In the field of robot technology, legged robots have gained significant attention due to their superior mobility and adaptability in complex environments. These systems can perform various locomotion modes such as walking, trotting, jumping, and climbing, which require diverse energy and power profiles across different motion cycles. However, existing legged robots often face challenges related to insufficient burst power and high energy consumption during dynamic movements like jumping. Traditional joint actuation methods, including hydraulic and electric drives, have limitations: hydraulic systems offer high power density but are bulky and nonlinear, while electric drives, though precise, struggle with mechanical losses and energy inefficiency under high-load conditions. To address these issues, this article introduces a novel drive joint based on magnetorheological (MR) technology, which enhances the motion capabilities of robotic systems by enabling variable stiffness and efficient energy storage. By integrating MR joints into a unipedal robot, we aim to improve jumping performance through dynamic locking mechanisms that facilitate energy transfer between elastic elements and electric actuators. This approach aligns with advancements in robot technology that seek to mimic biological systems, such as muscle-like energy storage and release. The following sections detail the design principles, kinematic and dynamic analyses, control strategies, and experimental validation of the MR joint-based unipedal robot, emphasizing the role of robot technology in achieving agile and efficient locomotion.
The core innovation in this study is the magnetorheological joint, which leverages the unique properties of MR fluids to provide controllable damping and locking capabilities. MR fluids consist of ferromagnetic particles suspended in a carrier fluid, and their rheological behavior changes rapidly in response to external magnetic fields. Specifically, when a magnetic field is applied, the particles form chain-like structures along the field lines, increasing the fluid’s shear stress and effectively “locking” the joint. This transition occurs within milliseconds and is reversible, allowing for precise control over joint stiffness. In the context of robot technology, such adaptability is crucial for handling varying dynamic demands, such as the high power needed for jumping versus the stability required for landing. The MR joint designed here incorporates an MR fluid-filled deep groove ball bearing, where the fluid acts as a lubricant in the absence of a magnetic field and as a locking mechanism when energized. The joint is positioned at the knee of the unipedal robot, with a torsion spring serving as an elastic energy storage unit. When the MR joint is locked via an electromagnetic coil, the relative motion between the thigh and shank compresses the spring, storing energy that can be released during takeoff to enhance jumping height. This design not only reduces the instantaneous current demand on the joint motors but also improves overall energy efficiency, showcasing the potential of MR-based solutions in advanced robot technology applications.
To model the unipedal robot with the MR joint, we first perform kinematic analysis to establish the relationship between joint angles and foot-end forces. The robot is simplified into a two-link system, with the hip and knee motors co-axially arranged and the MR joint at the knee. The virtual leg length, denoted as $\zeta$, represents the distance from the center of mass to the foot-end, and the angle $\psi$ defines the orientation relative to the ground. The kinematic equations are derived as follows:
$$ \zeta = 2l \cdot \cos\left(\frac{\theta_{1j}}{2}\right) $$
$$ \psi = \theta_{1j} + \frac{\theta_{2j}}{2} + \frac{\pi}{2} $$
where $l$ is the length of the thigh and shank links, $\theta_{1j}$ is the hip joint angle, and $\theta_{2j}$ is the knee joint angle. Using the Jacobian transpose, the static force relationship is expressed as:
$$ \begin{bmatrix} F_z \\ \tau_{\psi} \end{bmatrix} = \mathbf{J}^T \begin{bmatrix} \tau_{1j} \\ \tau_{2j} \end{bmatrix} $$
with $\mathbf{J}$ being the Jacobian matrix. For vertical jumping, the dynamics are simplified to a one-dimensional spring-loaded inverted pendulum model, where the virtual spring stiffness $k_z$ and damping $b_z$ describe the interaction during ground contact. The dynamic equations during the stance phase are:
$$ \ddot{z} = -\frac{k_z}{m}(z – z_0) – \frac{b_z}{m}\dot{z} – g $$
and during the flight phase:
$$ \ddot{z} = -g $$
Here, $m$ is the system mass, $g$ is gravity, $z$ is the vertical position, and $z_0$ is the rest length of the virtual spring. The MR joint allows $k_z$ to be decomposed into a fixed virtual stiffness $k_{vz}$ and a variable component $k_z’$ from the elastic element, enabling adjustable energy storage. The work done during the stance phase is calculated as:
$$ W = \int_{0}^{t_{des}/2} k_z (z – z_0) \dot{z} dt = W_u + W’ $$
where $W_u$ is the work done by the motor and $W’$ is the work from the elastic unit, with $t_{des}$ as the stance duration. This decouples the energy contributions, facilitating control strategies that minimize motor effort.
