In recent years, the field of robotics has seen significant advancements, with bionic robots drawing inspiration from biological systems to enhance performance in various applications. Among these, jumping robots have garnered attention due to their exceptional obstacle-crossing and risk-avoidance capabilities, making them ideal for exploration, rescue missions, and military operations. This paper presents the design and analysis of a bionic robot inspired by the jumping spider, focusing on a composite jumping motion that combines leg extension with a catapult mechanism to achieve superior jumping height and distance. The bionic robot leverages the spider’s leg structure and jumping mechanics to optimize energy utilization and stability during motion.
The jumping spider, known for its remarkable agility, employs a hydraulic-like mechanism to propel itself by rapidly extending its legs. By analyzing this motion, we designed a bionic robot with three pairs of legs, each simplified into four-link structures comprising hip, femur, knee-tibia, and tarsal segments. The tarsal segment incorporates a damping spring to absorb impact during landing. Additionally, a catapult device is integrated into the robot’s body to store and release energy, working in tandem with leg movements to execute jumps. This composite approach allows the bionic robot to achieve higher vertical and forward jumps compared to traditional designs.
To model the leg kinematics, the Denavit-Hartenberg (DH) method was applied, establishing coordinate frames for each joint. The transformation matrices between consecutive links are given by the general formula:
$$
^{i-1}T_i = \begin{bmatrix}
\cos\theta_i & -\sin\theta_i & 0 & a_{i-1} \\
\sin\theta_i \cos\alpha_{i-1} & \cos\theta_i \cos\alpha_{i-1} & -\sin\alpha_{i-1} & -d_i \sin\alpha_{i-1} \\
\sin\theta_i \sin\alpha_{i-1} & \cos\theta_i \sin\alpha_{i-1} & \cos\alpha_{i-1} & d_i \cos\alpha_{i-1} \\
0 & 0 & 0 & 1
\end{bmatrix}
$$
where $\theta_i$ represents the joint angle, $a_{i-1}$ is the link length, $\alpha_{i-1}$ is the twist angle, and $d_i$ is the link offset. For the bionic robot’s leg, the DH parameters are summarized in Table 1.
| Link | $\alpha_{i-1}$ (rad) | $a_{i-1}$ (mm) | $\theta_i$ (rad) | $d_i$ (mm) |
|---|---|---|---|---|
| 1 | 0 | 0 | $\theta_1$ | 0 |
| 2 | -$π/2$ | 0 | $\theta_2$ | 0 |
| 3 | 0 | $L_2$ | $\theta_3$ | 0 |
| 4 | 0 | $L_3$ | $\theta_4$ | 0 |
The position of the end-effector point M relative to the base frame is derived as:
$$
\begin{bmatrix}
X \\
Y \\
Z
\end{bmatrix}
=
\begin{bmatrix}
L_2 C_1 C_2 + L_3 C_1 C_{23} + L_4 C_1 C_{234} \\
L_2 S_1 C_2 + L_3 S_1 C_{23} + L_4 S_1 C_{234} \\
-L_2 S_2 – L_3 S_{23} – L_4 S_{234}
\end{bmatrix}
$$
where $C_i$ and $S_i$ denote $\cos\theta_i$ and $\sin\theta_i$, respectively, and subscripts like $C_{23}$ represent $\cos(\theta_2 + \theta_3)$. The workspace of the leg was analyzed using MATLAB, considering joint angle ranges of $(-π/2, π/2)$ for the hip, knee, and leg joints, and $(-3π/4, π/2)$ for the tarsal joint. This analysis ensures the bionic robot can achieve sufficient motion range for stable jumps.
Dynamics of the leg were modeled using the Lagrange method, where the Lagrangian $L$ is defined as the difference between kinetic energy $K$ and potential energy $P$. The torque $\tau_i$ for each joint is given by:
$$
\tau_i = \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}_i} \right) – \frac{\partial L}{\partial \theta_i}
$$
For the simplified leg model with masses concentrated at link ends, the torques are computed as follows:
$$
\tau_1 = (m_1 L_1^2 + m_2 L_2^2 C_2^2 + m_3 L_3^2 C_{23}^2 + m_4 L_4^2 C_{234}^2) \ddot{\theta}_1
$$
$$
\tau_2 = (m_2 L_2^2 + m_3 L_3^2 + m_4 L_4^2) \ddot{\theta}_2 + \frac{1}{2} (m_2 L_2^2 S_2^2 + m_3 L_3^2 S_{23}^2 + m_4 L_4^2 S_{234}^2) \dot{\theta}_1^2 + (m_2 g L_2 C_2 + m_3 g L_3 C_{23} + m_4 g L_4 C_{234})
$$
$$
\tau_3 = (m_3 L_3^2 + m_4 L_4^2) \ddot{\theta}_3 + (m_3 L_3^2 S_{23} + m_4 L_4^2 S_{234}) \dot{\theta}_1^2 + (m_3 g L_3 C_{23} + m_4 g L_4 C_{234})
$$
$$
\tau_4 = m_4 L_4^2 \ddot{\theta}_4 + m_4 L_4^2 S_{234} \dot{\theta}_1^2 + m_4 g L_4 C_{234}
$$
These equations help in designing the control system for the bionic robot, ensuring precise torque application during jumps.
