In the field of advanced robotics, the integration of wheel and leg mechanisms represents a significant leap in robot technology, enabling enhanced mobility across diverse terrains. Traditional wheeled robots excel on flat surfaces but struggle with obstacles, while legged robots offer superior obstacle negotiation at the cost of complexity and control challenges. To address these limitations, we propose a novel wheel-leg hybrid robot with a single-degree-of-freedom (DOF) composite linkage leg structure. This design simplifies control while maintaining high performance in both wheeled and legged modes, embodying the evolution of robot technology. The leg mechanism utilizes a six-bar linkage system, optimized for smooth foot-end trajectories and efficient motion. This article details the kinematic modeling, trajectory planning, parameter optimization using particle swarm optimization (PSO), and validation through simulations and experiments, all contributing to advancements in robot technology.
The robot’s overall structure comprises a four-legged configuration, with each leg group consisting of two identical single-DOF leg mechanisms. One leg in each group incorporates a self-locking hub motor for wheeled locomotion, while the other assists in obstacle crossing. This hybrid approach leverages the efficiency of wheeled movement on flat ground and the adaptability of legged motion over rough terrain, showcasing the versatility of modern robot technology. The mechanical design ensures stability and reduces the number of actuators, simplifying the control system. The leg mechanism is modeled as a planar six-bar linkage, with key dimensions optimized to achieve desired performance metrics. Below, we present the kinematic analysis and optimization framework that underpin this innovative robot technology.
Kinematic Analysis of the Leg Mechanism
The leg mechanism is a six-bar linkage with one DOF, determined using the Grübler’s formula: $$ F = 3(n-1) – 2P_L – P_H $$ where \( n = 5 \) (number of moving links), \( P_L = 7 \) (number of lower pairs), and \( P_H = 0 \) (number of higher pairs), resulting in \( F = 1 \). This simplicity is crucial for reliable robot technology. The linkage consists of a driving crank (AB), a connecting link (G-C-B), two rockers (CD and EF), and a leg link (EH), with fixed points A, D, and F on the mounting plate. A local coordinate system is established at point A, with the x-axis aligned along the mounting plate, and a global coordinate system with the X-axis horizontal. The vector loop equations for the four-bar linkage (AB-BC-CD-AD) are derived as:
$$ \begin{cases} l_1 \cos \theta_1 + l_2 \cos \theta_2 = l_4 + l_3 \cos \theta_3 \\ l_1 \sin \theta_1 + l_2 \sin \theta_2 = l_3 \sin \theta_3 \end{cases} $$
where \( \theta_1 \), \( \theta_2 \), and \( \theta_3 \) are the angles of the crank, connecting link, and rocker CD with the x-axis, respectively. Eliminating \( \theta_2 \) yields:
$$ a \sin \theta_3 + b \cos \theta_3 + c = 0 $$
with coefficients: $$ a = 2 l_1 l_3 \sin \theta_1 $$, $$ b = 2 l_3 (l_1 \cos \theta_1 – l_4) $$, and $$ c = l_2^2 – l_1^2 – l_3^2 – l_4^2 + 2 l_1 l_4 \cos \theta_1 $$. Solving for \( \theta_3 \) and \( \theta_2 \):
$$ \theta_2 = \sin^{-1} \left( \frac{l_3 \sin \theta_3 – l_1 \sin \theta_1}{l_2} \right) $$
$$ \theta_3 = 2 \tan^{-1} \left( \frac{-a \pm \sqrt{a^2 + b^2 – c^2}}{b – c} \right) $$
The coordinates of point G are then computed as:
$$ \begin{cases} x_G = l_1 \cos \theta_1 + l_2 \cos \theta_2 + l_7 \cos(180^\circ – \angle GCB + \theta_2) \\ y_G = l_1 \sin \theta_1 + l_2 \sin \theta_2 + l_7 \sin(180^\circ – \angle GCB + \theta_2) \end{cases} $$
where \( \angle GCB = \cos^{-1} \left( \frac{l_1^2 + l_7^2 – l_8^2}{2 l_1 l_7} \right) \). Connecting points G and F forms an auxiliary triangle, allowing the calculation of point E’s coordinates. The angle \( \theta_6 \) for rocker EF is derived as \( \theta_6 = \theta_4 – \theta_5 \), where \( \theta_4 = \cos^{-1} \left( \frac{\mathbf{x}_{GF} \cdot \mathbf{e}_x}{|\mathbf{x}_{GF}|} \right) \) and \( \theta_5 = \cos^{-1} \left( \frac{l_5^2 + l_{11}^2 – l_9^2}{2 l_5 l_{11}} \right) \). Thus, point E is:
$$ \begin{cases} x_E = x_F + l_5 \cos \theta_6 \\ y_E = l_5 \sin \theta_6 \end{cases} $$
Finally, the foot-end point H coordinates are obtained from the proportional relationship along link EG:
$$ \begin{cases} x_H = x_G + \frac{l_{10}}{l_9} (x_E – x_G) \\ y_H = y_G + \frac{l_{10}}{l_9} (y_E – y_G) \end{cases} $$
Transforming to the global coordinate system using the mounting plate angle \( \theta_0 \):
$$ \begin{cases} X_H = x_H \cos \theta_0 – y_H \sin \theta_0 \\ Y_H = x_H \sin \theta_0 + y_H \cos \theta_0 \end{cases} $$
This kinematic model forms the basis for trajectory planning and optimization in robot technology.
