In the field of robotics, the locomotion of legged systems, particularly quadrupedal platforms, represents a frontier of immense challenge and opportunity. I have long been fascinated by the elegant and efficient mobility exhibited by biological quadrupeds in nature. Their ability to traverse complex, unstructured terrains with a blend of static stability and dynamic agility is something I aimed to capture in a mechanical system. However, a critical survey of existing bionic robot designs reveals several persistent challenges: a high number of actuators leading to complex control architectures, significant energy consumption, and substantial leg inertia which impedes dynamic motion. To address these issues, I propose a novel leg mechanism for a bionic quadruped robot driven by a single hydraulic cylinder. This paper details the conceptual design, rigorous kinematic modeling, and simulation-based validation of this innovative leg structure, aiming to contribute a simpler, potentially more robust solution to the field of bionic robot locomotion.
1. Leg Mechanism Design and Principle
The foundational inspiration for my design stems from the biomechanical study of canine leg anatomy, a common reference point for bionic robot development. A canine leg primarily features a shoulder (scapulohumeral) joint and an elbow joint, creating a coupled motion between the upper arm (humerus) and forearm (radius/ulna). Traditional robotic implementations often mimic this by placing an independent actuator at each joint. While effective, this approach inherently leads to the problems of multiple drives, complex coordinated control, and increased mass concentrated distally on the limb.
My core innovation was to decouple the actuation from the joints directly. I conceived a planar linkage mechanism that translates the linear motion of a single hydraulic cylinder into the coordinated angular motion of both the thigh and shank links. The schematic of the proposed leg mechanism is illustrated below.
The mechanism consists of a fixed torso link (frame), a thigh link, a shank link, and two connecting rods (Rod 1 and Rod 2). The hydraulic cylinder is pinned to the frame at its base and to the midpoint of Rod 1 at its piston rod end. Rod 1 is connected to the thigh via a sliding bearing block, allowing relative translation, and is also hinged to Rod 2. Rod 2 is connected to the midpoint of the shank. The thigh and shank are connected at the “knee” joint, and the thigh is connected to the torso at the “hip” joint. The foot is located at the distal end of the shank.
The working principle is elegantly simple: the extension and retraction of the hydraulic cylinder cause Rod 1 to pivot. This pivoting motion, transmitted through the sliding block, forces the thigh to rotate about the hip. Simultaneously, the motion of Rod 1 is conveyed through Rod 2 to the shank, inducing its rotation about the knee. This coordinated movement results in the leg performing a “lifting” (protraction) and “stretching” (retraction) motion cycle, emulating the swing and stance phases of a biological leg in a bionic robot.
A fundamental step in validating the design’s feasibility was to analyze its mobility. Since the primary motion is confined to a plane, a planar degrees-of-freedom (DOF) analysis suffices. Using Grübler’s formula for planar mechanisms:
$$ F = 3(n – 1) – 2P_L – P_H $$
where \( F \) is the mechanism’s DOF, \( n \) is the number of links (including the fixed frame), \( P_L \) is the number of lower pairs (revolute and prismatic joints), and \( P_H \) is the number of higher pairs. In my mechanism: \( n = 8 \) (7 moving links + 1 fixed frame), \( P_L = 10 \) (all joints are revolute or prismatic), \( P_H = 0 \). Therefore:
$$ F = 3(8 – 1) – 2 \times 10 = 21 – 20 = 1 $$
This calculation confirms the mechanism has one degree of freedom. A single actuator (the hydraulic cylinder) is sufficient to control the entire leg’s posture, dramatically simplifying the驱动 and control system compared to multi-actuator designs for a bionic robot leg.
2. Kinematic Modeling and Analysis
To analyze and control the bionic robot leg, a precise mathematical model describing the relationship between the actuator input and the foot’s position (forward kinematics) is essential. I employed the Denavit-Hartenberg (D-H) convention to establish a systematic kinematic model.
2.1 Coordinate System Establishment and D-H Parameters
I attached coordinate frames to each link of the leg mechanism, following the standard D-H rules. The leg was modeled as a 2-revolute-joint serial chain from the hip to the foot for kinematic purposes, as the internal linkage constraints are handled separately. The established coordinate frames and the corresponding D-H parameters are summarized in the table below.
| Link \(i\) | Link Length \(a_{i-1}\) (mm) | Link Twist \(\alpha_{i-1}\) (rad) | Link Offset \(d_i\) (mm) | Joint Angle \(\theta_i\) (rad) |
|---|---|---|---|---|
| 1 (Thigh) | 0 | 0 | 0 | \(\theta_1\) |
| 2 (Shank) | \(L_1 = 450\) | 0 | 0 | \(\theta_2\) |
Here, \( \theta_1 \) is the rotation angle of the thigh relative to the torso (horizontal), and \( \theta_2 \) is the rotation angle of the shank relative to the thigh.