The catapult mechanism is a critical component of the bionic robot, designed to store energy in springs and release it rapidly. The传动 system employs a ratchet and bevel gear arrangement with a 1:1:1 transmission ratio. A drive motor compresses the springs via a soft steel cable, and upon release, the stored elastic energy propels the robot. The energy conversion follows the conservation law, where the spring’s potential energy $E_0$ transforms into kinetic energy $E_k$ and then into gravitational potential energy $E_g$:
$$
E_0 = E_k = E_g = m g h
$$
For a target jump height of 0.5 m and a robot mass of 2.4 kg, the required energy is:
$$
E_0 = 2.4 \times 9.8 \times 0.5 = 11.76 \, \text{J}
$$
Considering spring efficiency $\eta_{\text{spring}} = 0.8$ and gear transmission efficiency $\eta_{\text{gear}} = 0.9$, the initial energy needed is:
$$
E_1 = \frac{E_0}{\eta_{\text{spring}} \eta_{\text{gear}}} = \frac{11.76}{0.8 \times 0.9} = 16.29 \, \text{J}
$$
With a spring compression length of 0.45 m, the spring stiffness $k$ is calculated using Hooke’s law:
$$
E_1 = \frac{1}{2} k x^2 \implies k = \frac{2 E_1}{x^2} = \frac{2 \times 16.29}{0.45^2} \approx 160.9 \, \text{N/m}
$$
Two springs are used, each with stiffness $k_1 = k_2 = 80.45 \, \text{N/m}$. The force exerted by each spring is:
$$
F = k x = 80.45 \times 0.45 = 36.2 \, \text{N}
$$
Non-standard springs with a mean diameter of 12 mm, wire diameter of 2 mm, and 11 active coils are selected. The stiffness is verified using the formula:
$$
k = \frac{G d^4}{8 D^3 n}
$$
where $G$ is the shear modulus, $d$ is the wire diameter, $D$ is the mean diameter, and $n$ is the number of active coils. This ensures the springs meet the design requirements for the bionic robot.
The control system of the bionic robot utilizes an Arduino Uno R3 board paired with a PCA9685 16-channel servo driver to manage leg joint angles and catapult timing. Servos adjust the hip, leg, knee, and tarsal joints to predefined angles for takeoff, mid-air stability, and landing. A ranging module measures jump height, providing feedback for optimization. The hardware setup enables precise coordination between leg movements and energy release, crucial for the composite jumping motion of the bionic robot.
Simulations in Adams software validated the robot’s performance. For vertical jumps, the initial posture involved horizontal body alignment with leg joints set at specific angles: leg joint at 60°, knee joint at 20°, and tarsal joint at 50° relative to reference axes. The catapult released energy at 0.3 s, resulting in a maximum centroid height of 734.12 mm. Velocity and acceleration profiles showed abrupt changes during takeoff and landing, mitigated by damping springs. The energy ratio between leg motion and catapult was approximately 1:2, highlighting the mechanism’s efficiency. Table 2 summarizes the joint angle settings for vertical jumps.
| Leg Pair | Hip Joint (°) | Leg Joint (°) | Knee Joint (°) | Tarsal Joint (°) |
|---|---|---|---|---|
| Front | 0 | -60 | 60 | 50 to -30 |
| Middle | 0 | -60 | 60 | 50 to -30 |
| Rear | 0 | 60 | -60 | -50 to 30 |
For forward jumps, the body was tilted 10° forward, and joint angles were adjusted asymmetrically across leg pairs to achieve a horizontal displacement of 447.64 mm. The motion sequence included posture adjustment, catapult release, ascent, descent, and landing stabilization. Velocity in the Y-direction varied due to continuous leg adjustments, while acceleration spikes occurred at impact points. The simulation confirmed that the bionic robot could achieve significant forward jumps while maintaining balance.
A partial physical model of the bionic robot was fabricated using 3D printing, with a total weight of 901.3 g and an extended length of 866 mm. Controlled via Arduino, the model performed vertical jumps, and jump height was measured using a US-100 ultrasonic sensor. The experimental data showed a close match with simulation results, though minor discrepancies arose from spring stiffness tolerances and external noise. The jump posture aligned with theoretical predictions, demonstrating the feasibility of the composite jumping approach for bionic robots.
In conclusion, the bionic robot inspired by jumping spiders successfully integrates leg extension with a catapult mechanism to enhance jumping performance. Kinematic and dynamic analyses ensure optimal motion planning, while the energy-efficient catapult design amplifies jump height and distance. Simulations and experiments validate the composite jumping motion, achieving vertical jumps over 700 mm and forward jumps beyond 400 mm. This design opens avenues for advanced bionic robots in demanding environments, leveraging biological principles for robotic innovation.