 0 \\ s θ_{4} & c θ_{4} & 0 & 0 \\ 0 & 0 & 1 & - l_{4} \\ 0 & 0 & 0 & 1 \end{matrix}], {}^{4}A_{5} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - l_{5} \\ 0 & 0 & 0 & 1 \end{matrix}] .

(3)

The terms

c θ_{1}

and

s θ_{2}

denote

cos (θ_{1})

and

sin (θ_{2})

, respectively. We obtain the compound homogeneous transformation matrix

{}^{0}A_{5}

by pre-multiplying

{}^{2}A_{5}

{}^{0}A_{2}

as demonstrated in Equation (4), where the elements of the matrix

{}^{0}A_{5}

are specified in the Table A1 of Appendix B:

{}^{0}A_{5} = {}^{0}A_{2} \cdot {}^{2}A_{5} = {}^{0}A_{1} \cdot {}^{1}A_{2} \cdot {}^{2}A_{3} \cdot {}^{3}A_{4} \cdot {}^{4}A_{5} = [\begin{matrix} a 11 & a 12 & a 13 & P_{a x} \\ a 21 & a 22 & a 23 & P_{a y} \\ a 31 & a 32 & a 33 & P_{a z} \\ 0 & 0 & 0 & 1 \end{matrix}] .

(4)

The first three columns of the matrix

{}^{0}A_{5}

refer to the orientation, while the last column with

P_{a x}

P_{a y}

and

P_{a z}

indicates the positions of that matrix in the x, y and z directions, respectively.

The second kinematic chain corresponds to the pendulum section, defined by frames 0, 1, 2, 6, and 7 (see Equations (2) and (5)):

{}^{2}A_{6} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & c θ_{6} & - s θ_{6} & \frac{l_{3}}{2} \\ 0 & s θ_{6} & c θ_{6} & l_{6} \\ 0 & 0 & 0 & 1 \end{matrix}], {}^{6}A_{7} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & l_{7} \\ 0 & 0 & 0 & 1 \end{matrix}] .

(5)

The transformation matrix

{}^{0}A_{7}

can be written by pre-multiplying

{}^{2}A_{7}

{}^{0}A_{2}

as in Equation (6):

{}^{0}A_{7} = {}^{0}A_{2} \cdot {}^{2}A_{7} = {}^{0}A_{1} \cdot {}^{1}A_{2} \cdot {}^{2}A_{6} \cdot {}^{6}A_{7} = [\begin{matrix} b 11 & b 12 & b 13 & P_{b x} \\ b 21 & b 22 & b 23 & P_{b y} \\ b 31 & b 32 & b 33 & P_{b z} \\ 0 & 0 & 0 & 1 \end{matrix}],

(6)

where the elements of the matrix

{}^{0}A_{7}

in Equation (6) are described in the Table A2 of Appendix C.

2.1.2. Inverse Kinematics

To solve the inverse kinematics, the end-effector frame location and configuration are given, and the goal is to find each joint position. Similarly to the forward kinematics problem, demonstrated in Figure 3, both chains (swinging left leg and upper limb part) are considered in the inverse kinematic computation. Appendix B provides the details of the inverse kinematics solutions specific to the swinging left leg, with the final joint position results provided in Equations (7) and (8):

θ_{1} = - arctan (\frac{P_{a x}}{P_{a y}}) \pm π, θ_{2} = \pm a r c c o s (\frac{f_{a 1}^{2} + f_{a 2}^{2} + l_{3}^{2} - {(l_{4} + l_{5})}^{2}}{2 l_{3} \sqrt{f_{a 1}^{2} + f_{a 2}^{2}}}) + arctan (\frac{f_{a 2}}{f_{a 1}}),

(7)

f_{a 1} = \pm \sqrt{P_{a x}^{2} + P_{a y}^{2}}, f_{a 2} = - P_{a z} + l_{1} + l_{2}, θ_{3} = \pm a r c c o s (\frac{- f_{a 2} - l_{3} s θ_{2}}{l_{4} + l_{5}}) - θ_{2} .

(8)

Defining the yaw joint position

θ_{4}

of the swinging leg is optional, as it does not influence the position of the end-effector (frame 5 in Figure 1c).

In a similar manner, the inverse kinematics problem for the pendulum section (second kinematic chain) was solved in Appendix C, and the final results are presented in Equations (9) and (10):

θ_{1} = - arctan (\frac{P_{b x}}{P_{b y}}) \pm π, θ_{6} = \pm a r c c o s (\frac{- \frac{l_{3}}{2} s θ_{2} - l_{6} c θ_{2} - f_{b 2}}{l_{7}}) - θ_{2},

(9)

θ_{2} = \pm a r c c o s (\frac{l_{7}^{2} - f_{b 1}^{2} - \frac{l_{3}^{4}}{4} - l_{6}^{2} - f_{b 2}^{2}}{\sqrt{{(2 l_{6} f_{b 1} + l_{3} f_{b 2})}^{2} + {(l_{3} f_{b 1} - 2 l_{6} f_{b 2})}^{2}}}) \pm arctan (\frac{2 l_{6} f_{b 1} + l_{3} f_{b 2}}{l_{3} f_{b 1} - 2 l_{6} f_{b 2}}) .

(10)

The terms

f_{b 1}

and

f_{b 2}

are employed to simplify Equations (9) and (10), where

f_{b 1} = \pm \sqrt{P_{b x}^{2} + P_{b y}^{2}}

and

f_{b 2} = - P_{b z} + l_{1} + l_{2}

2.2. Static Architecture

The sketch of the proposed RRYY biped robot with a pendulum attachment is illustrated from the frontal, sagittal, and transverse plane viewpoints (see Figure 4). In this case, the robot moves forward by lightly touching the ground with its right leg, while all joints rotate at a low velocity. In static mode, accelerations and velocities are assumed to be negligible.

The locations of all frames are the same as those described in the kinematic architecture section, with the initial frame starting from the foot of the right leg. The same principle is applied when the left foot touches the ground as the right leg swings, with the initial frame located at the left foot. The total torque applied on the second frame joint is computed by summing the torques obtained from both frontal (Figure 4a) and sagittal planes (Figure 4b). In this scenario, Figure 4c is employed to determine the x and y coordinate locations of frames 3, 4, 5, 6, and 7, when the right leg’s yaw joint position

θ_{1}

and frame are available.

To maintain a foot in an elevated position, the amount of required torque at the lifting motor must be calculated first. This refers to the inverse statics problem. On the other hand, the pendulum actuator must apply the same amount of torque to the lifting motor, but in the opposite direction to maintain momentum equilibrium. Afterwards, the pendulum actuator position can be calculated depending on the counter torque that is required to apply to the lifting hip roll actuator. We can estimate the forward statics computations from the last notation.

2.2.1. Conditions for Equilibrium

Theoretically, to maintain the robot’s equilibrium while standing on one leg, the following three factors must be considered: (1) the weight attached to the pendulum, (2) the length of the pendulum link, and (3) the position of the pendulum actuator. These three variables can differ depending on the locations of frames 3, 4, 5 and 6 (see Figure 4), where the central pivot is at frame 2. In other words, these variables can be explained by representing the traditional weight balancer with a mass, link lengths and angles applied on both sides of the pivot (refer to Equation (11) and Figure 5). The objective is to achieve balanced momentum, with O representing the central pivot positioned on the horizontal dashed line:

F_{1} L_{1} cos θ_{1} = F_{2} L_{2} cos θ_{2} .

(11)

2.2.2. Conditions for Safety

All servomotor rotation angles and velocities are limited to prevent them from colliding and damaging the environment. Similarly, the pendulum’s link length is equal to 0.5 m with a maximum rotation angle of ±30 degrees to make it safe. In order to regulate the torque, we decided to adjust the weight applied to the pendulum link. The maximum weight attached to the top of the pendulum is about 4 kg, which is enough to maintain the weight equilibrium for the robot. The pendulum joint’s gearbox with a gear ratio of 1:100 is sufficient to work at the required torque in order to keep the robot’s weight equilibrium without overloading the motor.

2.2.3. Inverse Statics

According to the sketch presented in Figure 4, the swinging leg combined with a central link applies torque to the lifting roll actuator at the hip joint, which is located in frame 2 as a central point, and it is expressed in Equation (12):

τ = τ_{3} + τ_{4} + τ_{5} + τ_{6} = \sum_{i = 3}^{6} | P_{i} | | F_{i} | sin θ_{i} .

(12)

Alternatively, Equation (12) can be arranged in the vectorized form as shown in Equation (13):

τ = [\begin{matrix} x_{3} & y_{3} \end{matrix}] [\begin{matrix} F_{3 x} \\ F_{3 y} \end{matrix}] + [\begin{matrix} x_{4} & y_{4} \end{matrix}] [\begin{matrix} F_{4 x} \\ F_{4 y} \end{matrix}] + [\begin{matrix} x_{5} & y_{5} \end{matrix}] [\begin{matrix} F_{5 x} \\ F_{5 y} \end{matrix}] + [\begin{matrix} x_{6} & y_{6} \end{matrix}] [\begin{matrix} F_{6 x} \\ F_{6 y} \end{matrix}] .

