In the field of advanced robotics, the development of legged robots has garnered significant attention due to their superior adaptability to complex terrains compared to wheeled or tracked systems. Among these, quadruped robots stand out for their stability and mobility, drawing inspiration from biological counterparts like horses. This study focuses on the innovative design and optimization of an eight-bar leg mechanism for a quadruped robot, leveraging genetic algorithms to enhance performance metrics such as workspace and energy efficiency. The integration of cutting-edge robot technology enables these systems to navigate unstructured environments effectively, making them ideal for applications in search and rescue, exploration, and industrial automation. By combining biomechanical principles with advanced optimization techniques, we aim to push the boundaries of robot technology, contributing to more efficient and agile robotic systems.
The leg mechanism design begins with a thorough analysis of equine biomechanics, particularly the hind leg structure of horses, known for their powerful obstacle negotiation and运动 capabilities. We translate this biological model into a robotic leg configuration, starting with an open-chain structure that mimics the hip, knee, and ankle joints. However, open-chain designs often suffer from high inertia and increased actuator loads, limiting their efficiency. To address this, we transform the open-chain into a closed-chain mechanism, specifically an eight-bar linkage, which reduces joint torques by positioning actuators on the base. This transformation adheres to key principles: maintaining degrees of freedom, preserving joint dimensions, ensuring workspace equivalence, and avoiding singularities. The closed-chain design incorporates a five-bar mechanism for hip and knee joints and a linear actuator for the ankle, enhancing overall robot technology by improving energy transmission and motion flexibility.
To establish a parametric model, we derive initial dimensions from biological data of an adult Dutch Warmblood horse, as summarized in Table 1. These parameters include segment lengths and mass distributions, which serve as a baseline for subsequent optimization. The total leg length is normalized to unity for proportional analysis, ensuring that the design remains scalable and adaptable to various robot technology platforms. The closed-chain configuration consists of multiple links, such as L1 (frame), L2, L3 (hip), L45, L6, L67 (knee), L89 (ankle), and L10, with specific constraints to prevent extreme proportions that could lead to assembly issues or poor energy efficiency. This parametric approach allows for systematic variation and optimization, aligning with the goals of advanced robot technology to achieve robust and efficient designs.
| Joint | Mass (kg) | Length (mm) |
|---|---|---|
| Hip | 18.6 | 360 |
| Knee | 8.3 | 434 |
| Ankle | 2.84 | 353 |
Kinematic modeling is essential for understanding the motion characteristics of the eight-bar mechanism. We divide the system into two closed-loop five-bar mechanisms: OACEB and EDPQGF. Using vector loop equations, we derive the relationships between joint angles and link lengths. For the first five-bar mechanism OACEB, the vector equation is given by:
$$ \overrightarrow{OA} + \overrightarrow{AC} + \overrightarrow{CE} = \overrightarrow{OB} + \overrightarrow{BE} $$
This translates to the following component equations:
$$ L_1 \cos \theta_1 + L_3 \cos \theta_3 – L_6 \cos \theta_4 = L_2 \cos \theta_2 + L_{45} \cos \theta_{45} $$
$$ L_1 \sin \theta_1 + L_3 \sin \theta_3 + L_6 \sin \theta_4 = L_2 \sin \theta_2 + L_{45} \sin \theta_{45} $$
By applying trigonometric identities, we solve for the unknown angles, such as $\theta_2$ and $\theta_3$, using the formula:
$$ \theta_2 = 2 \arctan\left( \frac{M + \sqrt{M^2 + N^2 – P^2}}{N + P} \right) $$
where $M$, $N$, and $P$ are functions of the link lengths and known angles. Similarly, for the second five-bar mechanism EDPQGF, we compute the linear actuator length $L_{10}$ to control the ankle joint, ensuring precise foot trajectory planning. This kinematic analysis provides the foundation for dynamic modeling and optimization, crucial for advancing robot technology in motion planning and control.
Dynamic modeling employs the Lagrangian method to compute joint torques and energy consumption, which are vital for optimizing the robot’s efficiency. The Lagrangian function $L$ is defined as the difference between kinetic energy $T$ and potential energy $U$:
$$ L(\theta, \dot{\theta}) = T(\theta, \dot{\theta}) – U(\theta, \dot{\theta}) $$
The total kinetic energy accounts for translational and rotational components of each link:
$$ T = \frac{1}{2} \sum_{i=1}^{n} \left( m_i v_i^2 + I_i \dot{\theta}_i^2 \right) $$
where $m_i$ is the mass, $v_i$ is the velocity of the center of mass, $I_i$ is the moment of inertia, and $\dot{\theta}_i$ is the angular velocity of the $i$-th link. Potential energy is given by:
$$ U = \sum_{i=1}^{n} m_i g h_i $$
with $g$ as gravitational acceleration and $h_i$ as the height of the center of mass. The joint torques $\tau_i$ are derived from the Euler-Lagrange equation:
$$ \tau_i = \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\theta}_i} \right) – \frac{\partial T}{\partial \theta_i} + \frac{\partial U}{\partial \theta_i} $$
For the three actuated joints (hip, knee, and ankle), we compute the torques $\tau_1$, $\tau_2$, and $\tau_5$, which are used to evaluate energy consumption over a motion cycle. This dynamic model highlights the importance of minimizing torques to reduce energy usage, a key aspect of sustainable robot technology.