Trajectory Planning for Foot-End Motion
In robot technology, foot-end trajectory planning is critical for stable and efficient locomotion. The gait cycle consists of a stance phase (foot on ground) and a swing phase (foot in air). For this robot, an elliptical curve is chosen as the ideal foot-end trajectory due to its smooth, continuous derivatives and minimal impact at transition points. The ellipse parameters are set based on a step length \( L = 350 \, \text{mm} \) and step height \( H = 210 \, \text{mm} \), with the swing and stance phases equally divided along the ellipse. The ideal trajectory equation is:
$$ \frac{x^2}{a_0^2} + \frac{y^2}{b_0^2} = 1 $$
where \( a_0 = 175 \, \text{mm} \) and \( b_0 = 105 \, \text{mm} \). This curve ensures smooth acceleration and deceleration, reducing mechanical stress and enhancing the durability of robot technology systems.
To fit the actual foot-end trajectory to this ellipse, discrete points from the kinematic model are used in a least-squares ellipse fitting method. The general ellipse equation is:
$$ A x^2 + B x y + C y^2 + D x + E y + F = 0 $$
Defining the parameter vector \( \mathbf{R} = [A, B, C, D, E, F]^T \) and the data vector \( \mathbf{X} = [x^2, x y, y^2, x, y, 1]^T \), the optimization minimizes \( \min \mathbf{R}^T \mathbf{X} \mathbf{X}^T \mathbf{R} \) subject to the constraint \( \mathbf{R}^T \mathbf{U} \mathbf{R} = 1 \), where \( \mathbf{U} \) is a constraint matrix. Using Lagrange multipliers, the solution involves eigenvalue decomposition. The fitted ellipse parameters—semi-major axis \( a_1 \), semi-minor axis \( b_1 \), and orientation angle \( \alpha_1 \)—are calculated as:
$$ a_1 = \frac{2(A O_x^2 + C O_y^2 + B O_x O_y – F)}{(A + C) + \sqrt{(A – C)^2 + B^2}} $$
$$ b_1 = \frac{2(A O_x^2 + C O_y^2 + B O_x O_y – F)}{(A + C) – \sqrt{(A – C)^2 + B^2}} $$
$$ \alpha_1 = \frac{1}{2} \tan^{-1} \left( \frac{B}{A – C} \right) $$
where \( O_x \) and \( O_y \) are the ellipse center coordinates. This fitting process ensures the actual trajectory aligns with the ideal, improving the performance of robot technology.
Optimization Model Using Particle Swarm Optimization
To achieve optimal leg dimensions, we formulate an optimization problem with the design variables as the linkage lengths and mounting angle: \( \mathbf{X} = [l_1, l_2, l_3, l_5, l_7, l_8, l_9, l_{10}, \theta_0]^T \). The objective is to minimize the difference between the fitted ellipse and the ideal ellipse in terms of shape and orientation. The objective function is:
$$ W(\mathbf{x}) = \lambda_1 (a_1 – a_0)^2 + \lambda_2 (b_1 – b_0)^2 + \lambda_3 \alpha_1^2 $$
where \( \lambda_1 = 0.5 \), \( \lambda_2 = 0.5 \), and \( \lambda_3 = 0.01 \) are weight coefficients prioritizing shape similarity over orientation. Constraints include Grashof’s conditions for the crank-rocker mechanism and practical limits on dimensions and angles:
$$ \begin{cases}
\theta_0 \in [0, 90^\circ] \\
l_i^{\min} \leq l_i \leq l_i^{\max} \quad \text{for } i = 1,2,3,5,7,8,9,10 \\
l_1 + l_4 \leq l_2 + l_3 \\
l_1 + l_2 \leq l_3 + l_4 \\
l_1 + l_3 \leq l_2 + l_4
\end{cases} $$
Particle swarm optimization (PSO) is employed to solve this problem due to its efficiency in handling non-linear constraints. In PSO, each particle represents a candidate solution, and its velocity and position update as:
$$ \mathbf{V}_{id}^{t+1} = \omega \mathbf{V}_{id}^t + c_1 r_1 (\mathbf{p}_{id}^t – \mathbf{x}_{id}^t) + c_2 r_2 (\mathbf{g}_{d}^t – \mathbf{x}_{id}^t) $$
$$ \mathbf{x}_{id}^{t+1} = \mathbf{x}_{id}^t + \mathbf{V}_{id}^{t+1} $$
where \( \omega = 0.8 \) is the inertia weight, \( c_1 = c_2 = 1.4 \) are acceleration coefficients, and \( r_1, r_2 \in [0,1] \) are random numbers. The PSO parameters are summarized in the table below.