(13)

When the yaw motor of the support leg during the single support phase is activated (see Figure 4), each x parameter in the sagittal plane can be converted to a y parameter in the frontal plane, except frames 0, 1 and 2 (see Equation (14)):

x_{i} = y_{i} tan (θ_{1}), 2 < i < 8 .

(14)

Similarly, the forces

F_{i x}

and

F_{i y}

applied orthogonally in the x and y directions of the relative joint, are derived from the force magnitude

F_{i}

and yaw-oriented motor position

θ 1

(refer to Equations (15) and (16)):

F_{i x} = F_{i} s i n (θ_{1}) a n d F_{i y} = F_{i} c o s (θ_{1}), 2 < i < 8,

(15)

F_{i} = m_{i} g = \sqrt{F_{i x}^{2} + F_{i y}^{2}} = \sqrt{{(F_{i} s i n (θ_{1}))}^{2} + {(F_{i} c o s (θ_{1}))}^{2}}, 2 < i < 8 .

(16)

The axial locations of frame 3 in Equation (17) are demonstrated by taking Equation (14) into account:

y_{3} = L_{3} cos (θ_{2}) cos (θ_{1}) a n d x_{3} = L_{3} cos (θ_{2}) sin (θ_{1}) .

(17)

Subsequently, the remaining matrix elements along the y-axis, derived from Equation (13) and Figure 4, are organized as shown in Equations (18)–(20). The x-axis positions are determined using the conversion outlined in Equation (14):

y_{4} = y_{3} + d y_{4} = L_{3} cos (θ_{2}) cos (θ_{1}) + L_{4} sin (θ 2 + θ 3) cos (θ_{1}),

(18)

y_{5} = y_{3} + d y_{4} + d y_{5} = L_{3} cos (θ_{2}) cos (θ_{1}) + (L_{4} + L_{5}) sin (θ 2 + θ 3) cos (θ_{1}),

(19)

y_{6} = \frac{y_{3}}{2} - d y_{6} = \frac{L_{3} cos (θ_{2}) cos (θ_{1})}{2} - L_{6} sin (θ 2) cos (θ_{1}) .

(20)

2.2.4. Forward Statics

In the previous section, the torque applied to the right hip roll motor was computed. The task here is to determine the position of the pendulum servomotor that applies a counter-torque to achieve system equilibrium (see Figure 4 and Equation (21)):

τ = τ_{7} = - [\begin{matrix} x_{7} & y_{7} \end{matrix}] [\begin{matrix} F_{7 x} \\ F_{7 y} \end{matrix}], a n d [\begin{matrix} x_{7} & y_{7} \end{matrix}] = - τ {[\begin{matrix} F_{7 x} \\ F_{7 y} \end{matrix}]}^{- 1} .

(21)

The negative sign denotes that the coordinates of frame 7 are located at the left side of the reference frame 0. Subsequently, the matrix elements from Equation (21) are defined in Equations (14), (15) and (22):

y_{7} = \frac{y_{3}}{2} - d y_{6} - d y_{7} = (\frac{L_{3} cos (θ_{2})}{2} - L_{6} sin (θ_{2}) - L_{7} sin (θ_{2} + θ_{6})) cos (θ_{1}) .

(22)

Equation (21) can be arranged in the form demonstrated in Equation (23):

τ = - x_{7} F_{7 x} - y_{7} F_{7 y} = - y_{7} F_{7} (tan (θ_{1}) sin (θ_{1}) + cos (θ_{1})) .

(23)

Finally, by combining both Equations (22) and (23), we can obtain the pendulum actuator position

θ_{6}

that must keep a pose equilibrium at SSP (see Equation (24)):

θ_{6} = - θ_{2} + arcsin (\frac{L_{3} cos (θ_{2})}{2 L_{7}} - \frac{L_{6} sin (θ_{2})}{L_{7}} + \frac{τ}{F_{7} L_{7}}) .

(24)

3. Experimental Results

Before performing the gait sequence, the robot is tested at a single support phase. One of the major problems is foot inclination at the frontal plane due to the flexibility of the ankle (see Figure 6a). The goal is to design an optimized elastic and damping ankle mechanism that enhances posture stability and prevents the robot from falling during the stance phase. Additionally, the damping property of the ankle should attenuate the foot impact and reduce vibration in the foot structure. During the ankle inversion problem, the roll motor cannot lift the other leg in a vertical direction properly. Despite this, the robot shifts horizontally in a frontal plane, which results in failure to obtain a single support phase. The ankle inversion problem is minimized when the position of the pendulum motor rises.

In all experiments in the single support phase (SSP) configuration, the left leg was lifted successfully by applying the pendulum joint position obtained from Equation (24).

On the other hand, ankle eversion may cause the risk of the robot falling down on the ground due to increased momentum applied by the pendulum. In a word, the ankle inclination results in loss of the foot configuration, which restricts it from standing on one leg. Otherwise, the robot must remain upright while lifting the opposite leg along with the central link, as shown in Figure 6b. Because the foot’s width is smaller than its length, foot inclination is more critical in the frontal plane than in the sagittal plane (see Figure 6c). Therefore, we focus on the frontal plane more than the sagittal plane view in this paper.

The foot of the biped robot is split into three parts, which were designed and 3D printed: (1) front part; (2) ankle; and (3) back part. Both the front and back parts of the foot are made of PLA, while the ankle was printed with TPU material.

Alternatively to TPU material, Table 2 presents the chemical and physical properties of seven different materials. The most durable, chemically resistant, and tear-resistant options include carbon fiber with polyethylene terephthalate glycol (CF-PETg), nylon, and thermoplastic polyurethane (TPU). However, with its lower elastic modulus and melting temperature, TPU outperforms both CF-PETg and nylon materials in terms of shock and vibration absorption capabilities.

As a result, the underactuated RRYY bipedal robot features an ankle made from TPU material, functioning as a damper to help maintain posture stability.

3.1. Testing the Elastic Ankle Inclination

Figure 6d,e illustrates the parameters specific to the ankle design, including the ankle height h, half-width thickness a, and half-length thickness b. The objective was to determine the optimal dimensions for the TPU ankle, namely, a, b, and h, to minimize the inclination angle. Another question was whether doubling each of these parameters could further reduce the inclination angle. Prototypes of the TPU ankle were 3D-printed with various dimensions, all using the same infill density (see Figure 7).

A 500 mm aluminum extrusion profile was tightly installed inside the elastic ankle with full infill density, and an external force was applied at the edge of the aluminum profile by bending the ankle in the width direction. The bending angle

α

of the elastic ankle was measured at variable external forces F. Five ankle prototypes with different dimensions were compared. The amount of applied pulling external force is measured with a force sensor that is attached to the aluminum extrusion profile through the cable line (refer to Figure 8). Meanwhile, the IMU sensor measures the bending angle of the TPU ankle. In addition, the method of Finite Element Analysis (FEA) was used to test the inclination angle of the elastic ankle with Young’s modulus

σ

= 90 MPa.

In Figure 8, a sample ankle prototype with a dimension of 10 mm × 20 mm × 20 mm was tested. The graphs in Figure 9 indicate the results obtained from the ankle inclination test implemented in Figure 8. Among the five prototypes with different dimensions, the smallest ankle model, measuring 10 × 10 × 20 mm, exhibited the highest inclination angle due to its limited resistance to external forces. In the FEA analysis, it reached an inclination of 4.24 degrees, while in real-world testing, it reached 6.4 degrees. On the other hand, due to the higher physical resistance compared to other and smaller ankle prototypes, the biggest ankle structure with a 20 × 20 × 40 mm size exhibits the lowest inclination angle, about 2.25 deg in the FEA and 2.3 deg in the real prototype.

It is possible to observe that the ankle inclination angle rises linearly with respect to an applied external tension using FEA (Figure 9a), while this prediction is estimated with a nonlinear graph for each ankle prototype in the real model (Figure 9b).

According to the graphs in Figure 9 and the a, b and h parameters presented in Figure 6, five different ankle prototypes were compared. The smallest prototype has dimensions of 10 × 10 × 20 mm. The other prototypes include variations with doubled half-width thickness a (20 × 10 × 20 mm), doubled half-length thickness b (10 × 20 × 20 mm) and doubled height h (10 × 10 × 40 mm). The doubled half-width tends to have the highest resistance against the external force, when the tension is applied along the width direction. The doubled height is subjected to the lower resistance followed by the doubled half-length with external force applied along the width direction. Finally, Table 3 illustrates the experimental data obtained from the realistic setup vs. the simulation method at a constant modulus elasticity

σ

= 90 MPa, applied tension force F = 50 N and link length L = 500 mm. In columns 1 and 2 of Table 3, five different TPU ankle prototypes with variable geometries from Figure 9 are characterized, where each ankle prototype refers to each experimental trial. Table 3 demonstrates the full results of the experiment including three additional ankle prototypes (4, 6 and 7).