2.2 Forward Kinematics
The transformation from frame \(\{i-1\}\) to frame \(\{i\}\) is given by the homogeneous transformation matrix \( ^{i-1}T_i \):
$$ ^{i-1}T_i = \begin{bmatrix}
\cos\theta_i & -\sin\theta_i & 0 & a_{i-1}\\
\sin\theta_i \cos\alpha_{i-1} & \cos\theta_i \cos\alpha_{i-1} & -\sin\alpha_{i-1} & -\sin\alpha_{i-1} d_i\\
\sin\theta_i \sin\alpha_{i-1} & \cos\theta_i \sin\alpha_{i-1} & \cos\alpha_{i-1} & \cos\alpha_{i-1} d_i\\
0 & 0 & 0 & 1
\end{bmatrix} $$
Given our parameters (\( \alpha_{i-1}=0, d_i=0 \)), the matrices simplify to:
$$ ^{0}T_1 = \begin{bmatrix}
\cos\theta_1 & -\sin\theta_1 & 0 & 0\\
\sin\theta_1 & \cos\theta_1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}, \quad ^{1}T_2 = \begin{bmatrix}
\cos\theta_2 & -\sin\theta_2 & 0 & L_1\\
\sin\theta_2 & \cos\theta_2 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix} $$
The overall transformation from the hip frame \(\{0\}\) to the foot frame \(\{2\}\) is:
$$ ^{0}T_2 = ^{0}T_1 \cdot ^{1}T_2 = \begin{bmatrix}
\cos(\theta_1+\theta_2) & -\sin(\theta_1+\theta_2) & 0 & L_1 \cos\theta_1\\
\sin(\theta_1+\theta_2) & \cos(\theta_1+\theta_2) & 0 & L_1 \sin\theta_1\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix} $$
The position of the foot tip \( P \) in the hip coordinate frame, assuming the shank length \( L_2 = 380 \) mm extends along the x-axis of frame \(\{2\}\), is given by:
$$ ^{0}P = ^{0}T_2 \cdot [L_2, 0, 0, 1]^T $$
Thus, the forward kinematics equations are:
$$ \begin{cases}
x_f = L_1 \cos\theta_1 + L_2 \cos(\theta_1 + \theta_2) \\
y_f = L_1 \sin\theta_1 + L_2 \sin(\theta_1 + \theta_2)
\end{cases} $$
These equations define the workspace of the bionic robot leg’s foot.
2.3 Foot Tip Velocity Analysis
The velocity of the foot tip is crucial for assessing the dynamic performance of the bionic robot. It is obtained by differentiating the position equations with respect to time. This can be expressed in matrix form using the geometric Jacobian \( J \):
$$ \begin{bmatrix} \dot{x}_f \\ \dot{y}_f \end{bmatrix} = J(\theta_1, \theta_2) \begin{bmatrix} \dot{\theta}_1 \\ \dot{\theta}_2 \end{bmatrix} $$
Where the Jacobian matrix is:
$$ J = \begin{bmatrix}
-L_1 \sin\theta_1 – L_2 \sin(\theta_1+\theta_2) & -L_2 \sin(\theta_1+\theta_2) \\
L_1 \cos\theta_1 + L_2 \cos(\theta_1+\theta_2) & L_2 \cos(\theta_1+\theta_2)
\end{bmatrix} $$
This relationship is vital for motion planning and force control of the bionic robot.
2.4 Relationship Between Actuator Displacement and Joint Angles
Since the leg is driven by a single hydraulic cylinder, the core of its control lies in mapping the cylinder’s piston displacement \( \Delta L_c \) to the joint angles \( \theta_1 \) and \( \theta_2 \). This requires an analysis of the internal linkage geometry. By applying the law of cosines to the triangles formed within the mechanism (as shown in the conceptual diagrams), the following key relationships were derived.
Let \( L_a, L_b, L_{r1}, L_{r2} \) be the lengths of the hydraulic cylinder’s fixed mount arm, Rod 1, and other relevant linkage segments as defined in the mechanism schematic. Let \( \phi \) be the angle of Rod 1 relative to the torso. The cylinder displacement \( \Delta L_c \) is related to \( \phi \) by:
$$ (\Delta L_c)^2 = L_a^2 + L_{r1}^2 – 2 L_a L_{r1} \cos(\phi_0 – \phi) $$
$$ \phi = \phi_0 – \arccos\left( \frac{L_a^2 + L_{r1}^2 – (\Delta L_c)^2}{2 L_a L_{r1}} \right) $$
Subsequently, the thigh angle \( \theta_1 \) is related to \( \phi \) through the geometry of the quadrilateral formed by the torso, thigh, Rod 1, and the sliding block connection:
$$ \theta_1 = \arcsin\left( \frac{e}{L_{th}} \right) – \psi $$
where \( e \) is the offset of the sliding block, \( L_{th} \) is a relevant distance on the thigh, and \( \psi \) is an angle dependent on \( \phi \) and fixed lengths.
