Humanoid robots have been an active research field for decades because they closely resemble human morphology and are expected to operate in human-centered environments. Achieving stable locomotion is one of the most fundamental and challenging capabilities for a humanoid robot. The lower limbs of a humanoid robot must be designed with a proper number of degrees of freedom (DOF), sufficient structural strength, appropriate actuation systems, and a gait planner that yields human-like walking behavior. In this paper, I present a complete study of the lower limb structure design, gait planning, and dynamic simulation of a humanoid robot. The proposed humanoid robot has fourteen lower-limb DOF, including toe joints that are often neglected in conventional twelve-DOF designs. By introducing toe joints, the walking posture of the humanoid robot becomes much closer to that of a real human. A virtual prototype is built in Pro/E and imported into ADAMS for dynamic simulation, and a joint controller is developed in Simulink through the ADAMS/Controls interface. The simulation results confirm the feasibility of the toe-joint configuration and provide detailed joint torque profiles that validate the motor and reducer selection.
1 Introduction
The development of humanoid robots has progressed significantly since the early biped walking machines. Research groups in Japan, Korea, Europe, and China have produced notable platforms such as ASIMO, HRP-2, KHR, and others. Most existing humanoid robots adopt twelve DOF for the lower limbs: three at the hip, one at the knee, and two at the ankle per leg. This arrangement enables stable walking on flat ground, but it does not fully reproduce the human gait because the human foot contains a toe joint that plays an important role during the push-off phase. Therefore, in this work I extend the conventional lower-limb configuration by adding one pitch DOF at each toe, yielding a total of fourteen lower-limb DOF. This modification allows the humanoid robot to exhibit the heel-off and push-off phases during walking, making the gait more anthropomorphic.
The contributions of this paper are threefold. First, a detailed mechanical design of the lower limbs is presented, including the DOF layout, joint mechanisms, strength verification, and selection of motors and harmonic drives. Second, kinematic and dynamic models are derived using the Denavit–Hartenberg (D–H) convention and the Lagrangian formulation. Third, a gait planner based on the inverted pendulum model is developed for both the forward and lateral directions, and the planned trajectories are used to drive a virtual prototype in ADAMS. The results show that the toe joints are beneficial for reproducing human-like walking posture.
2 Mechanical Design of Lower Limbs
2.1 DOF Configuration
The lower limb system of the humanoid robot is designed with a total of fourteen DOF. Each leg has seven DOF: three at the hip (pitch, roll, yaw), one at the knee (pitch), two at the ankle (pitch, roll), and one at the toe (pitch). The distribution is listed in Table 1. This configuration enables the robot to perform forward/backward walking, side stepping, turning, and stair climbing. The toe-joint DOF is the key extension compared with conventional designs.
| Joint | Roll | Pitch | Yaw | Total per leg |
|---|---|---|---|---|
| Hip | 1 | 1 | 1 | 3 |
| Knee | 0 | 1 | 0 | 1 |
| Ankle | 1 | 1 | 0 | 2 |
| Toe | 0 | 1 | 0 | 1 |
| Total | 2 | 4 | 1 | 7 |
2.2 Joint Motion Ranges
The range of each joint must be specified so that the humanoid robot can imitate human actions without mechanical interference. The joint ranges are selected by referring to the anatomical data of a real human. Table 2 compares human joint ranges with the designed robot ranges. It can be seen that the robot’s ranges are generally equal to or slightly larger than those of a human, ensuring that the humanoid robot can perform a wide variety of movements.
| Joint | Direction | Human range (deg) | Robot range (deg) |
|---|---|---|---|
| Hip | Roll | −45 to 20 | −40 to 20 |
| Pitch | −125 to 15 | −130 to 40 | |
| Yaw | −45 to 45 | −45 to 45 | |
| Knee | Pitch | 0 to 130 | 0 to 150 |
| Ankle | Roll | −20 to 30 | −30 to 40 |
| Pitch | −20 to 45 | −50 to 45 | |
| Toe | Pitch | −45 to 30 | −30 to 0 |
2.3 Joint Mechanisms
The hip joint adopts a cantilever structure composed of three orthogonal axes. The cantilever type is chosen because it leaves a small distance between the two thighs and allows the legs to cross each other, which is useful for turning and for executing certain human-like motions. The knee joint uses a pitch axis driven by a DC motor through a timing belt and a harmonic drive. This arrangement helps to keep the joint compact while allowing a suitable reduction ratio. The ankle joint consists of two orthogonal axes. The pitch axis is driven by a motor and harmonic reducer through a belt, while the roll axis is directly driven by the motor and harmonic reducer to save space. The toe joint is unique: its input motor axis is orthogonal to the toe rotation axis, and a bevel gear pair transmits the torque. The foot structure is shown in the design, with the toe and heel lengths approximately in a 2:3 ratio. A curved toe and heel are used to reduce ground impact, and rubber bushings are embedded in the sole to act as low-pass filters for shock absorption.