The displacement dy can be converted to the inclination angle

α

and vice-versa depending on each trial number i with Equation (25):

α_{i} = arcsin (\frac{d y_{i}}{l}) a n d {d y}_{i} = l \dot{sin (α_{i})},

(25)

where l is the length of the link in mm.

The minimized inclination angle

Z_{i}

% is defined for both the real and FEA method results by considering the highest inclination angle

α_{m a x}

and inclination angle

α_{i}

of the ankle being tested (see Equation (26)):

Z_{i} % (F E A) = \frac{α_{m a x} (F E A) - α_{i} (F E A)}{α_{m a x} (F E A)} \cdot 100 % a n d Z_{i} % (R e a l) = \frac{α_{m a x} (R e a l) - α_{i} (R e a l)}{α_{m a x} (R e a l)} \cdot 100 % .

(26)

In this case,

α_{m a x}

= 4.24 deg for the FEA and

α_{m a x}

= 6.39 deg for the real experiment. These angles refer to the worst-case scenario with the smallest ankle prototype 1 (10 × 10 × 20 mm size), which are taken as the reference maximum inclination angles (highlighted in red in Table 3). Conversely, the largest sample, measuring 20 × 20 × 40 mm, demonstrates the most reduced inclination angle Zi%, achieving approximately 47% in FEA and 64% in the real experiment. This prototype, designated as prototype 8, is highlighted in green in Table 3 as the optimized structure. In other words, this biggest size and optimized ankle structure in our case absorbs the inclination angle 2–3 times higher than the smallest size prototype, and it helps stabilize the robot on both the frontal and sagittal planes due to its square shape. The average minimized inclination angle Zi% is computed in Equation (27) by considering the minimized inclination angles Zi% obtained from both FEA and experimental results:

\bar{Z_{i} %} = \frac{Z_{i} % (F E A) + Z_{i} % (R e a l)}{2} .

(27)

The diagrams in Figure 10 represent the outcomes of the FEA and real model tests; their average results are obtained from Equations (26) and (27) and Table 3. The smallest ankle prototype is located at the edge, while the biggest ankle prototype is at the center of each diagram. The tested ankle prototypes with doubled parameters a, b and h separately are highlighted in blue, red and green. But the dimensions a, b and h are doubled together depending on the mixture of the circles.

In most cases, the external bending force is absorbed by increasing the half-width thickness a (prototype 2, 4, 6, 8 in Table 3 and Figure 10) due to increased wall disturbance thickness against the direction of the bending force. Interestingly, increased height h and half-length thickness b together absorb the bending moment (prototype 7) more than growing only the height h (prototype 5) or the half-length thickness b (prototype 3). As the ankle dimensions increase, the results obtained from the FEA method and real experiment become closer to each other. In this case, we can approximate the precise predictions of the ankle inclination angle. Otherwise, at smaller geometries, the realistic model and simulation method exhibit big mismatches in the results, while applying the external bending forces.

3.2. Single Support Phase Stability Control

The roll motor of the right leg, together with the pendulum motor, was activated to lift the left leg and the central link, placing the robot in a single support phase.To test stability and assess the kinematic and dynamic behaviors, five different TPU ankle structures were compared by analyzing the position measurements from the magnetic encoder on the right leg’s roll motor and an IMU sensor mounted on the central link (see Figure 1 and Figure 6).

In the block diagram shown in Figure 11, the input is represented by the IMU reference angle (

ϕ_{r e f}

= 7.8 deg). The difference between the reference and the measured roll orientation is then sent to the roll motor of the support leg at each 0.01 s sample interval, over a 14 s period. Two outputs are measured: (1) the position of the activated servo motor and (2) the roll angle from the IMU sensor. As the servo motor is already equipped with a PID controller, a simple proportional (P) control action is sufficient for adjusting orientation based on the IMU measurement.

Initially, during the DSP, the pendulum motor rotated to 25 degrees towards the right leg within 1.8 s (see Algorithm 1). To configure the robot for SSP, an optimal angle, calculated using Equation (24), was applied to lift the left leg while minimizing bending issues in the TPU ankle. The terms

t_{m i n}

t_{s u p p o r t}

and

t_{m a x}

in the Algorithm 1 indicate 0.01 s, 1.8 s and 14 s, respectively. When the pendulum actuator reaches its set point, a slight backlash of ±0.4 degrees may occur in the system, as detected by the IMU sensor over a period of 1.8 s (see Figure 12a).

Algorithm 1: Testing the robot’s equilibrium stability at a single support phase.

Input: Input IMU roll reference position,

ϕ_{r e f}

(deg)

Output: Roll motor positions,

θ_{2}

[t] (deg), IMU roll actual positions,

ϕ

[t] (deg)

Next, the task was to rotate the roll motor of the right leg until the IMU sensor’s angle reached the setpoint of 7.8 degrees in the frontal plane. Before the IMU position reached the setpoint at the 4th second, the right leg’s roll motor position peaked, indicating a system overshoot (Figure 12b), which was necessary to compensate for the ankle’s angle of inversion (Figure 6a). After the 4th second, the ankle returned to a straight position (Figure 6b) due to the momentum balance between the pendulum and left leg acting on the right hip roll motor, though the IMU sensor’s roll angle began to fluctuate (Figure 12a). To stabilize the system, the roll motor reduced its peak value and maintained the IMU roll angle at

ϕ_{r e f}

= 7.8 deg. Smaller ankle structures tended to show significant ankle inversion and system overshoot (Figure 12), risking balance failure between the pendulum motor, left leg, and right hip’s roll actuator. The smallest ankle, measuring 10 mm × 10 mm × 20 mm, exhibited the highest IMU roll fluctuation with a maximum tolerance of 4 degrees, followed by ankles with dimensions of 10 mm × 10 mm × 40 mm, 10 mm × 20 mm × 20 mm, and 20 mm × 10 mm × 20 mm, with maximum tolerances of 3.4, 2.8, and 1.6 degrees, respectively (see Figure 12a).

In contrast, the larger ankle (20 mm × 20 mm × 40 mm) helped minimize vibration amplitudes measured by the IMU sensor, showing only a maximum of 0.8 degrees of tolerance, contributing to the stabilization of the right hip roll motor’s position.

3.3. Single Support Phase

At the SSP configuration, an additional experiment was executed to test the biped robot lifting the swinging leg by 45 degrees. In the first and second screenshots of Figure 13, the RRYY bipedal robot’s pendulum position changes from the home position to 27 degrees while standing in the DSP configuration. Both legs are aligned on the ground. The subsequent screenshots depict the beginning of the SSP phase, with each screenshot representing an action that lasts over 1.5 s, involving a five-degree rotation (

Δ

θ_{2}

= 5 deg) of the right hip roll actuator.

In the experiment, the right hip roll joint reaches its maximum lift angle of 45 degrees at the 15 s mark. (refer to Figure 13 and Figure 14). After 15 s, the swinging leg approaches the ground as the right hip roll motor rotates backward. During the whole phase, no other joints rotate except for the roll actuator of the support leg and the pendulum actuator.

This movement sequence was also tested on the real robot prototype (see Figure 14 and Figure 15). Both legs were aligned with the white pieces of tape that were stuck on the ground. The velocity graph for the real model (Figure 14d) is rougher than that from the virtual environmental model (Figure 14b), due to mechanical backlash in the gearbox. Notably, both the CoppeliaSim model and the real prototype exhibited symmetrical behavior between the right hip roll and pendulum actuators, which worked together to maintain the robot’s posture equilibrium over time. Even though the applied torque from the swinging leg to the support leg increased at a greater angle of the right hip roll actuator (

θ_{2}

= 45 deg), the pendulum actuator’s position decreased from 27 deg to −30 deg instead of increasing. This occurred because the pendulum actuator’s location shifted to the left as the right hip roll actuator elevated the robot’s entire body (refer to Figure 13 and Figure 15).

3.4. Performance of a Lateral Movement

In this experiment, the robot moved to the left in a sequence comprising two cycles over 18 s, with only the pendulum actuator and the left leg roll servomotor operated.

Initially, the pendulum actuator rotated counterclockwise to 25 degrees over 1.5 s, pushing on the right leg, while the left leg slid by 7 degrees, moving the left foot closer to the right leg over the next 1.5 s (see Figure 16 and Figure 17). Following this, the pendulum actuator rotated clockwise to −25 degrees by the 6th second, aiding the left hip roll motor in turning back to the home position and pressing the left leg onto the ground. This completed the first cycle of lateral movement, with the robot’s central link shifting to the left, while the right leg slid accordingly.