The jumping gait is divided into four phases: descent, compression, extension, and ascent. During descent, the robot falls freely under gravity, converting potential energy to kinetic energy. Upon ground contact, the virtual spring and elastic element compress, storing energy (compression phase). In the extension phase, the stored energy is released, propelling the robot upward. Finally, during ascent, the robot coasts to the peak height. A time-based trajectory for the center of mass is designed to achieve periodic jumping:
$$ z(t) = z_0 – \Delta z \sin\left(\frac{\pi t}{t_{des}}\right) $$
where $\Delta z$ is the maximum compression. This trajectory ensures smooth energy transfer and allows for height control by adjusting $t_{des}$ and $\Delta z$. The control system employs a proportional-derivative (PD) controller for joint motor tracking, with gains tuned to maintain stability. The integration of robot technology in this control framework enables real-time adaptation to dynamic conditions, enhancing the robot’s performance.
To validate the MR joint’s locking capability, magnetic field simulations were conducted using COMSOL. The MR fluid, with an iron powder mass fraction of 86.3%, was modeled within the bearing assembly. The torque $T$ of the MR bearing is given by:
$$ T = \frac{2}{\sin \theta} \left[ \mu_B P_{ax} D_p + K_0 \left( f_0 D_p^3 \nu \omega \right)^{1/2} \right] $$
where $\theta$ is the contact angle, $P_{ax}$ is the axial preload, $D_p$ is the pitch diameter, $\nu$ is the viscosity, $\omega$ is the angular velocity, and $\mu_B$, $K_0$, $f_0$ are empirical parameters. The simulation results showed a uniform magnetic field distribution across the MR bearing at a coil current of 1.3 A, sufficient for effective locking. Experimental tests on a unipedal robot platform measured the equivalent stiffness of the MR joint with different torsion springs. The platform consisted of a single leg mounted on a vertical rail, with loads applied to determine vertical displacement. The results are summarized in Table 1, demonstrating the variable stiffness achievable through MR joint control.
| Load (kg) | Spring Wire Diameter 6.0 mm | Spring Wire Diameter 6.5 mm | Spring Wire Diameter 7.0 mm | |||
|---|---|---|---|---|---|---|
| Vertical Compression (mm) | Equivalent Stiffness (N/m) | Vertical Compression (mm) | Equivalent Stiffness (N/m) | Vertical Compression (mm) | Equivalent Stiffness (N/m) | |
| 3.0 | 53 | – | 15 | – | 10 | – |
| 3.5 | 63 | 490.7 | 23 | 612.5 | 15 | 980.0 |
| 4.0 | 73 | 490.7 | 30 | 700.7 | 20 | 980.0 |
| 4.5 | 85 | 408.8 | 37 | 700.7 | 25 | 980.0 |
| 5.0 | 96 | 445.8 | 46 | 544.5 | 30 | 980.0 |
Jumping experiments were conducted to evaluate the performance of the MR joint-enabled unipedal robot. The control system used an STM32F767 microcontroller operating at 1000 Hz, with joint motors (Unitree GO-M8010) communicating via RS485. A laser rangefinder measured jump height, and a Hall effect sensor monitored motor current. The robot executed jumps under two conditions: with the MR joint locked and unlocked. Key parameters for the experiments are listed in Table 2.
| Parameter | Symbol | Value |
|---|---|---|
| Motor Stiffness | $k_p$ | 0.55 |
| Motor Damping | $k_d$ | 0.01 |
| Jump Period | $t_{des}$ | 100 ms |
| Compression Amount | $\Delta z$ | 0.18 m |
| Virtual Spring Rest Length | $z_0$ | 0.36 m |
| System Mass | $m$ | 2.6 kg |
| Link Length | $l$ | 0.23 m |
| Gravity | $g$ | 9.8 m/s² |
The results from 50 jump trials per condition showed that with the MR joint locked, the maximum motor torque decreased significantly compared to the unlocked state. For instance, with spring wire diameters of 6.0 mm, 6.5 mm, and 7.0 mm, the reduction in foot-end force was 23.8%, 31.6%, and 38.5%, respectively, leading to a proportional decrease in instantaneous motor current. Additionally, the locked joint condition resulted in a shorter stance phase and longer flight phase, increasing jump height by up to 10% relative to traditional electric drives. These findings highlight the effectiveness of the MR joint in decoupling motor and elastic contributions, thereby enhancing energy efficiency and performance in robot technology applications.
In conclusion, the integration of magnetorheological joints in legged robots represents a significant advancement in robot technology, addressing limitations in burst power and energy consumption. The MR joint’s dynamic locking mechanism enables efficient energy storage and release, as demonstrated through kinematic and dynamic modeling, magnetic simulations, and experimental validation. The unipedal robot achieved higher jumps with reduced motor current, underscoring the potential of variable stiffness actuation in agile locomotion. Future work will focus on optimizing control algorithms for more complex terrains and extending this technology to multi-legged systems. This research contributes to the ongoing evolution of robot technology, paving the way for more adaptive and efficient robotic platforms.