2.4 Strength and Stiffness Verification
Finite element analysis (FEA) is performed on the thigh and shank components using Pro/MECHANICA Structure. Two extreme loading conditions are considered: (1) the maximum joint torque of 140 Nm applied at the joint, and (2) an impact force of 1000 N acting on the foot, which is approximately 1.5 times the body weight. The material chosen is LY12 aluminum alloy with a tensile strength of 420 MPa and a compressive strength of 280 MPa. For the shank under the utmost torque, the maximum deformation is 0.736 mm and the maximum stress is 71.38 MPa, which is far below the allowable strength. Under the impact force, the deformation is 0.379 mm and the stress is 51.12 MPa. For the thigh under the utmost torque, the maximum deformation is 0.118 mm and the maximum stress is 29.77 MPa; under impact, the deformation is 0.628 mm and the stress is 31.22 MPa. All results satisfy the strength and stiffness requirements, confirming that the structural design is safe.
2.5 Motor and Harmonic Reducer Selection
The actuators are crucial because they contribute more than 60% of the total mass of a humanoid robot. High power-to-weight ratio, low rotor inertia, and reliable performance are the main criteria. MAXON RE series motors with coreless rotors are selected due to their low inertia and high dynamic response. Harmonic drives are chosen for their large reduction ratio, high efficiency, compactness, and smooth operation.
The selection process begins with the knee joint because it experiences the highest torque and angular speed. The required knee torque is 140 Nm and the maximum joint speed is 65 rpm. The MAXON RE40 (150 W) motor has a stall torque of 2.50 Nm and a maximum speed of 8200 rpm. If the total reduction ratio is 125, the motor can provide \(2.50 \times 125 \times 0.60 = 187.5\) Nm, which satisfies the requirements. The harmonic reducer is selected as CSG25-100, and the belt reduction ratio is 1.25. Similar procedures are applied to the other joints. The final selections are summarized in Table 3.
| Joint | Motor | Harmonic reducer | Total ratio | Reducer ratio | Belt ratio |
|---|---|---|---|---|---|
| Hip Pitch | RE40 | CSG20-100 | 125 | 100 | 1.25 |
| Hip Roll | RE40 | CSG20-120 | 220 | 120 | 22/12 |
| Hip Yaw | RE35 | CSG20-120 | 300 | 120 | 2.5 |
| Knee Pitch | RE40 | CSG25-100 | 125 | 100 | 1.25 |
| Ankle Pitch | RE35 | SHD20-100 | 125 | 100 | 1.25 |
| Ankle Roll | RE35 | CSG20-160 | 160 | 160 | — |
| Toe Pitch | RE25 | GP 32 A | 159 | 159 | — |
3 Kinematic and Dynamic Modeling
3.1 Kinematic Model with D–H Parameters
To analyze the motion of the humanoid robot, I assign a body-fixed coordinate frame to each link and use the Denavit–Hartenberg convention. The robot is modeled as a serial chain from the left toe (base) through the body to the right toe (end-effector). The base frame is fixed to the left toe, and the right toe is the terminal link. In total, there are 15 rigid bodies including the toes, feet, shanks, thighs, hip elements, and the upper body. Since the walking motion is cyclic, the left leg is considered the support leg and the right leg is the swing leg at the beginning of a gait cycle. The D–H parameters are listed in Table 4. The transformation matrix from frame \(i-1\) to frame \(i\) is given by:
$$
A_i^{i-1} = \begin{bmatrix}
c_i & -s_i c\alpha_i & s_i s\alpha_i & a_i c_i\\
s_i & c_i c\alpha_i & -c_i s\alpha_i & a_i s_i\\
0 & s\alpha_i & c\alpha_i & d_i\\
0 & 0 & 0 & 1
\end{bmatrix}
$$
where \(c_i = \cos\theta_i\), \(s_i = \sin\theta_i\), and \(\theta_i\) is the joint variable. The translation distances are \(a_i\) and \(d_i\), and \(\alpha_i\) is the link twist. The full set of homogeneous transformation matrices is obtained by substituting the D–H parameters into the above formula.