Initially, the feet were aligned with the level of the white lines, which were attached to the ground (see Figure 18). Within two cycles, the robot moved about 10 cm in the left direction with the left leg displacing from the right to the central white line. As shown in Figure 17a,c, the pendulum joint position reached a maximum angle of 25 degrees as the robot’s center of mass (COM) aligned with its center during the double support phase (DSP). However, the pendulum actuator’s minimum angle was about −9 degrees, as the COM shifted to the left due to the left leg tilting 7 degrees to the central part of the robot. If the pendulum rotated to a higher angle in the opposite direction of the COM, the robot could risk falling.

3.5. Performance of a Pseudo-Static Gait Sequence

The RRYY bipedal robot was tested to perform a walking sequence within two cycles with the optimized ankle structure (20 mm × 20 mm × 40 mm) obtained from the previous experiments. Generally, 20 screenshots were captured over two walking cycles starting from the home position, with a 1.5 s interval between each screenshot (refer to Figure 19 and Figure 20). Meanwhile, the robot performed different actions in each picture and moved forward.

Initially, the pendulum actuator rotated by 25 degrees in 1.5 s and the right hip’s roll motor started lifting the whole robot body by 7 degrees in the 3rd second (first row of Figure 20 and Figure 21). In the next motion, the yaw motor of each leg rotated by 12 degrees in order to step forward. At the 6 s mark, the roll motor of the right hip returned to the home position. Similarly, in the next step, the pendulum motor returned to its initial position, marking the beginning of the second half of the first cycle.

The bipedal robot, modeled with identical geometry, mass, and positions as the physical prototype, was tested in CoppeliaSim (see Figure 20). The robot’s kinematic behavior closely resembles that of the real model. However, the graph appears smoother due to the absence of frictional effects between the motors, links, and the ground (refer to Figure 19b,d).

In the real prototype, the robot completed two walking cycles in 30 s, moving forward 15 cm (Figure 21). At this time period, the robot displaced from the first to the second white line.

The kinematic graphs in Figure 19 illustrate each joint’s position and velocity. Figure 19d shows some vibrations due to friction and slight backlash in the actuator gearboxes.

The joint positions in both the real and virtual environmental models are closely matched, though there are slight differences in velocities. In the CoppeliaSim model, the pendulum actuator achieved the highest velocity, reaching ±16 deg/s, followed by the roll actuators of both legs at ±5 deg/s (Figure 19b). In the physical model, both pendulum and yaw actuators for the left and right legs reached the highest velocities at ±20 deg/s, with the roll actuators of both legs reaching ±6 deg/s (Figure 19d).

All joint torques were computed from Equations (13) and (21), except the joints with yaw direction where the gravitational forces do not have an effect (see Figure 22a). The joint torques in Figure 22b were measured during the experiment with Equation (28), which was developed by the team of ODrive manufacturers:

T = 8.27 \cdot I \cdot K V .

(28)

The term I represents the current measured by the motor controller (model: ODrive 3.5), and KV is the motor’s speed constant (KV = 600 rpm/V).

The torque behaviors shown in Figure 22a,b are nearly identical, with friction impacting motor speed and current in real-world measurements. Additionally, during the DSP configuration (5 to 7 s, 14 to 16 s, 20 to 22 s, and 28 to 30 s), when both legs of the robot were in contact with the ground, the motor generated extra torque due to frictional forces from the ground.

Both mechanical and electrical power of each joint are computed with Equation (29):

P_{m e c h} = \frac{T \cdot ω \cdot 2 \cdot π}{60} a n d P_{e l e c t r i c a l} = I \cdot V,

(29)

where T—torque, computed in Equation (28), Nm;

ω

—angular velocity of the actuator, rpm; V—motor’s voltage, V. As shown in Figure 22c,d, the pendulum actuator consumes the most mechanical power among other joints. However, the mechanical power decreases as the velocity drops. This indicates that while lifting the body, the motor draws significant electrical power, but the mechanical power is reduced.

4. Discussion

Various researchers have advanced the design of bipedal robots by making adjustments to joint numbers, configurations, foot structures, and other features (see Table 4).

Most robots equipped with six DOFs per leg are capable of spatial motion due to the inclusion of yaw actuators paired with actuators in the roll or pitch direction. In contrast, most reduced-DOF robots primarily exhibit motion in a single plane.

Additionally, numerous research studies have investigated modified foot structures. Some papers examine point foot designs [30,31,32], while others delve into spring-integrated feet [7,33,34], along with various other prototypes (see Table 4). However, there is still potential to develop a bipedal robot featuring an elastic ankle structure.

In our team’s recent study [35], we tested the stiffness and energy absorption characteristics of a TPU ankle by applying vertical force. In addition, the TPU ankle structure was optimized by changing only the ankle base thickness and testing how it withstands the external bending forces in sagittal plane.

In this study, our aim was to optimize the TPU ankle geometry in the 3D view (refer to Table 3) and test its ability to withstand the horizontal pulling force in the frontal plane. Moreover, the robot’s stability in 2D plane was predicted by considering the statics calculation (see Figure 12), and the flexible ankle structure was refined by comparing it at various dimensions.

In comparison to other robots with four-DOF motion outlined in Table 4, the proposed RRYY kinematic architecture attains spatial motion thanks to the addition of a yaw-oriented servomotor in each leg. According to the experimental results, the advantages of the proposed reduced-DOF bipedal robot include its ability to lift the leg to 45 degrees, execute lateral motion, and step forward with the aid of a yaw motor pair that enables turning, assisted by the pendulum actuator. Another benefit is the integrated and optimized elastic ankle pair, which acts as a damper in the system while the robot lifts its legs.

With a height of 1.12 m and a weight of 8 kg (excluding the 4 kg weight attachment on the upper limb), the proposed mechanism offers one of the best height-to-weight ratios among other robots (see Table 4), followed by the Athlete [44] and BRUCE robots [43].

Conversely, the kinematic design of the proposed bipedal robot is constrained by the absence of additional joints per leg. With one foot lifted, the forward kinematics has four DOFs, enabling one to follow the trajectory, but it is challenging to have a precise foot orientation. In order to achieve the ability to control the foot orientations, a six-DOF structure is required per leg. A robot with twelve DOFs (six DOFs in each leg) is generally not considered DOF-deficient, as six DOFs are sufficient to achieve any possible position and orientation of the feet. In this case, when both feet are in contact with the floor (assuming no sliding), the robot forms a parallel structure with a mobility F = 6 (F = 6 (12 − 12 − 1) + 12 = 6), allowing the body to stabilize by moving in three-dimensional space (three positions + three orientations). When one foot is lifted, it is true that the robot becomes redundant with 12 DOFs. However, the six DOFs in the supporting leg are used for body stabilization, while the other six DOFs in the moving leg are used to perform the step.

Another key challenge for the RRYY bipedal robot is its slow forward movement, caused by the integration of roll and yaw-oriented motors. The legs must move laterally and rotate in a yaw direction before stepping forward. However, the current design allows the robot to maintain stable posture control in the frontal plane. There is potential to develop an RRPP structure with pitch motors, allowing for faster forward movement compared to the RRYY configuration.

5. Conclusions

In this study, we proposed a reduced-DOF bipedal robot capable of performing three-dimensional gaits. By minimizing the number of actuators, the mechanical backlash, energy consumption, and maintenance costs are reduced. This structure enables the robot to achieve a successful gait with a lightweight and compact design. In this configuration, the proposed RRYY bipedal robot can maintain a stable SSP configuration, walk sideways, and step forward.

To improve stabilization control, a new elastic ankle was designed and integrated into the robot architecture. Five ankle prototypes made from TPU material, each with varying dimensions but identical infill densities, were compared. In the first experiment, the largest ankle prototype with 20 mm × 20 mm × 40 mm dimensions demonstrated the highest resistance to external forces. This ankle prototype is sufficient to withstand up to 50 N, which approximately matches the stress exerted by the robot on the support foot during the SSP configuration. In the second experiment, this same ankle structure effectively minimized fluctuations and inclination issues, enhancing stability in the SSP configuration.

In general, as the elastic ankle structure increases in width and height, the robot is better able to minimize vibrations, which helps prevent falls. Additionally, kinematic and static computations were performed to achieve equilibrium using an assistive inverted pendulum attachment.

Looking ahead, our goal is to develop an omnidirectional walking pattern for rough terrains. To further reduce backlash and friction, roll-oriented motors will be replaced with harmonic drive actuators. Additionally, the robot is expected to operate autonomously, equipped with an onboard Jetson Nano computer and a compact motherboard integrating various chips and microcontrollers. Rechargeable batteries will be mounted at the top of the inverted pendulum, providing both the necessary balancing weight and electrical power for the robot.