Finally, the knee joint angle \( \theta_2 \) is determined by the geometry involving the thigh, shank, Rod 1, and Rod 2. Another application of the law of cosines yields:
$$ L_{r2}^2 = L_{th_m}^2 + L_{sh_m}^2 – 2 L_{th_m} L_{sh_m} \cos(\pi – \theta_2) $$
$$ \theta_2 = \pi – \arccos\left( \frac{L_{th_m}^2 + L_{sh_m}^2 – L_{r2}^2}{2 L_{th_m} L_{sh_m}} \right) $$
where \( L_{th_m} \) and \( L_{sh_m} \) are distances from the hip/knee to the connection points on the thigh and shank, respectively. These equations complete the kinematic chain from the single actuator input \( \Delta L_c \) to the foot position \( (x_f, y_f) \), which is the cornerstone for controlling this bionic robot leg.
3. Simulation Analysis and Validation
To verify the theoretical model and the motion characteristics of the proposed leg mechanism for the bionic robot, I conducted a dynamic simulation using Adams software.
3.1 Simulation Model Setup
Based on the proportions of an adult canine forelimb, I defined the key dimensions for the bionic robot leg, as listed below.
| Component | Length (mm) |
|---|---|
| Thigh (L1) | 450 |
| Shank (L2) | 380 |
| Torso (Frame) | 300 |
| Connecting Rod 1 | 315 |
| Connecting Rod 2 | 330 |
A detailed 3D model was created in SolidWorks, simplifying non-essential features for simulation efficiency. The model was exported to Adams. The torso was fixed to ground. All pin joints were modeled as revolute joints, the cylinder connection as a translational joint, and the slider connection as a planar joint. A translational motion driver was applied to the hydraulic cylinder joint. The prescribed motion followed a periodic function to cycle the leg through a full stride: starting from the fully retracted position, lifting the foot (protraction), and then extending it forward and down (retraction). The leg material was set to aluminum (density 2700 kg/m³). To test load-bearing capacity, a 250 N downward force was applied to the torso (simulating part of the robot’s weight), and a 250 N upward ground reaction force was applied at the foot during the stance phase.
3.2 Simulation Results and Discussion
The simulation was run for several gait cycles. Data from three consecutive cycles was analyzed to ensure consistency and avoid transient effects.
Foot Trajectory: The foot’s trajectory in the vertical-lateral plane successfully reproduced the biologically inspired motion sequence. The foot lifted off the ground, reached a peak height, swung forward, and then extended downwards and backwards. The maximum lift height (vertical displacement from the lowest point) was approximately 150 mm.
Foot Velocity: The foot-tip velocity curve showed smooth and periodic behavior. Key velocity points were observed: a peak negative (backward) velocity of about -91.9 mm/s during late stance, and peak positive (forward) velocities of about 159.1 mm/s during the swing phase. The transitions near the stance-swing and swing-stance boundaries were relatively smooth, indicating stable motion without abrupt jerks, a desirable characteristic for a bionic robot.
Joint Torque: The torque at the hip (thigh-torso joint) was analyzed. The maximum torque of approximately 4.46 N·m was recorded at the instant when the foot was at its highest point in the swing phase. Conversely, the minimum torque occurred when the leg was almost fully extended near the end of the stance phase. This data is critical for actuator sizing, indicating the maximum load the hydraulic cylinder must overcome.
Consistency Check: A direct comparison between the foot displacement and velocity profiles confirmed their correct phasic relationship. The velocity peaked when the displacement curve had its steepest slope (mid-swing), and crossed zero when the foot reversed direction at the top and bottom of the trajectory. This internal consistency strongly validates the correctness of the kinematic model and the functional design of the leg mechanism for the envisioned bionic robot.
4. Conclusion
In this work, I have presented the complete design and analysis cycle for a novel leg mechanism intended for a bionic quadruped robot. The primary motivation was to overcome the drawbacks of multi-actuator designs by proposing a single-cylinder-driven linkage system. The key achievements are:
- Innovative Design: A planar, one-degree-of-freedom leg mechanism was conceived, transforming linear hydraulic motion into coordinated, biologically inspired thigh and shank rotation.
- Comprehensive Kinematic Model: A precise mathematical model was established using the D-H convention, providing forward kinematics and Jacobian for motion analysis. Crucially, the geometric relationship linking the single actuator’s displacement to the two primary joint angles was derived.
- Successful Simulation Validation: Dynamic simulations in Adams confirmed the feasibility of the design. The leg executed a realistic stepping motion with smooth velocity profiles. The simulated joint torques provide essential data for future actuator selection and strength validation of physical prototypes.
This study lays a solid theoretical and simulation foundation for the proposed bionic robot leg. The results confirm that the mechanism performs as intended, offering a simplified yet effective alternative for quadrupedal locomotion. Future work will focus on the detailed mechanical design for manufacturability, the construction of a physical prototype, and the integration of four such legs into a complete bionic quadruped robot platform to experimentally validate its walking and terrain adaptation capabilities.