| Link \(i\) | \(\theta_i\) | \(\alpha_i\) (rad) | \(a_i\) (m) | \(d_i\) (m) | \(\theta_i\) range (deg) |
|---|---|---|---|---|---|
| 1 | \(\theta_1\) | \(-\pi/2\) | 0.108 | 0 | −5 to 15 |
| 2 | \(\theta_2\) | \(\pi/2\) | 0 | 0 | −40 to 30 |
| 3 | \(\theta_3\) | 0 | 0.230 | 0 | −45 to 80 |
| 4 | \(\theta_4\) | 0 | 0.230 | 0 | −150 to 0 |
| 5 | \(\theta_5\) | \(-\pi/2\) | 0 | 0 | −130 to 40 |
| 6 | \(\theta_6\) | \(\pi/2\) | 0 | 0 | −20 to 40 |
| 7 | \(\theta_7\) | 0 | 0.120 | 0 | −45 to 45 |
| 8 | \(\theta_8\) | \(-\pi/2\) | 0 | 0 | −45 to 45 |
| 9 | \(\theta_9\) | \(\pi/2\) | 0 | 0 | −40 to 20 |
| 10 | \(\theta_{10}\) | \(\pi\) | 0.230 | 0 | −40 to 130 |
| 11 | \(\theta_{11}\) | 0 | 0.230 | 0 | 0 to 150 |
| 12 | \(\theta_{12}\) | \(\pi/2\) | 0 | 0 | −80 to 45 |
| 13 | \(\theta_{13}\) | \(-\pi/2\) | 0.108 | 0 | −30 to 40 |
| 14 | \(\theta_{14}\) | 0 | 0 | 0 | −15 to 5 |
With the above transformations, the position and orientation of every link can be computed for given joint angles. This kinematic model is used to generate the desired joint trajectories during walking and to provide feedback for the controller in simulation.
3.2 Dynamic Model with Lagrangian Formulation
The dynamic equations of the humanoid robot are derived using the Lagrangian approach. The Lagrangian \(L\) is defined as the difference between the kinetic energy \(K\) and the potential energy \(P\):
$$
L = K – P
$$
The generalized force (torque) \(T_i\) for joint \(i\) is obtained from:
$$
T_i = \frac{d}{dt}\frac{\partial L}{\partial \dot{\theta}_i} – \frac{\partial L}{\partial \theta_i}, \quad i=1,2,\dots,n
$$
To keep the model tractable, the forward walking plane is simplified as a nine-link mechanism. Each link has mass \(m_i\), length \(l_i\), distance \(r_i\) from the joint to the center of mass, and moment of inertia \(I_i\) about its center of mass. The center-of-mass coordinates of each link are expressed as functions of the joint angles. For example, the coordinates of the second link (right foot) are:
$$
x_2 = r_2 \cos\theta_1, \quad z_2 = r_2 \sin\theta_1
$$
The total potential energy is:
$$
P = g\sum_{i=1}^{9} m_i z_i
$$
The total kinetic energy is:
$$
K = \frac{1}{2}\sum_{i=1}^{9} \left( m_i (\dot{x}_i^2 + \dot{z}_i^2) + I_i \dot{\theta}_{i-1}^2 \right)
$$
Substituting these expressions into the Lagrangian equation yields the joint torques as functions of \(\theta_i\), \(\dot{\theta}_i\), and \(\ddot{\theta}_i\). This dynamic model provides a theoretical basis for verifying the torque limits of the actuators and for interpreting the simulation results.
4 Gait Planning
4.1 Walking Posture Transitions
Unlike conventional twelve-DOF humanoid robots, which walk with three main phases, the proposed robot with toe joints goes through a more natural sequence of postures. Assume the left leg is the support leg and the right leg is the swing leg. The walking cycle is divided as follows: from 0 to \(T_d\), the left foot is fully on the ground and the right foot is on its toe; from \(T_d\) to \(T_m\), the left foot remains flat while the right foot swings; from \(T_m\) to \(T\), the left heel lifts and only the left toe supports the body while the right foot continues swinging; from \(T\) to \(T+T_d\), the right foot lands fully and the left foot is on its toe. This sequence includes the single-support phase on the forefoot, which is absent in traditional robots, and makes the gait closer to that of a human.