Author Contributions

Conceptualization, S.Y.; methodology, S.Y.; software, T.U., S.Y. and M.F.; validation, S.Y. and M.F.; formal analysis, M.F.; investigation, S.Y., A.A., T.U. and M.F.; resources, M.F.; data curation, S.Y., A.A. and M.F.; writing—original draft preparation, S.Y.; writing—review and editing, S.Y. and M.F.; visualization, S.Y., T.U., A.A. and M.F.; supervision, S.Y. and M.F.; project administration, M.F.; funding acquisition, M.F. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by Nazarbayev University under the Faculty Development Competitive Research Grants Program through the Project “Designing an Energy-efficient Full-sized Humanoid Robot with Fast Adaptive Model based Neuromorphic Control Architecture”, under award 201223FD8812.

Data Availability Statement

The data collected from the experiments and simulations are available upon request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

TPU	Thermoplastic Polyurethane
FEA	Finite Element Analysis
DOF	Degrees of freedom
SLIP	Spring-Loaded Inverted Pendulum
LIPM	Linear Inverted Pendulum Model
LPM	Linear Pendulum Model
PDAC	Passive Dynamic Autonomous Control
RVFLNN	Random Vector Function-link Neural Network
ZMP	Zero moment point
COM	Center of mass
COP	Center of Pressure
MPC	Model Predictive Control
BLDC	Brushless direct current
CAN	Controller area network
IMU	Inertial measurement units
PLA	Polylactic acid
SSP	Single support phase
DSP	Double support phase
CPG	Central pattern generator
TPE	Thermoplastic elastomer
CF-PETg	Carbon fiber with polyethylene terephthalate glycol

Appendix A. Electrical System Design

Each leg of the robot is fitted with a dual motor controller board (Model: Odrive 3.5), which operates on 24 V DC from the power supply unit and manages the position, velocity, and torque of both the roll and yaw motors (refer to Figure 1b and Figure A1). The pendulum mechanism has a separate motor controller board dedicated to controlling the pendulum actuator’s roll direction. The control commands generated by algorithms executed on the PC are converted from USB to the CAN bus protocol by a microcontroller (ATmega 328) and then transmitted to all motor controllers via a CAN bus interface board (MCP 2515). The second USB port on the PC is used to connect the IMU sensor (Model: WitMotion), which measures the roll angle of the central link and assesses the robot’s stability. All device parameters are programmed in Python 3.10.10 and run at 100 Hz.

Figure A1. Block diagram of electrical and software connections.

Appendix B. Solving the Kinematics Problem for Chain 1 (Swinging Leg Side)

Table A1. Description of matrix

{}^{0}A_{5}

elements.

Table A1. Description of matrix

{}^{0}A_{5}

elements.

Matrix Elements	Specification
a11	$c θ_{1} c θ_{4} + s θ_{4} (s θ_{1} s θ_{2} s θ_{3} - c θ_{2} c θ_{3} s θ_{1})$
a12	$c θ_{4} (s θ_{1} s θ_{2} s θ_{3} - c θ_{2} c θ_{3} s θ_{1}) - c θ_{1} s θ_{4}$
a13	$c θ_{2} s θ_{1} s θ_{3} + s θ_{1} s θ_{2} c θ_{3}$
$P_{a x}$	$- (l_{4} + l_{5}) (c θ_{2} s θ_{1} s θ_{3} + s θ_{1} s θ_{2} c θ_{3}) - l_{3} c θ_{2} s θ_{1}$
a21	$c θ_{4} s θ_{1} - s θ_{4} (c θ_{1} s θ_{2} s θ_{3} - c θ_{1} c θ_{2} c θ_{3})$
a22	$- s θ_{1} s θ_{4} - c θ_{4} (c θ_{1} s θ_{2} s θ_{3} - c θ_{1} c θ_{2} c θ_{3})$
a23	$- c θ_{1} c θ_{2} s θ_{3} - c θ_{1} s θ_{2} c θ_{3}$
$P_{a y}$	$(l_{4} + l_{5}) (c θ_{1} c θ_{2} s θ_{3} + c θ_{1} s θ_{2} c θ_{3}) + l_{3} c θ_{1} c θ_{2}$
a31	$s θ_{4} (c θ_{2} s θ_{3} + s θ_{2} c θ_{3})$
a32	$c θ_{4} (c θ_{2} s θ_{3} + s θ_{2} c θ_{3})$
a33	$c θ_{2} c θ_{3} - s θ_{2} s θ_{3}$
$P_{a z}$	$l_{1} + l_{2} - (l_{4} + l_{5}) (c θ_{2} c θ_{3} - s θ_{2} s θ_{3}) + l_{3} s θ_{2}$

The end-effector frame data at the swinging left leg were obtained from the forward kinematics computation in Equation (4) and Table A1. The simplest way to compute the inverse kinematics is to divide

P_{a x}

by the

P_{a y}

term of the matrix

{}^{0}A_{5}

in order to define the support leg yaw motor position

θ_{1}

(see Equation (A1)):

\frac{P_{a x}}{P_{a y}} = - \frac{s θ_{1}}{c θ_{1}} a n d θ_{1} = - arctan (\frac{P_{a x}}{P_{a y}}) \pm π .

(A1)

As a result, the terms

P_{a y}

and

P_{a z}

of the matrix

{}^{0}A_{5}

can be combined to define two unknown variables,

θ_{2}

and

θ_{3}

, as shown in Equation (A2):

\{\begin{matrix} s (θ_{2} + θ_{3}) \cdot (l_{4} + l_{5}) = \frac{P_{a y}}{c θ_{1}} - l_{3} c θ_{2} \\ c (θ_{2} + θ_{3}) \cdot (l_{4} + l_{5}) = - P_{a z} + l_{1} + l_{2} + l_{3} s θ_{2} \end{matrix}

(A2)

By squaring both sides of Equation (A2) and summing them, the term

s (θ_{2} + θ_{3})

is eliminated, which simplifies the solution as presented in Equation (A3):

{(l_{4} + l_{5})}^{2} = {(\frac{P_{a y}}{c θ_{1}} - l_{3} c θ_{2})}^{2} + {(- P_{a z} + l_{1} + l_{2} + l_{3} s θ_{2})}^{2}

(A3)

In Equation (A3), all parts are squared by having the terms

s θ_{2}

and

c θ_{2}

left in the Equation (A4):

2 (- P_{a z} + l_{1} + l_{2}) \cdot l_{3} s θ_{2} + \frac{2 P_{a y} l_{3} c θ_{2}}{c θ_{1}} = \frac{P_{a y}^{2}}{c θ_{1}} + {(- P_{a z} + l_{1} + l_{2})}^{2} + l_{3}^{2} - {(l_{4} + l_{5})}^{2}

(A4)

From Equation (A4), the right leg roll motor’s position

θ_{2}

is defined, where two unknown terms s

θ_{2}

and c

θ_{2}

exist. When the support leg’s yaw motor of the reduced-DOFs RRYY bipedal robot is activated, the robot’s end-effector frame displaces to the

P_{x}

and

P_{y}

directions. The trigonometric term

c θ_{1}

in Equation (A4) can be substituted with an expression demonstrated in Equation (A5) that is employed to represent the magnitude of the end-effector frame between the

P_{a x}

and

P_{a y}

terms:

c θ_{1} = \pm \frac{P_{a y}}{\sqrt{P_{a x}^{2} + P_{a y}^{2}}} .

(A5)

Equation (A6) expresses the simplified form of Equation (A4) by taking Equation (A5) into account, where

f_{a 1} = \pm \sqrt{P_{a x}^{2} + P_{a y}^{2}}

and

f_{a 2} = - P_{a z} + l_{1} + l_{2}

2 f_{a 2} l_{3} s θ_{2} + 2 f_{a 1} l_{3} c θ_{2} = f_{a 1}^{2} + f_{a 2}^{2} + l_{3}^{2} - {(l_{4} + l_{5})}^{2}

(A6)

In order to solve the right hip roll angle

θ_{2}

in Equation (A6), a right triangle is implemented that has an angle

γ

located opposite to the altitude [46], where the altitude and base of the triangle are equal to

2 f_{a 2} l_{3}

and

2 f_{a 1} l_{3}

, respectively (see Figure A2a). The hypotenuse is equal to

\sqrt{4 f_{a 2}^{2} l_{3}^{2} + 4 f_{a 1}^{2} l_{3}^{2}}

. Alternatively, Equation (A6) can be substituted with Equation (A7):

\frac{2 f_{a 2} l_{3} s θ_{2}}{\sqrt{4 f_{a 2}^{2} l_{3}^{2} + 4 f_{a 1}^{2} l_{3}^{2}}} + \frac{2 f_{a 1} l_{3} c θ_{2}}{\sqrt{4 f_{a 2}^{2} l_{3}^{2} + 4 f_{a 1}^{2} l_{3}^{2}}} = \frac{f_{a 1}^{2} + f_{a 2}^{2} + l_{3}^{2} - {(l_{4} + l_{5})}^{2}}{\sqrt{4 f_{a 2}^{2} l_{3}^{2} + 4 f_{a 1}^{2} l_{3}^{2}}} .