| Parameter | Value |
|---|---|
| Population Size | 100 |
| Iterations | 100 |
| Inertia Weight \( \omega \) | 0.8 |
| Cognitive Coefficient \( c_1 \) | 1.4 |
| Social Coefficient \( c_2 \) | 1.4 |
The optimization converges after 39 iterations with a minimum objective value of 0.0118, indicating high similarity between the actual and ideal trajectories. The optimal parameters are listed in the following table, which are essential for fabricating the robot technology prototype.
| Parameter | Lower Bound (mm) | Upper Bound (mm) | Optimal Value (mm or rad) |
|---|---|---|---|
| \( l_1 \) | 90 | 150 | 106.2262 |
| \( l_2 \) | 312 | 450 | 450.0000 |
| \( l_3 \) | 320 | 450 | 334.2193 |
| \( l_5 \) | 570 | 682 | 570.0000 |
| \( l_7 \) | 243 | 350 | 294.4857 |
| \( l_8 \) | 432 | 550 | 432.0000 |
| \( l_9 \) | 250 | 360 | 250.0000 |
| \( l_{10} \) | 250 | 377 | 340.4565 |
| \( \theta_0 \) | 1.9198 rad | 2.5307 rad | 2.3932 rad |
These optimized dimensions ensure the leg mechanism meets the desired kinematic and dynamic performance, advancing robot technology capabilities.
Simulation and Experimental Validation
To validate the optimized leg structure, we conducted simulations using MATLAB and ADAMS software. The kinematic model was implemented in MATLAB to plot the foot-end trajectory, comparing the pre- and post-optimization curves. The results show that the optimized trajectory closely matches the ideal ellipse, with a step length of approximately 340 mm and step height of 207 mm, confirming the effectiveness of the optimization in robot technology design.
Dynamic simulations in ADAMS analyzed the velocity and acceleration profiles over two gait cycles. The driving crank was set to rotate at 30°/s. The foot-end velocity and acceleration curves are smooth and continuous, without abrupt changes, indicating stable motion and reduced mechanical shocks. This is crucial for the longevity and reliability of robot technology systems. The velocity curve peaks at around 0.5 m/s, and acceleration remains within ±2 m/s², demonstrating efficient dynamics.
An experimental prototype was built using 3D-printed resin parts and a motion control system comprising a Leadshine CL3C EtherCAT bus closed-loop stepper driver and a 57CME23 stepper motor. A red LED was attached to the foot-end to capture the trajectory via high-speed camera. The measured step length and height align with simulations, with minor errors due to manufacturing tolerances. The experimental trajectory overlays well with the optimized theoretical curve, verifying the practical applicability of this robot technology.
Conclusion
In this work, we designed and optimized a novel wheel-leg hybrid robot leg structure using a single-DOF composite linkage. The kinematic model was derived via vector loop equations, and an elliptical trajectory was planned for smooth foot-end motion. Through PSO-based optimization, we obtained optimal leg dimensions that minimize the deviation from the ideal trajectory. Simulations and experiments confirmed that the optimized leg achieves the desired kinematic performance with smooth velocity and acceleration profiles. This contribution enhances the field of robot technology by providing a simplified, efficient solution for hybrid locomotion, paving the way for future innovations in adaptive mobile robots. The integration of wheel and leg mechanisms in a single DOF design reduces control complexity while maintaining high obstacle-crossing ability, making it suitable for various applications in search, rescue, and exploration.