4.2 Forward Gait Planning
The forward gait is planned using the inverted pendulum model. The entire mass of the humanoid robot is concentrated at the center of mass (COM), located at a distance \(l\) from the toe joint. The equation of motion is:
$$
\ddot{\theta} – \frac{g}{l}\sin\theta = 0
$$
For small angles, \(\sin\theta \approx \theta\), giving the solution:
$$
\theta(t) = C_1 e^{kt} + C_2 e^{-kt}, \quad k = \sqrt{\frac{g}{l}}
$$
The boundary conditions are:
$$
\theta(0) = -\arcsin\left(\frac{S/2}{l}\right), \quad \theta(t_1)=0, \quad \dot{\theta}(0)=0
$$
For the designed robot, the step length \(S = 0.35\) m, the single-step period \(T = 0.5\) s, and \(l = 0.55\) m. This yields:
$$
\theta(t) = 0.1283 \left( e^{4.22(t-0.25)} – e^{-4.22(t-0.25)} \right)
$$
The hip trajectory in the forward direction is:
$$
X_H = l\sin\theta, \quad Z_H = l\cos\theta – Z_{cg}^{hip}
$$
The swing ankle trajectory is planned using cubic spline interpolation through selected key points. The key points are chosen to ensure zero velocity at liftoff and landing, thus avoiding ground impact. The resulting piecewise polynomial functions for \(X_{ankle}(t)\) and \(Z_{ankle}(t)\) are obtained. For instance, on the first interval the X-coordinate is constant at 0.431 m, and on later intervals it follows cubic polynomials.
The toe joint angle during the heel-off phase is planned separately. To avoid abrupt velocity changes, the angular velocity of the toe is prescribed as a half-sine profile:
$$
\dot{\theta}_{foot}(t) = A \sin\left(\frac{\pi(t-T_m)}{T-T_m}\right)
$$
Integrating and applying the boundary conditions \(\theta_{foot}(T_m)=0\) and \(\theta_{foot}(T)=0.3\) rad gives:
$$
\theta_{foot}(t) = 0.3 \sin^2\left(\frac{\pi(t-0.25)}{0.25}\right)
$$
With the hip and ankle positions known, the knee and hip pitch angles can be computed by solving the two-link inverse kinematics of the leg. The time functions of all joints are thus determined.
4.3 Lateral Gait Planning
The lateral motion is also modeled using the inverted pendulum. The lateral angle \(\theta_{ce}\) has the same general solution as the forward angle, but with different boundary conditions. At the beginning of the lateral step, the body moves from one side to the center. The lateral hip displacement is \(Y_H = l_{ce}\sin\theta_{ce}\). The maximum lateral displacement is half of the hip width \(W_b/2 = 0.103\) m. The boundary conditions give:
$$
\theta_{ce}(t) = 0.0585 \left( e^{4.22t} + e^{-4.22t} \right), \quad t \in [0,0.25]
$$
For the other half of the step, the sign changes to produce a periodic motion. The lateral hip and ankle joint angles are then computed from the geometry of the robot. This completes the lateral trajectory planning.
4.4 Stability Check via ZMP
The zero moment point (ZMP) criterion is used to verify the stability of the planned gait. The ZMP position is computed as:
$$
X_{ZMP} = \frac{\sum m_i (\ddot{Z}_i + g) X_i – \sum m_i \ddot{X}_i Z_i}{\sum m_i (\ddot{Z}_i + g)}
$$
$$
Y_{ZMP} = \frac{\sum m_i (\ddot{Z}_i + g) Y_i – \sum m_i \ddot{Y}_i Z_i}{\sum m_i (\ddot{Z}_i + g)}
$$
For the inverted pendulum model, substituting \(X = l\sin\theta\), \(Z = l\cos\theta\) and the corresponding lateral expressions yields \(X_{ZMP}=0\) and \(Y_{ZMP}=0\), which means the ZMP is always at the toe support point. Since this point lies inside the support polygon during the entire single-support phase, the planned gait is stable. The ZMP trajectory is plotted and it always remains inside the convex hull of the left and right foot trajectories.
5 Virtual Prototyping and Dynamic Simulation
A virtual prototype of the humanoid robot is essential to verify the mechanical design and gait planning before manufacturing. ADAMS (Automatic Dynamic Analysis of Mechanical Systems) is used for the dynamic simulation. Since ADAMS’ solid modeling capability is limited for complex parts, the three-dimensional solid model is built in Pro/E and then imported into ADAMS through the MECH/Pro interface. In ADAMS, revolute joints are defined at the fourteen DOF, and contact constraints are added between the feet/toes and the ground. Four contact pairs are defined for the left heel, left toe, right heel, and right toe. The friction coefficient is set to a sufficiently high value to prevent slipping during the walking simulation.