(A7)

In order to solve Equation (A7), the additional terms s

γ

and c

γ

are employed in Equation (A8):

s γ s θ_{2} + c γ c θ_{2} = c (θ_{2} - γ) = \frac{f_{a 1}^{2} + f_{a 2}^{2} + l_{3}^{2} - {(l_{4} + l_{5})}^{2}}{\sqrt{4 f_{a 2}^{2} l_{3}^{2} + 4 f_{a 1}^{2} l_{3}^{2}}},

(A8)

where

θ_{2} = \pm arccos (\frac{f_{a 1}^{2} + f_{a 2}^{2} + l_{3}^{2} - {(l_{4} + l_{5})}^{2}}{2 l_{3} \sqrt{f_{a 2}^{2} + f_{a 1}^{2}}}) + γ .

(A9)

The term

γ

in Equation (A9) is substituted with

arctan (\frac{f_{a 2}}{f_{a 1}})

by considering the right triangle represented in Figure A2a, The final expression of the term

θ_{2}

was indicated in Equation (2).

By arranging the

P_{a z}

element from the matrix

{}^{0}A_{5}

and Equation (A2), the left leg roll joint position

θ_{3}

is defined (refer to Equation (A10)):

θ_{3} = \pm a r c c o s (\frac{- f_{a 2} - l_{3} s θ_{2}}{l_{4} + l_{5}}) - θ_{2} .

(A10)

In this case, the

θ_{3}

term defined from Equation (A10) is expressed in Equation (3).

Figure A2. Right triangles were used to determine the right hip roll angle

θ_{2}

during the inverse kinematics (IK) calculation: (a) Swinging leg part. (b) Pendulum actuator part.

Figure A2. Right triangles were used to determine the right hip roll angle

θ_{2}

during the inverse kinematics (IK) calculation: (a) Swinging leg part. (b) Pendulum actuator part.

Appendix C. Solving the Kinematics Problem for Chain 2 (Upper Limb Part)

Table A2. Description of matrix

{}^{0}A_{7}

elements.

Table A2. Description of matrix

{}^{0}A_{7}

elements.

Matrix Elements	Specification
b11	$c θ_{1}$
b12	$s θ_{1} s θ_{2} s θ_{6} - c θ_{2} c θ_{6} s θ_{1}$
b13	$c θ_{2} s θ_{1} s θ_{6} + s θ_{1} s θ_{2} c θ_{6}$
$P_{b x}$	$l_{7} (c θ_{2} s θ_{1} s θ_{6} + s θ_{1} s θ_{2} c θ_{6}) - \frac{l_{3}}{2} c θ_{2} s θ_{1} + l_{6} s θ_{1} s θ_{2}$
b21	$s θ_{1}$
b22	$c θ_{1} c θ_{2} c θ_{6} - c θ_{1} s θ_{2} s θ_{6}$
b23	$- c θ_{1} c θ_{2} s θ_{6} - c θ_{1} s θ_{2} c θ_{6}$
$P_{b y}$	$\frac{l_{3}}{2} c θ_{1} c θ_{2} - l_{7} (c θ_{1} c θ_{2} s θ_{6} + c θ_{1} s θ_{2} c θ_{6}) - l_{6} c θ_{1} s θ_{2}$
b31	0
b32	$c θ_{2} s θ_{6} + s θ_{2} c θ_{6}$
b33	$c θ_{2} c θ_{6} - s θ_{2} s θ_{6}$
$P_{b z}$	$l_{1} + l_{2} + l_{7} (c θ_{2} c θ_{6} - s θ_{2} s θ_{6}) + l_{6} c θ_{2} + \frac{l_{3}}{2} s θ_{2}$

The forward kinematics solution for the pendulum section was developed in Equation (6) and Table A2. By extracting elements from the matrix

{}^{0}A_{7}

in Table A2, the simplest approach to solving the inverse kinematics begins with the ratio of terms

P_{b x}

and

P_{b y}

, where the yaw joint position

θ_{1}

of the supporting leg is determined (refer to Equation (A11)):

\frac{P_{b x}}{P_{b y}} = - \frac{s θ_{1}}{c θ_{1}} a n d θ_{1} = - arctan (\frac{P_{b x}}{P_{b y}}) \pm π

(A11)

Both

P_{b y}

and

P_{b z}

are combined in Equations (A12) and (A13) in order to get rid of the

θ_{2} + θ_{6}

terms and simplify the equations, which are similar to Equations (A2) and (A3) respectively. In this case,

f_{b 1} = \pm \sqrt{P_{b x}^{2} + P_{b y}^{2}}

and

f_{b 2} = - P_{b z} + l_{1} + l_{2}

\{\begin{matrix} - l_{7} s (θ_{2} + θ_{6}) = \frac{P_{b y}}{c θ_{1}} - \frac{l_{3}}{2} c θ_{2} + l_{6} s θ_{2} \\ l_{7} c (θ_{2} + θ_{6}) = - f_{b 2} - l_{6} c θ_{2} - \frac{l_{3}}{2} s θ_{2} \end{matrix},

(A12)

l_{7}^{2} = {(\frac{P_{b y}}{c θ_{1}} - \frac{l_{3}}{2} c θ_{2} + l_{6} s θ_{2})}^{2} + {(- f_{b 2} - l_{6} c θ_{2} - \frac{l_{3}}{2} s θ_{2})}^{2} .

(A13)

From Equation (A13), the unknown terms

s θ_{2}

and

c θ_{2}

can be separated to the left-hand side as it is arranged in Equation (A14):

s θ_{2} (\frac{2 l_{6} P_{b y}}{c θ_{1}} + l_{3} f_{b 2}) - c θ_{2} (\frac{l_{3} P_{b y}}{c θ_{1}} - 2 l_{6} f_{b 2}) = l_{7}^{2} - \frac{P_{b y}^{2}}{c^{2} θ_{1}} - \frac{l_{3}^{4}}{4} - l_{6}^{2} - f_{b 2}^{2}

(A14)

From Equation (A14), the position of the right leg roll motor

θ_{2}

is defined, where the unknown terms s

θ_{2}

and c

θ_{2}

appear. When the yaw motor of the support leg in the reduced-DOFs RRYY bipedal robot rotates, the robot’s end-effector frame moves in the

P_{x}

and

P_{y}

directions. The trigonometric term

c θ_{1}

in Equation (A14) can be replaced with the expression shown in Equation (A15), which represents the magnitude of the end-effector frame in terms of

P_{b x}

and

P_{b y}

c θ_{1} = \pm \frac{P_{b y}}{\sqrt{P_{b x}^{2} + P_{b y}^{2}}} .

(A15)

Equation (A16) represents the simplified version of Equation (A14), incorporating the expression from Equation (A15), where

f_{b 1} = \pm \sqrt{P_{b x}^{2} + P_{b y}^{2}}

and

f_{b 2} = - P_{b z} + l_{1} + l_{2}

s θ_{2} (2 l_{6} f_{b 1} + l_{3} f_{b 2}) - c θ_{2} (l_{3} f_{b 1} - 2 l_{6} f_{b 2}) = l_{7}^{2} - f_{b 1}^{2} - \frac{l_{3}^{4}}{4} - l_{6}^{2} - f_{b 2}^{2}

(A16)

To solve for the right hip roll angle

θ_{2}

in Equation (A16), a right triangle is used with an angle

β

opposite the altitude [46]. In this triangle, the altitude and base are represented by

2 l_{6} f_{b 1} + l_{3} f_{b 2}

and

l_{3} f_{b 1} - 2 l_{6} f_{b 2}

, respectively (see Figure A2b). The hypotenuse is equal to

\sqrt{{(2 l_{6} f_{b 1} + l_{3} f_{b 2})}^{2} + {(l_{3} f_{b 1} - 2 l_{6} f_{b 2})}^{2}}

. Equation (A17) is a modified version of Equation (A16), incorporating the additional terms

s β

and

c β

s β s θ_{2} - c β c θ_{2} = c (θ_{2} + β) = \frac{l_{7}^{2} - f_{b 1}^{2} - \frac{l_{3}^{4}}{4} - l_{6}^{2} - f_{b 2}^{2}}{\sqrt{{(2 l_{6} f_{b 1} + l_{3} f_{b 2})}^{2} + {(l_{3} f_{b 1} - 2 l_{6} f_{b 2})}^{2}}},

(A17)

θ_{2} = \pm arccos (\frac{l_{7}^{2} - f_{b 1}^{2} - \frac{l_{3}^{4}}{4} - l_{6}^{2} - f_{b 2}^{2}}{\sqrt{{(2 l_{6} f_{b 1} + l_{3} f_{b 2})}^{2} + {(l_{3} f_{b 1} - 2 l_{6} f_{b 2})}^{2}}}) - β .