To validate the kinematic and dynamic models, we conduct simulations using ADAMS software. The theoretical foot trajectory and joint torques are compared against simulation results, showing high consistency. For instance, the foot trajectory derived from the closed-chain mechanism matches the biological model, confirming the equivalence of workspace. The joint torque profiles from dynamics align with ADAMS outputs, with minor discrepancies due to friction effects. This verification ensures the reliability of our models for optimization purposes, demonstrating the integration of simulation tools in robot technology development.
Optimization of the leg mechanism focuses on two primary objectives: maximizing the foot-end workspace and minimizing single-cycle energy consumption. We employ genetic algorithms, a popular method in robot technology for handling multi-objective problems. The design variables are the link lengths, subject to constraints to avoid impractical proportions. For workspace optimization, we use the Monte Carlo method to estimate the area enclosed by the foot trajectory, as it handles irregular shapes effectively. The objective function for workspace is to maximize the area $A$, computed by generating random points within the reachable space and calculating the enclosed region.
For energy optimization, the goal is to minimize the total energy $E$ consumed per cycle, defined as the sum of the absolute products of torque and angular displacement for each joint:
$$ E = \sum_{i=1}^{n} | \tau_i \Delta \theta_i | $$
where $\tau_i$ is the torque and $\Delta \theta_i$ is the angular displacement of the $i$-th joint. The genetic algorithm parameters include a population size of 50, 200 generations, crossover probability of 0.8, and Pareto front retention of 0.35. These settings balance exploration and exploitation, ensuring diverse and high-quality solutions. The constraints on link lengths are refined to prevent extreme values, such as:
$$ 0.2 \leq L_3 \leq 0.4 $$
$$ 0.2 \leq L_{67} \leq 0.4 $$
$$ 0.2 \leq L_{89} \leq 0.4 $$
$$ L_3 + L_{67} + L_{89} = 1 $$
Table 2 presents different combinations of normalized link lengths and their corresponding workspace areas, showing that certain proportions, like $L_3=0.3$, $L_{67}=0.2$, $L_{89}=0.5$, yield the largest area, increasing workspace by 4.7% compared to the biological baseline.
| Combination | L3 | L67 | L89 | Area (m²) |
|---|---|---|---|---|
| 0 (Baseline) | 0.31 | 0.38 | 0.31 | 0.1243 |
| 1 | 0.2 | 0.2 | 0.6 | 0.1266 |
| 2 | 0.2 | 0.3 | 0.5 | 0.1281 |
| 3 | 0.2 | 0.4 | 0.4 | 0.1234 |
| 4 | 0.3 | 0.2 | 0.5 | 0.1301 |
| 5 | 0.3 | 0.3 | 0.4 | 0.1267 |
| 6 | 0.3 | 0.4 | 0.3 | 0.1230 |
| 7 | 0.4 | 0.2 | 0.4 | 0.1248 |
| 8 | 0.4 | 0.3 | 0.3 | 0.1239 |
| 9 | 0.4 | 0.4 | 0.2 | 0.1187 |
In single-objective optimization for energy minimization, the genetic algorithm adjusts link lengths such as L1, L2, L3, L45, L6, L67, and L89. The optimized parameters reduce the total energy consumption from 2059.3 N·m to 1374 N·m, a 33.27% decrease, as shown in Table 3. This significant improvement underscores the potential of optimization in enhancing the sustainability of robot technology.
| Link | Original Length (mm) | Optimized Length (mm) |
|---|---|---|
| L1 | 400 | 370 |
| L2 | 320 | 383 |
| L3 | 360 | 337 |
| L45 | 518 | 427 |
| L6 | 151 | 245 |
| L67 | 434 | 461 |
| L89 | 353 | 376 |
For multi-objective optimization, we combine both workspace and energy goals, generating a Pareto front that represents trade-offs between the two objectives. The optimized link lengths from this approach, listed in Table 4, result in a 2.30% increase in workspace area (from 0.1242 m² to 0.1271 m²) and a 20.78% reduction in energy consumption (from 2059.27 N·m to 1631.45 N·m). This balanced solution demonstrates the efficacy of genetic algorithms in robot technology for achieving compromised performance enhancements.
| Link | Original Length (mm) | Optimized Length (mm) |
|---|---|---|
| L1 | 400 | 441 |
| L2 | 320 | 370 |
| L3 | 360 | 400 |
| L45 | 518 | 628 |
| L6 | 151 | 185 |
| L67 | 434 | 273 |
| L89 | 353 | 426.5 |
The results highlight the critical role of link proportions in determining robot performance. For instance, a longer knee segment (L67) tends to expand workspace, while optimized lengths reduce joint torques, particularly for the hip and knee actuators. The linear actuator for the ankle shows minimal torque changes, indicating its robustness to dimensional variations. These findings provide valuable insights for designing energy-efficient and high-mobility quadruped robots, advancing the field of robot technology.
In conclusion, this study successfully designs and optimizes an eight-bar leg mechanism for a quadruped robot using genetic algorithms. The kinematic and dynamic models are rigorously derived and validated through simulations, ensuring accuracy. Single-objective optimization achieves substantial improvements in workspace and energy efficiency, while multi-objective optimization offers a balanced solution. This work underscores the importance of biomechanical inspiration and computational optimization in robot technology, paving the way for future developments in agile and sustainable robotic systems. Future research will involve physical prototyping and experimental validation to further refine these designs.