5.1 Co-Simulation Setup
The virtual prototype is controlled through a co-simulation framework. ADAMS provides the mechanical model and sends the actual joint angles to Simulink. Simulink runs the gait trajectory generator and PD controllers, and returns the joint torques to ADAMS. The interface is established using the ADAMS/Controls module. The control system for each joint is an independent PD controller:
$$
M = K_p (\theta_0 – \theta) + K_d (\dot{\theta}_0 – \dot{\theta})
$$
where \(K_p\) and \(K_d\) are the proportional and derivative gains, \(\theta_0\) is the desired joint angle, and \(\theta\) is the measured joint angle. The desired trajectories are generated from the gait planner described in Section 4.
5.2 PD Parameter Tuning
The PD gains are tuned experimentally in the simulation environment. For each joint, a step input of 1 rad is commanded, and the response is observed. The goal is to achieve fast response, small overshoot, and small steady-state error. For example, for the knee joint, an initial set of \(K_p = 700{,}000\), \(K_d = 15{,}000\) gives a steady-state error of 0.0038 rad, an overshoot of 0.0207 rad, and a settling time of about 0.05 s. Increasing \(K_d\) to 20,000 removes the overshoot but increases the settling time to 0.12 s. Increasing \(K_p\) to 900,000 and keeping \(K_d = 20{,}000\) yields a good compromise: no overshoot, a steady-state error of 0.0049 rad, and a settling time of 0.069 s. Similar experiments are performed for all joints. The final PD parameters are listed in Table 5.
| Joint | \(K_p\) | \(K_d\) |
|---|---|---|
| Hip Pitch | 2,500,000 | 70,000 |
| Hip Roll | 5,000,000 | 250,000 |
| Knee Pitch | 900,000 | 20,000 |
| Ankle Pitch | 400,000 | 4,000 |
| Ankle Roll | 500,000 | 3,500 |
| Toe Pitch | 320,000 | 8,000 |
5.3 Simulation Results
With the tuned PD controller, the co-simulation is executed for two gait cycles. The desired and actual joint angles for the left and right legs are compared. The PD controllers track the desired trajectories well for all twelve active joints. The hip yaw joints are not commanded because turning is not considered in this study.
The joint torques are recorded during the simulation. Table 6 summarizes the peak and RMS torque values for the major joints. The knee pitch torque has a peak of 83.543 Nm and an RMS of 29.097 Nm. The hip pitch peak torque is 74.369 Nm. The toe pitch joint, which is the new addition, shows a peak torque of 17.160 Nm and an RMS of 5.118 Nm. These values are all within the output capability of the selected harmonic reducers, thus validating the actuator selection.
| Joint | Peak torque (Nm) | RMS torque (Nm) |
|---|---|---|
| Ankle Pitch | 61.453 | 20.355 |
| Ankle Roll | 2.955 | 0.760 |
| Hip Pitch | 74.369 | 28.530 |
| Hip Roll | 35.364 | 13.620 |
| Knee Pitch | 83.543 | 29.097 |
| Toe Pitch | 17.160 | 5.118 |
The simulation animation clearly shows the walking posture of the humanoid robot. Unlike a conventional robot, the simulated robot exhibits a forefoot-only support phase near the end of the single-support phase, which is a characteristic of human walking. The toe joint allows the heel to lift while the toe remains in contact with the ground, generating a natural push-off. This confirms that the introduction of the toe joint improves the anthropomorphic quality of the gait.
6 Conclusion
In this paper, I have presented a comprehensive study of the lower limb design, gait planning, and dynamic simulation of a humanoid robot with toe joints. The main conclusions are as follows:
- The proposed fourteen-DOF lower limb configuration, which adds one toe pitch DOF to each leg, is mechanically feasible. The detailed joint mechanisms and component selections provide a solid basis for building a physical prototype.
- The kinematic and dynamic models derived from the D–H convention and Lagrangian formulation accurately describe the motion and torque relationships of the humanoid robot.
- The gait planner based on the inverted pendulum model generates stable and human-like walking trajectories. The toe joint trajectory is smoothly integrated into the gait cycle, producing a heel-off phase that is absent in traditional twelve-DOF designs.
- The co-simulation between ADAMS and Simulink with PD controllers demonstrates excellent trajectory tracking performance. The simulated joint torques are within the capacity of the selected motors and harmonic reducers, validating the design choices.
Future work will extend the gait planning to stair climbing and obstacle crossing, where the toe joints are expected to have a more significant effect on the required joint torques. Additionally, adaptive PD gains for the support and swing phases will be explored to improve tracking accuracy. The results of this study provide a valuable reference for the development of humanoid robots that can walk more naturally in human environments.