(A18)

As the angle

β

is located in the opposite direction to the altitude

2 l_{6} f_{b 1} + l_{3} f_{b 2}

of a sample triangle in Figure A2b, it can be simplified by substituting with the

\pm arctan (\frac{2 l_{6} f_{b 1} + l_{3} f_{b 2}}{l_{3} f_{b 1} - 2 l_{6} f_{b 2}})

term.

The right leg roll joint position

θ_{2}

was finalized in Equation (A19), which was also expressed in Equation (5):

θ_{2} = \pm a r c c o s (\frac{l_{7}^{2} - f_{b 1}^{2} - \frac{l_{3}^{4}}{4} - l_{6}^{2} - f_{b 2}^{2}}{\sqrt{{(2 l_{6} f_{b 1} + l_{3} f_{b 2})}^{2} + {(l_{3} f_{b 1} - 2 l_{6} f_{b 2})}^{2}}}) \pm arctan (\frac{2 l_{6} f_{b 1} + l_{3} f_{b 2}}{l_{3} f_{b 1} - 2 l_{6} f_{b 2}}) .

(A19)

By combining the element

P_{b y}

from the matrix

{}^{0}A_{7}

and Equation (A12), the pendulum actuator position

θ_{6}

is defined in Equation (A20):

θ_{6} = \pm a r c c o s (\frac{- \frac{l_{3}}{2} s θ_{2} - l_{6} c θ_{2} - f_{b 2}}{l_{7}}) - θ_{2} .

(A20)

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Figure 1. A novel 5-DOF biped robot kinematic architecture with a pendulum balancer: (a) CAD design developed by FreeCAD 0.19.3 software. (b) Real prototype. (c,d) Numerical model. 1—Weight applied by the end-effector frame of a pendulum; 2—Pendulum link; 3—Inverted pendulum actuator; 4—Central link; 5—Left hip roll servo motor; 6—Left leg link; 7—Left leg yaw servo motor. 8—IMU sensor (Model: WitMotion); 9—Dual motor controllers (Model: Odrive 3.5).

θ_{1}

—Right leg yaw actuator’s position;

θ_{2}

—Right hip roll joint’s position;

θ_{3}

—Left hip roll joint’s position;

θ_{4}

—Left leg yaw actuator’s position;

θ_{6}

—Upper limb pendulum actuator’s position.

θ_{1}

—Right leg yaw actuator’s position;

θ_{2}

—Right hip roll joint’s position;

θ_{3}

—Left hip roll joint’s position;

θ_{4}

—Left leg yaw actuator’s position;

θ_{6}

—Upper limb pendulum actuator’s position.

Figure 2. CAD design of a novel reduced-DOF bipedal robot with an RRYY kinematic architecture performing a half gait cycle: (a) Home position. (b) Rotation of the upper limb part to the right side. (c) Stance phase. (d) Forward movement. (e) Double support phase. (f) Returning the upper limb part to the home position. (g) Closeup view of the feet. Δx—Displacement of the robot.

Figure 3. Description of the chains used in kinematic calculations for the RRYY bipedal robot: (a) first kinematic chain (frames 1, 2, 3, 4 and 5); (b) second kinematic chain (frames 1, 2, 6 and 7); (c) intersection between the first and second kinematic chains (frames 1 and 2).

L_{i}

—link length that corresponds to frame i, m.

L_{i}

—link length that corresponds to frame i, m.

Figure 4. Sketch of the bipedal robot stepping forward while balancing on its right leg: (a) Frontal plane. (b) Sagittal plane. (c) Transverse plane.

Figure 5. Sketch of a weight balancer with weights applied on both sides.

Figure 6. TPU -based ankle inclination on the frontal plane and its geometric structure: (a) abnormal case; (b) normal case; (c) foot structure; (d) 3D view of an ankle; (e) 2D Surface view of an ankle. 1—IMU sensor; 2—aluminum extrusion 20 mm × 20 mm; 3—foot cover; 4—TPU-based ankle; 5—back part; 6—front part of a foot.

Figure 7. Pictures of the 3D printed sample TPU ankle prototypes with a modulus elasticity of

σ

= 90 MPa at various geometries a × b × h: (a) 10 mm × 10 mm × 20 mm; (b) 20 mm × 10 mm × 20 mm; (c) 10 mm × 10 mm × 40 mm; (d) 20 mm × 10 mm × 40 mm; (e) 20 mm × 20 mm × 40 mm.

Figure 7. Pictures of the 3D printed sample TPU ankle prototypes with a modulus elasticity of

σ

= 90 MPa at various geometries a × b × h: (a) 10 mm × 10 mm × 20 mm; (b) 20 mm × 10 mm × 20 mm; (c) 10 mm × 10 mm × 40 mm; (d) 20 mm × 10 mm × 40 mm; (e) 20 mm × 20 mm × 40 mm.

Figure 8. Pictures of the tested TPU ankles with a modulus elasticity of

σ

= 90 MPa and link length of 500 mm at different geometries. Graphs of inclination angle

α

(deg) vs. external tension force F (N): (a) FEA method; (b) Real prototypes. 1—IMU sensor. 2—Force sensor.

Figure 8. Pictures of the tested TPU ankles with a modulus elasticity of

σ

= 90 MPa and link length of 500 mm at different geometries. Graphs of inclination angle

α

(deg) vs. external tension force F (N): (a) FEA method; (b) Real prototypes. 1—IMU sensor. 2—Force sensor.

Figure 9. Graphs of the tested TPU ankles with a modulus elasticity of

σ

= 90 MPa and link length of 500 mm at different geometries with axes representing inclination angle

α

(deg) vs. tension force F (N): (a) FEA method; (b) Real prototypes.

Figure 9. Graphs of the tested TPU ankles with a modulus elasticity of

σ

= 90 MPa and link length of 500 mm at different geometries with axes representing inclination angle

α

(deg) vs. tension force F (N): (a) FEA method; (b) Real prototypes.

Figure 10. Diagrams of estimated inclination angle minimization: (a) FEA method; (b) real prototypes; (c) average result.

Figure 11. Block diagram dedicated to the RRYY biped robot single support phase stability control.

Figure 12. Elastic ankle stability control graphs during the single support phase: (a) Central link positions measured with an IMU sensor. (b) Right hip roll joint angles measured with a position encoder.

Figure 13. RRYY bipedal robot lifting the left leg by 45 degrees and putting it down at a single-support stance (developed in the CoppeliaSim environment).

Figure 14. Kinematic results of RRYY bipedal robot’s actuators lifting the left leg by 45 degrees and putting down in the single-support stance: (a) Joint positions (virtual environment). (b) Joint velocities (virtual environment). (c) Joint positions (real prototype). (d) Joint velocities (real prototype).

Figure 15. Real model of the RRYY bipedal robot lifting the left leg by 45 degrees and putting it down in the single support stance.

Figure 16. RRYY bipedal robot performing lateral motion lasting 2 cycles (developed in CoppeliaSim environment).

Figure 17. Graphs of biped robot’s joints were obtained during the execution of a lateral movement: (a) Joint positions (virtual environment). (b) Joint velocities (virtual environment). (c) Joint positions (real prototype). (d) Joint velocities (real prototype).

Figure 18. Testing an RRYY bipedal robot based on a lateral motion with a duration of 2 cycles.

Figure 19. Kinematic results of 5 joints were obtained during the execution of a gait sequence: (a) Joint positions (virtual environment). (b) Joint velocities (virtual environment). (c) Joint positions (real prototype). (d) Joint velocities (real prototype).

Figure 20. Pseudo-static walking sequence with 2 cycles (CoppeliaSim environment).

Figure 21. Pseudo-static walking sequence with 2 cycles (Physical prototype).

Figure 22. Experimental results of 5 joints obtained from numerical and real models after the execution of two gait cycles: (a) Torque computed from the numerical model. (b) Torque measured. (c) Joint mechanical power. (d) Joint electrical power.

Table 1. Overview of the reduced-DOF bipedal robot utilizing two motors per leg, each positioned at different joint configurations.

Lower Limb Joint Orientation	Geometric Motion Type and Direction	Possible Actions	Advantages	Drawbacks	Upper Limb Joint Orientation (Min. Requirement)
RRRR (Roll joints)	Planar motion (y and z direction)	- SSP configuration; - Lateral motion	- Simplicity to control SSP stability; - Minimized foot sliding problems; - Simplified upper limb structure	- Absence of effective walking sequence	Roll joint
PPPP (Pitch joints) [6,32,34]	Planar motion (x and z direction)	- SSP configuration; - Squatting; - Gait plan	- Fast forward movement	- Difficulty to control SSP stability; - Foot sliding problems; - Complicated upper limb design	Roll and pitch joints
YYYY (Yaw joints)	Planar motion (x and y direction)	- Turning; - Sliding gait	- Simplicity to stabilize posture equilibrium; - Simplified upper limb structure	- Poor motion quality due to sliding feet on the ground; - Slow forward movement	Roll joint
PPYY (Pitch and yaw joints)	Spatial motion (x, y and z direction)	- SSP configuration; - Turning; - Gait plan	- Fast forward movement	- Difficulty to control SSP stability; - Foot sliding problems; - Complicated upper limb structure	Roll and pitch joints
YYPP (Yaw and pitch joints)	Spatial motion (x, y and z direction)	- SSP configuration; - Turning; - Gait plan; - Squatting	- Fast forward movement; - Simplicity to control SSP stability	- Foot sliding problems; - Complicated upper limb design	Roll and pitch joints
RRPP (Roll and pitch joints)	Spatial motion (x, y and z direction)	- SSP configuration; - Gait plan; - Squatting; - Lateral motion	- Fast forward movement; - Simplicity to control SSP stability; - Minimized foot sliding problems	- Complicated upper limb design	Roll and pitch joints
PPRR (Pitch and roll joints)	Spatial motion (x, y and z direction)	- SSP configuration; - Gait plan; - Lateral motion	- Fast forward movement; - Minimized foot sliding problems	- Difficulty to control SSP stability; - Complicated upper limb design	Roll and pitch joints
YYRR (Yaw and roll joints)	Spatial motion (x, y and z direction)	- SSP configuration; - Gait plan; - Turning; - Lateral motion	- Minimized foot sliding problems; - Simplified upper limb structure	- Difficulty to control SSP stability; - Slow forward movement	Roll joint
RRYY ¹ (Roll and yaw joints)	Spatial motion (x, y and z direction)	- SSP configuration; - Gait plan; - Turning; - Lateral motion	- Simplicity to control SSP stability; - Minimized foot sliding problems; - Simplified upper limb structure	- Slow forward movement	Roll joint

¹ Proposed bipedal robot.

Table 2. Characterization of the physical and chemical properties, as well as possible applications of various materials.

Material Type	Durability, Chemical and Tear Resistance	Young’s Modulus, MPa	Melting Temperature, Deg	Applications
Polylactic acid (PLA)	Low and brittle	2500–3200	170–200	Food containers, biodegradable medical implants and drug delivery system.
CF-PETg ¹	High	2340–2800	230–250	Bike handles, protective cases, gears and load-bearing parts of machines.
Thermoplastic elastomer (TPE)	Average	4–120	180–250	Toy industry, cellphone cases, insulators of electrical cables.
Polyjet rubber	Low [38]	61 [39]	50–62	Gaskets, wearables, masks, covers [38].
Flexible resin	Low [38]	2 [40]	115–120	Shoe manufacturing; wearable devices; padding elements [38].
Nylon	High [41]	2700	270	Textile, ropes and tendon lines.
Thermoplastic polyurethane (TPU)	High[38]	60–100 [42]	200–220	Gaskets, shock absorbers, vibration isolators, seals [38]

¹ Carbon fiber with polyethylene terephthalate glycol.

Table 3. Results of the tested TPU ankles with

σ

= 90 MPa, F = 50 N, L = 500 mm at different dimensions.

Table 3. Results of the tested TPU ankles with

σ

= 90 MPa, F = 50 N, L = 500 mm at different dimensions.

Trial i	Ankle Size (a × b × h), mm	dy(FEA), mm	$α$ (FEA), deg	Z(FEA), %	dy(Real), mm	$α$ (Real), deg	Z(Real), %	$\bar{Z %}$
1	10 × 10 × 20	37	4.24	0	60	6.39	0	0
2	20 × 10 × 20	29.6	3.4	19.8	34.3	3.93	38.5	29.2
3	10 × 20 × 20	33.7	3.86	9	49.2	5.65	11.6	10.3
4	20 × 20 × 20	27.7	3.18	25	28.78	3.3	48.4	36.7
5	10 × 10 × 40	34.8	4	5.7	52.3	6	6.1	5.9
6	20 × 10 × 40	23.1	2.65	37.5	24	2.75	57	47.2
7	10 × 20 × 40	28.8	3.3	22.2	29.7	3.4	46.8	34.5
8	20 × 20 × 40	19.66	2.25	46.9	20.1	2.3	64	55.5

Table 4. List of sample state-of-the-art bipedal robots with various leg designs and research works.

Robot Name (Research Group)	Motion of the Legs (DOF)	Foot Structure	Yaw Joints	Total Mass, kg	Total Heigh, m	Walking Speed (m/s)	Research Works
Oda et al. [33]	12	Spring integrated	Present	35	1.06	N/A ¹	Vision-based vibration controller; visual stabilization; ankle deformation
L04 Robot [9]	6	Rigid telescopic legs	Absent	N/A	Adjustable	0.5	Dynamic analysis of the bipedal motion
Mir-Nasiri et al. [6]	4	Rigid flat	Absent	N/A	1	0.6	Trajectory planning strategy; pulley-based parallelogram mechanism
Christine Chevallereau et al. [30]	8	Point feet	Absent	9	0.6	0.45	Minimized energy consumption; asymptotically stable periodic walking
BRUCE [43]	10	4 bar linkage mechanism	Present	3.6	0.5	0.1	Pulley added on each leg to reduce the inertia; real-time dynamic motion controller
Zhang et al. [7]	4	Spring integrated	Absent	N/A	N/A	N/A	Time-invariant gait planning; gait stability; virtual constraint method
CRANE robot [31]	6	Point feet	Absent	8	0.96	0.1	Velocity tracking and a posture balance strategy; feedforward torque controller
Sadati et al. [32]	4	Point feet	Absent	32	1.44	N/A	CPG network, PI and feedback controller developed to regulate the hip and knee joints
Vu et al. [34]	4	Spring integrated	Absent	13.9	0.8	N/A	Floating trunk stabilization developed for a walking sequence
Aoyama et al. [17]	12	Rigid flat	Present	24	1	0.26	Prediction of the walking robot’s dynamic properties with PDAC concept
Athlete robot [44]	6	Elastic blade	Absent	10	1.86	2.1	Testing the running and jumping actions; Pneumatic driven joints
Zhenkun Lin et al. [45]	4	Pneumatic control unit integrated	Absent	11	0.56	0.35	Locomotion architecture; ankle stiffness analyzed
NU-Biped-4.5 [35]	12	TPU ankle integrated	Present	15	1.1	0.16	ankle stiffness, energy absorption analyzed at an optimized height
RRYY bipedal robot ²	4	TPU ankle integrated	Present	8–12	1.12	0.005	Ankle geometry optimized; robot’s SSP posture stability analyzed

¹ Not assigned. ² Proposed underactuated robot.

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Yessirkepov, S.; Folgheraiter, M.; Abakov, A.; Umurzakov, T. Development of a Reduced-Degree-of-Freedom (DOF) Bipedal Robot with Elastic Ankles. Robotics 2024, 13, 172. https://doi.org/10.3390/robotics13120172

AMA Style

Yessirkepov S, Folgheraiter M, Abakov A, Umurzakov T. Development of a Reduced-Degree-of-Freedom (DOF) Bipedal Robot with Elastic Ankles. Robotics. 2024; 13(12):172. https://doi.org/10.3390/robotics13120172

Chicago/Turabian Style

Yessirkepov, Sharafatdin, Michele Folgheraiter, Arman Abakov, and Timur Umurzakov. 2024. "Development of a Reduced-Degree-of-Freedom (DOF) Bipedal Robot with Elastic Ankles" Robotics 13, no. 12: 172. https://doi.org/10.3390/robotics13120172

APA Style

Yessirkepov, S., Folgheraiter, M., Abakov, A., & Umurzakov, T. (2024). Development of a Reduced-Degree-of-Freedom (DOF) Bipedal Robot with Elastic Ankles. Robotics, 13(12), 172. https://doi.org/10.3390/robotics13120172

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Development of a Reduced-Degree-of-Freedom (DOF) Bipedal Robot with Elastic Ankles

Abstract

1. Introduction

1.1. Trajectory Generation Using an Inverted Pendulum Model

1.2. Stability Control

1.3. Foot Slip Mitigation

1.4. Adjustment on Rough Terrains

2. Robot Design and Computations

2.1. Kinematic Architecture

2.1.1. Forward Kinematics

2.1.2. Inverse Kinematics

2.2. Static Architecture

2.2.1. Conditions for Equilibrium

2.2.2. Conditions for Safety

2.2.3. Inverse Statics

2.2.4. Forward Statics

3. Experimental Results

3.1. Testing the Elastic Ankle Inclination

3.2. Single Support Phase Stability Control

3.3. Single Support Phase

3.4. Performance of a Lateral Movement

3.5. Performance of a Pseudo-Static Gait Sequence

4. Discussion

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Abbreviations

Appendix A. Electrical System Design

Appendix B. Solving the Kinematics Problem for Chain 1 (Swinging Leg Side)

Appendix C. Solving the Kinematics Problem for Chain 2 (Upper Limb Part)

References

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